Modelling strain softening of structured soils

Strain softening response of structured soil is more complicated than that of reconstituted soil. After comparatively introducing the unified hardening model for over-consolidated soil (UH model), and the unified hardening model for structured soil (structured UH model), the simulations of strain softening by the 2 models are analyzed from the visual angle of mathematic description. Comparisons between model simulations and test data of strain softening for different soils are illustrated at last, and the reasonability of model descriptions for different strain softening are verified.


Introduction
Shear strength of structured soil will finally decrease with soil structure decay owing to loading, resulting in a strain softening response [1] , which is an expressional behaviour of soil [2] . As strain softening occurs, the deviatoric stress decreasing with soil deformations increasing usually leads to engineering accident. Thus reproducing strain softening response is an important work in soil constitutive model building. According to the generating mechanism, there are 3 kinds of strain softening: strain softening with confining stress decrease, strain softening with dilatancy and strain softening with soil structure decay [3] . So far, many researches have attempted different methods to modelling strain softening. An extensional hyperbolic function is applied to analytically describe strain softening [4] . By applying the relationship between potential peak stress ratio and current soil density, strain softening due to dilatancy is reproduced [5] . Through defining an evolution of stress ratio with deviatoric strain, a bonding surface model reproducing strain softening is built. Based on Camclay model, Yin built an elastic-visco-plastic model, which is qualified to express soil structure, anisotropy, time effects and strain softening [6] [7] .
Also based on Camclay model [8][9] , Yao et al. proposed a unified hardening model (UH model), which is qualified in smoothly and continuously describing strain hardening following strain softening and positive dilatancy following negative dilatancy [10] of over-consolidated soils (reconstituted soils). When the over-consolidation ratio equals to 1, the UH model degrades to Camclay model. Extending the static normal compression line (NCL) in the UH model to be moving with soil structure decay, a moving NCL (MNCL) is presented [11] . The vertical distance between MNCL and NCL in void ratio ~isotropic stress coordinates is applied to express soil structure level [11] [12] . Afterwards, constitutive models in turn describing isotropic compression, triaxial compression and bonding structure were developed [13] , and the a three-dimensional structured UH model was built [14] . Both of the UH model and the structured UH model are qualified in reflecting strain softening smoothly [15] . In this paper, the focus is analysing the description mechanism of strain softening of structured soils comparing with that of reconstituted soils.

Strain softening response
For reconstituted soil, strain softening mainly related to soil density variation in shearing. In drained shearing, sample density increase results in strain hardening, and sample density decrease results in strain softening. As illustrated in Fig. 1, for Fujinomori clay in drained triaxial compression [16] , the deviatoric stress of normally consolidated soil increases monotonically with sample volume compression (the hollow points in the figure), while significant postpeak reduction of deviatoric stress and sample volume expansion are observed in shearing on over-consolidated soil (the solid points in Fig. 1). In undrained shearing, the normally consolidated soil exerts strain hardening, thus other denser soil of course exerts strain hardening. As illustrated in Fig. 2, for Kaolin clay in undrained triaxial compression [17] , the deviatoric stress for all the over-consolidated soil samples increase monotonically with shearing.
For structured soil, besides density variation, soil structure decay in shearing also lead to strain softening. In drained shearing, greater soil structure decay results in sample density increase observably and strain hardening occures; while less soil structure decay result in sample density decrease overall (dilatancy) and strain softening occures. As illustrated in Fig. 3, for strong structured Corinth Canal marl in drained triaxial compression [18] , strain hardening is observed with sample volume compression, and strain softening is observed with sample volume expansion. In undrained shearing, soil structure decay results in pore water pressure increase. Thus greater soil structure decay may lead sharply decrease of effective confining stress, resulting in strain softening response. This is a significant difference between structured and reconstituted soil in strain softening. As illustrated in Fig.  4, for Eastern Osaka clay in undrained triaxial compression [19] , strain softening is observed in shearing with higher confining stress owing to sharply soil structure decay.

Strain softening reproduction
The UH model and the subsequent structured UH model are qualified in describing strain softening of reconstituted and structured soil respectively, whose common yield function is: where 1 ⁄ ; and are respectively isotropic compression and swelling index; is initial void ratio; stress ratio ⁄ ; is the initial value of ; is the critical state stress ratio; is the intercept of right end of yielding surface on axle. Applying the associated flow rule, the relationship between plastic volumetric and deviatoric strain increment d and d is: (2) The hardening law of the UH model and the structured UH model is expressed in the same form as: where is the potential failure stress ratio, expressed as: In equation (4), 12 3 ⁄ ; is the ratio of current stress and reference stress. At shearing beginning, according to equation (3), → 0 and → 3 ; In the shearing end, → 1, → , and a critical state is achieved. In the shearing, when 0 1, 3 ; and when 1, . The essential difference between UH model and structured UH model is the evolution of the internal variable . Beginning from a initial variable , the evolutions of in the UH model and the structured UH model respectively are : Obviously, an internal variable named structure potential Δ in equation (6) is applied to describe current soil structure, and its evolution law is: d Δ ⋅ ⋅ Δ 〈 d ln 〉 (7) In equation (7) (2), (4) and (5), evolutions of in isotropic compression and shearing are respectively: For the structured UH model, considering equation (2), (4), (6) and (7), evolutions of in isotropic compression and shearing are respectively: In equation (9) and (10), 3 and 3 are always satisfied, thus the first factor in the righthand of equation (9) and (10) are always negative (i.e., 3 6 ⁄ 0). Therefore, the sign of d in equation (9) and (10) depends on the second factor in the right-hand of the equations. In the following, strain softening descriptions in drained and undrained shearing will be discussed separately.

Strain softening description in drained shearing
In this paper, for simplicity, only drained shearing with d d ⁄ 0 is implemented. By totally differentiating equation (1), the increment d can either be expressed as increments d and d (11a) or be expressed as increments d and d (11b): In drained shearing, , thus 2 0 and ⁄ 2 0 。 Therefore, the sign of d , d and d are the same in drained shearing. Totally differentiating hardening law equation (3) and considering equation (1), (2) and (11), the increments d and d are respectively given by: In the UH model and the structured UH model, is an internal variable, which is the key point to describe strain softening. In the beginning of shearing 0, and is satisfied. Thus, according to equation (12), the signs of of d and d depend on the sign of the factor . Intially 0, both and increase with at shearing beginning. In the UH model for reconstituted soil, 1 is always satisfied in the whole loading procedure, thus is satisfied all the time. Therefore, according to equation (9), d 0 is satisfied either in isotropic compression or in shearing. When → , d → 0 . Until meeting , according to equation (12), and increase with shearing. Whereafter and 0, and begin to decrease with shearing. Finally, a critical state is reached with → 0 , → and → . The decreasing in shearing reflects the strain softening of reconstituted soil in drained shearing.
In the structured UH model, the adding structure potential Δ influences the evolution of as illustrated in equation (10). Even when , d 0 is also satisfied. This makes come to be possible in isotropic compression and shearing. Thus either stress ratio or internal variable might firstly meet critical state stress ratio . (A) firstly meets when . In this case, , according to equation (10b) d 0 and 0. Thus, according to equation (12),  (1) and (3), and considering equation (2), (8) and (13) In the structured UH model, according to equation (10a), even decays to , d 0 is satisfied due to Δ 0. Then will continues to decrease with loading and come to be less than . In undrained shearing: if soil structure is strong or confining stress is low, is satisfied in the shearing beginning; otherwise if soil structure is weak or confining stress is high, is satisfied in the shearing beginning. Thus either Stress ratio or internal variable might firstly meet critical state stress ratio . (A) firstly meets when . At this time, according to equation (10b), d 0 is satisfied due to . and increase with shearing according to equation (14). According to equation (14b), as ⁄ 0 (i.e., ⁄ ⁄ , when is satisfied right now), d 0 and stops to increase. For right now, according to equation (10b), d 0 。 d 0 and d 0 appearing simultaneously makes the factor ⁄ 0 in the next step, and begins to decrease according to equation (14b). Since then, and decrease together, keeping . Finally and tend to the critical state stress ratio . According to equation (14a), makes d 0 to be always satisfied. Thus, for the strong soil structure or low confining pressure, the