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An Efficient Method for Integrating von-Mises Plasticity with Mixed Hardening

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Abstract

Integrating the constitutive equations in plasticity in order to update the stresses is the most important part of an elastoplastic finite element analysis. Here, a von-Mises plasticity model along with the isotropic and kinematic hardenings is taken into account in the small strain realm. An accurate integration is formulated by converting the differential constitutive equations into an augmented stress space and utilizing the return mapping algorithm. Subsequently, a broad range of numerical tests are performed to investigate the accuracy and performance of the proposed integration. The results demonstrate the robustness of the new integration scheme.

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Author information

Correspondence to Mehrzad Sharifian.

Appendix

Appendix

For the sake of the convenience in using the presented algorithm in the computational implementations, the entries of the vectors and matrixes are given in the following:

$${\mathbf{\sigma = }}\left[ {\begin{array}{*{20}c} {\sigma_{11} } & {\sigma_{12} } & {\sigma_{13} } & {\sigma_{21} } & {\sigma_{22} } & {\sigma_{23} } & {\sigma_{31} } & {\sigma_{32} } & {\sigma_{33} } \\ \end{array} } \right]^{\text{T}}$$
(53)
$${\mathbf{i}} = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 \\ \end{array} } \right]^{\text{T}}$$
(54)
$${\mathbf{s}} = {\varvec{\upsigma}} - p\,{\mathbf{i}} = \left[ {\begin{array}{*{20}c} {s_{11} } & {s_{12} } & {s_{13} } & {s_{21} } & {s_{22} } & {s_{23} } & {s_{31} } & {s_{32} } & {s_{33} } \\ \end{array} } \right]^{\text{T}}$$
(55)
$${\mathbf{\varepsilon = }}\left[ {\begin{array}{*{20}c} {\varepsilon_{11} } & {\varepsilon_{12} } & {\varepsilon_{13} } & {\varepsilon_{21} } & {\varepsilon_{22} } & {\varepsilon_{23} } & {\varepsilon_{31} } & {\varepsilon_{32} } & {\varepsilon_{33} } \\ \end{array} } \right]^{\text{T}}$$
(56)
$${\mathbf{e}} = {\varvec{\upvarepsilon}} - \frac{1}{3}\theta \,{\mathbf{i}}{\mathbf{ = }}\left[ {\begin{array}{*{20}c} {e_{11} } & {e_{12} } & {e_{13} } & {e_{21} } & {e_{22} } & {e_{23} } & {e_{31} } & {e_{32} } & {e_{33} } \\ \end{array} } \right]^{\text{T}}$$
(57)
$${\mathbf{\alpha = }}\left[ {\begin{array}{*{20}c} {\alpha_{11} } & {\alpha_{12} } & {\alpha_{13} } & {\alpha_{21} } & {\alpha_{22} } & {\alpha_{23} } & {\alpha_{31} } & {\alpha_{32} } & {\alpha_{33} } \\ \end{array} } \right]^{\text{T}}$$
(58)
$${\mathbf{\varSigma = }}{\mathbf{s}} - {\varvec{\upalpha}} = \left[ {\begin{array}{*{20}c} {\varSigma_{11} } & {\varSigma_{12} } & {\varSigma_{13} } & {\varSigma_{21} } & {\varSigma_{22} } & {\varSigma_{23} } & {\varSigma_{31} } & {\varSigma_{32} } & {\varSigma_{33} } \\ \end{array} } \right]^{\text{T}}$$
(59)
$${\mathbf{n}} = {{\varvec{\Sigma}} \mathord{\left/ {\vphantom {{\varvec{\Sigma}} R}} \right. \kern-0pt} R} = \left[ {\begin{array}{*{20}c} {{{\varSigma_{11} } \mathord{\left/ {\vphantom {{\varSigma_{11} } R}} \right. \kern-0pt} R}} & {{{\varSigma_{12} } \mathord{\left/ {\vphantom {{\varSigma_{12} } R}} \right. \kern-0pt} R}} & {{{\varSigma_{13} } \mathord{\left/ {\vphantom {{\varSigma_{13} } R}} \right. \kern-0pt} R}} & {{{\varSigma_{21} } \mathord{\left/ {\vphantom {{\varSigma_{21} } R}} \right. \kern-0pt} R}} & {{{\varSigma_{22} } \mathord{\left/ {\vphantom {{\varSigma_{22} } R}} \right. \kern-0pt} R}} & {{{\varSigma_{23} } \mathord{\left/ {\vphantom {{\varSigma_{23} } R}} \right. \kern-0pt} R}} & {{{\varSigma_{31} } \mathord{\left/ {\vphantom {{\varSigma_{31} } R}} \right. \kern-0pt} R}} & {{{\varSigma_{32} } \mathord{\left/ {\vphantom {{\varSigma_{32} } R}} \right. \kern-0pt} R}} & {{{\varSigma_{33} } \mathord{\left/ {\vphantom {{\varSigma_{33} } R}} \right. \kern-0pt} R}} \\ \end{array} } \right]^{\text{T}}$$
(60)
$${\mathbf{\mathbb{I}}}_{\text{dev}} = \left[ {\begin{array}{*{20}c} {{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0pt} 3}} & 0 & 0 & 0 & { - {1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-0pt} 3}} & 0 & 0 & 0 & { - {1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-0pt} 3}} \\ {} & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ {} & {} & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ {} & {} & {} & 1 & 0 & 0 & 0 & 0 & 0 \\ {} & {} & {} & {} & {{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0pt} 3}} & 0 & 0 & 0 & { - {1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-0pt} 3}} \\ {} & {} & {} & {} & {} & 1 & 0 & 0 & 0 \\ {} & {} & {\text{sym}} & {} & {} & {} & 1 & 0 & 0 \\ {} & {} & {} & {} & {} & {} & {} & 1 & 0 \\ {} & {} & {} & {} & {} & {} & {} & {} & {{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0pt} 3}} \\ \end{array} } \right]_{9 \times 9}$$
(61)
$${\mathbf{X}} = \left[ {\begin{array}{*{20}c} {X^{0} \overline{{\varvec{\Sigma}}} } \\ {X^{0} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {{\mathbf{X}}^{\text{s}} } \\ {X^{0} } \\ \end{array} } \right] = X^{0} \left[ {\begin{array}{*{20}c} {{{\varSigma_{11} } \mathord{\left/ {\vphantom {{\varSigma_{11} } R}} \right. \kern-0pt} R}} & {{{\varSigma_{12} } \mathord{\left/ {\vphantom {{\varSigma_{12} } R}} \right. \kern-0pt} R}} & {{{\varSigma_{13} } \mathord{\left/ {\vphantom {{\varSigma_{13} } R}} \right. \kern-0pt} R}} & {{{\varSigma_{21} } \mathord{\left/ {\vphantom {{\varSigma_{21} } R}} \right. \kern-0pt} R}} & {{{\varSigma_{22} } \mathord{\left/ {\vphantom {{\varSigma_{22} } R}} \right. \kern-0pt} R}} & {{{\varSigma_{23} } \mathord{\left/ {\vphantom {{\varSigma_{23} } R}} \right. \kern-0pt} R}} & {{{\varSigma_{31} } \mathord{\left/ {\vphantom {{\varSigma_{31} } R}} \right. \kern-0pt} R}} & {{{\varSigma_{32} } \mathord{\left/ {\vphantom {{\varSigma_{32} } R}} \right. \kern-0pt} R}} & {{{\varSigma_{33} } \mathord{\left/ {\vphantom {{\varSigma_{33} } {R\begin{array}{*{20}c} {\begin{array}{*{20}c} {} & 1 \\ \end{array} } \\ \end{array} }}} \right. \kern-0pt} {R\begin{array}{*{20}c} {\begin{array}{*{20}c} {} & 1 \\ \end{array} } \\ \end{array} }}} \\ \end{array} } \right]^{\text{T}}$$
(62)
$${\mathbf{\mathbb{A}}}_{\text{e}} = \frac{2G}{R}\left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\dot{e}_{11} } \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\dot{e}_{12} } \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\dot{e}_{13} } \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\dot{e}_{21} } \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\dot{e}_{22} } \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\dot{e}_{23} } \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\dot{e}_{31} } \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\dot{e}_{32} } \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\dot{e}_{33} } \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]_{10 \times 10}$$
(63)
$${\mathbf{\mathbb{A}}}_{\text{e}} = \frac{2G}{R}\left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\dot{e}_{11} } \\ {} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\dot{e}_{12} } \\ {} & {} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\dot{e}_{13} } \\ {} & {} & {} & 0 & 0 & 0 & 0 & 0 & 0 & {\dot{e}_{21} } \\ {} & {} & {} & {} & 0 & 0 & 0 & 0 & 0 & {\dot{e}_{22} } \\ {} & {} & {} & {} & {} & 0 & 0 & 0 & 0 & {\dot{e}_{23} } \\ {} & {} & {} & {} & {} & {} & 0 & 0 & 0 & {\dot{e}_{31} } \\ {} & {} & {\text{sym}} & {} & {} & {} & {} & 0 & 0 & {\dot{e}_{32} } \\ {} & {} & {} & {} & {} & {} & {} & {} & 0 & {\dot{e}_{33} } \\ {} & {} & {} & {} & {} & {} & {} & {} & {} & 0 \\ \end{array} } \right]_{10 \times 10}$$
(64)
$${\mathbf{\bar{\mathbb{G}}}}_{\text{e}} = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{{2G\Delta e_{11} } \mathord{\left/ {\vphantom {{2G\Delta e_{11} } {R_{n} }}} \right. \kern-0pt} {R_{n} }}} \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{{2G\Delta e_{12} } \mathord{\left/ {\vphantom {{2G\Delta e_{12} } {R_{n} }}} \right. \kern-0pt} {R_{n} }}} \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & {{{2G\Delta e_{13} } \mathord{\left/ {\vphantom {{2G\Delta e_{13} } {R_{n} }}} \right. \kern-0pt} {R_{n} }}} \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & {{{2G\Delta e_{21} } \mathord{\left/ {\vphantom {{2G\Delta e_{21} } {R_{n} }}} \right. \kern-0pt} {R_{n} }}} \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & {{{2G\Delta e_{22} } \mathord{\left/ {\vphantom {{2G\Delta e_{22} } {R_{n} }}} \right. \kern-0pt} {R_{n} }}} \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & {{{2G\Delta e_{23} } \mathord{\left/ {\vphantom {{2G\Delta e_{23} } {R_{n} }}} \right. \kern-0pt} {R_{n} }}} \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & {{{2G\Delta e_{31} } \mathord{\left/ {\vphantom {{2G\Delta e_{31} } {R_{n} }}} \right. \kern-0pt} {R_{n} }}} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & {{{2G\Delta e_{32} } \mathord{\left/ {\vphantom {{2G\Delta e_{32} } {R_{n} }}} \right. \kern-0pt} {R_{n} }}} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & {{{2G\Delta e_{33} } \mathord{\left/ {\vphantom {{2G\Delta e_{33} } {R_{n} }}} \right. \kern-0pt} {R_{n} }}} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ \end{array} } \right]_{10 \times 10}$$
(65)
$$\Delta {\hat{\mathbf{e}}} = {{\Delta {\mathbf{e}}} \mathord{\left/ {\vphantom {{\Delta {\mathbf{e}}} {\left\| {\Delta {\mathbf{e}}} \right\|}}} \right. \kern-0pt} {\left\| {\Delta {\mathbf{e}}} \right\|}} = \left[ {\begin{array}{*{20}c} {{{e_{11} } \mathord{\left/ {\vphantom {{e_{11} } {\left\| {\Delta {\mathbf{e}}} \right\|}}} \right. \kern-0pt} {\left\| {\Delta {\mathbf{e}}} \right\|}}} & {{{e_{12} } \mathord{\left/ {\vphantom {{e_{12} } {\left\| {\Delta {\mathbf{e}}} \right\|}}} \right. \kern-0pt} {\left\| {\Delta {\mathbf{e}}} \right\|}}} & {{{e_{13} } \mathord{\left/ {\vphantom {{e_{13} } {\left\| {\Delta {\mathbf{e}}} \right\|}}} \right. \kern-0pt} {\left\| {\Delta {\mathbf{e}}} \right\|}}} & {{{e_{21} } \mathord{\left/ {\vphantom {{e_{21} } {\left\| {\Delta {\mathbf{e}}} \right\|}}} \right. \kern-0pt} {\left\| {\Delta {\mathbf{e}}} \right\|}}} & {{{e_{22} } \mathord{\left/ {\vphantom {{e_{22} } {\left\| {\Delta {\mathbf{e}}} \right\|}}} \right. \kern-0pt} {\left\| {\Delta {\mathbf{e}}} \right\|}}} & {{{e_{23} } \mathord{\left/ {\vphantom {{e_{23} } {\left\| {\Delta {\mathbf{e}}} \right\|}}} \right. \kern-0pt} {\left\| {\Delta {\mathbf{e}}} \right\|}}} & {{{e_{31} } \mathord{\left/ {\vphantom {{e_{31} } {\left\| {\Delta {\mathbf{e}}} \right\|}}} \right. \kern-0pt} {\left\| {\Delta {\mathbf{e}}} \right\|}}} & {{{e_{32} } \mathord{\left/ {\vphantom {{e_{32} } {\left\| {\Delta {\mathbf{e}}} \right\|}}} \right. \kern-0pt} {\left\| {\Delta {\mathbf{e}}} \right\|}}} & {{{e_{33} } \mathord{\left/ {\vphantom {{e_{33} } {\left\| {\Delta {\mathbf{e}}} \right\|}}} \right. \kern-0pt} {\left\| {\Delta {\mathbf{e}}} \right\|}}} \\ \end{array} } \right]^{\text{T}}$$
(66)

At the end, the entries of the matrix \({\mathbf{\bar{\mathbb{G}}}}_{\text{p}}\) can be easily obtained by using the above vector.

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Haji Aghajanpour, N., Sharifian, M. & Sharifian, M. An Efficient Method for Integrating von-Mises Plasticity with Mixed Hardening. Iran J Sci Technol Trans Mech Eng 44, 47–59 (2020). https://doi.org/10.1007/s40997-018-0248-8

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Keywords

  • Integration
  • Explicit exponential map
  • Backward Euler method
  • von-Mises plasticity
  • Mixed hardening