A reciprocal theorem in the linearized theory of couple-stresses

Abstract

Within the framework of the linearized theory of couple-stresses for perfectly elastic, non-homogeneous, anisotropic materials, a general reciprocal theorem is derived and applied in the proof of other general theorems in both elastostatics and elastodynamics. In particular, a general work theorem is derived; for small periodic vibrations it is shown that the normal mode functions are orthogonal; and these results are combined to prove that if the strain energy function is positive definite, then the normal mode frequencies are real and non-vanishing. In addition,Castigliano's theorem and several other theorems that are common to structural analysis are shown to follow in a rigorous and natural way from the theory of couple-stresses.