Multi-surface Cyclic Plasticity Sand Model with Lode Angle Effect

Abstract

A Drucker-Prager J ₂ multi-surface-plasticity sand model is modified to employ the Lade-Duncan failure criterion as the yield function. This function includes the first and third stress invariants to account for the dependence of cyclic shear stress–strain behavior on confining pressure and the Lode angle. Related modifications to the flow rule and hardening rule are described. Dependence of dilatancy on confinement is also included. Salient features of the model performance are presented under general three-dimensional (3D) loading conditions, where the yield function provides a more accurate representation of nonlinear shear response. Dynamic response analyses of a mildly inclined infinite slope are performed to illustrate the influence of excitation direction on the accumulation of liquefaction-induced lateral ground deformation.

Appendices

Appendix A: Incremental Yield Surface Translation Procedure

As shown in Fig. 19, we define S ^T as the intersection of the outer yield surface f _m + 1 and the vector connecting the active surface center p′α _m and the updated stress state S:

$$ \user2{S}^{T} = x(\user2{S} - {p}\ifmmode{'}\else$'$\fi\varvec{\alpha}_{m} ) + {p}\ifmmode{'}\else$'$\fi\varvec{\alpha}_{m} $$

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where x is a scalar (≥1) to be solved for. Considering that S ^T satisfies the yield function of the outer surface f _m+1, we obtain the following equation:

$$ f_{{m + 1}} = (\ifmmode\expandafter\bar\else\expandafter\=\fi{J}_{3} )^{T} - \frac{1} {3}(\eta _{{m + 1}} {\text{ }}I_{1} ){\text{ }}(\ifmmode\expandafter\bar\else\expandafter\=\fi{J}_{2} )^{T} + a_{1} {\text{ }}(\eta _{{m + 1}} {\text{ }}I_{1} )^{3} = 0 $$

(13)

where $(\bar{J}_{2} )^{T}$ and $(\bar{J}_{3})^{T}$ are respectively the second and third invariants of the stress vector S ^T  − p′α _m + 1. Equation 13 is a cubic equation in x with three real roots, of which the smallest positive root is the desired solution. Now that S ^T is defined, the direction of surface translation is given by:

$$ \varvec{\mu} = {\left( {\user2{S}^{T} - {p}\ifmmode{'}\else$'$\fi\varvec{\alpha}_{m} } \right)}{\text{ }} - {\text{ }}\frac{{\eta _{m} }} {{\eta _{{m + 1}} }}{\left( {\user2{S}^{T} - {p}\ifmmode{'}\else$'$ \fi\varvec{\alpha}_{{m + 1}} } \right)} $$

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Fig. 19

Definition of active surface translation rule in deviatoric stress space

This definition of surface translation follows the original hardening rule of Mroz (1967), with slight modification in order to enhance computational efficiency in the incremental numerical implementation (Elgamal et al. 2003).

The amount of surface translation dμ can then be obtained by enforcing the consistency condition. After updating the active surface f _m , all inner surfaces f _m−1, f _m−2, ..., f ₁ are translated to be tangential at the stress state S on f _m (Prevost 1985).

Appendix B: Evaluation of Stress Ratio

The stress ratio η associated with a stress state S is defined in such a way that a virtual yield surface passing by S (with α = 0) satisfies the yield function:

$$ J_{3} - \frac{1} {3}(\eta {\text{ }}I_{1} ){\text{ }}J_{2} + a_{1} {\text{ }}(\eta {\text{ }}I_{1} )^{3} = 0 $$

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Thus, η varies from 0.0 (on the hydrostatic line) to 1.0 (on the failure surface). Evaluation of η involves solving a cubic equation η ³ + aη ² + bη + c = 0 with:

$$ a = 0,\,\,b = \frac{{ - I_{1} {\text{ }}J_{2} }} {{3{\text{ }}a_{1} (I_{1} )^{3} }},\,\,c = \frac{{J_{3} }} {{a_{1} (I_{1} )^{3} }} $$

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This equation has three real roots, with the desired solution for η being the largest root.

Appendix C: Modification to Flow Rule

There are two modifications to the flow rule described in Yang et al. (2003), one for the contractive phase and the other for the dilative phase. The contractive phase was originally defined by (Eq. 6 in Yang et al. 2003):

$$ {\mathbf{P}}\ifmmode{''}\else$''$\fi = \left(1 - \hbox{sign}(\ifmmode\expandafter\dot\else\expandafter\.\fi{\eta })\frac{\eta } {{\eta _{{PT}} }}\right){\text{ }}(c_{1} + c_{2} {\text{ }}\varepsilon _{c} ) $$

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where c ₁ and c ₂ are positive calibration constants, and ε _c is a non-negative scalar. It was found that, for certain loading paths, the above expression results in an undesirable discontinuity in P″ (due to the $\hbox{sign}(\ifmmode\expandafter\dot\else\expandafter\.\fi{\eta })$ term). This defect is eliminated by employing the following alternative expression:

$$ {\mathbf{P}}^{\prime\prime} = \left(1 - \frac{\user2{n}\text{:}\dot{\user2{S}}}{\|\dot{\user2{S}}\|}\ \frac{\eta} {\eta _{PT}}\right)\ (c_{1} + c_{2} \ \varepsilon_{c}) ({p}^{\prime}/p_{atm})^{c_{3}} $$

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As shown in Fig. 20, n is unit outer normal to an imaginary surface passing by the stress point S (with α = 0). The new parameter c ₃ in Eq. 18 is introduced to represent dependence of pore pressure buildup on confinement (Kramer 1996, Manzari and Dafalias 1997), where p _atm is atmospheric pressure for the purpose of normalization.

Fig. 20

Schematic for modified flow rule

In the current model, the dilative phase (Eq. 8 in Yang et al. 2003) is modified to include a pressure-dependent term:

$$ {\mathbf{P}^{\prime\prime}} = (1 - \eta /\eta_{{PT}})d_{1}(\gamma _{d} )^{{d_{2} }} ({p^{\prime}}/p_{{atm}} )^{{d_{3}}}$$

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where d ₁, d ₂, and d ₃ are calibration constants, γ _d is the octahedral shear strain accumulated during this dilation phase (see Yang et al. 2003 for more details). The d ₃ term was included above to address the dependence of dilation tendency on confinement (Manzari and Dafalias 1997). It is noted that alternative approaches are available that combine contraction and dilation flow rules and ensure smooth transition under arbitrary 3D loading conditions (e.g., Manzari and Dafalias 1997; Dafalias and Manzari 2004).