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

• [General behaviour of data type conversion]({% link api/assignments.md %})

Operators

• [Dot Product]({% link api/operators.md %})
• [Double Dot Product]({% link api/operators.md %})
• [Dyadic Product]({% link api/operators.md %})
• [Crossed-dyadic Product]({% link api/operators.md %})
• [Division]({% link api/operators.md %})
• [Addition]({% link api/operators.md %})
• [Subtraction]({% link api/operators.md %})

Scalar-valued functions

• [Trace]({% link api/functions/trace.md %})
• [Determinant]({% link api/functions/determinant.md %})
• [Norm]({% link api/functions/norm.md %})

Tensor-valued functions

• [Deviator]({% link api/functions/deviator.md %})
• [Unimodular]({% link api/functions/unimodular.md %})
• [Inverse]({% link api/functions/inverse.md %})
• [Transpose]({% link api/functions/transpose.md %})
• [Permute]({% link api/functions/permute.md %})
• [Square root]({% link api/functions/squareroot.md %})
• [Power]({% link api/functions/power.md %})
• [Identity]({% link api/functions/identity.md %})
• [Piola Operation]({% link api/functions/piola.md %})
• [Rotation Matrix]({% link api/functions/rotation.md %})

Helper Functions

• [As Array]({% link api/functions/asarray.md %})
• [As Abaqus Array]({% link api/functions/asabqarray.md %})
• [As Voigt]({% link api/functions/asvoigt.md %})
• [As Tensor]({% link api/functions/astensor.md %})
• [Import Strain]({% link api/functions/voigtstrain.md %})