viscoNeoHookeanElastic

This law combines a finite-strain nonlinear deviatoric response with a generalised Maxwell relaxation model and an elastic volumetric response. The runtime type is:

viscoNeoHookeanElastic

User Guide

What it computes

The law first forms the total deformation gradient, its determinant and its volume-preserving part:

J    = det(F)
Fbar = J^(-1/3)*F
C    = F^T*F
lambdaBar_a = J^(-1/3)*sqrt(eigenvalue_a(C))

For m = 0, 1, 2, the three supplied modulus/exponent pairs define a diagonal tensor D and the current deviatoric response S:

D_aa = sum_m mu_m*lambdaBar_a^(alpha_m - 1)
S    = dev(2*Fbar*D*Fbar^T)
s    = inv(Fbar)*S*inv(Fbar)^T

For each Maxwell branch i, the internal variables are updated from their old-time values using the current time-step deltaT:

H_i = exp(-deltaT/tau_i)*h_i.oldTime()
    - exp(-deltaT/(2*tau_i))*s.oldTime()

h_i = H_i + exp(-deltaT/(2*tau_i))*s

The relative moduli, relaxation factor, pressure and stress assigned by the code are:

E0       = EInfinity + sum_i E_i
gammaInf = EInfinity/E0
gamma_i  = E_i/E0
g        = gammaInf + sum_i gamma_i*exp(-deltaT/(2*tau_i))
p        = k/(beta*J)*(1 - J^(-beta))

sigma = J*p*I + (1 + g)*S
      + sum_i gamma_i*dev(Fbar*H_i*Fbar^T)

The class implements both correct(volSymmTensorField&) and correct(surfaceSymmTensorField&). The inherited point-centred overload aborts with notImplemented. On OpenFOAM.com and OpenFOAM.org, the inherited CompactListList quadrature-point overload also aborts with notImplemented; that overload is not compiled for foam-extend.

Model options

The E and relaxationTimes lists must have the same non-zero length. Every entry of each list must be positive. The values in E are plain scalars, so they must use the same numerical modulus unit as EInfinity.

The implementation indexes the first three alpha values without checking the list length. Supply at least three values; later values are not used. The constructor requires k, warns that it is read directly, and uses it instead of the bulk modulus derived from EInfinity and nu.

planeStress is read from the enclosing mechanicalProperties dictionary. It changes the derived lambda, but the supplied k still replaces the derived bulk modulus. The pressure-equation entries are constructed by the base class, but this law does not call updateSigmaHyd().

Recommended dictionary setup

The following values define an illustrative three-branch silicone-like material:

planeStress     no;

mechanical
(
    siliconeLike
    {
        type            viscoNeoHookeanElastic;
        rho             rho [1 -3 0 0 0 0 0] 1100;
        EInfinity       EInfinity [1 -1 -2 0 0 0 0] 1.0e6;
        E               (5.0e5 2.0e5 1.0e5);
        relaxationTimes (0.1 1 10);
        nu              nu [0 0 0 0 0 0 0] 0.49;
        mu0             mu0 [1 -1 -2 0 0 0 0] 3.0e5;
        mu1             mu1 [1 -1 -2 0 0 0 0] 1.0e5;
        mu2             mu2 [1 -1 -2 0 0 0 0] 5.0e4;
        alpha           (2 4 6);
        beta            beta [0 0 0 0 0 0 0] 2;
        k               k [1 -1 -2 0 0 0 0] 5.0e7;

        // Optional base-class entries
        // solvePressureEqn             no;
        // pressureSmoothingScaleFactor 100;
        // regionName                   region0;
    }
);

Field glossary

• F_, Ff_: total deformation gradients at cell centres and faces, created by the base class and read on restart when present.
• relF_, relFf_: relative deformation gradients maintained by the base class while updating F_ and Ff_.
• s, sf: pull-backs of the current deviatoric response at cells and faces; their old-time values enter the relaxation update.
• h<i>, hf<i>: internal stress variables for Maxwell branch i; their old-time values are stored.
• H<i>, Hf<i>: intermediate branch-history fields calculated during each stress correction.
• transformH<i>, transformHf<i>: transformed branch-history fields used in the final stress sum.
• transformNeeded, transformNeededf: diagonal nonlinear response tensors assembled from the modified principal stretches.
• transformFbar, transformFbarf: transformed nonlinear response tensors whose deviatoric parts give S.
• impK: implicit stiffness supplied to the solid model's Laplacian term.

Developer Notes

Class role

viscoNeoHookeanElastic derives directly from mechanicalLaw and is registered in the nonlinear-geometry mechanical-law selection table. It stores the long-term modulus and Poisson's ratio, Maxwell moduli and times, relative weights, three nonlinear moduli, the exponent list, the volumetric exponent, the bulk modulus, and cell- and face-centred history fields.

The class is listed in both Make/files.openfoam and Make/files.foamextend. The header includes surfaceFields.H only under OPENFOAM_NOT_EXTEND. The source uses realEigenValues under OPENFOAM_COM and fork-specific field accessors under OPENFOAM_NOT_EXTEND; the constitutive algorithm is otherwise present for all three forks.

Construction

The base constructor first reads or inserts solvePressureEqn and pressureSmoothingScaleFactor, then selects regionName. The derived constructor reads EInfinity, E, relaxationTimes, nu, mu0, mu1, mu2, alpha, beta and k, in that order, and creates the cell and face history fields.

It then performs the following operations:

• a nu outside [-1, 0.5] causes a fatal error;
• nu == 0.5 prints an incompressibility message, sets lambda and k to GREAT, and then encounters the fatal independent-k check;
• otherwise it derives lambda and a bulk modulus, warns that k is read directly, and replaces the derived bulk modulus with the supplied k;
• unequal E and relaxationTimes list lengths cause a fatal error;
• it computes gammaInf and each gamma_i from the total modulus;
• a relaxation time below SMALL causes a fatal error;
• an E value below SMALL causes a fatal error;
• it prints the relative moduli, allocates one set of history fields per Maxwell branch, and stores the old-time h, hf, s and sf fields.

The constructor does not validate EInfinity, beta, the total modulus, or the length of alpha.

Key methods

• impK(): returns 2*mu when nu == 0.5 or a cell field named p exists; otherwise it returns 2*mu + 4*k/3. It is the diffusivity of the solid model's Laplacian term and affects the outer-iteration convergence rate rather than the converged answer.
• K(): returns a uniform cell field containing the stored k value.
• correct(): updates F or Ff, evaluates the nonlinear deviatoric response from the three modulus/exponent pairs, advances the Maxwell histories, and assigns the stress shown above. If the solid model enforces linear material behavior, the base-class updateF() assigns an elastic stress using mu and k and the method returns immediately.
• materialTangent(): is not overridden; the inherited implementation aborts with notImplemented.
• setRestart(): sets F and Ff to AUTO_WRITE.

Extension points

A related law can copy this class and replace the nonlinear response and history updates in the two correct() methods. New branch properties should be read and validated before allocating matching cell and face history lists. Implement the point and quadrature-point overloads, and materialTangent(), if the new law must support vertex-centred or tangent-based solid models. Keep the source in both build lists when all three OpenFOAM forks are supported.

The source is at viscoNeoHookeanElastic.C.

Tutorials

No tutorial currently uses viscoNeoHookeanElastic.