Material damping

The ${\alpha }_R$ factor defines a damping contribution proportional to the mass matrix for an element. The damping forces that are introduced are caused by the absolute velocities of nodes in the model. The resulting effect can be likened to the model moving through a viscous fluid so that any motion of any point in the model triggers damping forces. Reasonable mass proportional damping does not reduce the stability limit significantly.

The ${\beta }_R$ factor defines damping proportional to the elastic material stiffness. A damping stress, ${\sigma }_d$, proportional to the total strain rate is introduced, using the following formula:

where $\dot{\epsilon }$ is the strain rate. For hyperelastic and hyperfoam materials ${\tilde{D}}^{el}$ is defined as the initial elastic stiffness. For all other materials ${\tilde{D}}^{el}$ is the material's current elastic stiffness. This damping stress is added to the stress caused by the constitutive response at the integration point when the dynamic equilibrium equations are formed, but it is not included in the stress output. Damping can be introduced for any nonlinear analysis and provides standard Rayleigh damping for linear analyses. For a linear analysis stiffness proportional damping is exactly the same as defining a damping matrix equal to ${\beta }_R$ times the stiffness matrix. Stiffness proportional damping must be used with caution because it may significantly reduce the stability limit. To avoid a dramatic drop in the stable time increment, the stiffness proportional damping factor, ${\beta }_R$, should be less than or of the same order of magnitude as the initial stable time increment without damping.