viscousHookeanElastic

This page documents the small-strain generalised Maxwell law, with an elastic volumetric response and a relaxing deviatoric response. The runtime type is:

viscousHookeanElastic

User Guide

What it computes

The law places a relaxed spring in parallel with a list of Maxwell arms. It 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

For a non-incremental solid model, the trial deviatoric stress and elastic volumetric stress are:

e       = dev(symm(grad(D)))
s       = 2*mu*e
sigmaVol = k*tr(grad(D))*I

For an incremental solid model, the implemented update is:

De       = dev(symm(grad(DD)))
s        = s.oldTime() + 2*mu*De
sigmaVol = tr(sigma.oldTime())*I + k*tr(grad(DD))*I

Each Maxwell-arm stress is then advanced using the midpoint exponential update. With aT = 1 unless temperature shifting is enabled, the update and total stress are:

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

h[i] = exp(-deltaT/(aT*tau[i]))*h[i].oldTime()
       + exp(-deltaT/(2*aT*tau[i]))*(s - s.oldTime())

sigma = sigmaVol + gammaInf*s + sum(gamma[i]*h[i])

Only the deviatoric response relaxes; k is constant. The law implements correct(volSymmTensorField&) and correct(surfaceSymmTensorField&). The point-symmetric-tensor overload inherits notImplemented. On OpenFOAM.com and OpenFOAM.org, the CompactListList quadrature-point overload also inherits notImplemented; that overload is not compiled for foam-extend.

Model options

E and relaxationTimes are read as scalar lists and must have equal lengths. The code uses each relaxation time with deltaTValue(), so the values have the case time unit. Enabling WilliamsLandelFerry also requires a cell-centred T field for the cell correction and a face-centred T field for the face correction.

The base class selects solid as regionName when that mesh exists, then region0; otherwise regionName must be supplied. Although the base class reads the two pressure-equation options, this law does not call updateSigmaHyd(), so neither option changes its stress update.

Recommended dictionary setup

The following polymer values are used by 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] 39.58e9;
        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;
        // }
    }
);

Field glossary

• s, sf: instantaneous deviatoric trial stress at cell centres and faces. Their old-time values are stored, but the fields are neither read nor written.
• h0, h1, and so on: cell-centred Maxwell-arm internal stresses. Each field stores old-time and previous-iteration values and is neither read nor written.
• hf0, hf1, and so on: face-centred counterparts of the Maxwell-arm stresses, with the same time and iteration storage.
• T: temperature looked up when WLF shifting is enabled. Its location must match the selected cell- or face-centred correction.
• K: temporary cell field containing the constant relaxed bulk modulus k.
• impK: temporary cell field containing the implicit stiffness used by the solid model's Laplacian term.

Developer Notes

Class role

viscousHookeanElastic derives directly from mechanicalLaw and is registered in the linGeomMechLaw runtime-selection table. It stores EInfinity, the Maxwell moduli and relaxation times, their relative moduli, nu, lambda, mu, k, the cell and face internal-stress lists, the cell and face trial deviatoric stresses, and the WLF switch and coefficients.

The class uses #ifdef OPENFOAM_NOT_EXTEND for OpenFOAM-specific field headers and for primitiveField() in its residual calculation; foam-extend uses internalField(). It appears in both Make/files.openfoam and Make/files.foamextend.

Construction

The base constructor first reads solvePressureEqn and pressureSmoothingScaleFactor, adding their defaults to the stored dictionary, and selects the base region from regionName, solid, or region0. Failure to find a region is fatal. Density is read later by rhoScalar(), rather than by the constructor.

The derived constructor then reads EInfinity, E, relaxationTimes, and nu. It reads WilliamsLandelFerry with a default of false; when enabled, it requires C1, C2, and Tref from WilliamsLandelFerryCoeffs. It checks that the two lists have equal lengths, calculates E0 and the relative moduli, and rejects a relaxation time or Maxwell modulus below SMALL with a fatal error.

It creates zero-valued s, sf, h[i], and hf[i] fields and stores their old-time values. Finally, it calculates mu, rejects nu outside [-1, 0.5], and calculates lambda and k. For nu == 0.5, it prints an incompressibility message and sets lambda and k to GREAT. Construction also prints the relative moduli and the state of WLF shifting. It emits no warnings.

Key methods

• impK() forms a time-step-dependent algorithmic stiffness. Its scale factor is gammaInf + sum(gamma[i]*exp(-deltaT/(2*tau[i]))); the returned field is that factor times 2*mu, plus lambda unless nu == 0.5 or a cell field named p exists. This diffusivity affects the outer-iteration convergence rate rather than the converged answer. WLF shifting is not included in this scale factor.
• correct(volSymmTensorField&) and its face-field counterpart update the trial stress, Maxwell-arm stresses, and total stress. They select grad(D) or grad(DD) according to the solid model's incremental setting and apply WLF shifting when enabled.
• residual() returns the largest relative previous-iteration change over all Maxwell-arm stresses. It uses face fields when the base mesh has an Ff surface field and cell fields otherwise.
• bulkModulus() delegates to K(), which returns a uniform field containing k.
• materialTangent() is not overridden. The base implementation aborts with notImplemented, so solid models requiring a material tangent cannot use this law.

Extension points

A related linear viscoelastic law can copy this class and replace the Maxwell-arm update in both correct overloads. New internal variables need matching cell and face fields, old-time storage, previous-iteration storage if they contribute to residual(), and corresponding terms in impK(). A law needed by point- or quadrature-based solid models must also implement those correct overloads, and a law needed by Newton-based solid models must provide materialTangent().

The source is at viscousHookeanElastic.C.

Tutorials

• solids/viscoelasticity/viscoTube