HolzapfelGasserOgdenElastic

The Holzapfel-Gasser-Ogden (HGO) anisotropic hyperelastic law for fibre-reinforced soft tissue, in particular arterial wall. The runtime type is:

HolzapfelGasserOgdenElastic

The constitutive model is due to G. A. Holzapfel, T. C. Gasser and R. W. Ogden (2000).

User Guide

What it computes

This is a finite-strain law: it implements the finite-strain update only, so it is used with a nonlinear geometry solid model. A linear geometry solid model asks for the small-strain update, and the run stops with an error.

The response is a neo-Hookean ground substance plus two exponential fibre families, wound symmetrically about the circumferential direction Ec:

W = 0.5*mu*(I1bar - 3)
  + (k1/(2*k2))*sum_i (exp(k2*sqr(I_i - 1)) - 1)
  + U(J)

N4 =  cos(fibreAngle)*Ec + sin(fibreAngle)*Ea
N6 = -cos(fibreAngle)*Ec + sin(fibreAngle)*Ea

All invariants are taken on the isochoric deformation Fbar = J^(-1/3)*F:

bbar = symm(Fbar & Fbar^T)
Cbar = symm(Fbar^T & Fbar)
I4   = Cbar && symm(N4*N4),   n4 = Fbar & N4
I6   = Cbar && symm(N6*N6),   n6 = Fbar & N6

T     = mu*bbar
      + 2*k1*(I4 - 1)*exp(k2*sqr(I4 - 1))*symm(n4*n4)
      + 2*k1*(I6 - 1)*exp(k2*sqr(I6 - 1))*symm(n6*n6)

sigma = dev(T)/J + dU/dJ*I

U(J)  = 0.25*K*(J^2 - 1 - 2*ln(J))
dU/dJ = 0.5*K*(J^2 - 1)/J

where K is bulkModulus, a penalty enforcing near incompressibility.

A deformation gradient with J <= sqrt(SMALL) stops the run.

Model options

k2 and fibreAngle are read as dimensioned scalars, e.g. k2 k2 [0 0 0 0 0 0 0] 1.465;, and only their values are used. mu and bulkModulus must be positive; otherwise construction stops with an error.

Local basis

Ec and Ea are prescribed state variables. When the manager sets up the law's state it looks, for each, for a volVectorField of that name:

1. a file in the current (start) time directory;
2. a file in 0;
3. a field already registered by another model.

If one is found it is used, mapped from the cells to the law's integration points. Otherwise the dictionary vector is used when uniformLocalBasis is yes, and a zero vector when it is not. A zero-length Ec or Ea stops the run at the first evaluation. Both are normalised before use, so they need not be unit vectors. A field in the time directory takes precedence over the dictionary vector even when uniformLocalBasis is yes.

In the ratCarotid tutorial the fields are written by the case-local calcLocCoordinates utility, which Allrun compiles and runs before the solver.

State variables

Prescribed variables are read once and never written back, so they are not written with the results or to the restart files. The law has no persistent history.

Tangents

• scalar: (4/3)*mu + K. The fibre stiffness is not included; it is a constant stiffness estimate for the segregated solver, not a tangent of the energy.
• scalarDeviatoric: (4/3)*mu, for mixed displacement-pressure formulations.
• fourthOrderFiniteDifference: supported, through the base class.
• fourthOrder (analytical): not implemented; requesting it is a fatal error.

Mixed displacement-pressure formulation

The law provides the volumetric split, so it can be used with a mixed formulation such as coupledPressureDisplacementSolid or nonLinearGeometryTotalLagrangianTotalDisplacement with solvePressure true. The solid model asks for the isochoric stress and dU/dJ separately and replaces dU/dJ with its solved pressure. A large bulkModulus approaches the incompressible limit; ratCarotid uses one thousand times mu. The law can also be run in the displacement-only formulation, where the penalty alone controls the volume change.

Differences from the legacy law

• The shear modulus is given as mu only. E and nu are no longer read, and the nu = 0.5 requirement is gone. The given mu is used as it is: the legacy law overwrote it at start-up with a perturbation estimate.
• bulkModulus is new and required. The legacy law was incompressible only and took the spherical stress from the solid model's p and pf fields; this law never reads p, and the pressure enters through the volumetric split instead, so it is no longer tied to coupledPressureDisplacementSolid.
• The invariants are taken on Fbar; the legacy law assumed J = 1.
• Only Ec and Ea are needed, as cell fields. Er, Ecf, Eaf and Erf are no longer read. A uniform basis can be given in the dictionary with uniformLocalBasis.
• impKcoeff is no longer read, and the per-step effective shear modulus used for the implicit stiffness is replaced by the constant scalar tangents above. Unused legacy entries are ignored rather than reported.

Example

From the ratCarotid tutorial:

planeStress no;

mechanical
(
    pipe
    {
        type        HolzapfelGasserOgdenElastic;
        rho         rho [1 -3 0 0 0 0 0] 1200;
        mu          mu [1 -1 -2 0 0 0 0] 44.23e3;
        fibreAngle  fibreAngle [0 0 0 0 0 0 0] 39.76;
        k1          k1 [1 -1 -2 0 0 0 0] 0.206e3;
        k2          k2 [0 0 0 0 0 0 0] 1.465;
        bulkModulus bulkModulus [1 -1 -2 0 0 0 0] 44.23e6;
    }
);

Ec and Ea are read from the fields written by calcLocCoordinates. For a uniform basis instead:

        uniformLocalBasis yes;
        Ec                (1 0 0);
        Ea                (0 1 0);

Tutorials

• solids/hyperelasticity/ratCarotid: by default with nonLinearGeometryTotalLagrangianTotalDisplacement and solvePressure true; ./Allrun pressureDisplacement selects coupledPressureDisplacementSolid (foam-extend only).

References

• G. A. Holzapfel, T. C. Gasser and R. W. Ogden, A new constitutive framework for arterial wall mechanics and a comparative study of material models, Journal of Elasticity, 61 (2000) 1-48.