Non-Local Plastic-Damage Model for Failure Analysis of Sheet-Metals

Abstract

A new damage model which takes account of void shape effect and anisotropy of the matrix material is integrated into the dynamic explicit finite element framework to predict the damage evolution which occurs under crash or stamping process. For the strain localization and failure, the pathological mesh dependence has been overcome by a non-local approach where the evolution equation for the porosity and the equivalent plastic strain is modified by an additional term containing a characteristic internal length. The non-local plastic-damage potential is written as: $$ \Phi = C\frac{{q^2 }} {{\sigma _y^2 (\bar \varepsilon _{loc}^p ,\nabla ^2 \bar \varepsilon ^p )}} + 2q_1 f\cosh \left( {\frac{{\kappa \sigma _{\rm H} }} {{\sigma _y \left( {\bar \varepsilon _{loc}^p ,\nabla ^2 \bar \varepsilon ^p } \right)}}} \right) - (1 + q_3 f^2 ) = 0 $$ (1) The damage model can take into account the three main phases of damage evolution: growth, nucleation and coalescence. To determine the critical porosity f c, the void coalescence failure mechanism by internal necking is considered by using a modified Thomason’s plastic limit-load model on the reference volume element such as: $$ \left\{ {F/\left( {\frac{{R_Z }} {{X - R_X }}} \right)^N + G/\left( {\frac{{R_X }} {X}} \right)^M } \right\}A_n \left[ {1 - T\frac{{\sigma _1 }} {{\sigma _y }}} \right] \leqslant \frac{{\sigma _1 }} {{\sigma _y }} $$ (2) Consistent with (2), the plastic-damage potential (1) is used to calculate the void and matrix geometry changes using the current strain, void volume fraction f and shape factor S. Once the inequality (2) is satisfied, the void coalescence starts to occur and the void volume fraction at this point is the critical value f c.