Products: ABAQUS/Standard ABAQUS/Explicit
All the solid elements in ABAQUS allow for finite strain and rotation in large-displacement analysis. For kinematically linear analysis the strain is defined as
For cases where the strains and/or rotations are no longer small, two ways of measuring strain are used in the solid elements in ABAQUS. When the hyperelastic or hyperfoam material definition is used with an element, ABAQUS internally uses the stretch values calculated directly from the deformation gradient matrix, , to compute the material behavior. With any other material behavior it is assumed that any elastic strains are small compared to unity, so the appropriate reference configuration for the elasticity is only infinitesimally different from the current configuration and the appropriate stress measure is, therefore, the Cauchy (“true”) stress. (More precisely, the appropriate stress measure should be the Kirchhoff stress defined with respect to the elastic reference configuration, but the assumption that this reference configuration and the current configuration are only infinitesimally different makes the Kirchhoff and Cauchy stress measures almost the same: the differences are on the order of the elastic strains compared to unity). The conjugate strain rate to Cauchy stress is the rate of deformation,
For example, the stress is integrated by this method as
As shown in Procedures: overview and basic equations, Section 2.1.1, the contribution of the internal work terms to the Jacobian of the Newton method that is often used in ABAQUS/Standard is
where and are evaluated at the end of the increment.Using the integration definition above, it can be shown that
However, rather than computing the tangent matrix for the Newton method on this basis, we approximate this by using
This Jacobian is the tangent stiffness of the rate form of the problem. Experience with practical cases suggests that this approximation provides an acceptable rate of convergence in the Newton iterations in most applications with real materials.
The strain and rotation measures described above are approximations. Probably the most limiting aspect of these approximations is the definition of the rotation increment . While this measure does give a representation of the rotation of the material at a point in some average sense (both in ABAQUS/Standard and ABAQUS/Explicit), it is clear that each of the individual material fibers at a point has a different rotation (unless the material point undergoes rigid body motion only or, as an approximate extension, if the strains at the point are small). This suggests that the formulation described above will not be suitable for applications where the strains and rotations are large and where the material exhibits some form of anisotropic behavior. A common example of such cases is the induction of anisotropy through straining, as in “kinematic hardening” plasticity models. The integration methods described above are not suitable for such material models at large strains (for practical purposes with typical material parameters this means that the solutions will be quite wrong when the strains are greater than 20%–30%). Therefore, the use of the kinematic hardening model in ABAQUS at such strain levels is not recommended. There is extensive literature on this subject; for example, see Agah-Tehrani et al. (1986).