where ηb and ηv are the bulk and shear viscosity coefficients, respectively.

where Cel is the elastic Cauchy–Green tensor, is the rate of the elastic Green–Lagrange strain, and Jel is the elastic volume ratio.

When using artificial viscosity for shock capturing (Ref. 52), the bulk and shear viscosities are nonlinear functions of the strain rate

where cL and cQ are linear and quadratic tuning parameters, Le is the typical length of the element, ρ is the density, and c is the wave speed. The variable