Damping

The prescribed percentage of damping is a mass proportional damping defined in the form:

C =α*M

Or in given the diagonal form of the mass matrix:

The lowest frequency ( Ωmin) of the current deformation mode can be estimated by the Raleigh quotient  defined as:

The upper bound of the oscillation frequencies is defined by the critical time step as:

The frequency used in the evaluation of mass proportional damping is the minimum frequency evaluated by the two methods above. This constraint is applied to prevent inappropriate frequencies that may result if the approximation fort Kij is poor (e.g. due to local instabilities). The proportionality constant ( α) is evaluated every time step so that the fluctuating effective stiffness of the system is accounted for. Higher frequencies are damped more strongly than lower frequencies and so diminish first and consequently α tends to a constant value as the quasi-static solution is approached.

The standard dynamic bulk viscosity concept can also be used to provide volume damping in quasi-static simulations solved using an explicit solver. In this case the objective is to minimise oscillations in the effective mean stress and only the linear term of the standard bulk viscosity concept is required; i.e. the dynamic bulk viscosity ( q) for each element is defined as:

Damping is only applied in compression, so assuming elasticity, the contribution of the volumetric stress to the element internal forces is evaluated as:

•1 The model does not damp rigid body motion, so that the gross downward movement resulting from compaction of a column, or upward movement on a thrust, remain little affected.

•2 The damping term is transitory (i.e. it is not accumulated) so if the overall deformation pattern is compressive then this mode of deformation is damped but not prevented.

Notes

•Bulk viscosity damping increases the stiffness of the system and consequently reduces the critical time step necessitating more time steps

•Qlin =0.5 is a suitable value for many quasi-static problems