imoothness of the Scalar Coefficients in the Representation H(A) = αI + βA + γA~2 for Isotropic Tensor Functions of Class C~r

摘要

Many general representation theorems [1-4] have been proved for isotropic tensor functions in continuum mechanics. Almost all of these theorems, however, are algebraic in character. Besides the special instance of polynomial isotropic tensor functions, the algebraic theorems often leave two problems outstanding, namely: (ⅰ) the nonuniqueness of the representation, and (ⅱ) the unknown continuity and differentiability properties of a specific representation. An isotropic tensor function can often be expressed in different ways that conform to the same representation formula. Algebraically there is no basis for us to select one expression over another (unless we consider polynomial tensor functions and demand polynomial dependence in the representation). An arbitrary choice would likely result in a representation with terms which are discontinuous, even if the isotropic tensor function in question is smooth. As a result of these problems, much attention has been devoted to polynomial isotropic tensor functions in the study of representation theorems, especially during the development of the subject in the fifties and early sixties.