The Shield transform W*(F) of an isotropic elastic strain energy function W(F) is
W*(F) = (detF)W(F—1).
This transformation is reflexive, it preserves objectivity and material symmetry for isotropic materials, and it also preserves infinitesimal strain response, ellipticity and Hadamard stability, and the Baker – Ericksen condition. In view of Shield’s Inverse Deformation Theorem, the transforms of three classes of compressible, isotropic solids that afford both universal deformation solutions and reductions to quadrature define two new classes that afford similar exact solutions.
The supplementary strain energy w, defined as
w = a(i — 3) + b(j — 3) + c(k —1),
where (i, j, k) are the principal invariants of the stretch tensor and (a, b, c) are material constants, is not positive-definite in infinitesimal deformation. However, it has the unusual property that it supports every irrotational deformation without body force. This implies that, if a strain energy W supports a particular irrotational deformation, then the augmented strain energy W + w supports the same deformation. The usefulness of this idea is illustrated with respect to the solutions of Horgan et al for Blatz-Ko materials.
Two families of deformations, described by
r = f(X), q = g(X) + CY + DZ, z = h(X) + EY + FZ
and
x = f(R), y = g(R) + CQ + DZ, z = h(R) +EQ + FZ
are such that the governing system of second-order nonlinear ordinary differential equations reduces to a first order system for both isotropic and anisotropic materials.
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