The following modeling topics are available:
The strain energy potential for a nearly incompressible hyperelastic material is:
 | (38–1) |
where:
The 2nd Piola-Kirchhoff stress is:
 | (38–2) |
The material stiffness tensor is:
 | (38–3) |
For a Neo-Hookian model:
 | (38–4) |
where:
The 1st-order derivatives in the above equations are:
 | (38–5) |
where:
The 2nd-order derivatives in the above equations are:
 | (38–6) |
Substituting the 1st-order derivatives of the potential from Equation 38–5 into Equation 38–2, the 2nd Piola-Kirchhoff stress becomes:
 | (38–7) |
The Cauchy stress is:
 | (38–8) |
where:
Substituting the 1st- and 2nd-order derivatives from Equation 38–5 and Equation 38–6 into Equation 38–3, the material tangent stiffness is:
 | (38–9) |
which can be simplified to:
 | (38–10) |
Using the Piola transform, the spatial tangent is:
 | (38–11) |
where:
Large deformation is formulated in a co-rotated frame given by the rotation, R, from the polar decomposition of the deformation gradient:
 | (38–12) |
where:
The rotation of the co-rotated frame is then:
 | (38–13) |
The stress and tangent stiffness tensors are returned in the co-rotated frame.
Given the deformation gradient, the rotation is determined from the closed-form solution of the inverse stretch:[1]
 | (38–14) |
where IU, IIU, and IIIU are the principle invariants of U. The stretch tensor is given in closed form by:
 | (38–15) |
The eigenvalues of U are the square root of the eigenvalues of C. The principal invariants of U in terms of the eigenvalues of C are:
where λ1, λ2, and λ3 are the eigenvalues of C given by the roots of the characteristic polynomial:
 | (38–16) |
where the principle invariants are:
C is a symmetric, positive-definite, rank 3 matrix. The characteristic polynomial has three positive real roots given by:
 | (38–17) |
The constitutive model defined by the Neo-Hookean hyperelastic material gives the stress in the spatial configuration by direct evaluation of Equation 38–8. No rate form is used and no integration is required. The Cauchy stress in the co-rotated frame is returned by the user-defined material subroutine.
The co-rotational rate relative to the co-rotated frame is called the Green-Naghdi rate. The constitutive response of the Neo-Hookean material described in the previous sections is in terms of the Lie derivative, or Truesdell rate, of the Kirchhoff stress:
 | (38–18) |
where:
The superimposed dot indicates the time derivative in the reference configuration.
The Jaumann rate is a convenient approximation of the Green-Naghdi rate:
 | (38–19) |
Using the Jaumann rate, the constitutive response in the co-rotated frame is given by:
 | (38–20) |
where cJ is the Jaumann tangent stiffness tensor and is implicitly defined by:
 | (38–21) |
Voigt notation is a reduced order representation of tensors and is the form
required for the stress and tangent stiffness tensors. Voigt notation is given by
converting pairs of indices to a single index via
and redefining the rate of deformation tensor as:
 | (38–22) |
In component form:
 | (38–23) |
where:
Minor symmetry with respect to the index pairs is required for this conversion. The 4th-order identity i and the Jaumann tangent stiffness lack minor symmetry. To ensure minor symmetry, they are redefined as:
 | (38–24) |
Converting to Voigt notation:
 | (38–25) |
The terms
and
in Voigt notation are:
 | (38–27) |
The spatial tangent in Voigt notation is:
 | (38–28) |
The stress terms in the Jaumann tangent stiffness are:
 | (38–29) |
With simplification, the Jaumann tangent stiffness becomes:
 | (38–30) |
For an arbitrary change of basis given by the orthogonal tensor T, the components of the 2nd-order stress tensor and the 4th-order tangent stiffness tensor transform as:
 | (38–31) |
In Voigt notation, Equation 38–31 becomes:
 | (38–32) |
Comparing the tensor and Voigt notation transformations, the change of basis tensor Q is given by:
 | (38–33) |
where:
For a change of basis from the spatial frame to the co-rotated frame, T = RT.