A Rate-Dependent Damage Model and its Application to Uniaxial Strain

Our analysis is based on a damage model discussed in [1] in which the internal energy density $W$ depends on strain \textbf{E} and damage \textit{$\kappa $} : $W({\rm {\bf E}},\kappa )=\phi (\kappa ){\kern 1pt}{\kern 1pt}{\kern 1pt}\mu {\kern 1pt}{\kern 1pt}{\kern 1pt}\left( {\frac{\nu }{1-2\nu }E_{kk} E_{ll} +E_{ij} E_{ij} } \right)$; \textit{$\mu $} is elastic shear modulus, \textit{$\nu $} is Poisson's ratio. The factor $\phi (\kappa )=1-\left( {1-\phi _{\min } } \right)\frac{\kappa }{\kappa _{\max } }$ describes degradation of elastic modulus due to damage; \textit{$\phi $}$_{min}$ and \textit{$\kappa $}$_{max }$are material constants.$_{ }$ The system of evolution includes \[ \rho \frac{\partial ^2u}{\partial t^2}=\nabla \frac{\partial W}{\partial {\rm {\bf E}}},\quad \;\frac{\partial \kappa }{\partial t}=-K\frac{\partial W}{\partial \kappa } \] where$ K$ is (for now) a material constant. The above model was installed into LS-DYNA using the User Material Interface. The model was applied to a finite-element simulation of a rod under uniaxial strain, with a prescribed-velocity boundary condition at one end and a stress-free condition at the other. The resulting initial-value boundary-value problem was scaled to reveal the presence of the dimensionless group $\Pi =\frac{\rho _0 }{2}\sqrt {\frac{(1-2\nu )\rho _0 }{2{\kern 1pt}{\kern 1pt}{\kern 1pt}(1-\nu )\mu }} \cdot \frac{\left( {1-\phi _{\min } } \right)K}{\kappa _{\max } ^2}\cdot L\cdot \dot {u}_0 ^2$, where $\rho _0 $ is the material density, $L$ is the length of the rod, and $\dot {u}_0 $is the prescribed velocity. Solutions were obtained for a range of $\Pi $ values. The progression of contours of \textit{$\kappa $}($x$,$t)$ was observed. [1] Grinfeld, M.A., and Wright, T.W., \textit{Metallurgical and Materials Transactions A}, Vol. 35A, 2651-2661, 2004.