A robust numerical framework for simulating localized failure and fracture propagation in frictional materials

Abstract

A computationally robust framework for simulating geomaterial failure patterns is presented in this paper. Finite element simulations which feature the use of embedded discontinuities to track material failure are known to suffer from convergence issues due to a lack of robustness. Oftentimes, complex time step-cutting schemes or arc-length methods are required in order to achieve convergence. This may invariably limit the complexity of constitutive models available for use in tracking nonlinear material behavior. To this end, we use an implicit–explicit integration scheme [Impl–Ex (Oliver et al. in Comput Methods Appl Mech Eng 195(52):7093–7114, 2006)] coupled with a novel constitutive model which allows for combined opening and shearing displacement in tension, as well as frictional sliding in compression. We show that this framework is suitable for capturing complex fracture patterns in geomaterial structures without the need for elaborate continuance schemes.

Appendices

Appendix 1: Slip algorithms for both newly localized band and preexisting slip on band

See Boxes 1 and 2.

Box 1 Slip algorithm for a newly localized element

Box 2 Slip algorithm for an element which has preexisting slip on the band

Appendix 2: Derivatives of the yield functions

The gradients of the yield functions are as follows:

$$\begin{aligned} \frac{ \partial \Phi _1}{\partial \zeta _n}& = \frac{\partial \sigma }{\partial \zeta _n} - \left[ \zeta _n\frac{\partial k_n}{\partial \zeta _n} + k_n \right] \end{aligned}$$

(10.1)

$$\begin{aligned} \frac{ \partial \Phi _1}{\partial \zeta _s}& = \frac{\partial \sigma }{\partial \zeta _s} - \zeta _n\frac{\partial k_n}{\partial \zeta _s} \end{aligned}$$

(10.2)

$$\begin{aligned} \frac{\partial \Phi _2}{\partial \zeta _n}& = \frac{\partial \tau }{\partial \zeta _n} - \zeta _s\frac{\partial k_s}{\partial \zeta _n} \end{aligned}$$

(10.3)

$$\begin{aligned} \frac{\partial \Phi _2}{\partial \zeta _s}& = \frac{\partial \tau }{\partial \zeta _n} - \left[ \zeta _s\frac{\partial k_s}{\partial \zeta _s} + k_s \right] \end{aligned}$$

(10.4)

where

$$\begin{aligned} \frac{\partial k_s}{\partial \zeta _n} = \left\{ \begin{array}{lll} - \dfrac{ \alpha _{\zeta }^{2} c_0}{\zeta _c^3} \zeta _n &\quad \texttt {if} & \zeta _{\mathrm{eq}} = \zeta _c \\ 0 &\quad \texttt {if} & \zeta _{\mathrm{eq}} < \zeta _c \end{array} \right. \end{aligned}$$

(10.5)

$$\begin{aligned} \frac{\partial k_s}{\partial \zeta _s} = \left\{ \begin{array}{lll} - \dfrac{ c_0}{\zeta _c^3} \zeta _s & \quad \texttt {if} & \zeta _{\mathrm{eq}} = \zeta _c \\ 0 & \quad \texttt {if} & \zeta _{\mathrm{eq}} < \zeta _c \end{array} \right. \end{aligned}$$

(10.6)

$$\begin{aligned} \frac{\partial k_n}{\partial \zeta _n}& = \frac{\alpha _\zeta }{\alpha _\sigma }\frac{\partial k_s}{\partial \zeta _n} \end{aligned}$$

(10.7)

$$\begin{aligned} \frac{\partial k_n}{\partial \zeta _s}& = \frac{\alpha _\zeta }{\alpha _\sigma }\frac{\partial k_s}{\partial \zeta _s} \end{aligned}$$

(10.8)

To determine the derivatives of the bulk stresses with respect to the jumps, recall

$$\begin{aligned} {\varvec{\sigma }}_{n+1} = {\varvec{\sigma }}_n + {\varvec{c}}^e : \Delta {\varvec{\epsilon }}^{\mathrm {conf}} - {\varvec{c}}^e : \left( \Delta {\varvec{\zeta }}\otimes \nabla f^h \right) ^s \end{aligned}$$

(10.9)

Hence,

$$\begin{aligned} \frac{\partial {\varvec{\sigma }}}{\partial {\varvec{\zeta }}_n}& = - {\varvec{c}}^e : \left( {\varvec{n}}\otimes \nabla f^h \right) ^s \end{aligned}$$

(10.10)

$$\begin{aligned} \frac{\partial {\varvec{\sigma }}}{\partial {\varvec{\zeta }}_s}& = - {\varvec{c}}^e : \left( {\varvec{l}}\otimes \nabla f^h \right) ^s \end{aligned}$$

(10.11)

and therefore

$$\begin{aligned} \frac{\partial \sigma }{\partial {\varvec{\zeta }}_n}& = - \left( {\varvec{n}}\otimes {\varvec{n}}\right) : {\varvec{c}}^e : \left( {\varvec{n}}\otimes \nabla f^h \right) ^s \end{aligned}$$

(10.12)

$$\begin{aligned} \frac{\partial \sigma }{\partial {\varvec{\zeta }}_s}& = - \left( {\varvec{n}}\otimes {\varvec{n}}\right) : {\varvec{c}}^e : \left( {\varvec{l}}\otimes \nabla f^h \right) ^s \end{aligned}$$

(10.13)

$$\begin{aligned} \frac{\partial \tau }{\partial {\varvec{\zeta }}_n}& = - \left( {\varvec{n}}\otimes {\varvec{l}}\right) ^s : {\varvec{c}}^e : \left( {\varvec{n}}\otimes \nabla f^h \right) ^s \end{aligned}$$

(10.14)

$$\begin{aligned} \frac{\partial \tau }{\partial {\varvec{\zeta }}_s}& = - \left( {\varvec{n}}\otimes {\varvec{l}}\right) ^s : {\varvec{c}}^e : \left( {\varvec{l}}\otimes \nabla f^h \right) ^s \end{aligned}$$

(10.15)

Appendix 3: Derivation of the effective algorithmic operator

The effective algorithmic operator is defined as

$$\begin{aligned} {\varvec{C}}^{\mathrm {eff}}_{n+1} = \frac{\partial \tilde{{\varvec{\sigma }}}_{n+1}}{\partial {\varvec{\epsilon }}_{n+1}} \end{aligned}$$

(11.1)

by taking the derivative of the semi-implicit stress in Eq. (6.15) with respect to the implicit strain, we have the following:

$$\begin{aligned} \frac{\partial \tilde{{\varvec{\sigma }}}_{n+1}}{\partial {\varvec{\epsilon }}_{n+1}} &=\frac{\partial {\varvec{\sigma }}_{n}}{\partial {\varvec{\epsilon }}_{n+1}} + {\varvec{c}}^{e}:\frac{\partial \left( {\varvec{\epsilon }}_{n+1} - {\varvec{\epsilon }}_{n}\right) }{\partial {\varvec{\epsilon }}_{n+1}}\\&\quad- \Delta \tilde{\lambda }_{n+1}{\varvec{c}}^{e}:\frac{\partial }{\partial {\varvec{\epsilon }}_{n+1}}\left( \frac{\partial g\left( \tilde{{\varvec{\sigma }}}_{n+1}\right) }{\partial \tilde{{\varvec{\sigma }}}_{n+1} }\right) \\\\ &={\varvec{c}}^{e}:\frac{\partial {\varvec{\epsilon }}_{n+1}}{\partial {\varvec{\epsilon }}_{n+1}} - \Delta \tilde{\lambda }_{n+1}{\varvec{c}}^{e}:\tilde{{\varvec{A}}}_{n+1}:\frac{\partial \tilde{{\varvec{\sigma }}}_{n+1}}{\partial {\varvec{\epsilon }}_{n+1}} \end{aligned}$$

(11.2)

where

$$\begin{aligned} \tilde{{\varvec{A}}}_{n+1} = \frac{\partial ^{2} g\left( \tilde{{\varvec{\sigma }}}_{n+1}\right) }{\partial \tilde{{\varvec{\sigma }}}_{n+1}\otimes \partial \tilde{{\varvec{\sigma }}}_{n+1}} \end{aligned}$$

(11.3)

Noting that ${\varvec{c}}^{e}:\frac{\partial {\varvec{\epsilon }}_{n+1}}{\partial {\varvec{\epsilon }}_{n+1}} = {\varvec{c}}^{e}:{\varvec{I}}= {\varvec{c}}^{e}$, we group terms and factor:

$$\begin{aligned} \frac{\partial \tilde{{\varvec{\sigma }}}_{n+1}}{\partial {\varvec{\epsilon }}_{n+1}}&= {\varvec{c}}^{e} - \Delta \tilde{\lambda }_{n+1}{\varvec{c}}^{e}:\tilde{{\varvec{A}}}_{n+1}:\frac{\partial \tilde{{\varvec{\sigma }}}_{n+1}}{\partial {\varvec{\epsilon }}_{n+1}}\\ \\ {\varvec{c}}^{e}&= \frac{\partial \tilde{{\varvec{\sigma }}}_{n+1}}{\partial {\varvec{\epsilon }}_{n+1}} + \Delta \tilde{\lambda }_{n+1}{\varvec{c}}^{e}:\tilde{{\varvec{A}}}_{n+1}:\frac{\partial \tilde{{\varvec{\sigma }}}_{n+1}}{\partial {\varvec{\epsilon }}_{n+1}}\\ \\ {\varvec{c}}^{e}&= \left( {\varvec{I}}+ \Delta \tilde{\lambda }_{n+1}{\varvec{c}}^{e}:\tilde{{\varvec{A}}}_{n+1}\right) :\frac{\partial \tilde{{\varvec{\sigma }}}_{n+1}}{\partial {\varvec{\epsilon }}_{n+1}} \end{aligned}$$

finally

$$\begin{aligned} \frac{\partial \tilde{{\varvec{\sigma }}}_{n+1}}{\partial {\varvec{\epsilon }}_{n+1}} = \left( {\varvec{I}}+ \Delta \tilde{\lambda }_{n+1} {\varvec{c}}^{e}:\tilde{{\varvec{A}}}_{n+1}\right) ^{-1}:{\varvec{c}}^{e} = {\varvec{C}}^{\mathrm {eff}}_{n+1} \end{aligned}$$

(11.4)

The calculation in Eq. 11.4 is possible only if the quantities are in matrix form, and hence, the fourth-order symmetric identity tensor, ${\varvec{I}}$, must be written as the identity matrix given by

$$\begin{aligned} {\varvec{I}}= \begin{bmatrix} 1&\quad 0&\quad 0&\quad 0&\quad 0&\quad 0\\ 0&\quad 1&\quad 0&\quad 0&\quad 0&\quad 0\\ 0&\quad 0&\quad 1&\quad 0&\quad 0&\quad 0\\ 0&\quad 0&\quad 0&\quad \frac{1}{2}&\quad 0&\quad 0\\ 0&\quad 0&\quad 0&\quad 0&\quad \frac{1}{2}&\quad 0\\ 0&\quad 0&\quad 0&\quad 0&\quad 0&\quad \frac{1}{2}\\ \end{bmatrix} \end{aligned}$$

(11.5)

and additionally, ${\varvec{c}}^{e}$ and $\tilde{{\varvec{A}}}_{n+1}$ should be written as $6 \times 6$ matrices. After the quantities are multiplied, the result can be reduced to a $3 \times 3$ matrix which is suitable for the case of plane strain.