A Representation Formula For Time Rate And Conjugate Stress Of An Arbitrary Hill's Strain

In the theory of finite deformations and in constitutive modeling, Hill's strain measures, their time rate and their conjugate stresses are basic. The problem of finding expressions for these basic quantities has attracted many researchers' attention in past decades, especially in the last decade. By using the principal axis method, Hillderived component expressions for E(C) and some conjugate stresses. However,component expressions as given in a principal basis, are valid only in this basis. Indeed, only very recently the smoothness properties of the representative coefficients have been known that even the continuous differentiability property of the represented isotropic tensor function cannot ensure the continuity property of each representative coefficient in the representation. For the different cases of coalescence of the eigenvalues of the argument tensor, the existing formulas either supply different expressions for the representative coefficients or require a limiting process resulting from the assumption of the continuous differentiability property of each representative coefficient which might not be suitable for practical computations.In this paper, the representation formula for isotropic tensor functions in Xiao,Bruhnsand and Meyers(2002) are used to study time rate and conjugate stress of an arbitrary Hill—Seth's strain. A new representation formula is obtained for time rate and conjugate stress of an arbitrary Hill—Seth's strain, such as continuity and differentiability properties of the representation, determination of the representative coefficients in terms of experimental data for stress and deformation tensors, and computations of finite strain measures.Moreover, we apply this representation formulas to derive new tensorial expressions for Hill—Seth's elastic strain.