Coupled deformation-composition evolution

These examples illustrate the interaction of composition distribution and stress field in a deformed solid solution. The mathematical structure exists for analysis of such phenomena, but the governing equations are inherently nonlinear analysis is very difficult if shape changes are taken into account and systematic study has not yet been undertaken. The purpose here is to formulate and analyze a physical situation involving coupled deformation and composition evolution that serves as a reasonably transparent vehicle for presenting the underl5dng ideas, but that avoids the complexity of evolution of shape. 

Consider an elastic layer occup5ung the region —oo x,y oo, — h z j/i. Attention is limited to fields that vary only through the thickness of the layer. The slab material is a Ai B solid solution as described in Section 9.6.1. The composition is assumed to vary through the thickness according to 

The layer is flat for any uniform composition and it remains so if there is no lattice mismatch between constituents A and B. Assume that constituent B has an isotropic extensional mismatch strain ei with respect to constituent A. As a result, the midplane of the layer is curved, in general, for a nonuniform distribution From (2.58), it is known that the state of stress inducing this curvature is 

The chemical potential representing the tendency for redistribution of composition is defined in (9.92). The mean normal stress Ujj appearing in this expression is axx + (Tyy. The stress component Gzz and all shear stress components vanish throughout the layer in this case as a result of symmetry, translational invariance and vanishing of traction on the faces 2 = of the layer. The parameter Kch in the chemical potential comes into play only when spatial gradients in are very steep, and its effect is neglected for the distribution given in (9.111). 

The first term in the expression (9.92) for chemical potential is essentially the derivative of the free energy of mixing of the solution with respect to composition. In the present example, this concentration dependent mea- 

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The coupling between deformation and driving force for change in compositional distribution was considered for the case of a layer with through-the-thickness variation in composition in Section 9.6.5. For the case of composition evolution represented by the solution in (9.118), determine the corresponding dependence of curvature of the layer on time. 

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