anisotropicBiotElastic

Small-strain orthotropic linear elasticity, with the stiffness given as Young's moduli, Poisson's ratios and shear moduli in the coordinate directions. Pore pressure is included by nesting it inside poroMechanicalLaw. The runtime type is:

anisotropicBiotElastic

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

User Guide

What it computes

With epsilon = symm(grad(D)) from the solid model, the stress is sigma = A : epsilon, component by component. The form is chosen by the mesh, not by the dictionary.

On a 3-D mesh:

sigma_xx = A11*epsilon_xx + A12*epsilon_yy + A31*epsilon_zz
sigma_yy = A12*epsilon_xx + A22*epsilon_yy + A23*epsilon_zz
sigma_zz = A31*epsilon_xx + A23*epsilon_yy + A33*epsilon_zz
sigma_xy = A44*epsilon_xy
sigma_yz = A55*epsilon_yz
sigma_xz = A66*epsilon_xz

with

nuyx = nuxy*Ey/Ex
nuxz = nuzx*Ex/Ez
nuzy = nuyz*Ez/Ey
J    = (1 - nuxy*nuyx - nuyz*nuzy - nuzx*nuxz
        - 2*nuyx*nuzy*nuxz)/(Ex*Ey*Ez)
A11  = (1 - nuyz*nuzy)/(J*Ey*Ez)
A22  = (1 - nuxz*nuzx)/(J*Ex*Ez)
A33  = (1 - nuyx*nuxy)/(J*Ey*Ex)
A12  = (nuxy + nuzy*nuxz)/(J*Ex*Ez)
A31  = (nuzx + nuyx*nuzy)/(J*Ey*Ez)
A23  = (nuyz + nuyx*nuxz)/(J*Ex*Ey)
A44  = 2*Gxy,  A55 = 2*Gyz,  A66 = 2*Gzx

On a 2-D mesh whose empty direction is z, the plane stress reduction is used:

sigma_xx = A11*epsilon_xx + A12*epsilon_yy
sigma_yy = A21*epsilon_xx + A22*epsilon_yy
sigma_xy = A44*epsilon_xy
sigma_zz = sigma_yz = sigma_xz = 0

nuyx = nuxy*Ey/Ex
J    = 1/(1 - nuxy*nuyx)
A11  = J*Ex,  A22 = J*Ey,  A12 = J*nuyx*Ex,  A21 = J*nuxy*Ey
A44  = 2*Gxy

The law is small-strain only; a nonlinear geometry solid model stops with an error. It keeps no history.

Model options

The elastic constants are read as plain scalars, without dimension sets; use one consistent pressure unit for all moduli.

On a 2-D mesh the z entries are not read. When the law is nested inside poroMechanicalLaw, rho is taken from the enclosing law's dictionary and need not be repeated.

The following are fatal errors at construction:

• a 2-D mesh without planeStress yes; in mechanicalProperties: the 2-D reduction is plane stress, and there is no plane strain form;
• a 2-D mesh whose empty direction is x or y.

solutionD is supplied to the law by the framework from the mesh and must not be given in the dictionary. The moduli, Poisson's ratios and the determinant J are not checked.

Tangents and volumetric split

• Scalar tangent: max(A11, A22, A33) for both the scalar and the deviatoric scalar tangent; the same value is returned as the law's bulk modulus.
• 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

None. The law declares no state, so it writes no state fields or restart files.

Example

From the rodAndSeabed tutorial, a 3-D case:

planeStress     no;

mechanical
(
    steel
    {
        type            poroMechanicalLaw;
        rho             rho [1 -3 0 0 0 0 0] 2650;
        biotCoeff       biotCoeff [0 0 0 0 0 0 0] 1.0;
        effectiveStressMechanicalLaw
        {
            type            anisotropicBiotElastic;

            // Young's moduli (in Pa)
            Ex              1.2e7;
            Ey              1.2e7;
            Ez              2e7;

            // Poisson's ratios (dimensionless)
            nuxy            0.2;
            nuyz            0.24;
            nuzx            0.4;

            // Shear moduli (in Pa)
            Gxy             0.5e7;
            Gyz             1.2e7;
            Gzx             1.2e7;
        }
    }
);

Migrating from the legacy law

• The 2-D/3-D selection is fixed (issue #334). The legacy law took its reduced branch on 3-D meshes, silently ignoring Ez, nuyz, nuzx, Gyz and Gzx and leaving the out-of-plane stresses at zero. A 3-D case now uses all nine constants, so its results change.
• A 2-D case (empty z) now needs planeStress yes; and only the four in-plane constants; the legacy law required all nine there.
• In the 2-D form the out-of-plane stress components are now set to zero rather than left unassigned.
• rho is read at construction; regionName is no longer read. The legacy law silently ignored solvePressureEqn; the solid model now refuses it for this law.
• The law now serves every integration-point location, and a fourth-order tangent is available by finite differences.

Tutorials

• rodAndSeabed, inside the effectiveStressMechanicalLaw sub-dictionary of poroMechanicalLaw