TY - JOUR
AU1 - He, Fei
AU2 - Lei, Wanyu
AU3 - Mao, Erqing
AU4 - Liu, Qingquan
AU5 - Chen, Hangjie
AU6 - Wang, Xu
AB - 1 Introduction The issue of foundation settlement and deformation of cold region structures is a critical concern in the study of frozen soil engineering. It is essential for the normal functioning and safe service of the structure in permafrost regions. This problem has been exacerbated by the prolonged degradation of permafrost due to global climate change. Creep deformation in pile foundations is an important deformation mechanism during the operational phase of bridge piles in cold regions, and even relatively small loads can cause creep deformation in the piles [1]. Furthermore, frozen soil exhibits significant temperature sensitivity, adding complexity and instability to the creep behavior between piles and soil in cold regions. Consequently, there is an urgent need for research on the settlement and deformation patterns of pile foundations in frozen soil under different constant loads, as well as strategies for mitigating settlement. Research into shear creep tests at the frozen soil-structure interface and creep constitutive models plays a pivotal role in addressing such issues. The creep deformation patterns at the frozen soil-structure interface are similar to those of frozen soil creep deformation and can be assessed through stress-controlled shear testing apparatus or triaxial testing equipment [2–5]. Creep deformation in frozen soil refers to the process of elastic-plastic-viscous deformation under constant loading conditions. It is primarily influenced by the temperature, moisture (ice) content, and stress conditions of the frozen soil. There are two types of permafrost foundation creep, attenuation and non-attenuation. The three broad categories of permafrost creep modeling techniques include empirical models, stress-strain-time models, and rheological models [6]. The most commonly employed approach in rheological modeling involves combining various mechanical elements with distinct characteristics to effectively describe the relationship between soil creep deformation and time [7]. Within rheological models, classical mechanical element models include the viscoelastic Kelvin body, the viscoplastic Bingham body, the Maxwell body, the Burgers model, and the Nishihara model, among others [8]. The Nishihara model can explain the attenuation and isotropic creep process in permafrost creep experiments better at lower loads [9]. But at higher stresses, the linear element in this model is unable to adequately capture the nonlinear rapid creep stage. Therefore, in order to more accurately describe the whole process of creep, both domestic and international scholars introduced the nonlinear rheological model [10]. Song et al. [11], and Sun et al. [12] pointed out that the creep viscosity coefficient η is a function of the applied load and the duration of the load. Deng et al. [13] proposed a non-Newtonian fluid viscous damper element and combined it with a traditional model to obtain a new rheomechanical model. Qi et al. [14] realized the description of accelerated creep stage and landslide near-slip prediction by connecting nonlinear dashpot pots with strain-triggered function in series on the basis of traditional Nishihara model. Hou et al. [15] introduced hardening parameters and damage variables in the Nishihara model to describe the creep process of frozen coarse-grained pulverized clay at different coarse-grained contents. Hou et al. [16] presented a nonlinear creep damage model for rocks considering initial damage by introducing an improved viscous unit and a new nonlinear viscoplastic damage unit. Li et al. [17] proposed an improved Nishihara model considering temperature and stress by introducing hardening factor and damage factor. Zhu et al. [18] modified the elastic modulus of the elastic component in the Nishihara model as a function of stress and introduced a damage variable to enhance the viscoplastic unit. This modification enables the model to accurately describe the entire creep process under various shear stress levels and temperatures. Xu et al. [19] introduced a nonlinear viscoplastic unit and proposed a creep model for describing the whole process of creep in kilomagnetite. The aforementioned enhancements to the Nishihara model primarily focus on a specific type of soil or rock. However, there is a relative dearth of research on the shear creep characteristics at the interface between frozen soil and concrete, particularly in the context of pile foundations in permafrost regions subjected to elevated temperatures due to permafrost degradation. Currently, the long-term deformation of pile foundations in permafrost regions primarily considers the structural deformation of the piles themselves, with limited attention given to the creep behavior at the interface between the pile and the surrounding soil. Previous studies have shown that, Material surface roughness, soil density, moisture content, particle angle characteristics, structural material, and loading characteristics have important effects on the shear behavior of the material-soil interface [20–22]. Kishida et al [23] concluded that roughness plays an important role in the friction between steel and soil surface. S Quanbin et al. [24] constructed an empirical formula on adfreezing strengths incorporating temperatures, normal stress, and roughness. They concluded that relationship between peak adfreezing strength and roughness satisfied a logarithmic function, The fluctuation cycle about strength curves of residual adfreezing strength increases with increasing roughness. Chen et al. [25] investigated different types of red clay-concrete interfaces using large-scale direct shear tests. The results showed that the peak shear strength values were close to the residual shear strength values at different normal stress levels. The shear strength of the clay-concrete interface is both cohesive and frictional, and the interfacial shear strength increases with the increase of surface roughness. Karam et al. [26] conducted a series of laboratory large-scale direct shear tests on sand-concrete specimens and sound that the shear strength characteristics of the sand-concrete interface are mainly affected by the relative density of sand, normal stress level, concrete surface roughness and interface area ratio (Ar). The increase of concrete surface roughness leads to the increase of mobilization friction between sand and concrete. In summary, the aforementioned improvements to the Nishihara model have almost been proposed for specific soil or rock types. There is a relative lack of research on the shear creep characteristics of frozen soil-concrete contact surfaces, especially in the context of pile foundations in permafrost regions under high-temperature conditions resulting from permafrost degradation. Besides, the freezing strength test results of permafrost-structure contact surface obtained from the rapid shear test cannot be directly applied to the long-term deformation study of pile foundation, and cannot reveal the mechanism of the deterioration of the long-term bearing performance of pile foundation in the permafrost zone. This work is based on the Nishihara model and extends it by analogizing frozen soil creep behavior. The modification involves replacing the linear viscous Kelvin element with a nonlinear component, where the viscosity coefficient η1 becomes a function of time and stress. Additionally, a damage variable D is introduced within the viscoplastic Bingham model. These adaptations result in an improved Nishihara model that comprehensively accounts for time-dependent and stress-dependent behavior, specifically tailored for characterizing the creep phenomena at frozen soil-concrete interfaces. A large-scale stress-controlled shear instrument was used to carry out graded loading creep tests on concrete with different roughness and frozen soil at -1°C, and the rationality of the model was verified on the basis of the test data. Findings from current study can be useful in guiding the design and theoretical research of pile foundations in permafrost regions. 2 Creep model of permafrost-concrete contact surface The Nishihara model, which uses Kelvin and Bingham bodies in succession to describe the creep curve of the permafrost-concrete contact surface (Eq (1)), and the mechanical model is shown in Fig 1. (1) Where G0 and G1 are the Hooke body shear modulus and Kelvin body shear modulus, respectively. η1 and η2 are the Kelvin body viscosity coefficient and viscoplastic body viscosity coefficient, respectively. τ is the creep stress, τs is the long-term strength limit and γ is strain value. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 1. Nishihara model. https://doi.org/10.1371/journal.pone.0297824.g001 Since the Nishihara model cannot describe the accelerated phase of shear creep, its mechanical components were modified to obtain the improved Nishihara model, which is depicted mechanically in Fig 2. The following improvements are listed. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 2. Modified Nishihara model. https://doi.org/10.1371/journal.pone.0297824.g002 2.1 Viscoelastic Kelvin body The viscoelastic Kelvin model is obtained by connecting a Hooke elastic body with a Newton viscous body and is capable of describing the nonlinear creep decay phase of the frozen soil-concrete. According to the analysis of frozen Qinghai-Tibetan chalk-concrete contact surface test results, Creep has non-linear characteristics, the degree of nonlinearity is related to the creep time, and the creep process of the viscous coefficient of hardening with time increases the law of increasing. Assuming that the viscosity coefficient η1is a function of stress level and time [27, 28], it can be expressed as shown in Eq (2): (2) Where represents the initial viscosity coefficient of the Kelvin model; τ denotes the shear stress level, and t signifies the creep time. Considering that with the increase of shear stress, the longer the time for the specimen to reach the steady state creep stage, so the Kelvin body viscous coefficient will increase with the increase of shear stress, combined with the previous study, it is assumed that the relationship between the effect of shear stress level on the viscous coefficient of the Kelvin body is shown in Eq (3) [29, 30]: (3) Eq (2) is derived for t as shown in Eq (4): (4) In Eq (4), , which indicates that the viscous coefficient with the increase in time shows a monotonous increasing trend, in line with its rule of change. From Eq (2), it can be seen that if t = 0, η1(t,τ) = 0, if t→∞, , which show that with the continuous growth of creep time, the improved Kelvin body hysteresis coefficient monotonically increasing from 0 to . Since an increase in the hysteresis coefficient will prevent the shear rate from growing, it can be utilized to characterize the lower shear stress under a reduced creep stage. The improved Kelvin body constitutive equation is shown in Eq (5): (5) Where G1 is the Kelvin body shear modulus and η1 is the Kelvin body viscosity coefficient. γve is the viscoelastic strain. To get the Kelvin body shear creep equation illustrated in Eq (6), substitute Eq (2) into Eq (5), using t = 0 and γve(t) = 0 as the initial value: (6) This part is used to describe the deformation characteristics of the frozen soil-concrete contact surface during the decay creep stage. 2.2 Viscoplastic Bingham body Вялов introduced the concept of "damage" into the study of permafrost mechanics for the first time [31, 32]. Damage and viscoplastic flow are the causes of the material’s energy consumption under specific circumstances [33]. Damage mechanics is more suited to understanding the deformation and damage of permafrost because it can explain the full process of structural destruction. In order to characterize the developmental (accelerated) creep phase, the damage variable D is included. The Weibull distribution is used to define the damage variable D as shown in Eq (7) and the viscoplastic strain as shown in Eq (8) [34, 35]: (7) (8) (9) Where is the initial viscous coefficient of Bingham body, τs is the long-term strength limit and τu is the long-term strength, n, m are the damage factor parameters, which can be obtained by the least squares method. The improved model is composed of Hooke elastomer, Kelvin body and Bingham body in series, and the total strain is the sum of the strains in each part, which is obtained by substituting into Eq (3) and Eq (9): (10) Eq (10) represents the improved constitutive model for the creep behavior of the frozen soil-concrete interface. Its applicability has been validated through a series of creep tests on the frozen silt-concrete interface using specific conditions (as described in Section 3) with a moisture content of 22%, normal stress of 150 kPa, and a roughness parameter of R = 0.538 mm. The creep curves under various shear stress levels were individually fitted to the experimental data. The fitting results are illustrated in Fig 3, and the coefficients of determination are presented in Table 1. The correlation coefficients squared values exceed 0.99, demonstrating that this model effectively characterizes the creep behavior of the frozen soil-concrete interface under different shear stress levels. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 3. Validation of the applicability of creep constitutive model for the interface of frozen soil and concrete. https://doi.org/10.1371/journal.pone.0297824.g003 Download: PPT PowerPoint slide PNG larger image TIFF original image Table 1. Evaluation of model goodness of fit. https://doi.org/10.1371/journal.pone.0297824.t001 In order to unify the factor of shear stress level and establish the contact surface creep model considering the shear stress level, according to the above conclusions and synthesize the results of the frozen silt-concrete contact surface creep test, the relationship between G0, G1, and the shear stress τ can be defined as an exponential function form as shown in Eq (11) ~ (14) respectively: (11) (12) (13) Where a, b, c, d, e, f, g, h and i are model parameters, τ is the shear stress. firstly increases, then decreases, and is expressed as a quadratic function: (14) Where j, k, l are model parameters and τ is shear stress. The Eq (10)–Eq (14) above illustrate the improved creep intrinsic model of the permafrost-concrete contact surface taking the stress level into account. 2.1 Viscoelastic Kelvin body The viscoelastic Kelvin model is obtained by connecting a Hooke elastic body with a Newton viscous body and is capable of describing the nonlinear creep decay phase of the frozen soil-concrete. According to the analysis of frozen Qinghai-Tibetan chalk-concrete contact surface test results, Creep has non-linear characteristics, the degree of nonlinearity is related to the creep time, and the creep process of the viscous coefficient of hardening with time increases the law of increasing. Assuming that the viscosity coefficient η1is a function of stress level and time [27, 28], it can be expressed as shown in Eq (2): (2) Where represents the initial viscosity coefficient of the Kelvin model; τ denotes the shear stress level, and t signifies the creep time. Considering that with the increase of shear stress, the longer the time for the specimen to reach the steady state creep stage, so the Kelvin body viscous coefficient will increase with the increase of shear stress, combined with the previous study, it is assumed that the relationship between the effect of shear stress level on the viscous coefficient of the Kelvin body is shown in Eq (3) [29, 30]: (3) Eq (2) is derived for t as shown in Eq (4): (4) In Eq (4), , which indicates that the viscous coefficient with the increase in time shows a monotonous increasing trend, in line with its rule of change. From Eq (2), it can be seen that if t = 0, η1(t,τ) = 0, if t→∞, , which show that with the continuous growth of creep time, the improved Kelvin body hysteresis coefficient monotonically increasing from 0 to . Since an increase in the hysteresis coefficient will prevent the shear rate from growing, it can be utilized to characterize the lower shear stress under a reduced creep stage. The improved Kelvin body constitutive equation is shown in Eq (5): (5) Where G1 is the Kelvin body shear modulus and η1 is the Kelvin body viscosity coefficient. γve is the viscoelastic strain. To get the Kelvin body shear creep equation illustrated in Eq (6), substitute Eq (2) into Eq (5), using t = 0 and γve(t) = 0 as the initial value: (6) This part is used to describe the deformation characteristics of the frozen soil-concrete contact surface during the decay creep stage. 2.2 Viscoplastic Bingham body Вялов introduced the concept of "damage" into the study of permafrost mechanics for the first time [31, 32]. Damage and viscoplastic flow are the causes of the material’s energy consumption under specific circumstances [33]. Damage mechanics is more suited to understanding the deformation and damage of permafrost because it can explain the full process of structural destruction. In order to characterize the developmental (accelerated) creep phase, the damage variable D is included. The Weibull distribution is used to define the damage variable D as shown in Eq (7) and the viscoplastic strain as shown in Eq (8) [34, 35]: (7) (8) (9) Where is the initial viscous coefficient of Bingham body, τs is the long-term strength limit and τu is the long-term strength, n, m are the damage factor parameters, which can be obtained by the least squares method. The improved model is composed of Hooke elastomer, Kelvin body and Bingham body in series, and the total strain is the sum of the strains in each part, which is obtained by substituting into Eq (3) and Eq (9): (10) Eq (10) represents the improved constitutive model for the creep behavior of the frozen soil-concrete interface. Its applicability has been validated through a series of creep tests on the frozen silt-concrete interface using specific conditions (as described in Section 3) with a moisture content of 22%, normal stress of 150 kPa, and a roughness parameter of R = 0.538 mm. The creep curves under various shear stress levels were individually fitted to the experimental data. The fitting results are illustrated in Fig 3, and the coefficients of determination are presented in Table 1. The correlation coefficients squared values exceed 0.99, demonstrating that this model effectively characterizes the creep behavior of the frozen soil-concrete interface under different shear stress levels. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 3. Validation of the applicability of creep constitutive model for the interface of frozen soil and concrete. https://doi.org/10.1371/journal.pone.0297824.g003 Download: PPT PowerPoint slide PNG larger image TIFF original image Table 1. Evaluation of model goodness of fit. https://doi.org/10.1371/journal.pone.0297824.t001 In order to unify the factor of shear stress level and establish the contact surface creep model considering the shear stress level, according to the above conclusions and synthesize the results of the frozen silt-concrete contact surface creep test, the relationship between G0, G1, and the shear stress τ can be defined as an exponential function form as shown in Eq (11) ~ (14) respectively: (11) (12) (13) Where a, b, c, d, e, f, g, h and i are model parameters, τ is the shear stress. firstly increases, then decreases, and is expressed as a quadratic function: (14) Where j, k, l are model parameters and τ is shear stress. The Eq (10)–Eq (14) above illustrate the improved creep intrinsic model of the permafrost-concrete contact surface taking the stress level into account. 3 Experimental procedure 3.1 Testing apparatus Creep Shear Instrument. The aforementioned improved Nishihara model was validated using experimental results conducted on the frozen silt-concrete interface. The testing apparatus employed was a large-scale stress-controlled shear device developed in-house at the Geotechnical Laboratory of Lanzhou Jiaotong University (as depicted in Fig 4), which mainly consists of three parts: the main frame, the shear box and the stress loading system. The dimensions of both the upper and lower shear boxes were 200×200×100 mm3 (length × width × height). The normal pressure was applied by a hydraulic jack through a reaction beam, while horizontal shear stress was incrementally applied using 25 kg bags of iron sand through pulleys. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 4. Large-scale stress-controlled shearing apparatus. https://doi.org/10.1371/journal.pone.0297824.g004 Refrigeration and temperature control systems. XT5701LTB-450 high and low temperature cold bath system was selected to control the temperature of the cryogenic chamber, which can realize the temperature control accuracy of ± 0.1°C. Pt-100 Platinum resistance sensor was used for temperature measurement. 3.2 Specimen production The concrete portion of the shear specimen was pre-fabricated with vertical grooves on its surface to create different roughness levels. The mix proportions of concrete are detailed in Table 2. In the experiments, four different concrete specimens with distinct surface roughness were designed, and their detailed geometric dimensions are illustrated in Fig 5. Surface roughness for these four concrete structures was measured using the sand-pouring method, resulting in values of 0 mm, 0.538 mm, 0.775 mm, and 1.225 mm, respectively. The principle behind this method involves using the volume replacement technique with standard sand to measure the volume of depressions within the concrete surface area. This measurement provides the average depth of these depressions, which serves as a quantification of the concrete surface roughness, as shown in Eq (15): (15) Where R is the concrete surface roughness; Vs is the volume of standard sand in the groove; A is the surface area of the contact surface. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 5. Surface roughness design of concrete test block (unit: mm). (a) R = 0 mm; (b) R = 0.538 mm; (c) R = 0.775 mm; (d) R = 1.225 mm. https://doi.org/10.1371/journal.pone.0297824.g005 Download: PPT PowerPoint slide PNG larger image TIFF original image Table 2. Mix proportion of concrete (unit: kg /m3). https://doi.org/10.1371/journal.pone.0297824.t002 The test soil is Lanzhou silt, and the physical properties of the soil samples are shown in Table 3. The particle size distribution of silt is shown in Table 4, and the moisture content of the soil samples is 22%. Inside the sample box with the precast concrete structure at the bottom, the silt was layered and compacted to form an integrated silt-concrete structure, with temperature sensors (Pt100) embedded at the center and edges of the contact surface, as well as at the height center of the soil sample. Download: PPT PowerPoint slide PNG larger image TIFF original image Table 3. Silt physical index. https://doi.org/10.1371/journal.pone.0297824.t003 Download: PPT PowerPoint slide PNG larger image TIFF original image Table 4. Particle size distribution of silt. https://doi.org/10.1371/journal.pone.0297824.t004 3.3 Test method After sample preparation, the entire assembly, along with the sample, was rapidly frozen at -20°C for 24 hours in a low-temperature chamber. Subsequently, it was subjected to constant temperature and pressure treatment in a low-temperature test chamber according to the target conditions (temperature of -1°C, normal pressure of 150 kPa). When conditions stabilized, the horizontal shear stress was applied in a graded manner. Creep loading conditions are shown in Table 5. The entire shear process took place within the low-temperature test chamber, with shear displacement data automatically collected using an electronic dial gauge connected to a computer. The creep data obtained during the hierarchical loading process couldn’t directly describe the shear creep behavior of the frozen silt-concrete interface under individual shear stress levels. Therefore, the "Chen’s deformation superposition method" was employed to transform the creep test data from the entire loading process into creep data corresponding to different individual shear stress levels [36]. Download: PPT PowerPoint slide PNG larger image TIFF original image Table 5. Creep test conditions. https://doi.org/10.1371/journal.pone.0297824.t005 In this creep test, the stability criteria are as follows: during the creep decay phase, the strain rate is less than or equal to 0.001 mm/h; in the steady-state creep phase, the shear creep strain rate reaches a stable value; and during the accelerated creep phase, specimen failure occurs. 3.1 Testing apparatus Creep Shear Instrument. The aforementioned improved Nishihara model was validated using experimental results conducted on the frozen silt-concrete interface. The testing apparatus employed was a large-scale stress-controlled shear device developed in-house at the Geotechnical Laboratory of Lanzhou Jiaotong University (as depicted in Fig 4), which mainly consists of three parts: the main frame, the shear box and the stress loading system. The dimensions of both the upper and lower shear boxes were 200×200×100 mm3 (length × width × height). The normal pressure was applied by a hydraulic jack through a reaction beam, while horizontal shear stress was incrementally applied using 25 kg bags of iron sand through pulleys. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 4. Large-scale stress-controlled shearing apparatus. https://doi.org/10.1371/journal.pone.0297824.g004 Refrigeration and temperature control systems. XT5701LTB-450 high and low temperature cold bath system was selected to control the temperature of the cryogenic chamber, which can realize the temperature control accuracy of ± 0.1°C. Pt-100 Platinum resistance sensor was used for temperature measurement. 3.2 Specimen production The concrete portion of the shear specimen was pre-fabricated with vertical grooves on its surface to create different roughness levels. The mix proportions of concrete are detailed in Table 2. In the experiments, four different concrete specimens with distinct surface roughness were designed, and their detailed geometric dimensions are illustrated in Fig 5. Surface roughness for these four concrete structures was measured using the sand-pouring method, resulting in values of 0 mm, 0.538 mm, 0.775 mm, and 1.225 mm, respectively. The principle behind this method involves using the volume replacement technique with standard sand to measure the volume of depressions within the concrete surface area. This measurement provides the average depth of these depressions, which serves as a quantification of the concrete surface roughness, as shown in Eq (15): (15) Where R is the concrete surface roughness; Vs is the volume of standard sand in the groove; A is the surface area of the contact surface. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 5. Surface roughness design of concrete test block (unit: mm). (a) R = 0 mm; (b) R = 0.538 mm; (c) R = 0.775 mm; (d) R = 1.225 mm. https://doi.org/10.1371/journal.pone.0297824.g005 Download: PPT PowerPoint slide PNG larger image TIFF original image Table 2. Mix proportion of concrete (unit: kg /m3). https://doi.org/10.1371/journal.pone.0297824.t002 The test soil is Lanzhou silt, and the physical properties of the soil samples are shown in Table 3. The particle size distribution of silt is shown in Table 4, and the moisture content of the soil samples is 22%. Inside the sample box with the precast concrete structure at the bottom, the silt was layered and compacted to form an integrated silt-concrete structure, with temperature sensors (Pt100) embedded at the center and edges of the contact surface, as well as at the height center of the soil sample. Download: PPT PowerPoint slide PNG larger image TIFF original image Table 3. Silt physical index. https://doi.org/10.1371/journal.pone.0297824.t003 Download: PPT PowerPoint slide PNG larger image TIFF original image Table 4. Particle size distribution of silt. https://doi.org/10.1371/journal.pone.0297824.t004 3.3 Test method After sample preparation, the entire assembly, along with the sample, was rapidly frozen at -20°C for 24 hours in a low-temperature chamber. Subsequently, it was subjected to constant temperature and pressure treatment in a low-temperature test chamber according to the target conditions (temperature of -1°C, normal pressure of 150 kPa). When conditions stabilized, the horizontal shear stress was applied in a graded manner. Creep loading conditions are shown in Table 5. The entire shear process took place within the low-temperature test chamber, with shear displacement data automatically collected using an electronic dial gauge connected to a computer. The creep data obtained during the hierarchical loading process couldn’t directly describe the shear creep behavior of the frozen silt-concrete interface under individual shear stress levels. Therefore, the "Chen’s deformation superposition method" was employed to transform the creep test data from the entire loading process into creep data corresponding to different individual shear stress levels [36]. Download: PPT PowerPoint slide PNG larger image TIFF original image Table 5. Creep test conditions. https://doi.org/10.1371/journal.pone.0297824.t005 In this creep test, the stability criteria are as follows: during the creep decay phase, the strain rate is less than or equal to 0.001 mm/h; in the steady-state creep phase, the shear creep strain rate reaches a stable value; and during the accelerated creep phase, specimen failure occurs. 4 Result and discussion 4.1 Long-term strength Based on the observed patterns in the experimental data, the strain value corresponding to the maximum damage rate during the accelerated phase is defined as the strain at specimen failure. The time corresponding to the maximum damage rate is determined by Eq (16), and this time can be used to calculate the strain at failure: (16) then (17) Where n, m are the damage factor parameters, and tf is the damage time of the specimen. According to the above discriminating criteria, the curve of the long-term intensity is determined by fitting the first-order decay exponential function as shown in Eq (18): (18) Where τf is the shear stress causing damage at time tf; τs is the ultimate long-term strength;4, tf →∞, τf = τs; B and C are ultimate long-term strength parameters. Assuming that the shear stress is less than the shear stress associated with the accelerated creep phase, steady-state creep occurs. Based on the resulting failure strain values, long-term strength curves under different conditions were plotted, as depicted in Fig 6, with relevant parameters detailed in Table 6. The ultimate long-term strength τs for specimens with roughness values of 0 mm, 0.538 mm, 0.775 mm, and 1.225 mm were found to be 32.84 kPa, 33.45 kPa, 33.59 kPa, and 34.57 kPa, respectively. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 6. Long-term strength curve of frozen soil-concrete interface under different conditions. https://doi.org/10.1371/journal.pone.0297824.g006 Download: PPT PowerPoint slide PNG larger image TIFF original image Table 6. Long-term strength curve parameters under different conditions. https://doi.org/10.1371/journal.pone.0297824.t006 4.2 Model verification The degree of nonlinear creep of the contact surface is related to the creep time, and the viscous coefficient in the creep process shows a hardening increase law with time, and the increase of viscous coefficient will inhibit the growth of shear rate. The Kelvin body shear creep equation of Eq (6) can be used to describe the decaying creep stage at lower shear stresses. The introduction of the damage variable D can be used to describe the developmental (accelerated) creep stage. Therefore, the improved model can more accurately simulate the creep curves of the frozen silt -concrete surface under different shear stress levels. Fig 7 presents a comparison between the shear creep test results of the frozen silt-concrete interface and the calculated values obtained using the improved Nishihara model. In this study, different roughness was formed by setting vertical grooves. The roughness of the experimental setup has a certain gap with the actual engineering, and the test results will have a certain difference. The data in Figs 7–10 show that when the roughness R is 1.225 mm, the difference between the experimental data and the simulation results is larger compared to the other roughness, Therefore, the effect of roughness on the creep modeling of permafrost-concrete contact surfaces is yet to be thoroughly investigated. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 7. Comparison of experimental and theoretical values of the interface at R = 0 mm. https://doi.org/10.1371/journal.pone.0297824.g007 Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 8. Comparison of experimental and theoretical values of the interface at R = 0.538 mm. https://doi.org/10.1371/journal.pone.0297824.g008 Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 9. Comparison of experimental and theoretical values of the interface at R = 0.775mm. https://doi.org/10.1371/journal.pone.0297824.g009 Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 10. Comparison of experimental and theoretical values of the interface at R = 1.225mm. https://doi.org/10.1371/journal.pone.0297824.g010 Fig 11 illustrates the conformity of the calculated values from the improved model and the traditional Nishihara model with the experimental data. It is evident that the improved model provides a better description of the shear creep test results at the accelerated phase of the frozen silt-concrete interface. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 11. Comparison of effect before and after model improvement. https://doi.org/10.1371/journal.pone.0297824.g011 4.3 Parameters analysis The relevant parameters in the improved model can be obtained from the physical interpretation and mathematical relationships within the creep curves. These model-related parameters are presented in Table 7, determined through the application of the Levenberg-Marquardt nonlinear least-squares optimization algorithm for fitting. Download: PPT PowerPoint slide PNG larger image TIFF original image Table 7. Long-term strength curve parameters under different conditions. https://doi.org/10.1371/journal.pone.0297824.t007 The effects of roughness R and shear stress τ on the parameters of the improved Nishihara model are shown in Figs 12–15. It can be concluded that the shear elastic modulus G0 of the Hooke body, the shear elastic modulus G1of the Kelvin body, and the viscous hysteresis coefficient of the Bingham body in the improved model grow as the increase of the roughness. However, the reverse is true for the variation of in Kelvin bodies; With the increase of shear stress, G0 and G1 demonstrate a tendency to decline and tend toward a constant value; decreases and then increases, whereas increases and then decreases. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 12. Variation of G0 with shear stress change rule. https://doi.org/10.1371/journal.pone.0297824.g012 Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 13. Variation of G1 with shear stress change rule. https://doi.org/10.1371/journal.pone.0297824.g013 Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 14. Variation of with shear stress change rule. https://doi.org/10.1371/journal.pone.0297824.g014 Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 15. Variation of with shear stress change rule. https://doi.org/10.1371/journal.pone.0297824.g015 Pile foundations in permafrost regions are usually located deep in the foundation, and the contact surface between concrete and soil may contain different soil layers, and the physical and mechanical properties of different soils may also have an effect on the creep characteristics of the contact surface. The experimental apparatus and numerical simulations also differ in the setting of roughness from the actual engineering, which has an effect on the creep characteristics of the shear surface. These limitations were not considered in this study. The long-term stability of pile foundations is important for the safe service of bridges in permafrost regions. Foundations in frozen soil regions exhibit significant viscoelastic behavior. Currently, the primary means of studying this behavior involve field monitoring and model testing. By defining the contact relationship using a creep constitutive model for the frozen soil-concrete interface, it is possible to numerically simulate the rheological properties of foundations in frozen soil areas. This approach offers a solution to some of the limitations that other methods face in the temporal dimension. It is instructive for the study of long-term creep deformation of pile foundations in permafrost regions. 4.1 Long-term strength Based on the observed patterns in the experimental data, the strain value corresponding to the maximum damage rate during the accelerated phase is defined as the strain at specimen failure. The time corresponding to the maximum damage rate is determined by Eq (16), and this time can be used to calculate the strain at failure: (16) then (17) Where n, m are the damage factor parameters, and tf is the damage time of the specimen. According to the above discriminating criteria, the curve of the long-term intensity is determined by fitting the first-order decay exponential function as shown in Eq (18): (18) Where τf is the shear stress causing damage at time tf; τs is the ultimate long-term strength;4, tf →∞, τf = τs; B and C are ultimate long-term strength parameters. Assuming that the shear stress is less than the shear stress associated with the accelerated creep phase, steady-state creep occurs. Based on the resulting failure strain values, long-term strength curves under different conditions were plotted, as depicted in Fig 6, with relevant parameters detailed in Table 6. The ultimate long-term strength τs for specimens with roughness values of 0 mm, 0.538 mm, 0.775 mm, and 1.225 mm were found to be 32.84 kPa, 33.45 kPa, 33.59 kPa, and 34.57 kPa, respectively. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 6. Long-term strength curve of frozen soil-concrete interface under different conditions. https://doi.org/10.1371/journal.pone.0297824.g006 Download: PPT PowerPoint slide PNG larger image TIFF original image Table 6. Long-term strength curve parameters under different conditions. https://doi.org/10.1371/journal.pone.0297824.t006 4.2 Model verification The degree of nonlinear creep of the contact surface is related to the creep time, and the viscous coefficient in the creep process shows a hardening increase law with time, and the increase of viscous coefficient will inhibit the growth of shear rate. The Kelvin body shear creep equation of Eq (6) can be used to describe the decaying creep stage at lower shear stresses. The introduction of the damage variable D can be used to describe the developmental (accelerated) creep stage. Therefore, the improved model can more accurately simulate the creep curves of the frozen silt -concrete surface under different shear stress levels. Fig 7 presents a comparison between the shear creep test results of the frozen silt-concrete interface and the calculated values obtained using the improved Nishihara model. In this study, different roughness was formed by setting vertical grooves. The roughness of the experimental setup has a certain gap with the actual engineering, and the test results will have a certain difference. The data in Figs 7–10 show that when the roughness R is 1.225 mm, the difference between the experimental data and the simulation results is larger compared to the other roughness, Therefore, the effect of roughness on the creep modeling of permafrost-concrete contact surfaces is yet to be thoroughly investigated. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 7. Comparison of experimental and theoretical values of the interface at R = 0 mm. https://doi.org/10.1371/journal.pone.0297824.g007 Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 8. Comparison of experimental and theoretical values of the interface at R = 0.538 mm. https://doi.org/10.1371/journal.pone.0297824.g008 Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 9. Comparison of experimental and theoretical values of the interface at R = 0.775mm. https://doi.org/10.1371/journal.pone.0297824.g009 Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 10. Comparison of experimental and theoretical values of the interface at R = 1.225mm. https://doi.org/10.1371/journal.pone.0297824.g010 Fig 11 illustrates the conformity of the calculated values from the improved model and the traditional Nishihara model with the experimental data. It is evident that the improved model provides a better description of the shear creep test results at the accelerated phase of the frozen silt-concrete interface. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 11. Comparison of effect before and after model improvement. https://doi.org/10.1371/journal.pone.0297824.g011 4.3 Parameters analysis The relevant parameters in the improved model can be obtained from the physical interpretation and mathematical relationships within the creep curves. These model-related parameters are presented in Table 7, determined through the application of the Levenberg-Marquardt nonlinear least-squares optimization algorithm for fitting. Download: PPT PowerPoint slide PNG larger image TIFF original image Table 7. Long-term strength curve parameters under different conditions. https://doi.org/10.1371/journal.pone.0297824.t007 The effects of roughness R and shear stress τ on the parameters of the improved Nishihara model are shown in Figs 12–15. It can be concluded that the shear elastic modulus G0 of the Hooke body, the shear elastic modulus G1of the Kelvin body, and the viscous hysteresis coefficient of the Bingham body in the improved model grow as the increase of the roughness. However, the reverse is true for the variation of in Kelvin bodies; With the increase of shear stress, G0 and G1 demonstrate a tendency to decline and tend toward a constant value; decreases and then increases, whereas increases and then decreases. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 12. Variation of G0 with shear stress change rule. https://doi.org/10.1371/journal.pone.0297824.g012 Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 13. Variation of G1 with shear stress change rule. https://doi.org/10.1371/journal.pone.0297824.g013 Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 14. Variation of with shear stress change rule. https://doi.org/10.1371/journal.pone.0297824.g014 Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 15. Variation of with shear stress change rule. https://doi.org/10.1371/journal.pone.0297824.g015 Pile foundations in permafrost regions are usually located deep in the foundation, and the contact surface between concrete and soil may contain different soil layers, and the physical and mechanical properties of different soils may also have an effect on the creep characteristics of the contact surface. The experimental apparatus and numerical simulations also differ in the setting of roughness from the actual engineering, which has an effect on the creep characteristics of the shear surface. These limitations were not considered in this study. The long-term stability of pile foundations is important for the safe service of bridges in permafrost regions. Foundations in frozen soil regions exhibit significant viscoelastic behavior. Currently, the primary means of studying this behavior involve field monitoring and model testing. By defining the contact relationship using a creep constitutive model for the frozen soil-concrete interface, it is possible to numerically simulate the rheological properties of foundations in frozen soil areas. This approach offers a solution to some of the limitations that other methods face in the temporal dimension. It is instructive for the study of long-term creep deformation of pile foundations in permafrost regions. 5. Conclusions In this paper, the improved Nishihara model is validated using the results of creep tests on frozen chalk-concrete contact surfaces. The main research results are as follows: It can be obtained from the test data that, in the same time range, the specimen with larger contact surface roughness corresponds to a larger destructive shear stress and a larger long-term strength limit. When the roughness R is 0 mm~1.225 mm, the specimen corresponds to a long term strength of 32.84 kPa~34.57 kPa. At the same roughness, the creep deformation of the contact surface is more significant with the increase of the shear stress τ. The traditional Nishihara model has been improved by transforming its stationary components into non-stationary elements and introducing a damage factor. It can provide a relatively accurate description of the time-dependent creep deformation at the interface between frozen soil and concrete under different roughness conditions, especially during the acceleration phase of creep. The calculated values of the model agree well with the experimental results and have only minimal errors An analysis of the parameters of the improved model reveals the following trends: (i) As the shear stress continues to increase, both the Hooke body shear modulus (G0), and the Kelvin body shear modulus (G1), exhibit a first-order exponential decay. Parameter decreases and then increases, while parameter follows the opposite trend of increasing first and then decreasing. (ii) With an increase in the roughness of the contact interface, G0, G1, and gradually increase, while parameter progressively decreases. (iii) The damage factor (D) shows a monotonically increasing trend with time and eventually converges to the numerical value of 1.
TI - Creep constitutive modeling of the shear strength of the permafrost-concrete interface considering the stress level at -1°C
JF - PLoS ONE
DO - 10.1371/journal.pone.0297824
DA - 2024-04-30
UR - https://www.deepdyve.com/lp/public-library-of-science-plos-journal/creep-constitutive-modeling-of-the-shear-strength-of-the-permafrost-JPdZrUQWBr
SP - e0297824
VL - 19
IS - 4
DP - DeepDyve
ER -