Matrices and Tensors - Continuum Mechanics of Anisotropic Materials - page 382

Problems
A.10.1 Construct the two-dimensional Mohr's circle for the matrix A given in
problem A.7.3.
A.10.2 Construct the three-dimensional Mohr's circles for the matrix T given in
problem A.7.2.
A.11 Special Vectors and Tensors in Six Dimensions
The fact that the components of a second order tensor in n dimensions can be
represented as an n -by- n square matrix allows the powerful algebra of matrices to
be used in the analysis of second order tensor components. In general this use of the
powerful algebra of matrices is not possible for tensors of other orders. For example
in the case of the third order tensor with components A ijk one could imagine a
generalization of a matrix from an array with rows and columns to one with rows,
columns and a depth dimension to handle the information of the third index. This
would be like an n -by- n -by- n cube sub-partitioned into n 3 cells that would each
contains an entry similar to the entry at a row/column position in a matrix. Modern
symbolic algebra programs might be extended to handle these n -by- n -by- n cubes
and to represent them graphically. By extension of this idea, fourth order tensors
would require an n -by- n -by- n -by- n hypercube with no possibility of graphical
representation. Fortunately for certain fourth order tensors (a case of special interest
in continuum mechanics) there is a way to again employ the matrix algebra of n -by-
n square matrices in the representation of tensor components. The purposes of this
section it to explain how this is done.
The developments in this text will frequently concern the relationship between
symmetric second order tensors in three dimensions. The symmetric second order
tensors of interest will include stress and stress rate and strain and strain rate, among
others. The most general form of a linear relationship between second order tensors
in three dimensions involves a three-dimensional fourth order tensor. In general the
introduction of tensors of order higher than two involves considerable additional
notation. However, since the interest here is only in three-dimensional fourth order
tensors that relate three-dimensional symmetric second order tensors, a simple
notational scheme can be introduced. The basis of the scheme is to consider a
three-dimensional symmetric second order tensor also as a six-dimensional vector,
and then three-dimensional fourth order tensors may be associated with second
order tensors in a space of six dimensions. When this association is made, all of the
algebraic machinery associated with the linear transformations and second order
tensors is available for the three-dimensional fourth order tensors.
The first point to be made is that symmetric second order tensors in three
dimensions may also be considered as vectors in a six-dimensional space. The
one-to-one connection between the components of the symmetric second order
tensors T and the six-dimensional vector T is described as follows. The definition of