linearElastic

The isotropic Hookean linear elastic law, for small strains. The runtime type is:

linearElastic

It is the most widely used law in the solids4foam tutorials, and the law most often nested inside thermoMechanicalLaw and poroMechanicalLaw. The dictionary layout common to all laws is described in the material models page, and the framework itself in the framework README.

User Guide

What it computes

The stress is computed from the displacement gradient supplied by the solid model, with epsilon = symm(grad(D)):

sigma = 2*mu*epsilon + lambda*tr(epsilon)*I + sigma0

where sigma0 is an optional initial stress (see below). The law is small-strain only: it implements the small-strain update and a nonlinear geometry solid model, which asks for the finite-strain update, stops with an error.

The constants are derived from whichever pair of elastic entries was given:

mu     = E/(2*(1 + nu))
lambda = nu*E/((1 + nu)*(1 - 2*nu))        // plane strain / 3-D
lambda = nu*E/((1 + nu)*(1 - nu))          // planeStress yes
K      = lambda + (2/3)*mu

If mu and K are given, E and nu are first recovered from them with E = 9*K*mu/(3*K + mu) and nu = (3*K - 2*mu)/(2*(3*K + mu)).

Model options

The elastic constants are given as either E and nu or mu and K. The E/nu pair is checked first; if neither pair is complete the law stops at construction.

Entry Required Default Description
rho yes - Density, [1 -3 0 0 0 0 0]
E with nu - Young's modulus, [1 -1 -2 0 0 0 0]
nu with E - Poisson's ratio, dimensionless
mu with K - Shear modulus, [1 -1 -2 0 0 0 0]
K with mu - Bulk modulus, [1 -1 -2 0 0 0 0]
sigma0 no zero Uniform initial stress, [1 -1 -2 0 0 0 0]

The dimensions of E, mu and K (pressure) and of nu (dimensionless) are checked. nu must satisfy -1 < nu < 0.5: a value at or above 0.5 (to within SMALL) is a fatal error, because the bulk modulus would be ill-conditioned.

planeStress is a top-level entry of mechanicalProperties; it changes lambda as shown above and must not be given inside the law's dictionary.

The optional solvePressureEqn and pressureSmoothingScaleFactor entries are read by the solid model, not by this law. This law separates its deviatoric and volumetric responses, so, when selected directly, it can be used with hydrostatic stress smoothing and with the mixed displacement-pressure formulations. The thermoMechanicalLaw and poroMechanicalLaw wrappers do not provide the split, so neither is available through them.

Initial stress

An initial (residual) stress can be supplied in two ways:

  • a uniform value, as the sigma0 entry in the law's dictionary; or
  • a spatially varying volSymmTensorField named sigma0 in the start time directory (or in 0), for example written by setFields.

If both are present, the dictionary value wins and the field is ignored. The initial stress is added unchanged to every evaluation.

Tangents and volumetric split

  • Scalar tangent: 2*mu + lambda; the deviatoric scalar tangent is (4/3)*mu.
  • Fourth-order tangent: the analytical isotropic 6x6 elasticity matrix. The finite-difference tangent is also available.
  • Volumetric split: yes. The volumetric response is K*tr(epsilon), and the deviatoric part returned with it keeps any initial stress, including the spherical part of sigma0.

State variables

Name Type Role Description
sigma0 symmTensor prescribed Initial stress, read from a field

sigma0 is prescribed when it comes from a field and fixed when the dictionary gives it. Neither role is history, so this law writes no restart files and no state fields.

Example

From the plateHole tutorial:

planeStress     no;

mechanical
(
    steel
    {
        type            linearElastic;
        rho             rho [1 -3 0 0 0 0 0] 7854;
        E               E [1 -1 -2 0 0 0 0] 200e+9;
        nu              nu [0 0 0 0 0 0 0] 0.3;
    }
);

With a uniform initial stress:

        sigma0          sigma0 [1 -1 -2 0 0 0 0] (10e6 2e6 -3e6 15e6 0 -5e6);

Migrating from the legacy law

  • regionName is no longer read.
  • nu = 0.5 is now a fatal error; the legacy law accepted it and set K and lambda to GREAT. There is no longer a warning for nu > 0.49. For a nearly incompressible material use a mixed displacement-pressure solid model or solvePressureEqn.
  • solvePressureEqn combined with planeStress yes is no longer refused.
  • One evaluation serves every integration-point location (cells, faces, points, quadrature points). The legacy point overload refused a non-zero sigma0; this law does not.
  • E(), nu(), mu() and lambda() are still available to solid models: coupledUnsLinGeomLinearElasticSolid and kirchhoffPlateSolid require this law for that reason.

Tutorials

linearElastic is used by most of the solids/linearElasticity cases, for example:

It is also the law nested inside thermoMechanicalLaw in the solids/thermoelasticity tutorials, for example hotCylinder and hotSphere.