Large Strain Viscoelasticity

The implementation for large strain viscoelasticity follows the derivation by Holzapfel (Ref. 1).

The generalized Maxwell model is based on the splitting of the strain energy density into volumetric, isochoric, and the contribution from the viscoelastic branches

The strain energy in the main hyperelastic branch is normally denoted with the superscript ∞ to denote the long-term equilibrium (as t → ∞).

The second Piola–Kirchhoff stress is computed from

where the auxiliary second Piola–Kirchhoff stress tensors Qm are defined as

The time evolution of the auxiliary stress tensor Qm in each viscoelastic branch is given by the rate

here, Siso,m is the isochoric second Piola–Kirchhoff stress tensor in the branch m. These tensors are derived from the strain energy density in the main hyperelastic branch and the energy factors βm as

so the time evolution of the auxiliary stress tensor Qm is given by

This equation is not well suited for modeling prestressed bodies. Applying the change of variables

the time evolution of the auxiliary stress tensor qm reads

For the The Generalized Maxwell Model, it is possible to write the Prony series in terms of weights wm with respect to the instantaneous stiffness G0

Here, w∞ is the ratio between the long term and instantaneous stiffness, w∞ = G∞/G0, and the weight wm relates the branch stiffness to the instantaneous stiffness through wm = Gm/G0. The weights are bounded by

Use this expression to relate the instantaneous stiffness in the small strain regime to the large strain formulation. The instantaneous response for the second Piola–Kirchhoff stress is computed from

In this expression, the isochoric response represents the long term stiffness, and the sum of the energy factors the contribution to the overall stiffness from the Maxwell branches.

The weights in the Generalized Maxwell model for the small strain formulation are related to the energy factors in the large strain formulation by

equivalently, the energy factors represent the ratio between the branch stiffness and the long term stiffness in the main branch.


	The volumetric contribution from the auxiliary stress tensor qm is removed before adding these to the viscoelastic stresses per branch Qm = βmSiso − qm, so the stresses Qm are isochoric.

Temperature Effects

The same options for defining Temperature Effects as described for Linear Viscoelasticity are available for large strain viscoelasticity.