Development of Constitutive Model for Simulation of Cemented Soil

Abstract

Classical soil constitutive models fail to describe the strain-softening response of cemented soil at a low stress level. This paper proposes a constitutive model that incorporates the Modified Cam Clay model to describe pressure-dependent soil stiffness and strength corresponding to volume changes into (1) empirical functions to recognize the mobilized friction and initial cementation as evolutions of the deviatoric plastic strain and (2) a power-type compressibility function to track the effective stress-dependent volumetric response. In this case, the no additional complexity related to the model formulation is added. The cement-treated constitutive model is implemented to simulate the drained triaxial test. The results indicate that the initial state of isotropic stress and the cement degradation rate defines the stress–strain-volume response, whereas the deviatoric stress eventually approaches the critical state strength.

Appendix: Mathematical Formulation of the Constitutive Model

A summary of the proposed constitutive model is presented here. The plastic strain increments are employed following standard procedures of the theory of hardening plasticity.

1.1 Elasto-plastic Constitutive Relation

The general consistency equation related to the state parameter can be expressed as follows:

$$\partial {\text{f}} = \left( {\frac{{\partial {\text{f}}}}{{\partial\upsigma}}} \right)^{\text{T}} \partial\upsigma + \left( {\frac{{\partial {\text{f}}}}{{\partial\upalpha}}} \right)^{\text{T}} \partial\upalpha = 0$$

(10)

Here σ (p′_c and p′_cto) is the stress state parameter updated with the evolution of the yield surface and the state parameter α is characterized by the plastic deformations consisting of the plastic multiplier δχ and hardening function h:

$$\partial\upalpha = {\text{h}}\left( {\upsigma,\upalpha} \right)\partial\upchi$$

(11)

By inserting Eq. (11) into Eq. (10), the consistency equation can be reformulated with the hardening modulus H:

$$\partial {\text{f}} = \left( {\frac{{\partial {\text{f}}}}{{\partial\upsigma}}} \right)^{T} \partial\upsigma - {\text{H}}\partial\upchi = 0 \quad {\text{where}}\quad {\text{H}} = {\text{H}}_{1} \cdot {\text{H}}_{2} = - \left( {\frac{{\partial {\text{f}}}}{{\partial\upalpha}}} \right)^{\text{T}} \cdot \frac{{\partial\upalpha}}{{\partial\upchi}}$$

(12)

The elastic constitutive relation can be expressed by the elastic constitutive matrix and plastic strain:

$$\updelta \upsigma = {\text{C}}\left( {\updelta \upvarepsilon -\updelta \upvarepsilon ^{\text{p}} } \right) = {\text{C}}\left( {\updelta \upvarepsilon - \partial\upchi\frac{{\partial {\text{f}}}}{{\partial\upsigma}}} \right)$$

(13)

Inserting the stress increment from Eq. (12) into the constitutive relation Eq. (13), the plastic multiplier can be derived:

$$\partial\upchi = \frac{{\left( {{\text{C}}\frac{{\partial {\text{f}}}}{{\partial\upsigma}}} \right)^{\text{T}} }}{{{\text{H}} + \left( {\frac{{\partial {\text{f}}}}{{\partial\upsigma}}} \right)^{\text{T}} {\text{C}}\left( {\frac{{\partial {\text{f}}}}{{\partial\upsigma}}} \right)}}\updelta \upvarepsilon$$

(14)

The continuum tangent modulus related to incremental stress and strain level is derived with the plastic multiplier that involves the hardening response and variation of state parameter

$$\updelta \upsigma = {\text{C}}\left( {\updelta \upvarepsilon - \partial\upchi\frac{{\partial {\text{f}}}}{{\partial\upsigma}}} \right) = \left[ {{\text{C}} - \frac{{\left( {{\text{C}}\frac{{\partial {\text{f}}}}{{\partial\upsigma}}} \right)\left( {{\text{C}}\frac{{\partial {\text{f}}}}{{\partial\upsigma}}} \right)^{\text{T}} }}{{{\text{H}} + \left( {\frac{{\partial {\text{f}}}}{{\partial\upsigma}}} \right)^{\text{T}} {\text{C}}\left( {\frac{{\partial {\text{f}}}}{{\partial\upsigma}}} \right)}}} \right]\updelta \upvarepsilon$$

(15)

By taking the derivatives of yield function into account with the stress state parameters, several components of the constitutive matrix can be defined:

$$\frac{{\partial {\text{f}}}}{{\partial {\text{p}}^{'} }} = 2\left( {{\text{p}}^{'} -\upbeta} \right){\text{M}}_{\text{cs}}^{2}$$

(16)

$$\frac{{\partial {\text{f}}}}{{\partial {\text{q}}}} = 2{\text{q}}$$

(17)

$$\frac{{\partial {\text{f}}}}{{\partial {\text{p}}_{\text{c}}^{'} }} = - \left( {{\text{p}}^{'} -\upbeta} \right){\text{M}}_{\text{cs}}^{2} -\upalpha{\text{M}}_{\text{cs}}^{2}$$

(18)

$$\frac{{\partial {\text{f}}}}{{\partial {\text{p}}_{\text{ct}}^{'} }} = - \left( {{\text{p}}^{'} -\upbeta} \right){\text{M}}_{\text{cs}}^{2} +\upalpha{\text{M}}_{\text{cs}}^{ 2}$$

(19)

Thus, the components of hardening modulus H can be determined with the following equation:

$${\text{H}}_{1} = - \left[ {\begin{array}{*{20}c} {\mathop {\frac{{\partial {\text{f}}}}{{\partial {\text{p}}_{\text{c}}^{'} }}}\limits_{{}} } \\ {\mathop {\frac{{\partial {\text{f}}}}{{\partial {\text{p}}_{\text{ct}}^{'} }}}\limits^{{}} } \\ \end{array} } \right]^{\text{T}} = - \left[ {\begin{array}{*{20}c} {\mathop {\left( {{\text{p}}^{'} -\upbeta} \right){\text{M}}_{\text{cs}}^{ 2} -\upalpha{\text{M}}_{\text{cs}}^{ 2} }\limits_{{}} } \\ {\mathop {\left( {{\text{p}}^{'} -\upbeta} \right){\text{M}}_{\text{cs}}^{ 2} +\upalpha{\text{M}}_{\text{cs}}^{ 2} }\limits^{{}} } \\ \end{array} } \right]^{\text{T}}$$

(20)

$${\text{H}}_{2} = - \left[ {\begin{array}{*{20}c} {\mathop {\frac{{\partial {\text{p}}_{\text{c}}^{'} }}{{\partial\upchi}}}\limits_{{}} } \\ {\mathop {\frac{{\partial {\text{p}}_{\text{ct}}^{'} }}{{\partial\upchi}}}\limits^{{}} } \\ \end{array} } \right]^{T} = - \left[ {\begin{array}{*{20}c} {\mathop {\frac{{\upnu{\text{p}}_{\text{c}}^{'} }}{{\uplambda -\upkappa}} \cdot \frac{{\partial {\text{f}}}}{{\partial {\text{p}}^{'} }}}\limits_{{}} } \\ {\mathop {{\text{p}}_{\text{cto}}^{'} \cdot {\text{k}}_{\text{p}} \cdot {\text{e}}^{{ - {\text{k}}_{\text{p}}\upvarepsilon_{\text{q}}^{\text{p}} }} \cdot \frac{{\partial {\text{f}}}}{{\partial {\text{q}}}}}\limits^{{}} } \\ \end{array} } \right]^{T}$$

(21)