Hypoelastic behavior

In a hypoelastic material the rate of change of stress is defined as a tangent modulus matrix multiplying the rate of change of the elastic strain:

where $d\mathit{\sigma }$ is the rate of change of the stress (the “true,” Cauchy, stress in finite-strain problems), ${\mathbf{D}}^{el}$ is the tangent elasticity matrix, and $d{\mathit{\epsilon }}^{el}$ is the rate of change of the elastic strain (the log strain in finite-strain problems).

The entries in ${\mathbf{D}}^{el}$ are provided by giving Young's modulus, E, and Poisson's ratio, $\nu$, as functions of strain invariants. The strain invariants are defined for this purpose as

You can define the material parameters directly or by using a user subroutine.

For plane stress and uniaxial stress states Abaqus/Standard does not compute the out-of-plane strain components. For the purpose of defining the above invariants, it is assumed that $I_1=\mathrm{\hspace{0.25em}0}$; that is, the material is assumed to be incompressible. For example, in a uniaxial stress case (such as a truss element) this assumption implies that

For large-displacement analysis the strain measure in Abaqus is the integration of the rate of deformation. This strain measure corresponds to log strain if the principal directions do not rotate relative to the material. The strain invariant definitions should be interpreted in this way.