On the Simulation of the Cyclic Mobility Effect with an ISA-Hypoplastic Model

Abstract

The ISA-Hypoplasticity corresponds to an extended version of conventional Hypoplasticity to enable the simulation of some observed effects on cyclic loading. This extension offers novel features compared to the intergranular strain theory by Herle and Niemunis [4], including the incorporation of an elastic strain amplitude, to separate the elastic and plastic response, and the ability to reduce the plastic accumulation rate upon a larger number of cycles ($N>10$). In the present work, a modification to the ISA-hypoplastic model is described in order to enable the simulation of cyclic mobility effects exhibited by granular materials. The modification is based on a new state variable, able to detect paths at which the cyclic mobility effect is activated. With this information, some factors of the ISA-hypoplastic model are modified to deliver the proper response on paths showing cyclic mobility effects. Simulations examples are given to illustrate the new mechanism and a short analysis of the new parameters is also included.

Appendices

Appendix 1

Appendix 1 presents a summary of the constitutive equations of the ISA-hypoplastic model. Details of the equations below are found in [2, 5, 9].

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$$\begin{aligned} \mathsf {M} = \left\{ \begin{array}{ll} [ m_{R} + (1-m_{R})y_{h} ] (\mathsf {L}^\mathrm{hyp} + \rho ^{\chi } \mathbf {N}^\mathrm{hyp}\otimes \mathbf {N}) &{} \text {for } F_H\ge 0\\ m_{R}\mathsf {L}^\mathrm{hyp} &{} \text {for } F_H < 0 \end{array} \right. \end{aligned}$$

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$$\begin{aligned} \dot{\lambda }_H=\dfrac{\langle \mathbf {N}:\dot{\varepsilon }\rangle }{1-\left( \dfrac{\partial F_H}{\partial \mathbf {c}}\right) :\bar{\mathbf {c}}} \end{aligned}$$

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$$\begin{aligned} \rho =1-\dfrac{\Vert \mathbf {h}_b-\mathbf {h}\Vert }{2R} ,\quad \text {with}\quad \mathbf {h}_b= R\mathbf {N}\end{aligned}$$

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$$\begin{aligned} m=m_R+(1-m_R)y_h \end{aligned}$$

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$$\begin{aligned} \chi = \chi _0+\varepsilon _\mathrm{acc}(\chi _\mathrm{max}-\chi _0) \end{aligned}$$

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The set of parameters are R, $\chi _0$, $\chi _\mathrm{max}$, $m_R$, $\beta _0$, $\beta _\mathrm{max}$ and $C_a$.

Appendix 2

In the present appendix, the remaining equations of the reference hypoplastic model [9] are given:

$$\begin{aligned} {\mathsf{L}}^\mathrm{hyp}=f_bf_e\dfrac{1}{\hat{{\varvec{\sigma }}} :\hat{{\varvec{\sigma }}}} (F^2\mathsf{I}+a^2\hat{{\varvec{\sigma }}}\hat{{\varvec{\sigma }}}) \end{aligned}$$

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$$\begin{aligned} {\mathbf {N}}^\mathrm{hyp}=f_df_bf_e\dfrac{Fa}{\hat{{\varvec{\sigma }}}:\hat{{\varvec{\sigma }}}}(\hat{{\varvec{\sigma }}}+\hat{{\varvec{\sigma }}}^\mathrm{dev}) \end{aligned}$$

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$$\begin{aligned} F=\sqrt{\dfrac{1}{8}\tan ^2(\psi )+\dfrac{2-\tan ^2(\psi )}{2+2\sqrt{2}\tan (\psi )\cos (3\theta )}}-\dfrac{1}{2\sqrt{2}\tan (\psi )} \end{aligned}$$

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The set of parameters are $\varphi _c$, $h_s$, $n_B$, $e_{i0}$, $e_{c0}$, $e_{d0}$, $\alpha$ and $\beta$.