MooneyRivlinElastic

Three-parameter Mooney-Rivlin (Rivlin-Saunders) hyperelasticity, with a volumetric-isochoric split. The runtime type is:

MooneyRivlinElastic

The model adds a dependence on the second invariant to the neo-Hookean form. The dictionary layout common to all laws is described in the material models page.

User Guide

What it computes

At each integration point, from F and J = det(F):

isoB = J^(-2/3)*symm(F & F.T())
I1   = tr(isoB)
I2   = 0.5*(I1^2 - tr(isoB & isoB))
s    = 2*(c10 + c11*(I2 - 3))*isoB - 2*(c01 + c11*(I1 - 3))*inv(isoB)
p    = 0.5*K*(J^2 - 1)
sigma = (dev(s) + p*I)/J

which is the Cauchy stress of

Psi = c10*(I1 - 3) + c01*(I2 - 3) + c11*(I1 - 3)*(I2 - 3) + U(J)
U(J) = 0.25*K*(J^2 - 1 - 2*ln(J))

c11 = 0 gives the two-parameter Mooney-Rivlin model, and c01 = c11 = 0 a neo-Hookean model with mu = 2*c10. The law is finite-strain only; a linear geometry solid model stops with an error. A point with J <= sqrt(SMALL) is a fatal error. The law holds no history.

Model options

Entry Required Default Description
rho yes - Density, [1 -3 0 0 0 0 0]
c10 yes - First coefficient, [1 -1 -2 0 0 0 0]
c01 yes - Second coefficient, [1 -1 -2 0 0 0 0]
c11 yes - Coupled coefficient, [1 -1 -2 0 0 0 0]
K K or nu - Bulk modulus, [1 -1 -2 0 0 0 0]
nu K or nu - Poisson's ratio, used if K is absent
solvePressureEqn no no Hydrostatic smoothing, see below
pressureSmoothingScaleFactor no 100 Scale of that smoothing

All three coefficients are required; give c11 c11 [1 -1 -2 0 0 0 0] 0; if the coupled term is not wanted. Their dimensions, and those of K and nu, are checked.

Giving neither K nor nu is a fatal error. If both are given, K is used and a warning is printed. When only nu is given, K follows from the linearised Young's modulus:

E = 6*(c10 + c01),   K = E/(3*(1 - 2*nu))

and nu >= 0.5 - SMALL is a fatal error; give K directly for a nearly incompressible material. The linearised shear modulus is always mu = 2*(c10 + c01). The top-level planeStress entry is not used by this law: there is no plane-stress reduction.

Tangents and volumetric split

  • Scalar tangent: (4/3)*mu + K; the deviatoric scalar tangent is (4/3)*mu. These are constant, from the linearised moduli.
  • Fourth-order tangent: fourthOrderFiniteDifference only. Asking for the analytical fourthOrder tangent is a fatal error.
  • Volumetric split: yes. The isochoric stress is dev(s)/J and the volumetric response is p/J = dU/dJ. The isochoric stress is invariant under a superposed dilation.

The split lets the law be used with solvePressureEqn and with a mixed displacement-pressure formulation, as for neoHookeanElastic. The U(J) above is the one the pressure equation of nonLinearGeometryTotalLagrangianTotalDisplacement assumes.

Hydrostatic stress smoothing

solvePressureEqn yes asks the solid model to replace the explicit hydrostatic stress p = 0.5*K*(J^2 - 1) with a smoothed field, sigmaHyd, which is written. It is done by the solid model, not the law, and only with nonLinearGeometryUpdatedLagrangian, nonLinearGeometryTotalLagrangianTotalDisplacement or linearGeometryTotalDisplacement, the implicitSegregated algorithm, a single material, and without solvePressure. Any other combination is refused at start-up.

State variables

None. The law writes no restart files and no state fields.

Example

From the cylinderCrush hyperelasticity tutorial:

planeStress     no;

mechanical
(
    rubber
    {
        type            MooneyRivlinElastic;
        rho             rho [1 -3 0 0 0 0 0] 1000;
        K               K [1 -1 -2 0 0 0 0] 1.410e+9;
        c10             c10 [1 -1 -2 0 0 0 0] 0.293e+6;
        c01             c01 [1 -1 -2 0 0 0 0] 0.177e+6;
        c11             c11 [1 -1 -2 0 0 0 0] 0;
        solvePressureEqn yes;
        pressureSmoothingScaleFactor 100;
    }
);

Differences from the legacy law

  • The coefficients are uniform. c10, c01 and c11 fields in the start time directory are no longer read, and are no longer written.
  • No initial stress is carried: a sigma0 field is not read, and symm(F & sigma0 & F.T()) is no longer added to the stress.
  • regionName is no longer read.
  • Giving both K and nu now prints a warning; K is still used. A nu at or above 0.5 - SMALL is now a fatal error.
  • The hydrostatic smoothing is done by the solid model rather than the law, and is refused where it is not supported rather than ignored.
  • A fourth-order tangent is available by finite differences.

Tutorials

References

  • M. Mooney, A theory of large elastic deformation, Journal of Applied Physics, 11, 582-592, 1940, 10.1063/1.1712836.
  • R.S. Rivlin and D.W. Saunders, Large elastic deformations of isotropic materials VII. Experiments on the deformation of rubber, Philosophical Transactions of the Royal Society of London A, 243, 251-288, 1951, 10.1098/rsta.1951.0004.