Stress invariance and redundant moduli in three-dimensional elasticity

A three-dimensional framework is established for generating invariant stress configurations and associated shifts in the elastic compliance. Under these shifts the stress throughout an elastic body is unaltered, while the compatibility equations for the strain are automatically satisfied. The types of invariant stress fields and translations of the compliance identified here generalize the results of Cherkaev, Lurie & Milton (CLM) for planar elasticity. The key to the classification is the partitioning of the fourth-order compliance tensor into symmetric and antisymmetric components. The CLM theorem and its generalization are closely linked to the six-dimensional antisymmetric part of the compliance, and several examples are given of stress invariance under shifts of these elements of the compliance tensor.