Module Function Overview

This module provides several useful assignments, operators and functions for the derived Tensor Data Types.

A word on Voigt notation

Symmetric 3x3 Tensors may be stored as 6x1 column vectors with the help of the well known Voigt notation. In an analogous way a transformation may be obtained for at least minor symmetric fourth order tensors (3x3x3x3) which may be stored as a 6x6 matrix. If the fourth order tensor is also major symmetric the reduced 6x6 matrix is also symmetric.

There is no unique method for the storage ordering of tensor components - this toolbox uses the following approach: 11,22,33,12,23,31. If you use Abaqus please use asabqarray to export tensor components.

To ensure consistency in calculating the virtual work shear terms are treated differently in Voigt notation: “stress”-like tensors or, to be more precisely, contra-variant tensor components are stored “as they are” whereas “strain”-like tensors or, again, co-variant tensor components are stored with doubled shear components. As this method is quite complicate for mixed-variant tensors and not really straight-forward to implement inside a whole toolbox this module uses a different “Voigt”-like approach:

All tensors, whether they are “stress”- or “strain”-like are stored with original (no doubled) shear components. Instead all dot- and double-dot-products are modified to take virtual work consistency into account. To be more precise, the function asvoigt is not really a Voigt storage - it is more a synonym for storing symmetric tensors as vectors and matrices. The user does have to take care of that storage if the strain in a user subroutine is used as an input for a Tensor2s data type and divide all shear components by a factor of 2.

Assignments

Operators

Scalar-valued functions

Tensor-valued functions

Helper Functions

Table of contents