By Gerard G. Emch
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Extra resources for Algebraic Methods in Statistical Mechanics and Quantum Field Theory (Physics & Astronomical Monograph)
9) of the elasticity and stress-composition tensors are consistent with the chain-rule calculation _ ¼ CðE; r~Þ½E _ þ T N X Aa ðE; r~Þr_a : ð7:11Þ a¼1 Note that, by Eq. 9)2, Fick’s law becomes 0 1 N N 2 ^ X X › C ðE; r ~ Þ g b A Mab ðE; r~Þ@ a ¼ 2 b ›rg 7r þ A ðE; r~Þ7E ; › r b¼1 g¼1 ð7:12Þ where, using Cartesian components, def ðAb 7EÞj ¼ Abkl ›Ekl ; ›x j ð7:13Þ so that jai ¼ N X 0 N X 2 Mijab ðE; r~Þ@ b¼1 g¼1 1 ›2 C^ ðE; r~Þ ›rg ›Ekl A b þ Akl ðE; r~Þ : ›x j ›rb ›rg ›x j ð7:14Þ Thus both density gradients and strain gradients may drive atomic diffusion.
Gurtin defined over ›RðtÞ for all t; which we view as the velocity with which an external agency adds material at ›RðtÞ: Compatibility then requires that v›R have V›R as its normal component, v›R ·n ¼ V›R ; ð11:3Þ but v›R is otherwise arbitrary. We refer to any such field v›R as a velocity field for ›R: Non-normal velocity fields, while not intrinsic, are important. For example, given an arbitrary time-dependent parametrization x ¼ rðp1 ; p2 ; tÞ of ›R; the field defined by v›R ¼ ›r=›t (holding ðp1 ; p2 Þ fixed) generally represents a nonnormal velocity field for ›R: But while it is important that we allow for the use of non-normal velocity fields, it is essential that the theory itself not depend on the particular velocity field used to describe a given migrating control volume.
E. Gurtin 36 IX. Constitutive Theory for a Substitutional Alloy The lattice constraint N X ra ¼ rsites a¼1 renders the constitutive theory for a substitutional alloy more difficult than that for an unconstrained material. In many respects the substitutional theory mirrors that for unconstrained materials; in particular, the theory is based on constitutive equations in which the density-list r~ ¼ ðr1 ; r2 ; …; rN Þ appears as an independent variable. Difficulties arise because each such list r~ must be admissible; that is, must satisfy the lattice constraint and must have 0 # ra , 1 for all atomic species a: Thus, since varying one of the densities while holding the others fixed, violates the lattice constraint, standard partial differentiation of the constitutive response functions with respect to the atomic densities is not well-defined.