Constitutive Modelling of Wetting Deformation of Rockfill Materials

Constitutive Modelling of Wetting Deformation of Rockfill Materials In this paper, the constitutive framework of hypoplasticity is used to model long-term deformations and stress relaxations of weathered and moisture sensitive rockfill materials. The state of weathering of the material is represented by a so-called solid hardness in the sense of a continuum description. The time-dependent degradation of the solid hardness is a result of progressive weathering caused for instance by hydro-chemical reactions of fluid with the solid material. The degradation of the solid hardness can lead to collapse settlements and creep deformations, which are also called wetting deformations. In contrast to a previous version, a new evolution equation for a more refined modelling of the degradation of the solid hardness is proposed. With respect to a pressure-dependent relative density, the influence of the pre-compaction of the material and also the influence of the pressure level on the stiffness can be modelled in a unified manner using a single set of constants. The performance of the new model is validated by comparison of the numerical simulations with experiments data. Keywords Rockfill material · Wetting deformation · Solid hardness · Creep · Hypoplasticity 1 Introduction effect of capillary forces between neighbouring particles can be neglected. Thus, the understanding of the physical and Wetting deformations in the form of collapse and long- thermo-chemical mechanisms of the time-dependent process term creep settlements are observed in rockfill dams and of weathering is important for an accurate interpretation of laboratory experiments. For coarse-grained and weathered the data obtained from monitoring and experiments as well rockfills, a change in the moisture content can lead to an as for numerical modelling. acceleration of the propagation of micro-cracks of weath- The focus of this paper is on modelling the properties of ered grains and consequently to grain fragmentation and a weathered rockfill materials using a continuum approach. reorientation of the grain skeleton into a denser state. The In particular, a hypoplastic model is used which takes into mechanical behaviour of rockfill material can be different account the influence of pressure, density, state of weath- for the dry state and the wet state of the material [1–10]. ering and the rate of deformation on the time-dependent, The mechanical properties not only depend on the type of non-linear and inelastic stress–strain behaviour. The state the rock material and its mineralogical composition, but also of weathering and the influence of a change in the moisture on the grain size distribution, the compaction, the moisture on the stiffness of the material is related to the solid hard- content, the stress state at contact areas between the grains, ness. To take into account solid hardness as a state vari- and the time-dependent degradation of the solid hardness able in the constitutive model, the solid hardness is defined due to the chemical reaction of the grain material with mois- in the sense of a continuum description. In particular, the ture. Experiments with coarse-grained and weathered rock- solid hardness is a measure of the pressure level of the grain fills show that wetting deformations cannot be explained assembly under isotropic compression where grain crushing based on the concept of effective stresses. This is also obvi- becomes dominant [11, 12]. While in the version by Bauer ous for partly saturated coarse-grained materials where the [13], the time-dependent degradation of the solid hardness is modelled in a simplified manner using only three mate - rial parameters, a more sophisticated evolution equation is * Erich Bauer proposed in this paper. The benefit of the extended evolution erich.bauer@TUGraz.at equation is a refined distinction of the degradation of the Institute of Applied Mechanics, Graz University of Technology, Technikerstrasse 4/II, 8010 Graz, Austria Vol.:(0123456789) 1 3 International Journal of Civil Engineering solid hardness related to collapse settlements and to long- For weathered rockfills, the solid hardness is usually lower term deformations. for the wet material than for the dry material as illustrated in Fig. 1b. Wetting of the dry material leads to a degradation of the solid hardness accompanied by collapse settlements, creep 2 Constitutive Modelling of Wetting and/or stress relaxation. The following evolution equation for Deformation the irreversible degradation of the solid hardness was proposed by Bauer [13]: In the present constitutive model, the solid hardness is a key parameter to reflect the different stiffnesses of weathered and h =− (h − h ), (2) st st sw moisture-sensitive rockfill materials under dry and wet con- ditions. With respect to a continuum description, the solid where h = dh /dt denotes the change of the solid hardness st st hardness is related to the grain assembly under monotonic with the time t, h denotes the current value of the solid st isotropic compression. More precisely, the solid hardness hardness, h is the value of the solid hardness in the asymp- sw is a parameter in the following compression law by Bauer totical state for time t → ∞ , and c has the dimension of time [11, 12]: and is related to the degradation velocity. With respect to an initial solid hardness of h , the integration of Eq. (2) so 3p leads to: e = e exp − , (1) i io h = h +(h − h ) exp − . (3) st sw so sw where p denotes the mean pressure, i.e. p =− ∕3 , e kk io denotes the void ratio for p ≈ 0 and h is the solid hardness, For rockfill materials with pronounced deformation imme- which is defined for the mean pressure where the compres- diately after changing the moisture content or after full satura- sion curve in a semi-logarithmic representation shows the tion, it could be convenient to distinguish between short-term point of inflection as illustrated in Fig.  1a. Parameter n is deformations, also called collapse settlements, and long-term related to the inclination of the compression curve at the deformations called creep. To distinguish these two effects, point of inflection. the degradation of the solid hardness is decomposed into a As the solid hardness is related to a pressure level where part h , which is related mainly to collapse settlements, and st1 grain crushing becomes dominant, isotropic compression a part h , which is related mainly to the long-term deforma- sw2 experiments have to be carried out up to rather high pres- tions. In the modified model, the two parts are combined to sures to calibrate the parameters h and n. However, in stand- obtain a smooth transition between collapse deformations and ard laboratories, isotropic compression tests under high long-term deformations. The corresponding evolution equa- pressures are usually more difficult to carry out than high tions read: pressure oedometer experiments. In this context, it can be ̇ ̇ ̇ h = h + h , (4) st st1 st2 noted that the data of the compression curve obtained from with oedometric compression can alternatively be used for the calibration of the solid hardness as outlined in [12]. h =− (h − h ), st1 st1 sw1 (5) Fig. 1 Illustration of the isotropic compression curve: a semi-logarithmic representation of the compression curve and definition of the solid hardness h ; b compression curves for the dry and wet material and anintermediate state during the time-dependent process of degradation 1 3 International Journal of Civil Engineering and in which the term h is obtained from a consistency condition [13]. The pressure-dependent maximum void ratio e and h =− (h + h ). critical void ratio e are modelled as: (6) st2 st2 sw2 3p Herein, the constitutive parameters c ≤ c and h ≥ h 1 2 sw1 sw2 e = e exp − , (10) i io can be obtained by back analysis of creep curves obtained in h st experiments. For modelling wetting deformations, i.e. long- and term deformations, depending on the state of weathering, the solid hardness was implemented into particular models based 3p on the concept of hypoplasticity [14] and generalized plastic- e = e exp − . (11) c co ity [15]. The performance of the concept of degradation of h sc the solid hardness based on the evolution Eq. (2) has been Herein, e and e are the corresponding values for the demonstrated for different rockfill materials by comparing the io co nearly stress-free state. In contrast to previous versions of results obtained from numerical simulations with laboratory the hypoplastic model for weathered rockfill materials, the experiments, e.g. [16–19]. For this paper, the extended evolu- solid hardness h for isotropic compression is die ff rent from tion Eq. (4) was implemented into a simplified hypoplastic st the solid hardness h for the critical void ratio. In particu- constitutive model, which is similar to the one shown by Li sc lar, h < h , which can be explained by a more pronounced et al. [14]. The main constitutive relations can be summarized sc st grain damage under deviatoric loading. This is also justified as follows. by experimental data for sandstone considered in [14]. The objective stress rate tensor,  , depends on the current With respect to the balance equation of mass, the follow- void ration, e , the current state of the solid hardness, h , the st ̇ ing relation between the rate of the void ratio and the volume rate of the solid hardness, h , the effective Cauchy stress ten- st strain rate holds: sor,  , and the rate of deformation tensor, . With respect to a Cartesian co-ordinate system, the components of the stress ė =(1 + e) . (12) rate read: For calibration of the constitutive constants, the following � � 2 ∗ strategy can be used. In a first step, the mechanical behav - 𝜎 = f a ̂ +(𝜎 ̂ )𝜎 ̂ + f a ̂ (𝜎 ̂ + 𝜎 ̂ ) ij s ij kl kl ij d ij kl kl ij iour of the dry material, i.e. the state of the material before � � st wetting takes place, is considered. For this state the constitu- + 𝜎 𝛿 ∕3 + 𝜅𝜎 . kk ij ij (7) tive Eq. (7) reduces to: st � � o 2 ∗ 𝜎 = f a ̂ +(𝜎 ̂ ) 𝜎 ̂ + f a ̂ (𝜎 ̂ + 𝜎 ̂ ) . Herein, 𝜎 ̂ = 𝜎 𝜎 and 𝜎 ̂ = 𝜎 − 𝛿 ∕3 denote the nor- ij s ij kl kl ij d ij kl kl ij ij ij kk ij ij ij (13) malized Cauchy stress  , and the normalized deviatoric part ij Based on Eq. (13) and with respect to the experimental of  , respectively. Factors f and f are functions of the pres- ij s d data for the initial dry material, the parameters  , n , h , c so sure-dependent relative density. The scalar function a ̂ is related e , e , h and can be calibrated as outlined in more detail io co sc to the stress limit condition given by Masuoka and Nakai [20] for a similar version of the hypoplastic model by Bauer and depends on the critical friction angle  [21]. With respect [12] and Gudehus [22]. In the second step, the parameters to the last term on the right hand side of Eq. (7), creep and h , c and  in Eq. (2), and for the extended creep model sw stress relaxation can be modelled in a unified manner [13].  is ij the parameters h , c , h and c in Eqs.  (5) and (6) can sw1 1 sw2 2 the Kronecker delta, and parameter  controls the magnitude be adapted from creep experiments. For the special case and direction of the creep strain [18]. In contrast to the model of pure creep, i.e. no stress relaxation, the stress rate in proposed in [13], simplied fi representations of factors f and Eq. (7) becomes zero and the corresponding constitutive f are used in this paper, i.e. equation reads: (1−n) � � (1 + e )(3p∕h ) i st 2 ∗ f = h , (8) 0 = f a ̂ +(𝜎 ̂ ) 𝜎 ̂ + f a ̂ (𝜎 ̂ + 𝜎 ̂ ) s st s ij kl kl ij d ij kl kl ij en h 𝜎 ̂ 𝜎 ̂ i kl kl � � (14) st + 𝜎 𝛿 ∕3 + 𝜅𝜎 . kk ij ij and h st On the other hand, under constant volume the degrada- tion of the solid hardness leads to the following equation for f = , (9) stress relaxation: 1 3 ̇𝜀 ̇𝜀 ̇𝜀 ̇𝜀 ̇𝜀 ̇𝜀 ̇𝜀 ̇𝜀 ̇𝜀 ̇𝜀 ̇𝜀 ̇𝜀 ̇𝜀 ̇𝜀 International Journal of Civil Engineering particular, the model takes into account the influence of the o confining stress on the stiffness at the beginning of devia- st 𝜎 = 𝜎 𝛿 ∕ 3 + 𝜅𝜎 . (15) ij kk ij ij toric loading, the peak friction angle, strain softening and st the volume strain behaviour. For mixed boundary conditions, usually a combination When wetting of the rockfill material takes place, the of creep deformation and stress relaxation will take place, incremental stiffness is also influenced by the time-depend - which can be computed using the full representation of the ent process of degradation of the solid hardness. In the constitutive Eq. (7). following, the performance of the proposed model (7) is investigated for two die ff rent evolution equations for model - ling the degradation of the solid hardness. In particular, the 3 Verification of the Proposed Model evolution Eq. (2) is compared with the extended version (4). The calibration leads to the following two sets of constants: For verification of the proposed hypoplastic model, the Constants related to the evolution Eq. (2): constitutive parameters were calibrated based on data from h = 78.5 MPa; c = 4.0 h;  = 0.6. sw laboratory tests with broken sandstone [10]. For the dry state Constants related to the evolution Eq. (4): of the material, the solid hardness is constant and the val- h = 90.5  MPa; c = 0.6  h; h = 12.0  MPa; c = 8.0; sw1 1 sw2 2 ues of the corresponding constitutive parameters are sum-  = 0.6. marized in Table  1. More detailed information about the In Fig. 3, the numerical results of simulation of creep calibration can be found in [14]. It should be noted that the tests under different isotropic stress states and different model can capture the influence of the pressure level on the deviatoric stress states are compared with the experimental incremental stiffness using only a single set of constants. data. In particular, the dashed curves are related to the evo- A comparison of the results of numerical simulations with lution Eq. (2), the solid curves are related to the extended experimental data obtained from triaxial compression tests evolution Eq. (4) and the markers represent the experi- is shown in Fig. 2. As can be seen, the proposed hypoplastic mental data. A comparison of the creep curves shows that model simulates well the essential mechanical properties for the amount of creep deformation is much higher under different confining stresses with the same set of constants. In higher confining stress states. Furthermore, the creep Table 1 Constitutive parameters (°) n h (MPa) e e h (MPa) c so io co sc for dry broken sandstone 40.0 0.82 120.0 0.3 0.24 73.5 0.75 50 -1.5 Num. results σ =3000kPa -1.0 22 σ =2000kPa 40 22 σ =1200kPa -0.5 σ =800 kPa σ =400 kPa 0.5 Num. results σ =3000 kPa 1.0 σ =2000 kPa σ =1200 kPa 1.5 σ = 800 kPa σ = 400 kPa 0 2.0 0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16 ε [%] ε [%] 11 11 (a) (b) Fig. 2 Comparison of numerical simulations with experimental data of triaxial compression tests under different confining stresses: a mobilized friction angle against axial strain; b volume strain against axial strain  1 3 ϕ [°] ε [%] v International Journal of Civil Engineering 1.0 0.8 Num. sim. usingEqs.(4) and(14) Num. sim. usingEqs.(4) and (14) Num. sim. usingEqs.(2) and(14) Num. sim. usingEqs.(2) and (14) Exp.: σ =4087kPa Exp.: σ =4087kPa 11 11 0.8 Exp.: σ =2443kPa Exp.: σ =2443kPa 11 11 0.6 Exp.: σ =800 kPa Exp.: σ =800 kPa 11 11 0.6 0.4 0.4 0.2 0.2 0 0 0 10 20 30 40 50 0 10 20 30 40 50 t [h] t [h] (a) 1.0 0.8 Num. sim. usingEqs.(4) and(14) Num. sim. usingEqs.(4) and(14) Num. sim. usingEqs.(2) and(14) Num. sim. usingEqs.(2) and(14) Exp.: σ =5723kPa Exp.: σ =5723kPa 0.8 11 Exp.: σ =3461kPa 11 Exp.: σ =3461kPa 0.6 Exp.: σ =1200kPa 11 Exp.: σ =1200kPa 0.6 0.4 0.4 0.2 0.2 0 0 0 10 20 30 40 50 0 10 20 30 40 50 t [h] t [h] (b) Fig. 3 Creep curves under different confining stresses and different vertical stress states: a confining stress 800 kPa; b confining stress 1200 kPa  deformation is also more pronounced under higher verti- 4 Conclusions cal stresses. Although the evolution Eq. (2) can capture the dependency of the amount of creep strain for different This paper proposes a constitutive model for simulation confining stresses and vertical stresses, the extended evo- of wetting deformations and stress relaxation in which the lution equation (4) obviously allows a significant refined solid hardness is a key parameter to reflect the state of modelling. In particular, the deformation during the initial weathering of rockfill material. In this model, the solid creep period can be better adapted. The latter becomes hardness is defined as a measure of the stress level of the important for rockfill materials after wetting that exhibit grain assembly under isotropic compression where grain pronounced collapse settlements which occur within a crushing becomes dominant. The time-dependent and irre- rather short time. versible reduction of the solid hardness caused by wetting or other environmental effects is described by an evolu- tion equation depending on the current state of weathering 1 3 ε [%] ε [%] 11 11 ε [%] ε [%] v International Journal of Civil Engineering 7. Li GX (1988) Triaxial wetting experiments on rockfill materi- and the degradation velocity. To simulate the evolution als used in Xiaolangdi earth dam. Research report 17-1-2-88031. of creep deformation and/or stress relaxation, the rate of Tsinghua University, Beijing degradation of the solid hardness is linked to a hypoplastic 8. Oldecop LA, Alonso EE (2007) Theoretical investigation of the constitutive equation. In this paper, the performance of time-dependent behavior of rockfill. Geotechnique 57(3):289–301 9. Fang XS (2005) Test study and numerical simulation on wetting two different evolution equations for describing the degra- deformation of gravel sand. Dissertation for the Master Degree, dation of the solid hardness is investigated and compared Hohai University, Nanjing (in Chinese) with data from creep experiments. It is shown that the 10. Fu H, Ling H (2009) Experimental research on the engineering extended evolution equation allows a refined modelling properties of the fill materials used in the Cihaxia concrete faced rockfill dam. Research report. Nanjing Hydraulic Research Insti- of collapse and long-term settlements in a unified manner. tute, Nanjing (in Chinese) 11. Bauer E (1995) Constitutive modelling of critical states in hypo- Acknowledgements Open access funding provided by Graz University plasticity. In: Pande GN, Pietruszczak S (eds) Proceedings of the of Technology. The author wishes to thank Dr. Z. Z. Fu from the Nan- 5th international symposium on numerical models in geomechan- jing Hydraulic Research Institute for providing experimental data. Mr. ics. Balkema Press, Rotterdam, pp 15–20 L. Li and Mr. M. Khosravi supported this paper with some drawings 12. Bauer E (1996) Calibration of a comprehensive hypoplastic model and numerical calculations. for granular materials. Soils Found 36(1):13–26 13. Bauer E (2009) Hypoplastic modelling of moisture-sensitive Open Access This article is distributed under the terms of the Crea- weathered rockfill materials. Acta Geotech 4:261–272 tive Commons Attribution 4.0 International License (http://creat iveco 14. Li L, Wang Z, Liu S, Bauer E (2016) Calibration and performance mmons.or g/licenses/b y/4.0/), which permits unrestricted use, distribu- of two different constitutive models for rockfill materials. Water tion, and reproduction in any medium, provided you give appropriate Sci Eng 9(3):227–239 (pp. 1–12) credit to the original author(s) and the source, provide a link to the 15. Wang ZJ, Chen SS, Fu ZZ (2015) Dilatancy behaviors and a gen- Creative Commons license, and indicate if changes were made. eralized plasticity model of rockfill materials. Rock Soil Mech 36(7):1931–1938 (in Chinese) 16. Fu Z, Bauer E (2009) Hypoplastic constitutive modeling of the long term behaviour and wetting deformation of weathered granu- References lar materials. In: Bauer E, Semprich S, Zenz G (eds) Proceedings of the 2nd international conference on long-term behaviour of 1. Brauns J, Kast K, Blinde A (1980) Compaction effects on the dams. Graz University of Technology, Graz, pp 437–478 mechanical and saturation behavior of disintegrated Rockfill. Proc 17. Bauer E, Fu ZZ, Liu SH (2010) Hypoplastic constitutive model- Int Conf Compaction Paris 1:107–112 ling of wetting deformation of weathered rockfill materials. Front 2. Ovalle C, Dano C, Hicher PY, Cisternas M (2015) An experimen- Archit Civ Eng China 4(1):78–91 tal framework for evaluating the mechanical behaviour of dry and 18. Bauer E, Fu ZZ, Liu SH (2012) Influence of pressure and density wet crushable granular materials based on the particle breakage on the rheological properties of rockfills. Front Struct Civ Eng ratio. Can Geotech J 52:1–12 6(1):25–34 3. Alonso E, Oldecop LA (2000) Fundamentals of rockfill collapse. 19. Bauer E, Fu Z, Liu S (2011) Constitutive modelling of rheologi- In: Rahardjo H, Toll DG, Leong EC (eds) Proceedings of the 1st cal properties of materials for rockfill dams. In: Pina E, Portela J, Asian conference on unsaturated soils. Balkema Press, Rotterdam, Gomes J (eds) Proceedings of the 6th international conference on pp 3–13 dam engineering, Lisbon, Portugal (CD ROM). CI-Premier Pte 4. Alonso E, Cardoso R (2009) Behaviour of materials for earth Ltd, Singapore, pp 1–14 and rockfill dams. Perspective from unsaturated soil mechanics. 20. Matsuoka H, Nakai T (1997) Stress–strain relationship of soil In: Bauer E, Semprich S, Zenz G (eds) Proceedings of the 2nd based on ‘SMP’. In: Proceedings of Specialty Session 9, IX inter- international conference on long term behaviour of dams. Graz national conference on soil mechanics and foundation engineer- University of Technology, Graz, pp 1–38 ing, Tokyo, pp 153–162 5. Fang XS (2005) Test study and numerical simulation on wetting 21. Bauer E (2000) Conditions for embedding Casagrande’s critical deformation of gravel sand. Master thesis, Hohai University, Nan- states into hypoplasticity. Mech Cohes Frict Mater 5:125–148 jing (in Chinese) 22. Gudehus G (1996) A comprehensive constitutive equation for 6. Oldecop LA, Alonso E (2004) Testing rockfill under relative granular materials. Soils Found 36(1):1–12 humidity control. 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Constitutive Modelling of Wetting Deformation of Rockfill Materials

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Abstract

In this paper, the constitutive framework of hypoplasticity is used to model long-term deformations and stress relaxations of weathered and moisture sensitive rockfill materials. The state of weathering of the material is represented by a so-called solid hardness in the sense of a continuum description. The time-dependent degradation of the solid hardness is a result of progressive weathering caused for instance by hydro-chemical reactions of fluid with the solid material. The degradation of the solid hardness can lead to collapse settlements and creep deformations, which are also called wetting deformations. In contrast to a previous version, a new evolution equation for a more refined modelling of the degradation of the solid hardness is proposed. With respect to a pressure-dependent relative density, the influence of the pre-compaction of the material and also the influence of the pressure level on the stiffness can be modelled in a unified manner using a single set of constants. The performance of the new model is validated by comparison of the numerical simulations with experiments data. Keywords Rockfill material · Wetting deformation · Solid hardness · Creep · Hypoplasticity 1 Introduction effect of capillary forces between neighbouring particles can be neglected. Thus, the understanding of the physical and Wetting deformations in the form of collapse and long- thermo-chemical mechanisms of the time-dependent process term creep settlements are observed in rockfill dams and of weathering is important for an accurate interpretation of laboratory experiments. For coarse-grained and weathered the data obtained from monitoring and experiments as well rockfills, a change in the moisture content can lead to an as for numerical modelling. acceleration of the propagation of micro-cracks of weath- The focus of this paper is on modelling the properties of ered grains and consequently to grain fragmentation and a weathered rockfill materials using a continuum approach. reorientation of the grain skeleton into a denser state. The In particular, a hypoplastic model is used which takes into mechanical behaviour of rockfill material can be different account the influence of pressure, density, state of weath- for the dry state and the wet state of the material [1–10]. ering and the rate of deformation on the time-dependent, The mechanical properties not only depend on the type of non-linear and inelastic stress–strain behaviour. The state the rock material and its mineralogical composition, but also of weathering and the influence of a change in the moisture on the grain size distribution, the compaction, the moisture on the stiffness of the material is related to the solid hard- content, the stress state at contact areas between the grains, ness. To take into account solid hardness as a state vari- and the time-dependent degradation of the solid hardness able in the constitutive model, the solid hardness is defined due to the chemical reaction of the grain material with mois- in the sense of a continuum description. In particular, the ture. Experiments with coarse-grained and weathered rock- solid hardness is a measure of the pressure level of the grain fills show that wetting deformations cannot be explained assembly under isotropic compression where grain crushing based on the concept of effective stresses. This is also obvi- becomes dominant [11, 12]. While in the version by Bauer ous for partly saturated coarse-grained materials where the [13], the time-dependent degradation of the solid hardness is modelled in a simplified manner using only three mate - rial parameters, a more sophisticated evolution equation is * Erich Bauer proposed in this paper. The benefit of the extended evolution erich.bauer@TUGraz.at equation is a refined distinction of the degradation of the Institute of Applied Mechanics, Graz University of Technology, Technikerstrasse 4/II, 8010 Graz, Austria Vol.:(0123456789) 1 3 International Journal of Civil Engineering solid hardness related to collapse settlements and to long- For weathered rockfills, the solid hardness is usually lower term deformations. for the wet material than for the dry material as illustrated in Fig. 1b. Wetting of the dry material leads to a degradation of the solid hardness accompanied by collapse settlements, creep 2 Constitutive Modelling of Wetting and/or stress relaxation. The following evolution equation for Deformation the irreversible degradation of the solid hardness was proposed by Bauer [13]: In the present constitutive model, the solid hardness is a key parameter to reflect the different stiffnesses of weathered and h =− (h − h ), (2) st st sw moisture-sensitive rockfill materials under dry and wet con- ditions. With respect to a continuum description, the solid where h = dh /dt denotes the change of the solid hardness st st hardness is related to the grain assembly under monotonic with the time t, h denotes the current value of the solid st isotropic compression. More precisely, the solid hardness hardness, h is the value of the solid hardness in the asymp- sw is a parameter in the following compression law by Bauer totical state for time t → ∞ , and c has the dimension of time [11, 12]: and is related to the degradation velocity. With respect to an initial solid hardness of h , the integration of Eq. (2) so 3p leads to: e = e exp − , (1) i io h = h +(h − h ) exp − . (3) st sw so sw where p denotes the mean pressure, i.e. p =− ∕3 , e kk io denotes the void ratio for p ≈ 0 and h is the solid hardness, For rockfill materials with pronounced deformation imme- which is defined for the mean pressure where the compres- diately after changing the moisture content or after full satura- sion curve in a semi-logarithmic representation shows the tion, it could be convenient to distinguish between short-term point of inflection as illustrated in Fig.  1a. Parameter n is deformations, also called collapse settlements, and long-term related to the inclination of the compression curve at the deformations called creep. To distinguish these two effects, point of inflection. the degradation of the solid hardness is decomposed into a As the solid hardness is related to a pressure level where part h , which is related mainly to collapse settlements, and st1 grain crushing becomes dominant, isotropic compression a part h , which is related mainly to the long-term deforma- sw2 experiments have to be carried out up to rather high pres- tions. In the modified model, the two parts are combined to sures to calibrate the parameters h and n. However, in stand- obtain a smooth transition between collapse deformations and ard laboratories, isotropic compression tests under high long-term deformations. The corresponding evolution equa- pressures are usually more difficult to carry out than high tions read: pressure oedometer experiments. In this context, it can be ̇ ̇ ̇ h = h + h , (4) st st1 st2 noted that the data of the compression curve obtained from with oedometric compression can alternatively be used for the calibration of the solid hardness as outlined in [12]. h =− (h − h ), st1 st1 sw1 (5) Fig. 1 Illustration of the isotropic compression curve: a semi-logarithmic representation of the compression curve and definition of the solid hardness h ; b compression curves for the dry and wet material and anintermediate state during the time-dependent process of degradation 1 3 International Journal of Civil Engineering and in which the term h is obtained from a consistency condition [13]. The pressure-dependent maximum void ratio e and h =− (h + h ). critical void ratio e are modelled as: (6) st2 st2 sw2 3p Herein, the constitutive parameters c ≤ c and h ≥ h 1 2 sw1 sw2 e = e exp − , (10) i io can be obtained by back analysis of creep curves obtained in h st experiments. For modelling wetting deformations, i.e. long- and term deformations, depending on the state of weathering, the solid hardness was implemented into particular models based 3p on the concept of hypoplasticity [14] and generalized plastic- e = e exp − . (11) c co ity [15]. The performance of the concept of degradation of h sc the solid hardness based on the evolution Eq. (2) has been Herein, e and e are the corresponding values for the demonstrated for different rockfill materials by comparing the io co nearly stress-free state. In contrast to previous versions of results obtained from numerical simulations with laboratory the hypoplastic model for weathered rockfill materials, the experiments, e.g. [16–19]. For this paper, the extended evolu- solid hardness h for isotropic compression is die ff rent from tion Eq. (4) was implemented into a simplified hypoplastic st the solid hardness h for the critical void ratio. In particu- constitutive model, which is similar to the one shown by Li sc lar, h < h , which can be explained by a more pronounced et al. [14]. The main constitutive relations can be summarized sc st grain damage under deviatoric loading. This is also justified as follows. by experimental data for sandstone considered in [14]. The objective stress rate tensor,  , depends on the current With respect to the balance equation of mass, the follow- void ration, e , the current state of the solid hardness, h , the st ̇ ing relation between the rate of the void ratio and the volume rate of the solid hardness, h , the effective Cauchy stress ten- st strain rate holds: sor,  , and the rate of deformation tensor, . With respect to a Cartesian co-ordinate system, the components of the stress ė =(1 + e) . (12) rate read: For calibration of the constitutive constants, the following � � 2 ∗ strategy can be used. In a first step, the mechanical behav - 𝜎 = f a ̂ +(𝜎 ̂ )𝜎 ̂ + f a ̂ (𝜎 ̂ + 𝜎 ̂ ) ij s ij kl kl ij d ij kl kl ij iour of the dry material, i.e. the state of the material before � � st wetting takes place, is considered. For this state the constitu- + 𝜎 𝛿 ∕3 + 𝜅𝜎 . kk ij ij (7) tive Eq. (7) reduces to: st � � o 2 ∗ 𝜎 = f a ̂ +(𝜎 ̂ ) 𝜎 ̂ + f a ̂ (𝜎 ̂ + 𝜎 ̂ ) . Herein, 𝜎 ̂ = 𝜎 𝜎 and 𝜎 ̂ = 𝜎 − 𝛿 ∕3 denote the nor- ij s ij kl kl ij d ij kl kl ij ij ij kk ij ij ij (13) malized Cauchy stress  , and the normalized deviatoric part ij Based on Eq. (13) and with respect to the experimental of  , respectively. Factors f and f are functions of the pres- ij s d data for the initial dry material, the parameters  , n , h , c so sure-dependent relative density. The scalar function a ̂ is related e , e , h and can be calibrated as outlined in more detail io co sc to the stress limit condition given by Masuoka and Nakai [20] for a similar version of the hypoplastic model by Bauer and depends on the critical friction angle  [21]. With respect [12] and Gudehus [22]. In the second step, the parameters to the last term on the right hand side of Eq. (7), creep and h , c and  in Eq. (2), and for the extended creep model sw stress relaxation can be modelled in a unified manner [13].  is ij the parameters h , c , h and c in Eqs.  (5) and (6) can sw1 1 sw2 2 the Kronecker delta, and parameter  controls the magnitude be adapted from creep experiments. For the special case and direction of the creep strain [18]. In contrast to the model of pure creep, i.e. no stress relaxation, the stress rate in proposed in [13], simplied fi representations of factors f and Eq. (7) becomes zero and the corresponding constitutive f are used in this paper, i.e. equation reads: (1−n) � � (1 + e )(3p∕h ) i st 2 ∗ f = h , (8) 0 = f a ̂ +(𝜎 ̂ ) 𝜎 ̂ + f a ̂ (𝜎 ̂ + 𝜎 ̂ ) s st s ij kl kl ij d ij kl kl ij en h 𝜎 ̂ 𝜎 ̂ i kl kl � � (14) st + 𝜎 𝛿 ∕3 + 𝜅𝜎 . kk ij ij and h st On the other hand, under constant volume the degrada- tion of the solid hardness leads to the following equation for f = , (9) stress relaxation: 1 3 ̇𝜀 ̇𝜀 ̇𝜀 ̇𝜀 ̇𝜀 ̇𝜀 ̇𝜀 ̇𝜀 ̇𝜀 ̇𝜀 ̇𝜀 ̇𝜀 ̇𝜀 ̇𝜀 International Journal of Civil Engineering particular, the model takes into account the influence of the o confining stress on the stiffness at the beginning of devia- st 𝜎 = 𝜎 𝛿 ∕ 3 + 𝜅𝜎 . (15) ij kk ij ij toric loading, the peak friction angle, strain softening and st the volume strain behaviour. For mixed boundary conditions, usually a combination When wetting of the rockfill material takes place, the of creep deformation and stress relaxation will take place, incremental stiffness is also influenced by the time-depend - which can be computed using the full representation of the ent process of degradation of the solid hardness. In the constitutive Eq. (7). following, the performance of the proposed model (7) is investigated for two die ff rent evolution equations for model - ling the degradation of the solid hardness. In particular, the 3 Verification of the Proposed Model evolution Eq. (2) is compared with the extended version (4). The calibration leads to the following two sets of constants: For verification of the proposed hypoplastic model, the Constants related to the evolution Eq. (2): constitutive parameters were calibrated based on data from h = 78.5 MPa; c = 4.0 h;  = 0.6. sw laboratory tests with broken sandstone [10]. For the dry state Constants related to the evolution Eq. (4): of the material, the solid hardness is constant and the val- h = 90.5  MPa; c = 0.6  h; h = 12.0  MPa; c = 8.0; sw1 1 sw2 2 ues of the corresponding constitutive parameters are sum-  = 0.6. marized in Table  1. More detailed information about the In Fig. 3, the numerical results of simulation of creep calibration can be found in [14]. It should be noted that the tests under different isotropic stress states and different model can capture the influence of the pressure level on the deviatoric stress states are compared with the experimental incremental stiffness using only a single set of constants. data. In particular, the dashed curves are related to the evo- A comparison of the results of numerical simulations with lution Eq. (2), the solid curves are related to the extended experimental data obtained from triaxial compression tests evolution Eq. (4) and the markers represent the experi- is shown in Fig. 2. As can be seen, the proposed hypoplastic mental data. A comparison of the creep curves shows that model simulates well the essential mechanical properties for the amount of creep deformation is much higher under different confining stresses with the same set of constants. In higher confining stress states. Furthermore, the creep Table 1 Constitutive parameters (°) n h (MPa) e e h (MPa) c so io co sc for dry broken sandstone 40.0 0.82 120.0 0.3 0.24 73.5 0.75 50 -1.5 Num. results σ =3000kPa -1.0 22 σ =2000kPa 40 22 σ =1200kPa -0.5 σ =800 kPa σ =400 kPa 0.5 Num. results σ =3000 kPa 1.0 σ =2000 kPa σ =1200 kPa 1.5 σ = 800 kPa σ = 400 kPa 0 2.0 0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16 ε [%] ε [%] 11 11 (a) (b) Fig. 2 Comparison of numerical simulations with experimental data of triaxial compression tests under different confining stresses: a mobilized friction angle against axial strain; b volume strain against axial strain  1 3 ϕ [°] ε [%] v International Journal of Civil Engineering 1.0 0.8 Num. sim. usingEqs.(4) and(14) Num. sim. usingEqs.(4) and (14) Num. sim. usingEqs.(2) and(14) Num. sim. usingEqs.(2) and (14) Exp.: σ =4087kPa Exp.: σ =4087kPa 11 11 0.8 Exp.: σ =2443kPa Exp.: σ =2443kPa 11 11 0.6 Exp.: σ =800 kPa Exp.: σ =800 kPa 11 11 0.6 0.4 0.4 0.2 0.2 0 0 0 10 20 30 40 50 0 10 20 30 40 50 t [h] t [h] (a) 1.0 0.8 Num. sim. usingEqs.(4) and(14) Num. sim. usingEqs.(4) and(14) Num. sim. usingEqs.(2) and(14) Num. sim. usingEqs.(2) and(14) Exp.: σ =5723kPa Exp.: σ =5723kPa 0.8 11 Exp.: σ =3461kPa 11 Exp.: σ =3461kPa 0.6 Exp.: σ =1200kPa 11 Exp.: σ =1200kPa 0.6 0.4 0.4 0.2 0.2 0 0 0 10 20 30 40 50 0 10 20 30 40 50 t [h] t [h] (b) Fig. 3 Creep curves under different confining stresses and different vertical stress states: a confining stress 800 kPa; b confining stress 1200 kPa  deformation is also more pronounced under higher verti- 4 Conclusions cal stresses. Although the evolution Eq. (2) can capture the dependency of the amount of creep strain for different This paper proposes a constitutive model for simulation confining stresses and vertical stresses, the extended evo- of wetting deformations and stress relaxation in which the lution equation (4) obviously allows a significant refined solid hardness is a key parameter to reflect the state of modelling. In particular, the deformation during the initial weathering of rockfill material. In this model, the solid creep period can be better adapted. The latter becomes hardness is defined as a measure of the stress level of the important for rockfill materials after wetting that exhibit grain assembly under isotropic compression where grain pronounced collapse settlements which occur within a crushing becomes dominant. The time-dependent and irre- rather short time. versible reduction of the solid hardness caused by wetting or other environmental effects is described by an evolu- tion equation depending on the current state of weathering 1 3 ε [%] ε [%] 11 11 ε [%] ε [%] v International Journal of Civil Engineering 7. Li GX (1988) Triaxial wetting experiments on rockfill materi- and the degradation velocity. To simulate the evolution als used in Xiaolangdi earth dam. Research report 17-1-2-88031. of creep deformation and/or stress relaxation, the rate of Tsinghua University, Beijing degradation of the solid hardness is linked to a hypoplastic 8. Oldecop LA, Alonso EE (2007) Theoretical investigation of the constitutive equation. In this paper, the performance of time-dependent behavior of rockfill. Geotechnique 57(3):289–301 9. Fang XS (2005) Test study and numerical simulation on wetting two different evolution equations for describing the degra- deformation of gravel sand. Dissertation for the Master Degree, dation of the solid hardness is investigated and compared Hohai University, Nanjing (in Chinese) with data from creep experiments. It is shown that the 10. Fu H, Ling H (2009) Experimental research on the engineering extended evolution equation allows a refined modelling properties of the fill materials used in the Cihaxia concrete faced rockfill dam. Research report. Nanjing Hydraulic Research Insti- of collapse and long-term settlements in a unified manner. tute, Nanjing (in Chinese) 11. Bauer E (1995) Constitutive modelling of critical states in hypo- Acknowledgements Open access funding provided by Graz University plasticity. In: Pande GN, Pietruszczak S (eds) Proceedings of the of Technology. The author wishes to thank Dr. Z. Z. Fu from the Nan- 5th international symposium on numerical models in geomechan- jing Hydraulic Research Institute for providing experimental data. Mr. ics. Balkema Press, Rotterdam, pp 15–20 L. Li and Mr. M. Khosravi supported this paper with some drawings 12. Bauer E (1996) Calibration of a comprehensive hypoplastic model and numerical calculations. for granular materials. Soils Found 36(1):13–26 13. Bauer E (2009) Hypoplastic modelling of moisture-sensitive Open Access This article is distributed under the terms of the Crea- weathered rockfill materials. Acta Geotech 4:261–272 tive Commons Attribution 4.0 International License (http://creat iveco 14. Li L, Wang Z, Liu S, Bauer E (2016) Calibration and performance mmons.or g/licenses/b y/4.0/), which permits unrestricted use, distribu- of two different constitutive models for rockfill materials. Water tion, and reproduction in any medium, provided you give appropriate Sci Eng 9(3):227–239 (pp. 1–12) credit to the original author(s) and the source, provide a link to the 15. Wang ZJ, Chen SS, Fu ZZ (2015) Dilatancy behaviors and a gen- Creative Commons license, and indicate if changes were made. eralized plasticity model of rockfill materials. Rock Soil Mech 36(7):1931–1938 (in Chinese) 16. Fu Z, Bauer E (2009) Hypoplastic constitutive modeling of the long term behaviour and wetting deformation of weathered granu- References lar materials. In: Bauer E, Semprich S, Zenz G (eds) Proceedings of the 2nd international conference on long-term behaviour of 1. Brauns J, Kast K, Blinde A (1980) Compaction effects on the dams. Graz University of Technology, Graz, pp 437–478 mechanical and saturation behavior of disintegrated Rockfill. Proc 17. Bauer E, Fu ZZ, Liu SH (2010) Hypoplastic constitutive model- Int Conf Compaction Paris 1:107–112 ling of wetting deformation of weathered rockfill materials. Front 2. Ovalle C, Dano C, Hicher PY, Cisternas M (2015) An experimen- Archit Civ Eng China 4(1):78–91 tal framework for evaluating the mechanical behaviour of dry and 18. Bauer E, Fu ZZ, Liu SH (2012) Influence of pressure and density wet crushable granular materials based on the particle breakage on the rheological properties of rockfills. Front Struct Civ Eng ratio. Can Geotech J 52:1–12 6(1):25–34 3. Alonso E, Oldecop LA (2000) Fundamentals of rockfill collapse. 19. Bauer E, Fu Z, Liu S (2011) Constitutive modelling of rheologi- In: Rahardjo H, Toll DG, Leong EC (eds) Proceedings of the 1st cal properties of materials for rockfill dams. In: Pina E, Portela J, Asian conference on unsaturated soils. Balkema Press, Rotterdam, Gomes J (eds) Proceedings of the 6th international conference on pp 3–13 dam engineering, Lisbon, Portugal (CD ROM). CI-Premier Pte 4. Alonso E, Cardoso R (2009) Behaviour of materials for earth Ltd, Singapore, pp 1–14 and rockfill dams. Perspective from unsaturated soil mechanics. 20. Matsuoka H, Nakai T (1997) Stress–strain relationship of soil In: Bauer E, Semprich S, Zenz G (eds) Proceedings of the 2nd based on ‘SMP’. In: Proceedings of Specialty Session 9, IX inter- international conference on long term behaviour of dams. Graz national conference on soil mechanics and foundation engineer- University of Technology, Graz, pp 1–38 ing, Tokyo, pp 153–162 5. Fang XS (2005) Test study and numerical simulation on wetting 21. Bauer E (2000) Conditions for embedding Casagrande’s critical deformation of gravel sand. Master thesis, Hohai University, Nan- states into hypoplasticity. Mech Cohes Frict Mater 5:125–148 jing (in Chinese) 22. Gudehus G (1996) A comprehensive constitutive equation for 6. Oldecop LA, Alonso E (2004) Testing rockfill under relative granular materials. Soils Found 36(1):1–12 humidity control. Geotech Test J 27(3):1–10 1 3

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International Journal of Civil EngineeringSpringer Journals

Published: Jun 4, 2018

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