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13) A "'(T,F) = - T r 11- 1 (T - ll)c(Il,F)d ll + Th(F) + g(F), TO where c( T,F) is experimentally obtained (cf. 5). 14) "'(T,F) = M(T) + Th(F) + g(F). For a material of Mooney-Rivlin type, h(F) principal invariants of B. 15) T = -pI + f(T) -1 ( T - TO a1 + a2 ~ )B T - TO - f(T) (b 1 + b - 2 TO 1 )8- , with aI' a2 , bi > 0 and b2 ) O. 15). 16) 3. Boundary Value Problems for Spherically Symmetric Bodies We consider a cartesian system and a set of spherical coordinates IE 3. 1) where R1 and R2 denote the radii of the inner and the outer surfaces of the shell, respectively.
1) E ~ £ W~,m(E;Rn) such that < m. Idea of proof We 'blow up' the minimization problem in a neighbourhood of xo ' so that u becomes linear. This is done by defining, for £ > 0 sufficiently small, x- x u£(x) = u(x) + £+( ~ ), where + is extended by zero outside E, Making the change of variables ~ x - Xo = £y, and letting £ + 0 in the inequality 0 I(u£)) I(ul. 1, including treatment of the case when are given in [RJ. 1) says roughly that W is quasi convex at and this follows if W does not take the value +m.
1). This approach is consistent with the point of view that a constrained material is, in fact, the limiting case of a family of unconstrained ones (cf. [A]) and is also compatible with the variational formulation of the problem. In this framework, we seek deformations and temperature fields that maximize the entropy (cf. , if the 'reaction entropy' integrates to zero on the body) (cf. [G2], [C3]). In this article, I study radial equilibrium solutions to the equations of balance of mass, linear momentum and energy for a spherically symmetry body with the previously described The ~ain ~aterial properties.