Damage Models

Damage index

The damage index $\phi$ of a material point is computed as [1] $\phi = \frac{\int_{\mathcal{H}}(1-\chi\langle\boldsymbol{\xi}\rangle)dV_{\xi}}{\int_{\mathcal{H}}dV_{\xi}}=\frac{\sum^n_{i=1}(1-\chi_i dV_i}{\sum^n_{i=1} dV_i}$ with $\chi$ is the bond damage between 0 (broken) and 1 (unbroken), $V$ is the volume and $n$ is the number of neighbors.

Critical Stretch

The critical value correspondends to the critical stretch for this model, defined in the theory manual.

Critical Energy

Interblock Damage

Interlaminar behaviour between different material blocks can be defined using the Interblock Critical Value parameter. If a Bond is crossing a block interface, a user defined critical damage value is applied to the bonds.

As bonds are bidirectional, the critical damage value can be defined for both orientations, for example:

• Interblock Damage:
  • Interblock Critical Value 1_2: 0.1
  • Interblock Critical Value 2_1: 0.1

Bonds that aren't crossing a block interface are not affected by the interblock damage model.

Local damping

Silling proposed a local damping to reduce waves induced by cracked bonds in [1] in (EQ 2.34). This proposed algorithm is introduced in Material_Basis.jl.

The bond force due to damping is computed as

\[\mathbf{t}_{damp}\langle\boldsymbol{\xi}\rangle = \overline{\phi}d c\frac{|\eta_{i}|-|\eta_{i-1}|}{dt v_0}\frac{\eta_i}{|\eta_i|}\]

with the numerical local damping coefficient $d$, the bond stiffness $c$, the rate of bond extension $\dot{e}=\frac{|\eta_{i}|-|\eta_{i-1}|}{dt},$ where $|\eta_{i}|$ and $|\eta_{i-1}|$ the length of the deformed bond vector at iteration step $i$ and $i-1$ and $dt$ is the time increment.

The average damage index between point $i$ and it's neighbor $j$ can be computed as $\overline{\phi}=\frac{\phi_i+\phi_j}{2}$ and using the Young's modulus $E$ and the mass density $\rho$ the dilatation wave speed [7] as

\[v_0=\frac{E}{\rho}\]