neoHookeanElastic

This page documents the compressible neo-Hookean hyperelastic law. The runtime type is:

neoHookeanElastic

The law uses the volumetric-isochoric split of Simo and Hughes (1998), Eqn 9.2.6. It is the default choice for large-strain rubber-like materials in solids4foam, and it is also the elastic backbone of several other laws in the toolbox.

Reference: J. C. Simo and T. J. R. Hughes, Computational Inelasticity, Springer, 1998, 10.1007/b98904.

User Guide

Strain energy function

The stored energy is split into an isochoric and a volumetric part:

Psi = 0.5*mu*(tr(bBar) - 3) + U(J)

bBar = J^(-2/3)*b,   b = F & F.T(),   J = det(F)

with the default volumetric part

U(J) = 0.5*K*(0.5*(J^2 - 1) - ln(J))

The resulting Cauchy stress, which is what the code evaluates, is

sigma = (1/J)*(0.5*K*(J^2 - 1)*I + mu*dev(bBar))

Setting alternatePressureDefinition replaces the Kirchhoff hydrostatic term 0.5*K*(J^2 - 1) by K*(J - 1), which corresponds to U(J) = K*(J - ln(J) - 1).

The law is registered in the nonlinear-geometry table, so it works with both total-Lagrangian and updated-Lagrangian solid models; the base class updateF() selects the right deformation-gradient update. It is not available to linear-geometry solid models.

Model options

Elastic constants must be given as either E and nu or mu and K. Mixing the two pairs is a fatal error.

When E and nu are given, mu = E/(2*(1 + nu)) and K follows from the plane-stress or plane-strain relation, as set by the top-level planeStress entry of constant/mechanicalProperties.

Optional entries:

tangentEps is read with lookup() inside materialTangentField(), so it is only required by solid models that ask for the material tangent; it is not read at construction.

Pressure-displacement mode

Setting pressureDisplacement true switches the law into the form expected by coupledPressureDisplacementSolid. In that mode:

• the deviatoric stress becomes mu*(b - I)/J, i.e. the full left Cauchy-Green tensor is used rather than its isochoric part;
• the hydrostatic part is taken from the separately solved p field (pf on faces) and multiplied by pressureDisplacementCoeff;
• impK() returns mu alone rather than (4/3)*mu + K;
• if E and nu are supplied with nu at or above 0.5 - SMALL, K is set to GREAT, which allows a truly incompressible material to be specified.

pressureDisplacementCoeff defaults to 3*nu/(1 + nu), with nu recovered from mu and K, or taken as 0.5 when K has been set to GREAT.

Recommended dictionary setup

Minimal example for constant/mechanicalProperties:

planeStress     no;

mechanical
(
    rubber
    {
        type            neoHookeanElastic;
        rho             rho [1 -3 0 0 0 0 0] 1000;
        E               E [1 -1 -2 0 0 0 0] 2e+06;
        nu              nu [0 0 0 0 0 0 0] 0.45;
    }
);

For a nearly incompressible material solved with a pressure equation, add solvePressureEqn yes; and, if needed, pressureSmoothingScaleFactor.

Field glossary

• F, Ff: total deformation gradient at cell centres and faces.
• relF, relFf: relative deformation gradient, used in the updated Lagrangian formulation.
• sigmaHyd: hydrostatic Cauchy stress, -p; written when solvePressureEqn is enabled.
• sigma: Cauchy stress tensor returned to the solid model.
• impK: implicit stiffness used as the Laplacian diffusivity.

Developer Notes

Class role

neoHookeanElastic inherits from mechanicalLaw and is added to the nonLinGeomMechLaw runtime selection table. It stores mu_ and K_ as uniform dimensionedScalar values, so the law is homogeneous within a given material entry; heterogeneity is obtained by using several mechanical entries with the multi-material machinery.

Construction

The constructor reads the two switches and the coefficient first, then decides between the (E, nu) and (mu, K) branches. It finishes by calling F().storeOldTime() and Ff().storeOldTime(), which forces the old-time deformation gradients to exist from the first time step.

correct() walkthrough

correct(volSymmTensorField&):

1. calls updateF(sigma, mu_, K_); if that returns true the solid model has requested enforced linearity for stability, the base class has already set a Hooke's law stress, and the function returns immediately;
2. materialises FT explicitly before contracting, to avoid a tmp<> evaluation problem documented in StVenantKirchhoffElastic.C;
3. forms J, b, and bBar, then the deviatoric stress s;
4. in pressure-displacement mode looks up p and returns -pressureDisplacementCoeff*p*I + s directly;
5. otherwise calls updateSigmaHyd() with either K*(J - 1) or 0.5*K*(J^2 - 1), then assembles sigma = (1/J)*(sigmaHyd*I + s).

The surface overloads route through the private correctF() helper, which is also used by the correct(sigma, gradD) overload that builds F from a supplied displacement gradient.

Implicit stiffness

impK() returns (4/3)*mu + K, which equals 2*mu + lambda and is the standard choice for the segregated Laplacian operator. In pressure-displacement mode it returns mu only, because the volumetric response is carried by the pressure equation.

bulkModulus() and shearModulus() return uniform fields built from K_ and mu_.

Material tangent

materialTangentField() builds a face-based 6x6 tangent by finite differences: each component of grad(D)f is perturbed by tangentEps and the resulting change in sigmaf is divided by the perturbation. It looks up sigmaf and grad(D)f from the registry, so it only works with solid models that create those surface fields.

Extension points

• correctF() is the single place where the constitutive relation is evaluated for surface fields; a new volumetric-isochoric split can be added there and in correct(volSymmTensorField&).
• setRestart() sets AUTO_WRITE on F and Ff so a restart reproduces the same deformation state.

Tutorials

Cases that select neoHookeanElastic:

• solids/hyperelasticity/blockPunch
• solids/hyperelasticity/cantileverBeam
• solids/hyperelasticity/cylindricalPressureVessel, in the three caseOptions/pressureDisplacement variants (cylinder, cylinderLin and cylinderUnsteady)
• solids/linearElasticity/plateHole, in both caseOptions/pressureDisplacement variants (compressible and incompressible)
• fluidSolidInteraction/beamInCrossFlow
• fluidSolidInteraction/fillingElasticContainer
• fluidSolidInteraction/flexibleDamBreak
• fluidSolidInteraction/HronTurekFsi3