The finite-strain viscoelasticity theory implemented in ABAQUS is a time domain generalization of either the hyperelastic or hyperfoam constitutive models. It is assumed that the instantaneous response of the material follows from the hyperelastic constitutive equations:

for a compressible material and for an incompressible material. In the above, and are, respectively, the deviatoric and the hydrostatic parts of the instantaneous Kirchhoff stress . is the “distortion gradient” related to the deformation gradient by whereis the volume change.

Using integration by parts and a variable transformation, the basic hereditary integral formulation for linear isotropic viscoelasticity can be written in the form

or entirely in terms of stress quantities, where is the reduced time, , and . and are the instantaneous small-strain shear and bulk moduli, and and are the time-dependent small-strain shear and bulk relaxation moduli. Recall that the reduced time represents a shift in time with temperature and is related to the actual time through the differential equation where is the temperature and is the shift function.

A suitable generalization to finite strain of the hereditary integral formulation is obtained as follows:

where is the deformation gradient of the state at relative to the state at t. A transformation is performed on the stress relating the state at time to the state at time t. We also ensure the symmetry of the transformed stress. Observe that because is proportional to the identity tensor, we have It can also be observed that since is deviatoric. Hence the deviatoric and volumetric parts can be separated into two hereditary integrals:

As in small-strain viscoelasticity, we represent the relaxation moduli in terms of the Prony series

where and are the relative moduli of terms i. Note that . ABAQUS assumes that the relaxation times are the same so that from here on, we will sum on N terms for both bulk and shear behavior. In reality, the number of nonzero terms in bulk and shear, and , need not be equal, unless the instantaneous behavior is based on the hyperfoam model. In the latter case, the two deformation modes are closely related and are then assumed to relax equally and simultaneously.

Substituting Equation 4.8.2–2 and Equation 4.8.2–3 in Equation 4.8.2–1, we obtain

Next, we introduce the internal stresses, associated with each term of the series

These stresses are stored at each material point and are integrated forward in time. We will assume that the solution is known at time t, and we need to construct the solution at time .

The internal hydrostatic stresses at time follow from

With and , it follows thatwhich yields with Equation 4.8.2–6

To integrate the first integral in Equation 4.8.2–7, we assume that varies linearly with the reduced time over the increment

Substituting into Equation 4.8.2–7 yields

The integrals are readily evaluated, providing the solution at the end of the increment

or, in a slightly different form with Observe that for , and . For , and .

The internal deviatoric stresses at time follow from

Observe that

and the inverse of this is which—when substituted into Equation 4.8.2–10, with and —gives With and , it follows:

Now introduce the variable

Note that and

Then we can also introduce

Substitution of Equation 4.8.2–5, Equation 4.8.2–12, and Equation 4.8.2–15 into Equation 4.8.2–11 yields

To integrate the first integral in Equation 4.8.2–16, we assume that varies linearly with the reduced time over the increment:

which with Equation 4.8.2–14 becomes

Equation 4.8.2–16 and Equation 4.8.2–17 for the deviatoric stress have exactly the same form as Equation 4.8.2–7 and Equation 4.8.2–8 for the hydrostatic stress. Hence, after integration we obtain

with Equation 4.8.2–13, Equation 4.8.2–15, and Equation 4.8.2–18, thus, provide a straightforward integration scheme.

The total stress at the end of the increment becomes

which with Equation 4.8.2–9 and Equation 4.8.2–18 can also be written as

To solve the system of nonlinear equations generated by the constitutive equations, we need to generate the corotational constitutive rate equations. From Equation 4.8.2–20 it follows

where is the corotational (Jaumann) stress rate. Since and in Equation 4.8.2–20 are independent of the increment size, their derivatives vanish. The derivatives and follow from the hyperelastic equations being used and, thus, do not need to be considered here.

With Equation 4.8.2–13 it follows that

where is the velocity gradient.

Using the definition of the corotational (Jaumann) rate, it follows that

where is the spin tensor following from the increment. Note that hence, substitution of Equation 4.8.2–23 and Equation 4.8.2–24 into Equation 4.8.2–22 yields since both and are symmetric. Similarly for ,

Equation 4.8.2–21 then simplifies to

All equations have been worked out in terms of the Kirchhoff stress. However, the implementation in ABAQUS uses the Cauchy stress. To transform to Cauchy stress, we use the relations

With , this allows us to write Equation 4.8.2–9, Equation 4.8.2–13, Equation 4.8.2–15, Equation 4.8.2–18, Equation 4.8.2–19, and Equation 4.8.2–27 in the following form:

The virtual work and rate of virtual work equations are written with respect to the current volume. Therefore, the corotational stress rates are rates of Kirchhoff stress mapped into the current configuration and transformed in the same way as the stresses themselves.

This set of equations—combined with the expressions for , , and —describe the full implementation of the hyper-viscoelasticity model in a displacement formulation.

The rate equations can be written in a form similar to “Hyperelastic material behavior,” Section 4.6.1. Introduce

and where and are the instantaneous moduli, corresponding to and of “Hyperelastic material behavior,” Section 4.6.1. Thus, all rate equations can be obtained by substitution of by and by .

The in-plane deformation produces and , from which we can calculate only , , , and . , , and are zero. The deformation in the third direction, characterized by , is derived from the plane stress condition

Applying the condition to Equation 4.8.2–20 yields where stands for the projection along the 33 component. In the derivations it is convenient to express kinematic variables in terms of incremental values, such as and .