Damage Evolution

Damage Evolution Options

The Damage Evolution options appear when you select Use damage evolution during the definition of a damage criterion. Multiple forms of damage evolution may act on a material at the same time—one for each defined damage initiation criterion.

Damage evolution can be specified by Type (displacement or energy) and by Softening method. Your selections of type and softening method determine the equation used to evaluate damage evolution.

Damage Evolution Based on Displacement

When you define damage evolution based on effective plastic displacement (Type=Displacement), the effective plastic displacement, ${\overline{u}}^{pl}$ , is defined with the evolution equation

where $L$ is the characteristic length of the element.

The evolution of the damage variable with the relative plastic displacement can be specified in tabular, linear, or exponential form. Instantaneous failure occurs if the plastic displacement at failure, ${\overline{u}}_f^{pl}$ , is specified as 0. However, this choice is not recommended. Use it with care because it causes a sudden drop of the stress at the material point that can lead to dynamic instabilities.

• For the tabular form, you can specify the damage variable directly as a tabular function of equivalent plastic displacement, $d=d\left({\overline{u}}^{pl}\right)$ .
• For the linear form, you can specify the effective plastic displacement, ${\overline{u}}_f^{pl}$ , at the point of failure (full degradation). Then, the damage variable increases according to
  $$\dot{d}=\frac{L{\stackrel{\dot{¯}}{\epsilon }}^{pl}}{{\overline{u}}_f^{pl}}=\frac{{\stackrel{\dot{¯}}{u}}^{pl}}{{\overline{u}}_f^{pl}}\mathrm{.}$$
  This definition ensures that when the effective plastic displacement reaches the value ${\overline{u}}^{pl}={\overline{u}}_f^{pl}$ , the material stiffness fully degrades ( $d=1$ ). The linear damage evolution law defines a truly linear stress-strain softening response only if the effective response of the material is perfectly plastic (constant yield stress) after damage initiation.
• For the exponential form, you can specify the relative plastic displacement at failure, ${\overline{u}}_f^{pl}$ , and the exponent $\alpha$ . The damage variable is given as
  $$d=\frac{1-e^{-\alpha ({\overline{u}}^{pl}/{\overline{u}}_f^{pl})}}{1-e^{-\alpha }}\mathrm{.}$$

Damage Evolution Based on Energy

When you define damage evolution based on fracture energy (Type=Energy), you can specify the fracture energy ( $G_f$ ) per unit area to be dissipated during the damage process directly. Instantaneous failure occurs if $G_f$ is specified as 0. However, this choice is not recommended and must be used with care because it causes a sudden drop in the stress at the material point that can lead to dynamic instabilities.

The evolution in the damage can be specified in linear or exponential form.

• For the linear form, once the damage initiation criterion is met, the damage variable increases according to
  $$\dot{d}=\frac{L{\stackrel{\dot{¯}}{\epsilon }}^{pl}}{{\overline{u}}_f^{pl}}=\frac{{\stackrel{\dot{¯}}{u}}^{pl}}{{\overline{u}}_f^{pl}},$$
  where the equivalent plastic displacement at failure is computed as
  $${\overline{u}}_f^{pl}=\frac{2G_f}{{\sigma }_{y0}},$$
  and ${\sigma }_{y0}$ is the value of the yield stress at the time when the failure criterion is reached.
• For the exponential form, the damage variable given as
  $$d=1-\exp (-{\int }_0^{{\overline{u}}^{pl}}\frac{{\overline{\sigma }}_y{\stackrel{\dot{¯}}{u}}^{pl}}{G_f})\mathrm{.}$$
  The formulation of the model ensures that the energy dissipated during the damage evolution process is equal to $G_f$ .

Once the damage initiation criterion is reached, the effective plastic displacement, ${\stackrel{\dot{¯}}{u}}^{pl}$ , is defined with the evolution equation

When multiple damage criteria are active, the damage variable, $D$ , captures the combined effect of all active mechanisms and is computed in terms of individual damage variables, $d_i$ , for each mechanism. You can choose to combine some of the damage variables in a multiplicative sense to form an intermediate variable, $d_{mult}$ , as follows:

Then, the overall damage variable is computed as the maximum of $d_{mult}$ and the remaining damage variables: