$\def\rcontract{{\scriptscriptstyle\circ\bullet}} \def\lcontract{{\scriptscriptstyle\bullet\circ}}$

## 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, $$\label{eq:viscous helmhotz} \mu\dot\phi_{nl} = l^{2}\nabla \cdot \tnsr D \cdot \nabla\phi_{nl} + (\phi_{l} - \phi_{nl}),$$ 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,

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

where $\hat{n}$ is the unit normal to boundary.

### Initial condition

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

## Coupling

Damage needs to be coupled with mechanical problem to cause softening in the material. In current implementation, this has been done in different ways for each damage model.

### IsoBrittle

$\tnsrfour C_{eff} = \phi^2_{nl}\tnsrfour C$

### AnisoBrittle

$\tnsr F = \tnsr F_{e}\tnsr F_{d}\tnsr F_{p}$

### IsoDuctile

$\tnsrfour C_{eff} = \phi^2_{nl}\tnsrfour C$

### AnisoDuctile

$\tnsr F = \tnsr F_{e}\tnsr F_{d}\tnsr F_{p}$

## References

[1]
J. Lemaitre
A course on Damage mechanics
Second Edition, Springer
Topic revision: r5 - 30 Jan 2015, PratheekShanthraj

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