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sigma prime Subscript 23 Baseline equals m sigma 23 minus n sigma 13 equals sigma prime Subscript 32 Baseline comma 5th Row sigma prime Subscript 13 Baseline equals n sigma 23 plus m sigma 13 equals sigma prime Subscript 31 Baseline comma 6th Row sigma prime Subscript 12 Baseline equals minus m n sigma 11 plus m n sigma 22 plus left-parenthesis m squared minus n squared right-parenthesis sigma 12 equals sigma prime Subscript 21 Baseline period EndLayout"/>(2.179)
The relationships (2.178) and (2.179) are the standard transformations, arising from tensor theory, for the rotation of stress components about one axis of a right-handed rectangular set of Cartesian coordinates. Identical transformations apply when considering the strain tensor so that
with inverse relations
2.17 Transformations of Elastic Constants
On substituting the stress-strain relations (2.170) into (2.181)
Substitution of (2.178) into (2.182) leads to the relations
where
