Fracture energy of columnar freshwater ice: Influence of loading type, loading rate and size

I.E. Gharamti; J.P. Dempsey; A. Polojärvi; J. Tuhkuri

doi:10.1016/j.mtla.2021.101188

Gharamti, I.E.;Dempsey, J.P.;Polojärvi, A.;Tuhkuri, J.;

2021-12-01 00:00:00

Rat e an d size effec to s n th e deformatio n an d fractur e o w f ar m an d ﬂoatin g columna frreshwater I G En lhar ti A t o Universi Mecha ca E l ng eer g Rate and size effects on the deformation and fracture of warm and ﬂoating columnar freshwater ice Iman El Gharamti DOCTORAL DISSER TATIONS ec yt la ma am ni ni in Aalto University publication series DOCTORAL DISSERTATIONS 134/2021 Rate and size effects on the deformation and fracture of warm and ﬂoating columnar freshwater ice Iman El Gharamti A doctoral dissertation completed for the degree of Doctor of Science (Technology) to be defended, with the permission of the Aalto University School of Engineering, at a public examination held at the lecture hall 216 of the school on 29 October 2021 at 12:00. Remote connection link: https://aalto.zoom.us/j/66036070494 Aalto University School of Engineering Mechanical Engineering Solid Mechanics L Supervising professor Professor Jukka Tuhkuri, Aalto University, Finland Thesis advisor Professor John P. Dempsey, Clarkson University, USA Preliminary examiners Professor Sveinung Løset, Norwegian University of Science and Technology, Norway Dr. David Cole, Cold Regions Research and Engineering Lab, USA Opponent Professor Sveinung Løset, Norwegian University of Science and Technology, Norway Aalto University publication series DOCTORAL DISSERTATIONS 134/2021 © 2021 Iman El Gharamti ISBN 978-952-64-0533-9 (printed) ISBN 978-952-64-0534-6 (pdf) ISSN 1799-4934 (printed) ISSN 1799-4942 (pdf) http://urn.fi/URN:ISBN:978-952-64-0534-6 Unigrafia Oy Helsinki 2021 Finland Printed matter Printed matter 1234 5678 4041-0619 N Abstract Aalto University, P.O. Box 11000, FI-00076 Aalto www.aalto.fi Author Iman El Gharamti Name of the doctoral dissertation Rate and size effects on the deformation and fracture of warm and ﬂoating columnar freshwater ice Publisher School of Engineering Unit Mechanical Engineering Aalto University publication series DOCTORAL DISSERTATIONS 134/2021 Series Field of research Mechanical Engineering Manuscript submitted 25 May 2021 Date of the defence 29 October 2021 Permission for public defence granted (date) 7 September 2021 Language English Monograph Article dissertation Essay dissertation Abstract Climate change has led to warmer and fragmented ice, and thus has increased the interest in understanding and modelling the fracture behavior and creep deformations of warm ice. The thesis explores the fracture and creep response of warm and ﬂoating columnar freshwater S2 ice under different loading scenarios, by conducting large scale experiments in the Ice Tank of Aalto University. A program of nineteen mode I fracture tests, using deeply cracked edge-cracked rectangular plates, that varied the test sizes, loading types, and loading rates was conducted. The ice was very warm with a temperature of about -0.3 C at the top surface. The ice was loaded in the direction normal to the columnar grains, and the loading conditions divided the test program into two parts. In the ﬁrst part, fourteen tests were conducted in displacement control (DC) and loaded with different rates monotonically to fracture. The plates covered a size range of 1:39, the largest for ice tested under laboratory conditions, with three plate sizes: 0.5m x 1m, 3m x 6m and 19.5m x 36m. In the second part, ﬁve tests of 3m x 6m plates were loaded in load control (LC) under creep/cyclic-recovery loading and monotonic loading to fracture. For the DC tests, methods for both the linear elastic fracture mechanics (LEFM) and a non-linear viscoelastic ﬁctitious crack model (VFCM) were derived to analyze the data and calculate values for the apparent fracture toughness, crack opening displacement, stress-separation curve, fracture energy, and size of the process zone near a crack tip. Issues of notch sensitivity and minimum size requirements for polycrystalline homogeneity were addressed. Size and rate effects were interrelated as rate dependent size effects and size dependent rate effects. The loading rates applied led to test durations from fewer than 2 seconds to more than 1000 seconds, leading to an elastic response at the highest rates and a viscoelastic response at the lower rates. Under the LC tests' loading conditions, the ice response was overall elastic-viscoplastic; no signiﬁcant viscoelasticity or major recovery were detected. Moreover, there was no clear effect of the creep loading on the fracture properties at crack growth initiation: the failure load and crack opening displacements. Several factors were discussed as possibly contributing to the observed behavior, and the effect of the very warm ice temperature was highlighted. Schapery's model of nonlinear thermodynamics was tested and validated against the experimental response at the crack mouth. The VFCM and Schapery's model were coupled with the Nelder-Mead's optimization scheme to obtain the constitutive parameters, by matching the displacement records generated by the model and measured by the experiment. Further, different methods for computing the fracture energy were applied, and the values were compared regarding the effects of loading type, rate and scale. Keywords Freshwater ice, fracture, creep, size effect, rate effect, viscoelasticity ISBN (printed) 978-952-64-0533-9 ISBN (pdf) 978-952-64-0534-6 ISSN (printed) 1799-4934 ISSN (pdf) 1799-4942 Location of publisher Helsinki Location of printing Helsinki Year 2021 172 http://urn.ﬁ/URN:ISBN:978-952-64-0534-6 Pages urn … and say: “My Lord! Increase me in knowledge.” … we have no knowledge except what You have taught us. Indeed, it is You who is the Knowing and the Wise. The Holy Quran (20:114 - 2:32) 2 Preface As I write the preface today, I close my eyes and the years begin to move in reverse, slowly ticking backward, rolling back to Jan 4, 2016. Thinking back on that day of my life, I recall everything as clearly as if it were all still unfolding before my eyes. I remember standing at the airport holding two suitcases loaded with food from back home that my Mom packed. My mind vividly replays how I arrived to Finland and what it felt to be alone in cold and dark place. My life during PhD has not been all "rainbows and butterﬂies", but more or less a roller coast ride. I was riding waves of sadness, happiness, stress, relief, pain, homesickness, success ... all in one ride. There are so many people to thank as I wrap up the writing of my PhD thesis. My ﬁrst and foremost appreciation goes to my supervisor Prof. Jukka Tuhkuri and my advisor Prof. John Dempsey. Their trust in me since the very beginning granted me the doctoral position and paved my way towards a successful PhD life. Their cooperation and assistance were vital in conducting my research and helping me stay on track. Special thanks go to Jukka for welcoming me in his “Solid Mechanics" group and for granting me the opportunity to travel to Svalbard and work as a ﬁeldwork assistant with other researchers from NTNU (Trondheim, Norway) and UNIS (Svalbard, Norway). Besides the academic guide, Jukka and John have great characters. Jukka is a great listener and supporter; his door was always open for whatever issues I had. John has a great sense of humour; in most of the Skype calls we had, we started with what’s happening around the world and concluded in the last few minutes about research. Working with John, mostly remotely during the whole PhD, proved to me undoubtedly that the success of the work relies mainly on a mutual understanding between the student and the advisor, regardless of whether their oﬃces are in the same building or in diﬀerent continents. My work during the PhD years was funded though the Finland Distinguished Profes- sor programme "Scaling of Ice Strength: Measurements and Modeling", and through the ARAJÄÄ research project, both funded by Business Finland and the industrial partners Aker Arctic Technology, Arctech Helsinki Shipyard, Arctia Shipping, ABB 3 Preface Marine, Finnish Transport Agency, Suomen Hyötytuuli Oy, and Ponvia Oy. I was also funded by the Doctoral Program of the School of Engineering. The ﬁnancial support of all my funders is gratefully acknowledged. The great part of my PhD studies was experimental. I would like to sincerely thank people as Prof. Arttu Polojärvi, Kari Kantola, Teemu Päivärinta, Dr. Mikko Suominen, Veijo Laukkanen and every person who provided any kind of help in conducting my experiments at the Aalto Ice Tank. I want to thank Prof. Sven Bossuyt, Dr. Antti Forsström and Kim Widell who took care of the Digital Image Correlation (DIC) work in my experiments. I am thankful to Dr. Otto Puolakka who assisted my work in the laboratory cold room and always had great ideas to share. My gratitude extends to my group members and colleagues (esp. Dr. Hanyang Gong, Anastasia Markou and Dr. Ville-Pekka Lilja) who added a friendly touch to the stressful work atmosphere. I am grateful to Dr. Mingdong Wei who was always ready to help me at subzero temperatures whenever I was stuck with the ice; together we formed the "Panda and Jääkarhu" team. Special thanks go to Dr. David Cole for training me in thin sections techniques and polarized light procedures during his visit to Aalto University. My thankful words reach Profs. Arttu Polojärvi and Luc St-Pierre who assigned me as teaching assistant in their courses. Not to forget to thank Dr. Kari Santaoja for several useful and enlightening discussions. This dissertation could not have been completed without the valuable comments of the pre-examiners Prof. Sveinung Løset and Dr. David Cole. I will be forever indebted to the support of my brother Moha who acted as my numerical advisor; I greatly appreciate his useful advice and help on various numerical issues examined in the thesis. I would like to thank my brother Moustafa for getting me through tough analytical solutions. I am deeply thankful to my aunts and cousins who have tolerated my tense mood during my study years. My love reaches my friends in Finland (Dandoon, Rima, Diza, Nesma and Lubna) who made me feel right at home. Throughout these years, my mother Wafaa has been the buﬀer between me and the obstacles that were threatening to stop me. It was her support that allowed me to go along with the decision of coming to Finland. Since then, I have been like sailing in a boat in the middle of a sea. Sometimes, the sea is calm. Sometimes, the wind picks up. My mother’s voice, always echoing through my head, was the anchor that prevented my vessel from rocking from side to side and forced me to move forward. Her beautiful face had reassured me, more times than I could ever count, that I was always capable of doing better. You are the greatest gift from God to me and the most beautiful-hearted mother among all the women I met and those that I haven’t. Thank you for providing me with more-than-enough love. 4 Preface My dearest Murtaza, your presence has been my comfort and peace during the PhD days. You are a man who thinks with straight-line logic and weighs a situation without the cloud of worries that aﬀects my own judgement. I have always poured out my concerns to you; and patiently, you were ready to address any surfacing of my negative thoughts and weed out all the what-ifs I bring into the picture. You came into my life, unfroze me from the ice I have been studying, and showed me a new meaning of life that really mattered. A sincere thank you from my heart to a person who is good, all the way to his soul. Finally, I am dedicating this dissertation to the loving memory of my beloved family members (Souad, Jiji and Khalil) who departed this life but continue to mean so much to me. Until we meet again. . . With timeless love. . . Espoo, September 27, 2021, Iman El Gharamti 5 Preface 6 Contents Preface 3 Contents 7 List of Publications 11 Author’s Contribution 13 List of Figures 15 Symbols 25 1. Introduction 29 1.1 Motivation ............................. 29 1.2 Historical Background of Fracture Mechanics .......... 29 1.2.1 Linear Elastic Fracture Mechanics (LEFM) ...... 30 1.2.2 Cohesive Mechanics .................. 32 1.3 Time-Dependent Behavior ..................... 36 1.4 S2 Columnar Freshwater Ice .................... 38 1.5 Fracture Behavior of S2 Columnar Freshwater Ice ........ 39 1.6 Time-dependent Behavior of S2 Columnar Freshwater Ice .... 40 1.7 Objectives, Scope and Structure of the Thesis ............ 41 2. Grown Ice and Experimental Setup [PI and PII] 45 2.1 Experiments ............................ 45 2.1.1 Ice Spraying Trials ................... 46 2.1.2 Ice Growth, Thickness, and Temperature ......... 47 2.1.3 Grain Size and C-axis Orientation ........... 48 2.1.4 Specimen’s Geometry .................. 51 2.1.5 Specimen Preparation and Experimental Procedure . . 52 2.1.6 Fractographic Examinations .............. 55 7 Contents 3. Fracture of Warm S2 Columnar Freshwater ice Under Monotonic Loading: Size and Rate Eﬀects [PI] 59 3.1 Linear Elastic Fracture Mechanics Model ............. 60 3.2 Viscoelastic Fictitious Crack Model ................ 62 3.3 Results ............................... 66 3.3.1 Linear Elastic Fracture Mechanics Analysis ...... 66 3.3.2 Nominal Strength Analysis ............... 69 3.3.3 Viscoelastic Fictitious Crack Model Analysis ...... 71 3.3.4 Notch Sensitivity Analysis ............... 79 3.4 Discussion .............................. 81 4. Fracture of S2 Columnar Freshwater Ice Under Creep/Cyclic-Recovery Loading [PII] 85 4.1 Creep/Cyclic-Recovery Fracture Experiments .......... 86 4.1.1 Specimen ........................ 86 4.1.2 Creep-Recovery and Monotonic Loading Proﬁle . . . 86 4.1.3 Cyclic-Recovery and Monotonic Loading Proﬁle .... 87 Nonlinear Time-Dependent Modeling of S2 Columnar Freshwater Ice 88 4.2 4.3 Results ............................... 92 4.3.1 Experimental and Modelling Results .......... 92 4.3.2 Eﬀect of the Creep and Cyclic Sequences on the Fracture Properties ........................ 92 4.3.3 Ice Response Under the Testing Conditions ...... 94 4.3.4 Nonlinear Modelling Analysis ............. 96 4.4 Discussion ............................. 99 5. Fracture Energy of Columnar Freshwater Ice: Inﬂuence of Loading Type, Loading Rate and Size [PIII] 103 5.1 Fracture Energies ......................... 103 Fracture Energy at Crack Growth Initiation via the J- 5.1.1 integral (J ) ....................... 103 5.1.2 Work-of-Fracture (W ) ................. 104 5.1.3 Fracture Energy at Crack Growth Initiation via the VFCM (G ) ......................... 106 VFCM 5.2 Results and Discussion....................... 106 6. Conclusion 111 6.1 Grown Ice and Experimental Setup [PI, PII and PIII] ........ 111 8 Contents 6.2 Fracture of Warm S2 Columnar Freshwater Ice Under Monotonic Loading: Size and Rate Eﬀects [PI] ................ 112 6.2.1 Summary ........................ 112 6.2.2 Research Suggestions .................. 113 6.3 Fracture of S2 Columnar Freshwater Ice Under Creep/Cyclic- Recovery Loading [PII] ...................... 114 6.3.1 Summary ........................ 114 6.3.2 Research Suggestions .................. 115 6.4 Fracture Energy of Columnar Freshwater Ice: Inﬂuence of Loading Type, Loading Rate and Size [PIII] ................ 116 6.4.1 Summary ........................ 116 6.4.2 Research Suggestions .................. 116 A. Weight Function For An Edge-Cracked Rectangular Plate 117 Bibliography 119 Publications 127 9 Contents 10 List of Publications This thesis consists of an overview of the following publications which are referred to in the text by their Roman numerals. I Iman E. Gharamti, John P. Dempsey, Arttu Polojärvi and Jukka Tuhkuri. Fracture of warm S2 columnar freshwater ice: size and rate eﬀects. Acta Materialia, 22-34, 202 II Iman E. Gharamti, John P. Dempsey, Arttu Polojärvi and Jukka Tuhkuri. Creep and fracture of warm columnar freshwater ice. The Cryosphere, 2401–2413, 15 2021. III Iman E. Gharamti, John P. Dempsey, Arttu Polojärvi and Jukka Tuhkuri. Fracture energy of columnar freshwater ice: Inﬂuence of loading type, loading rate and size. Materialia, 101188, 20 2021. 11 List of Publications 12 Author’s Contribution Publication I: “Fracture of warm S2 columnar freshwater ice: size and rate effects” The author ran the models, analyzed the experimental and model results and wrote the ﬁrst draft of the paper. The other authors supervised the work and edited the manuscript. All the authors conducted the experiments. Publication II: “Creep and fracture of warm columnar freshwater ice” The author ran the model, analyzed the experimental and model results and wrote the ﬁrst draft of the paper. The other authors supervised the work and edited the manuscript. All the authors conducted the experiments. Publication III: “Fracture energy of columnar freshwater ice: Inﬂuence of loading type, loading rate and size” The author generated and analyzed the results and wrote the ﬁrst draft of the paper. The other authors supervised the work and edited the manuscript. 13 Author’s Contribution 14 List of Figures 1.1 (a) Kirsch stress distribution around a hole and (b) Inglis stress solution around an ellipse in an inﬁnite plate under uniaxial tension. 30 1.2 Westergaard stress solution around a crack in an inﬁnite plate under equibiaxial tension. The equation mentioned represents the stress along the crack plane. ........................ 31 1.3 linear elastic fracture mechanics (LEFM) stress solution near the tip of a crack under Mode I. K is the stress intensity factor. .... 32 1.4 Dugdale (left) and Barenblatt (right) cohesive models. In Dugdale’s model, the cohesive stress in the process zone (PZ) is constant at the yield strength (σ ) of the material. In Barenblatt model, the cohesive stress (σ(x)) is function of the distance from the traction- free crack tip. ........................... 33 1.5 FCM: (a) idealization of the cohesive crack; (b) stress distribution in the PZ before, and (c) after the development of a FDPZ; (d) stress-separation curve (). δ (2U ) and δ are the crack-opening- v v v displacement and the rate of crack opening, respectively. The subscript v indicates the viscoelastic characterization [PIII]. . . . 34 1.6 Strain responses to a (a) step stress applied between time t and t ; displaying (b) purely elastic, (c) elastic-viscoelastic, (d) elastic- viscoplastic, and (e) elastic-viscoelastic-viscoplastic behavior. The ﬁgure was edited as David Cole suggested. Deformation was changed into strain in Figs. 1.6d and 1.6e. The elastic recovery is now added to Fig. 1.6d. ....................... 37 1.7 General strain-time curve showing the three stages (primary, sec- ondary and tertiary) of time-dependent strain. Adapted from San- taoja (1990). ............................. 37 15 List of Figures 1.8 (a) Maxwell, (b) Kelvin-Voigt, and (c) Maxwell-Kelvin mechanical models. E and η are the elasticity and viscosity constants of the spring and dashpot, respectively. ................. 38 1.9 Chemical structure of ice Ih along the basal plane. ........ 38 1.10 Overview of the thesis structure. ................. 43 2.1 Aalto Ice Tank is a unique testing basin in size and shape. .... 45 2.2 Spraying setup consisting of pressure washers and barrels and placed inside the carriage. ..................... 46 2.3 Complete spraying setup attached to the moving carriage. .... 46 2.4 (a) The edge cracked rectangular plate of width H, length L, and crack length A used in the experiments. (b) The crack orientation and direction of crack propagation; also the columnar grains are shown. The dotted rectangular box represents the ice cut for storage and analysis (PI). .......................... 48 2.5 (a) Temperature proﬁle. Each data point represents the average of measurements taken at the same depth of diﬀerent ice cores throughout the one month duration of the test program. (b) Grain size distribution. Each data point is measured from one thin section (PII). ................................ 49 2.6 (a) Milling machine used to (b) mill the ice and (c) produce thin section. ............................... 50 2.7 Vertical thin sections photographed between crossed polaroids showing columnar freshwater ice (a) at the top of the ice sheet and (b) at the bottom of the ice sheet. The arrows indicate the growth direction. .............................. 50 2.8 Crystalline textures and associated c-axis orientation plots of hori- zontal thin sections photographed between crossed polaroids. The thin sections were taken from the (a) top, (b) middle, and (c) bot- tom of the ice sheet. In the type of axis projection plot used, a horizontal c-axis plots on the circumference and a vertical c-axis would be at the center. Each plot consists of 100 poles of the basal planes measured by a four-axis universal Rigsby stage. ....... 51 2.9 A sketch of the servo-controlled hydraulic loading system. .... 53 2.10 (a) Loading device; inductive displacement transducer (LVDT)s arrangement in a (b) 0.5m x 1m specimen and (c) 3m x 6m speci- men. The digital image correlation (DIC) pattern stamped on the ice surface is shown. ....................... 54 16 List of Figures 2.11 Crack path through the LVDTs in a (a) 0.5mx1m and (b) 3mx6m specimens, as observed directly on the ice surface. ........ 54 2.12 Vertical strain plots computed by DIC technique and showing clearly the crack path for (a) slow 0.5mx1m test (RP6), (b) fast 0.5mx1m test (RP3), (c) slow 3mx6m test (RP13), and (d) fast 3mx6m test (RP11) (see Table 5.1). The arrow indicates the direc- tion of crack propagation. ..................... 55 2.13 Horizontal thin sections showing a crack path (a) from 2 cm behind the initially-sharpened crack tip to 15 cm ahead of it and (b) from 15 cm to 35 cm ahead of the initial crack tip. Both thin sections are from a depth of 24 cm from the top of the ice sheet. The arrows indicate the direction of crack propagation. ............ 56 2.14 (a) Dried replica being peeled oﬀ the ice fracture surface and (b) its fractographic examination by scanning electron microscope (SEM). 56 3.1 The edge cracked rectangular plate of width H, length L, and crack length A (PI). ........................... 60 3.2 (a) Illustration of the ﬁctitious crack model. (b) Stress-separation curve describing the behavior in the PZ and showing the control points (see Eq. 3.25) used in the optimization procedure. (c) A ﬂowchart illustrating the forward and inverse problems used for matching the experimental and model results (PI). ........ 63 3.3 (a) Load-CMOD records for the 3m x 6m samples with loading −1 rates as indicated in the ﬁgure. The 39.3kPa ms and 1.83 −1 kPa ms data were smoothed by using moving averages (PI). (b) The pre-peak load-CMOD record. ................. 68 3.4 Load-displacement records for specimen RP4 loaded at rate 6.22 √ √ −1 −1 kPa ms (a) and for specimen RP3 loaded at rate 57.1kPa ms (b). Both specimen had the dimensions of 0.5m x 1m. Locations, where crack mouth opening displacement (CMOD), crack opening displacement at about half the length of the crack (COD), and near crack-tip opening displacement (NCOD)1 were measured, are shown in Fig. 3.1. ......................... 68 3.5 Measured apparent fracture toughness as a function of loading rate. First-order power-law ﬁts were applied separately to the data from the larger specimen (3m x 6m and 19.5m x 36m) and from the smaller specimen (0.5m x 1m) (PI). ................ 69 17 List of Figures 3.6 Measured initiation crack opening displacements at the crack mouth (a) and near the crack tip (b) as a function of loading rate. First- order power-law ﬁts were applied to the data. In (b), the power-law ﬁt was calculated separately for the larger specimen (both 3m x 6m and 19.5m x 36m) and from the smaller specimen (0.5m x 1m) (PI). Note that a small error in PI has been ﬁxed here: the power-law exponent in (b) for the small samples should be -0.03 instead of -0.01. ................................ 69 3.7 Nominal tensile strength σ as a function of loading rate (a) and specimen size (b). First-order power-law ﬁts were applied to the data. In (b), the data for the 0.5m x 1m and 3m x 6m specimen is obtained from the power law ﬁts at diﬀerent loading rates; the −1 loading rate of K = 0.871 kPa ms is indicated with the vertical dashed line in (a). ......................... 70 3.8 Optimization results of RP5. Convergence of (a) the cohesive stresses, (b) the critical crack opening displacement, (c) the creep compliance constant, and (d) the objective function after a certain number of iterations. ........................ 72 3.9 Experimental and model results for RP9 (3m x 6m). (a) Load at the crack mouth, see Fig. 3.1 and Eq. (3.1). (b) Displacement - time records. (c) Load - displacement records. The CMOD data was smoothed with moving average smoothing with a span of 10% successive data points. ....................... 72 3.10 Experimental and model results for RP14 (19.5m x 36m). (a) Load at the crack mouth, see Fig. 3.1 and Eq. (3.1). (b) Displacement - time records. (c) Load - displacement records. (The corresponding ﬁgures in PI are edited; (0,0) is added to the model plots.) .... 73 3.11 The crack opening displacement at crack growth initiation at X = A (a) and the actual fracture energy (b) as a function of loading rate. Viscoelastic ﬁctitious crack model was used in the analysis. First-order power-law ﬁts were applied separately to the data for the larger specimen (3m x 6m and 19.5m x 36m) and for the smaller specimen (0.5m x 1m) (PI). .................... 73 3.12 Stress-separation curves under FDPZ conditions for the 0.5m x 1m specimen (a) and for the 3m x 6m and 19.5m x 36m specimen (b) at diﬀerent loading rates. The number next to each curve reﬂects the index of the experiment in the Legend (PI). .......... 75 18 List of Figures 3.13 The attained stress-separation curves for the (a) 0.5m x 1m speci- mens and for the (b) 3m x 6m and 19.5m x 36m specimens. The number next to each curve reﬂects the index of the experiment in the Legend [PIII]. ......................... 75 3.14 (a) Experimental and VFCM results for the crack proﬁles and (b) PZ proﬁles by the VFCM for the 3m x 6m and 19.5m x 36m spec- imens. The total separation between the upper and lower crack surfaces is given by 2U ; U is half of δ which is the viscoelastic v v v crack-opening-displacement (see Fig. 1.5). X = A is the coordi- nate of the traction-free tip, and a is the normalized traction-free crack length (a = A /L). The number next to each curve reﬂects 0 0 the index of the experiment in the Legend [PIII]. .......... 77 3.15 Variation of the process zone size (PZ) with loading rate and speci- men size. First-order power-law ﬁts were applied separately to the data for the larger specimen (3m x 6m and 19.5m x 36m) and for the smaller specimen (0.5m x 1m) (PI). .............. 78 3.16 The growth of the normalized process zone with normalized time for the small, 0.5m x1m, specimens (a) and for the large, 3m x 6m and 19.5m x 36m, specimens (b). The process zone size was normalized by L and the time axis by the time to failure (t )of each experiment (Table 3.1). .................... 78 3.17 Notch sensitivity (σ /σ ) as a function of the normalized crack n t length for the 0.5m x 1m (a), 3m x 6m (b), and 19.5m x 36 m (c) specimens. Lines of constant brittleness numbers are shown. The notch sensitivity of σ /σ = 0.4, 1 are indicated with the horizontal n t dashed lines. The normalized crack length of each specimen size is shown by the vertical dashed line. (d) Notch sensitivity (σ /σ ) n t as a function of the loading rate. First-order power-law ﬁts were applied to the data for the 0.5m x 1m and 3m x 6m specimens (PI). 80 1/2 3.18 Variation of creep compliance (J = 1/E +Ct , given in Eq. 3.17) as a function of loading rate (K) for diﬀerent specimen sizes (PI). (J in PI was changed to J in the thesis to distinguish between the creep compliance (J ) and the J-integral (J)). ........... 83 4.1 (a) Specimen geometry, edge cracked rectangular plate of length L, width H, and crack length A (PII). ................ 86 4.2 Loading consisting of (a) creep-recovery and (b) cyclic sequences followed by a monotonic fracture ramp. The number above each segment indicates the duration in s (PII). .............. 87 19 List of Figures 4.3 Experimental results for the (a) peak load P , (b) crack mouth max opening displacement CMOD and (c) near crack tip opening dis- placement NCOD1 at crack growth initiation, as a function of time to failure t for the monotonically-loaded DC experiments (Chapter 3) and the creep/cyclic and monotonically-loaded LC experiments (PII). ................................ 93 4.4 Measured load versus CMOD for the (a) displacement control (DC) experiments (Chapter 3), (b) load control (LC) experiments, and (c) LC experiments up to the peak load (PII) ........... 93 4.5 Load versus CMOD over the (a) creep-recovery cycles for RP15 and the (b) cyclic-recovery sequences for RP17. (c) Schematic illustration of the hysteresis load-displacement diagram. The whole of the hysteresis loop area is the energy loss per cycle. The dashed area is the part of that total that is due to the viscoelastic mechanism and the rest is due to viscous processes (PII). ........... 94 4.6 Experimental results for RP16. (a) Load at the crack mouth, see Fig. 4.1. (b) Displacement - time records. (c) Load - displacement record. (d) Typical response of a Maxwell model, consisting of a nonlinear spring and nonlinear dashpot, to a constant load step. (e) CMOD vs time plot showing the ﬁrst creep sequence. [PII]. . . . 95 4.7 Experimental results for RP17. (a) Load at the crack mouth, see Fig. 4.1. (b) Displacement - time records. (c) Load - displacement record (PII). ............................ 96 4.8 Experimental and model results for RP16. (a) Load at the crack mouth, see Fig. 4.1. (b) CMOD - time records: Schapery’s model with and without viscoelasticity vs the experimental data (PII). . . 97 4.9 Experimental and model results for RP17. (a) Load at the crack mouth, see Fig. 4.1. (b) CMOD - time records (PII). ....... 98 4.10 Contribution of each individual model component to the total CMOD displacement for (a) RP16 and (b) RP17 (PII). ...... 98 5.1 Description of the area used for (a) J (A ) and (b) W (A ) Q J f W Q f calculations. Load versus crack mouth opening displacement for (c) RP4 and (d) RP12, illustrating the A and A areas [PIII]. . 105 J W Q f 5.2 Description of the area used for J (A ) calculations for metals, Q J as deﬁned by the ASTM E1820-20b. Adapted from ASTM E1820- 20b AST (2020). .......................... 105 5.3 Load-CMOD records for the (a) 0.5m x 1m DC (b) 3m x 6m DC, (c) 19.5m x 36m DC and (d) 3m x 6m LC experiments [PIII]. . . . 107 20 List of Figures 5.4 Variation of J and G as a function of the time to failure for Q VFCM the DC experiments [PIII]. ...................... 107 5.5 (a) The apparent fracture energy at crack growth initiation (J ) and (b) the work-of-fracture fracture energy (W ) as a function of the time to failure for the DC experiments. In (a), ﬁrst-order power-law ﬁts were applied separately to the data for the larger specimen (3m x 6m and 19.5m x 36m) and for the smaller specimen (0.5m x 1m) [PIII]. ................................ 108 5.6 (a) The apparent fracture energy at crack growth initiation (J ) and (b) the work-of-fracture fracture energy (W ) as a function of the time to failure for the 3m x 6m DC and LC experiments. First-order power-law ﬁts were applied to the data for the DC experiments. Note in (a), two LC experiments RP16 and RP17 gave the same J value [PIII]. ............................ 109 5.7 Fracture energy ratio W /J as a function of the time to failure for f Q the DC and LC experiments [PIII]. ................ 110 21 List of Figures 22 List of Acronyms CMOD crack mouth opening displacement COD crack opening displacement at about half the length of the crack CZ cohesive zone CZM cohesize zone model DC displacement control DIC digital image correlation ECRP edge-cracked rectangular plates FCM ﬁctitious crack model LC load control LEFM linear elastic fracture mechanics LVDT inductive displacement transducer N-M Nelder-Mead NCOD near crack-tip opening displacement PZ process zone SEM scanning electron microscope SIF stress intensity factor SSY small scale yielding VFCM viscoelastic ﬁctitious crack model 23 List of Acronyms 24 Symbols A total length of the crack, including the traction-free crack and the process zone A traction-free crack length A area under the load-CMOD curve up to crack growth initiation A area under the entire load-CMOD curve a normalized crack length by the crack-parallel dimension L (A/L) a normalized traction-free crack length by the crack-parallel dimension L (A /L) 0 0 a time scale shift factor in Schapery’s model C creep compliance constant C elastic compliance of Schapery’s model C viscoelastic compliance of Schapery’s model ve C viscoplastic compliance of Schapery’s model vp D length of contact between the ice and the loading device d normalized ice-loading device contact length by the crack-parallel dimension L (D/L) d average grain size av E short-term elastic modulus G fracture energy actually consumed ac G full fracture energy under fully-developed process zone conditions G fracture energy computed via the viscoelastic ﬁctitious crack model, the same VFCM as G ac 25 Symbols g , g , g , g nonlinear functions of the load in Schapery’s model 0 1 2 3 H width of the edge-cracked rectangular plate H weight function h thickness of the ice h normalized weight function J energy release rate given by the J-integral J creep compliance J apparent fracture energy at crack growth initiation via Rice’s J-integral K Mode I stress-intensity factor stress intensity factor (SIF) K Mode I stress-intensity factor due to the cohesive loading coh K critical SIF or fracture toughness IC K apparent fracture toughness K Mode I stress-intensity factor due to the external loading K loading rate L length, crack-parallel dimension, of the edge-cracked rectangular plate P applied force P peak or failure load max t time t time to failure U half of the crack-opening displacement W work-of-fracture X position along the crack x normalized position (X/L) β brittleness number δ crack-opening displacement, equals to 2U δ elastic component of the crack-mouth-opening displacement CMOD 26 Symbols ve δ viscoelastic component of the crack-mouth-opening displacement CMOD vp δ viscoplastic component of the crack-mouth-opening displacement CMOD δ critical crack-opening displacement, equals to 2U ,at X = A given a fully cr cr 0 developed δ viscoelastic crack-opening displacement δ viscoelastic crack-opening displacement due to the external loading vσ δ viscoelastic crack-opening displacement due to the cohesive loading process vcoh zone δ crack-opening displacement at crack growth initiation, at X = A 0 0 δ rate of crack-opening F objective function ν Poisson’s ratio ψ, ψ reduced times of Schapery’s model Σ crack face pressure on 0 ≤ X ≤ D σ normal tensile traction in the cohesive zone coh σ nominal tensile stress at the crack tip σ tensile strength σ /σ notch sensitivity n t σ yield strength 27 Symbols 28 1. Introduction 1.1 Motivation The main goal of this thesis is to gain a better understanding of the fracture behavior of columnar freshwater ice under diﬀerent loading scenarios. Understanding the deformation and fracture processes of columnar freshwater ice is important in many engineering problems. For example, freshwater ice sheets fracture when in contact with ships, river ice fractures during interaction with bridge piers, and thermal cracks form in lakes and reservoirs. In fact, climate change has led to a warmer, thinner, and broken ice. In addition, the applications like river ice breakup happen in late spring and the ice is very warm. Accordingly, the warming climate increases the importance of studying warm ice, and warming ice increases the importance of creep deformations. But, historically cold ice has been studied typically. It is well-known that the deformation and fracture processes of freshwater ice are highly dependent on temperature, strain rate, sample size, grain type and grain size. Qualitatively, high temperature and low strain rate lead to creep behaviour and ductile fracture; low temperature and high strain rate lead to elastic behaviour and brittle fracture. However, quantitatively these relations are not well known. Generally speaking, unless we restrict our interest on the short time scales where only elastic response is relevant, the creep deformations must be modeled to obtain the true fracture behavior. In materials with time-dependent properties, the fracture and creep deformations are coexistent. 1.2 Historical Background of Fracture Mechanics Fracture mechanics oﬀers a mathematical study of the science of cracks: nucleation, propagation, energy, stress ﬁelds, failures, etc. The ﬁeld started unoﬃcially at the turn of the 20th century with new analytical linear elastic solutions formulated for stresses 29 Introduction at holes in 1898, then at ellipses in 1913. Fracture mechanics research was oﬃcially born in 1920 with Griﬃth’s energy-based analysis of cracks. Later contributions were motivated by several crack-caused failures that happened during World War II and led to an expansion of research in the ﬁeld. Early research was focused on linear elastic fracture mechanics (LEFM). Nowadays, cohesive mechanics illustrated by the cohesize zone model (CZM) or the ﬁctitious crack model (FCM) is the basic nonlinear approach used to describe the fracture of quasibrittle materials. 1.2.1 Linear Elastic Fracture Mechanics (LEFM) Kirsch (1898) used linear elasticity in his study of stress concentrations and derived a linear elastic solution for stresses around a hole in an inﬁnite plate. His solution contains the well known factor-of-three stress concentration at the hole under uniaxial loading, as shown in Fig. 1.1a. ߪ ߪ ୫ୟ୶ ୶ ͵ߪ ͵ߪ ஶ ஶ ௬௬ ߪ ൌߪ ͳ൅ʹ ߪ ஶ (a) (b) Figure 1.1. (a) Kirsch stress distribution around a hole and (b) Inglis stress solution around an ellipse in an inﬁnite plate under uniaxial tension. The second step in the development of LEFM was done by Inglis (1913). Inglis derived a linear elastic solution for stress ﬁeld surrounding an ellipse in an inﬁnite plate under uniaxial tension, as portrayed in Fig. 1.1b. Inglis result predicts that the maximum stress at the ellipse’s tip goes to inﬁnity as the radius of curvature at the tip goes to zero forming a crack. Griﬃth (1921) used Inglis linear elastic solution to develop his energy-based failure criterion of cracks. He computed the strain energy release rate associated with crack growth and compared it with the work required to break the atomic bonds ୫ୟ୶ ୫ୟ Introduction and propagate the crack. Griﬃth’s criterion only applies to brittle materials because it doesn’t account for the energy dissipation associated with plastic deformation near the crack tip. This was later addressed by Irwin (1948) and Orowan (1949), who modiﬁed Griﬃth’s initial development by adding the energy due to plastic deformation at the crack tip. Westergaard (1939) applied an Airy stress function of complex numbers to derive a complete solution for the stress ﬁeld surrounding a crack in an inﬁnte plate under equibiaxial tension (Fig. 1.2). ߪ ߪ ஶ ஶ ʹܣ ܻൌͲǡܺ൐ܣ ׷ߪ ൌߪ ൌ ଴ ଶ ͳെሺ ሻ Figure 1.2. Westergaard stress solution around a crack in an inﬁnite plate under equibiaxial tension. The equation mentioned represents the stress along the crack plane. The stress values along the crack plane is given by a simple equation, as shown in Fig. 1.2. However, computing the stress at any other position near the crack requires a heavy work with complex functions. This hinders the general understanding of the stress state surrounding a crack. Irwin (1957) showed that Westergaard’s result could be greatly simpliﬁed in the area immediately surrounding the crack tip. Irwin’s approximate solution can replace Westergaard’s exact solution near the crack tip only. Irwin’s expressions lose its accuracy away from the crack tip. The importance of Irwin’s work lies in the fact that his solution led to the deﬁnition of the stress intensity factor (SIF), which is the most important parameter of LEFM. LEFM expressions for the crack tip stress ﬁeld under Mode I are shown compactly in Fig. 1.3, and it is clear that LEFM predicts an unavoidable singularity and inﬁnite stresses at the crack tip for any applied loading. In short, LEFM is a one-parameter fracture model, and it is based on the calculation of a critical stress intensity factor, fracture toughness, which measures a material’s resistance to fracture and is denoted by K . LEFM is limited to situations where IC two main conditions are satisﬁed: 1) an initial crack exists in the body and 2) the material behavior is linearly elastic and where any nonlinear zone at the crack tip is ୷୷ ୶୶ Introduction ʹܣ ܭ ܭ ߪ ൌ ݂ ߠǢ ߪ ൌ ݃ ߠǢ ʹߨݎ ʹߨݎ ߬ ൌ ݄ሺߠሻ ʹߨݎ Figure 1.3. LEFM stress solution near the tip of a crack under Mode I. K is the stress intensity factor. small compared to other dimensions of the specimen; a condition known as small scale yielding (SSY) (Rice, 1965). 1.2.2 Cohesive Mechanics As previously mentioned, LEFM has proven a useful tool for solving fracture problems provided that two conditions (a pre-existing crack and SSY) are satisﬁed. However, in a real material, the size of the nonlinear zone, due to creep and plasticity (ductile ma- terials) or microcracking (quasi-brittle materials), doesn’t always satisfy the condition of SSY. Besides, LEFM cannot be used to analyze bodies with no initial crack or with blunt notches [Elices et al., 2002]. Particularly speaking about quasi-brittle materials (such as ice, concrete, rock, and ceramic), the attempts to apply classical LEFM turned out to be misleading [Dempsey et al., 2018], and the fracture toughness results were very scattered and highly dependent on several factors; such as size, loading rate, grain size, temperature, tip acuity, etc. Thus, alternative nonlinear fracture models were needed. The cohesive zone model (CZM) or ﬁctitious crack model (FCM) proved successful to analyze the fracture of concrete, ice and other quasi-brittle materials. A review of the properties of the model was done by many researchers [Bazant and Planas, 1998; Bilby et al., 1963; Smith, 1974; Elices et al., 2002]. A brief presentation of the model development and its main ingredients is presented in the following paragraphs. LEFM claims that all fracture processes take place at the crack tip. On the other ୶୷ ୷୷ ୶୶ Introduction hand, cohesive mechanics eliminates the stress singularity of LEFM by assuming the presence of a cohesive zone (CZ) or process zone (PZ) ahead of the tip where all nonlinearity takes place. The physical crack is a traction-free crack, and the PZ is a ﬁctitious crack whose faces are bridged by unbroken ligaments and are able to transfer cohesive stresses. Dugdale (1960) developed a strip yield model for ductile materials in plane stress. He assumes the existence of a plastic zone at the crack tip. The plastic zone is a PZ whose behavior is represented by a uniform stress distribution equals to the yield strength of the material in a plane stress situation. In Dugdale’s PZ, the cohesive stress is maintained at a constant value irrespective of the crack opening displacement, as shown by the left side of Fig. 1.4. elastic Traction-free ߪሺݔሻ ߪ crack tips ʹܣ PZ PZ Figure 1.4. Dugdale (left) and Barenblatt (right) cohesive models. In Dugdale’s model, the cohesive stress in the PZ is constant at the yield strength (σ ) of the material. In Barenblatt model, the cohesive stress (σ(x)) is function of the distance from the traction-free crack tip. Later, Barenblatt (1962) developed a model for brittle materials analogous to the Dugdale model. However, he assumed that the cohesive stress inside the PZ is not constant, but varying with the distance from the traction-free crack tip, as shown by the right side of Fig. 1.4. Dugdale and Barenblatt models were restricted to the analysis of the fracture processes near the tip of a preexisting crack with the assumption of a small PZ in comparison to other dimensions in the structure. Hillerborg et al. (1976) built on the ideas of the pioneering work of Barenblatt and Dugdale and extended their approaches with two main generalizations. 1) The cohesive crack can develop anywhere in a specimen or a structure, and not only ahead of a preexisting crack tip. In other words, the model applies to all situations of short cracks, long cracks, and no cracks at all. 2) The small size requirement is relaxed, i.e. the process zone size can be large. Hillerborg developed the FCM or CZM. The FCM has proved successful in modeling the fracture behavior of several quasi-brittle materials. The key idea of the FCM is the concentration of all nonlinear fracture mechanisms into a localized process zone, 33 Introduction traction-free fictitious bulk crack crack material A PZ ߜൌʹܷ ߪ ൌ݂ߜሺ ǡ ߜ ሻ ୡ୭୦ ௩ ௩ ߜ ൏ߜ ௩ ௖ ߜ ൌߜ ௩ ௖ ݂ሺߜ ǡ ߜ ሻ ௩ ௩ ୡ୭୦ ݂ሺߜ ǡ ߜ ሻ ௩ ௩ Figure 1.5. FCM: (a) idealization of the cohesive crack; (b) stress distribution in the PZ before, and (c) after the development of a FDPZ; (d) stress-separation curve (). δ (2U ) and δ are the v v v crack-opening-displacement and the rate of crack opening, respectively. The subscript v indicates the viscoelastic characterization [PIII]. 34 Introduction located on the crack line, as shown in Fig. 1.5a. The length of a traction free crack is denoted as A and the total length of a crack including the process zone as A, giving a length of the process zone as PZ = A − A . The process zone initiates at a point where the maximum principal stress reaches the tensile strength (σ ) (Fig. 1.5a). The process zone develops with its surfaces being attracted to each other by cohesive stresses. The cohesive tensile stress (σ ) carried at a particular point within the PZ coh over the distance (A − A ) is a function of the separation of the PZ at that point. The points lying within the PZ ﬁrst transmit the full tensile strength, but then the stress transmitted softens with crack opening (Fig. 1.5b). The total separation between the upper and lower crack surfaces is denoted by δ = 2U; U is half of the crack opening. At some critical separation (δ , Fig. 1.5c), no tensile stress can be transmitted and the PZ is termed a fully developed process zone (FDPZ). The behavior inside the process zone is described by a constitutive relation, known as the stress-separation law (σ − δ law), which relates the softening stresses to the increasing crack opening displacements (Fig. 1.5d). The FCM assumes that there is a smooth closure of the surfaces within the cohesive zone, i.e. dδ/dX = 0at X = A (known as the closure condition). The main elements that characterize the cohesive model are shown in Fig. 1.5: the length of the process zone (PZ), the maximum cohesive stress (σ ) that the material coh in the process zone can transfer and which initiates the growth of the process zone (tensile strength, σ ), the critical displacement beyond which the material is fully separated and can transfer no more stresses (δ ), the shape of the σ − δ curve, and the fracture energy represented by the area under the σ − δ curve. Precisely speaking, Fig. 1.5d shows a σ−δ curve obtained under conditions in which a fully-developed-process-zone (FDPZ) has formed. The full fracture energy (G )is consumed. A FDPZ is achieved when two conditions are satisﬁed at X = A : σ = σ 0 coh t and δ = δ . Based on the testing conditions (load control versus displacement control, 0 c ice temperature, operating deformation mechanisms, etc.), either a FDPZ or a non- FDPZ is obtained before fracture occurs. In the case of a non-FDPZ, the cohesive stress at X = A does not drop to zero and the fracture occurs at δ <δ ; thus, only a 0 0 c portion of the σ − δ curve / G (Fig. 1.5d) is attained. Several approaches have been proposed to quantify the softening law and can be categorized into: direct (experimental) and indirect (experimental-numerical). A pure experimental method would involve stable tensile tests to determine the parameters of the σ − δ curve. However, this approach has proven diﬃcult and few reliable results have been obtained. As summarized by Elices et al. (2002) and Van Mier and Van Vliet (2002), several technical, material, and environmental problems arise and prevent the achievement of a stable tensile test. Some technical diﬃculties include: 1) 35 Introduction achieving the perfect alignment of the specimen in a system with very low eccentricity, 2) solving the gripping problem such that failure occurs far away from the platens, and 3) selecting a stiﬀ and stable test-control system. Other material and specimen related problems arise from 1) the heterogeneity of the material disrupting the uniform stress distribution across the specimen’s cross section and causing multiple cracking and 2) internal rotations of the specimen itself leading to asymmetric modes of fracture and nonuniform crack opening across the specimen. Environmental conditions of the specimen such as the temperature gradients can also cause deviation from uniform stress distribution. For these reasons, direct methods are replaced with indirect methods, typically through a combined experimental-numerical procedure. Indirect methods have been employed both in concrete and ice research and proved promising (Elices et al., 2002; Mulmule and Dempsey, 1997). These techniques apply inverse analysis in an iterative procedure to obtain the softening curve of the FCM. 1.3 Time-Dependent Behavior The material’s behavior is highly dependent on the loading scenario: loading rate, loading type (monotonic, creep, cyclic), and testing conditions (displacement vs load control). In fact, every scenario produces a new response of the same material. Fig. 1.6 displays diﬀerent strain responses to a step stress applied between time t and t 0 1 (Fig. 1.6a). Fig. 1.6b shows a purely elastic response, where the material behavior is independent of time. The material reacts instantaneously to the load application and removal. However, Figs. 1.6c, 1.6d, and 1.6e show a time-dependent behavior where the material response is not instantaneous but delayed with time. In these cases, the constitutive relation is given by the stress and the strain rate (instead of strain). General time-dependent deformation (Fig. 1.6e) consists of three components: elastic, viscoelastic (delayed elastic, anelastic), and viscoplastic (viscous). Elastic deformation is the instantaneous, time-independent and recoverable deformation. Vis- coelastic deformation is the time-dependent and recoverable deformation. Viscoplastic deformation is the time-dependent and permanent (unrecoverable) deformation. Costa Andrade (1910) developed the idea of dividing the time-dependent deforma- tion curve into three main stages. Each stage is characterized by diﬀerent deformation mechanisms. Division into stages allows to describe each stage separately and for- mulate an expression for each dominant deformation. Fig. 1.7 shows a general creep strain-time curve made up of four main stages: 1) Instantaneous stage is characterized by instantaneous purely elastic and recovered deformation (ε ), 2) primary (transient) stage is characterized by a decreasing strain rate and a time-dependent viscoelastic and ve recoverable deformation (ε ), 3) secondary (steady state) stage is characterized by a Introduction Figure 1.6. Strain responses to a (a) step stress applied between time t and t ; displaying (b) purely elas- 0 1 tic, (c) elastic-viscoelastic, (d) elastic-viscoplastic, and (e) elastic-viscoelastic-viscoplastic behavior. The ﬁgure was edited as David Cole suggested. Deformation was changed into strain in Figs. 1.6d and 1.6e. The elastic recovery is now added to Fig. 1.6d. constant strain rate and a time-dependent viscoplastic and unrecoverable deformation vp (ε ), and 4) tertiary stage is characterized by an increasing strain rate and accelerating te deformation (ε ) leading to complete failure of the material. Failure ௩௣ ௩௘ 2 3 4 Primary Secondary Tertiary Figure 1.7. General strain-time curve showing the three stages (primary, secondary and tertiary) of time-dependent strain. Adapted from Santaoja (1990). Creep behavior is represented by mechanical models of springs and dashpots, mod- elling the elastic and time-dependent behavior, respectively. The two simplest models are the Maxwell and Kelvin-Voigt models which consist of a spring and dashpot in series and parallel, respectively, as shown in Figs. 1.8a and 1.8b. The other general- ized models (for example, Fig. 1.8c) are constructed by considering more and more elements. The more elements the model has, the more accurate it will be in describing ௧௘ Introduction ܧ ߟ (a) (b) (c) Figure 1.8. (a) Maxwell, (b) Kelvin-Voigt, and (c) Maxwell-Kelvin mechanical models. E andη are the elasticity and viscosity constants of the spring and dashpot, respectively. the real response of real materials. 1.4 S2 Columnar Freshwater Ice Ice exists in a number of forms, but only the ordinary ice Ih is present in a signiﬁcant amount in nature. In ice Ih, the oxygen atoms occupy the points of a hexagonal lattice in which each oxygen atom is tetrahedrally coordinated with four other oxygen atoms. Each oxygen atom is associated with two hydrogen atoms, as shown in Fig. 1.9. The oxygen molecules are most closely packed on basal planes, situated perpendicular to the principal hexagonal axis, called the c-axis. The c-axis also coincides with the optical axis of each ice crystal. Figure 1.9. Chemical structure of ice Ih along the basal plane. Freshwater ice possesses important optical properties: birefringence, uniaxial (one optical axis), and transparent (especially when bubble-free). Birefringence means that the ice exhibits the property of double refraction when it is examined under polarized light. The interference colors under cross-polaroids depend on the orientation of the crystals and allows individual grains in polycrystalline ice to be identiﬁed. This property is crucial in the analysis of thin sections to investigate the crystalline texture, c-axis orientation, and the ice type. The ice type examined in this thesis is columnar freshwater S2 ice. As deﬁned by Michel and Ramseier (1971), S2 ice is columnar polycrystalline ice with a random and horizontal c-axis orientation in most of its grains. S2 ice individual crystals exhibit the 38 Introduction same color but with diﬀerent intensities under cross-polarized light [Shestov, 2018]. 1.5 Fracture Behavior of S2 Columnar Freshwater Ice Generally speaking, freshwater ice is a quasi-brittle material that tends to be fairly brittle and fails by microcracking instead of yielding. However, the exact failure mode depends critically on the temperature and loading conditions, including grain size, thickness, loading rate, crack orientation, loading direction, notch acuity, loading conﬁguration, and specimen size. Investigations of their eﬀects have been studied by Liu and Loop (1972), Goodman and Tabor (1978), Liu and Miller (1979), Hamza and Muggeridge (1979), Urabe and Yoshitake (1981), Timco and Frederking (1982), Danilenko (1985), Parsons and Snellen (1985), Timco and Frederking (1986), Nixon and Schulson (1986), Tuhkuri (1987), Bentley et al. (1988), Dempsey et al. (1989), Parsons et al. (1989), Wei et al. (1991), Weber and Nixon (1996a,b), and Xu et al. (2004). More than ﬁve decades ago, Weeks and Assur (1969) stated that "We feel that an understanding of the scale eﬀect in ice testing is essential before a thorough scientiﬁc basis can be developed for the utilization of small-scale testing in engineering design problems". This requires a large range in specimen size to be tested in order to reveal the scale eﬀect on the tensile fracture characteristics. However, most of the earlier studies with freshwater ice have used specimens with dimensions of tens of centimeters, and large scale in-situ fracture experiments examining the scaling problem with freshwater S2 ice have not been conducted before. Dempsey (1991) provided a summary of the experimental work on fracture of freshwater ice and suggested that many laboratory specimens may have been too small in terms of the requirement of LEFM. It is important to note that a scale eﬀect may not occur unless the specimen sizes are large enough (Dempsey et al., 1999a). In addition, the general focus of the previous work with freshwater ice has been to characterize the mode I fracture behavior in terms of fracture toughness using LEFM. Most of these toughness studies relied heavily on the principles of LEFM, without ensuring that the necessary requirements have been satisﬁed (Dempsey et al., 1992). By conducting large-scale in-situ fracture experiments with sea ice, Dempsey et al. (Mulmule and Dempsey, 1999; Dempsey et al., 2018) eventually questioned the applicability of a one parameter fracture mechanics for sea ice, ultimately favoring a viscoelastic ﬁctitious crack model. These, and other, observations led to the suggestion that the fracture of small and large ice specimen may not be similar, and that the applicability of LEFM to even freshwater ice may be limited. At the core of this discussion is the requirement of LEFM that the material behaviour is linearly elastic 39 Introduction except in a small area near a crack tip (Dempsey et al., 2018; Dempsey, 1991). Qualitatively, it can be argued that these conditions prevail when the loading rate is high and the specimen is large. Quantitatively, we know little as to what is a high rate, how large is a large specimen, and how these two questions are aﬀected by temperature, size and type of the ice (granular or columnar), and other parameters. While the viscoelastic ﬁctitious crack model had proved useful for ﬁrst-year sea ice (Dempsey et al., 2018), it is unknown if it can model the fracture of columnar freshwater ice. In addition, the stress-separation curve for this type of ice has not yet been determined. Another important concern relates to the requirements (number of grains and spec- imen size) needed to ensure notch sensitivity and homogeneous response of the polycrystal. For a specimen to be notch sensitive, the crack and uncracked ligament must both be long enough for the specimens to fail by fracture (Dempsey et al., 1992). The dependence of notch sensitivity on specimen size was discussed by (Carpinteri, 1982) for concrete fracture. However, the combined eﬀects of size and rate on notch sensitivity have not been discussed before. Polycrystalline homogeneity requires that the crack length and the uncracked ligament are signiﬁcantly larger than the grain size, and the specimen contains suﬃcient number of grains to be regarded as homo- geneous (Dempsey, 1991; Abdel-Tawab and Rodin, 1993, 1998). As discussed by Dempsey et al. (1999b), polycrystallinity is an issue with large grained ice. Mulmule and Dempsey (2000) speculated that sample homogeneity for ﬁrst-year sea ice would be obtained if the crack-parallel specimen size harboured at least 200 d , where d av av is the average grain size. For the tensile and compressive testing of un-notched ice cylinders, it is recommended that the cylinder diameter be 15 to 20 times d (Schwarz av et al., 1981). This highlights the huge diﬀerence posed by the testing of cracked and uncracked test samples. As soon as one is testing a precracked test sample, the requisite specimen size is measured not in the 10’s of grain size, but the 100’s of the grain size. The requisite test size for polycrystalline homogeneity and notch sensitivity for large grained and warm columnar freshwater ice, considering size and rate eﬀects, is unknown. 1.6 Time-dependent Behavior of S2 Columnar Freshwater Ice As discussed earlier, freshwater ice can exhibit a strong time-dependent behavior; and accordingly, a general constitutive model should incorporate elastic, viscoelastic, and viscoplastic components (Jellinek and Brill, 1956; Sinha, 1978). The importance of each component depends on the problem studied. For example, thermal deformations of ice in dams can have a time scale of a few days and viscous behaviour dominates. 40 Introduction In ice-structure interaction problems, the time scale of interest is often seconds and hours, so all three components of deformation need to be modeled. The time-dependent behavior of freshwater ice has been addressed with great atten- tion, and several constitutive models were developed (Michel, 1978a; Sinha, 1978; Le Gac and Duval, 1980; Ashby and Duval, 1985; Sunder and Wu, 1989; Mellor and Cole, 1983; Cole, 1990; Duval et al., 1991; Sunder and Wu, 1990; Abdel-Tawab and Rodin, 1997; Santaoja, 1990). Constitutive laws can be phenomenological or micromechanical. Micromechanical modeling in ice faces challenges because the characterization of the microscopic mechanisms of ice deformation is still inadequate (Abdel-Tawab and Rodin, 1997). Phenomenological laws are classiﬁed into two groups. The ﬁrst group are empirical- based relations (Sinha, 1978; Schapery, 1969). Their equations relate macroscopic variables: stress/load, strain/displacement, and time. They do not contain state vari- ables that describe the internal state of the material. The functions in these models can be easily calibrated to simulate the experiments. The second group of phenomeno- logical models starts from physically-based models involving internal state variables (dislocation density, internal stresses reﬂecting hardening, etc ...); they develop dif- ferential equations for the evolution of these variables with time and quantify the dependence of these variables on stress, temperature and strain (Le Gac and Duval, 1980; Sunder and Wu, 1989, 1990; Abdel-Tawab and Rodin, 1997). These models provide insights into the microscopic mechanisms taking place, and the state vari- ables describe the deformation resistance oﬀered by changes in the microstructure of the material. However, they require a proper identiﬁcation of the deformation mechanisms. The eﬀect of time-dependent loading on the strength of freshwater ice has been examined in the literature. Subjecting freshwater ice to cyclic loading apparently leads to a signiﬁcant increase in the tensile, compressive, and ﬂexural strength of that ice (Murdza et al., 2020; Iliescu et al., 2017; Iliescu and Schulson, 2002; Cole, 1990; Jorgen and Picu, 1998). On the other hand, no detailed investigation of the eﬀect of creep and cyclic loading on the fracture properties of freshwater ice has been conducted in the past. 1.7 Objectives, Scope and Structure of the Thesis The thesis explores the fracture and creep response of warm columnar freshwater S2 ice under diﬀerent loading scenarios under laboratory conditions. Large scale experiments were conducted in the Ice Tank of Aalto University. The work is directly relevant to a number of practical problems of freshwater ice (Ashton, 1986), but has 41 Introduction also general relevance in ice research by studying the coupled creep and fracture in a quasi-brittle material. Two main factors make the conducted study unique. The studied size range (1:39, largest size 19.5m x 36m) is the largest to any material under laboratory conditions. Another factor is the very warm temperature of the ice (-0.3 C at the top surface). As discussed, the behavior of warm ice is inevitably complicated by the presence of creep, especially for slower loading rates. The tensile fracture behavior of warm columnar freshwater ice is still a relatively unexplored topic, especially regarding the inﬂuence of loading type, loading rate, and specimen size. Sections 1.5 and 1.6 presented several limitations and unanswered questions in earlier research of columnar freshwater ice. To ﬁll the afore-mentioned gaps and to obtain a better understanding of the mechan- ical behavior of columnar freshwater ice, the thesis addresses the following research questions: • RQ1. What are the eﬀects of plate size and loading rate on the fracture behavior of columnar freshwater ice? Are these eﬀects independent or interrelated? No previous study of this ice examined both eﬀects simultaneously or their interrelation. (Chapter 3) • RQ2. Is LEFM a valid model to analyze the fracture data of columnar freshwater ice? (Chapter 3) • RQ3. Is viscoelastic ﬁctitious crack model (VFCM) applicable to study the fracture of columnar freshwater ice? (Chapter 3) • RQ4. What are the requirements and the requisite test size for notch sensitivity and polycrystalline homogeneity for large grained columnar freshwater ice? (Chapter 3) • RQ5. What is the inﬂuence of time-dependent loading (creep/cyclic-recovery) on the fracture properties i.e. the failure (peak) load and the crack opening displacements at crack growth initiation? (Chapter 4) • RQ6. How do the testing conditions aﬀect the creep stages (primary/transient and steady-state/secondary) present in the ice? What is the extent to which the elastic, viscoelastic and viscoplastic components contribute to the ice deformation? (Chapter 4) 42 Introduction • RQ7. What is the inﬂuence of loading type (monotonic, creep and cyclic), loading rate, and specimen size on the fracture energy of columnar freshwater S2 ice? (Chapter 5) In order to answer these questions, a program of nineteen mode I fracture exper- iments, using deeply cracked edge-cracked rectangular plates (ECRP), that varied the test sizes, loading types, and loading rates was conducted. The ice thickness was homogeneous throughout the parent ice sheet, but it grew from 340 mm to 410 mm during the test program. The ice was very warm; the ice surface was slightly cooler than -0.3 C while the bottom surface, the ice-water interface, manifested an ice temperature of 0 C . Each initial notch was fabricated and tip-sharpened to be 70− 75% of the crack-parallel dimension. The ice was loaded in the direction normal to the columnar grains. The test program was divided into two parts. In the ﬁrst part, fourteen experiments were conducted in displacement control (DC) and loaded with diﬀerent rates monotonically to fracture. The loading rates applied led to loading durations from fewer than 2 seconds to more than 1000 seconds. The plates covered a size range of 1:39 with three plate sizes: 0.5m x 1m, 3m x 6m and 19.5m x 36m. In the second part, ﬁve experiments of 3m x 6m ECRP were loaded in load control (LC) under creep/cyclic-recovery loading and monotonic loading to fracture. Creep and cyclic sequences were applied maintaining peak loads well below the failure loads, followed by monotonic ramps leading to complete fracture of each specimen. Chapter 3 Chapter 2 {RQ1, RQ2, RQ3, RQ4} Monotonic loading Ice growth/properties Loading rate effect Size effect Microstructural/ LEFM vs VFCM Fractographic Notch sensitivity characterization Lar Larg ge-scale in-situ e-scale in-situ Polycrystalline mode I fracture mode I fracture Experimental setup homogeneity experiments of experiments of warm warm columnar columnar Chapter 4 {RQ5, RQ6} freshwater ice freshwater ice Chapter 5 Creep/cyclic- {RQ7} Effect on fracture recovery loading properties Influence of loading type, loading rate, and test size Deformation Creep stages on the fracture energy components Figure 1.10. Overview of the thesis structure. The rest of the thesis is structured as shown in Fig. 1.10. Chapter 2 presents descriptions of the ice growth, ice properties, microstructural and fractographic char- acterization, and the experimental setup. Chapter 3 discusses the ﬁrst part of the test 43 Introduction program: the DC monotonic experiments. Linear elastic fracture mechanics is derived and applied as a one-parameter fracture mechanics approach to derive the expressions of the apparent fracture toughness and the crack opening displacement. Then, the chapter introduces a non-linear viscoelastic ﬁctitious crack model and explains how it is used to analyze the experiments. The experimental and model results with the notch sensitivity and homogeneity analysis are reported and analyzed. Chapter 4 discusses the second part of the test program: the LC creep/cyclic-recovery experiments. The ice response is examined under the testing conditions, and Schapery’s model is used to analyze the experimental data. Chapter 5 uses the results of Chapters 3 and 4 to quantify the fracture energy, of all the experiments using diﬀerent methods, and discuss its sensitivity to loading rate, loading type and plate size. The last chapter summarizes the main ﬁndings of the thesis and speculates on some possible future research. 44 2. Grown Ice and Experimental Setup [PI and PII] This chapter presents detailed descriptions of the ice growth, the ice properties, the microstructural and fractographic analysis, and the experimental setup of the test program. The chapter has been adapted from the author’s publications: PI and PII. 2.1 Experiments Large-scale in-situ experiments were conducted in the Aalto Ice Tank, a 40 m x 40 m, 2.8 m deep water basin equipped with a cooling system and a moving measuring carriage (Fig. 2.1). While ﬁeld work with ice provides results from a natural material in the real environment, laboratory studies allow a control of test parameters and ice texture not possible in the ﬁeld. Cooling fans Carriage Grown ice Figure 2.1. Aalto Ice Tank is a unique testing basin in size and shape. 45 Grown Ice and Experimental Setup [PI and PII] 2.1.1 Ice Spraying Trials Before the actual experiments, several ice spraying trials were carried out during two months. The spraying was conducted from the measurement carriage. The spraying system consisted of two pressure washers with a maximum pressure of 145 bar and a water ﬂow of 500 liters per hour. Both pressure washers were connected to four water barrels, 200 liters each, with a linking hub, as shown in Fig. 2.2. Carriage Pressure washer Linking hub Barrel Figure 2.2. Spraying setup consisting of pressure washers and barrels and placed inside the carriage. The two pressure washers were connected with rods to spraying nozzles and attached to the carriage, which runs along the length of the basin, as shown in Fig. 2.3 . The rods and the spraying heads were wrapped with foam-covered heating cables to prevent freezing. Spraying rods and head wrapped into Nozzle foam Pushing Scrapers plates Cutting Fishing rods line of the connecting the boom scrapers to the pushing plates Booms Figure 2.3. Complete spraying setup attached to the moving carriage. Basic guidance for growing S2 ice was provided by Gow et al. (1988). The spraying 46 Grown Ice and Experimental Setup [PI and PII] trials helped to understand the conditions needed to generate columnar S2 ice. Some of these factors are: the 1) temperature of sprayed water, 2) volume of sprayed water, and 3) spraying speed. The monitoring of these three parameters helps to control the resultant ice type and grain size. The large-grained ice that forms without spraying should be cleared away before the real spraying. This ice was pushed away using some pushing plates and installed booms (Fig. 2.3), and scrapers were used to cut the ice loose from the boom. Besides, enough water should be sprayed into the air to prevent the formation of open water spots which resulted in large grains’ areas. To do so, the spraying speed was reduced to increase the number of areas where the spraying units overlap; this generated areas of small grains. Moreover, the water droplets sprayed into the air should be ﬁne and cold enough to freeze in the air before touching the water surface. This requires the basin’s water and air temperature to be low enough. The sprayed water droplets were sprayed high enough to stay longer in the air. The spraying nozzles were approximately one meter above the water surface, and the pressure was high enough to push the sprayed water close to the roof. This prevented the remaining of open water areas and allowed the ice to nucleate once the frozen droplet hits the surface. Furthermore, the water in the basin should be as calm as possible without any current or circulation. This was achieved by stopping the air bubbling systems of the basin some time before the spraying started. 2.1.2 Ice Growth, Thickness, and Temperature A sheet of freshwater S2 ice was grown. Protective piping with warming cables was installed along the walls of the tank to protect the ice and the tank itself from any damage during the ice growth. The growth process was initiated by lowering the air temperature in the ice tank to −14 C and bubbling air into the water to obtain a uniform water temperature. When a water temperature of 0.2 C was reached, the air bubbling system was shut oﬀ and after approximately 15 minutes a ﬁne mist of water droplets was sprayed into the air above the basin. Once the droplets reached the water surface, they acted as nuclei for ice crystals to form. The temperature of the sprayed water was around 2 C and around 1200 liters of water was sprayed into the air. After four weeks at −14 C, the initial seeded ice layer had developed into a homogeneous sheet of bubble-free columnar S2 ice with a thickness of approximately 34 cm. For the experiments, the ambient temperature was raised to −2 C and maintained at that temperature. The ice thickness was homogeneous throughout the sheet. But, as the test program lasted about a month, the ice sheet grew from the 34 cm to 41 cm during the testing duration. The ice thickness at the crack tip was measured before each test and is reported in Table 3.1. Cutting all the specimens from a single 47 Grown Ice and Experimental Setup [PI and PII] H = 2L Dimensions in mm NCOD3 60-100 NCOD2 NCOD1 COD A = 0.7-0.75 L 0.5A D = 150 CMOD 50-80 (a) (b) Figure 2.4. (a) The edge cracked rectangular plate of width H, length L, and crack length A used in the experiments. (b) The crack orientation and direction of crack propagation; also the columnar grains are shown. The dotted rectangular box represents the ice cut for storage and analysis (PI). homogeneous ice sheet ensured the same grain size and structure for all the specimens. Except specimen size, loading rate, and the gradually increasing thickness, all ice and test parameters were constant. After completing each experiment, a through-the-thickness block of ice was cut from both sides of the crack path as illustrated by the dotted rectangular box in Fig. 2.4b. These blocks extended from 5 cm behind up to 15 − 35 cm ahead of an initial crack tip and allowed the measurement of thickness proﬁles along the crack paths: the ice thickness varied only a few millimeters along each crack path. The ice blocks studied were clear and transparent; there were practically no air bubbles. The ice blocks were stored in triple plastic bags to prevent sublimation and placed in freezers operating at -20 C Care was taken to mark the correct orientation of the blocks. The vertical temperature proﬁle of the ice sheet was measured daily, at diﬀerent locations in the ice sheet. Fig. 2.5a shows the temperature proﬁle throughout the whole test program. Each data point represents the average of several measurements taken at the same depth of diﬀerent ice cores. The ice temperature varied more or less linearly with depth but did not vary much during the test program. It is worth emphasizing that this ice temperature, −0.3 C at the top surface, is very warm in comparison to the majority of laboratory tests performed to date. 2.1.3 Grain Size and C-axis Orientation The microstructure of the ice was analysed by making and studying thin sections in a cold room. The conventional way of thinning ice with a microtome was replaced by a milling machine (Figs. 2.6a and 2.6b). The milling machine proved to be successful in producing a good quality surface ﬁnish (2.6c), saving the long working hours with a microtome, and generating larger thin sections than with a microtome. This helped to visualize the complete crack path. In this method, ice sections were frozen on a 48 Grown Ice and Experimental Setup [PI and PII] 0 0 -5 -10 -10 -15 -20 -20 -25 -30 -30 -35 -40 -40 02468 10 -0.4 -0.3 -0.2 -0.1 0 (a) (b) Figure 2.5. (a) Temperature proﬁle. Each data point represents the average of measurements taken at the same depth of diﬀerent ice cores throughout the one month duration of the test program. (b) Grain size distribution. Each data point is measured from one thin section (PII). plexiglass plate (2.6c), instead of a glass plate. The plexiglass used has as good optical properties as glass, if not better, and provide safer operations. The ice froze fast to the plexiglass and formed a strong bond. With the milling machine, the thickness of the ice sections was reduced to about 0.8 mm, as recommended by Shestov (2018). The thin sections were then examined and photographed under cross-polarized light. Figs. 2.7 and 2.8 show vertical and horizontal sections, respectively. The columnar structure of the ice, with the diameter of the grains increasing with depth, is evident. Several methods can be used to estimate the grain size of ice (Cole, 1986). In the present work, grain size was estimated by using the uniform-sphere-assumption method which relies on estimating the number of grains within a unit area and as- suming that the grains are of uniform size and spherical shape. Fig. 2.5b shows the diameter of the columnar grains in horizontal planes at diﬀerent depths. The grain size varied between 3 mm at the top and 10 mm at the bottom portion of the ice sheet. The mean grain diameter (d ) was 6.5 mm. av An examination of the c-axis orientation was carried out by analyzing hundreds of grains at diﬀerent locations and depths along the ice sheet. The crystalline texture and c-axis orientation plots from horizontal thin sections are shown in Fig. 2.8. Each Shmidt net in Fig. 2.8 consists of 100 poles of the basal planes measured with a four-axis universal Rigsby stage from the corresponding horizontal sections - top, middle, and bottom - randomly taken from the ice sheet. Refraction corrections, following Kamb (1962), were applied for the universal stage measurements. In the type of axis projection plot introduced by Langway (1958), a horizontal c-axis would be on the circumference and a vertical c-axis would be at the center. It can be seen that the c-axes of the columnar grains were randomly horizontal. The ice sheet had the same type of textural features throughout the depth and for the whole ice sheet: 49 Grown Ice and Experimental Setup [PI and PII] (a) (b) (c) Figure 2.6. (a) Milling machine used to (b) mill the ice and (c) produce thin section. 10 mm 10 mm (a) (b) Figure 2.7. Vertical thin sections photographed between crossed polaroids showing columnar freshwater ice (a) at the top of the ice sheet and (b) at the bottom of the ice sheet. The arrows indicate the growth direction. 50 Grown Ice and Experimental Setup [PI and PII] the ice was columnar S2 ice. 10 mm 10 mm 10 mm Figure 2.8. Crystalline textures and associated c-axis orientation plots of horizontal thin sections pho- tographed between crossed polaroids. The thin sections were taken from the (a) top, (b) middle, and (c) bottom of the ice sheet. In the type of axis projection plot used, a horizontal c-axis plots on the circumference and a vertical c-axis would be at the center. Each plot consists of 100 poles of the basal planes measured by a four-axis universal Rigsby stage. 2.1.4 Specimen’s Geometry The edge-cracked rectangular plate (ECRP) specimen was used to study the fracture of freshwater ice (Fig. 2.4a). Dempsey and Mu (2014) outlined the theoretical 51 Grown Ice and Experimental Setup [PI and PII] development of the weight function for this geometry with width to length (H/L) ratios of (Fig. 2.4a): 0.25, 0.5,1,1.5, 2, and 4. The weight function is a method that uses the knowledge of a two-dimensional elastic crack solution (the reference solution for any reference crack face pressure) to determine the expressions of the stress intensity factor and crack opening displacement for the same body under arbitrary loading. A wider plate with H/L = 2 was selected for the current study because it oﬀers a higher potential to obtain straight crack path stability and make the crack propagate in a self-similar manner (Lu et al., 2015). The geometry was self-equilibriating; the specimen was completely free-ﬂoating and no supports at the edges were required. 2.1.5 Specimen Preparation and Experimental Procedure The mode I fracture tests with loading at the crack mouth were conducted with diﬀerent loading rates and three specimen sizes: 36m x 19.5m, 6m x 3m, and 1m x 0.5m. The samples covered a size range of 1 : 39 which is the largest employed in laboratory ice experiments. The loading rates applied led to loading durations from fewer than 2 seconds to more than 1000 seconds. The ECRP specimens were cut from the parent ice sheet with an electric chain saw. The cuts were made suﬃciently large to prevent fast refreezing between the specimen and the parent ice sheet, and the kerfs around each specimen were kept clear of ice by using shovels. Cracks were cut with the electric chain saw giving a crack of width 8 mm. Directly before each test a sharp crack tip was produced using either a carpet knife with a blade width of 0.71 mm, or a hand saw with a blade width of 1.21 mm intended to be used for lightweight concrete. The length of the sharpened crack was A ≈ 0.7 − 0.75 L, where L is the crack-parallel side length. The ﬁnal sharpening can help to nucleate microcracks along the crack front, thus eﬀectively creating a very sharp and more realistic crack tip. No signiﬁcant blunting of the tip was visually observed during the test. The plane of the crack was perpendicular to the plane of the ice sheet; the crack front was vertical as shown in Fig. 2.4b. The ice was stored as a ﬂoating parent ice sheet until cut into a specimen, which was subsequently instrumented and tested, before the next specimen was cut out of the parent ice sheet. A closed-loop servo-controlled hydraulic loading device (Fig. 2.9) was installed in a rectangular slot cut at the crack mouth to apply pressure on the crack faces (Figs. 2.4a and 2.10a). The length of the contact between ice and the loading device (D = 150 mm, Figs. 2.4a) was the same for each specimen size and was chosen to keep the contact pressure below 0.5 MPa to avoid crushing of the crack faces. The contact pressure was estimated before the experiments by Equation (3.8) introduced in Chapter 3. Two hydraulic cylinders with maximum forces of 25 kN and 39 kN in compression and two load cells with maximum capacities of 25 kN and 50 kN were used. The 52 Grown Ice and Experimental Setup [PI and PII] feedback displacement-control measurement system control signal feedback coming from the measurement load-control system servo control connected to data-measurement servo system pressure under plate for servo connected to the valve (manifold) cable pressure unit outlet Force Transducer displacement transducer Figure 2.9. A sketch of the servo-controlled hydraulic loading system. smaller cylinder and load cell were used for the smallest sized specimens. Direct measurement of the crack opening displacements was carried out using LVDTs positioned at six locations (Figs. 2.4a, 2.10b and 2.10c): at the centerline of the loading device (the control displacement), at the crack mouth (CMOD), at about half the length of the crack (COD), 10 cm behind the initial sharpened crack tip (NCOD1), 6 − 10 cm ahead of the tip (NCOD2), and 20 cm ahead of it (NCOD3). NCOD denotes near-crack-tip opening displacement. LVDTs with a measuring range of 0 − 2 mm and a resolution of ±3 μm were used. In addition, parallel LVDTs with a larger range were used as secondary sources of data and as backup. No major discrepancies between the primary, high resolution transducers and the secondary transducers were observed. All the LVDTs were mounted on 14.5 mm diameter wooden sticks that were frozen into holes in the ice, on both sides of the crack. The positions of the LVDTs, including the elevation above the ice surface, were kept constant for all the tests. The data were sampled at 1000 Hz. The crack propagation was monitored using the Digital Image Correlation (DIC) method, which is a whole-ﬁeld measurement technique. The region in the vicinity of the initially sharpened tip was stamped with optimized pattern with a feature size of 3-5 pixels. Generation of the pattern was based on the methodology discussed by Bossuyt (2013). The objective was to generate a pattern with sharp features and high contrast. Taking advantage of the fact that freshwater ice is transparent, a thin hinge plate piston rod amplifier hoses pipes hinge amplifier cylinder plate Grown Ice and Experimental Setup [PI and PII] (a) (b) (c) Figure 2.10. (a) Loading device; LVDTs arrangement in a (b) 0.5m x 1m specimen and (c) 3m x 6m specimen. The DIC pattern stamped on the ice surface is shown. layer of white titanium oxide mixed into water was applied to the ice surface as the base layer, and black carbon powder mixed into water was used to stamp the black features of the pattern, as shown in Figs. 2.10c and 2.11. High speed cameras, placed normal to the deformation surface, have been used to record the images of the deformation surface. Frame rate was selected at 300 frames per second. (a) (b) Figure 2.11. Crack path through the LVDTs in a (a) 0.5mx1m and (b) 3mx6m specimens, as observed directly on the ice surface. In all the tests, the cracks propagated more or less straight, approximately along the x-axis and through the gauges NCOD2 and NCOD3, as shown in Figs. 2.4 and 2.11. Regardless of the specimen size and loading rate, straight cracking path characterized the fracture, as clearly illustrated by the vertical strain (perpendicular to the X-axis, Fig.2.4a) plots in Fig. 2.12, as computed by DIC on top of the ice sheet for diﬀerent tests’ conditions (Ahmad, 2019). This conﬁrms that the aspect ratio of H/L = 2 used in the tests supports a straight crack propagation. 54 Grown Ice and Experimental Setup [PI and PII] (a) (b) (c) (d) Figure 2.12. Vertical strain plots computed by DIC technique and showing clearly the crack path for (a) slow 0.5mx1m test (RP6), (b) fast 0.5mx1m test (RP3), (c) slow 3mx6m test (RP13), and (d) fast 3mx6m test (RP11) (see Table 5.1). The arrow indicates the direction of crack propagation. It is worth noting that the choice of the gauge setup (Fig. 2.4a) is linked to the modelling procedures and analysis in Chapters 3 and 4. The CMOD, COD and NCOD1 records are important for the implemented VFCM (Chapter 3) and Schapery’s model (Chapter 4). The matching of the measured records with the models’ results leads to the back-calculation of the models’ parameters. The NCOD1, NCOD2 and NCOD3 records are important for the validation of the DIC measurements in the vicinity of the tip. These experiments were the ﬁrst attempt to apply DIC to study the fracture of ice. Unfortunately, the trial was not very successful. The DIC nicely showed the growth of the crack; however, the resolution of the DIC system was not high enough to measure reliably the deformation. 2.1.6 Fractographic Examinations The ice samples cut from both sides of the crack path were used to study the crack propagation by making horizontal thin-sections as sketched in Fig. 2.4b. Some of the ice samples were cut immediately after an experiment, others were cut one or two days after an experiment. As a result, the crack healed in some samples, and in other samples, the right and left sides of the crack were unattached. It was easy to make thin sections with a healed crack. For unhealed cracks, the two halves of the specimen were matched, and the matched sections were analyzed. The thin sections covered almost the whole crack path and provided numerous grains to study if a crack propagated through a grain or along a grain boundary. It is obvious from Fig. 2.13 55 Grown Ice and Experimental Setup [PI and PII] 10 mm 10 mm (a) (b) Figure 2.13. Horizontal thin sections showing a crack path (a) from 2 cm behind the initially-sharpened crack tip to 15 cm ahead of it and (b) from 15 cm to 35 cm ahead of the initial crack tip. Both thin sections are from a depth of 24 cm from the top of the ice sheet. The arrows indicate the direction of crack propagation. (a) (b) Figure 2.14. (a) Dried replica being peeled oﬀ the ice fracture surface and (b) its fractographic examina- tion by SEM. 56 Grown Ice and Experimental Setup [PI and PII] that a transgranular fracture was dominant: the cracks propagated through the grain for almost all the grains. It is worth mentioning that the author did several trials to replicate the fracture surface. Struers silicon solution, made of silicon and hardener, was dropped on the selected areas of the fracture surface, as shown in Fig. 2.14a. After storing the ice sample in the freezer, the solution dried, and the resulting plastic ﬁlm replica was peeled oﬀ ready for examination. Unfortunately, when the dry replica was observed with scanning electron microscope (SEM), no visible fracture features were detected. However, signs of frozen water droplets were observed. Accordingly, this replica procedure of the fracture surface does not work with wet fracture experiments. Although some of the ice was saved immediately after each experiment, the water from the tank melted the fracture features. Water droplets were frozen to the fracture surface, preventing any further analysis of the fracture surface. 57 Grown Ice and Experimental Setup [PI and PII] 58 3. Fracture of Warm S2 Columnar Freshwater ice Under Monotonic Loading: Size and Rate Effects [PI] The following chapter covers the ﬁrst part of the experimental program. Fourteen displacement control (DC) experiments were conducted to study the fracture behavior of columnar freshwater S2 ice under monotonic loading scenarios. The experiments used three ECRP sizes: 19.5m x 36m, 3m x 6m, and 0.5m x 1m (Fig. 2.4). The crack-parallel dimension covered a size range of 1:39, the largest for ice tested under laboratory conditions. The loading was applied in the direction normal to the columnar grains. The loading rates applied led to test durations from fewer than 2 seconds to more than 1000 seconds (Table 3.1). Two analytical approaches were implemented: a one-parameter linear elastic fracture mechanics approach and a multi-parameter nonlinear viscoelastic approach. These methods were derived to analyze the data and calculate values for the apparent fracture toughness, crack opening displacement, notch sensitivity, stress-separation curve, fracture energy, and size of the process zone near a crack tip. The chapter addresses the research questions RQ1, RQ2, RQ3, and RQ4 (Section 1.7). Speciﬁcally, it investigates: 1) the eﬀects of size (scale) and loading rate on the fracture behavior of columnar freshwater ice, 2) the validity of LEFM as a one-parameter fracture tool, 3) the applicability of VFCM to analyze the fracture data, and 4) the requirements for polycrystalline homogeneity and notch sensitivity. The chapter is structured as follows. First, linear elastic fracture mechanics is introduced to derive the expressions of the apparent fracture toughness and the crack opening displacement. Second, a non-linear viscoelastic ﬁctitious crack model is explained and coupled with an optimization model. Finally, the experimental and modelling results are presented and discussed. The chapter has been adapted from the author’s publication (PI). 59 Fracture of Warm S2 Columnar Freshwater ice Under Monotonic Loading: Size and Rate Eﬀects [PI] H = 2L Dimensions in mm NCOD3 60-100 NCOD2 NCOD1 COD A = 0.7-0.75 L 0.5A D = 150 CMOD 50-80 P P Figure 3.1. The edge cracked rectangular plate of width H, length L, and crack length A (PI). 3.1 Linear Elastic Fracture Mechanics Model Expressions for stress intensity factor and crack opening displacement for the edge loaded, edge cracked rectangular plate used in the present experiments (Fig. 3.1) are derived below by following the weight function approach outined by Dempsey and Mu (2014) and reviewed in Appendix A. The crack face pressure Σ(X) by the loading device is ⎪ 0 < X ≤ D Dh Σ(X) = (3.1) 0 D ≤ X < A where P is the applied force, D the length of contact between ice and the loading device, h the ice thickness, X the position along the crack, and A the crack length. The stress intensity factor K(A )is K(A ) = Σ(X)H (A , X)dX (3.2) 0 r 0 where H (A , X) is the weight function (valid for X ≤ A ) given as r 0 0 i− 1 A X H (A , X) = √ G 1 − (3.3) r 0 i L A 2πA i=1 G (i = 1,2,...,5) are functions given by Eq. (A3) in Appendix A and L is the length of the rectangular plate (Fig. 3.1). Inserting Eqs. (3.1) and (3.3) into Eq. (3.2) leads to K(A ) = H (A , X)dX (3.4) 0 r 0 Dh where 2A H (A , X)dX = Z (d, a ), r 0 1 0 G (a ) d i 0 i− with Z (d, a ) = 1 − 1 − , 1 0 2i − 1 a i=1 D A d = , a = (3.5) L L 60 Fracture of Warm S2 Columnar Freshwater ice Under Monotonic Loading: Size and Rate Eﬀects [PI] Thus, P 2A K(A ) = Z (d, a ) (3.6) 0 1 0 Dh π If the critical stress intensity factor (the fracture toughness) is specimen size inde- pendent and specimen geometry independent, it can be denoted by K . Because IC of the lack of standards for determining the proper specimen size, loading rate, etc. for the fracture toughness testing of ice, Dempsey (1991) suggested the notation of apparent fracture toughness K instead of K . Then K(A ) = K when P = P Q IC 0 Q max and it follows from Eq. (3.6) that P 2a max 0 K = Z (d, a ) (3.7) Q 1 0 dh L The maximum contact pressure p = P /Dh follows as max max 2a K 0 p = √ (3.8) max Z (d, a ) 1 0 For the crack opening displacement at a position X along a crack with length A , Dempsey and Mu (2014) gave an expression δ(A , X) = K(T)H (T, X)dT (3.9) 0 r where E is Young’s modulus. For X < D < A , this equation can be put into form D A E δ(A , X) = 2 K(T)H (T, X)dT + K(T)H (T, X)dT (3.10) 0 r r X D An expression for K(T) when 0 ≤ T ≤ D, obtained by using Eq. (8) of Dempsey and Mu (2014), is P T K(T) = Tπ f (η) with η = (3.11) Dh L where f (η) is the dimensionless factor given by Eq. (5) in Dempsey and Mu (2014). The expression for K(T) when D ≤ T ≤ A can be obtained from Eq. (3.6) by replacing A with T and a by η. The expression for COD then reads 0 0 2P 2T π E δ = √ f (η)H (T, X)dT r r Dh π X 2 2T + Z (d,η)H (T, X)dT (3.12) 1 r When 0 < X < D: d a 2P π E δ = √ f (η)Z (x,η)dη + Z (d,η)Z (x,η)dη (3.13) r 2 1 2 πdh 2 x d 61 Fracture of Warm S2 Columnar Freshwater ice Under Monotonic Loading: Size and Rate Eﬀects [PI] in which 5 3 i− X x 2 x = , while Z (x,η) = G (η) 1 − (3.14) 2 i L η i=1 Similarly, when D < X < A : ⎪ A ⎪ ⎪ ⎪ 2P ⎨ 2T ⎬ E δ = Z (d,η)H (T, X)dT (3.15) ⎪ 1 r ⎪ ⎪ ⎪ ⎩ ⎭ Dh π and ﬁnally 2P E δ = Z (d,η)Z (x,η)dη (3.16) 1 2 πdh 3.2 Viscoelastic Fictitious Crack Model The ﬁctitious crack model (FCM) introduced in Section 1.2.2 is another modelling approach to analyze the experiments. According to this model, the length of a traction free crack is denoted as A and the total length of a crack including the process zone as A, giving a length of the process zone as PZ = A − A (Fig. 3.2a). The crack-opening displacement at X = A is denoted as δ . The behavior inside the process zone is 0 0 described by a constitutive relation, known as the stress-separation law (σ − δ law), as shown in Fig. 3.2b. The response outside the process zone is governed by an applicable constitutive relation of the bulk material. In the present work, the viscoelas- tic ﬁctitious crack model (VFCM) formulated by Mulmule and Dempsey (1997) is adopted to model the response of the freshwater fracture experiments conducted. The VFCM couples the FCM with the assumption that the behavior of the bulk material is viscoelastic. However, for the freshwater ice fracture experiments conducted, neither the stress-separation law nor the viscoelastic material parameters are known before the experiments, but need to be back-calculated by requiring that the results from the experiments and from the modeling match. An inverse analysis is applied in an iterative procedure to obtain the softening curve of the FCM. The method is outlined below. In this study, the creep compliance J is used to characterize the viscoelastic behavior of the bulk material: 1/2 J (t) = 1/E + Ct (3.17) where E is the short-term elastic modulus measured at the crack mouth, C is the creep compliance constant, and t is the time. Eq. (3.17) is based on laboratory-scale experiments conducted by (Cole, 1993; Schapery, 1993) on saline ice at -10 C under low compressive stress (0.5 MPa). Here, E is obtained from the initial linear portion 62 Fracture of Warm S2 Columnar Freshwater ice Under Monotonic Loading: Size and Rate Eﬀects [PI] (a) (c) traction-free fictitious bulk Choose C, ߪെߜ curve, and PZ size crack crack material Forward Solve the A PZ 0 problem VFCM (Eqs. 3.17-3.24) Obtain CMOD, COD, NCOD1, and PZ records Iterations continue until the solution converges (b) Solve an optimization Inverse scheme by minimizing problem between the model and data (Eqs. 3.25-3.26) Obtain C and ߪെߜ curve Figure 3.2. (a) Illustration of the ﬁctitious crack model. (b) Stress-separation curve describing the behavior in the PZ and showing the control points (see Eq. 3.25) used in the optimization procedure. (c) A ﬂowchart illustrating the forward and inverse problems used for matching the experimental and model results (PI). of the load - crack mouth opening displacement plot using Eq. 3.16 and C through the iterative procedure described below. According to the spirit of the FCM, the growth of the process zone is governed by the interaction of an externally applied load and the cohesive stress such that the resultant stress intensity factor vanishes at the ﬁctitious crack tip: K = K (σ, a, t) + K (σ , a, t) = 0 (3.18) σ coh coh where K, K , and K represent the resultant stress intensity factor (SIF) at the σ coh ﬁctitious crack tip, the SIF due to external loading and the SIF due to the cohesive loading, respectively. The normalized length of the resultant crack is denoted by a = A/L, Figs. 3.2a and 3.1), and t is the time. Using the deﬁnition in Eq. (3.2), K and K become coh K (σ, a, t) = L σ(x, t) h (a, x) dx σ r K (σ , a, t) = L σ (x, a, t) h (a, x) dx, a ≥ a (3.19) coh coh coh r 0 where h (a, x) is a normalized weight function, h (a, x) = LH (A, X). The viscoelas- r r r tic crack opening displacement δ is obtained similarly by superposition: δ (x, a, t) = δ (σ, x, a, t) + δ (σ (δ , δ , x), x, a, t) (3.20) v vσ v coh coh v v where δ and δ represent the viscoelastic crack opening displacement due to the vσ v coh external loading, and the viscoelastic crack opening displacement due to the cohesive loading, respectively. The normalized position along the crack is denoted by x = X/L. 63 Fracture of Warm S2 Columnar Freshwater ice Under Monotonic Loading: Size and Rate Eﬀects [PI] The functional dependencies in Eqs. 3.18 and 3.20 are shown to clarify the nonlinearity of the problem. Using Eq. (3.9), the weight function method gives t e dδ δ (x, a, t) = J (t − τ) (x, a(τ),τ) dτ (3.21) v c dτ where J is the creep compliance given by Eq. (3.17), and e e e δ (x, a,τ) = δ (σ, x, a,τ) + δ (δ , x, a,τ) (3.22) v vσ v coh with a(τ) δ (σ, x, a,τ) = 2 L K (s,τ)h (s, x)ds σ r vσ a(τ) δ (δ , x, a,τ) = 2 L K (s,τ)h (s, x)ds (3.23) v coh r v coh max(a ,x) Details of the time integrations are explained in Mulmule and Dempsey (1997). The relation between the cohesive stress, the viscoelastic crack opening displace- ment, and the rate of the crack opening displacement is given by the stress-separation curve, shown in Fig. 3.2b, as follows, f (δ , δ )0 ≤ δ <δ ⎨ v v v c σ (δ , δ ) = (3.24) coh v v ⎪ 0 δ ≥ δ v c where δ and δ represent the critical crack opening displacement and the rate of crack c v opening, respectively. To complete the viscoelastic ﬁctitious crack model for freshwater ice, an iterative procedure was implemented. The availability of the load history and the displacement histories at diﬀerent positions along the crack provides a way to conduct an inverse analysis and back-calculate the stress-separation (σ − δ) law, the value of the creep compliance constant (C), and the length of the process zone (PZ) (Figs. 3.2). At each iteration, two problems are solved, forward and inverse. The forward problem refers to the VFCM outlined above (Eqs. 3.17-3.24), and the inverse problem corresponds to the optimization procedure described hereinafter. In the forward problem, values for σ−δ curve, C, and PZ are initially assumed – and later obtained though optimization – and the viscoelastic ﬁctitious crack model (Eqs. 3.17-3.24) is solved to obtain the crack opening proﬁle and the length of the process zone. The measured load - time record is applied in discretized steps into the model. The traction-free crack length is known and is discretized into ﬁxed number of stations (5, 22 and 138 for the 0.5m x 1m, 3m x 6m and 19.5m x 36m specimens, respectively). However, the process zone mesh is updated with each load step because the process zone size is increasing. In each load step, new stations are added in the process 64 Fracture of Warm S2 Columnar Freshwater ice Under Monotonic Loading: Size and Rate Eﬀects [PI] zone to keep the mesh ﬁne while keeping the previous stations at constant locations. The governing equations are satisﬁed at these stations. In each iteration, the crack opening displacements are computed at all the stations. This provides displacement results at diﬀerent positions along the crack, including the process zone, through an experiment. The length of the process zone is obtained from the requirement that the stress intensity factor is zero at the ﬁctitious crack tip (Eq. 3.18). The solution of the VFCM is considered converged when the crack opening displacements don’t change by more than 0.2% at any of the stations. In the inverse problem, the crack opening proﬁle is assumed to be known, and an optimization scheme using the Nelder-Mead (N-M) algorithm (Nelder and Mead, 1965) is used to obtain the constitutive parameters (C and the σ−δ law) by minimizing the diﬀerence between the numerical data and the experimental data. Once the σ − δ law and C are solved, they are used as input in the forward problem and the iteration continues until the best attainable agreement is obtained. A ﬂowchart illustrating the implemented procedure is shown in Fig. 3.2c. In the optimization scheme, ﬁve control pairs, shown in Fig. 3.2b, are used to deﬁne the σ − δ curve: ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ δ = 0,δ ,δ ,δ ,δ ⎨ 1 2 3 c ⎬ (3.25) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ σ = σ ,σ ,σ ,σ , 0 t 1 2 3 in which δ , δ , and δ are assumed such that δ <δ <δ <δ , and the N-M 1 2 3 1 2 3 c scheme optimizes the control points, δ , σ , σ , σ , and σ . The σ − δ law was c t 1 2 3 back-calculated without any assumption about its shape. The optimization scheme minimizes an objective function F given by the norm of a residual N=45 F = arg min M (δ ,σ ,σ ,σ ,σ , C) − D (3.26) i c t 1 2 3 i δ ,σ ,... c t i=1 where M and D refer to the displacement values given by the model and obtained i i by using the experimental data, respectively. . is the Euclidean norm of a vector. The displacement values that are matched total to 45: CMOD, COD, and NCOD1 (Fig. 3.1) at 15 time instances during the loading. As no LVDT was placed at the initially sharpened crack tip, it was decided to match the displacement values given by NCOD1. 15 instances were chosen to capture the overall shape of the measured load- time record, so that the load given to the model is very similar to the experimental one. Similarly, 5 pairs were selected to capture the overall shape of the stress-separation curve. Choosing more or less points is possible; it is a trade-oﬀ between accuracy vs computational complexity. In the current problem, the objective function F is numerical, and accordingly, no closed form solution for the gradient of the objective function is available. Derivative-based optimization methods such as Newton solvers 65 Fracture of Warm S2 Columnar Freshwater ice Under Monotonic Loading: Size and Rate Eﬀects [PI] cannot be used. This is the main motive behind the selection of the Nelder-Mead (N-M) method, which is a derivative-free optimization method (Nelder and Mead, 1965). It starts with an initial guess of C and σ−δ curve. It constructs a simplex of N+1 vertices; N is the number of parameters to be optimized (σ ,σ ,σ ,σ ,δ , and C). In each t 1 2 3 c iteration, N-M computes the value of the objective function at each vertex, removes the vertex with the largest function value, and replaces it with a new vertex with smaller residual by following one of four schemes. The four schemes are reﬂection, expansion, outside contraction, and inside contraction. This algorithm applies an iterative update of the simplex in shape and size until convergence is reached. The N-M algorithm uses a termination criterion based both on the absolute size of the simplex with respect to the initial simplex and the diﬀerence of the function values in the simplex. The optimization problem has constraints: the cohesive stresses are tensile and have a softening behavior (Fig. 3.2b). Besides, upper and lower bounds were provided for the arguments. The bounds for the cohesive stresses were based on the tensile strength of columnar freshwater S2 ice (Carter, 1971; Gold, 1977; Michel, 1978b), for the critical crack opening displacement on the measured displacements at NCOD1 and NCOD2 in each experiment, and for the creep compliance constant on sea ice fracture studies performed by Dempsey et al. (2018). This optimization problem is typically called a least-squares problem when using the Euclidean norm (Eq. 3.26). It is a convex problem because F is a convex function and the feasible set is convex. Thus, the optimization algorithm will converge to the global optimal solution. 3.3 Results 3.3.1 Linear Elastic Fracture Mechanics Analysis Table 3.1 gives the dimensions of the ice samples together with the measured and computed results. The apparent fracture toughness (K ) was computed from the failure load and dimensions by using Eq. (3.7). The loading rate (K) was computed by dividing K by the time to failure (t ). The time to failure varied from about 2 Q f seconds to about 1000 seconds (17 minutes) giving a loading rate range of 0.18 to 57 −1 kPa ms . The elastic moduli (E ) and (E ) were computed by Eq. (3.16) CMOD COD from the initial linear portion of the load - CMOD displacement records and load - COD displacement records, respectively. Some of the E values are missing, caused COD by the fact that the initial portion of the associated load-COD curve was very noisy. No signiﬁcant diﬀerence in the moduli was found. Displacements at the crack mouth 66 Fracture of Warm S2 Columnar Freshwater ice Under Monotonic Loading: Size and Rate Eﬀects [PI] Table 3.1. Specimen dimensions, measured data, and results computed using linear elastic fracture mechanics. ˙ ˙ ˙ Experiment LH A hE E P t σ K K CMOD CMOD NCOD1 NCOD1 0 CMOD COD max f n Q √ √ −1 −1 −1 m m m cm GPa GPa kN s MPa kPa mkPa ms μm μms μm μms RP1 0.5 1 0.35 35 6.9 - 1.7 54.9 0.486 126.2 2.298 33.1 0.602 5.2 0.094 RP2 0.5 1 0.35 35 6.9 - 2.3 51.9 0.657 168.6 3.247 43.3 0.834 7.5 0.145 RP3 0.5 1 0.35 37.1 6.4 - 1.6 1.9 0.431 108.4 57.058 29.1 15.321 7.8 4.128 a a RP4 0.5 1 0.35 37.6 7.0 - 1.9 20.8 0.505 129.2 6.223 37.6 1.811 5.2 0.25 a a RP5 0.5 1 0.35 40.2 7.3 - 2.1 571.7 0.522 132.2 0.231 36.1 0.063 5.4 0.009 RP6 0.5 1 0.35 41.1 5.6 - 2.4 811.4 0.584 145 0.179 59.5 0.073 10.4 0.013 RP7 3 6 2.1 34.5 6.0 6.2 4.6 86.9 0.259 159.4 1.834 236 2.716 47.2 0.543 RP8 3 6 2.1 34.5 6.6 - 3.0 2.7 0.169 106 39.259 157 58.148 22.9 8.482 RP9 3 6 2.1 34.5 6.1 6.2 5.2 148.0 0.293 180.6 1.221 266 1.798 42.8 0.289 RP10 3 6 2.1 34.5 6.7 6.7 4.8 222.2 0.271 166.8 0.751 215.6 0.970 34.6 0.156 RP11 3 6 2.1 36 7.4 7.7 4.2 15.3 0.227 139.4 9.091 174.4 11.374 25.2 1.644 RP12 3 6 2.1 37.2 5.7 5.9 6.8 701.8 0.355 221.3 0.315 370.5 0.528 68.5 0.098 RP13 3 6 2.1 37.6 4.7 4.8 5.7 1027.1 0.295 184.3 0.179 420.3 0.409 63.4 0.062 RP14 19.5 36 14.6 35 5.7 5.7 10 213.5 0.127 186.0 0.871 961.7 4.505 37.2 0.174 Errors in Table 1 of PI have been ﬁxed. and near the crack tip at the crack growth initiation, CMOD and NCOD1 respectively, ˙ ˙ were measured at locations shown in Figure 3.1. CMOD and NCOD1 indicate the displacement rates and were obtained by dividing CMOD and NCOD1 by the time to failure. Attempts were made to measure displacements ahead of the crack also (Fig. 3.1), but those displacements were very small and the measurements failed especially for small specimens and fast experiments and are not analysed here. Fig. 3.3 shows the load-CMOD records for the 3m x 6m specimens. A decrease of the peak load with increase in the loading rate is observed. The pre-peak records (Fig. 3.3b) show an approximately linear load-displacement relation up to the peak load at high loading rates, but at low loading rates, a non-linear relation is observed. Fig. 3.4 shows the load-CMOD data for two 0.5m x 1m specimen: for a low rate in Fig. 3.4a (Experiment RP4) and for a high rate in Fig. 3.4b (Experiment RP3). For RP3, a period of crack growth took place after the peak load before complete fracture happened: the load-displacement plots show portions of approximately constant load and increasing displacement. The post-peak crack growth was observed for experiment RP3 only, all the other experiments fractured completely at the peak load. Fig. 3.5 shows the apparent fracture toughness (K ) as a function of the loading rate. Interestingly, the data suggests a loading rate dependent size eﬀect for the warm columnar freshwater S2 ice studied: there is a size eﬀect at low rates, but there is no size eﬀect at high rates. At low rates, K is higher for the large specimen (both 3m x 67 Fracture of Warm S2 Columnar Freshwater ice Under Monotonic Loading: Size and Rate Eﬀects [PI] (a) (b) Figure 3.3. (a) Load-CMOD records for the 3m x 6m samples with loading rates as indicated in the √ √ −1 −1 ﬁgure. The 39.3kPa ms and 1.83 kPa ms data were smoothed by using moving averages (PI). (b) The pre-peak load-CMOD record. NCOD1 COD 1.5 CMOD 0.5 0 1020304050 (a) (b) −1 Figure 3.4. Load-displacement records for specimen RP4 loaded at rate 6.22 kPa ms (a) and for −1 specimen RP3 loaded at rate 57.1kPa ms (b). Both specimen had the dimensions of 0.5m x 1m. Locations, where CMOD, COD, and NCOD1 were measured, are shown in Fig. 3.1. 6m and 19.5m x 36m) than for the small specimen (0.5m x 1m), but when K is higher −1 than about 10kPa ms , the specimen size does not have an eﬀect on K . For both the large and small specimen, K is decreasing with increasing loading rate, but this rate eﬀect is stronger for the large specimen. This observed size eﬀect is clear between specimen of size 0.5m x 1m and of size 3m x 6m, but there was no size eﬀect between the 3m x 6m specimen and the largest specimen of size 19.5m x 36m. The relation between the apparent fracture toughness and the loading rate is non-linear and can be well described with a power-law relation. Fig. 3.6 shows the crack opening displacements at the crack growth initiation as a function of the loading rate. The CMOD data shows diﬀerent values for each specimen size but the NCOD1 data suggests that the deformations near the crack tip are similar for the 3m x 6m and the 19.5m x 36m specimen. Both these specimen sizes appear large as compared with the small, 0.5m x 1m, specimen. This is in line with the 68 Fracture of Warm S2 Columnar Freshwater ice Under Monotonic Loading: Size and Rate Eﬀects [PI] -1 0 1 2 10 10 10 10 Figure 3.5. Measured apparent fracture toughness as a function of loading rate. First-order power-law ﬁts were applied separately to the data from the larger specimen (3m x 6m and 19.5m x 36m) and from the smaller specimen (0.5m x 1m) (PI). K data in Fig. 3.5. Also similarly to the K data, a rate dependent size eﬀect was Q Q observed: at low rates, NCOD1 is larger for the large specimen, but with increasing loading rate, NCOD1 for the large specimen decreases and approaches the NCOD1 of the small specimen. For NCOD1, the large specimen display a power-law type rate eﬀect, but the small specimens do not show a rate eﬀect. -1 0 1 2 -1 0 1 2 10 10 10 10 10 10 10 10 (a) (b) Figure 3.6. Measured initiation crack opening displacements at the crack mouth (a) and near the crack tip (b) as a function of loading rate. First-order power-law ﬁts were applied to the data. In (b), the power-law ﬁt was calculated separately for the larger specimen (both 3m x 6m and 19.5m x 36m) and from the smaller specimen (0.5m x 1m) (PI). Note that a small error in PI has been ﬁxed here: the power-law exponent in (b) for the small samples should be -0.03 instead of -0.01. 3.3.2 Nominal Strength Analysis In studies on size eﬀect, the maximum nominal tensile strength (σ ) at the crack tip location at failure has been used as a measure for the strength (Bazant, 1984; Dempsey 69 Fracture of Warm S2 Columnar Freshwater ice Under Monotonic Loading: Size and Rate Eﬀects [PI] et al., 1999b,a). For the edge-cracked rectangular geometry with the current loading conﬁguration, the nominal stress is given by 2P 2 + a − 3d/2 max 0 σ = g(a ), g(a ) = (3.27) n 0 0 hL (1 − a ) where d is deﬁned in Eq. 3.5. If a failure process was dominated by a strength criterion, such as yield, σ would be independent of the specimen size, but if a failure process was dominated by fracture, σ would decrease as a function of the size. In many quasi-brittle materials, like concrete, where a process zone in front of a crack can be identiﬁed, the failure of small specimen can be explained with a strength criterion and the failure of large specimen with a fracture criterion, as described by Bazant (1984). −0.5 For an LEFM type of fracture, σ is related to L . 1 0.2 0.8 -0.2 -0.4 0.6 -0.6 0.4 -0.8 -1 0.2 -1.2 -1 0 1 2 -1.4 10 10 10 10 -1 -0.5 0 0.5 1 1.5 (a) (b) Figure 3.7. Nominal tensile strength σ as a function of loading rate (a) and specimen size (b). First- order power-law ﬁts were applied to the data. In (b), the data for the 0.5m x 1m and 3m x 6m specimen is obtained from the power law ﬁts at diﬀerent loading rates; the loading rate −1 of K = 0.871 kPa ms is indicated with the vertical dashed line in (a). Fig. 3.7a shows how the nominal strength (σ ) of the ice specimen is decreasing both with increasing loading rate and with increasing specimen size. The decrease of σ with rate can be described with a power-law. Then, taking data points from these power-law ﬁts for the 0.5m x 1m and 3m x 6m specimen, Fig. 3.7b gives the relation between σ and specimen size (L) for the indicated loading rates. The decrease of σ with L can be described with a power-law where the exponent is decreasing with −0.37 −0.65 increasing rate: σ ∝ L is found for the lowest rate and σ ∝ L for the n n −0.49 highest rate. If averaging for all the data, σ ∝ L is obtained. The data in Fig. 3.7b suggests further that the rate eﬀect is decreasing with decreasing specimen size and appears to disappear if specimen smaller than 0.5m x 1m were studied; the same result was obtained in the LEFM analysis of the data. 70 Fracture of Warm S2 Columnar Freshwater ice Under Monotonic Loading: Size and Rate Eﬀects [PI] 3.3.3 Viscoelastic Fictitious Crack Model Analysis The viscoelastic ﬁctitious crack model (VFCM) and the optimization procedure out- lined in Section 3.2 were used to analyze the experimental data. Fig. 3.8 shows the optimization results from RP5: convergence of the cohesive stresses (σ ,σ ,σ , and t 1 2 σ ), critical crack opening displacement (δ ), creep compliance constant (C), and 3 c objective function (F ). Similar behavior was observed for all the experiments and the results are given in Table 3.2: the number of iterations needed to reach convergence, the ﬁve control points deﬁning the σ − δ relation (σ ,σ ,σ ,σ , δ ; Fig. 3.2b), t 1 2 3 c the crack opening displacement at crack growth initiation at X = A (δ ), the creep 0 0 compliance constant (C), the full fracture energy under FDPZ conditions (G ; Fig. 3.2b), the actual fracture energy (G ), and the process zone length (PZ; Fig. 3.2a). ac The VFCM analysis required successful measurements of CMOD, COD, and NCOD1 (Fig. 3.1) at 15 time instances during the loading. These displacements were small and challenging to measure at the initial, low load levels. Only those experiments, where the whole displacement history was measured successfully, are included in Table 3.2. Note that Table 3.1 includes data from all the 14 experiments, as the analysis in Table 3.1 is based on measured maximum values, not on the whole displacement-time record. Table 3.2. Optimization results computed using the viscoelastic ﬁctitious crack model. ˙ 11 Experiment size K Number of σt σ1 σ2 σ3 δc δ0 C x10 G f Gac PZ σn/σt √ √ −1 2 −1 −1 (m) (kPa ms ) iterations (MPa) (MPa) (MPa) (MPa) (μm) (μm) (m N s )(N/m) (N/m) (mm) RP2 0.5x1 3.247 243 1.29 1.29 1.24 1.06 7.09 4.07 5.8 6.54 4.97 6.9 0.509 RP3 0.5x1 57.058 148 1.30 1.30 1.23 0.30 7.57 1.30 3.4 5.48 1.69 2.3 0.332 RP4 0.5x1 6.223 247 1.26 1.26 1.26 1.26 4.44 2.66 6.7 3.42 3.35 4.6 0.401 RP5 0.5x1 0.231 167 1.30 1.28 1.27 1.11 5.27 0.77 4.1 3.63 1.00 1.4 0.402 RP6 0.5x1 0.179 203 1.28 1.28 1.22 0.80 6.73 2.85 3.3 6.49 3.65 5.1 0.456 RP7 3x6 1.834 161 1.30 1.30 1.17 0.54 33.86 3.25 5.2 16.89 4.24 5.9 0.199 RP8 3x6 39.259 688 1.24 1.00 1.00 1.00 18.00 1.44 5.9 14.60 1.78 2.7 0.136 RP9 3x6 1.221 229 1.32 1.22 1.09 0.92 32.80 4.23 5.6 22.30 5.54 6.4 0.222 RP12 3x6 0.315 211 1.02 1.02 1.02 1.02 37.00 8.14 6.2 27.54 8.17 18.4 0.348 RP13 3x6 0.179 130 1.00 1.00 1.00 1.00 40.10 5.69 8.8 30.00 5.69 13.2 0.295 RP14 19.5x36 0.871 476 1.05 1.05 1.05 0.80 36.30 5.58 9.0 20.30 5.83 12.3 0.121 Figs. 3.9 and 3.10 illustrate the correspondence between the experiments and the VFCM for two experiments. The ﬁgures show the load as measured and as applied to the model, together with the response of the ice samples as measured and as obtained from the model. Both experiments show a good agreement between the model results and experimental data. This agreement supports the ability of the VFCM to describe the response of columnar freshwater S2 ice: the VFCM is able to model the non-linear 71 Fracture of Warm S2 Columnar Freshwater ice Under Monotonic Loading: Size and Rate Eﬀects [PI] 5.4 1.25 1.2 5.2 1.15 1.1 1.05 0 1 2 3 0 2 10 10 10 10 10 10 1.8 -5 1.6 1.4 1.2 -8 0.8 -11 0 2 0 2 10 10 10 10 Figure 3.8. Optimization results of RP5. Convergence of (a) the cohesive stresses, (b) the critical crack opening displacement, (c) the creep compliance constant, and (d) the objective function after a certain number of iterations. displacement record not only at the crack mouth where the loading is applied (CMOD), but also along the crack (COD) and near the crack tip (NCOD1). The model does not account for crack propagation and thus the comparison is shown up to the peak load only. 6 300 Experiment Model 5 250 3 150 2 100 1 50 050 100 150 0 50 100 150 (a) (b) (c) Figure 3.9. Experimental and model results for RP9 (3m x 6m). (a) Load at the crack mouth, see Fig. 3.1 and Eq. (3.1). (b) Displacement - time records. (c) Load - displacement records. The CMOD data was smoothed with moving average smoothing with a span of 10% successive data points. Fig. 3.11 shows the crack opening displacement at crack growth initiation at X = A (δ ) and the actual fracture energy (G ) as functions of loading rate. Similar rate and 0 ac size eﬀects were observed than with the linear elastic fracture mechanics analysis above. The results from the large and mid-size specimen (3m x 6m and 19.5m x 36m) are again interchangeable and higher than the results from the small specimen (0.5m x 72 Fracture of Warm S2 Columnar Freshwater ice Under Monotonic Loading: Size and Rate Eﬀects [PI] Experiment Model 0 0 0 50 100 150 200 250 0 50 100 150 200 250 (a) (b) (c) Figure 3.10. Experimental and model results for RP14 (19.5m x 36m). (a) Load at the crack mouth, see Fig. 3.1 and Eq. (3.1). (b) Displacement - time records. (c) Load - displacement records. (The corresponding ﬁgures in PI are edited; (0,0) is added to the model plots.) 1m). This size eﬀect is loading rate dependent and decreases with increasing rate: δ and G for the larger specimen decrease with increasing loading rate and approach ac the values obtained for the small specimen. No signiﬁcant rate eﬀect was observed for the small specimen. Power-law relations can be used to describe the decrease of δ and G with rate. ac 4 4 0 0 -1 0 1 2 -1 0 1 2 10 10 10 10 10 10 10 10 (a) (b) Figure 3.11. The crack opening displacement at crack growth initiation at X = A (a) and the actual fracture energy (b) as a function of loading rate. Viscoelastic ﬁctitious crack model was used in the analysis. First-order power-law ﬁts were applied separately to the data for the larger specimen (3m x 6m and 19.5m x 36m) and for the smaller specimen (0.5m x 1m) (PI). Stress-separation curves were constructed by using the ﬁve control points shown in Fig. 3.2b, and straight lines connecting the points. Fig. 3.12 shows the back-calculated σ−δ curves under FDPZ conditions and illustrates the impact of the specimen size and loading rate. With increasing separation, the σ − δ curves show initially an approximately constant stress and then extend with a softening behavior, which is getting gradually steeper. This kind of concave stress-separation curve is similar to what has been observed for sea ice by Mulmule and Dempsey (1999). The data 73 Fracture of Warm S2 Columnar Freshwater ice Under Monotonic Loading: Size and Rate Eﬀects [PI] obtained with the larger specimen (Fig. 3.12b) illustrate that with increasing loading rate, the length of the constant part is decreasing while the slope of the softening part appears less sensitive to the loading rate. This suggest that the observed decrease in fracture energy with loading rate (Fig. 3.11) is due to a decrease in the constant part, not due to change in the softening part. At the highest rate studied, no constant part of the stress-separation can be observed, the curve shows more-or-less linear softening only. The stress-separation curves obtained with the small specimens (Fig. 3.12a) show similar behaviour, but not as clearly. The stress-separation curve includes the tensile strength (σ ) of the ice studied (Fig. 3.12); σ is given by σ at δ = 0. This value was not as strongly or clearly aﬀected t coh v by specimen size or loading rate as the fracture parameters: for the small specimen σ ≈ 1.3 MPa, for the large specimen σ = 1.0··· 1.3 MPa, higher for the higher rates. t t The change in the shape of the σ − δ curve with loading rate is an indication of changes in the deformation and fracture processes with rate. At low rates, the approximately constant initial part of the σ − δ curve indicates that the freshwater ice deforms initially without softening and the material inside the cohesive zone is able to transfer the full tensile strength for some time with increasing load, until the deformation becomes so large that softening starts and ﬁnally leads to extension of the physical, traction-free crack. At high rates, no constant part of the curve is observed and softening initiates immediately when the critical stress is reached. This explains why freshwater ice appears brittle at high loading rates and less brittle at low rates. In fact, Fig. 3.12 represents the unattained σ − δ curves of fully-developed process zones. Although the experiments were conducted under displacement control, none of them realized a FDPZ, i.e. consumed the full fracture energy, and the experiments fractured at peak load. As illustrated in Fig. 1.5, crack growth initiation occurred when the crack was in the state shown in Fig. 1.5b. The crack did not achieve the fully-developed state shown in Fig. 1.5c; the cohesive stress (σ ) did not drop to coh zero and the crack opening displacement (δ ) did not reach the critical displacement (δ ). Only a portion of the stress-separation curve (Fig. 3.12) and the fracture energy (G , Fig. 3.2b and Table 3.1) was attained prior to the initiation of crack growth. It is possible that the very high homologous test temperatures and the inevitable grain boundary melting (Dash et al., 2006) are the reason for not attaining a FDPZ. Fig. 3.13 displays the attained σ−δ curves describing the real process zone behavior of the 0.5m x 1m (Fig. 3.13a) and the 3m x 6m and 19.5m x 36m (Fig. 3.13b) specimens at the testing conditions. These curves were deduced by interpolation from the complete stress-separation curve (Fig. 3.12) and the process zone proﬁle deduced by the model. Generally, the specimens deformed without softening, and the process zone was transferring full tensile strength when fracture occurred. Thus, a 74 Fracture of Warm S2 Columnar Freshwater ice Under Monotonic Loading: Size and Rate Eﬀects [PI] (a) (b) Figure 3.12. Stress-separation curves under FDPZ conditions for the 0.5m x 1m specimen (a) and for the 3m x 6m and 19.5m x 36m specimen (b) at diﬀerent loading rates. The number next to each curve reﬂects the index of the experiment in the Legend (PI). (a) (b) Figure 3.13. The attained stress-separation curves for the (a) 0.5m x 1m specimens and for the (b) 3m x 6m and 19.5m x 36m specimens. The number next to each curve reﬂects the index of the experiment in the Legend [PIII]. fully developed process zone was not obtained: σ 0 and δ δ at crack growth coh 0 c initiation. The area under the σ−δ curves gives the fracture energy actually consumed, denoted as G in Table 3.2. ac Fig. 3.14a shows the upper-half crack proﬁles over the normalized crack length (A/L, Fig. 3.2a) for the 3m x 6m and 19.5m x 36m specimens from the experiment (deﬁned by CMOD, COD and NCOD1, Fig. 3.1) and the VFCM. The optimized values generated by the model were relatively close to the measured values. The modelling worked better for the faster experiments (RP7 and RP8), especially at the crack mouth. However, for slower experiments (RP12, RP13 and RP14), the model underestimated the CMOD, COD and NCOD1 values. This clearly gives evidence on some eﬀects that were more pronounced in slow experiments than fast experiments, and these eﬀects are not accounted for in the model. The data-model misﬁt is mainly 75 Fracture of Warm S2 Columnar Freshwater ice Under Monotonic Loading: Size and Rate Eﬀects [PI] due to the very high homologous test temperatures, liquidity on the gain boundaries, and the inevitable grain boundary melting (Dash et al., 2006) which have a signiﬁcant inﬂuence at slow loading rates (Bell and Langdon, 1967). As discussed by Bell and Langdon (1967), under conditions of slow rate of strain, grain boundary sliding can play a very important role in the overall deformation. This concludes that the VFCM, which does not account for grain boundary sliding, works well for a range of relatively fast loading rates; slower than which results in a misﬁt between the measured data and the model. In addition, the model generated a better ﬁt for the CMOD than the COD and NCOD1 values. This can be explained by the choice of the elastic modulus 1/2 in the model. In the viscoelastic characterization (Eq. 3.17, J (t) = 1/E + Ct ), E was computed as the short-time elastic modulus from the load-displacement record measured at the crack mouth. This probably led to more accuracy at the crack mouth. Another reason could be due to an increased viscoplasticity and minor viscoelasticity in the vicinity of tip. Therefore, the model worked better at the crack mouth than near the crack tip. Figs. 3.14b shows the upper-half PZ proﬁles by the VFCM model for the 3m x 6m specimens and the 19.5m x 36m specimen as function of the distance along the crack from the traction-free tip (X = A ). According to the VFCM, the crack-opening proﬁle has a cusp shape with a zero gradient at X = A, with X being the position along the crack. Fig. 3.14b also gives the process zone size achieved in each experiment, as reported in Table 3.2. Fig. 3.15 shows the variation of the process zone size (PZ) with loading rate. The data from the two larger specimens (3m x 6m and 19.5m x 36m) are again interchangable and show a loading rate eﬀect: the size of the process zone increased with decreasing loading rate. The data for the small specimens (0.5m x 1m) are scattered. Fig. 3.15 gives the same result as shown above: the size eﬀect is loading rate dependent; the diﬀerence of the PZ values between the specimen sizes was large at low loading rates, but vanished at high loading rates. Fig. 3.16 illustrates the calculated growth of the process zone in non-dimensional form for both the small, 0.5m x 1m, specimens and the large, 3m x 6m and 19.5m x 36m, specimens. The process zone size was normalized by L and the time axis by the time to failure (t ) of each experiment (Table 3.1). The PZ - time relation is non-linear; the growth rate is increasing with time. For both the small and large specimen, PZ is increasing with decreasing rate for the high rates, but due to scatter, any trend is less clear for the lower rates. Decrease of PZ with decreasing rate appears non-physical, and thus it is possible that PZ has reached a maximum size at the lower rates; about 6mm for the smaller specimens and about 15mm for the larger specimens. 76 Fracture of Warm S2 Columnar Freshwater ice Under Monotonic Loading: Size and Rate Eﬀects [PI] 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 (a) 4.5 3.5 2.5 1.5 0.5 0 5 10 15 20 (b) Figure 3.14. (a) Experimental and VFCM results for the crack proﬁles and (b) PZ proﬁles by the VFCM for the 3m x 6m and 19.5m x 36m specimens. The total separation between the upper and lower crack surfaces is given by 2U ; U is half of δ which is the viscoelastic crack- v v v opening-displacement (see Fig. 1.5). X = A is the coordinate of the traction-free tip, and a is the normalized traction-free crack length (a = A /L). The number next to each curve 0 0 0 reﬂects the index of the experiment in the Legend [PIII]. 77 Fracture of Warm S2 Columnar Freshwater ice Under Monotonic Loading: Size and Rate Eﬀects [PI] 1.5 0.5 -1 0 1 2 10 10 10 10 Figure 3.15. Variation of the process zone size (PZ) with loading rate and specimen size. First-order power-law ﬁts were applied separately to the data for the larger specimen (3m x 6m and 19.5m x 36m) and for the smaller specimen (0.5m x 1m) (PI). 0.016 0.014 0.012 0.01 0.008 0.006 0.004 0.002 0 0.2 0.4 0.6 0.8 1 (a) -3 3.5 2.5 1.5 0.5 0 0.2 0.4 0.6 0.8 1 (b) Figure 3.16. The growth of the normalized process zone with normalized time for the small, 0.5m x1m, specimens (a) and for the large, 3m x 6m and 19.5m x 36m, specimens (b). The process zone size was normalized by L and the time axis by the time to failure (t ) of each experiment (Table 3.1). 78 Fracture of Warm S2 Columnar Freshwater ice Under Monotonic Loading: Size and Rate Eﬀects [PI] 3.3.4 Notch Sensitivity Analysis Notch sensitivity is given by the ratio between the peak nominal tensile stress (σ ) at the crack tip and the tensile strength (σ ) of an uncracked specimen. This ratio is function of the brittleness number β which is used to test that a fracture test is notch sensitive and the validity of LEFM (Dempsey, 1991). Using P provided in Eq. (3.7) and σ deﬁned in Eq. (3.27), it follows that max n = C β (3.28) n t in which K d 2π Ic β = √ , C = √ g(a ) (3.29) t n 0 a Z (d, a ) 0 1 0 σ L Note that β as used in the thesis is expressed in terms of Dempsey’s notation of K t Q (Eq. 3.7) instead of K , given that no standard exists for K . Ic Ic Table 3.2 gives the notch sensitivity values for the experiments, computed using the σ and σ values listed in Table 3.1 and Table 3.2, respectively. n t Figs. 3.17a, 3.17b, and 3.17c show the σ /σ versus the normalized crack length for n t the 0.5m x 1m, 3m x 6m, and 19.5m x 36 m specimens, respectively. Lines of constant brittleness numbers are shown. The crack length of each specimen size is indicated by the vertical dashed line. It is evident that the fracture experiments completely lose their meaning if σ = σ : a strength failure occurs prior to the attainment of a n t critical SIF. For the 0.5m x 1m and 3m x 6m specimens, that occurs for β higher 0 0 than β ≈ 0.36; for the largest specimen β ≈ 0.33 (Fig. 3.17). This is shown by the t t intersection of the horizontal dashed line of σ /σ = 1 and the vertical dashed line. n t For values of β <β , the fracture experiments are more signiﬁcant for shorter cracks. However, the selection of the optimum crack length should ensure simultaneously a polycrystalline behavior and a high enough degree of brittleness (low enough σ /σ ) n t for a given specimen size. More importantly, the brittleness number can reveal the optimum specimen size so that the experiments are signiﬁcant (β <β ) and suitably notch sensitive (σ /σ << 1) t n t which further restricts β . Considering size eﬀect, previous analyses showed that the results of the two larger, 3m x 6m and 19.5m x 36m, samples were interchangeable, suggesting from Figs. 3.17b and 3.17c that a σ /σ < 0.4 is good enough to generate n t size-independent fracture results. Dempsey (1991) proposed the same ratio to ensure notch sensitivity and referred to it as the specimen shape-independent condition. It is worth noting that the 0.5m x 1m, 3m x 6m, and 19.5m x 36m plates are 77x, 462x, and 3000x the grain size, respectively. Mulmule and Dempsey (2000) speculated that sample homogeneity for ﬁrst-year sea ice would be obtained if the crack-parallel specimen size harboured at least 200 d , where d is the average grain size. These av av 79 Fracture of Warm S2 Columnar Freshwater ice Under Monotonic Loading: Size and Rate Eﬀects [PI] 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 (a) (b) 0.9 0.8 0.8 0.7 0.6 0.6 0.5 0.4 0.4 0.3 0.2 0.2 0.1 0 0.2 0.4 0.6 0.8 1 -1 0 1 2 10 10 10 10 (c) (d) Figure 3.17. Notch sensitivity (σ /σ ) as a function of the normalized crack length for the 0.5m x 1m n t (a), 3m x 6m (b), and 19.5m x 36 m (c) specimens. Lines of constant brittleness numbers are shown. The notch sensitivity of σ /σ = 0.4, 1 are indicated with the horizontal dashed n t lines. The normalized crack length of each specimen size is shown by the vertical dashed line. (d) Notch sensitivity (σ /σ ) as a function of the loading rate. First-order power-law n t ﬁts were applied to the data for the 0.5m x 1m and 3m x 6m specimens (PI). experiments for very warm S2 columnar ice suggest that a crack-parallel specimen size of about 460 d is enough to achieve polycrystalline homogeneity, while 77 d av av is not. Considering the rate eﬀect further restricts the notch sensitivity requirement. Fig. 3.17d extends this analysis by showing the variation of σ /σ as a function of loading n t rate. First-order power-law ﬁts were applied to the data for the 0.5m x 1m and 3m x 6m specimens. The 3m x 6m specimens displayed a higher rate eﬀect than the 0.5m x 1m specimens which satisﬁed the shape-independent condition only at high rates. For the 3m x 6m specimens: at low rate, the suggested σ /σ < 0.4 is valid; while at n t higher rates, a much lower ratio of σ /σ < 0.2 is necessary to ensure notch sensitivity. n t The discussion of notch sensitivity in terms of both size and rate further limits the brittleness condition to σ /σ < 0.2. A tentative conclusion is that the selection of n t the optimal specimen size and crack length, for LEFM applicability, should consider 80 Fracture of Warm S2 Columnar Freshwater ice Under Monotonic Loading: Size and Rate Eﬀects [PI] simultaneously polycrystalline homogeneity and notch sensitivity, including the eﬀects of size and rate. This is the ﬁrst fracture study to discuss the eﬀect of loading rate on the notch sensitivity of ice. Previous fracture studies focused only on the eﬀect of scale on notch sensitivity (Dempsey, 1991). 3.4 Discussion The observed fracture characteristics of S2 columnar freshwater ice are reminiscent of what is expected of quasi-brittle materials, such as concrete and rock (Hillerborg, 1983). The direct consequence is that there is no validity to plane stress and plane strain idealizations, which proved very important to the development of the metals-based fracture mechanics (Broek, 1974). This implies that the thickness of the specimen (i.e. the length of the crack front) is not an important variable for the fracture behavior, as has been shown for the fracture of concrete (Hillerborg, 1983; Mindess and Nadeau, 1976). Hillerborg (1983) stated: "The microcracking of concrete is not accompanied by any substantial contraction corresponding to that which occurs when metals yield. For this reason no essential diﬀerence exists between plain strain and plane stress conditions for concrete. For concrete we thus do not need to bother about the diﬀerence between plain strain and plain stress, which causes a lot of complexity for metals. One consequence of this is that the width of the specimen (the length of the crack front) cannot be expected to be of any major importance for concrete, like it is for metals." Hillerborg’s conclusion was conﬁrmed by experimental work done by Mindess and Nadeau (1976). Ice, being a quasi-brittle material, is expected to behave in a similar way. For long crack fronts, a triaxial state of stress does not materialize, as the tensile stress parallel to the crack front is relieved by microcracking parallel to the top and bottom surfaces or by creep, the latter taking place rapidly. The fracture experiments conducted with columnar freshwater S2 ice showed both size and rate eﬀects which can be expressed either as a rate dependent size eﬀect or as a size dependent rate eﬀect. There was a size eﬀect at the lower rates but no size eﬀect at the higher rates. There was a rate eﬀect for the larger specimens but a weak or no rate eﬀect for the smaller specimens. This interrelation of size and rate eﬀects for columnar freshwater ice is a novel observation; earlier experiments have observed rate eﬀects (Liu and Loop, 1972; Liu and Miller, 1979; Hamza and Muggeridge, 1979; Urabe and Yoshitake, 1981; Timco and Frederking, 1986; Nixon and Schulson, 1986; Tuhkuri, 1987; Weber and Nixon, 1996a,b) or size eﬀects (Mulmule and Dempsey, 1999; Dempsey et al., 2018, 1999b,a), but not how these two are related. In addition, the rate dependency of fracture energy has not been reported earlier: fracture energy increased with decreasing loading rate (Fig. 3.11b). Compared with the earlier studies 81 Fracture of Warm S2 Columnar Freshwater ice Under Monotonic Loading: Size and Rate Eﬀects [PI] with freshwater S2 ice, the specimens tested here were large (L ≥ 0.5)m, very warm (≥−0.3 C), and tested in-situ. The apparent fracture toughness (K ) measured – about 100 kPa m at high rates and about 100 − 200 kPa m at low rates – is within the range measured earlier for columnar S2 freshwater ice (Dempsey, 1991). While values for K are given in Table 3.1and Fig. 3.5, the validity of K as a material parameter for ice should be discussed. K is an LEFM parameter and thus limited to be used when the material response can be idealised as linearly elastic, except in a small process zone near the crack tip, where yielding – or in the case of ice, microcracking and creep – may occur (Riedel and Rice, 1980). If such a process zone is small enough compared with the other dimensions of the specimen, fracture toughness governs crack growth; if a specimen experiences extensive yielding or creep – near the crack tip or elsewhere in the specimen – fracture toughness is not a relevant parameter. The size of the process zone in ice has been estimated with diﬀerent methods. Following Riedel and Rice (1980), who studied tensile cracks in elastic-nonlinear viscous materials, a creep zone size for freshwater ice has been estimated to be in the −1 ◦ range of 0.01··· 0.5 mm at a rate of about 2 kPa ms and temperatures of −10 Cor lower (Timco and Frederking, 1986; Nixon and Schulson, 1986; Xu et al., 2004). It is important to note that the elastic power-law model of Riedel and Rice is applicable to materials where considerable ductility is present in the process zone. Another method used for ice is the method suggested by Veerman and Muller (1972), where the size of a plastic zone near a crack tip is related to the location of an apparent rotation axis of a specimen in bending. For specimen with the largest dimension of 0.4 m, a rate −1 ◦ of 2 kPa ms and temperature of −5 C, this method has suggested a process zone size of about 0.5 mm (Weber and Nixon, 1996a,b). These results of a small process zone have led to conclusions that LEFM is a suitable model for ice fracture at the rates and temperatures studied. However, the above values are an order of magnitude smaller than the about 5 mm measured here at the same rate but with larger specimens and warmer ice (Fig. 3.15). The PZ measured here for freshwater ice, about 5 − 15 mm depending on rate and specimen size, is again an order of magnitude smaller than the PZ measured for sea ice: about 25 mm for 1 m specimen and up to 150 mm for test sizes larger than 3 m (Mulmule and Dempsey, 1999). The size of the freshwater fracture process zone, as with many ice parameters, is aﬀected by scale, rate, and temperature. Rodin and coworkers (Rodin, 2001; Wang et al., 2008) studied similar polycrystalline S2 warm ice and modelled the plate as a single grain specimen governed by the crack tip’s location and loaded through a metal-ice composite testing machine. Their model anticipated smaller PZ size than the grain size. However, the current observed PZ sizes are signiﬁcant and are of the order of the grain size if not 82 Fracture of Warm S2 Columnar Freshwater ice Under Monotonic Loading: Size and Rate Eﬀects [PI] multiple times the grain size, especially as the specimen gets larger with the 3m x 6m and 19.5m x 36 m specimens. This leads to the conclusion that the one grain ice-metal system envisaged by Rodin and coworkers is not applicable in this case. 3.5 2.5 1.5 0.5 0 102030405060 1/2 Figure 3.18. Variation of creep compliance (J = 1/E + Ct , given in Eq. 3.17) as a function of loading rate (K) for diﬀerent specimen sizes (PI). (J in PI was changed to J in the thesis to distinguish between the creep compliance (J ) and the J-integral (J)). Freshwater ice is a viscoelastic material, and it can be hypothesised that the viscous deformation is less signiﬁcant when the loading rate is high, and the temperature is low. Quantitively the elastic and viscous responses can be analysed by using the non-linear viscoelastic model (VFCM) and especially the creep compliance (J = 1/E + Ct ). J , calculated for each experiment by using E = E , t = t , and C from Tables 3.1 c CMOD f and 3.2, is shown in Fig. 3.18 as a function of the loading rate. Note that both E and C show some scatter but not a relation with the loading rate. As J is a function of time, a rate eﬀect is observed, but interestingly, J appears not to have size eﬀect–at least the size eﬀect is much weaker than for the other parameters studied here. At high loading rates, J approaches the value of 1/E (elastic compliance), but with decreasing loading rate, the viscous component of J increases and the material response deviates −1 more and more from an elastic behavior. At loading rates K ≈ 20 kPa m s or less, J is twice or more of the elastic compliance. Thus, the warm freshwater ice studied −1 here can be considered elastic only when K is higher than about 50 kPa m s ; or the time to failure is less than 1 − 2 seconds. This strongly supports the notch sensitivity ratio (σ /σ < 0.2), suggested in Section 3.3.4, for LEFM applicability. For other n t types of ice and for other temperatures, diﬀerent limit rates for elastic response should be found. Earlier studies, based on analysis of the process zone size, have concluded −1 that LEFM is a valid fracture model when K > 10 kPa m s (Urabe and Yoshitake, 83 Fracture of Warm S2 Columnar Freshwater ice Under Monotonic Loading: Size and Rate Eﬀects [PI] 1981; Nixon and Schulson, 1986), suggesting a larger range for LEFM than suggested here. Finally, it is interesting to note that the tensile strength (σ ), obtained from the experimental data through the VFCM analysis, was about 1.2 MPa (Figs. 3.12 and 3.13) and not as strongly aﬀected by size or rate as the fracture parameters. In earlier experiments tensile strength of ice has been observed to be a function of temperature, grain size, orientation, porosity – all constants in the experiments reported here – but also of strain rate (Schulson and Duval, 2009). This raises the question of why the tensile strength, as deﬁned in the VFCM, does not show a clear rate eﬀect when it has been measured before, and when the other fracture parameters do show both rate and size eﬀects. 84 4. Fracture of S2 Columnar Freshwater Ice Under Creep/Cyclic-Recovery Loading [PII] The following chapter covers the second part of the experimental program. Load control (LC) experiments were conducted to characterize the time-dependent and fracture behavior of 3m x 6m warm and ﬂoating edge-cracked rectangular plates of warm columnar freshwater S2 ice. A program of ﬁve load control (LC) experiments was completed in the test basin (40 m square and 2.8 m deep) at Aalto University. The loading was applied in the direction normal to the columnar grains and consisted of creep/cyclic-recovery sequences followed by a monotonic ramp to fracture. The LC results were compared with the fracture results of monotonically loaded displacement control (DC) experiments of the same ice (Chapter 3), and the eﬀect of the creep and cyclic sequences on the fracture properties were analyzed. To characterize the nonlinear displacement-load relation, Schapery’s constitutive model of nonlinear thermodynamics Schapery (1969) was applied to analyze the experimental data. A numerical optimization procedure using Nelder-Mead’s (N-M) method was implemented to evaluate the model functions by matching the displace- ment record generated by the model and measured by the experiment. The accuracy of the constitutive model is checked and validated against the experimental response at the crack mouth. This study presents the ﬁrst attempt to use Schapery’s model for freshwater ice. The choice of this model for freshwater ice is motivated by the fact that the model was successfully applied to saline ice (Schapery, 1997; Adamson and Dempsey, 1998; LeClair et al., 1999, 1996) with encouraging results. The model accurately described the deformation response during load/unload applications over varying load proﬁles. This chapter aims to assess the time-dependent nature of warm columnar freshwater S2 ice and addresses the research questions RQ5 and RQ6 (Section 1.7). Precisely, the experiments in this study examine: 1) the extent to which the elastic, viscoelastic and viscoplastic components contribute to the ice deformation observed at the crack mouth, 2) the eﬀects of the testing conditions on the creep stages (primary/transient 85 Fracture of S2 Columnar Freshwater Ice Under Creep/Cyclic-Recovery Loading [PII] H = 2L= 6000 Dimensions in mm NCOD3 NCOD2 L= 3000 NCOD1 COD A = 0.7L 0.5A D = 150 CMOD P P Figure 4.1. (a) Specimen geometry, edge cracked rectangular plate of length L, width H, and crack length A (PII). and steady-state/secondary) present in the ice, 3) the eﬀects that creep and cyclic sequences have on the fracture properties; i.e. failure load and crack growth initiation displacements, and 4) the ability of Schapery’s nonlinear constitutive model to predict the experimental response. The rest of the chapter is structured as follows. In Section 4.1, a description of the applied loading proﬁle is presented. Section 4.2 introduces Schapery’s model that is used to analyze the experiments. In section 4.3, the experimental and model results are summarized. Finally, a discussion of the results is provided in Section 4.4. The chapter has been adapted from the author’s publication (PII). 4.1 Creep/Cyclic-Recovery Fracture Experiments 4.1.1 Specimen The ice specimens tested were 3m x 6m edge-cracked rectangular plates (Fig. 4.1). 4.1.2 Creep-Recovery and Monotonic Loading Proﬁle In two experiments, ice specimens were subjected to creep-recovery loading followed by a monotonic fracture ramp. The creep-recovery sequences consisted of four constant load applications, separated by zero load recovery periods. Each sequence was composed of alternating load/hold and release/recovery periods. Creep phases were applied at load levels of 0.4 kN, 0.8 kN, 1.2 kN, and 0.4 kN, as given by the loading signal in Fig. 4.2a. The loads were chosen low enough to avoid crack propagation and failure of the specimen. Each load-hold-unload was applied in the form of a trapezoidal wave function to avoid instantaneous load jump and drop; the 86 Fracture of S2 Columnar Freshwater Ice Under Creep/Cyclic-Recovery Loading [PII] 0 1000 2000 3000 4000 0 1000 2000 3000 4000 (a) (b) Figure 4.2. Loading consisting of (a) creep-recovery and (b) cyclic sequences followed by a monotonic fracture ramp. The number above each segment indicates the duration in s (PII). load up was applied in approximately 10 seconds and released in approximately 10 seconds. The slopes of the wave on load up and load release were 0.04kN/s, 0.08 kN/s, and 0.12 kN/s for the 0.4kN, 0.8 kN, and 1.2 kN load levels, respectively. Once at the desired hold level, the load was kept constant for a predetermined time interval. The load intervals were multiples of the hold interval for the 0.4 kN load level, Δt = 126 seconds. For the 0.8 kN and 1.2 kN load levels, the time interval was doubled and quadrupled: 2Δt = 252 seconds and 4Δt = 504 seconds, respectively. 1 1 The four zero load recovery periods, separating the creep load periods, were also function of Δt . Three recovery periods were held at zero load level for 5Δt = 630 1 1 seconds, but the last recovery period was maintained for a longer interval of 10Δt = 1260 seconds. Immediately following the creep and recovery loading sequences, the specimen was loaded monotonically to failure on a load-controlled linear ramp. The ramp up to the peak load and unloading were each applied over an interval of Δt . 4.1.3 Cyclic-Recovery and Monotonic Loading Proﬁle In three experiments, ice specimens were loaded with cyclic-recovery sequences followed by a fracture ramp, as shown in Fig. 4.2b. The cyclic-recovery loading consisted of 3 sequences, each being composed of four ﬂuctuating loads, at the levels of 0.4 kN, 0.8kN, and 1.2 kN. Each cyclic sequence continued for a constant time interval Δt = 480 seconds. The slopes of the wave on the load up and load release were 1/150 kN/s, 1/75 kN/s, and 1/50 kN/s for the 0.4kN, 0.8kN, and 1.2 kN load levels, respectively. The 0.4kN, 0.8kN, and 1.2 kN cyclic load periods were followed by zero load recovery periods of 1.25Δt = 600 seconds, 1.25Δt = 600 seconds, and 2 2 2.5Δt = 1200 seconds, respectively. 87 Fracture of S2 Columnar Freshwater Ice Under Creep/Cyclic-Recovery Loading [PII] At the completion of the cyclic-recovery loading sequences, the specimen was loaded to failure by a monotonic linear ramp. The ramp up to the peak load and unloading were each applied over an interval of 0.25Δt = 120 seconds. 4.2 Nonlinear Time-Dependent Modeling of S2 Columnar Freshwater Ice The model applied in this section to characterize the nonlinear elastic/ viscoelastic/ viscoplastic response of S2 columnar freshwater ice was presented by Schapery; it was used to model the time-dependent mechanical response of polymers in the nonlinear range under uniaxial stress-strain histories (Schapery, 1969). Schapery’s stress-strain constitutive equations are derived from nonlinear thermodynamic principles, and are very similar to the Boltzmann superposition integral form of linear theory (Flügge, 1975). Schapery’s model represents the material as a system of an arbitrarily large number of nonlinear springs and dashpots. The equations in this section are presented in terms of load and displacement instead of the original stress-strain relations. The notations of the original equations in (Schapery, 1969) are modiﬁed to bring out similarity between all the equations in the paper. When the applied loads are low enough, the material response is linear. For an arbitrary load input, P = P(t) applied at t = 0, Boltzmann’s law approximates the load by a sum of a series of constant load inputs and describes the linear viscoelastic displacement response of the material using the hereditary integral in a single integral constitutive equation. The Boltzmann superposition principle states that the sum of the displacement outputs resulting from each load step is the same as the displacement output resulting from the whole load input. If the number of steps tends to inﬁnity, the total displacement is given as: dP δ(t) = C P + ΔC(t − τ) dτ, (4.1) dτ where C is the initial, time-independent compliance component and ΔC(t) is the transient, time-dependent component of compliance. Turning now to nonlinear viscoelastic response, Schapery developed a simple single-integral constitutive equation from nonlinear thermodynamic theory, with either stresses or strains entering as independent variables (Schapery, 1969). Using load as the independent variable, the displacement response under isothermal and uniaxial loading takes the following form: d(g P) δ(t) = g C P + g ΔC(ψ − ψ ) dτ, (4.2) 0 0 1 dτ 88 Fracture of S2 Columnar Freshwater Ice Under Creep/Cyclic-Recovery Loading [PII] where C and ΔC are the previously deﬁned components of Boltzmann principle, ψ and ψ are the so-called reduced times deﬁned by: t τ dt dt ψ = and ψ = ψ(τ) = (4.3) a [P(t )] a [P(t )] 0 P 0 P and g , g , g , and a are nonlinear functions of the load. Each of these functions 0 1 2 P represents a diﬀerent nonlinear inﬂuence on the compliance: g models the elastic response, g the transient response. g the loading rate, and a is a time scale shift 1 2 P factor. These load-dependent properties have a thermodynamic origin. Changes in g , g , and g reﬂect third and higher order stress-dependence of the Gibbs free energy, 0 1 2 and changes in a are due to similar dependence of both entropy production and the free energy. These functions can also be interpreted as modulus and viscosity factors in a mechanical model representation. In the linear viscoelastic case, g = g = g = 0 1 2 a = 1, and Schapery’s constitutive equation (4.2) reduces to Boltzmann’s equation (4.1). Equation (4.2) contains one time-dependent compliance property, from linear vis- coelasticity theory, ΔC and four nonlinear load-dependent functions g , g , g , and a , 0 1 2 P which reﬂect the deviation from the linear viscoelastic response, that need to be eval- uated. Schapery’s model uses experimental data to evaluate the material property functions in Eq. (4.2). Lou and Schapery (1971) outlined a combined graphical and numerical procedure to evaluate these functions. In their work, a data-reduction method was applied to evaluate the properties from the creep and recovery data. Pa- panicolaou et al. (1999) proposed a method capable of analytically evaluating the material functions using only limiting values of the creep-recovery test. Numerical methods are also employed and are the most commonly used techniques; they are based on ﬁtting the experimental data to the constitutive equation (LeClair et al., 1999). In the current study, a numerical-experimental procedure is adopted. An optimization procedure is applied using the Nelder-Mead (N-M) method (Nelder and Mead, 1965) to back-calculate the values that achieve the best ﬁt between the model and the experi- mental data. To avoid multiple ﬁtting treatments of data and account for the mutual dependence of the functions, the properties were determined from the full data. This avoided errors that may result from separating the data into parts and estimating the functions independently from diﬀerent parts. This approach of ﬁtting a model with experimental data is common in fracture models with several parameters. Schapery (1997) later updated his formulation. He added a viscoplastic term to account for the viscoplastic response of the material and stated that the total compli- ance can be represented as the summation of elastic, viscoelastic, and viscoplastic components. Adamson and Dempsey (1998) applied Schapery’s updated constitutive equation to model the crack mouth opening displacement of saline ice in an experi- mental setup similar to the current study. The theory represents the displacement at the 89 Fracture of S2 Columnar Freshwater Ice Under Creep/Cyclic-Recovery Loading [PII] crack mouth (δ ) as the sum of elastic, viscoelastic, and viscoplastic components: CMOD e ve vp δ = δ + δ + δ (4.4) CMOD CMOD CMOD CMOD where δ = g C P (4.5) 0 e CMOD d g P ve δ = g C (ψ − ψ ) dτ (4.6) 1 ve CMOD dτ vp δ = C g Pdτ (4.7) vp 3 CMOD In the above equations, ψ and ψ are deﬁned in (4.3). g , g , g , g , and a are nonlinear 0 1 2 3 P load functions to be determined. The coeﬃcients C , C , and C are the elastic, e ve vp viscoelastic, and viscoplastic compliances, respectively. Schapery’s equation has been developed for uniaxial loading. The response of the test specimen is dominated by the normal stresses at the direction normal to the X-axis, ahead of the crack (Fig. 4.1). This stress state can be approximated as uniaxial in the same way as in beam bending; the stress is uniaxial tension at the crack tip and then changes linearly. Thus, Schapery’s equations are used to analyze the experimental data. Few assumptions are applied at this point and are based on the choices made by Adamson and Dempsey (1998). For ice, the elastic displacement is linear with load; this immediately leads to g = 1. Schapery stated that g = a = 1 if the instantaneous jump and drop in the 0 1 P displacement are equal (Schapery, 1969). Examination of the current data shows that this condition is not valid, and the functions need to be evaluated. Accordingly, the following approximations are employed: a b−1 c−1 d g ∝ P ; g ∝ P ; g ∝ P ; a ∝ P (4.8) 1 2 3 P From Eq. (4.3): dt ψ − ψ = (4.9) a [P(t )] The viscoelastic compliance is assumed to follow a power law in time with a fractional exponent n. This gives: C (β) ≈ κβ (4.10) ve Incorporating each of these conditions, the total displacement is expressed as t t b t dt d[P(τ)] a c δ = C P + κP dτ + C P dτ (4.11) CMOD e vp [P(t )] dτ 0 τ 0 where δ , P and t are in m, N, and seconds, respectively. It follows from Eq. CMOD 4.11 that two unknown parameters (C , and C ), one unknown constant (κ), and ﬁve e vp unknown exponents (a, b, c, d, and n) need to be determined. As previously mentioned, 90 Fracture of S2 Columnar Freshwater Ice Under Creep/Cyclic-Recovery Loading [PII] the problem is optimized through the N-M technique, by minimizing the objective function F given by the diﬀerence between the model and data, as shown in Eq. 4.12. The components of the total displacement were computed and optimized using MATLAB. A positive constraint was applied to the model variables. Initial guesses of the exponents on the load and time functions were assumed based on previous work on saline ice. The optimized values were then obtained by comparing the model response and the experimentally measured response over the full length of the test up to crack growth initiation. F = arg min M (C , C ,κ, a, b, c, d, n) − D (4.12) i e vp i C ,C ,a,b,... e vp i=1 where M and D refer to the CMOD values given by the model (Eq. 4.11) and the i i experimental data, respectively. . is the Euclidean norm of a vector. N is the number of data points (≈ 2e6 points). This problem is typically called a least-squares problem when using the Euclidean norm. It is a convex problem because F is a convex function and the feasible set is convex. Thus, the optimization algorithm will converge to the global optimal solution. As mentioned earlier, Schapery’s model originated from the thermodynamic theory. The model is not physically-based, and its parameters are not linked to the microstruc- tural properties of the ice (dislocation density, grain size, ...). In addition, the analysis does not account for the formation of fracture process zone in the vicinity of the crack tip. Schapery’s formulation models the experimental response until crack growth initiation and does not account for crack propagation. Table 4.1. Measured experimental data and computed results for the LC experiments. ˙ ˙ Experiment Type LH A hE E E E E P t CMOD CMOD NCOD1 NCOD1 0 1 2 3 4 f max f −1 −1 (m) (m) (m) (mm) (GPa) (GPa) (GPa) (GPa) (GPa) (kN) (s) (μm) (μms )(μm) (μms ) RP15 creep 3 6 2.1 364 6.6 6.7 7.3 7.4 6.9 5.8 68.2 320.1 4.7 53.6 0.8 RP16 creep 3 6 2.1 385 5.6 5.8 7.6 - 6.0 3.8 42.8 228.2 5.3 49.1 1.1 RP17 cyclic 3 6 2.1 407 6.5 - 7.6 - 6.6 4.5 49.3 173.7 3.5 30.0 0.6 RP18 cyclic 3 6 2.1 408 - - - - 5.3 3.9 40.1 143.7 3.6 28.5 0.7 RP19 cyclic 3 6 2.1 412 - 7.0 6.6 - 6.3 6.3 52.5 221.4 4.2 44.0 0.8 91 Fracture of S2 Columnar Freshwater Ice Under Creep/Cyclic-Recovery Loading [PII] 4.3 Results 4.3.1 Experimental and Modelling Results This section presents the results measured and computed for the LC experiments. The current results are compared with the fracture results of monotonically loaded DC experiments of the same ice and same specimen size (3m x 6m, Chapter 3). The main aim is to elucidate the eﬀect of creep and cyclic sequences on the fracture properties. 4.3.2 Eﬀect of the Creep and Cyclic Sequences on the Fracture Properties Table 4.1 shows the measured and computed parameters for the LC experiments. P is the measured peak load which is also the failure load. t represents the time max f to failure, computed from the fracture ramp. CMOD is measured at crack growth initiation. CMOD indicates the displacement rate at the crack mouth and is obtained by dividing CMOD by the failure time. Similarly, NCOD1 (see Fig. 4.1) represents the displacement at crack growth initiation near the initially sharpened crack tip. NCOD1 indicates the displacement rate in the vicinity of the tip and is obtained by dividing NCOD1 by the failure time. Fig. 4.3 gives the results of the peak load P , crack mouth opening displacement max CMOD, and near crack-tip opening displacement NCOD1 as a function of the loading time for the DC experiments (Chapter 3) and the current LC experiments. In these subplots, ﬁrst-order power-law ﬁts were applied to the data of the DC experiments. The LC values lie above, below, and along the DC ﬁt. No clear eﬀect of creep and cyclic loading on the fracture properties was detected. Figs. 4.4a and 4.4b show the experimental load versus the crack opening displace- ment at the crack mouth for the DC and the LC experiments, respectively. Fig. 4.4c displays a zoomed view of the fracture ramp of the LC experiments. Comparing the failure loads of the DC and LC experiments indicates that the failure loads, of experi- ments with comparable loading rates, were similar. Therefore, in these experiments, the creep and cyclic sequences had no inﬂuence on the failure load. Table 4.1 presents several elastic moduli for each experiment. The elastic moduli were calculated from the load-CMOD record usin Eq. 3.16 (Chapter 3, Section 3.1). For the creep experiments (RP15 and RP16), this procedure is repeated for the four creep cycles, resulting in E , E , E , E , and for the fracture ramp, resulting in E . 1 2 3 4 f Similarly for the cyclic experiments (RP17, RP18, and RP19), the moduli calculation was done for the last cycle of each cyclic sequence, giving steady state moduli E , 92 Fracture of S2 Columnar Freshwater Ice Under Creep/Cyclic-Recovery Loading [PII] 1 50 0 1 2 3 0 1 2 3 10 10 10 10 10 10 10 10 (a) (b) 0 1 2 3 10 10 10 10 (c) Figure 4.3. Experimental results for the (a) peak load P , (b) crack mouth opening displacement max CMOD and (c) near crack tip opening displacement NCOD1 at crack growth initiation, as a function of time to failure t for the monotonically-loaded DC experiments (Chapter 3) and the creep/cyclic and monotonically-loaded LC experiments (PII). (a) (b) (c) Figure 4.4. Measured load versus CMOD for the (a) DC experiments (Chapter 3), (b) LC experiments, and (c) LC experiments up to the peak load (PII) 93 Fracture of S2 Columnar Freshwater Ice Under Creep/Cyclic-Recovery Loading [PII] E , E , and for the fracture ramp, resulting in E . Some of the values are missing, 2 3 f caused by the fact that the initial portion of the associated load-CMOD curve was very noisy. The values of the elastic moduli calculation for the creep/cyclic sequences and fracture ramps were fairly constant upon load application, as shown by the linear load-delection curves in Figs. 4.4c, 4.5a, and 4.5b. This linearity justiﬁes the choice of g = 1 in the elastic CMOD component in Eq. 4.5. Table 3.1 in Chapter 3 presents the elastic modulus (E ) calculated at the crack CMOD mouth for the DC experiments; E is similar to E in Table 4.1 here; both values CMOD f lie within the same range. Therefore, the creep and cyclic sequences preceding the fracture ramp did not aﬀect the load-CMOD prepeak behavior. However, the sequences aﬀected the post-peak response as can be distinguished from Fig. 4.4b which displays a longer decay behavior than Fig. 4.4a. The gradual decay of the load portrays the time dependency in the behavior of freshwater ice. Viscoelastic response Viscoplastic response ௩௣ CMOD େ୑୓ୈ (a) (b) (c) Figure 4.5. Load versus CMOD over the (a) creep-recovery cycles for RP15 and the (b) cyclic-recovery sequences for RP17. (c) Schematic illustration of the hysteresis load-displacement diagram. The whole of the hysteresis loop area is the energy loss per cycle. The dashed area is the part of that total that is due to the viscoelastic mechanism and the rest is due to viscous processes (PII). 4.3.3 Ice Response Under the Testing Conditions Fig. 4.6 shows the experimental results for RP16: the applied load and the crack opening displacements at the crack mouth (CMOD), halfway along the crack (COD), and 10 cm behind the tip (NCOD1) (see Fig. 4.1). Similarly, Fig. 4.7 shows the experimental response for RP17. The time-dependent nature of of the ice response is evident. A complete load-CMOD curve was obtained during loading and unloading for each experiment of Table 4.1, indicating stable crack growth. It is clear from Figs. 4.6b and 4.7b that the CMOD, COD, and NCOD1 displace- ments were composed mainly of elastic and viscoplastic components. No signiﬁcant viscoelasticity was detected in the displacement-time records for all the experiments. The primary (transient) creep stage was almost absent or instantaneous. The load Load Fracture of S2 Columnar Freshwater Ice Under Creep/Cyclic-Recovery Loading [PII] 4 250 0 0 0 1000 2000 3000 4000 0 1000 2000 3000 4000 (a) (b) (c) 10.3 10.4 10.5 10.6 Time (d) (e) Figure 4.6. Experimental results for RP16. (a) Load at the crack mouth, see Fig. 4.1. (b) Displacement - time records. (c) Load - displacement record. (d) Typical response of a Maxwell model, consisting of a nonlinear spring and nonlinear dashpot, to a constant load step. (e) CMOD vs time plot showing the ﬁrst creep sequence. [PII]. sequences were characterized by a non-decreasing displacement rate at all levels. The displacement-time slope was linear and constant, indicating that the secondary/steady- state creep regime dominated during each load application. Although the recovery time was longer than the loading time, ≥ 1.25Δt (Creep experiment, Fig. 4.2a and Section 4.1.2) and ≥ 1.25Δt (Cyclic experiment, Fig. 4.2b and Section 4.1.3), the recovery (unload) phases consisted mainly of an elastic recovery (instantaneous drop) and unrecovered viscoplastic displacement. The behavior as observed resembles the response of a Maxwell model composed of a series combination of a nonlinear spring and nonlinear dashpot (Fig. 4.6d). There is no delayed elastic recovery, but there is the elastic response and a permanent deformation. It is important to note that the vertical-looking jumps in the creep and cyclic sequences (Figs. 4.6b and 4.7b) are real but not vertical. It is only a scaling issue; the scale makes them appear as instantaneous with zero displacement. However, zooming in (Fig. 4.6e) reveals that the jumps are just steep, reﬂecting the immediate response to load. Figs. 4.5a and 4.5b support the same analysis. Unlike the viscoelastic response (Fig. 4.5c) which displays no residual displacement in the loading and unloading hysteresis diagram, the current load-CMOD plots showed large permanent displacement after each loading cycle. This concludes that the response of columnar freshwater S2 ice in these experiments was overall elastic-viscoplastic. Displacement Fracture of S2 Columnar Freshwater Ice Under Creep/Cyclic-Recovery Loading [PII] 0 1000 2000 3000 4000 0 1000 2000 3000 4000 (a) (b) (c) Figure 4.7. Experimental results for RP17. (a) Load at the crack mouth, see Fig. 4.1. (b) Displacement - time records. (c) Load - displacement record (PII). 4.3.4 Nonlinear Modelling Analysis The nonlinear theory, outlined in Section 4.2, was used to analyze the experiments. Modelling the visoelastic term (second term of Eq. 4.11) proved to be very challenging. Instead, a simpliﬁed version was modelled by setting a = g = 1. The results of the p 2 initial optimization trials conﬁrmed the previous analysis; the viscoelastic component ve δ had no eﬀect on the ﬁnal ﬁt between the data and the model. The optimization CMOD −18 algorithm ﬁne-tunedκ (Eq. 4.11) to a very small number (10 ), indicating that the best model-data ﬁt is attained when the viscoelastic term goes to zero (Fig. 4.8b). The ﬁnal optimization runs were carried out by considering the elastic and viscoplas- tic components (ﬁrst and last terms of Eq. (4.11)) only. This resulted in 2 parameters, C and C , and one exponent c, that need to be optimized. The optimization con- e vp verged results are given in Table 4.2: C , C and c. For all the experiments, the e vp %reduction of the objective function exceeded 95% and about 110 iterations were needed to reach convergence. A value of c = 1 for the viscoplastic load function provided the best ﬁt between the model and the experiment at all load levels over the total experimental time up to the peak load. The ﬁnal compliance values of the −8 −1 elastic and viscoplastic components were in the ranges 1.8-3.8 x10 mN and 0.2-1 −10 −1 −1 x10 mN s . respectively. Table 4.2. Optimization results computed using Schapery’s model. 8 10 experiment t C x10 C x10 c f e vp −1 −1 −1 (s) (mN ) (mN s ) RP15 68.2 3.330 1.061 1 RP16 42.8 3.845 0.974 1 RP17 49.3 2.637 0.512 1 RP18 40.1 1.861 0.209 1 RP19 52.5 2.775 0.938 1 96 Fracture of S2 Columnar Freshwater Ice Under Creep/Cyclic-Recovery Loading [PII] Figs. 4.8 and 4.9 give the model results, obtained using Eqs. (4.4-4.11), and the experimental results for experiments RP16 and RP17, respectively. Figs. 4.8a and 4.8b show the measured load and the load applied to the model and the measured CMOD-time record compared to the response of the model, respectively for RP16. Figure 4.9 shows similar plots for experiment RP17. experiment RP17 showed a rather good model-experiment ﬁt for the three cyclic-recovery sequences over the load and unload periods. The model succeeded to follow the increasing and decreasing load levels and the corresponding recovery phases. The experimental response for the creep-recovery experiment RP16 appeared to generally conform to the model results, but the model overestimated the recovery displacement in the ﬁrst two cycles. It is unclear to the authors why the model did a better job in ﬁtting the cyclic-recovery than the creep-recovery sequences. This probably hints at some mechanisms that took place in the creep-recovery experiments and are not accounted for by Schapery’s model. Schapery’s model has been tested for creep-recovery sequences of saline ice with an increasing load proﬁle (Schapery, 1997; Adamson and Dempsey, 1998; LeClair et al., 1999, 1996). This is the ﬁrst application of the model with a load proﬁle of increasing and decreasing load levels (Fig. 4.2). Experiment Model with viscoelasticity Model with no viscoelasticity 0 1000 2000 3000 4000 0 1000 2000 3000 4000 5000 (a) (b) Figure 4.8. Experimental and model results for RP16. (a) Load at the crack mouth, see Fig. 4.1. (b) CMOD - time records: Schapery’s model with and without viscoelasticity vs the experimental data (PII). Considering the fracture ramp, Schapery’s nonlinear equation succeeded to model the monotonic displacement response up to crack growth initiation perfectly well for all the experiments. As previously mentioned, the model does not account for crack propagation, so modeling was applied until the peak load. The model was also successful in predicting the critical crack opening displacement values at the failure load. Thus, the model gives a very close prediction of the experimental data over the whole loading proﬁle up to the failure load. The other experiments displayed the same experiment-model agreement. In this study, Schapery’s constitutive model is tested for the ﬁrst time for freshwater 97 Fracture of S2 Columnar Freshwater Ice Under Creep/Cyclic-Recovery Loading [PII] Experiment 160 Model 0 1000 2000 3000 4000 0 1000 2000 3000 4000 (a) (b) Figure 4.9. Experimental and model results for RP17. (a) Load at the crack mouth, see Fig. 4.1. (b) CMOD - time records (PII). ice. The match between the model and the measured data, especially for the cyclic- recovery experiments, provides a ﬁrm support of the ability of Schapery’s constitutive model to describe the time-dependent response of columnar freshwater S2 ice up to crack growth initiation. Figs. 4.10a and 4.10b show the contribution of each individual model component, elastic and viscoplastic, to the total CMOD displacement, for RP16 and RP17, respectively. As mentioned earlier, the elastic and viscoplastic components account for the total deformation. For RP16, the viscoplastic component dominated over the elastic component. For RP17, the elastic and viscoplastic components contributed equally to the total displacement. The applicability of the proposed model and the ﬁtted parameters are limited to the studied ice type, geometry, specimen size, ice temperature, and the current testing conditions. Variation in the operating conditions will change the dominant deformation mechanisms and the ice behavior; and accordingly, new model parameters would be needed to adapt to the new response. 100 80 0 0 0 1000 2000 3000 4000 0 1000 2000 3000 4000 (a) (b) Figure 4.10. Contribution of each individual model component to the total CMOD displacement for (a) RP16 and (b) RP17 (PII). 98 Fracture of S2 Columnar Freshwater Ice Under Creep/Cyclic-Recovery Loading [PII] 4.4 Discussion Interestingly, the ice behavior in the current study diﬀers from previous experimen- tal creep and cyclic work on freshwater ice. Large delayed elastic or recoverable component has been previously observed. Several researchers performed creep ex- periments on granular freshwater ice at lower temperatures (Mellor and Cole, 1981, 1982, 1983; Cole, 1990; Duval et al., 1991) and reported considerable recovery. Duval conducted torsion creep experiments on glacier ice at a similar testing temperature of -1.5 C (Duval, 1978). When unloaded, the ice exhibited creep recovery. According to his analysis: during loading, the internal stresses opposing the dislocation motion increases; upon unloading, the movement of dislocations produced the reversible defor- mation and is caused by the relaxation of internal stresses. Sinha (1978, 1979) studied columnar-grained freshwater ice and concluded that the high-temperature creep is associated with grain boundary sliding. Cole (1995) developed a physically-based constitutive model in terms of dislocation mechanics and quantiﬁed two mechanisms of anelasticity: dislocation and grain boundary relaxations. He demonstrated that the increased temperature sensitivity of the creep properties of ice within a few degrees of the melting point is due to a thermally induced increase in the dislocation density (Cole, 2020). The question then arises as to why warm columnar freshwater ice tested here showed no signiﬁcant delayed elastic eﬀect, and the microstructural changes were mainly irreversible upon unloading? The measured ice response is a novel result for any type of ice. It is important to emphasize that in comparison with earlier freshwater ice studies, the tested samples are very warm and large. Viscoelasticity normally happens due to the elastically- accommodated grain boundary sliding. Upon loading, internal stresses build up at local stress concentrations in the grain boundary geometry (triple points and grain boundary ledges). Assuming there is no microcracking, the growing stress impedes further grain boundary sliding and causes sliding in the reverse direction, giving rise to the recoverable component after unloading. However, in the present case, the measurements showed large unrecoverable deformation. Several reasons can be discussed, related to the ice temperature, microstructure, and nonlinear mechanisms in the process zone. Concerning the eﬀect of temperature: the warmer the temperature, the more liquid on the grain boundary. The high homologous test temperature (top ice surface ≈ -0.3 C) causes liquidity on the gain boundaries (Dash et al., 2006). The intergranular melt phase on the grain boundary renders the ice as two-phase polycrystal and signiﬁcantly inﬂuences the creep and recovery response. In fact, the grain boundary sliding then consists theoretically of two processes: 1) the sliding of grains over one another 99 Fracture of S2 Columnar Freshwater Ice Under Creep/Cyclic-Recovery Loading [PII] and 2) the squeezing-in/out of the liquid between adjacent grains (Muto and Sakai, 1998). The shear behavior of the liquid ﬁlm is function of its properties (thickness and amount). The presence of this liquid at the triple points and the boundary acted as a resisting obstacle for the grains to shear and deform back to their original form, causing the viscoplastic deformation. The microstructure (grain size, crystalline texture) could be another contributing factor. Sinha (1979) developed a nonlinear viscoelastic model, incorporating the grain size eﬀect, to describe the high-temperature creep of polycrystalline materials. He concluded that delayed elastic strain exhibits an inverse proportionality with grain size. This suggests that the grain size (3-10 mm, Fig. 2.5b) of the ice samples is coarse enough not to produce any measurable viscoelastic deformation under the testing conditions. It is also probable that for this grain size, there was not enough local concentration points to arrest the grain boundary sliding and drive the recoverable and reverse sliding. In addition, Gasdaska (1994) discussed that regularly ordered and packed microstructures limit the amount of sliding and rearrangement and lead to less anelastic strain. The ice growth in the Aalto Ice tank was very controlled and resulted in homogeneous ice sheet. Knauss presented a thorough review of the time-dependent fracture models available to date (Knauss, 2015). The essence of the models is based on modelling the behavior in a ﬁnite cohesive/process zone which is attached to the traction-free crack tip. The one-parameter fracture mechanics encompassed by the apparent fracture toughness is not applicable (Dempsey et al., 2018). It is believed that the mechanisms taking place in the process zone play an inﬂuencing role in the current experiments. The nonlinearity in the fracture zone relieved the internal stresses that would ordinarily accommodate the grain boundary sliding and generate some viscoelastic deformation upon unloading. Thus, any microstructural damage that occurred during loading manifested as permanent deformation at the end of the experiment. It is noteworthy that the earlier studies used test sizes which are smaller than the plate size used here. It was shown in the DC fracture experiments (Chapter 3) that scale had an eﬀect at the tested loading rates. It is probable that the specimen size inﬂuenced the time-dependent deformation of freshwater ice. The current experiments suggest that for the large sample size and the kind of ice studied (very warm freshwater ice) under the loading applied, viscoelasticity is not an important deformation component. The experimental results support this prediction, but more experiments are needed to make more general conclusions. All the above-mentioned factors might have contributed to the measured elastic- viscoplastic response. However, the question as to which factor inﬂuenced mostly the behavior is an important research question that requires more testing programs. 100 Fracture of S2 Columnar Freshwater Ice Under Creep/Cyclic-Recovery Loading [PII] Testing the eﬀect of each factor separately requires a set of experiments that considers this factor while keeping all the other conditions ﬁxed. 101 Fracture of S2 Columnar Freshwater Ice Under Creep/Cyclic-Recovery Loading [PII] 102 5. Fracture Energy of Columnar Freshwater Ice: Inﬂuence of Loading Type, Loading Rate and Size [PIII] This chapter addresses the research question RQ7 (Section 1.7). It investigates the inﬂuence of loading type, loading rate, and specimen size on the fracture energy of columnar freshwater S2 ice for the DC (Chapter 3) and LC (Chapter 4) experiments. Diﬀerent methods for computing the fracture energy were applied and compared. The apparent fracture energy at crack growth initiation was obtained via Rice’s J-integral expression (J ) modiﬁed to be applicable to the special case of a deeply cracked edge-cracked plate as well as via a viscoelastic ﬁctitious crack analysis (G ). The VFCM work-of-fracture (W ) was also evaluated. Both J and W were measured from f Q f the load-displacement record at the crack mouth. G was obtained from the VFCM back-calculated stress-separation (σ − δ) relation within the fracture process zone. The rest of the chapter is structured as follows. Section 5.1 introduces the fracture energies and explains how they are obtained. In section 5.2, the computed results from the DC and LC experiments are summarized and analyzed. The chapter has been adapted from the author’s publication (PIII). 5.1 Fracture Energies 5.1.1 Fracture Energy at Crack Growth Initiation via the J-integral (J ) The J-integral, formulated by Rice (1968), approximates the crack-tip stress-strain ﬁelds in a linear or nonlinear elastic material, and thus becomes more diﬃcult to interpret once unloading occurs in the specimen. Consequently, the J-integral is used for specimens at the initiation of crack growth (Mindess et al., 1977; Homeny et al., 1980). Therefore, Rice’s J-integral gives the apparent fracture energy at crack growth initiation (J ), i.e. the energy required to initiate fracture. Later, Rice et al. (1973) derived the J-integral expression for a deeply-notched edge-cracked plate. His approximation is essentially exact if the fabricated notch is 103 Fracture Energy of Columnar Freshwater Ice: Inﬂuence of Loading Type, Loading Rate and Size [PIII] deep enough, with just a short ligament subjected to bending: crack J = Pdδ (5.1) Q crack h(L − A ) 0 0 where h is the plate thickness, L − A the remaining uncracked ligament (see Fig. 3.1), P the applied load, and δ is the crack-mouth-opening-displacement (CMOD). crack The integral is simply the area (A ) under the load-CMOD curve up to crack growth initiation. It represents the work done in loading; where the elastic deformations with no crack present should be eliminated from the calculations in Eq. 5.1. J has been used extensively in metals (Begley and Landes, 1972; Landes and Begley, 1972; Bucci et al., 1972; Landes and Begley, 1974; Clarke et al., 1976; Logsdon, 1976; Yoder and Griﬃs, 1976; Griﬃs and Yoder, 1976; Homeny et al., 1980), concrete (Mindess et al., 1977), and ceramics (Sakai et al., 1988). Apparently, few attempts have been made to estimate this quantity for any ice type. To my knowledge, the ﬁrst eﬀort for freshwater ice was reported by Li et al. (1996). They used the photoelastic method and the speckle technique to measure the stress and displacement ﬁelds necessary to obtain the J-integral. Unfortunately, they could not test for the critical J because of limitations of the testing device. Eq. 5.1 is a good approximation for the current experiments because the initially- sharpened crack length is suﬃciently long: A ≈ 70%L (0.5m x 1m and 3m x 6m plates) and ≈ 75%L (19.5m x 36m plate). The elastic displacements of the load points of the uncracked specimen are negligible compared to the displacements due to the crack (Rice et al., 1973). Therefore, the total raw area under the load-CMOD curve up to the maximum load is considered, as portrayed in Fig. 5.1a. 5.1.2 Work-of-Fracture (W ) The work-of-fracture (W ), as used by Nakayama and others (Nakayama, 1964; Tattersall and Tappin, 1966; Homeny et al., 1980; Sakai et al., 1988; Bažant, 1996; Dempsey et al., 2012), represents the fracture energy required to propagate a crack completely through a specimen. It is calculated from the total work done in splitting the plate, as determined from the area under the entire load-CMOD record (A , see Fig. 5.1b), i.e. W = (5.2) h(L − A ) It is worth noting that J and W provide a more accurate analysis than the apparent Q f fracture toughness (K , Section 3.1 and Eq. 3.7). The K value depends only on Q Q the peak load; however, evaluation of J and W depends on the peak load and the Q f shape of the load-CMOD curve. The ratio of the work-of-fracture to the apparent 104 Fracture Energy of Columnar Freshwater Ice: Inﬂuence of Loading Type, Loading Rate and Size [PIII] (a) (b) (c) (d) Figure 5.1. Description of the area used for (a) J (A ) and (b) W (A ) calculations. Load versus Q J f W Q f crack mouth opening displacement for (c) RP4 and (d) RP12, illustrating the A and A J W Q f areas [PIII]. Figure 5.2. Description of the area used for J (A ) calculations for metals, as deﬁned by the ASTM Q J E1820-20b. Adapted from ASTM E1820-20b AST (2020). Fracture Energy of Columnar Freshwater Ice: Inﬂuence of Loading Type, Loading Rate and Size [PIII] fracture energy at crack growth initiation (W /J ) gives information about the energy f Q absorbed during crack extension. 5.1.3 Fracture Energy at Crack Growth Initiation via the VFCM (G ) VFCM G represents the fracture energy at crack growth initiation obtained via a vis- VFCM coelastic cohesive analysis. A viscoelastic ﬁctitious crack model (VFCM) was applied to model the experimental data of the monotonic experiments (Chapter 3). G is VFCM measured from the area under the σ − δ curve (Fig. 3.13) attained in each experiment. G corresponds to G in Table 3.2 of Chapter 3. VFCM ac 5.2 Results and Discussion Table 1 gives the dimensions of the ice samples together with the measured (L, H, A , h, P , t ) and computed (G , J , W , W /J ) results. max f VFCM Q f f Q Table 5.1. Specimen dimensions, measured data, and results computed for the DC and LC experiments. Experiment Type LH A hP t G J W W /J 0 max f VFCM Q f f Q m m m mm kN s N/mN/mN/m RP1 DC 0.5 1 0.35 350 1.7 55 - 1.2 3.3 2.6 RP2 DC 0.5 1 0.35 350 2.3 52 5.0 2.4 3.7 1.5 RP3 DC 0.5 1 0.35 371 1.6 2 1.7 1.4 3.3 2.3 RP4 DC 0.5 1 0.35 376 1.9 21 3.4 1.3 3.9 2.9 RP5 DC 0.5 1 0.35 402 2.1 572 1.0 1.2 3.2 2.7 RP6 DC 0.5 1 0.35 411 2.4 811 3.6 2.0 4.9 2.4 RP7 DC 3 6 2.1 345 4.6 87 4.2 3.8 4.1 1.1 RP8 DC 3 6 2.1 345 3.0 3 1.8 1.5 3.2 2.1 RP9 DC 3 6 2.1 345 5.2 148 5.5 5.1 7.8 1.5 RP10 DC 3 6 2.1 345 4.8 222 - 3.7 4.2 1.1 RP11 DC 3 6 2.1 360 4.2 15 - 2.4 5.4 2.3 RP12 DC 3 6 2.1 372 6.8 702 8.2 8.3 8.8 1.1 RP13 DC 3 6 2.1 376 5.7 1027 5.7 7.3 5.9 0.8 RP14 DC 19.5 36 14.6 350 10 214 5.8 5.7 5.3 0.9 RP15 LC creep 3 6 2.1 364 6.0 68 - 6.5 19.1 2.9 RP16 LC creep 3 6 2.1 385 3.8 43 - 2.5 14.0 5.6 RP17 LC cyclic 3 6 2.1 407 4.6 49 - 2.5 16.4 6.5 RP18 LC cyclic 3 6 2.1 408 4.0 40 - 2.5 14.2 5.7 RP19 LC cyclic 3 6 2.1 412 6.4 53 - 3.1 1.6 0.5 Fig. 5.3 shows the load-CMOD records for the 0.5m x 1m DC experiments (Fig. 5.3a), 3m x 6m DC experiments (Fig. 5.3b), 19.5m x 36 m DC experiment (Fig. 5.3c), and the 3m x 6m LC experiments (Fig. 5.3d). The area under the load-CMOD curve changed with size, rate, and loading type. The area under the monotonic ramp up to maximum load and up to complete fracture was used to compute J and W , Q f 106 Fracture Energy of Columnar Freshwater Ice: Inﬂuence of Loading Type, Loading Rate and Size [PIII] (a) (b) 0 500 1000 1500 2000 (c) (d) Figure 5.3. Load-CMOD records for the (a) 0.5m x 1m DC (b) 3m x 6m DC, (c) 19.5m x 36m DC and (d) 3m x 6m LC experiments [PIII]. 0 1 2 3 4 10 10 10 10 10 Figure 5.4. Variation of J and G as a function of the time to failure for the DC experiments [PIII]. Q VFCM 107 Fracture Energy of Columnar Freshwater Ice: Inﬂuence of Loading Type, Loading Rate and Size [PIII] 16 10 0 3 0 1 2 3 4 0 1 2 3 4 10 10 10 10 10 10 10 10 10 10 (a) (b) Figure 5.5. (a) The apparent fracture energy at crack growth initiation (J ) and (b) the work-of-fracture fracture energy (W ) as a function of the time to failure for the DC experiments. In (a), ﬁrst-order power-law ﬁts were applied separately to the data for the larger specimen (3m x 6m and 19.5m x 36m) and for the smaller specimen (0.5m x 1m) [PIII]. respectively (see Fig. 5.1). It is important to note that the area used here for evaluating J diﬀers from the area deﬁned in Fig. A1.2 of ASTM E1820-20b AST (2020) and used for the J calculation of metallic materials (Fig. 5.2). The used area (Figs. 5.1c and 5.1d) matches the area applied for other quasi brittle materials such as concrete and ceramics (Mindess et al., 1977; Homeny et al., 1980). There is a remarkable diﬀerence between the load-CMOD plots measured for columnar freshwater ice and quasi-brittle materials in general and those that typically occur for metallic materials. For metals, yielding occurs, followed by strain hardening then complete fracture (Fig. 5.2). On the other hand, yielding and strain hardening do not happen for quasi-brittle materials (Fig. 5.1). Fig. 5.4 shows the apparent fracture energies at crack growth initiation J and G as a function of the time to failure for the DC experiments. Interestingly, VFCM good agreement was obtained between J and G , especially for the 3m x 6m and Q VFCM 19.5m x 36m specimens. Although J and G are computed with diﬀerent inherent Q VFCM assumptions, they generated similar results. J assumes a nonlinear elastic material behavior to characterize the crack tip deformation. G is based on cohesive crack VFCM mechanics coupled with a linear viscoelastic bulk behavior. This indicates that the fracture energy of a deeply cracked ECRP can be analyzed by two methods: Rice J-integral approximation and VFCM back-calculation. Fig. 5.5 shows the apparent fracture energy at crack growth initiation (J ) and the work-of-fracture (W ) as a function of the time to failure for the DC experiments. The J data indicates interrelated size and rate eﬀects. There is a size eﬀect at long failure times, but there is no size eﬀect for fast experiments. The results from the larger plates (3m x 6m and 19.5m x 36m) are interchangeable and higher than the results from the smallest specimen (0.5m x 1m). For the larger specimens, J is 108 Fracture Energy of Columnar Freshwater Ice: Inﬂuence of Loading Type, Loading Rate and Size [PIII] decreasing with faster time to failure, but no signiﬁcant rate eﬀect was observed for the smallest specimen. The J − t relations are non-linear and can be well described Q f with a power-law relation. Fig. 5.5b displays similar size and rate eﬀects as J ,but less clearly due to more scatter displayed by the values. Fig. 5.6 shows the J and W for the 3m x 6m DC and LC experiments as a function Q f of the time to failure. No clear eﬀect of the creep-recovery loading on J was observed. However, the creep-recovery loading caused a signiﬁcant increase in W for all the LC experiments excluding RP19. This can be seen from Figs. 5.3b and 5.3d. The LC experiments displayed a time-dependent behavior characterized by the gradual post-peak decline and large area under the load-CMOD curve. Therefore, while the creep and cyclic sequences did not aﬀect the fracture energy at crack growth initiation (J ), they had a more pronounced eﬀect on the work-of-fracture (W ). Q f 8 10 0 0 0 1 2 3 4 0 1 2 3 4 10 10 10 10 10 10 10 10 10 10 (a) (b) Figure 5.6. (a) The apparent fracture energy at crack growth initiation (J ) and (b) the work-of-fracture fracture energy (W ) as a function of the time to failure for the 3m x 6m DC and LC experiments. First-order power-law ﬁts were applied to the data for the DC experiments. Note in (a), two LC experiments RP16 and RP17 gave the same J value [PIII]. Fig. 5.7 displays the fracture energy ratio (W /J ) as a function of the time to f Q failure. The W /J values for the DC experiments lay in a narrow range of 1 − 3. f Q The larger Experiment sizes showed a rate eﬀect: At low rates, the ratio was 1, but it increased at high rates. The 0.5m x 1m experiments gave a higher ratio with no rate eﬀect. Fig. 5.7 reveals also the eﬀect of loading type. LC experiments RP16, RP17, and RP18 resulted in high W /J values. However, RP15 and RP19 were almost in the f Q same range as the DC experiments. This shows that the creep-recovery loading can increase the W /J in some cases. This leads to the conclusion that in the absence of f Q any creep or cyclic loading, the W of columnar freshwater S2 ice ranges between J f Q and 3J . 109 Fracture Energy of Columnar Freshwater Ice: Inﬂuence of Loading Type, Loading Rate and Size [PIII] -1 0 1 2 3 4 10 10 10 10 10 10 Figure 5.7. Fracture energy ratio W /J as a function of the time to failure for the DC and LC experi- f Q ments [PIII]. 110 6. Conclusion The thesis examined the fracture behavior of warm and ﬂoating columnar freshwater S2 ice under diﬀerent loading (monotonic, creep, cyclic) scenarios, by varying the specimen size and loading rate/type in an in-situ experimental program. Chapter 2 introduced the reader to the ice growth procedure, the ice characterization, and the experimental setup. Chapter 3 covered the ﬁrst part of the experimental program and examined the fracture behavior of warm columnar freshwater S2 ice under monotonic loading. Chapter 4 covered the second part of the experimental program and examined the fracture behavior of columnar freshwater ice under creep/cyclic- recovery and monotonic loading. Chapter 5 discussed the eﬀect of loading rate and type and scale on the fracture energy of all the conducted experiments. The next sections summarize the main conclusions of each chapter and state potential future work. The text has been adapted from the author’s corresponding publications. 6.1 Grown Ice and Experimental Setup [PI, PII and PIII] Large-scale in-situ laboratory mode I fracture experiments of ﬂoating columnar (S2) freshwater ice were completed at the Ice Tank of Aalto University. The ice was very warm; the top surface was slightly cooler than -0.3 C. The ice thickness was very homogeneous throughout the ice sheet and increased from 34 cm to 41 cm during the test program that lasted four weeks. The specimen used was the edge-cracked rectangular plates (ECRP) loaded at the crack mouth up to fracture. The fracture mode was transgranular in all the experiments. 111 Conclusion 6.2 Fracture of Warm S2 Columnar Freshwater Ice Under Monotonic Loading: Size and Rate Effects [PI] 6.2.1 Summary Fourteen experiments on size (scale) and rate eﬀects on the fracture behavior of warm, columnar (S2), freshwater ice were completed. The ECRP covered a size range of 1 : 39; the largest sample had dimensions of 19.5m x 36m. This size and the size range are to my knowledge the largest to any material under laboratory conditions. The loading rates applied led to loading durations from fewer than 2 seconds to more than 1000 seconds leading to an elastic response at the highest rates and a viscoelastic response at the lower rates. The experiments were displacement controlled (DC) and monotonically loaded up to fracture. The experiments were analysed by using the linear-elastic fracture mechanics (LEFM) and the non-linear viscoelastic ﬁctitious crack model (VFCM) approach. For the LEFM analysis, expressions for the apparent fracture toughness (K ) and crack opening displacement were derived by following the weight function method by Dempsey and Mu (2014). For the VFCM, the approach formulated by Mulmule and Dempsey (1997) was adopted and combined with the Nelder-Mead optimization scheme (Nelder and Mead, 1965) to back-calculate the constitutive parameters: stress- separation curve, fracture energy, creep compliance constant, and size of the fracture process zone (PZ). The main observation from the experiments is that the size and rate eﬀects were interrelated. There was a size eﬀect at low loading rates, but no size eﬀect was observed at the higher rates. This applied to the crack opening displacement near the crack tip (NCOD1) as measured, to K as an LEFM parameter, as well as to the stress-separation curve, the fracture energy (G ) and PZ as VFCM parameters. The ac rate dependent size eﬀect is a novel result for any type of ice. Earlier both rate eﬀects and size eﬀects have been measured, but not how these two are related. Similar to earlier studies, the rate eﬀects – when observed – followed power law type relations. The rate dependent size eﬀect can be related with the fracture process at the vicinity of the crack tip: Both the measured NCOD1 and the calculated PZ of the larger specimen decrease with increasing rate and approach the values for the smaller specimen. As was illustrated by using the VFCM, the ice studied can be considered elastic only at the highest rates applied, the rates where the size eﬀect vanishes. The measured size and rate eﬀects can be expressed also as size dependent rate eﬀects. For the larger specimen (3m x 6m and 19.5m x 36m), a clear rate eﬀect was observed, while for the small specimen (0.5m x 1m) the rate eﬀect was weak or absent. 112 Conclusion The results of the two larger, 3m x 6m and 19.5m x 36m, samples were interchange- able suggesting that the 3m x 6m sample size is large enough to give size-independent fracture results for this type of warm ice. From the experimental results and analysis presented, the following requirements are recommended. L/d ≥ 460 enough for polycrystalline homogeneity, av σ /σ < 0.2 notch sensitivity, n t in which L is the specimen size in the cracking direction and d is a measure of the av average grain size. The crack length must be selected to optimize the brittleness of the test plate. This study proved that the viscoelastic ﬁctitious crack model is successful in treating the fracture of S2 columnar freshwater ice. 6.2.2 Research Suggestions • In future studies, it would be useful to do similar experiments but scale the ice- loading device contact length (D) with the specimen size (L, Fig. 3.1). It would be interesting to see if this scaling will aﬀect the measured and the observed results. • Another interesting topic is the eﬀect of the length of the crack front (i.e. plate thickness) on the fracture behavior. For metals, it is known that the state of stress near the tip of a crack tends to change from plane strain to plane stress as the ratio of width-to-thickness of a plate increases (Broek, 1974). However, it was proven for other quasi-brittle materials such as concrete that the thickness is not an important variable (Hillerborg, 1983; Mindess and Nadeau, 1976). This implies that there is no validity to the concepts of triaxiality and plane stress and plane strain idealizations in the fracture of quasi-brittle materials. It is believed that ice exhibits similar behavior as other quasi-brittle materials because a triaxial state of stress does not materialize. The tensile stress parallel to the crack front is relieved by microcracking parallel to the top and bottom surfaces or by creep, the latter taking place rapidly. However, it is important to prove this idea conclusively in an experimental program which vary the thickness of the specimen. To the author’s knowledge, these kind of experiments have not been conducted for ice. • The measured, LEFM and VFCM results showed a size dependent rate eﬀect. There was a rate eﬀect for the large specimens (3m x 6m and 19.5m x 36m) but weak or no rate eﬀect for the smallest specimen (0.5m x 1m). Fig. 3.7b supported this conclusion further and portrayed that the rate eﬀect is decreasing with decreasing specimen size and appears to disappear if specimen smaller than 0.5m x 1m were 113 Conclusion studied. It would be good to conduct experiments with specimens smaller than the smallest size tested here (0.5m x 1m) in order to conﬁrm the lack of rate eﬀect for small specimens. • The experiments suggested that a crack-parallel specimen size of about 460 d is av enough to achieve polycrystalline homogeneity, while 77 d is not. Finding the av exact requirement necessitates the testing of sample sizes which are larger than 0.5m x 1m and smaller than 3m x 6m. 6.3 Fracture of S2 Columnar Freshwater Ice Under Creep/Cyclic-Recovery Loading [PII] 6.3.1 Summary Five 3m x 6m warm freshwater S2 ice specimens were tested under creep/cyclic- recovery sequences followed by a monotonic ramp. The experiments were load controlled (LC) and led to complete fracture of the specimen. The purpose of this study was to examine the time-dependent behavior of freshwater ice using a joint experimental-modeling approach. In the experimental part, the experiments aimed to (1) measure and examine the time- dependent response of columnar freshwater S2 ice through the applied creep/cyclic- recovery sequences and (2) investigate the eﬀect of creep and cyclic sequences on the fracture parameters/behavior through the fracture monotonic ramp. The current experiments were compared with the monotonically loaded DC experiments. The results showed that the creep and cyclic sequences had no clear eﬀect on the failure load and the crack opening displacements at crack growth initiation. The ice response at the testing conditions was overall elastic-viscoplastic. The loading phases displayed an instantaneous transformation from the primary (transient) stage to the steady-state regime, which resulted in permanent (unrecoverable) displacement. The conducted experiments provided a novel observation for the time-dependent behavior of fresh- water ice. Though the delayed elastic component has been reported as a major creep component in freshwater ice, no signiﬁcant viscoelasticity was detected in this study. Several factors were discussed as possibly contributing to the observed behavior: the very warm columnar freshwater ice, liquidity on the grain boundary, large sample size, coarse grain size, and nonlinear mechanisms in the fracture zone. Testing the eﬀect of each factor on the ice response requires a diﬀerent set of experiments that varies this 114 Conclusion factor only while keeping the other conditions ﬁxed. In the modeling part, Schapery’s nonlinear constitutive model was applied for the displacement response at the crack mouth. The elastic-viscoplastic formulation generated a good ﬁt with the experimental response of columnar freshwater S2 ice over the applied loading proﬁle up to crack growth initiation, especially for the cyclic- recovery experiments. The model parameters were obtained via an optimization procedure using the N-M method by comparing the model and experimental CMOD values. The proposed model parameters are valid only for the studied ice type, geometry, specimen size, ice temperature, and the range of applied load experienced in the experiments. Schapery’s model was selected in this study, as it is able to capture the sort of time dependent behavior known to occur in ice and produces a simple and expedient way to help understand the observed behavior. 6.3.2 Research Suggestions • More thorough analysis of the experiments can be done with a physically-based approach that can give insight about the dislocation density, grain boundary sliding and the operating mechanisms. • It would be useful to try longer recovery period to see if this will aﬀect the accumu- lation of unrecoverable deformation. • The results showed no eﬀect of the creep/cyclic-recovery loading on the fracture properties: the failure load and the crack opening displacements at crack growth initiation. It would be interesting to try higher creep and cyclic load levels for similar testing conditions and observe again the eﬀects on the fracture properties. • The current experiments suggest that for the large sample size and the kind of ice studied (very warm freshwater ice) under the loading applied, viscoelasticity is not an important deformation component. The experimental results support this prediction; but more tests, that vary the temperature and size, are needed to make more general conclusions. 115 Conclusion 6.4 Fracture Energy of Columnar Freshwater Ice: Inﬂuence of Loading Type, Loading Rate and Size [PIII] 6.4.1 Summary The mode I fracture energies of columnar freshwater S2 ice were quantiﬁed using diﬀerent methods and the inﬂuence of the type of loading, test size and loading rate on the fracture energy was investigated. The apparent fracture energy at crack growth initiation was computed in two ways: 1) J , via Rice’s J-integral expression for deeply-cracked edge-cracked plates using the load-CMOD record and 2) G , via VFCM a viscoelastic ﬁctitious crack model using the back-calculated stress-separation curve. The work-of-fracture (W ) was obtained from the load-CMOD record. Values of J , f Q G , and W were analyzed considering the eﬀects of scale and loading rate and VFCM f type. A good match was obtained between the J and G values, especially for the Q VFCM 3m x 6m and 19.5m x 36m specimens. J displayed a size dependent rate eﬀect and a rate dependent size eﬀect. There was a size eﬀect at low loading rates, but no size eﬀect was observed at the higher rates. For the larger specimens (3m x 6m and 19.5m x 36m), a clear rate eﬀect was observed, while for the small specimen (0.5m x 1m) the rate eﬀect was almost absent. W showed similar rate and size eﬀects. The results of the DC experiments were compared against the LC experiments to indicate the eﬀect of creep-recovery loading on the fracture energy. No clear eﬀect of the creep and cyclic sequences on J was detected. However, the creep-recovery loading caused a noticeable increase in W for most of the LC experiments. The W /J values for the DC experiments lay in a narrow range (1-3). The larger f Q experiments (3m x 6m and 19.5m x 36m) gave a ratio of 1 at low rates and a higher ratio at higher rates. For the 0.5m x 1m experiments, the ratio was rate independent and was higher than 1. The creep-recovery loading led to higher W /J in some cases. f Q This study proves the signiﬁcance of the deep-notched edge-cracked rectangular plate geometry. Several ways of analysis can be applied; Rice J-integral and VFCM were implemented with similar results. 6.4.2 Research Suggestions The experiments showed an interesting similarity between the J and G values. Q VFCM It is important to test this observation by conducting more experiments using diﬀerent sample sizes and testing conditions. 116 A. Weight Function For An Edge-Cracked Rectangular Plate The weight function method is used in fracture mechanics to determine the expressions of the stress intensity factor and the crack opening displacement [Wu and Carlsson, 1991]. Knowledge of a two-dimensional elastic crack solution (the reference solution) as a function of crack length A for any reference crack face pressure Σ (X) enables the 0 r determination of the stress intensity factor K(A ) and crack opening displacement U(A , X) for the same body under arbitrary loading Σ(X): The derivation of the weight function for an ECRP was given by Dempsey and Mu (2014). E ∂U K(A ) = Σ(X)H (A , X)dX, H (A , X) = (A , X) 0 r 0 r 0 0 K (A ) ∂A 0 r 0 0 E U(A , X) = K(T)H (T, X)dT (A1) 0 r in which H (A , X) denotes the weight function (valid for X ≤ A ) and E is r 0 0 the elastic modulus. The coordinate X is along the crack line with its origin at the crack mouth (see Fig. 2.4a, Chapter 2). K (A ) is the known or reference stress r 0 intensity factor while 2U (A , X) represents the known or reference crack face opening r 0 dis- placement.Chapter 2 introduced the reader to the ice growth procedure, the ice characterization, and the experimental setup. Chapter 3 covered the ﬁrst part of the experimental program and examined the fracture behavior of warm columnar freshwater S2 ice under monotonic loading. Chapter 4 covered the second part of the experimental program The ﬁnal expression for the weight function is 1 x 3 i− h (a , x) = √ G (a )(1 − ) (A2) r 0 i 0 2πa i=1 117 Weight Function For An Edge-Cracked Rectangular Plate The G (i = 1, 2, 3, 4, 5) functions are given by G (a ) = 2.0 1 0 6 105E(a ) aF (a ) 0 0 G (a ) = + √ + 4 − 30 2 0 (1 − a ) F(a ) 0 1 − a 0 S (a ) 26 455E(a ) 52a F (a ) 3 0 0 0 0 G (a ) = − − √ − + 86 3 0 3/2 (1 − a ) (1 − a ) 3F(a ) 1 − a 0 0 0 S (a ) 154 539E(a ) 308a F (a ) 434 4 0 0 0 0 G (a ) = + + √ + − 4 0 3/2 5(1 − a ) 15F(a ) 5 (1 − a ) 1 − a 0 0 0 S (a ) 54 189E(a ) 36a F (a ) 144 5 0 0 0 0 G (a ) = − − − + (A3) 5 0 √ 3/2 5(1 − a ) 5F(a ) 5 (1 − a ) 0 0 1 − a 0 where E(a ) = [3πΦ(a ) − V(a )] /8 2F(a ) 0 0 0 0 35 V(a ) S (a ) = 3πF(a ) − 3 0 0 F(a ) 4 2 0 7 V(a ) S (a ) = √ −12πF(a ) + 5 4 0 0 F(a ) 2 2 0 9 V(a ) S (a ) = √ 7πF(a ) − 3 5 0 0 F(a ) 4 2 V(a ) = (1 + a )V(a ) + a (1 − a )V (a ) (A4) 0 0 0 0 0 0 The ( ) denotes diﬀerentiation with repect to a . Functions F(a ), V(a ), and Φ(a ) 0 0 0 0 are related to the reference SIF K (a ), the reference CMOD V (a ), and the reference r 0 r 0 normalized crack opening area φ(a ), respectively, by F(a ) V(a ) Φ(a ) 0 0 0 K (A ) = σ πA , V (a ) = ,φ(a ) = (A5) r 0 0 r 0 0 2 2 (1 − a ) (1 − a ) (1 − a ) 0 0 where σ is the reference uniform crack face pressure. The actual expressions for F(a ), V(a ), and Φ(a ) were ﬁtted with a seventh-order 0 0 0 polynomial as follows 7 7 7 i i i F(a ) = α s , V(a ) = γ s , Φ(a ) = ϕ s (A6) 0 i 0 i 0 i i=0 i=0 i=0 where the coeﬃcients α , γ „ and ϕ (i = 0, 1,... 7), consecutively, were found to be i i i α:1.12150, −1.59550, 6.83570, −16.4318, 26.2079, −26.3156, 14.8991, −3.59980 γ:2.90860, −5.56710, 19.3629, −38.1097, 56.5619, −52.6474, 27.4002, −5.9579 ϕ:0.62890, −1.20010, 4.1109, −8.6992, 12.7842, −11.6705, 5.8923, −1.2176 (A7) 118 Bibliography 1. Santaoja K. Ph.D. Thesis: Mathematical modelling of deformation mechanisms in ice. 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'W e nee d t o stud y warm ic e becaus e it 'w s ha w t e'r e seein g i n nature; glob a w l arm ih g sappe g T.h m e echa c a plropertie o s ic f a en h dow rie t spon d t fo s or c m e a b fy u e ndamental d ly iffere n w the i n tw 'sarm rath e th r col d a w t s ,r eaditio ls ly tu d i s t y,a ' y Im s El Gh mti .A . .w s m e crondition s ie ncreasing le yxpecte id n previous lf yr i d region ls ik t eh G e rea L take o s B r alt iS ce – a on o etfhe world s 'busies m t arin e area s – Pro fJ . ukk a Tuhkurs iay s ii s tcrucia tlo u ersta t h m eechan i w o cs far i m c o 'I e . u . f s ..h r i p as i rastructu rle ik b eridg an w d in t d urbin hav b eee d n igne fd or fair lp yredictab lse eason w s ,n eee td k ono w wh a htappen w she g nlobal warmi n bg rin g ns e c w diti lIo st.o k lis k t h e o e r ld ul e m s a y ht old up ,T ' uhkur i says.. . P ss leas e W : ar m ic m e a fy ractu diffe nt lty ha c no lid c e K ,atrina Jurva/Aalt U o versi ty BUSINESS + ECONOMY IS - - - -9 (print ) AR T + IS - - - -6 ( f) DESIGN + IS - (print ) ARCHITECTURE IS - ( f) SCIENCE + TECHNOLOGY A la t o Universi yt CROSSOVER Scho o o Elf gn ineeri gn Mecha in c aE l ng ni eer ni g DOCTORAL .aalto .ﬁ DISSER TATIONS dd fa MF SH 202 43 DD aA www dp 2494 9971 NS de 4394 9971 NS dp 4350 46 259 879 NB de 3350 46 259 879 NB in er er er er on no no se se se fn dn dn dn gi ra ra ara na an na in nin ni en en ni nin ni gn gn gn ro or ro si re is si is

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Fracture energy of columnar freshwater ice: Influence of loading type, loading rate and size

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Publisher: Unpaywall
ISSN: 2589-1529
DOI: 10.1016/j.mtla.2021.101188
Publisher site: See Article on Publisher Site

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Published: Dec 1, 2021

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Fracture energy of columnar freshwater ice: Influence of loading type, loading rate and size

Fracture energy of columnar freshwater ice: Influence of loading type, loading rate and size

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Fracture energy of columnar freshwater ice: Influence of loading type, loading rate and size

Fracture energy of columnar freshwater ice: Influence of loading type, loading rate and size

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