A modified elastoplastic damage structural constitutive model of unsaturated Q₃ undisturbed loess

Abstract

The weakening of loess structure under hydro-mechanical effect is an important reason for collapsible deformation of loess. Therefore, when establishing the constitutive model of unsaturated loess, it is necessary to consider the influence of loess structure to truly reflect the mechanical characteristics. A modified elastoplastic damage structural constitutive model (MEDSCM) is proposed for unsaturated Q₃ undisturbed loess based on the modified Barcelona basic model (MBBM), assuming that the yield stress of undisturbed loess is a coupling of remolded loess and structure, and adopting the non-associated flow rule. The structural evolution equations of unsaturated Q₃ loess in loading and collapsing are introduced respectively, and obtained the constitutive model of soil skeleton in loading and collapsing. There are a total of 18 parameters for the loading model and 15 parameters for the collapsibility model, which are determined by mechanical tests on unsaturated loess. By comparing the triaxial compression and collapsible test data with the model calculation results, the accuracy of the model is verified, and progress has been made in describing the weak hardening characteristics of unsaturated loess. The research results provide a new attempt for further understanding the mechanical properties of loess.

Introduction

Unsaturated loess has typical structural characteristics¹. Under the action of hydraulic and mechanical, its original structure weakens or even disappears, and accompanied by the generation of new structure, which is an important reason for collapsibility^2,3,4,5,6. It can be seen that the effect of structural properties plays an important role in the study of mechanical deformation characteristics of unsaturated loess. Therefore, establishment a constitutive model that can reflect both the disappearance of the original structure and the generation of the new structure is necessary.

At present, scholars have achieved many excellent results in the exploration of the structural constitutive model of soil. Desai et al.⁷ first proposed a method for establishing soil structural constitutive model based on the concept of disturbed state. Since the proposal of the disturbed state provides a new idea to establish the constitutive model of unsaturated soil, some scholars establish new constitutive models on this basis. Liu et al.⁸ established a new constitutive model of artificial structural soil, which is considering structural loess as a binary medium composed of bonding blocks and weakening bands. Dong et al.⁹ used a nonlinear elastic constitutive model to characterize the RI undisturbed soil, and the fully adjusted state combined with the strength criterion on the basis of the modified Cambridge model to form a structural constitutive model. However, the existence of two reference states in the structural model based on the disturbed state allows for many parameters, which limits the practical application of the model.

Rouainia¹⁰ took the structural yield surface as the boundary surface and determined the reference yield surface of the Cambridge model. Gajo¹¹ proposed to comprehensively consider isotropic hardening, kinematic hardening and rotational hardening on the basis of ' normalized stress space '. By defining the dissipation potential and the back stress as the stress function, Chen et al.¹² realized the non-associated flow in the real stress space, and based on the boundary surface model framework, accurately simulated the elastoplastic response model in the initial yield surface. Based on the concept of critical state and boundary surface, Yang et al.¹³ established a structural boundary surface model considering the influence of structural disturbance on elastic deformation, structural yield surface size and cementation suction. These models can well describe the mechanical problems of structured soils, but their application to describe the mechanical characteristics of unsaturated loess still needs further validation.

Xie et al.^14,15 believed that it was difficult to find structural parameters from the methods of microstructure and solid mechanics, so they took an alternative way to directly seek structural parameters reflecting structural nature, and proposed a method of establishing soil structural constitutive model based on comprehensive structure potential. Luo¹⁶, Shao¹⁷ and Chen¹⁸ determined the structural parameters based on the structural change characteristics of loess, and established the relevant constitutive models by using the comprehensive structure potential theory. Based on the idea of damage mechanics, Chu et al.¹⁹ conducted an in-depth study on the structural damage deformation characteristics in the macroscopic mechanical test of loess, and constructed a constitutive relationship accordingly. Jiang²⁰, Zhang²¹ and Hu²² introduced the equivalent plastic strain into the hardening law to explore the damage law of the microscopic mechanism, and established constitutive models considering the structural evolution of loess. The structural constitutive model established by Yao et al.²³, its damage evolution equation adopted the research results of Zhu²⁴, and the associated flow rule adopted is still insufficient in describing the loading softening characteristics of loess.

In summary, the structural model of loess is still further improved and developed. How to accurately describe the mechanical response of unsaturated loess in the process of loading and collapsing, needs to be carried out on the basis of reasonable description of loess structure. Thus, we²⁵ carried out triaxial compression and triaxial collapsible tests of unsaturated loess based on CT scanning, introduced structural parameters reflecting the structural strength, established the structural damage variables during loading and collapsing, used the newly established damage evolution equations to further modify the previous research, and established a reasonable structural constitutive model of unsaturated loess. The model describes the loading and collapsible process respectively, which deepens the understanding of the mechanical characteristics and collapse deformation characteristics of loess.

Modeling approach

Given that matric suction and structural characteristics significantly influence soil compression and collapsible deformation, this study emphasizes these factors in its modeling approach^26,27. Through CT scanning of the meso-structure changes of unsaturated Q₃ undisturbed loess and remolded loess during loading and collapsing, the structural parameters are calculated, and the evolution equation describing structural damage is introduced. This equation can well reflect the disappearance of the original structure and the generation of new structures. Then, based on the modified Barcelona unsaturated loess elastoplastic model, the yield stress p_c of undisturbed loess composed of remolded loess and structure is taken as the hardening parameter H, and the structural constitutive model of unsaturated Q₃ loess is established by using the non-associated flow rule to describe the loading and collapsing respectively.

The model is based on the following assumptions: collapsible deformation is plastic deformation, and the collapsible process is accompanied by structural damage; the increase in moisture content causes structural damage with or without collapsible deformation; the structural damage is isotropic damage; remolded loess ignores structure.

Elastoplastic stress–strain relationship of unsaturated soil

Based on the modified Barcelona unsaturated soil elastoplastic model, the structural evolution equations of unsaturated Q₃ loess during loading and collapsing are introduced respectively, and the structural model describing the soil skeleton during loading and collapsing is established. The modified part includes the correction of the suction increase (SI) yield line. The MBBM can be used as a constitutive model for unsaturated remolded loess without considering the structure.

Elastic properties

The expressions for elastic volumetric strain increment $\varepsilon_{v}^{e}$ and deviatoric strain increment $\varepsilon_{s}^{e}$ are shown in Eqs. (1) and (2) :

$${\text{d}}\varepsilon_{v}^{e} = {\text{d}}\varepsilon_{vp}^{e} + {\text{d}}\varepsilon_{vs}^{e} = \frac{\kappa }{\nu }\frac{{{\text{d}}p}}{p} + \frac{{\kappa_{s} }}{\nu }\frac{{{\text{d}}s}}{{s + p_{atm} }}$$

(1)

$${\text{d}}\varepsilon_{s}^{e} = \frac{{{\text{d}}q}}{3G}$$

(2)

where ${\text{d}}\varepsilon_{vp}^{e}$ and ${\text{d}}\varepsilon_{vs}^{e}$ are the elastic strain increments resulting from the net mean stress and suction, respectively; $\kappa$ and $\kappa_{s}$ are the elastic stiffness coefficients associated with the net mean stress loading and suction increase, respectively; v is the specific volume of the soil; p_atm is the atmospheric pressure; G is the shear modulus. s is the suction; and q is the deviatoric stress.

Plastic properties

Yield surface equations

The load-collapse (LC) yield surface is shown in Eq. (3)

$$f_{1} \left( {p,q,s,p_{0}^{*} } \right) \equiv q^{2} - M^{2} \left( {p + p_{s} } \right)\left( {p_{0} - p} \right) = 0$$

(3)

$$p_{s} = k_{c} s$$

(4)

$$\left( {\frac{{p_{0} }}{{p^{c} }}} \right) = \left( {\frac{{p_{0}^{*} }}{{p^{c} }}} \right)^{{\frac{\lambda \left( 0 \right) - \kappa }{{\lambda \left( s \right) - \kappa }}}}$$

(5)

$$\lambda \left( s \right) = \lambda \left( 0 \right)\left[ {\left( {1 - \gamma } \right)\exp \left( { - \beta s} \right) + \gamma } \right]$$

(6)

Replacing the maximum suction s₀ with the remolded loess yield suction s_y, the modified SI yield surface^28,29 is shown in Eq. (7):

$$f_{2} \left( {s,s_{0} } \right) \equiv s - s_{y} = 0$$

(7)

where q is the deviatoric stress, $p_{0}^{*}$ is the net mean yield stress in the saturated state, $p_{0}$ is the net mean yield stress in the unsaturated loess at a particular suction, $p_{s}$ is the intercept of the critical state line (CSL) on the axis p at a particular suction, and $p^{c}$ is the reference stress; M is the slope of CSL at saturation; $k_{c}$ is a parameter reflecting the increase in cohesion with suction; $\kappa$ is the elastic stiffness coefficient associated with the net mean stress; $\lambda (s)$ is the compression exponent after yielding of the net mean stress loading for a given suction, which is equal to $\lambda (0)$ at saturation; $\gamma$ is the constant associated with the maximum stiffness for the same soil sample, $\gamma = \lambda (s \to \infty )/\lambda \left( 0 \right)$; $\beta$ is the parameter controlling the rate of increase in soil stiffness with suction; and s_y is the yielding suction.

Flow rule and hardening law

The associated flow rule will be too large when predicting the expansion of friction materials such as loess, while the non-associated flow rule can accurately describe the deformation by specifying the amount of expansion. At the same time, the associated flow rule has some shortcomings in describing the softening of soil structure. Therefore, the model adopts the non-associated flow rule. The plastic strain increments associated with the LC yield surface are plastic volumetric strain $d\varepsilon_{vp}^{p}$ and plastic deviatoric strain $d\varepsilon_{s}^{p}$, which are expressed as:

$$d\varepsilon_{vp}^{p} = \mu_{1} n_{p}$$

(8)

$$d\varepsilon_{s}^{p} = \mu_{1} n_{q}$$

(9)

$$n_{p} = 1$$

(10)

$$n_{q} = \left[ {2q\alpha /M^{2} \left( {2p + p_{s} - p_{0} } \right)} \right]$$

(11)

α is a constant reflecting the characteristics of non-associated flow. When k₀ is loaded, there is $\varepsilon_{2} = \varepsilon_{3} = 0$, then:

$$\alpha = \frac{{M\left( {M - 9} \right)\left( {M - 3} \right)}}{{9\left( {6 - M} \right)}}\left\{ {1/\left[ {1 - {\kappa \mathord{\left/ {\vphantom {\kappa {\lambda \left( 0 \right)}}} \right. \kern-0pt} {\lambda \left( 0 \right)}}} \right]} \right\}$$

(12)

The plastic strain increment associated with the SI yield surface is the plastic volumetric strain $d\varepsilon_{vs}^{p}$, and the plastic deviatoric strain is 0, then the volumetric strain expression is:

$$d\varepsilon_{vs}^{p} = \mu_{2}$$

(13)

The LC and SI yield surface equations are differentiated separately according to the plastic consistency condition and are obtained:

$$\frac{{\partial f_{1} }}{\partial p}{\text{d}}p + \frac{{\partial f_{1} }}{\partial q}{\text{d}}q + \frac{{\partial f_{1} }}{\partial s}{\text{d}}s + \frac{{\partial f_{1} }}{{\partial p_{0} }}{\text{d}}p_{0} = 0$$

(14)

$$\frac{{\partial f_{2} }}{\partial s}{\text{d}}s + \frac{{\partial f_{2} }}{{\partial s_{y}^{{}} }}{\text{d}}s_{y}^{{}} = 0$$

(15)

Differentiating Eq. (5) yields:

$${\text{d}}p_{0} = \frac{\lambda \left( 0 \right) - \kappa }{{\lambda \left( s \right) - \kappa }} \cdot \left( {\frac{{p_{0}^{*} }}{{p^{c} }}} \right)^{{\frac{\lambda \left( 0 \right) - \kappa }{{\lambda \left( s \right) - \kappa }} - 1}} {\text{d}}p_{0}^{*}$$

(16)

The hardening law determines the magnitude of the incremental plastic strain induced by a given stress increment. The modeling process assumes that the soil will undergo volume hardening, and the hardening parameter H is the plastic volumetric strain ${\text{d}}\varepsilon_{v}^{p}$. The corresponding hardening law is:

$$\frac{{{\text{d}}p_{0}^{*} }}{{p_{0}^{*} }} = \frac{\nu }{\lambda \left( 0 \right) - \kappa }{\text{d}}\varepsilon_{vp}^{p}$$

(17)

$$\frac{{{\text{d}}s_{y} }}{{s_{y} + p_{atm} }} = \frac{\nu }{{\lambda_{s} - \kappa_{s} }}{\text{d}}\varepsilon_{vs}^{p}$$

(18)

where v is the specific volume of the soil; $\kappa_{s}$ and $\lambda_{s}$ are the shrinkage coefficients related to the increase of suction before and after suction yield, respectively.

The total plastic volumetric strain and elastoplastic volumetric strain are:

$${\text{d}}\varepsilon_{v}^{p} = \frac{{\frac{{\partial f_{1} }}{\partial p}{\text{d}}p + \frac{{\partial f_{1} }}{\partial q}{\text{d}}q + \frac{{\partial f_{1} }}{\partial s}{\text{d}}s}}{{\frac{{\partial f_{1} }}{{\partial p_{0}^{{}} }} \cdot \frac{\lambda \left( 0 \right) - \kappa }{{\lambda \left( s \right) - \kappa }} \cdot \left( {\frac{{p_{0}^{*} }}{{p^{c} }}} \right)^{{\frac{\lambda \left( 0 \right) - \kappa }{{\lambda \left( s \right) - \kappa }} - 1}} \cdot \frac{{\nu p_{0}^{*} }}{\kappa - \lambda \left( 0 \right)}}} + \frac{{\lambda_{s} - \kappa_{s} }}{{\nu \left( {s_{y} + p_{atm} } \right)}}{\text{d}}s$$

(19)

$${\text{d}}\varepsilon_{v}^{{}} = \frac{\kappa }{\nu }\frac{{{\text{d}}p}}{p} + \frac{{\kappa_{s} }}{\nu }\frac{{{\text{d}}s}}{{s + p_{atm} }} + \frac{{\frac{{\partial f_{1} }}{\partial p}{\text{d}}p + \frac{{\partial f_{1} }}{\partial q}{\text{d}}q + \frac{{\partial f_{1} }}{\partial s}{\text{d}}s}}{{\frac{{\partial f_{1} }}{{\partial p_{0}^{{}} }} \cdot \frac{\lambda \left( 0 \right) - \kappa }{{\lambda \left( s \right) - \kappa }} \cdot \left( {\frac{{p_{0}^{*} }}{{p^{c} }}} \right)^{{\frac{\lambda \left( 0 \right) - \kappa }{{\lambda \left( s \right) - \kappa }} - 1}} \cdot \frac{{\nu p_{0}^{*} }}{\kappa - \lambda \left( 0 \right)}}} + \frac{{\lambda_{s} - \kappa_{s} }}{{\nu \left( {s_{y} + p_{atm} } \right)}}{\text{d}}s$$

(20)

The plastic deviatoric strain and elastoplastic deviatoric strain are:

$$\begin{aligned} & {\text{d}}\varepsilon _{s}^{p} = {\text{d}}\varepsilon _{{vp}}^{p} \cdot \frac{{2q\alpha }}{{M^{2} \left( {2p + p_{s} - p_{0} } \right)}} \\ & \quad = - \frac{{\left( {\frac{{\partial f_{1} }}{{\partial p}}{\text{d}}p + \frac{{\partial f_{1} }}{{\partial q}}{\text{d}}q + \frac{{\partial f_{1} }}{{\partial s}}{\text{d}}s} \right)2q\alpha }}{{\left[ {\frac{{\partial f_{1} }}{{\partial p_{0}^{{}} }} \cdot \frac{{\lambda \left( 0 \right) - \kappa }}{{\lambda \left( s \right) - \kappa }} \cdot \left( {\frac{{p_{0} ^{*} }}{{p^{c} }}} \right)^{{\frac{{\lambda \left( 0 \right) - \kappa }}{{\lambda \left( s \right) - \kappa }} - 1}} \cdot \frac{{\nu p_{0}^{*} }}{{\lambda \left( 0 \right) - \kappa }}} \right]M^{2} \left( {2p + p_{s} - p_{0} } \right)}} \\ \end{aligned}$$

(21)

$${\text{d}}\varepsilon_{s}^{{}} = \frac{{{\text{d}}q}}{3G} - \frac{{\left( {\frac{{\partial f_{1} }}{\partial p}{\text{d}}p + \frac{{\partial f_{1} }}{\partial q}{\text{d}}q + \frac{{\partial f_{1} }}{\partial s}{\text{d}}s} \right)2q\alpha }}{{\left[ {\frac{{\partial f_{1} }}{{\partial p_{0}^{{}} }} \cdot \frac{\lambda \left( 0 \right) - \kappa }{{\lambda \left( s \right) - \kappa }} \cdot \left( {\frac{{p_{0}^{*} }}{{p^{c} }}} \right)^{{\frac{\lambda \left( 0 \right) - \kappa }{{\lambda \left( s \right) - \kappa }} - 1}} \cdot \frac{{\nu p_{0}^{*} }}{\lambda \left( 0 \right) - \kappa }} \right]M^{2} \left( {2p + p_{s} - p_{0} } \right)}}$$

(22)

Structural constitutive model in loading process

Assuming that the structural damage is generated only in the plastic part, so the effects of structure damage is considered in the plastic part, while the elastic part is calculated according to the elastic properties.

Yield surface equation

The initial yield line shapes of the initial pristine loess in planes $p - s$ and $p - q$ are similar to those of the Barcelona basic model (BBM) in the corresponding planes³⁰. The yield stress $p_{c}$ is assumed to consist of a suction-related part and a structural-related part. The suction-related part is calculated using the proposed change rule in the BBM, while the structural-related part is described by the damage rule. The yield stress $p_{c}$ of undisturbed loess is shown in Eq. (23):

$$p_{c} = p_{0} + p^{s} m_{\sigma }$$

(23)

where $p_{0}$ is the net mean stress at yield of unsaturated remolded loess; $p^{s}$ is the difference between the net mean stresses at initial yield of undisturbed loess and remolded loess with the same dry density and the same initial suction; and $m_{\sigma }$ is the structural parameter during the loading process. Where, $p^{s}$ and $m_{\sigma }$ expressions are shown in Eqs. (24) and (25), respectively:

$$p^{s} = p_{u0i} - p_{d0i}$$

(24)

$$m_{\sigma } = m_{o} \left( {1 - D_{1} } \right)$$

(25)

where $p_{u0i}$ and $p_{d0i}$ are the initial yield net average stresses of the undisturbed loess and remolded loess, respectively, and their values can be calculated by the LC yield line equation of the undisturbed loess and remolded loess; $m_{o}$ is the initial structural parameter; and D₁ is the structural damage variable produced by the soil during the loading process.

The structure of the undisturbed loess gradually undergoes damage under loading and its mechanical properties gradually transition to remolded loess. The slope of the CSL is determined based on the complex concept, as shown in Eq. (26):

$$M_{f} = \left( {1 - D_{1} } \right)M + D_{1} M^{*}$$

(26)

where, M is the slope of CSL for undisturbed loess; $M^{*}$ is the slope of CSL for remolded loess. From the above equation:

$$M_{f} = \left\{ \begin{gathered} \, M, D_{1} { = 0 } \hfill \\ \left( {1 - D_{1} } \right)M + D_{1} M^{*}, {0 < }D_{1} < 1 \hfill \\ \, M^{*}, D_{1} { = 1} \hfill \\ \end{gathered} \right. \,$$

(27)

when $D_{1} = 0$ indicates that the structure of undisturbed loess sample is not damaged; $0 < D_{1} < 1$ indicates that the soil sample is subjected to loading, the structure is damaged but not completely destroyed, and the soil sample is in the coupled state of undisturbed and remolded loess; $D_{1} = 1$ indicates that the structure of the remolded loess sample is completely destroyed.

Substituting the $p_{c}$ as well as M_f determined above into the LC yield surface of the MBBM, the LC yield surface equation during loading considering structural effects can be obtained as shown in the following equations:

$$f_{1} \left( {p,q,s,p_{0}^{*} } \right) \equiv q^{2} - M_{f}^{2} \left( {p + p_{s} } \right)\left( {p_{c} - p} \right) = 0$$

(28)

$$\lambda \left( s \right) = \lambda \left( 0 \right)\left[ {\left( {1 - \gamma } \right)\exp \left( { - \beta s} \right) + \gamma } \right]$$

(29)

where $\lambda (0)$ is the slope of the isotropic compression curve of the saturated undisturbed loess after yielding in the plane $e - \ln p$; $\gamma$ is a constant related to the maximum stiffness of the undisturbed loess; and $\beta$ is a parameter controlling the growth rate of undisturbed loess stiffness with suction.

Flow rule and hardening law

The non-associated flow rule is still used in the subsequent loading process. The evolution process of LC yield surface considering structural influence is controlled by hardening parameters: yield stress $p_{c}$ and initial yield suction s_y of undisturbed loess considering structural influence. The incremental calculation of Eq. (23) can be obtained:

$${\text{d}}p_{c} = {\text{d}}\left[ {p_{0} + p^{s} m_{\sigma } } \right] = {\text{d}}p_{0} + p^{s} dm_{\sigma }$$

(30)

From Eq. (16) for an applied to remolded loess:

$${\text{d}}p_{0} = \frac{{\lambda \left( 0 \right)^{*} - \kappa^{*} }}{{\lambda \left( s \right)^{*} - \kappa^{*} }} \cdot \left( {\frac{{p_{0}^{*} }}{{p^{c*} }}} \right)^{{\frac{{\lambda \left( 0 \right)^{*} - \kappa^{*} }}{{\lambda \left( s \right)^{*} - \kappa^{*} }} - 1}} {\text{d}}p_{0}^{*}$$

(31)

Substituting Eq. (31) into Eq. (23) gives:

$${\text{d}}p_{c} = \frac{{\lambda \left( 0 \right)^{*} - \kappa^{*} }}{{\lambda \left( s \right)^{*} - \kappa^{*} }} \cdot \left( {\frac{{p_{0}^{*} }}{{p^{c*} }}} \right)^{{\frac{{\lambda \left( 0 \right)^{*} - \kappa^{*} }}{{\lambda \left( s \right)^{*} - \kappa^{*} }} - 1}} {\text{d}}p_{0}^{*} + p^{s} {\text{d}}m_{\sigma }$$

(32)

$${\text{d}}p_{0}^{*} = \frac{{\nu p_{0}^{*} }}{{\lambda \left( 0 \right)^{ * } - \kappa^{ * } }}{\text{d}}\varepsilon_{vp}^{p}$$

(33)

$${\text{d}}m_{\sigma } = - m_{o} {\text{d}}D_{1}$$

(34)

Equation (30) can reflect the process of softening and hardening of soil samples: the degree of structural damage of soil samples under the action of the load gradually increases, making $p_{c}$ decrease, which reflects the undisturbed loess structure is damaged during the loading process and the soil sample softens. $p_{0}$ increases with the increase of the strain of the soil sample, reflecting the destruction of the original structure and the formation of the new structure, resulting in the hardening of the soil sample.

The formula for the remolded loess sample $\lambda (s)^{*}$ is the same as the undisturbed loess Eq. (29), and the main parameters can be obtained by adopting the corresponding parameters of the remolded soil sample:

$$\lambda \left( s \right)^{ * } = \lambda \left( 0 \right)^{ * } \left[ {\left( {1 - \gamma^{ * } } \right)\exp \left( { - \beta^{ * } s} \right) + \gamma^{ * } } \right]$$

(35)

where $\lambda (0)^{ * }$ is the slope of the saturated remolded soil sample after yielding in the plane $e - \ln p$; $\gamma^{ * }$ is a constant related to the maximum stiffness of the remolded loess; and $\beta^{ * }$ is a parameter controlling the rate of increase of the stiffness of the remolded loess with suction.

The hardening law corresponding to the SI yield surface is following in Eq. (18). Differential calculation of the yield surface during loading according to the plastic consistency condition can be obtained:

$$\frac{{\partial f_{1} }}{\partial p}{\text{d}}p + \frac{{\partial f_{1} }}{\partial q}{\text{d}}q + \frac{{\partial f_{1} }}{\partial s}{\text{d}}s + \frac{{\partial f_{1} }}{{\partial p_{c}^{{}} }}{\text{d}}p_{c}^{{}} = 0$$

(36)

The plastic volumetric strain related to suction does not change due to structural damage^31,32. Thus, the plastic volumetric strain and deviatoric strain of the undisturbed loess sample during the loading can be obtained. It should be noted that the above solving procedure is aimed at the loading yield surface f₁ shown in Eq. (28). Then the elastoplastic volumetric strain ${\text{d}}\varepsilon_{v}^{{}}$ and deviatoric strain ${\text{d}}\varepsilon_{s}^{{}}$ of undisturbed loess in the loading process considering structural effects are as follows:

$${\text{d}}\varepsilon_{v}^{{}} = \frac{\kappa }{\nu }\frac{{{\text{d}}p}}{p} + \frac{{\kappa_{s} }}{\nu }\frac{{{\text{d}}s}}{{s + p_{atm} }} + \frac{{\lambda_{s} - \kappa_{s} }}{{\nu \left( {s_{y} + p_{atm} } \right)}}{\text{d}}s + \frac{{ - \frac{{\frac{{\partial f_{1} }}{\partial p}{\text{d}}p + \frac{{\partial f_{1} }}{\partial q}{\text{d}}q + \frac{{\partial f_{1} }}{\partial s}{\text{d}}s}}{{{{\partial f_{1} } \mathord{\left/ {\vphantom {{\partial f_{1} } {\partial p_{c}^{{}} }}} \right. \kern-0pt} {\partial p_{c}^{{}} }}}} + p^{s} m_{o} {\text{d}}D_{1} }}{{\frac{{\lambda \left( 0 \right)^{*} - \kappa^{*} }}{{\lambda \left( s \right)^{*} - \kappa^{*} }} \cdot \left( {\frac{{p_{0}^{*} }}{{p^{c*} }}} \right)^{{\frac{{\lambda \left( 0 \right)^{*} - \kappa^{*} }}{{\lambda \left( s \right)^{*} - \kappa^{*} }} - 1}} \cdot \frac{{p_{0}^{*} \nu }}{{\lambda \left( 0 \right)^{*} - \kappa^{*} }}}}$$

(37)

$${\text{d}}\varepsilon_{s}^{{}} = \frac{{{\text{d}}q}}{3G} + \frac{{\left( { - \frac{{\frac{{\partial f_{1} }}{\partial p}{\text{d}}p + \frac{{\partial f_{1} }}{\partial q}{\text{d}}q + \frac{{\partial f_{1} }}{\partial s}{\text{d}}s}}{{{{\partial f_{1} } \mathord{\left/ {\vphantom {{\partial f_{1} } {\partial p_{c}^{{}} }}} \right. \kern-0pt} {\partial p_{c}^{{}} }}}} + p^{s} m_{o} {\text{d}}D_{1} } \right)2q\alpha_{f} }}{{\left[ {\frac{{\lambda \left( 0 \right)^{*} - \kappa^{*} }}{{\lambda \left( s \right)^{*} - \kappa^{*} }} \cdot \left( {\frac{{p_{0}^{*} }}{{p^{c*} }}} \right)^{{\frac{{\lambda \left( 0 \right)^{*} - \kappa^{*} }}{{\lambda \left( s \right)^{*} - \kappa^{*} }} - 1}} \cdot \frac{{p_{0}^{*} \nu }}{{\lambda \left( 0 \right)^{*} - \kappa^{*} }}} \right]M_{f}^{2} \left( {2p + p_{s} - p_{c} } \right)}}$$

(38)

Structural damage evolution equation in loading process

The structural damage evolution equation during the loading process is based on whether the load is applied. Using the yield stress as a threshold, the elastic phase prior to yielding is defined as the non-destructive state, whereas the undisturbed soil sample that exhibits plastic deformation is classified as being in the damaged state. The structural damage variable D₁ generated during the loading process is defined as:

$$D_{1} = \frac{{m_{0} - m_{\sigma } }}{{m_{o} }}$$

(39)

The CT-triaxial test facilitates comprehensive monitoring of stress–strain behavior, moisture content, and microstructural changes in the specimen during the loading process³³. The structural parameter m of the undisturbed sample is expressed by the microscopic data measured in the CT-triaxial test as follows:

$$m = \frac{{ME_{f} - ME_{{}} }}{{ME_{f} - ME_{i} }}$$

(40)

where $ME_{i}$ is the CT value of the relatively complete undisturbed sample, which can be considered as no structural damage; $ME_{f}$ is the CT value of the completely adjusted undisturbed sample, which can be regarded as the complete structural damage of the sample after the load is applied. $ME$ is the CT value of the undisturbed sample corresponding to any time during the loading process. When the sample structure is in a non-destructive state, $m = 1$, $ME = ME_{i}$; when the sample structure is completely destroyed and the readjustment state occurs, $m = 0$, $ME = ME_{f}$.

According to the CT results²⁵, it can be seen that at the initial scan, the CT number of the sample at the net confining pressure of 50 kPa is $ME_{i}$, i.e., $ME_{i} = 732.78{\text{ HU}}$; the CT number of the sample at the net confining pressure of 300 kPa is selected as $ME_{f}$, i.e., $ME_{f} = 975.14{\text{ HU}}$, and other CT values are within this range. For the initial structural parameter m₀, the sample is scanned by CT after the consolidation confining pressure is stable. At this time, the CT number ME is substituted into Eq. (40) to obtain m₀.

The CT-triaxial compression test shows that the relationship between the structural damage variable D₁ and each strain during the loading process satisfies the Eq. (41), and the increment of the structural damage variable dD₁ during the loading process can be obtained by differentiating it:

$$D_{1} = 1 - e^{{\left[ { - \left( {A_{1} \varepsilon_{s} + A_{2} \varepsilon_{v} } \right)} \right]}}$$

(41)

$${\text{d}}D_{1} = {\text{d}}\left[ {1 - e^{{\left[ { - \left( {A_{1} \varepsilon_{s} + A_{2} \varepsilon_{v} } \right)} \right]}} } \right] = e^{{\left[ { - \left( {A_{1} \varepsilon_{s} + A_{2} \varepsilon_{v} } \right)} \right]}} \cdot \left( {A_{1} {\text{d}}\varepsilon_{s} + A_{2} {\text{d}}\varepsilon_{v} } \right)$$

(42)

Structural constitutive model in collapsing process

Yield surface equation

During the collapsing process, the suction of the sample is continuously reduced due to the continuous increase of the water immersion of the sample, and the sample structure is damaged during this process, resulting in a structural damage variable D₂. The structural damage caused by the loading stage before the collapse has ended, so the structural damage variable D₁ is fixed to the structural damage variable D_1σ caused at the end of the loading. From the above analysis, it can be seen that the structural damage variable D generated by the sample at any time during the collapsing process can be expressed as:

$$D = D_{1\sigma } + D_{2}$$

(43)

The structural parameter $m_{\sigma sh}$ in the collapsing process corresponding to the structural damage variable D can be expressed as:

$$m_{\sigma sh} = m_{o} \left( {1 - D_{1\sigma } - D_{2} } \right)$$

(44)

In the process of collapsing, the structure of undisturbed loess is further destroyed, and its mechanical properties are further close to those of remolded loess. The slope $M_{f\sigma sh}$ of the CSL is determined based on the composite concept, as shown in Eq. (45):

$$M_{f\sigma sh} = \left( {1 - D} \right)M + DM^{*}$$

(45)

The LC yield surface equation in the collapsing process considering the structural influence is shown in Eq. (46):

$$f_{1} \left( {p,q,s,p_{0}^{*} } \right) \equiv q^{2} - M_{f\sigma sh}^{2} \left( {p + p_{s} } \right)\left( {p_{c} - p} \right) = 0$$

(46)

If the structural damage during the loading process is not considered, the structural damage variable is the structural damage variable D₂ generated by the sample in the collapsing process. The structural parameters $m_{sh}$ corresponding to the structural damage variable D₂ in the collapsing process can be expressed as:

$$m_{sh} = m_{1o} \left( {1 - D_{2} } \right)$$

(47)

where $m_{1o}$ is the structural parameter before collapsing; D₂ is the structural damage variable generated by the soil during the collapsing process.

The slope of the CSL is determined based on the complex concept during collapsing, as shown in Eq. (48):

$$M_{fsh} = \left( {1 - D_{2} } \right)M + D_{2} M^{*}$$

(48)

The LC yield surface equation in the collapsing process considering the structural influence is shown in Eq. (49):

$$f_{1} \left( {p,q,s,p_{0}^{*} } \right) \equiv q^{2} - M_{fsh}^{2} \left( {p + p_{s} } \right)\left( {p_{c} - p} \right) = 0$$

(49)

Flow rule and hardening law

The non-associated flow rule is still used in the collapsing process. If only the structural damage caused by the collapsing process is considered, the hardening law is modified by substituting the structural parameter $m_{sh}$ into Eq. (32):

$${\text{d}}p_{c} = \frac{{\lambda \left( 0 \right)^{*} - \kappa^{*} }}{{\lambda \left( s \right)^{*} - \kappa^{*} }} \cdot \left( {\frac{{p_{0}^{*} }}{{p^{c*} }}} \right)^{{\frac{{\lambda \left( 0 \right)^{*} - \kappa^{*} }}{{\lambda \left( s \right)^{*} - \kappa^{*} }} - 1}} {\text{d}}p_{0}^{*} + p^{s} {\text{d}}m_{sh}$$

(50)

$${\text{d}}m_{sh} = - m_{1o} {\text{d}}D_{2}$$

(51)

Then the total collapse volumetric strain increment during the collapsing process can be expressed as:

$${\text{d}}\varepsilon_{v}^{sh} = \frac{\kappa }{\nu }\frac{{{\text{d}}p}}{p} + \frac{{\kappa_{s} }}{\nu }\frac{{{\text{d}}s}}{{s + p_{atm} }} + \frac{{\lambda_{s} - \kappa_{s} }}{{\nu \left( {s_{y} + p_{atm} } \right)}}{\text{d}}s + \frac{{ - \frac{{\frac{{\partial f_{1} }}{\partial p}{\text{d}}p + \frac{{\partial f_{1} }}{\partial q}{\text{d}}q + \frac{{\partial f_{1} }}{\partial s}{\text{d}}s}}{{{{\partial f_{1} } \mathord{\left/ {\vphantom {{\partial f_{1} } {\partial p_{c}^{{}} }}} \right. \kern-0pt} {\partial p_{c}^{{}} }}}} + p^{s} m_{1o} {\text{d}}D_{2} }}{{\frac{{\lambda \left( 0 \right)^{*} - \kappa^{*} }}{{\lambda \left( s \right)^{*} - \kappa^{*} }} \cdot \left( {\frac{{p_{0}^{*} }}{{p^{c*} }}} \right)^{{\frac{{\lambda \left( 0 \right)^{*} - \kappa^{*} }}{{\lambda \left( s \right)^{*} - \kappa^{*} }} - 1}} \cdot \frac{{p_{0}^{*} \nu }}{{\lambda \left( 0 \right)^{*} - \kappa^{*} }}}}$$

(52)

The increment of collapse deviatoric strain can be expressed as:

$${\text{d}}\varepsilon_{s}^{sh} = \frac{{{\text{d}}q}}{3G} + \frac{{\left( { - \frac{{\frac{{\partial f_{1} }}{\partial p}{\text{d}}p + \frac{{\partial f_{1} }}{\partial q}{\text{d}}q + \frac{{\partial f_{1} }}{\partial s}{\text{d}}s}}{{{{\partial f_{1} } \mathord{\left/ {\vphantom {{\partial f_{1} } {\partial p_{c}^{{}} }}} \right. \kern-0pt} {\partial p_{c}^{{}} }}}} + p^{s} m_{1o} {\text{d}}D_{2} } \right)2q\alpha_{fsh} }}{{\left[ {\frac{{\lambda \left( 0 \right)^{*} - \kappa^{*} }}{{\lambda \left( s \right)^{*} - \kappa^{*} }} \cdot \left( {\frac{{p_{0}^{*} }}{{p^{c*} }}} \right)^{{\frac{{\lambda \left( 0 \right)^{*} - \kappa^{*} }}{{\lambda \left( s \right)^{*} - \kappa^{*} }} - 1}} \cdot \frac{{p_{0}^{*} \nu }}{{\lambda \left( 0 \right)^{*} - \kappa^{*} }}} \right]M_{fsh}^{2} \left( {2p + p_{s} - p_{c} } \right)}}$$

(53)

At this point, the LC yield surface equation f₁ is expressed in Eq. (49). Through the above calculation, the increment of collapse volumetric strain and collapse deviatoric strain in the collapsing process can be obtained.

Structural damage evolution equation in collapsing process

The load applied to the sample before immersion is small, and there is no significant change in CT images and CT data after consolidation and before immersion. Therefore, the structural damage variable D₂ during immersion can be approximated as:

$$D_{2} = \frac{{m_{10} - m_{sh} }}{{m_{1o} }}$$

(54)

where $m_{10}$ is the structural parameter of the sample before immersion in water after load stability; $m_{sh}$ is the structural parameter of the sample at any moment during the collapsing process; and $m_{o}$ is the initial structural parameter of the sample.

Each structural parameter is calculated in the same way as the loading part. The increment of structural damage variable dD₂ can be obtained by fitting the relationship between structural damage variable D₂ and the collapse volumetric strain and saturation increment, and differentiating the above equation:

$$\begin{aligned} & {\text{d}}D_{2} = d\left[ {1 - e^{{\left[ { - \left( {A_{1} \varepsilon _{v}^{{sh}} + A_{2} } \right)\Delta S_{r} } \right]}} } \right] \\ & \quad = e^{{\left[ { - \left( {A_{1} \varepsilon _{v}^{{sh}} + A_{2} } \right)\Delta S_{r} } \right]}} \cdot \left[ {(A_{1} \cdot \Delta S_{r} ){\text{d}}\varepsilon _{v}^{{sh}} + \left( {A_{1} \varepsilon _{v}^{{sh}} + A_{2} } \right){\text{d}}\Delta S_{r} } \right] \\ \end{aligned}$$

(55)

Model parameter determination and model verification

Model parameter determination

The model contains the initial variables $s_{y}$ and $p_{0}^{*}$, which can be obtained from the yield curve of unsaturated Q₃ loess on the p-s plane.

All the relevant parameters in the model can be obtained through tests, of which 18 parameters of the constitutive model in the loading process and 15 parameters of the constitutive model in the collapsing process are totaled. The relevant parameters are summarized as follows:

Parameters related to the CSL of undisturbed loess (2): $M$, $k_{c}$, which can be obtained by the triaxial drained shear test controlling s and $(\sigma_{3} - u_{a} )$ as constants; parameters related to CSL of remolded loess (2) : $M^{ * }$, $k_{c}^{ * }$, the same as above; parameters related to the initial yield surface of remolded loess LC (5) : $\kappa^{ * }$, $\lambda (0)^{ * }$, $\gamma^{ * }$, $\beta^{ * }$, $p^{c * }$, can be obtained by controlling the isotropic compression test of s, the acquired parameters are calibrated using Eqs. (32), (33), (35), (52), and (53); parameters related to the SI yield surface of undisturbed loess (2) : $\kappa_{s}$, $\lambda_{s}$, which can be obtained by controlling the triaxial shrinkage test of p; shear modulus related to the elastic deformation of undisturbed loess (1) : G, which can be obtained by controlling the triaxial drained shear test with s and $\sigma_{3} - u_{a}$ as constants ; parameters related to the structural damage evolution equation of intact loess (6 CT-triaxial shear tests and 3 CT-triaxial immersion tests): $\alpha_{1}$ ~ $\alpha_{6}$, can be determined by CT-triaxial shear tests with controlled axial pressure and confining pressure; $\alpha_{1}^{ * }$ ~ $\alpha_{3}^{ * }$, which can be determined by CT-triaxial immersion test with controlled axial pressure and confining pressure.

The initial state variables and related model parameters determined based on triaxial shear test and CT-triaxial loading-collapsible test of undisturbed and remolded loess are shown in Tables 1, 2.

Table 1 Initial state variables of soil samples.

Table 2 Parameters of the improved BBM.

LC yield line parameters and SI yield line parameters can refer to the results of isotropic compression test and triaxial shrinkage test of Q₃ undisturbed and remolded loess in Heping Township, Lanzhou³⁴. Through the above analysis and generalization, the MBBM parameters are obtained.

Preliminary verification of the model

The relevant parameters are substituted into the elastoplastic damage constitutive model, and the triaxial compression and immersion tests are calculated to verify the accuracy of the model.

During the calculation, the load is applied step by step at the corresponding value of the stress increment dp (dq or ds). The first step is to determine whether yielding has occurred based on the yield surface equation f₁ during loading or collapsing, and to determine the yield surface where yielding has occurred. If yield, the elastic volumetric strain increment $\varepsilon_{v}^{e}$ and deviatoric strain increment $\varepsilon_{s}^{e}$ are calculated according to Eq. (1) and Eq. (2), and the plastic volumetric strain increment $\varepsilon_{v}^{p}$ and deviatoric strain increment $\varepsilon_{s}^{p}$ are calculated according to Eqs. (19) and (21). Finally, the structural evolution variable values D₁ and D₂ are calculated according to Eqs. (40) and (55), respectively. The above steps are used as a calculation cycle, and the calculated structural damage variable value and hardening variable value in the new loading or collapsing process are substituted into the next step to carry out the next load calculations until the calculation at all levels is completed.

Test 1: Triaxial compression test

The model is used to calculate the Consolidation drainage (CD) test with control s and $(\sigma_{3} { - }u_{a} )$ as constants. According to the results of CT-triaxial compression test, the initial structural parameters of undisturbed samples with net confining pressure of 50 kPa, 100 kPa and 200 kPa are 0.980, 0.916 and 0.842, respectively. Since s is a constant, there is no ds increment, only dp and dq increments. The increment is applied step by step at 5 kPa during the calculation process, and the original test data and model calculation results can be obtained as shown in Fig. 1. As illustrated in the figure, under different suction conditions, when the axial strain is less than 2% to 4%, the axial strain of the undisturbed sample increases gradually with the increase in deviatoric stress. Subsequently, a distinct inflection point appears in the curve, and the axial strain exhibits rapid growth within a small range of deviatoric stress increment, indicating that the specimen demonstrates hardening and shear contraction characteristics. Specifically, at a matrix suction of 50 kPa, the specimen shows a weak hardening trend; while within the range of 100 to 200 kPa, it exhibits a strong hardening trend. By comparing the model calculation results with the experimental data, it is found that all models can reflect the general trend of structural damage in undisturbed loess during shearing. However, the elastoplastic constitutive model that does not account for structural effects shows a more pronounced strong hardening trend. The constitutive model developed in this study, which considers the influence of structure during the loading process, better captures the stress–strain behavior of Q₃ loess during shearing, yielding satisfactory results. Additionally, in the low-strain state, the fitting performance of both models is suboptimal, likely due to the fact that at this stage, the undisturbed specimen undergoes both elastic deformation and structural damage, leading to poor model fit.

Fig. 1

Comparison of model fitting curves and triaxial compression test data (UL = undisturbed loess).

Test 2: triaxial pressure immersion test

The model is used to calculate the triaxial immersion test with control $(\sigma_{3} { - }u_{a} )$ as a constant. Since the structure of the sample is less disturbed in the consolidation stage, the structure influence can not be taken into account, so the elastic increment part in the model is used to calculate the volumetric strain. During the immersion, with the increase of confining pressure, the structure of the sample is gradually damaged under hydro-mechanical effect. Therefore, it is necessary to calculate the collapse volumetric strain according to the collapse constitutive model considering the structural influence. According to the results of CT-triaxial collapse test, the initial structural parameters of the undisturbed samples with net confining pressures of 50, 100, 200 and 300 kPa are 0.985, 0.962, 0.930 and 0.872, respectively. Since the influence of s and q is not considered during the immersion process, the ds and dq increments do not exist, only the dp increment. The dp increment is applied step by step according to 20 kPa during the calculation process, and the original test data and model calculation results can be obtained as shown in Fig. 2. As illustrated in Fig. 2, there is a minimal discrepancy between the experimental results and the model predictions. The coefficient of determination for the experimental curves consistently exceeds 94%, indicating that the established model exhibits high reliability and validity.

Fig. 2

Comparison of model fitting curves and triaxial pressure immersion test data.

Li³⁵ conducted a triaxial pressure immersion test on the undisturbed loess Q₃ located in Lanzhou. Taking s = 100 kPa as an example, and applied the model to calculate the collapse volumetric strain under the same immersion amount. Samples with net confining pressures of 100 and 200 kPa were selected, and the initial structural parameters corresponding to the two samples were calculated to be 0.960 and 0.899 according to the measured CT number. The relevant parameters of the model are shown in³⁵.

The other parameters are the same as the previous ones. In the process of immersion, because s and q are constants, ds and dq increments do not exist, only dp increments are available, which are applied step by step according to 20 kPa during the calculations. Under the same amount of immersion water, the collapse volumetric strain calculated by the model is compared with the test results, as shown in Fig. 3. It can be seen from the curve trend in the figure that the model calculation results are basically close to the test results.

Fig. 3

Comparison of model fitting curves and triaxial pressure immersion test data.

From the above test examples, it can be seen that the constitutive model considering the structural effects can better reflects the trend of stress–strain changes caused by structural damage of the sample during the test than the model without considering the structural effects. The model considering the structural influence can basically fit the actual variation law of the sample but there are still some errors, mainly in the process of compression and immersion. Under the same suction, the calculation results of the structural model under low confining pressure are lower than the actual test data, and the coincidence degree under high confining pressure is relatively high. Combined with the CT-triaxial scans of the samples, it is found that the sample damage during compression is regional damage, and the deformation of the soil is also partial volume deformation. During the immersion process, the immersion deformation of the sample occurs in the immersion region. With the increase of the immersion amount, the collapsible deformation of the sample will gradually increase. Whether in compression or immersion process, the model is always to the sample as a whole as the object of calculation, thus resulting in the calculation of m selected value is large, so that m is small and D is large, that is, the calculated value of the sample structural damage is greater than the actual value, which makes the final simulation results too small.

Conclusion

Based on the modified Barcelona elastoplastic constitutive model of unsaturated soil, the structural evolution equations of unsaturated Q₃ loess during loading and collapsing are introduced to obtain a structural constitutive model describing the soil skeleton during loading and collapsing; the model can effectively reflect the original structural damage of the undisturbed loess during the process and also reflect the effects of the new structure generation on the mechanical deformation of undisturbed loess in the later stage of loading and immersion.

The new structural model adopts the non-associative flow rule and optimizes the authors’ established elastoplastic constitutive model, which is a big progress in describing the weak hardening mechanical characteristics of unsaturated loess in triaxial compression.

Taking the triaxial shear and collapse tests of unsaturated Q₃ undisturbed and remolded soils as examples, the structural constitutive model of unsaturated Q₃ loess is verified. The calculated results of the model are in good agreement with the experimental data, indicating that the model has certain accuracy in reflecting the mechanical deformation response of the structure to the unsaturated Q₃ loess during loading and collapsing process.