A rate-dependent model for sand to predict constitutive response and instability onset

Abstract

A constitutive model has been proposed for predicting rate-dependent stress–strain response of sand and further implemented in finite element code to explore the influence of strain rate on the localization behavior of sand. The proposed model simulates various constitutive features of sand subjected to higher strain rates, e.g., enhanced shear strength, early peak followed by a softening response, reduced compression for loose sand, etc., which have been reported in the literature. Numerical simulations predict a delayed onset of strain localization and increase in the band angle with increasing strain rate. Strains are noticed to localize in the hardening regime for loose sand, whereas for denser state localization emerges in the post-peak regime.

Appendix

In case of plastic or viscoplastic material response, the nonlinear stress–strain relations are generally evaluated incrementally. Following the approach given in Wang et al. [52], this section presents a one-step Euler stress-update algorithm for the proposed Perzyna-type viscoplastic model. In case of small deformation formulation, the incremental strain $\Delta \varepsilon$ can be decomposed into an elastic part $\Delta \varepsilon^{e}$ and a viscoplastic part $\Delta \varepsilon^{vp}$ according to

$$\Delta \varepsilon = \Delta \varepsilon^{e} + \Delta \varepsilon^{vp}.$$

(8)

In case of proposed model, flow rule is defined by

$$\dot{\varepsilon }^{vp} = \dot{\mathcal{\varepsilon }}_{\text{ref}} \mathcal{\phi }\bar{n},{\text{ where}}\,\bar{n} = {{\partial g_{d} } \mathord{\left/ {\vphantom {{\partial g_{d} } {\partial \sigma^{'} }}} \right. \kern-0pt} {\partial \sigma^{'} }}\,{\text{and}}\,\phi = \left( {{{\eta_{yd} } \mathord{\left/ {\vphantom {{\eta_{yd} } {\eta_{ys} }}} \right. \kern-0pt} {\eta_{ys} }}} \right)^{n}.$$

(9)

The incremental stress is related to the elastic response by

$$\Delta \sigma^{'} = E(\Delta \varepsilon - \Delta \varepsilon^{vp} )\,\,,$$

(10)

where $E$ represents the elastic stiffness tensor. A generalized trapezoidal rule can be applied to estimate the incremental viscoplastic strain and change in the internal state variable

$$\begin{aligned} \Delta \varepsilon^{vp} & = \left[ {(1 - \bar{\mathcal{\theta }})\dot{\varepsilon }_{t}^{vp} + \bar{\mathcal{\theta }}\dot{\varepsilon }_{t + \Delta t}^{vp} } \right]\Delta t\,\,, \\ \Delta \bar{\mathcal{\kappa }} & = \left[ {(1 - \bar{\mathcal{\theta }})\mathcal{\dot{\bar{\kappa }}}_{\mathcal{t}} + \mathcal{\bar{\theta }\dot{\bar{\kappa }}}_{t + \Delta t} } \right]\Delta t\,\,, \\ \end{aligned}$$

(11)

where $\bar{\kappa }$ denotes the internal variable of the viscoplastic model, and for the proposed model, two internal variables are considered, i.e., $\Delta \bar{\mathcal{\kappa }}_{{1}} = \varepsilon_{q}^{vp}$ and $\Delta \bar{\mathcal{\kappa }}_{{2}} = \varepsilon_{p}^{vp}$. The interpolation parameter, $\bar{\theta }$, is such that $0 \le \bar{\theta } \le 1$, where $\bar{\theta } = 0$ implies a complete explicit method and $\bar{\theta } = 1$ stands for the fully implicit method. In the one-step Euler integration scheme, the viscoplastic strain rate at the end of the time interval $\Delta t\,$ is expressed in a limited Taylor series expansion as

$$\begin{aligned} \dot{\varepsilon }_{t + \Delta t}^{vp} & \varvec{ = }\dot{\varepsilon }_{t}^{vp} + \left[ {\frac{{\partial \dot{\varepsilon }^{vp} }}{{\partial \sigma^{'} }}} \right]_{t} \Delta \sigma^{'} + \left[ {\frac{{\partial \dot{\varepsilon }^{vp} }}{{\partial \bar{\mathcal{\kappa }}}}} \right]_{t} \Delta \bar{\mathcal{\kappa }} \\ & \varvec{ = }\dot{\varepsilon }_{t}^{vp} + \hat{G}_{t} \Delta \sigma^{'} + h_{t} \Delta \bar{\mathcal{\kappa }}\,\,, \\ \end{aligned}$$

(12)

$${\text{where}}\,\hat{G}_{t} = \dot{\mathcal{\varepsilon }}_{\text{ref}} \left[ {\frac{{\partial \mathcal{\phi }}}{{\partial \sigma^{'} }}\bar{n}^{{T}} + \mathcal{\phi }\frac{{\partial \bar{n}}}{{\partial \sigma^{'} }}} \right]_{{t}}, \quad {}h_{t} = \dot{\mathcal{\varepsilon }}_{\text{ref}} \left[ {\frac{{\partial \mathcal{\phi }}}{{\partial \bar{\mathcal{\kappa }}}}\bar{n}} \right]_{{t}}.$$

(13)

Substitution of Eq. 12 into Eq. 11 yields

$$\Delta \varepsilon^{vp} \varvec{ = }\left( {\dot{\varepsilon }_{t}^{vp} + \bar{\mathcal{\theta }}\hat{G}_{{t}} \Delta \sigma^{'} + \bar{\mathcal{\theta }}h_{{t}} \Delta \bar{\mathcal{\kappa }}} \right)\Delta {t}\,.$$

(14)

Further substitution of Eq. 14 into the incremental stress–strain relation of Eq. 10 leads to the following relation:

$$\Delta \sigma^{'} = D_{c} \Delta \varepsilon - \Delta \bar{q}\,\,,$$

(15)

$${\text{where}}\,D_{c} = \left[ {E^{ - 1} + \bar{\mathcal{\theta }}\Delta {t}\hat{G}_{{t}} } \right]^{ - 1} \,\,,\,\,\Delta \bar{q} = E\left( {\dot{\varepsilon }_{t}^{vp} \Delta {t + \bar{\theta }}\Delta {t}h_{{t}} \Delta \bar{\mathcal{\kappa }}} \right)\,\,.$$

(16)

The tensor $D_{c}$ is the algorithmic tangent stiffness tensor. Following are the expressions derived for updating the stress–strain relation of the proposed model

$$\begin{aligned} \frac{{\partial g_{d} }}{{\partial \sigma_{ij}^{'} }} & = \sqrt {\frac{3}{2}} \frac{{S_{ij} }}{{\sqrt {S_{kl} S_{kl} } }} + \frac{{\delta_{ij} }}{3}\left( {M_{\text{c}} - \eta_{\text{yd}} } \right),{\text{where}}\,S_{ij} = \sigma_{ij}^{'} - \frac{{\sigma_{kk}^{'} \delta_{ij} }}{3},\eta_{\text{yd}} = \frac{{3\sqrt {3J_{2} } }}{{I_{1}^{'} }}\,\,, \\ \frac{{\partial^{2} g_{d} }}{{\partial \sigma_{ij}^{'} \partial \sigma_{pq}^{'} }} & = \sqrt {\frac{3}{2}} \left[ { - \frac{{S_{ij} S_{pq} }}{{2\left( {S_{kl} S_{kl} } \right)^{3/2} }} + \frac{{\delta_{pi} \delta_{qj} - {{\delta_{ij} \delta_{pq} } \mathord{\left/ {\vphantom {{\delta_{ij} \delta_{pq} } 3}} \right. \kern-0pt} 3}}}{{\sqrt {S_{kl} S_{kl} } }}} \right] - \frac{{\delta_{ij} }}{3}\frac{{\partial \eta_{\text{yd}} }}{{\partial \sigma_{pq}^{'} }}\,\,, \\ \frac{{\partial \eta_{\text{yd}} }}{{\partial \sigma_{pq}^{'} }} & = \sqrt {\frac{3}{2}} \frac{{3S_{pq} }}{{I_{1}^{'} \sqrt {S_{kl} S_{kl} } }} - \frac{{3\sqrt {3J_{2} } \delta_{pq} }}{{\left( {I_{1}^{'} } \right)^{2} }}\,\,,\hat{G}_{{{ijpq}}} { = n}\dot{\mathcal{\varepsilon }}_{\text{ref}} \frac{{\left( {\eta_{\text{yd}} } \right)^{n - 1} }}{{\left( {\eta_{\text{ys}} } \right)^{n} }}\frac{{\partial \eta_{\text{yd}} }}{{\partial \sigma_{pq}^{'} }}\frac{{\partial g_{d} }}{{\partial \sigma_{ij}^{'} }} + \dot{\mathcal{\varepsilon }}_{{\text{ref}}} \left( {\frac{{\eta_{\text{yd}} }}{{\eta_{\text{ys}} }}} \right)^{n} \frac{{\partial^{2} g_{d} }}{{\partial \sigma_{ij}^{'} \partial \sigma_{pq}^{'} }}\,\,, \\ {h}_{{{ij}}} \Delta \bar{{\kappa }} & { = - n}\dot{\mathcal{\varepsilon }}_{\text{ref}} \frac{{\left( {\eta_{\text{yd}} } \right)^{n} }}{{\left( {\eta_{\text{ys}} } \right)^{n + 1} }}\left[ {\frac{{\partial \eta_{\text{ys}} }}{{\partial \bar{\mathcal{\kappa }}_{{1}} }}\Delta \bar{\mathcal{\kappa }}_{{1}} + \frac{{\partial \eta_{\text{ys}} }}{{\partial \bar{\mathcal{\kappa }}_{{2}} }}\Delta \bar{\mathcal{\kappa }}_{{2}} } \right]\frac{{\partial g_{d} }}{{\partial \sigma_{ij}^{'} }}\,\,, \\ {\text{and}}\,\frac{{\partial \eta_{\text{ys}} }}{{\partial \bar{\mathcal{\kappa }}_{{1}} }} & = \frac{{\left( {\eta_{\text{ps}} - \eta_{\text{ys}} } \right)^{2} }}{{a\eta_{\text{ps}} }}\,,\,\,\frac{{\partial \eta_{\text{ys}} }}{{\partial \bar{\mathcal{\kappa }}_{{2}} }} = \frac{{\eta_{\text{ys}} }}{{\eta_{\text{ps}} }}\kappa v_{0} \,\,. \\ \end{aligned}$$

(17)

The stress increase for a given strain increment can be calculated from Eq. 16 along with Eq. 17, and subsequently, the viscoplastic strain increment can be estimated from Eq. 12. A complete explicit stress-point integration scheme has been adopted in the present work with $\bar{\theta } = 0$. It is important to note that such an explicit integration scheme, also known as forward Euler scheme, is conditionally stable and requires smaller discretization step while performing the time integration. Alternatively, an implicit-type time integration scheme, which is usually more robust and unconditionally stable, can also be employed for this purpose [10].