Products: ABAQUS/Standard ABAQUS/Explicit
Many of the constitutive models in ABAQUS require tensors to be stored to define the state at a material calculation point. Such “material state tensors” are stored as their components in a local, orthonormal, system at the material calculation point. The orientation of that system with respect to the global spatial system is stored as a rotation from the global axis system. The purpose of this section is to define the manner in which such tensors are stored and updated.
Three types of local basis are used in ABAQUS for material calculations. For isotropic materials in continuum elements the global, spatial, system is used—the material basis is fixed in time. For isotropic materials in structural surface elements (shells and membranes) the local system is defined by the standard ABAQUS convention described in Conventions, Section 1.2.2 of the ABAQUS Analysis User's Manual; and for beams and trusses it is defined with the 1-direction along the axis of the member and the 2- and 3-directions in material directions in the cross-section. Thus, with isotropic materials the material basis is always the same as the element basis, although for structural elements the material basis changes with time. For anisotropic materials the material basis must be defined by the user and rotates with the average rigid body spin of the material. In this case the material basis and the element basis are not the same.
We refer to this local material basis at time t as , where the superscript indicates that the basis is associated with material calculations and means that the basis is taken at time t. In this section Latin subscripts (like the above) take the range 1–3, while Greek subscripts will take the range 1–2.
Any tensor associated with the material's state, , say (such as the stress tensor ), is stored in terms of its components along the material basis:
The increment from time t to time of local motion at the material calculation point is defined by the incremental deformation gradient,
The polar decomposition of is
During an increment any material state tensor changes according to
For isotropic materials the have been chosen for geometric convenience only, so the are quite general.
For anisotropic materials the material basis system, , rotates with the average rigid body rotation of the material, , and so is updated by