Nonlinear elastoplastic analysis of pressure sensitive materials

Abstract

When structures undergo extreme loading conditions, the materials pass the elastic limits. Therefore, to preserve economy as well as safety, it is essential to perform a realistic elastoplastic analysis using the constitutive equations in plasticity. On the other hand, computing the stress alongside its associated variables on Gauss points is a delicate process and virtually the most important part of these analyses. In this study, an efficient stress-updating technique is presented for the constitutive rate equations of the pressure sensitive materials such as concrete, rock, soil and some kind of metals. Accordingly, the Drucker–Prager plasticity is utilized to consider the hydrostatic pressure in addition to the J ₂-invariant of the deviatoric stress. Moreover, the isotropic and kinematic hardenings are used to take into account more realistic behavior of the materials. Finally, a wide range of numerical tests is carried out to show the performance of the presented method together with the application of the suggested formulations in elastoplastic analysis of a gravity dam.

Appendices

Appendix A: Computing the factor to determine the complete-elastic and elastic–plastic parts of the load step

The scalar factor $\alpha$ addressed in stress-computing method section is computed by solving a second-order equation obtained from $F({\mathbf{s^{\prime}}}_{n} + \alpha 2G\Delta {\mathbf{e}},p^{\prime}_{n} + \alpha K\Delta {\varvec{{\varepsilon}}}_{\text{v}} ,\tau_{\text{y,n}} ) = 0$ as follows

$$ \begin{aligned} \alpha = \frac{{\sqrt {B^{2} - 4AC} - B}}{2A} \hfill \\ A = \left\| {2G\Delta {\mathbf{e}}} \right\|^{2} - 2\left( {\beta K\Delta {\varvec{{\varepsilon}}}_{\text{v}} } \right)^{2} \hfill \\ B = 4G\Delta {\mathbf{e}}^{\text{T}} {\mathbf{s^{\prime}}}_{n} + 4\beta K\left( {\tau_{\text{y,n}} - \beta p^{\prime}_{n} } \right)\Delta {\varvec{{\varepsilon}}}_{\text{v}} \hfill \\ C = \left\| {{\mathbf{s^{\prime}}}_{n} } \right\|^{2} - 2\left( {\tau_{\text{y,n}} - \beta p^{\prime}_{n} } \right)^{2} \hfill \\ \end{aligned} $$

(A1)

Appendix B: Computing the derivatives addressed in consistent tangent operator section

Here, it is aimed to derive the derivatives ${{{\partial} {\mathbf{X}}_{n + \alpha }^{\text{s}} } \mathord{\left/ {\vphantom {{{\partial} {\mathbf{X}}_{n + \alpha }^{\text{s}} } \partial }} \right. \kern-0pt} \partial } {\varvec{\upvarepsilon}}$, ${{\partial X_{n + 1}^{0} } \mathord{\left/ {\vphantom {{\partial X_{n + 1}^{0} } \partial }} \right. \kern-0pt} \partial } {\varvec{\upvarepsilon}}$, ${{\partial\Delta {\mathbf{e}}^{\text{p}} } \mathord{\left/ {\vphantom {{\partial\Delta {\mathbf{e}}^{\text{p}} } \partial }} \right. \kern-0pt} \partial }\upvarepsilon$ and ${{\partial \left\| {\Delta {\mathbf{e}}^{\text{p}} } \right\|} \mathord{\left/ {\vphantom {{\partial \left\| {\Delta {\mathbf{e}}^{\text{p}} } \right\|} \partial }} \right. \kern-0pt} \partial } {\varvec{\upvarepsilon}}$ appeared in Eqs. (54) to (56). To compute ${{\partial {\mathbf{X}}_{n + \alpha }^{\text{s}} } \mathord{\left/ {\vphantom {{\partial {\mathbf{X}}_{n + \alpha }^{\text{s}} } \partial }} \right. \kern-0pt} \partial }{\varvec{{\varepsilon}}}$, it is needed to take the derivatives of Eqs. (35) and (38) given as,

$$ \frac{{\partial {\mathbf{X}}_{n + 1}^{\text{s}} }}{\partial {\varvec\upvarepsilon} } = \frac{{\partial {\mathbf{X}}_{n + \alpha }^{\text{s}} }}{\partial {\varvec\upvarepsilon} } + \left( {\Delta \hat{{\varvec{\psi}} }^{\text{T}} {\mathbf{X}}_{n + \alpha }^{\text{s}} } \right)\Delta \hat{{\varvec{\psi}} }\frac{\partial a}{\partial {\varvec\upvarepsilon} } + (a - 1)\Delta \hat{{\varvec{\psi}} }\left( {\frac{{\partial\Delta \hat{{\varvec{\psi}} }}}{\partial {\varvec\upvarepsilon} }{\mathbf{X}}_{n + \alpha }^{\text{s}} } \right)^{\text{T}} + (a - 1)\Delta \hat{{\varvec{\psi}} }\left( {\frac{{\partial {\mathbf{X}}_{n + \alpha }^{\text{s}} }}{\partial {\varvec\upvarepsilon} }\Delta \hat{{\varvec{\psi}} }} \right)^{\text{T}} + (a - 1)\left( {\Delta \hat{{\varvec{\psi}} }^{\text{T}} {\mathbf{X}}_{n + \alpha }^{\text{s}} } \right)\frac{{\partial\Delta \hat{{\varvec{\psi}} }}}{\partial \varvec {\upvarepsilon} } + X_{n + 1}^{\text{R}}\Delta \hat{{\varvec{\psi}} }\frac{\partial b}{\partial \varvec {\upvarepsilon} } + b\Delta \hat{{\varvec{\psi}} }\frac{{\partial X_{n + \alpha }^{\text{R}} }}{\partial \varvec {\upvarepsilon} } + bX_{n + 1}^{\text{R}} \frac{{\partial\Delta \hat{{\varvec{\psi}} }}}{\partial {\varvec \upvarepsilon} } $$

(A65)

In the above equation, ${{\partial {\mathbf{X}}_{n + \alpha }^{\text{s}} } \mathord{\left/ {\vphantom {{\partial {\mathbf{X}}_{n + \alpha }^{\text{s}} } \partial }} \right. \kern-0pt} \partial } {\varvec{\upvarepsilon}}$ can be calculated by

$$ \frac{{\partial {\mathbf{X}}_{n + \alpha }^{\text{s}} }}{\partial {\varvec {\upvarepsilon}} } = X_{n + \alpha }^{0} \frac{{\partial {\mathbf{s^{\prime}}}_{n + \alpha } }}{\partial {\varvec {\upvarepsilon}} }{\text{ with }}\frac{{\partial {\mathbf{s^{\prime}}}_{n + \alpha } }}{\partial \varvec {\upvarepsilon} } = 2G\Delta {\mathbf{e}}\frac{\partial \alpha }{\partial {\varvec {\upvarepsilon}} } + 2G\alpha \frac{{\partial\Delta {\mathbf{e}}}}{\partial {\varvec\upvarepsilon} } $$

(A66)

where,

$$ \begin{aligned} \frac{{\partial\Delta {\mathbf{e}}}}{\partial \varvec\upvarepsilon } = {{\mathbb{I}}\ominus }_{\text{dev}} \hfill \\ \frac{\partial \alpha }{\partial \varvec\upvarepsilon } = - \left( {\frac{C}{{A\sqrt {B^{2} - 4AC} }} + \frac{{\sqrt {B^{2} - 4AC} - B}}{{2A^{2} }}} \right)\left( {8G^{2}\Delta {\mathbf{e}} - 4\beta^{2} K^{2}\Delta {\varvec{{\varepsilon}}}_{\text{v}} {\mathbf{i}}} \right) + \left( {\frac{B}{{2A\sqrt {B^{2} - 4AC} }} - \frac{1}{2A}} \right)\left( {4G{\mathbb{I}}_{\text{dev}} {\mathbf{s^{\prime}}}_{n} + 4\beta K\left( {\tau_{\text{y,n}} - \beta p^{\prime}_{n} } \right){\mathbf{i}}} \right) \hfill \\ \end{aligned} $$

(A67)

Also, the derivatives ${{\partial a} \mathord{\left/ {\vphantom {{\partial a} \partial }} \right. \kern-0pt} \partial }{\varvec{\upvarepsilon}}$ and ${{\partial b} \mathord{\left/ {\vphantom {{\partial b} \partial }} \right. \kern-0pt} \partial }{\varvec{\upvarepsilon}}$ are obtained through

$$ \begin{aligned} \frac{\partial a}{\partial {\varvec {\upvarepsilon}} } = \sinh (g)\frac{\partial g}{\partial {\varvec {\upvarepsilon}} } \hfill \\ \frac{\partial b}{\partial {\varvec {\upvarepsilon}} } = \cosh (g)\frac{\partial g}{\partial {\varvec {\upvarepsilon}} } \hfill \\ \end{aligned} $$

(A68)

$$ \frac{\partial g}{\partial {\varvec {\upvarepsilon}} } = \frac{1}{{R_{n + \alpha } }}\frac{{\partial\Delta {\varvec{\psi}} }}{\partial {\varvec {\upvarepsilon}} }\Delta \hat{\varvec{\psi}} - \frac{{\left\| {\Delta {\varvec{\psi}} } \right\|}}{{\left( {R_{n + \alpha } } \right)^{2} }}\frac{{\partial R_{n + \alpha } }}{\partial {\varvec {\upvarepsilon}} } $$

(A69)

where the derivatives appeared in the preceding relations are produced in the following,

$$ \frac{{\partial\Delta {\varvec{\psi}} }}{\partial {\varvec {\upvarepsilon}} } = 2G(1 - \alpha ){\mathbb{I}}_{\text{dev}} - 2G\Delta {\mathbf{e}}\frac{\partial \alpha }{\partial {\varvec {\upvarepsilon}} } $$

(A70)

$$ \frac{{\partial R_{n + \alpha } }}{\partial {\varvec {\upvarepsilon}} } = - \sqrt 2 \beta K(\Delta {\upvarepsilon}_{\text{v}} \frac{\partial \alpha }{\partial {\varvec {\upvarepsilon}} } + \alpha \,{\mathbf{i}}) $$

(A71)

The rest of the derivatives in Eq. (A65) are ${{\partial\Delta \hat{{\varvec{\psi}} }} \mathord{\left/ {\vphantom {{\partial\Delta \hat{{\varvec{\psi}} }} \partial }} \right. \kern-0pt} \partial } {\varvec{\upvarepsilon}}$ and ${{\partial X_{n + \alpha }^{\text{R}} } \mathord{\left/ {\vphantom {{\partial X_{n + \alpha }^{\text{R}} } \partial }} \right. \kern-0pt} \partial } {\varvec{\upvarepsilon}}$ which can be computed via,

$$ \frac{{\partial\Delta \hat{\varvec{\psi}}}}{\partial {\varvec\upvarepsilon} } = \frac{1}{{\left\| {\Delta {\varvec{\psi}} } \right\|}}\left[ {\frac{{\partial\Delta {\varvec{\psi}} }}{\partial {\varvec\upvarepsilon} } - \frac{{\Delta {\varvec{\psi}} }}{{\left\| {\Delta {\varvec{\psi}}} \right\|}}\left( {\frac{{\partial\Delta {\varvec{\psi}}}}{\partial {\varvec\upvarepsilon} }\Delta \hat{{\varvec{\psi}} }} \right)} \right] $$

(A72)

$$ \frac{{\partial X_{n + \alpha }^{\text{R}} }}{\partial {\varvec {\upvarepsilon}} } = - \sqrt 2 \beta KX_{n + \alpha }^{0} (\Delta {\upvarepsilon}_{\text{v}} \frac{\partial \upalpha }{\partial {\varvec {\upvarepsilon}} } + \upalpha \,{\mathbf{i}}) $$

(A73)

At this stage, the derivative of $X_{n + 1}^{0}$ with respect to strain is expressed reading,

$$ \frac{{\partial X_{n + 1}^{0} }}{\partial {\varvec {\upvarepsilon}} } = \frac{1}{{R_{n + 1} }}\frac{{\partial X_{n + 1}^{\text{R}} }}{\partial {\varvec {\upvarepsilon}} } - \frac{{X_{n + 1}^{\text{R}} }}{{\left( {R_{n + 1} } \right)^{2} }}\frac{{\partial R_{n + 1} }}{\partial {\varvec {\upvarepsilon}} } $$

(A74)

where ${{\partial X_{n + 1}^{\text{R}} } \mathord{\left/ {\vphantom {{\partial X_{n + 1}^{\text{R}} } \partial }} \right. \kern-0pt} \partial }\varvec\upvarepsilon$ and ${{\partial R_{n + 1} } \mathord{\left/ {\vphantom {{\partial R_{n + 1} } \partial }} \right. \kern-0pt} \partial }\varvec\upvarepsilon$ are calculated by taking their derivatives with respect to strain from Eqs. (35), (38), (45) and their related equations as follows,

$$ \frac{{\partial X_{n + 1}^{\text{R}} }}{\partial {\varvec\upvarepsilon} } = \left( {\Delta \hat{{\varvec{\psi}} }^{\text{T}} {\mathbf{X}}_{n + \alpha }^{\text{s}} } \right)\frac{\partial b}{\partial {\varvec {\upvarepsilon}} } + b\frac{{\partial\Delta \hat{{\varvec{\psi}} }}}{\partial {\varvec {\upvarepsilon}} }{\mathbf{X}}_{n + \alpha }^{\text{s}} + b\frac{{\partial {\mathbf{X}}_{n + \alpha }^{\text{s}} }}{\partial {\varvec {\upvarepsilon}} }\Delta \hat{{\varvec{\psi}} } + X_{n + \alpha }^{\text{R}} \frac{\partial a}{\partial {\varvec {\upvarepsilon}} } + a\frac{{\partial X_{n + \alpha }^{\text{R}} }}{\partial {\varvec \upvarepsilon} } $$

(A75)

$$ \frac{{\partial R_{n + 1} }}{\partial \varvec\upvarepsilon } = \sqrt 2 H_{\text{iso}} \frac{\partial \lambda }{\partial {\varvec \upvarepsilon} } - \sqrt 2 \beta \left[ {\frac{1}{{1 - 2\beta^{2} \bar{K}\lambda }}\left( {\frac{{\partial p_{n + 1}^{{{\prime }{\text{TR}}}} }}{\partial {\varvec \upvarepsilon} } - 2\beta \bar{K}\tau_{y,n} \frac{\partial \lambda }{\partial {\varvec \upvarepsilon} } - 4\beta \bar{K}H_{\text{iso}} \lambda \frac{\partial {\lambda} }{\partial {\varvec \upvarepsilon} }} \right) + \left( {\frac{{p_{n + 1}^{{{\prime }{\text{TR}}}} - 2\beta \bar{K}\tau_{y,n} \lambda - 2\beta \bar{K}H_{\text{iso}} \lambda^{2} }}{{\left( {1 - 2\beta^{2} \bar{K}\lambda } \right)^{2} }}} \right)2\beta^{2} \bar{K}\frac{\partial \lambda }{\partial {\varvec \upvarepsilon} }} \right] $$

(A76)

All the parameters in Eq. (A75) have already been presented and the derivatives of Eq. (A76) are computed as,

$$ \frac{\partial \lambda }{\partial {\varvec \upvarepsilon} } = - \frac{1}{{C_{2} }}\left( { - \frac{1}{\sqrt 2 }\frac{{\partial \left\| {{\mathbf{s}}_{n + 1}^{{{\prime TR}}} } \right\|}}{\partial {\varvec {\upvarepsilon}} } - \beta \frac{{\partial p_{n + 1}^{{{\prime TR}}} }}{\partial {\varvec {\upvarepsilon}} }} \right) + \frac{{C_{3} }}{{C_{2}^{2} }}\left( {\sqrt 2 \beta^{2} \bar{K}\frac{{\partial \left\| {{\mathbf{s}}_{n + 1}^{{{\prime TR}}} } \right\|}}{\partial \varvec \upvarepsilon } + 2\bar{G}\beta \frac{{\partial p_{n + 1}^{{{\prime TR}}} }}{\partial {\varvec\upvarepsilon} }} \right) $$

(A77)

$$ \frac{{\partial p_{n + 1}^{{{\prime TR}}} }}{\partial {\varvec {\varepsilon}} } = K\,{\mathbf{i}} $$

(A78)

Finally, ${{\partial\Delta {\mathbf{e}}^{\text{p}} } \mathord{\left/ {\vphantom {{\partial\Delta {\mathbf{e}}^{\text{p}} } \partial }} \right. \kern-0pt} \partial } {\varvec{\upvarepsilon}}$ and ${{\partial \left\| {\Delta {\mathbf{e}}^{\text{p}} } \right\|} \mathord{\left/ {\vphantom {{\partial \left\| {\Delta {\mathbf{e}}^{\text{p}} } \right\|} \partial }} \right. \kern-0pt} \partial } {\varvec{\upvarepsilon}}$, which are required to compute the consistent tangent operator addressed in consistent tangent operator section, are obtained by the succeeding relations,

$$ \frac{{\partial\Delta {\mathbf{e}}^{\text{p}} }}{\partial {\varvec {\upvarepsilon}} } = = \frac{1}{{2\bar{G}}}\left( {\frac{{\partial {\mathbf{s}}_{n + 1}^{{{\prime TR}}} }}{\partial {\varvec {\upvarepsilon}} } - \frac{{\partial {\mathbf{s^{\prime}}}_{n + 1} }}{\partial {\varvec {\upvarepsilon}} }} \right) $$

(A79)

$$ \frac{{\partial \left\| {\Delta {\mathbf{e}}^{\text{p}} } \right\|}}{\partial {\varvec {\upvarepsilon}} } = \frac{1}{{2\bar{G}}}\left( {\frac{{\partial {\mathbf{s}}_{n + 1}^{{{\prime TR}}} }}{\partial {\varvec {\upvarepsilon}} } - \frac{{\partial {\mathbf{s^{\prime}}}_{n + 1} }}{\partial {\varvec {\upvarepsilon}} }} \right)\frac{{\Delta {\mathbf{e}}^{\text{p}} }}{{\left\| {\Delta {\mathbf{e}}^{\text{p}} } \right\|}} $$

(A80)

where ${{\partial {\mathbf{s}}_{n + 1}^{{{\prime TR}}} } \mathord{\left/ {\vphantom {{\partial {\mathbf{s}}_{n + 1}^{{{\prime TR}}} } \partial }} \right. \kern-0pt} \partial }{\varvec{\upvarepsilon}} = 2{\text G}{\mathbb{I}}_{\text{dev}}$ and ${{\partial {\mathbf{s^{\prime}}}_{n + 1}^{{}} } \mathord{\left/ {\vphantom {{\partial {\mathbf{s^{\prime}}}_{n + 1}^{{}} } \partial }} \right. \kern-0pt} \partial } {\varvec{\upvarepsilon}}$ were previously presented in Eq. (A66).