Stress Independence of Poisson's Ratio and Divergence-Free Body Forces

Abstract

In simply connected bodies, the compatibility conditions in 2D elasticity are independent of Poisson's ratio (and hence there is a reduction in the number of constants) for divergence-free body forces for the boundary-value problem of traction. It is shown that that for divergence-free body forces in two fields with different compliances and the same stress (hence, inducing a reduction in constants and independence of Poisson's ratio) the displacements differ by a field that is a solution for λ + 2μ = 0, where λ and μ are the Lame constants for the elastic material, which is a constant rotation. Seen from another aspect, in a Cosserat spectral orthogonal and complete decomposition of the stress [12], the terms associated with Poisson's ratio in the stress representation have as coefficient the inner product of the body force with a displacement which is an eigenfunction at λ + 2μ = 0. This displacement is orthogonal to a divergence- free body force vector, which results in the vanishing of the coefficient of the Poisson's ratio dependence term in the orthogonal decomposition. Due to the completeness and orthogonality of the eigenfunctions, divergence-free body force is a necessary and sufficient condition for stress independence of Poisson's ratio in simply connected 2D bodies, while for multiply connected ones, the Michell generalized conditions also have to be satisfied, as shown here as well. Thus, both approaches, the CLM [1] theorem, which deals with reduction of constants, and the Cosserat spectral decomposition, which shows explicitly the material dependence of the solution, lead to the same conditions for stress independence of Poisson's ratio both for simply connected and multiply connected 2D bodies.