Alloy separation of a binary mixture in a stressed elastic sphere

We consider the static state of a spherical isotropic binary elastic solid mixture whose boundary is given a uniform radial displacement. The elastic volumetric strain energy is given by the classical quadratic form from linear elasticity theory,% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqef0uAJj3BZ9Mz0bYu% H52CGmvzYLMzaerbd9wDYLwzYbItLDharqqr1ngBPrgifHhDYfgasa% acOqpw0xe9v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8Wq% Ffea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dme% GabaqaaiGacaGaamqadaabaeaafiaakeaacaqGxbGaeyypa0ZaaSaa% aeaacaGGXaaabaGaaiOmaaaacaGG7bGaaCOUdiaacIcacaqGJbGaam% ykaiaadIcacaWG0bGaamOCaiaadwgacaWGPaWaaWbaaSqabeaacaGG% YaaaaOGaey4kaSIaaiOmaiabeY7aTjaacIcacaqGJbGaaiykaiaacY% hacaWGLbGaeyOeI0IaaiiiamaalaaabaGaamymaaqaaiaadodaaaGa% aiikaiaadshacaWGYbGaamyzaiaacMcacaWGXaGaaiiFamaaCaaale% qabaGaaiOmaaaakiaac2hacaGGUaaaaa!63E0!\{\text{W}} = \frac{1}{2}\{ {\mathbf{\kappa }}({\text{c}})(tre)^2 + 2\mu ({\text{c}})|e - \frac{1}{3}(tre)1|^2 \} .\]Here, e is the infinitesimal strain tensor, c0, 1] is the volumetric concentration of the mixture, and (·) and (·) are the (positive) bulk and shear material moduli, respectively, which are given functions of the concentration. As a function of c and e, the strain energy function is generally nonconvex. Thus, we consider the nonconvex problem of minimizing the potential energy of the body, among all spatial concentration and displacement fields, subject to a given boundary displacement and a fixed amount of component materials. Assuming spherical symmetry, we find that the two component materials must be separated in the optimal state of minimum potential energy. The harder material forms the central core of the sphere, and the softer material is segregated into a surrounding shell. This behavior is remindful of a general notion in metallurgy that in the casting of materials the harder material tends to migrate toward the center.Partial support of the NSF under grant MSS-9024637 and Alliant Techsystems Inc. is gratefully acknowledged.Professor R. Bartelletti of the Università di Pisa is gratefully acknowledged.