A micropolar isotropic plasticity formulation for non‐associated flow rule and softening featuring multiple classical yield criteria

Abstract

The Cosserat continuum is very effective in regularizing the ill‐posed governing equations of the Cauchy/Maxwell continuum. An elasto‐plastic constitutive model for the linear formulation of the Cosserat continuum is here presented, which features non‐associated flow, hardening/softening behaviour and multiple yield and plastic potential surfaces, whilst linear hyper‐elasticity is adopted to reproduce the recoverable response. For the definition of the yield and plastic potential functions, the equivalent von Mises stress is formulated using energy considerations and the theory of representations, the latter being also used to retrieve an expression for the Lode's angle. The recent Generalized classical criterion is then used to define the yield and plastic potential functions so that Lode's angle‐dependent deviatoric sections can be used.

An implicit integration algorithm based on invariants for isotropic elasto‐plastic models of the Cosserat continuum

“…It should be noted that, following Panteghini and Lagioia, 2 the approach described in this paper is based on the decomposition of the strain, curvature, stress and couple tensors into their symmetric and skew‐symmetric parts $$\begin{eqnarray} \undertilde {{\bm {\gamma }}}&=&\text{sym} \, \undertilde {{\bm {\gamma }}} +\text{skew} \, \undertilde {{\bm {\gamma }}}= \undertilde {{\bm {\uvarepsilon }}} + \undertilde {{\bm {\omega }}}\nonumber \nonumber \\ \undertilde {{\bm {\chi }}}&=&\text{sym} \, \undertilde {{\bm {\chi }}} +\text{skew} \, \undertilde {{\bm {\chi }}}= \undertilde {{\bm {\chi }}}_{sym}+\undertilde {{\bm {\chi }}}_{skw}\nonumber \nonumber \\ \undertilde {{\bm {\sigma }}}&=&\text{sym} \, \undertilde {{\bm {\sigma }}}+\text{skew} \, \undertilde {{\bm {\sigma }}} = \undertilde {{\bm {\sigma }}}_{sym}+\undertilde {{\bm {\sigma }}}_{skw}\nonumber \nonumber \\ \undertilde {{\bm {\mu }}}&=&\text{sym} \, \undertilde {{\bm {\mu }}}+\text{skew} \, \undertilde {{\bm {\mu }}} = \undertilde {{\bm {\mu }}}_{sym}+\undertilde {{\bm {\mu }}}_...$$…”

“…The stress and couple‐stress tensors conjugated in the meaning of Hill 15 to the strain $$\undertilde {{\bm {\gamma }}}$$ and wryness $$\undertilde {{\bm {\chi }}}$$ tensors are indicated as $$\undertilde {{\bm {\sigma }}}$$ and $$\undertilde {{\bm {\mu }}}$$, respectively. The strong form of the balance equations is (for its derivation see Panteghini and Lagioia 2 ) $$\begin{equation*} { \def\eqcellsep{&}\begin{array}{c}\undertilde {{\bm {\sigma }}} \cdot \underline{\bm {\bm {\nabla }}} + \underline{\bm {f}}=0 \\ \undertilde {{\bm {\mu }}} \cdot \underline{\bm {\bm {\nabla }}} + 2 \underline{\bm {\sigma }}_{skw} + \underline{\bm {c}} =0 \end{array} } \nonumber \end{equation*}$$the latter showing that due to the couple‐stress tensor, the stress $$\undertilde {{\bm {\sigma }}}$$ is not symmetric and that its skew component is $$\begin{equation} \undertilde {{\bm {\sigma }}}_{skw}= \underline{\undertilde {{\bm {e}}}} \cdot \frac{1}{2} {\left(\undertilde {{\bm {\mu }}} \cdot \underline{{\bm {\bm {\nabla }}}} + \underline{{\bm {c}}} \right)} \end{equation}$$whilst on the surface of the continuum …”

“…In the case of the Cosserat continuum, the generalization of the yield functions of Equations (4) and (5) is based on the definition of the equivalent von Mises stress and of the Lode's angle proposed by Panteghini and Lagioia 2 $$\begin{equation} \begin{split} q= {\left\lbrace \frac{3}{2} {\left[ \undertilde {{\bm {s}}}_{sym} : \undertilde {{\bm {s}}}_{sym} + \frac{G}{G_c} \undertilde {{\bm {s}}}_{skw} : \undertilde {{\bm {s}}}_{skw} + \frac{G}{B} \undertilde {{\bm {m}}}_{sym} : \undertilde {{\bm {m}}}_{sym} +\frac{G}{B_c} \undertilde {{\bm {m}}}_{skw} : \undertilde {{\bm {m}}}_{skw} \right]} \right.} {\left.$$…”

“…Constitutive models for the Cosserat continuum can be integrated using a fully implicit backward Euler scheme, but its implementation in a finite element programme is laborious. However, since it is the only unconditionally stable method, even with very non‐linear yield and plastic potential surfaces such as those adopted in the constitutive model formulated by Panteghini and Lagioia, 2 its use is particularly recommended and it is indeed chosen in this study.…”

“…The algorithm has been implemented in a proprietary FE programme and numerical analyses have been conducted to simulate a biaxial compression test on a specimen with a defect inclusion. The material behaviour has been reproduced by means of the constitutive model recently proposed by Panteghini and Lagioia 2 for the Cosserat continuum, which, thanks to the formulation of the Lode's angle for the micro‐polar medium, allows the use of the Generalized Classical Criterion of Lagioia and Panteghini 11 (this name was given by Lester and Sloan 12 ).…”