Diffusion-induced stress in inhomogeneous materials: concentration-dependent elastic modulus

By incorporating the contribution of solute atoms to the Helmholtz free energy of solid solution, a linear relation is derived between Young's modulus and the concentration of solute atoms. The solute atoms can either increase or decrease Young's modulus of solid solution, depending on the first-order derivative of the Helmholtz free energy with respect to the concentration of solute atoms. Using this relation, a closed-form solution of the chemical stress in an elastic plate is obtained when the diffusion behavior in the plate can be described by the classical Fick's second law with convection boundary condition on one surface and zero flux on the other surface. The plate experiences tensile stress after short diffusion time due to asymmetrical diffusion, which will likely cause surface microcracking. The results show that the effect of the concentration dependence of Young's modulus on the evolution of chemical stress in elastic plates is negligible if the change of Young's modulus due to the diffusive motion of solute atomsis is not compatible in magnitude with Young's modulus of the pure material. Also, a new diffusion equation is developed for strictly regular binary solid solution. The effective diffusivity is a nonlinear function of the concentration of solute atoms.