Introduction

A fully coupled non local analysis is necessary to model damage that spreads over parts of representative volume element or the structural component. The type of damage laws used are selected by subroutines inside constitutive.f90

Governing Equation

The governing p.d.e for fully coupled damage analysis is a viscous enhanced Helmoltz type equation, \begin{equation} \label{eq:viscous helmhotz} \mu\dot\phi_{nl} = l^{2}\nabla \cdot \tnsr D \cdot \nabla\phi_{nl} + (\phi_{l} - \phi_{nl}), \end{equation} where, $\phi_{l}$ represents local damage while $\phi_{nl}$ is its non local counterpart. The first term on the left in \eqref{eq:viscous helmhotz} is a time regularization term with $\mu$ being the viscosity. $\tnsr D$ and $l$ are diffusion tensor and length scale parameter respectively, which are both material dependent properties. Here the local damage acts a driving force and the Laplacian term diffuses the solution over the length scale.

Boundary condition

The boundary condition for \eqref{eq:viscous helmhotz} is a flux-free condition,i.e,

\begin{equation} \label{eq:viscous helmhotz bc} \nabla\phi_{nl} \cdot \hat{n} = 0 \end{equation}

Initial condition

The initial condition for \eqref{eq:viscous helmhotz} is that material is damage free($\phi_{nl}$ =1). \begin{equation} \label{eq:viscous helmhotz ic} \phi_{nl}(x,t=0) = 1 \end{equation} $\phi_{nl} $= 0 would imply a completely damaged material point. The value of $\phi_{nl}$ is bounded between 0 and 1.

Coupling

IsoBrittle

AnisoBrittle

IsoDuctile

AnisoDuctile

References

[1]

J. Lemaitre
A course on Damage mechanics
Second Edition, Springer