Convexity Conditions for Isotropic Functions

Abstract

The constitutive inequalities of the preceding chapter gain much more concrete forms for the thermodynamic potentials of isotropic solids; the goal of this chapter is to describe them. Sects. 18.1, 18.2, and 18.3 deal with the convexity of symmetric, isotropic, and objective-isotropic functions, respectively, and each section uses the results of the preceding sections. Using the results on objective-isotropic functions, Sect. 18.5 exhibits important special classes of polyconvex functions. The final section deals with the positivity properties of the second differential, the Legendre—Hadamard condition, Baker—Ericksen inequalities, and the inequalities of Coleman & Noll and Hill.