Table 1 VE–VP algorithm.

From: Publisher Correction: A viscoelastic–viscoplastic constitutive model for polymer bonded explosives under low impact loading

Input: \(\Delta e_{ij}^{ve}\), \(\Delta \varepsilon_{V}^{ve}\), \(\Delta t\); Params: \(G^{\left( n \right)}\), \(K^{\left( n \right)}\), \(g^{\left( n \right)}\), \(k^{\left( n \right)}\), \(\sigma_{y}\), h, \(\beta\), and a

Output: \({{\varvec{\upvarepsilon}}}^{vp}\), \({{\varvec{\upvarepsilon}}}^{ve}\), \(S_{ij}^{{}} \left( {t_{u + 1} } \right)\), \(\sigma_{V}^{{}} \left( {t_{u + 1} } \right)\)

1: Read the strain tensor at the (u + 1)th step

2: Compute the trial stress tensor using Eqs. (25) and (26)

3: Compute the incremental relaxation modulus:

\(\left\{ \begin{array}{*{20}l} \tilde{G}^{{_{\left( n \right)} }} = \left[ {1 - \exp \left( { - \frac{\Delta t}{{\tau^{\left( n \right)} }}} \right)} \right]\frac{{g^{\left( n \right)} }}{\Delta t} \hfill \\ \tilde{K}^{\left( n \right)} = \left[ {1 - \exp \left( { - \frac{\Delta t}{{\tau^{\left( n \right)} }}} \right)} \right]\frac{{k^{\left( n \right)} }}{\Delta t} \hfill \\ \tilde{G} = G^{(\infty )} + \sum\limits_{n = 1}^{N} {G^{\left( n \right)} \tilde{G}^{{^{\left( n \right)} }} } \hfill \\ \tilde{K} = K^{(\infty )} + \sum\limits_{n = 1}^{N} {K^{\left( n \right)} \tilde{K}^{{^{\left( n \right)} }} } \hfill \\ \end{array} \right.\)

3: Compute the von Mises yield function f:

\(f = \sigma_{eq}^{trial} - \sigma_{y} = \left( {\frac{3}{2}S_{ij}^{trial} :S_{ij}^{trial} } \right) - \sigma_{y}\)

4: If \(f \ge 0\), compute the plastic strain rate; otherwise, go to step (1)

5: Use New iteration to compute the effective plastic strain increment as follows:

\(\left\{ \begin{array}{*{20}l} \psi_{\alpha } = \sigma_{eq}^{trial} - 3\tilde{G}\Delta p - \sigma_{eq} \hfill \\ \psi_{\gamma } = \Delta p - \phi \Delta t \hfill \\ \phi_{\Delta p} = - 3\tilde{G}a\beta \cosh \left[ {\beta \left( {\sigma_{eq}^{trial} - 3\tilde{G}\Delta p - hp - \sigma_{y} } \right)} \right] \hfill \\ \phi_{\sigma } = - a\beta \cosh \left[ {\beta \left( {\sigma_{eq}^{trial} - 3\tilde{G}\Delta p - hp - \sigma_{y} } \right)} \right] \hfill \\ d\Delta p = \frac{{\phi_{eq} \psi_{\sigma } \Delta t - \psi_{\varepsilon } }}{{1 - \phi_{\Delta p} \Delta t + 3\tilde{G}\phi_{eq} \Delta t}} \hfill \\ \Delta p^{{\left( {r + 1} \right)}} = \Delta p^{(r)} + d\Delta p \hfill \\ \end{array} \right.\)

6: Compute the nominal VP strain tensor as follows:

\(\left\{ \begin{array}{*{20}l} {{\varvec{\upvarepsilon}}}^{vp} = \frac{3}{2}\Delta p\frac{{{\mathbf{S}}^{trial} }}{{\sigma_{eq}^{trial} }} \hfill \\ {{\varvec{\upvarepsilon}}}^{ve\_new} = {{\varvec{\upvarepsilon}}} - {{\varvec{\upvarepsilon}}}^{vp} \hfill \\ \end{array} \right.\)

7: Compute the nominal stress tensor:

\( \begin{aligned} S_{{ij}}^{{new}} \left( {t_{{u + 1}} } \right) = & S_{{ij}}^{{\left( \infty \right)}} \left( {t_{u} } \right) + 2G^{{\left( \infty \right)}} \Delta e_{{ij}}^{{ve\_new}} \\ & + \sum\limits_{{n = 1}}^{N} {\exp \left( {\frac{{ - \Delta t}}{{\tau ^{{\left( n \right)}} }}} \right)S_{{ij}}^{{\left( n \right)}} \left( {t_{u} } \right)} \\ & + \sum\limits_{{n = 1}}^{N} {2g^{{\left( n \right)}} \frac{{g^{{\left( n \right)}} }}{{\Delta t}}\left[ {1 - \exp \left( {\frac{{ - \Delta t}}{{\tau ^{{\left( n \right)}} }}} \right)} \right]\Delta e_{{ij}}^{{ve\_new}} } \\ \end{aligned} \)

\( \begin{aligned} \sigma _{V}^{{new}} \left( {t_{{u + 1}} } \right) = & \sigma _{V}^{{\left( \infty \right)}} \left( {t_{u} } \right) + 3K^{{\left( \infty \right)}} \Delta \varepsilon _{V}^{{ve\_new}} \\ & + \sum\limits_{{n = 1}}^{N} {\exp \left( {\frac{{ - \Delta t}}{{\tau ^{{\left( n \right)}} }}} \right)\sigma _{V}^{{\left( n \right)}} \left( {t_{u} } \right)} \\ & + \sum\limits_{{n = 1}}^{N} {3K^{{\left( n \right)}} \frac{{\tau ^{{\left( n \right)}} }}{{\Delta t}}\left[ {1 - \exp \left( {\frac{{ - \Delta t}}{{\tau ^{{\left( n \right)}} }}} \right)} \right]\Delta \varepsilon _{V}^{{ve\_new}} } \\ \end{aligned} \)

Back to article page