viscousHookeanElastic

Small-strain linear viscoelasticity: a generalised Maxwell (Prony series) model with an elastic volumetric response and a relaxing deviatoric response. The runtime type is:

viscousHookeanElastic

The dictionary layout common to all laws is described in the material models page.

User Guide

What it computes

A relaxed spring (EInfinity) acts in parallel with a list of Maxwell arms (E[i], relaxationTimes[i]). The law first forms the instantaneous modulus and relative moduli:

E0        = EInfinity + sum(E[i])
gammaInf  = EInfinity/E0
gamma[i]  = E[i]/E0
mu        = E0/(2*(1 + nu))

lambdaInf = nu*EInfinity/((1 + nu)*(1 - 2*nu))  // plane strain / 3-D
lambdaInf = nu*EInfinity/((1 + nu)*(1 - nu))    // planeStress yes
k         = lambdaInf + (2/3)*gammaInf*mu

At each integration point, with epsilon = symm(grad(D)) from the solid model and deltaT the time step:

s     = 2*mu*dev(epsilon)
h[i]  = exp(-deltaT/tau[i])*h[i]_0 + exp(-deltaT/(2*tau[i]))*(s - s_0)
sigma = k*tr(epsilon)*I + gammaInf*s + sum(gamma[i]*h[i])

following Simo and Hughes (1998), eq. 10.3.12. Only the deviatoric response relaxes; the bulk response k is the long-term one and constant.

With the Williams-Landel-Ferry (WLF) shift enabled, each relaxation time is multiplied by

aT = 10^(-C1*(T - Tref)/(C2 + (T - Tref)))

evaluated with the temperature T at the integration point.

The law is small-strain only; a nonlinear geometry solid model stops with an error.

Model options

WilliamsLandelFerryCoeffs contains:

The following are fatal errors at construction:

• EInfinity without pressure dimensions, or nu not dimensionless;
• E and relaxationTimes of different lengths, or empty;
• any E[i] or relaxationTimes[i] below SMALL;
• nu at or above 0.5 (to within SMALL).

E and relaxationTimes are plain scalar lists, without dimensions; the relaxation times are in the case's time unit.

With WilliamsLandelFerry yes, the law asks the framework for a live temperature input T. It is taken from a registered T field (for example one solved by a thermal solid model), from a case directory given under inputCaseDirectories in the law's dictionary, or from a T file in the current or initial time directory; if none is found the run stops. See the framework README for the lookup order.

Tangents and volumetric split

• Scalar tangent: factor*2*mu + lambdaInf, and factor*(4/3)*mu for the deviatoric scalar tangent, with factor = gammaInf + sum(gamma[i]*exp(-deltaT/(2*tau[i]))). It depends on the time step. The WLF shift is not included in it.
• Fourth-order tangent: fourthOrderFiniteDifference only. Asking for the analytical fourthOrder tangent is a fatal error.
• Volumetric split: no. The law cannot be used with solvePressureEqn or with the mixed displacement-pressure formulations.

State variables

There is one h<i> per entry of E. At each write time, for cell-centred solid models, each persistent variable is written as a volField named after it (s, h0, h1, …) for viewing. The restart data are written alongside as <material>_<topology>_<variable>, for example polymer_cellCentredIntegrationPointTopology_h0, together with a <material>_<topology>_integrationPoints file. To restart a run from a later time, set restart yes; in the solid model's coefficients dictionary from the start of the run.

Example

From the viscoTube tutorial:

planeStress     no;

mechanical
(
    polymer
    {
        type            viscousHookeanElastic;
        rho             rho [1 -3 0 0 0 0 0] 7850;
        EInfinity       EInfinity [1 -1 -2 0 0 0 0] 39.58e+9;
        nu              nu [0 0 0 0 0 0 0] 0.33;
        E               (2.9318518519e9 5.8637037037e9
                         6.5966666667e9 18.3240740741e9);
        relaxationTimes (30 300 3000 12000);

        // Optional temperature shift
        // WilliamsLandelFerry yes;
        // WilliamsLandelFerryCoeffs
        // {
        //     C1   C1 [0 0 0 0 0 0 0] 17.44;
        //     C2   C2 [0 0 0 1 0 0 0] 51.6;
        //     Tref Tref [0 0 0 1 0 0 0] 293.15;
        // }
    }
);

Migrating from the legacy law

• regionName is no longer read. The legacy law silently ignored solvePressureEqn; the solid model now refuses it for this law.
• nu = 0.5 is now a fatal error; the legacy law accepted it and set lambda and k to GREAT.
• The update is always written on the total displacement gradient and its old-time value. The legacy law had a separate incremental form, based on grad(DD) and the old-time total stress.
• The WLF temperature is gathered by the framework from any of the sources above; the legacy law required a T field at cells or at faces, matching the stress location.
• The scalar tangent no longer drops lambda when a field named p exists.
• The Maxwell-arm stresses are now written (h0, …) and restartable; the legacy h, hf, s and sf fields were neither read nor written.
• The law now serves every integration-point location, and a fourth-order tangent is available by finite differences.

Tutorials

• viscoTube, whose regression tests also cover restart

References

• J.C. Simo and T.J.R. Hughes, Computational Inelasticity, Springer, 1998.