An experimental investigation of the mechanical behavior and a hyperplastic constitutive model of frozen loess

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International Journal of Engineering Science 84 (2014) 29–53

Contents lists available at ScienceDirect

International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci

An experimental investigation of the mechanical behavior and a hyperplastic constitutive model of frozen loess Yuanming Lai a,⇑, Xiangtian Xu b, Wenbing Yu a, Jilin Qi a a State Key Laboratory of Frozen Soil Engineering, Cold and Arid Regions Environmental and Engineering Research Institute, Chinese Academy of Sciences, Lanzhou 730000, China b Institute of Transportation, Inner Mongolia University, Hohhot 010070, China

a r t i c l e

i n f o

Article history: Received 31 January 2014 Received in revised form 21 June 2014 Accepted 24 June 2014

Keywords: Frozen loess Hyperplasticity theory Stress path Thermodynamic function Constitutive model

a b s t r a c t For the engineering construction in cold regions, it is essential to understand the mechanical properties of frozen soil. In order to investigate the mechanical characteristics of frozen loess, three kinds of laboratory tests were carried out at 6 °C. These included constant-conﬁning pressure triaxial compression tests, isotropic compression tests, and constant-slope stress path tests. The test results show that the direction of the plastic strain increment is inﬂuenced by stress path. The inﬂuence of the mechanism of the conﬁning pressures on the strength and deformation of frozen loess is investigated in detail. An analytical method to determine pressure crushing and melting of the ice in frozen soil is proposed. In order to apply the hyperplasticity theory to modeling constitutive behavior of frozen soil, a new systematical approach to derive the yield criterion and ﬂow rule from dissipation function is proposed based on the properties of homogenous function. From the mechanical characteristics of frozen loess, Gibbs free energy function and dissipation function of frozen loess are established by applying the hyperplasticity theory. An elasto-plastic incremental constitutive model for frozen loess is derived from the two thermodynamic functions and a method of determining the corresponding parameters is also given. Simulated results show that the constitutive model, proposed in this paper, describes well the deformation behavior of frozen loess under different stress levels and stress paths. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Frozen soil, including permafrost and seasonally freezing soil, accounts for about 50% of the land area on the earth (Xu, Wang, & Zhang, 2010). Loess is widely distributed in the world, occupying 1/10 of the world’s land area, in both permafrost and seasonally frozen regions. For instance, in the permafrost region of Alaska loess is widely distributed in the piedmont zone. While in the seasonally frozen region of Lanzhou city, China, loess deposits with a thickness of up to 409.9 m, which is the largest loess thickness and its geological proﬁle is the most typical in the world. Also, there is a lot of loess or loess-like deposits in cold regions in Russia, Canada, America, China etc. With the increase in infrastructure construction in cold regions with loess, such as highways, railways, tunnels and irrigation work with frozen soil as their foundations, it is essential to study the mechanical behavior of frozen loess for engineering design, especially to establish a reasonable constitutive model for frozen loess. ⇑ Corresponding author. Tel.: +86 931 4967288; fax: +86 931 8279968. E-mail addresses: [email protected] (Y. Lai), [email protected] (X. Xu), [email protected] (W. Yu), [email protected] (J. Qi). http://dx.doi.org/10.1016/j.ijengsci.2014.06.011 0020-7225/Ó 2014 Elsevier Ltd. All rights reserved.

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Frozen soil consists of four phases, i.e., soil skeleton, ice, unfrozen water and air. Compared with unfrozen soil, the mechanical properties of frozen soil is much more complex. Besides the factors which are also important for unfrozen soils, such as conﬁning pressure, water content and loading rate, temperature plays a dominant role in mechanical behavior of frozen soils. In order to understand the mechanical properties of frozen soils, a large number of experiments have been carried out (Bragg & Andersland, 1981; Chamberlain, Groves, & Perham, 1972; Fish, 1991; Lai, Long, & Chang, 2009; Zhang, Lai, Sun, & Gao, 2007). The factors inﬂuencing the strength and deformation characteristics of frozen soil are investigated using both uniaxial compression and triaxial compression with constant conﬁning pressure conditions. Some empirical formulas are proposed. Furthermore, Azmatch, Sego, Arenson, and Biggar (2011) investigated tensile strength and stress–strain behavior of Devon silt under frozen fringe conditions. For more information on experimental studies, the readers can refer to the most up to date summarization on the state-of-the-art of frozen soils by Lai, Xu, Dong, and Li (2013). However, few previous studies were found which made an extensive investigation on mechanical properties of frozen loess. The constitutive model is a mathematical tool for describing the basic mechanical properties of materials. In the past decades, many constitutive models for unfrozen soils and other solid materials have been successfully developed by introducing plasticity theory developed for metal materials. When investigating the shear deformation of soils, Drucker, Gibson, and Henkel (1957) proposed a cap yield surface and a hardening law for the ﬁrst time. According to the concept of yield locus for clay proposed by Calladine (1963) and Roscoe and Poorooshasb (1963) established an elasto-plastic constitutive model for normally consolidated soils considering critical state of soil, and later on Roscoe and Burland (1968) modiﬁed the CamClay Model. Lade and Duncan (1975, 1977) established the so-called Lade–Duncan Model, in which non-associated ﬂow rule is adopted and plastic work is taken as the hardening parameter to describe the failure and dilatancy effects of sandy soil. Huang, Pu, and Chen (1981) established the Tsinghua elasto-plastic model for soils based on an associated ﬂow rule. In this model, the directions of plastic strain increments under various stress states were ﬁrst determined from test results, and then the yield criterion was determined according to the associated ﬂow rule (Li, 2004). Considering the effects of tangential stress/strain rate on soils and granular materials, a class of sub loading surface models for these materials was proposed and developed (Hashiguchi, 1980; Hashiguchi, Ozaki, & Okayasu, 2005; Hashiguchi & Protasov, 2004; Hashiguchi & Tsutsumi, 2001, 2007; Tsutsumi & Hashiguchi, 2005; Tsutsumi, Hashiguchi, Okayasu, Saitoh, & Sugimoto, 2001). Cleja-Tigoiu (2000) established a constitutive equation for anisotropic ﬁnite elastoplasticity in the framework of materials with relaxed conﬁgurations and internal state variables (Cleja-Tigoiu, 1990a, 1990b). By idealizing the materials as a random array of identical, elastic, frictional spheres and assuming that the particles move with an average strain, Ragione, Prantil, and Sharma (2008) proposed a model for inelastic response of an idealized granular material. Based on the deﬁnition of critical state, classical elastoplasticity and bounding surface, Anandarajah (2008) proposed a multi-mechanism anisotropic model for granular materials. A number of elasto-plastic constitutive models were established based on the friction and compression characteristics of geotechnical materials or compressible porous materials (Cleja-Tigoiu, Cazacu, & Tigoiu, 2008; Hashiguchi, 2005; Hashiguchi, Saitoh, Okayasu, & Tsutsumi, 2002; Shen, Shao, Kondo, & Gatmiri, 2012). When the plastic strain is calculated using a single yield surface theory, it cannot explain the phenomenon that the direction of plastic strain increment varies with the stress path for some geotechnical materials. On the other hand, applying an associated ﬂow rule in single yield surface model usually results in excessive plastic dilatancy (Chen & Saleeb, 2001). To overcome these defects, a series of multiple yield surfaces or multiple plastic potential models for geotechnical materials were proposed (Calladine, 1971; Desai, 1980; Gens & Nova, 1993; Ottosen & Ristinmaa, 2013; Peric & Ayari, 2002; Seiki, 1979; Xie & Shao, 2012; Zhu, Shao, & Mainguy, 2010). These constitutive models based on the conventional elasto-plastic theory can describe the mechanical behaviors of geotechnical materials to some extent, and some are applied in engineering practice (Chen & Saleeb, 2001). The laws of thermodynamics are universally applicable. In the study of constitutive theory for continuum or deformable solid, the early researchers realized that the thermodynamic laws should also be introduced to describe the mechanical process of continuum. Based on the irreversible thermodynamic theory, Ziegler (1977) proposed a frame for establishing constitutive models and its foundation—Ziegler’s orthogonality principle (Houlsby & Puzrin, 2000). This principle has ever been criticized on a number of grounds (Collins & Houlsby, 1997). Srinivasa (2010) also pointed out Ziegler’s orthogonality principle was based on an incorrect assumption. However, Collins and Houlsby (1997) proposed that the principle is a weak assumption and can be used to deﬁne a very wide class of constitutive equations. Furthermore, by applying convex analysis, Srinivasa (2010) showed that a same orthogonality principle can be derived from maximum rate of dissipation criterion both in rate-dependent case and rate-independent case (in this case, the ﬁrst-order homogeneity of dissipation function was required). Therefore, in this study, we adopt the orthogonality principle proposed by Srinicasa to modeling the constitutive behavior of frozen loess. The approach to constitutive modeling based on thermodynamics was termed as ‘‘hyperplasticity’’ (Puzrin & Houlsby, 2001a, 2001b). In hyperplasticity, the constitutive model of a material is directly derived from two thermodynamic functions, i.e., energy potential function and dissipation function. At the same time, the elastic law, yield criterion, ﬂow rule and hardening law corresponding to the conventional elasto-plastic theory were formulated (Collins & Houlsby, 1997; Houlsby & Puzrin, 2000) Later on, hyperplasticity was further developed and widely used in constitutive modeling of soils (Chiarelli, Shao, & Hoteit, 2003; Cleja-Tigoiu, 2003; Maugin, 1992, 1999; Rosakis, Rosakis, Ravichandran, & Hodowany, 2000; Rousselier, 2001; Vorobiev, 2008; Zhu et al., 2010). Houlsby (1981, 1982) introduced hyperplasticity to geotechnical materials and showed that the constitutive model of some geotechnical materials could be derived within the frame of irreversible thermodynamics. Collins and Houlsby (1997) further developed and improved the constitutive theory for geotechnical materials

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within the same frame. They presented a new method for deriving constitutive models from the Gibbs free energy function and dissipation function and theoretically showed the non-associated behavior of the single yield surface model for frictional materials. Houlsby and Puzrin (2000) suggested that the constitutive model for dissipative materials can be established from the four energy functions according to Legendre transformation. Using the multiple yield surface models in conventional plastic theory, Puzrin and Houlsby (2001a, 2001b) extended the single internal variable to multiple internal variables and internal functions. Collins (2005) put forward the concept of frozen elastic energy in soils. Based on this, the thermodynamic compatibility of constitutive models for some soils was investigated (Collins, 2003; Collins & Hilder, 2002; Collins & Kelly, 2002b). There are many advantages to establish constitutive models based on irreversible thermodynamic theory. The ﬁrst is that the laws of thermodynamics are automatically satisﬁed. Secondly, the formulation is compact. It is necessary to establish some competitive constitutive models in a uniform theory frame, which makes comparison of different models more convenient so as to select the most promising one (Houlsby & Puzrin, 2000). The complexities mentioned above means that the constitutive models for unfrozen soils are not suitable for frozen soils. Thus a constitutive model which describes the main mechanical properties of frozen soils should be established (Lai, Yang, Chang, & Li, 2010; Lai, Zhang, & Li, 2009). In frozen soil engineering, frozen soil is affected by both mechanical and thermal loads. The deformation characteristics of frozen soils are inﬂuenced by the soil temperature change, ice–water phase change and moisture migration. It is therefore a typical hydro-thermo-mechanic coupling problem (Lai et al., 2009). This feature of frozen soil shows that the deformation process under a surcharge load is actually a complex thermodynamic process. It is necessary to establish a constitutive model within the framework of irreversible thermodynamic theory which can automatically couple the constitutive model and other differential equations, making the hydro-thermo-mechanic coupling model more consistent and systematical. In this study, the frozen Lanzhou loess is taken as study object. A series of triaxial compression tests were carried out under constant conﬁning pressure, to study the inﬂuence of conﬁning pressure on the strength and deformation characteristics from a new viewpoint. Isotropic compression tests and constant-slope tests were used to investigate the inﬂuence of the stress path on the deformation characteristics of frozen soil. In this paper, based on the test results and the hyperplasticity theory, speciﬁc expressions of Gibbs free energy function and dissipation function for frozen Lanzhou loess are proposed. An incremental elasto-plastic constitutive model for frozen Lanzhou loess, within the frame of irreversible thermodynamics, is established and the method to determine the model parameters is presented. In addition, to make the hyperplasticity more applicable and comprehensible, a method of deriving yield criterion and ﬂow rules from dissipation function without Legendre transformation and convex analysis is proposed. 2. Mechanical test on frozen loess All the mechanical tests mentioned in this study were conducted under axisymmetric conditions. First of all, the mechanical symbols used in this study are deﬁned and their transformational relations are given. They are as follows:

r1 – maximum axial stress (maximum principle stress), r3 – conﬁning pressure, p – mean stress, q – deviator stress or equivalent stress, pa – atmospheric pressure, pa = 0.101325 MPa, ea – axial strain, e3 – lateral strain, ev – volumetric strain, ec – equivalent strain or shear strain, Under axisymmetric condition,

p¼

r1 þ 2r3 3

;

q ¼ r1 r3 ;

2 3

ev ¼ ea þ 2e3 ; ec ¼ ðea e3 Þ:

2.1. Test material and sample preparation The soil tested in this study was loess collected from Donggang town, Lanzhou City, a seasonally frozen soil region. The grain size distribution of the tested loess is shown in Table 1. To ensure the uniformity of the samples, the loess was mixed

Table 1 Grain size distribution of the loess (%). <0.005 mm

0.005–0.05 mm

0.05–0.075 mm

0.075–0.10 mm

0.10–0.25 mm

>0.25 mm

11.21

68.46

14.03

4.54

1.76

0

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with water at a moisture content of about 17%, and then kept for 6 h without evaporation to allow the water to be uniformly distributed in the soil. After this, the mixed loess was ﬁlled in a cylindrical mold, to make cylindrical samples of 12.5 cm (height) 6.18 cm (diameter) with the desired dry densities. The basic physical parameters of the loess are listed in Table 2. 2.2. Sample freezing and test temperature To avoid large frost heaving during freezing, the samples were put in rigid molds and were quickly frozen at 30 °C. After 48 h of freezing, the samples were taken from the molds and mounted with epoxy resin platens on both ends and covered with rubber sleeve. Finally, the samples were kept in an incubator for over 12 h until the samples reached the target testing temperature of 6.0 °C. 2.3. Test equipment The test equipment used in this study is a cryogenic triaxial apparatus modiﬁed from the MTS-810 material test machine, which can measure volumetric strain of sample in triaxial test under constant conﬁning pressure. Fig. 1 shows a photo and schematics of the equipment. The cryogenic triaxial apparatus can be controlled by load, displacement and time, respectively. The test is automatically controlled by a computer program, and the data recorded automatically. The maximum axial load is 100 KN. The axial displacement variation range is 85 cm 85 cm. The available conﬁning pressure range is 0 20 MPa. The liquid used in conﬁning pressure loading is aeronautic hydraulic oil. The medium in the cooling system is alcohol. The test temperature range is 30 °C 25 °C. 2.4. Three types of mechanical tests for frozen loess In this study, three types of mechanical tests on frozen Lanzhou loess were carried out under 6.0 °C, (a) triaxial compression test under constant conﬁning pressure (including the case of loading–unloading–reloading test), (b) isotropic compression test (including the case of loading- unloading–reloading test) and (c) constant-slope stress path test. These tests can not only be used to study the mechanical behavior of frozen soil under different loading modes (or different stress paths), but also to obtain the parameters for constitutive model and to verify the constitutive models. 2.4.1. Constant conﬁning pressure triaxial compression–CCPTC test In these tests the conﬁning pressure is kept constant during the test. In this study, a series of CCPTC tests with conﬁning pressures ranging from 0.5 MPa to 17.0 MPa were carried out. Before testing, the temperature in the pressure chamber was set at 6.0 °C, and the samples were mounted into the apparatus. After the temperature in the pressure chamber was stable, the conﬁning pressure was applied and kept constant for 5 min before the axial load was applied. The axial strain rate was 1.67 104/s. The tests were stopped when the axial train reached 20%. To obtain the elastic shear module of the samples, a series of loading–unloading–reloading tests (under CCPTC condition) were carried out with conﬁning pressures ranging from 0.5 MPa to 9.0 MPa. The ﬁrst unloading began when the axial strain reached to ea1. After unloading the deviator stress equaled to zero (r1 r3 = 0). Then another loading was applied until the axial strain reached to ea2, followed by an unloading. In this study, 7 loading–unloading cycles were carried out with a strain rate of 1.67 104/s. 2.4.2. Isotropic compression–IC test To investigate mechanical behavior of frozen soil samples under hydrostatic pressure and determine the elastic volumetric module of the samples, isotropic compression tests with loading–unloading–reloading cycles (under IC condition) on frozen Lanzhou loess were carried out. In these tests, the axial strain was controlled. The displacement of axial pressure piston was limited to zero, and the tests were controlled by conﬁning pressure loading system. The reaction force of the axial pressure piston generated by the conﬁning pressure was used to make the constant hydrostatic pressure in the pressure chamber. The curves of volumetric strain versus hydrostatic pressure were obtained. In the loading–unloading–reloading tests (isotropic compression), four different stress levels were used to investigate the inﬂuence of conﬁning pressures (or stress level) on elastic volumetric module of frozen soil. In addition, in the isotropic compression tests loading rates ranged from 0.1 MPa/min to 10 MPa/min. It was found that the loading rate had little inﬂuence on the isotropic compression curve when lower than 0.1 MPa/min. Therefore, in the isotropic compression tests a loading rate of 0.1 MPa/min was used.

Table 2 Basic physical parameters of the loess. Density

Water content (%)

Speciﬁc gravity

Liquid limit (%)

Plastic limit (%)

2.01 g/cm3

16.9

2.69

27

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Fig. 1. Cryogenic triaxial apparatus. (a) Photo, (b) Schematics.

2.4.3. Constant slope stress path–CSSP test To investigate the inﬂuence of stress path on the deformation of the frozen loess, in addition to the constant conﬁning pressure triaxial compression tests, four kinds of constant-slope stress path tests were carried out on the frozen loess. The constant-slope means that the slope of stress path d q/d p was kept constant during the whole test process. The constant conﬁning pressure compression test actually refers to the constant-stress path test of dq/dp = 3. The four stress paths SP1, SP2, SP3 and SP4, with corresponding slopes 98.24, 113.68, 99.74 and 77.90, respectively, as shown in Fig. 2 were carried out. In the four CSSP tests, the conﬁning pressures of 2 MPa, 3 MPa, 4 MPa and 5 MPa, respectively, were applied. Then the axial loading began with an axial loading rate of 0.006KN/s and the conﬁning pressure loading began with a loading rate of 0.001 MPa/s. 2.5. Test results and analyses 2.5.1. Effects of conﬁning pressure on the mechanical behaviors of frozen loess The CCPTC test results for frozen loess under different conﬁning pressures are shown in Figs. 3 and 4. Fig. 3 shows the inﬂuence of conﬁning pressure on the stress–strain curves. When conﬁning pressure is below 2 MPa, the stress–strain curves

Fig. 2. Four constant-slope stress paths in p–q plane.

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Fig. 3. Stress–strain curves of frozen loess from CCPTC tests.

Fig. 4. Volumetric strain curves of frozen loess from CCPTC tests.

show softening, which decreases with the increase of conﬁning pressure. When the conﬁning pressure exceeds 2 MPa, the stress–strain curves show hardening. Fig. 4 shows how the conﬁning pressure inﬂuences the volumetric strain of frozen loess during shear process. Under all conﬁning pressures there is ﬁrst of all contraction followed by dilatancy. When the conﬁning pressure is lower than 2 MPa, the contraction stage of frozen loess is relatively small. Dilatancy of frozen loess occurs at axial strains less than 10%, and the ﬁnal dilatancy is relatively large. The reason for this is that the radial conﬁnement is low when the conﬁning pressure is low, and the relative movement between solid particles in the sample after shear failure results in volumetric expansion. When the conﬁning pressure is higher than 2 MPa, only small volumetric expansion occurs during the shear process of the sample. In particular when the conﬁning pressure is more than 7 MPa, the volumetric strain is small, between 0.037% to 0.642%. In order to describe more concretely the inﬂuence of conﬁning pressure on the mechanical behavior of frozen loess more accurately, the strength qf and the initial slope of stress–strain curve, Ei, are introduced. The strength is deﬁned according to the standard for test methods of soils GB/T 50123–1999 (People’s Republic of China National Standard, 1999). That is to say, for the softening curves shown in Fig. 3, the peak deviation stress is taken as strength, while for hardening curves, the strength is equal to the deviation stress when axial strain is 15%. qf and Ei are the two main variables determining the shape of stress–strain curves. qf and Ei can therefore be used to determine the inﬂuence of conﬁning pressure on the mechanical behavior of frozen loess. The strength-conﬁning pressure relationship and initial slope-conﬁning pressure relationship are shown in Figs. 5 and 6, respectively.

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Fig. 5. Relationship between strength and conﬁning pressure.

Fig. 6. Relationship between Initial slope and conﬁning pressure.

It can be seen in Fig. 5 that the variation of the strength of frozen loess with conﬁning pressures can be divided into three phases. Under relatively low conﬁning pressure, i.e., from 0.5 MPa to 4 MPa, pressure melting does not occur, so the increase of conﬁning pressure only makes the solid particles (soil particles and ice particles) more compact. This increases the sliding and interlocking frictional forces, which enhance the strength of frozen loess. In conﬁning pressure ranges from 4 MPa to 13 MPa, the ice in the sample begins to be crushed and pressure melting occurs. The ice crushing reduces the ice bonding force between the particles, while the pressure melting of ice increases the unfrozen water ﬁlm on the granular surfaces, both of which decrease the angle of internal friction of the frozen loess. At the same time, the extent of crushing and pressure melting of the ice increase with the increase in conﬁning pressure. The strength of frozen loess decreases with the increase of conﬁning pressure. When the conﬁning pressure exceeds 13 MPa (a critical value in this study), the pressure melting produces enough porous water to saturate the sample. The conﬁning pressure increment is almost totally carried by the porous water. The effective conﬁning pressure affecting on the solid skeleton no longer increases. The studies on other types of frozen soil show similar tendencies (Chamberlain et al., 1972; Parameswaran & Jones, 1981; Xu, Lai, Dong, & Qi, 2011). From Fig. 6, it can be seen that the initial slope of stress–strain curve with varying conﬁning pressure can also be divided into three phases. In the ﬁrst phase, the increase of conﬁning pressure makes the solid granular more compact, which results

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in the initial slope increasing with the increase of conﬁning pressure. It should be noted that the conﬁning pressure range for the initial slope (from 0.5 MPa to 13 MPa) is larger than that for strength (from 0.5 MPa to 4 MPa). This indicates that in the range from 4 MPa to 13 MPa the extent of ice crushing is greater than that of pressure melting. The reason for this is that the ice crushing decreases the strength of frozen loess other than the initial slope. The variation difference between initial slope and strength with conﬁning pressure also exists in frozen sandy soils (Xu et al., 2011). In the second phase, the pressure melting of ice increases with the increase in conﬁning pressure, so the initial slope decreases with the increase of conﬁning pressure in this phase. In the third phase, it is similar to the variation of strength with conﬁning pressure. The initial slope is not inﬂuenced obviously by conﬁning pressure, but remains constant. It further shows that when conﬁning pressure reaches a critical value, the pore water generated by the pressure melting of ice carries almost all the conﬁning pressure increment, and the mechanical behavior of frozen loess is not inﬂuenced. 2.5.2. The inﬂuence of stress path on the direction of plastic strain increment of frozen loess Numerous studies show that the direction of plastic strain increment of unfrozen soil is related to not only stress state, but also stress increment direction (Hambly, 1972; Lade & Duncan, 1976; Lewin & Burland, 1970; Tatsuoka & Ishihara, 1974). This implies that the deformation behavior of unfrozen soil is inﬂuenced by the stress path. To describe this characteristic of unfrozen soils, some constitutive models for unfrozen soils, considering the inﬂuence of stress path, have been established (Seiki, 1979; Shen, 1980; Yin & Duncan, 1984). However, for frozen soils, almost no reports on the inﬂuence of stress path on the direction of plastic strain increment have been published. In this study, two directions of plastic strain increment under stress paths with slopes of dq/dp = 3.00 and dq/dp = -113.68 were investigated, as shown in Fig. 7. In order to identify how the direction of plastic strain increments vary with the change of stress path from geometry, the stress space (p, q) in Fig. 7 is chosen together with the plastic strain increment space depv ; depc , namely, the two sets of coordinates coincide. It can be seen from Fig. 7 that, after stress states A and B reach point C along stress paths AC and BC, respectively, the plastic strain increments generated by the stress increments along these two stress paths are not of the same direction. This indicates that for frozen loess, the direction of plastic strain increment is not alone determined by the stress state, and the inﬂuence of stress path should also be taken into account. 2.5.3. Variation of elastic parameters with conﬁning pressures When calculating the elastic strain for elasto-plastic materials, the elastic parameters should be determined. In this study, the elastic volumetric module is determined using isotropic compression loading–unloading–reloading tests. The elastic shear module is determined by the loading–unloading–reloading tests under constant conﬁning pressure compression. The ev ln (1 + p/pa) curve, obtained from isotropic compression loading–unloading–reloading tests, is shown in Fig. 8. It can be seen in Fig. 8 that an elastic after-effect exists in the unloading phase in isotropic compression loading–unloading–reloading tests on frozen loess, and that hysteresis loop of energy dissipation is formed during unloading–reloading process. The elastic strain of frozen soil accounts for small amount of the total strain (Lai et al., 2010), so for calculation convenience, the hysteresis loop of ev ln (1 + p/pa) in unloading–reloading phase is approximated to a straight line. The slope of each straight line can be calculated by the following expression:

Fig. 7. Directions of plastic strain increment under different stress paths.

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Fig. 8. Isotropic compression loading–unloading–reloading curve of frozen loess.

k¼

deev d lnð1 þ p=pa Þ

ð1Þ dee

According to Eq. (1), one has d lnð1 þ p=pa Þ ¼ kv . For convenience, a dimensionless parameter K = 1/k is introduced which has the following relationship with the conventional elastic volumetric module K:

K¼

dp ¼ Kðp þ pa Þ deev

It can be seen from Fig. 8 that k varies with the conﬁning pressure at the unloading points, as does K. From the test results, the relationship between K and conﬁning pressure r3, as shown in Fig. 9, can be ﬁtted by:

K ¼ Ak ðr3 =pa Þ þ Bk

ð2Þ

where Ak = 10.244, Bk = 38.25. Fig. 10 shows the q ec curve obtained from loading–unloading–reloading tests under CCPTC conditions. Similar to the test curve of isotropic compression loading–unloading–reloading test, elastic after-effect also exists in unloading phase for

Fig. 9. Relationship between elastic volumetric module and conﬁning pressure.

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Fig. 10. Loading–unloading–reloading curve of frozen loess under CCPTC conditions.

loading–unloading–reloading tests under CCPTC conditions, and closed hysteresis loops are also formed by unloading curve and reloading curve. It can be seen that the hysteresis loops in the q ec curve can be approximated as straight lines with slope of C. The relationship between deviator stress and elastic shear strain can be expressed by:

q ¼ Ceec

ð3Þ

In Eq. (3), C is deﬁned as elastic shear modulus. It should be noted that the slope of hysteresis loop, under a given conﬁning pressure, decreases with increase in strain. This is elasto-plastic coupling. The elastic shear modulus for each loading– unloading cycle is shown in Fig. 10. Since the elastic shear modulus decreases gradually with the increase in the strain, the elasto-plastic coupling of frozen loess is considered to be a kind of damage behavior. As the strain increases, the microdefects and micro-cracks in frozen loess are continually merged and expanded, resulting in the continuous deterioration of elastic properties of frozen loess. The stress concentrations at the original defects of the soil may induce the pressure melting of ice during loading process. This is the reason that the shear elastic modulus of frozen loess decreases with the increase of the strain. Since the effect of plastic strain in each cycle is far greater than that of elastoplastic coupling, the elastoplastic coupling phenomenon is ignored. The slope of the ﬁrst hysteresis loop is taken as the elastic shear modulus. That is to say, the plastic strain is the main part in the total strain for each cycle, and is much larger than the elastic strain. From Fig. 10, it can be seen that the elastic strain only accounts for 4.1% of the total strain in the seventh cycle. Therefore, the contribution of elastic strain to the total strain can be almost ignored, and the effect of elastoplastic coupling phenomenon on both plastic strain and elastic strain also be ignored. Similarly, when the elastic strain accounts for a large proportion in the total strain for an elastoplastic coupling materials, the stress–strain relationship can also be simpliﬁed without considering the elastoplastic coupling effect. Furthermore, it is shown by the test results that the elastic shear modulus is inﬂuenced by conﬁning pressure. The relationship between elastic shear modulus C and conﬁning pressure r3, as shown in Fig. 11, can be expressed by

C ¼ AC pa ðr3 =pa ÞBC

ð4Þ

where AC = 6001.18, BC = 0.1603. 3. Hyperplasticity theory In this study, a constitutive model for frozen loess was derived by applying the hyperplasticity theory which was developed by researchers (Collins and Houlsby, 1997; Houlsby and Puzrin, 2000; Puzrin and Houlsby, 2001). This theory includes two types: single internal variable hyperplasticity (Collins & Houlsby, 1997; Houlsby & Puzrin, 2000) and multiple internal variable hyperplasticity (Puzrin & Houlsby, 2001a, 2001b), corresponding to the single yield surface model and the multiple yield surface model in classical plasticity, respectively. In the hyperplasticity, the orthogonality principle and ﬁrst-order homogeneity are two important conditions that the dissipation function needs to meet besides the second law of thermodynamics. These two conditions are very important in constructing the constitutive model for rate-independent materials, since they restrict the selective range of the dissipation function, which also in turn, provides the theoretical basis for its speciﬁc form. The basis was extensively discussed by many researchers (Collins, 2005; Collins & Houlsby, 1997; Houlsby

Y. Lai et al. / International Journal of Engineering Science 84 (2014) 29–53

39

Fig. 11. Relationship between elastic shear module and conﬁning pressure.

& Puzrin, 2000; Puglisi & Truskinovsky, 2005; Srinivasa, 2010; Ziegler, 1958, 1963). Collins and Houlsby (1997) applied the degenerate Legendre transformation of dissipation function to derive the yield criterion and ﬂow rule. Srinivasa (2010) provided the existence proof of yield function and ﬂow rule from positively homogenous dissipation function by using gauge function and convex analysis. Although these approaches established the relationship between dissipation function and yield criterion, it relied on the use of complicated mathematical operations which are unfamiliar to most geotechnical engineers. According to the properties of homogeneity function, a systematic method is proposed to derive the yield criterion and the ﬂow rule from dissipation function. This method is more accessible to geotechnical specialist and more convenient to apply. 3.1. Single internal variable hyperplasticity (corresponding to the single yield surface model) In irreversible processes, internal variables are introduced and used to describe internal structure, deformation histories and dissipation behavior of materials (Ziegler, 1977). They can be scalars, vectors and tensors. More details about internal variables were introduced by Maugin (1992). The hyperplasticity using only one internal variable in thermodynamic functions to describe dissipative characteristics is named as single internal variable hyperplasticity. According to the frame of single internal variable hyperplasticity developed by Collins and Houlsby (1997) and Houlsby and Puzrin (2000), if the plastic strain tensor epij is chosen as internal variable, the Gibbs free energy G and the dissipation function D have the following forms:

G rij ; epij ¼ G1 ðrij Þ þ rij epij þ G2 epij D rij ; epij ; e_ pij ¼ hs_ i

ð5Þ ð6Þ

where rij is stress tensor, h is temperature and s_ i is the rate of entropy production per unit volume. For various materials, it should be noted that the dissipation function in Eq. (6) may be either stress-dependent or stress-independent. If the dissipation function is stress-independent, then the normal ﬂow rule applies in the true stress space. Otherwise, the ﬂow rule is non-associated in true stress space. The connection between ﬁrst-order homogenous dissipation function, yield criterion and ﬂow rule was detailed by Rajagopal and Srinivasa (2005) and Srinivasa (2010). By virtue of Eqs. (5) and (6), the ﬁrst law of thermodynamics and the second law of thermodynamics can be written as follows:

_ rij ; ep Þ ¼ @G r_ ij þ @G e_ p ¼ r_ ij eij þ D Gð ij @ rij @ epij ij D rij ; epij ; e_ pij ¼ hs_ i P 0

ð7Þ ð8Þ

Substituting Eq. (5) into Eq. (7), gives

eij ¼

@G ; @ rij

D ¼ hs_ i ¼

eeij ¼ eij epij ¼ @G p e_ @ epij ij

@G1 @ rij

ð9Þ ð10Þ

40

Y. Lai et al. / International Journal of Engineering Science 84 (2014) 29–53

In the irreversible process, the rate of the entropy production s_ i was expressed as the sum of products of thermodynamic forces and thermodynamic ﬂows (Li, 1986). Then, in Eq. (10), the internal variable rate e_ pij is referred to as the thermodynamic is taken as a thermodynamic force. vij is deﬁned as a generalized stress in hyperplasticity theory (Collins & ﬂow and vij ¼ @@G ep ij

Houlsby, 1997; Houlsby & Puzrin, 2000). For rate-independent material, much researchers assumed that the dissipation function is a ﬁrst-order homogeneous function in rate of internal variable e_ pij , and gave various explanations (Collins & Houlsby, 1997; Houlsby & Puzrin, 2000; Collins, 2005; Puglisi & Truskinovsky, 2005; Puzrin and Houlsby 2001; Srinivasa, 2010). According to the ﬁrst-order homogeneity of dissipation function, owing to Euler’s theorem, one has

D¼

@D p e_ @ e_ pij ij

ð11Þ

Srinivasa (2010) proposed the orthogonality principle for rate-independent materials, based on the maximum rate of dissipation criterion and ﬁrst-order homogeneity of dissipation function, as following:

@D

vij ¼ _ p @ eij

ð12Þ

According to the ﬁrst-order homogeneity of D, following yield criterion and ﬂow rule in generalized stress space can be derived based on Srinivasa’s method (Srinivasa, 2010):

Y

rij ; vij ; epij ¼ 0

e_ pij ¼ k

ð13Þ

@Y @ vij

ð14Þ

where Y is the yield criterion in generalized stress space, and k is a multiplier. Furthermore, by differentiating Eq. (5) with respect to epij , one has

vij ¼ rij þ

@G2 @ epij

ð15Þ

2 qij ðepij Þ ¼ @G p was deﬁned as back stress (Collins & Houlsby, 1997). It is the central point of yield surface in true stress space @e ij

and is used to describe kinematic hardening in conventional plasticity, while isotropic hardening is deﬁned by a dependency of yield criterion (13) on internal variable epij . In order to derive the incremental constitutive model, the following consistency condition of yield criterion should be introduced:

_ rij ; v ; ep Þ ¼ @Y r_ ij þ @Y v_ ij þ @Y e_ p ¼ 0 Yð ij ij @ rij @ vij @ epij ij

ð16Þ

From Eq. (15), one has

v_ ij ¼ r_ ij þ

@ 2 G2 p e_ @ epij @ epkl kl

ð17Þ

The multiplier k can be determined by substituting Eqs. (14) and (17) into Eq. (16):

k ¼

@Y @ rij

þ @@Y v

ij

2 @Y @ G2 @ vkl @ ep @ ep kl ij

r_ ij

þ @@Yep

ij

ð18Þ @Y @ vij

By differentiating the second part of Eq. (9) with respect to time, the expression of the elastic strain rate is

e_ eij ¼

@ 2 G1 r_ kl @ rij @ rkl

ð19Þ

Then, from Eqs. (14), (18) and (19), the elasto-plastic incremental constitutive model can be given by:

0

_ pij

_ eij

e_ ij ¼ e þ e ¼ 2

@ G1 where C ep ijkl ¼ @ rij @ r kl

C ep ijkl

r_ kl

@ 2 G1 ¼@ @ rij @ rkl

@Y @Y @Y @Y @ vij @ rkl þ@ vij @ vkl 2G @ @Y @Y þ @Y @Y 2 @ vst @ ep @ ep @ vmn @ ep @ vmn mn st mn

@Y @Y @ vij @ rkl

1

@Y þ @@Y vij @ vkl

@ 2 G2 @Y @Y @ vst @ ep @ epmn @ vmn st

þ @@Y ep

mn

@Y

Ar_ kl

@ vmn

is the elasto-plastic compliance tensor.

ð20Þ

41

Y. Lai et al. / International Journal of Engineering Science 84 (2014) 29–53

3.2. Hyperplasticity of multiple internal variables (corresponding to the multiple yield surface model) For some geomaterials, single yield surface model can not reasonably simulate the dependency of plastic strain increment direction on stress path (Zheng, Shen, & Gong, 2002). In order to improve some of the defects in single yield surface model, a multiple yield surface model should be introduced to describe the effect of stress path on the mechanical characteristic of geomaterials. To build a multiple yield surface model based on hyperplasticity theory, a ﬁnite number of internal variables should be introduced to two thermodynamic functions and the number of internal variables is equal to the number of yield surfaces. For consistency, the hyperplasticity using the ﬁnite number of internal variables in thermodynamic functions is named as the hyperplasticity of multiple internal variables. This idea was ﬁrst developed by Puzrin and Houlsby (2001a, 2001b). According to their approach, the Gibbs free energy function and the dissipation function can be expressed as: n n X X ð1Þ ðnÞ G rij ; aij ; . . . ; aij ¼ G1 ðrij Þ þ rij aijðkÞ þ Gk2 aðkÞ ij k¼1

ð21Þ

k¼1

n X ð1Þ ðnÞ ð1Þ ðnÞ ð1Þ ðnÞ ðkÞ Dk rij ; aij ; . . . ; aij ; a_ ij D rij ; aij ; . . . ; aij ; a_ ij ; . . . ; a_ ij ¼

ð22Þ

k¼1

The dissipation function D in Eq. (22) is divided into the sum of n sub-dissipation functions, and it is still a ﬁrst-order homogeneous function in the rate of internal variables. In decoupled case, each sub-dissipation function Dk is a ﬁrst-order ðkÞ homogeneous function in the corresponding rate of internal variable a_ ij . As a single internal variable hyperplasticity, the constitutive model with n yield surfaces can be deduced from thermodynamic functions (21) and (22), which automatically obeys thermodynamic principles. In the irreversible process, we assume that total plastic strain of geomaterials is produced in n kinds of mechanisms. If the inter-coupling effect is neglected, the total plastic strain can be split up into additive sub-plastic strains each involving indiP pðkÞ pðkÞ vidual mechanism, i.e. epij ¼ nk¼1 eij , where epij is the total plastic strain, and eij is the sub-plastic strain corresponding to the kth mechanism. Therefore, for constitutive modeling of geomaterials in multiple internal variable hyperplasticity, all of the sub-plastic strains

epðkÞ ij ðk ¼ 1; . . . ; nÞ are chosen as internal variables. Then, Eqs. (21) and (22) can be rewritten as:

n n X X pð1Þ pðnÞ ¼ G1 ðrij Þ þ G rij ; eij ; . . . ; eij rij eijpðkÞ þ Gk2 eijpðkÞ k¼1

ð21aÞ

k¼1

n X pð1Þ pðnÞ pð1Þ pðnÞ pðkÞ pðkÞ D rij ; eij ; . . . ; eij ; e_ ij ; . . . ; e_ ij Dk rij ; eij ; e_ ij ¼

ð22aÞ

k¼1

From Eqs. (21a) and (22a), the multiple yield surface model for geomaterials can be derived by using the approach proposed by Puzrin and Houlsby (2001a). By differentiating Eq. (21a) with respect to rij, decomposition of strain tensor can be deduced:

eij ¼

n @G @G1 X ¼ eeij þ epij ¼ þ epðkÞ @ rij @ rij k¼1 ij

ð23Þ

So that the rate of elastic strain e_ eij can be obtained from Eq. (19). The rate of plastic strain is expressed by sum of n rates of sub-plastic strains corresponding to each mechanism:

e_ pij ¼

n X

e_ ijpðkÞ

ð24Þ

k¼1 pðkÞ Eq. (24) shows that to solve for e_ pij it is ﬁrst necessary to obtain e_ ij ðk ¼ 1; . . . ; nÞ. From orthogonality principle and ﬁrstðkÞ order homogeneity of dissipation function D, each thermodynamic force vij ðk ¼ 1; . . . ; nÞ corresponding to thermodynamic pðkÞ ﬂow e_ ij can be given by

pð1Þ

vijðkÞ ¼

pðnÞ

pð1Þ

pðnÞ

@Dðrij ; eij ; . . . ; eij ; e_ ij ; . . . ; e_ ij Þ pðkÞ

@ e_ ij

pðkÞ

¼

pðkÞ

@Dk ðrij ; eij ; e_ ij Þ pðkÞ

@ e_ ij

ð25Þ

Based on Srinivasa’s approach (Srinivasa, 2010) the kth yield criterion Y(k)(k = 1, . . . ,n) and ﬂow rule corresponding to Dk can be determined in the following form:

Y ðkÞ

pðkÞ rij ; vðkÞ ¼0 ij ; eij ðkÞ

e_ pðkÞ ¼ kðkÞ ij

ð26Þ

ðkÞ pðkÞ ij ; ij ; ij Þ ; ðkÞ @ ij

@Y ðr

v v

e

k ¼ 1; . . . ; n:

ð27Þ

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Y. Lai et al. / International Journal of Engineering Science 84 (2014) 29–53

Where k(k) is a multiplier corresponding to Dk. Similar to Eq. (15), the kth thermodynamic force

vijðkÞ ¼ rij þ

@Gk2

eijpðkÞ

;

pðkÞ

@ eij

vðkÞ ij is written as

k ¼ 1; . . . ; n

ð28Þ

Furthermore, the consistency condition of each yield criterion is

Y_ ðkÞ

¼ rij ; vijðkÞ ; epðkÞ ij

@Y ðkÞ @Y ðkÞ @Y ðkÞ r_ ij þ ðkÞ v_ ijðkÞ þ pðkÞ e_ ijpðkÞ ¼ 0; @ rij @v @e ij

k ¼ 1; . . . ; n

ð29Þ

ij

Combining Eqs. (27)–(29), we have

ðkÞ þ @Y ðkÞ r_ ij @ vij ; @ 2 Gk2 @Y ðkÞ @Y ðkÞ pðkÞ pðkÞ þ pðkÞ ðkÞ

@Y ðkÞ @ rij

kðkÞ ¼

@Y ðkÞ ðkÞ @ vst @ est @ eij

@ eij

k ¼ 1; . . . ; n

ð30Þ

@ vij

Substituting Eqs. (24), (27) and (30) into Eq. (23), the multiple yield surface model, based on irreversible thermodynamics, can be derived as follows

_e _p _e _ e_ ij ¼ C ep ijmn rmn ¼ eij þ eij ¼ eij þ

n X

e_ ijpðkÞ

k¼1

2

@ G1 where C ep ijmn ¼ @ rij @ rmn

Pn

k¼1

0

ðkÞ

ðkÞ

ðkÞ

@Y ðkÞ @Y ðkÞ þ@Y ðkÞ @Y ðkÞ ðkÞ @ rmn ðkÞ ðkÞ @v @v @v mn ij ij @ 2 Gk ðkÞ ðkÞ @Y @Y @Y ðkÞ @Y ðkÞ 2 þ ðkÞ pðkÞ pðkÞ ðkÞ pðkÞ ðkÞ @e @v @e @v @e @v pq pq pq pq st st

ðkÞ

@ vst

pðkÞ

pðkÞ

@ est @ epq

ðkÞ

@ vpq

1

ðkÞ

@Y @Y þ @Y ðkÞ @YðkÞ ðkÞ n 2 X @ vij @ rmn @ vij @ vmn B @ G1 ¼@ @ rij @ rmn k¼1 @Y ðkÞ @ 2 Gk2 @Y ðkÞ þ @Y ðkÞ pðkÞ

@ epq

@Y ðkÞ ðkÞ @ vpq

C_ Armn

ð31Þ

is the elasto-plastic compliance tensor in the multiple yield surface model.

3.3. A systematic approach for deriving the yield criterion and ﬂow rule from the dissipation function For the rate-independent materials, the existence proof for yield criterion and ﬂow rule and their connections with dissipation function were proposed by Srinivasa (2010) applying the gauge function and convex analysis based on the maximum rate of dissipation criterion. However, the mathematical notions and tools involving convex analysis are usually unfamiliar to most geotechnical engineers and is inconvenient to applications. In this study, a new systematic approach, based on the properties of homogeneous functions, is proposed for solving the yield criterion and ﬂow rule from the dissipation function. The new approach can be speciﬁed by three lemmas about homogeneous function as follows: Lemma 1. If f(x, y) is a homogeneous function of order n, then

@f @x

and

@f @y

is a homogeneous function of order n1.

@f @f Lemma 2. If f(x, y) is a ﬁrst-order homogeneous function, then partial derivatives @x and @y can be expressed as the functions which depend on a single variable yx or yx.

Lemma 3. If f(x, y) is a ﬁrst-order homogeneous function, then yx can be a function of two variables p and s by solving the equation ¼ ff10 ðy=xÞ y=x, here f1(y/x) = f(1, y/x), and f0 1(y/x) is a derivative of f1 with respect to independent variable yx. ðy=xÞ 1 Lemmas 1–3 can be extended to a homogeneous function with n independent variables. Then, in a six dimensional stress space (or generalized stress space), the yield criterion and ﬂow rule from dissipation function can be derived by applying the approach deﬁned by the Lemmas. First, the second-order symmetric tensors vij and a_ ij can be regarded as vectors with six dimensions. Thus, all of vectors ðv11 ; v12 ; v13 ; v22 ; v23 ; v33 Þ form a six dimensional vector space (generalized stress space). The dissipation function D is a ﬁrst-order homogeneous function in ða_ 11 ; a_ 12 ; a_ 13 ; a_ 22 ; a_ 23 ; a_ 33 Þ, i.e., D ¼ Dðrij ; aij ; a_ 11 ; a_ 12 ; a_ 13 ; a_ 22 ; a_ 23 ; a_ 33 Þ, here rij and aij are regarded as given parameters. From orthogonality principle and Lemma 2, the generalized stress vij can be expressed as p s

@D a_ a_ a_ a_ a_ vij ¼ _ ¼ vij rij ; aij ; _ 12 ; _ 13 ; _ 22 ; _ 23 ; _ 33 ; i; j ¼ 1; 2; 3 @ aij a11 a11 a11 a11 a11

ð32Þ

In fact, Eq. (32) contains six equations and it shows that each component of the generalized stress is a function of ﬁve a_ 12 a_ 13 a_ 22 a_ 23 a_ 33 a_ 11 ; a_ 11 ; a_ 11 ; a_ 11 ; a_ 11 . Eliminating these ﬁve variables

independent variables, in which the set of independent variables is

between the six equations gives the yield criterion in six-dimensional generalized stress space:

Yðvij ; rij ; aij Þ ¼ 0 Eq. (33) deﬁnes a hyper-surface in generalized stress space, i.e., yield surface in generalized stress space.

ð33Þ

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Y. Lai et al. / International Journal of Engineering Science 84 (2014) 29–53

Similarly, an extension of Lemma 3 for functions with six independent variables can deﬁne the ﬂow rule in six-dimensional stress space. By the ﬁrst-order homogeneity of D, one has

Dðrij ; aij ; a_ 11 ; a_ 12 ; a_ 13 ; a_ 22 ; a_ 23 ; a_ 33 Þ ¼ a_ 11 D1

a_ a_ a_ a_ a_ rij ; aij ; _ 12 ; _ 13 ; _ 22 ; _ 23 ; _ 33 a11 a11 a11 a11 a11

ð34Þ

a_ a_ a_ a_ a_ a_ a_ a_ a_ a_ rij ; aij ; _ 12 ; _ 13 ; _ 22 ; _ 23 ; _ 33 ¼ D rij ; aij ; 1; _ 12 ; _ 13 ; _ 22 ; _ 23 ; _ 33 . Thus, the orthogonality principle implies a11 a11 a11 a11 a11 a11 a11 a11 a11 a11 a_ 12 a_ 13 a_ 22 a_ 23 a_ 33 @D1 a_ 12 @D1 a_ 13 @D1 a_ 22 @D1 a_ 23 v11 ¼ D1 rij ; aij ; _ ; _ ; _ ; _ ; _ a11 a11 a11 a11 a11 @ða_ 12 =a_ 11 Þ a_ 11 @ða_ 13 =a_ 11 Þ a_ 11 @ða_ 22 =a_ 11 Þ a_ 11 @ða_ 23 =a_ 11 Þ a_ 11 @D1 a_ 33 ð35Þ @ða_ 33 =a_ 11 Þ a_ 11

where D1

vij ¼

@D1 ðwhere i and j cannotbeequalto1atthesametimeÞ @ða_ ij =a_ 11 Þ

ð36Þ

From the extension of Lemma 3 and Eqs. (35) and (36), ﬁve equations about the set of

a_ 12 a_ 13 a_ 22 a_ 23 a_ 33 a_ 11 ; a_ 11 ; a_ 11 ; a_ 11 ; a_ 11

are as follows

D1 ðrij ; aij ; a_ ij =a_ 11 Þ a_ 12 @D1 =@ða_ 13 =a_ 11 Þ a_ 13 @D1 =@ða_ 22 =a_ 11 Þ a_ 22 @D1 =@ða_ 23 =a_ 11 Þ a_ 23 @D1 =@ða_ 33 =a_ 11 Þ a_ 33 v11 ¼ @D1 =@ða_ 12 =a_ 11 Þ a_ 11 @D1 =@ða_ 12 =a_ 11 Þ a_ 11 @D1 =@ða_ 12 =a_ 11 Þ a_ 11 @D1 =@ða_ 12 =a_ 11 Þ a_ 11 @D1 =@ða_ 12 =a_ 11 Þ a_ 11 v12 ð37Þ D1 ðrij ; aij ; a_ ij =a_ 11 Þ @D1 =@ða_ 12 =a_ 11 Þ a_ 12 a_ 13 @D1 =@ða_ 22 =a_ 11 Þ a_ 22 @D1 =@ða_ 23 =a_ 11 Þ a_ 23 @D1 =@ða_ 33 =a_ 11 Þ a_ 33 v11 ¼ @D1 =@ða_ 13 =a_ 11 Þ @D1 =@ða_ 13 =a_ 11 Þ a_ 11 a_ 11 @D1 =@ða_ 13 =a_ 11 Þ a_ 11 @D1 =@ða_ 13 =a_ 11 Þ a_ 11 @D1 =@ða_ 13 =a_ 11 Þ a_ 11 v13 ð38Þ D1 ðrij ; aij ; a_ ij =a_ 11 Þ @D1 =@ða_ 12 =a_ 11 Þ a_ 12 @D1 =@ða_ 13 =a_ 11 Þ a_ 13 a_ 22 @D1 =@ða_ 23 =a_ 11 Þ a_ 23 @D1 =@ða_ 33 =a_ 11 Þ a_ 33 v11 ¼ @D1 =@ða_ 22 =a_ 11 Þ @D1 =@ða_ 22 =a_ 11 Þ a_ 11 @D1 =@ða_ 22 =a_ 11 Þ a_ 11 a_ 11 @D1 =@ða_ 22 =a_ 11 Þ a_ 11 @D1 =@ða_ 22 =a_ 11 Þ a_ 11 v22 ð39Þ D1 ðrij ; aij ; a_ ij =a_ 11 Þ @D1 =@ða_ 12 =a_ 11 Þ a_ 12 @D1 =@ða_ 13 =a_ 11 Þ a_ 13 @D1 =@ða_ 22 =a_ 11 Þ a_ 22 a_ 23 @D1 =@ða_ 33 =a_ 11 Þ a_ 33 v11 ¼ @D1 =@ða_ 23 =a_ 11 Þ @D1 =@ða_ 23 =a_ 11 Þ a_ 11 @D1 =@ða_ 23 =a_ 11 Þ a_ 11 @D1 =@ða_ 23 =a_ 11 Þ a_ 11 a_ 11 @D1 =@ða_ 23 =a_ 11 Þ a_ 11 v23 ð40Þ D1 ðrij ; aij ; a_ ij =a_ 11 Þ @D1 =@ða_ 12 =a_ 11 Þ a_ 12 @D1 =@ða_ 13 =a_ 11 Þ a_ 13 @D1 =@ða_ 22 =a_ 11 Þ a_ 22 @D1 =@ða_ 23 =a_ 11 Þ a_ 23 a_ 33 v11 ¼ @D1 =@ða_ 33 =a_ 11 Þ @D1 =@ða_ 33 =a_ 11 Þ a_ 11 @D1 =@ða_ 33 =a_ 11 Þ a_ 11 @D1 =@ða_ 33 =a_ 11 Þ a_ 11 @D1 =@ða_ 33 =a_ 11 Þ a_ 11 a_ 11 v33

a_ 12 a_ 13 a_ 22 a_ 23 a_ 33 a_ 11 ; a_ 11 ; a_ 11 ; a_ 11 ; a_ 11

ð41Þ

, while the right of Eqs. (37)–(41) are expresThe left sides of Eqs. (37)–(41) are expressions related to sions about components of generalized stress. Solving this set of equations gives,

a_ kl ¼ Lkl ðvij ; rij ; aij Þðwhere k and l cannot be equal to 1 at the same timeÞ a_ 11

ð42Þ

Eq. (42) deﬁnes the ratios between the components of internal variable rate, and naturally the ﬂow rule in the generalized stress space. Example. Consider the model proposed by Collins and Hilder (2002) as an example to further illustrate the application of this new approach. The speciﬁc form of dissipation function in the model is

D¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 ðAe_ pv Þ þ Be_ pc

ð43Þ

where

A ¼ a1 p þ a2 q þ a3 pc B ¼ b1 p þ b2 q þ b3 pc

ð44Þ

ai and bi(i = 1,2,3) in Eq. (44) are the parameters related to materials, pc is normal consolidation pressure related to plastic volumetric strain. From orthogonality principle, the generalized mean stress vp and generalized equivalent stress vq can be obtained as follows:

@D

@D

vp ¼ _ p ; vq ¼ _ p @ ev @ ec

ð45Þ

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Y. Lai et al. / International Journal of Engineering Science 84 (2014) 29–53

Substituting Eq. (43) into Eq. (45) gives

8 A2 e_ pv =e_ pc > ﬃ > v ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ > p > p p 2 < A2 þðBe_ v =e_ c Þ

ð46Þ

> B2 > ﬃ v ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ > > 2 : q ðAe_ p =e_ p Þ þB2 v

c

e_ p

e_ p

In Eq. (46), functions vp and vq depend on a single independent variable e_ vp . Eliminating e_ vp from these two functions gives c c the yield criterion in generalized stress space as

v2p A

2

þ

v2q B2

¼1

ð47Þ

In order to deduce the ﬂow rule, Eq. (43) is rewritten as

D ¼ e_ pc D1 ¼ e_ pc

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ p p 2 Ae_ v =e_ c þ B2 ;

D1 ¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ p p 2 Ae_ v =e_ c þ B2

ð48Þ

Thus, the ﬂow rule in generalized stress space can be obtained from Eq. (48) based on Lemma 3.

e_ pv vp B2 ¼ e_ pc vq A2

ð49Þ

Eqs. (47) and (49) are the yield criterion and ﬂow rule derived by applying the new approach proposed by this paper, respectively. The results are similar to these of previous method. 4. Constitutive model of frozen loess The hyperplasticity theory (in Section 3) indicates that constitutive model of material can be derived from the energy potential function and the dissipation function. In this section, the thermodynamic potential function for frozen loess based on experimental results is derived. In this study, the constitutive model of frozen loess is restricted to small deformation hypothesis and isothermal conditions. 4.1. Gibbs free energy function and dissipation function of frozen loess The experimental results show that the direction of plastic strain increment of frozen loess is inﬂuenced by the stress path. Thus, this should be characterized by multiple internal variable hyperplasticity. Both shear mechanism and compressive mechanism have effect on plastic deformation of soils (Xie & Shao, 2006). The total plastic strain epij can be considered as pð1Þ the sum of two sub-plastic strains. One is related to shear mechanism eij while the other is related to compressive mech pð2Þ pð1Þ pð2Þ , namely epij ¼ eij þ eij . As a geotechnical material, frozen loess is also governed by the two mechanisms. It is anism eij assumed that the volumetric expansion of frozen loess is produced by shear mechanism and the volumetric contraction produced by compressive mechanism. According to multiple internal variable hyperplasticity, the Gibbs free energy function of frozen loess can be expressed as: pð1Þ

pð2Þ

Gðrij ; eij ; eij Þ ¼ G1 ðrij Þ þ

2 X

2 X pðkÞ Gk2 eij

k¼1

k¼1

rij eijpðkÞ þ

ð50Þ

Under the axisymmetric condition, Eq. (50) can be simpliﬁed by

1 pð1Þ pð2Þ pð2Þ pð1Þ pð2Þ pð2Þ pð1Þ pð1Þ pð2Þ ¼ G1 ðp; qÞ þ pepð1Þ þ G22 epð2Þ G p; q; epð1Þ v ; ec ; ev ; ec v þ qec þ pev þ qec þ G2 ev ; ec v ; ec pð1Þ

Here ev

¼e

pð1Þ a

þ 2e

pð1Þ 3

pð1Þ

ec

¼ 23

e

pð1Þ a

e

pð1Þ 3

ð51Þ

are plastic volumetric strain and plastic shear strain under shear mecha pð1Þ pð2Þ pð2Þ pð2Þ pð2Þ pð2Þ pð2Þ nism, respectively. It should be noted that the value of ev is negative. ev ¼ ea þ 2e3 and ec ¼ 23 ea e3 are plasand

pð2Þ

tic volumetric strain and plastic shear strain under compressive mechanism, respectively. The value of ev is positive. For frozen loess, G1(p, q) is corresponding to the elastic law, which can be determined from constant-conﬁning pressure loading–unloading–reloading test and isotropic compression loading–unloading–reloading test. From Eqs. (1) and (3), the expression of G1(p, q) is as follows:

G1 ðp; qÞ ¼

ðpa þ pÞ lnð1 þ p=pa Þ p q2 þ K 2C

ð52Þ

Y. Lai et al. / International Journal of Engineering Science 84 (2014) 29–53

45

Considering the connection between thermodynamic functions and the formulations in conventional plasticity, pð1Þ pð2Þ pð2Þ and G22 ev ; ec for frozen loess can be determined from experimental results under the triaxial compression epð1Þ v ; ec

G12

test of constant-conﬁning pressure and isotropic compression test.

G12

G22

pð1Þ epð1Þ ¼ ðf3t þ 1Þepð1Þ v ; ec v

! pð1Þ 2 2lec þ m pa mpa pð1Þ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ p arctan þ ln l epð1Þ m e þ n c c 2l l 4nl m2 4nl m2 pð2Þ

pð2Þ epð2Þ ¼ v ; ec

1 pa ev f3t pð2Þ 2 T 1 T 2 ev

ð53Þ

!

epð2Þ v

ð54Þ

where f3t is isotropic tensile strength, T1 and T2 are material parameters relating to volumetric hardening and l, m, n are material parameters depending on conﬁning pressures. G12 and G22 are so called stored plastic work under shear mechanism and compressive mechanism, respectively. Similar to the stored plastic work, the dissipation function D for the frozen loess can be divided into two parts D1 and D2. 1 D and D2 are related to shear mechanism and compression mechanism, respectively. According to the experimental results of the triaxial compression tests with constant conﬁning pressures and the isotropic compression test, the dissipation function D of frozen loess can be expressed as:

8 D ¼ D1 þ D2 > > sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃqﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ > > pð1Þ > p a ec > pð1Þ pð1Þ < D1 ¼ 2 e_ v e_ c pð1Þ 2 pð1Þ lðec Þ þmec þn > > rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ > 2 2 > pð2Þ > pð2Þ > : D2 ¼ 12 f3t þ pa ev pð2Þ e_ pð2Þ þ M2 e_ c v

ð55Þ

T 1 T 2 ev

where M is material parameter related to conﬁning pressure. 4.2. Constitutive model based on thermodynamic functions When the Gibbs free energy function G and the dissipation function D are determined, the elasto-plastic incremental constitutive model for frozen loess can be derived. In the following, the parameters used in this model will be discussed in detail. 4.2.1. Elastic law Using the theories mentioned in Section 3, the rate of elastic strain can be determined by Eq. (52):

8 2 < e_ ev ¼ @ G21 p_ ¼ 1 p_ Kðpþp Þ @p a

ð56Þ

: e_ e ¼ @ 2 G1 q_ ¼ 1 q_ c C @q2

where the elastic parameters K and C are determined from constant-conﬁning pressure triaxial compression, loading– unloading–reloading tests and isotropic compression loading-unloading–reloading tests. 4.2.2. Yield criterion, ﬂow rule and hardening law According to the discussion in Section 3, the yield criterion, the ﬂow rule and the hardening law of frozen loess can be determined by the functions of G12 ; D1 ; G22 and D2, respectively. First of all, the yield criterion, the ﬂow rule and the hardening law of the frozen loess for shear mechanism are determined using G12 and D1, respectively. From the expression of D1 in Eq. (55), one has

8 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃrﬃﬃﬃﬃﬃﬃﬃﬃﬃ pð1Þ > pa ec e_ pð1Þ > ð1Þ c @D1 > v ¼ ¼ > p pð1Þ > pð1Þ 2 pð1Þ @ e_ v e_ pð1Þ < lðec Þ þmec þn v sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃrﬃﬃﬃﬃﬃﬃﬃﬃﬃ > pð1Þ pð1Þ > 1 p a ec e_ v ð1Þ > > ¼ > vq ¼ @@D pð1Þ 2 pð1Þ : e_ pð1Þ e_ pð1Þ lðec Þ þmec þn c c

ð57Þ

ð1Þ

ð1Þ

The yield criterion, the ﬂow rule in the generalized stress space (vp vq plane) are given by pð1Þ pa ec vpð1Þ vð1Þ 2 q ¼ pð1Þ pð1Þ l ec þ mec þ n

ð58Þ

vð1Þ e_ pð1Þ q v ¼ e_ pð1Þ vð1Þ p c

ð59Þ

46

Y. Lai et al. / International Journal of Engineering Science 84 (2014) 29–53

By Eq. (53), there is

8 @G12 ð1Þ > > ¼ ðf3t þ 1Þ < qp ¼ @ epð1Þ v @G12 ð1Þ > > ¼ : qq ¼ @ epð1Þ c

pð1Þ pa ec pð1Þ 2 pð1Þ

lðec

Þ

þmec

ð60Þ þn

Substituting Eq. (60) into Eq. (58), the yield criterion in true stress space (p–q plane) can be written as pð1Þ

pa ec Y ð1Þ ðp; q; epð1Þ ¼0 2 c Þ ¼ qðp þ f3t þ 1Þ ðp þ f3t Þ pð1Þ pð1Þ l ec þ mec þ n

ð61Þ

Combining Eqs. (58), (59) and (61) gives the ﬂow rule in p–q plane by

e_ pð1Þ q v ¼ ðp þ f3t þ 1Þðp þ f3t Þ e_ pð1Þ c

ð62Þ

From Eq. (61), the hardening law under shear mechanism can be obtained as follows

h1

epð1Þ c epð1Þ ¼ 2 c pð1Þ pð1Þ l ec þ mec þ n

ð63Þ

Similarly, the yield criterion, the ﬂow rule and the hardening law of the frozen loess under compressive mechanism can be pð2Þ pð2Þ determined using G22 and D2, respectively. Differentiating D2 with respect to e_ v and e_ c gives:

8 pð2Þ pa ev ð2Þ > @D2 1 1 > qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ v ¼ ¼ f þ > 3t p pð2Þ pð2Þ 2 > @ e_ v T 1 T 2 ev > pð2Þ pð2Þ 2 < 1þM2 ðe_ c =e_ v Þ pð2Þ > pa ev ð2Þ > @D2 1 M2 > qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ v ¼ ¼ f þ > 3t q pð2Þ pð2Þ 2 > @ e_ c T 1 T 2 ev : pð2Þ pð2Þ 2 ðe_ v =e_ c Þ þM2

ð64Þ

ð2Þ

ð2Þ

From Eq. (64), the yield criterion and the ﬂow rule in generalized stress space (vp vq plane) are given by

ð2Þ p

v

2

þ

vð2Þ q

2

M2

pð2Þ

1 pa ev ¼ f3t þ pð2Þ 4 T 1 T 2 ev

!2 ð65Þ

ð2Þ M 2 vp e_ pð2Þ v ¼ pð2Þ ð2Þ e_ c vq

ð66Þ pð2Þ

Differentiating Eq. (54) with respect to ev

8 @G22 ð2Þ > > ¼ 12 < qp ¼ @ epð2Þ T v

> > : qqð2Þ ¼

@G22 pð2Þ

@ ec

pð2Þ pa ev pð2Þ 1 T 2 ev

and

epð2Þ gives: c

f3t ð67Þ

¼0

Substituting Eq. (67) into Eq. (65), the yield criterion in true stress space (p–q plane) can be written as

Y ð2Þ ðp; q; epð2Þ v Þ¼pþ

q2 2

M ðp þ f3t Þ

pð2Þ

pa ev

pð2Þ

T 1 T 2 ev

¼0

ð68Þ

Combining Eqs. (65), (66) and (68) yields the ﬂow rule in true stress space (p–q plane) under compressive mechanism as follows 2 e_ pð2Þ M2 ðp þ f3t Þ q2 v ¼ pð2Þ 2ðp þ f3t Þq e_ c

ð69Þ

From Eq. (68), the hardening law under compressive mechanism can be obtained as follows:

h2 epð2Þ ¼ v

epð2Þ v pð2Þ T 1 T 2 ev

ð70Þ

4.2.3. The elasto-plastic incremental constitutive model for frozen loess From the thermodynamic functions G and D, the elastic law, yield criterion, ﬂow rule, and hardening law (both of shear mechanism and compressive mechanism) for frozen loess, respectively, can be obtained. Then the elasto-plastic incremental

Y. Lai et al. / International Journal of Engineering Science 84 (2014) 29–53

47

constitutive model can be derived from these formulations. The rate of strain (strain increment for the rate-independent case) is:

(

e_ v e_ c

)

( ¼

)

e_ ev e_ ec

(

e_ pð1Þ v þ e_ pð1Þ c

)

(

e_ pð2Þ v þ e_ pð2Þ c

) ð71Þ

Eq. (56) implies

(

e_ ev e_ ec

)

" ¼

1 Kðpþpa Þ

0

#

p_ 1 q_

0

ð72Þ

C

From Eqs. (61)–(63), the sub-plastic strain rate is given by

(

where L ¼

2

)

e_ pð1Þ v e_ pð1Þ c

¼4 nlS2 2

ðlS2 þmSþnÞ

q2 ðpþf3t þ1Þðpþf3t Þ3 pa L

q ðpþf3t Þ2 pa L

q ðpþf3t Þ2 pa L

pþf3t þ1 ðpþf3t Þpa L

3

p_ 5 q_

ð73Þ

, and S is a function of p and q

ðp þ f3t Þpa m S¼ þ 2lqðp þ f3t þ 1Þ 2l

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 m ðp þ f3t Þpa n 2l 2lqðp þ f3t þ 1Þ l

From Eqs. (68)–(70), the following formulae can be obtained.

(

)

e_ pð2Þ v e_ pð2Þ c

2

M 2 ðpþf3t Þ2 q2

M2 pa Vðpþf3t Þ

h

M 2 ðp2 þpf3t Þþq2 1þT 2 M 2 pa ðpþf3t Þ

i2

3

2q M2 pa Vðpþf3t Þ

6 M2 p Vðpþf3t Þ2 ¼4 a 2q

4q2 2

2

M pa V ½M ðpþf3t Þ

2

q2

7 p_ 5 q_

ð74Þ

where V ¼ . T1 Substituting Eqs. (72)–(74) into Eq. (71) yields

(

e_ v e_ c

)

where C ep ¼

¼

C 11 C

C 11 C 21 C 12 C

p_ C 22 q_ C 12

ð75Þ

is the elastoplastic ﬂexibility matrix. Its elements are given by

21 22 8 2 ðpþf3t Þ2 q2 q2 1 > C 11 ¼ Kðpþp þ ðpþf þ1Þðpþf þM > 3 Þ > M2 pa Vðpþf3t Þ2 a 3t 3t Þ pa L > < 2q C 12 ¼ C 21 ¼ ðpþfqÞ2 p L þ M2 p Vðpþf 3t Þ a 3t a > > > pþf3t þ1 4q2 > : C 22 ¼ C1 þ ðpþf þ 3t Þpa L M 2 pa V ½M2 ðpþf3t Þ2 q2

4.3. Determination of material parameters There are nine material parameters in the constitutive model of the frozen loess, namely, K, C, l, m, n, M, T1, T2 and f3t. According to previous investigations on frozen loess (Peng, 1998), the value of f3t is 1.721 MPa. The parameters K, C, l, m, n and M, depend on the conﬁning pressures, and characterize the pressure crushing and melting phenomena of frozen loess. All of these parameters can be determined by triaxial compression tests under constant-conﬁning pressures and isotropic compression tests. An exception is f3t which is obtained from isotropic tensile tests. In Section 2, the elastic parameters of K and C were determined using constant-conﬁning pressure triaxial compression loading–unloading–reloading tests and isotropic compression loading–unloading–reloading tests, respectively. Using the elastic shear module C, the plastic shear strain in constant-conﬁning pressure triaxial compression test can be separated from the total shear strain. And Eq. (61) implies pð1Þ

ec qðp þ f3t þ 1Þ ¼ 2 ðp þ f3t Þpa pð1Þ pð1Þ l ec þ mec þ n

ð76Þ

The values of l, m and n can be obtained by ﬁtting experimental data to Eq. (76). The dependency of l, m and n on conﬁning pressure is determined from the experimental results of constant-conﬁning pressure triaxial compression test. Their expressions are given by

48

Y. Lai et al. / International Journal of Engineering Science 84 (2014) 29–53

Fig. 12. Comparisons between simulated results and test results under constant-conﬁning triaxial compression conditions.

Y. Lai et al. / International Journal of Engineering Science 84 (2014) 29–53

Fig. 12 (continued)

49

50

Y. Lai et al. / International Journal of Engineering Science 84 (2014) 29–53

8 8 0:29464r3 =pa > < l ¼ 6:5741 10 þ 0:00381 e 0:067r3 =pa m ¼ 0:01292 e þ 3 105 r3 =pa þ 0:01426 > : n ¼ 0:01274 þ 0:03603 e0:04657r3 =pa

ð77Þ

Similarly, when the elastic volumetric module K is determined, the plastic volumetric strain of the frozen loess in isotropic compression tests can be separated from the total volumetric strain. The parameters T1 and T2 can be deﬁned by ﬁtting the experimental data to following equation

p0 =pa ¼

epð2Þ v pð2Þ T 1 T 2 ev

ð78Þ

where p0 is hydrostatic pressure in isotropic compressive condition. From the experimental results of isotropic compression tests and Eq. (78), T1 = 23.345 and T2 = 1.904 can be determined. In order to determine the parameter M, Eq. (68) can be rewritten as pð2Þ

p q2 ev þ 2 ¼ pa M pa ðp þ f3t Þ T 1 T 2 epð2Þ v

ð79Þ

Substituting the given values of T1, T2 into Eq. (79), the relationship between M and conﬁning pressure is deﬁned by:

M ¼ e0:02172r3 =pa þ

r3 =pa 0:32778r3 =pa þ 63:31

ð80Þ

4.4. Model veriﬁcation In order to check the validity of the proposed constitutive model for frozen loess, a calculation program based on the proposed model Eq. (75) in this paper was formulated. Using the parameters given in Section 4.3, the simulations of constantconﬁning pressure triaxial compression tests and constant-slope stress path tests on frozen loess have been performed. All of the simulated results are shown in Figs. 12 and 13, respectively. Fig. 12 shows a comparison between test results and simulated results under the constant-conﬁning triaxial compression condition. Fig. 13 shows the comparison of stress–strain curves between test results and simulated results under the constant-slope stress path condition. From Fig. 12, it can be seen that the proposed model, based on the hyperplasticity theory, has the capability of describing the mechanical behavior of frozen loess under various conﬁning pressures. This indicates that the proposed model can describe the inﬂuences of conﬁning pressure (or stress level) on the mechanical behavior of frozen soil, especially the pressure crushing and melting phenomena under high conﬁning pressures. Furthermore, the volumetric contraction and dilation are also well simulated. Using the multiple internal variable hyperplasticity (corresponding to the multiple yield surfaces in conventional plasticity) to construct the constitutive model, the simulated results, shown in Fig. 13, illustrate that the proposed model can also predict the stress–strain curves under constant-slope stress path condition ideally. Therefore, the proposed model can be used to investigate the effects of stress paths on the mechanical behaviors of frozen loess.

Fig. 13. Comparisons between simulated results and test results under constant-slope stress path conditions.

Y. Lai et al. / International Journal of Engineering Science 84 (2014) 29–53

51

5. Conclusions In this paper, a series of tests, such as the constant-conﬁning pressure triaxial compression tests (including loading– unloading–reloading), the isotropic compression tests (including loading–unloading–reloading) and the constant-slope stress path tests, were carried out on frozen loess to investigate the mechanical behavior under various loading conditions. Based on the test results, the mechanical characteristics for frozen loess are analyzed and summarized as follows: The mechanical behavior of frozen loess is inﬂuenced by the conﬁning pressures. When the conﬁning pressures are low (r3 < 2 MPa), the stress–strain curves show softening, which decreases with the increase in conﬁning pressure. When the conﬁning pressures are more than 2 MPa, the stress–strain curves show hardening. In addition, when the conﬁning pressures are lower than 2 MPa, frozen loess experience volumetric expansion during the shear process, and an increase in conﬁning pressure greatly reduces the dilatancy. The mechanical behavior of frozen soil under various conﬁning pressures is investigated through the inﬂuence of conﬁning pressures on the strength of frozen soil and its initial slope of stress–strain curve. It was found that the strength and the initial slope of stress–strain curve for frozen loess versus conﬁning pressures can be divided into three phases. In the ﬁrst phase, both of them increase with the increase in conﬁning pressure followed by a decrease in the second phase. In the third phase when a large enough conﬁning pressure is applied, the changes were relatively small. However, the inﬂuences of conﬁning pressure on the strength and the initial slope of stress–strain curve are not the same. In the ﬁrst stage, the inﬂuence of the conﬁning pressure range on the strength is much smaller than that corresponding to initial slope. This enables the identiﬁcation of the stress level when ice crushing happens in frozen soil using macroscopic tests. Using constant-conﬁning pressure triaxial compression tests and constant-slope stress path tests, it is found that the plastic strain incremental direction of frozen loess is inﬂuenced by stress path. This characteristic of frozen loess cannot be described by a single yield surface or a single plastic potential. In order to construct a constitutive model that can appropriately describe the mechanical characteristics of frozen loess, following conclusions based on the hyperplasticity theory are made: Based on the properties of a homogeneous function, an approach to deriving the yield criterion and ﬂow rule directly from orthogonality principle and ﬁrst-order homogeneity of dissipation function is proposed, in which the mathematical tools with regard to Legendre transformation and convex analysis need not be introduced. A dissipation function and Gibbs free energy function with two internal variables are constructed. According to these two thermodynamic functions, an elasto-plastic incremental constitutive model for frozen loess is set up and a method of determining the material parameters in the model is also given. The stress–strain curves for frozen loess under constant-conﬁning pressure triaxial compression and isotropic-slope stress path conditions are simulated by the constitutive model proposed in this study. The comparison of the simulated results with test results shows that the model can describe well the mechanical characteristics of frozen loess, such as ice crushing, pressure melting, dilatancy and the dependency of plastic strain direction on stress path.

Acknowledgements This research was supported by National key Basic Research Program of China (973 Program No. 2012CB026102), National Natural Science Foundation of China (41230630, 41301072), the Western Project Program of the Chinese Academy of Sciences (KZCX2-XB3-19), the Program of Higher-level talents of Inner Mongolia University (30105-125146), the foundation of State Key Laboratory of Frozen Soil Engineering (SKLFSE-ZY-03), and the Open Project Program of the State Key Laboratory of Frozen Soil Engineering (SKLFSE201208). The authors also thank Mr. Geoffrey Gay, a geotechnical engineer retired from Stuttgart University, for his kind help in revising the paper. References Anandarajah, A. (2008). Multi-mechanism anisotropic model for granular materials. International Journal of Plasticity, 24(5), 804–846. Azmatch, T. F., Sego, D. C., Arenson, L. U., & Biggar, K. W. (2011). Tensile strength and stress–strain behaviour of Devon silt under frozen fringe conditions. Cold Regions Science and Technology, 68(1-2), 85–90. Bragg, R. A., & Andersland, O. B. (1981). Strain rate, temperature, and specimen size effects on compression and tensile properties of frozen soil. Engineering Geology, 18(12), 35–46. Calladine, C. R. (1963). Correspondence on the yielding of clay. Geotechnique, 13(3), 250–255. Calladine, C. R. (1971). A microstructural view of the mechanical properties of saturated clay. Geotechnique, 21(4), 391–415. Chamberlain, E., Groves, C., & Perham, R. (1972). The mechanical behavior of frozen earth materials under high pressure triaxial test conditions. Geotechnique, 22(3), 469–483. Chen, W. F., & Saleeb, A. F. (2001). Constitutive equations for engineering materials. Wuhan: Huazhong University of Science and Technology Press [translated by Yu, T.Q., Wang, X.W., Liu, Z.H.]. Chiarelli, A. S., Shao, J. F., & Hoteit, N. (2003). Modeling of elastoplastic damage behavior of a claystone. International Journal of Plasticity, 19(1), 23–45. Cleja-Tigoiu, S. (1990a). Large elasto-plastic deformations of materials with relaxed conﬁgurations-I. Constitutive assumptions. International Journal of Engineering Science, 28(3), 171–180.

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An experimental investigation of the mechanical behavior and a hyperplastic constitutive model of frozen loess

An experimental investigation of the mechanical behavior and a hyperplastic constitutive model of frozen loess

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