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Nonlinear Field Theory of Stress Induced Diffusion in Crystalline Solids

Two different mathematical descriptions of the material drift based respectively on the concept of massand molar-averaged velocities are presented. In this way two different material derivatives over time are introduced: mass and molar, respectively. The first approach based on the overall mass motion is used mainly in the traditional continuum thermodynamics of deformable solids while the second approach developed here comes back to the constitutive modelling used mainly in the chemical physics. The problem of the stress induced diffusion is situated on the border of the two different approaches developed often separately. In this paper, taking into account the standard balance laws for molar motion, the driving forces for diffusion are determined. The forces obtained for diffusion of chemical components depend not only on gradients of chemical potentials (osmotic force) but also on some additional forces controlling the stress induced diffusion process. The driving forces derived for dislocation field comprise the Peach-Koehler force as well as an additional term resulting from the effect of dislocation movement on the concentration of vacancies. Mass balance vs. molar balance Mass balance for each component of solid structure satisfies the following balance law ρ̇i + ρi div ẋi = 0, (1) where ρi and ẋi denote the mass density and velocity of the i-th material particle while ρ̇i denotes the material derivative of mass density of the i-th component, i.e. ρ̇i = ∂ρi ∂t + grad ρi ẋi. (2) Dividing the first equation by the molar mass of the i-th component, Mi = const, it is easy to prove that the molar concentration also satisfies the master balance law ċi + ci div ẋi = 0, (3) where ρi = Mici and ċi = ∂ci ∂t + grad ci ẋi. (4) This means that to determine an overall motion function for a material as a whole we can use alternatively the mass densities or molar concentrations. Namely, let us define two mutually different overall material velocities u df = 1 ρ N ∑ i=1 ρiẋi, v df = 1 c N ∑