Compatibility constraint at interfaces with elastic, crystalline solids–I: Theory

Abstract

Starting with an extended Gibbs–Duhem equation and an expression for stress-deformation behavior derived by Oh and Slattery for elastic crystalline solids, we derive a new compatibility constraint on stress at coherent interfaces. Its use is demonstrated in determining the residual stresses developed during oxidation on the surface of a cylinder.

Keywords:

• coherent interface
• compatibility condition
• Hadamard condition
• oxidation
• residual stress

Acknowledgements

The authors are grateful for the discussions with and suggestions by D.C. Lagoudas and J.R. Walton.

Notes

Notes

1. The choice of reference configuration is clearly important in computing displacements, strains, and stresses. In fact, there is considerable precedent that suggests that the metal and oxide must have different reference configurations 6, but let us examine this question further. The oxide and metal are different phases. A phase is a body with a distinct description of material behavior, but this simple definition of a phase does not say anything about the choice of reference configuration. As is true for most materials, the metal and the oxide can almost certainly be described as simple materials, materials for which stress is a functional of the history of the deformation gradient 7. When multiple reference configurations can be used for a phase, this functional takes the same form in all cases, the deformation gradient simply being replaced by the deformation gradient with respect to the new reference configuration operating on ∇λ, where λ is the map of the original reference configuration onto the new reference configuration 7. The important point here is that λ is a mapping of material particles for a specific phase from one reference configuration to another. It does not represent a mapping between two different phases. For example, it is not a mapping of metal particles onto oxide particles. This simply confirms the summary of this point given by Rajagopal and Srinivasa 6.

2. It is important to recognize that the approach of Slattery and Lagoudas 2, in which the mass density is included as one of independent variables, and our use of Equation (11) are fundamentally equivalent as the result of Equation (10). The results derived by these two approaches can be used somewhat interchangeably, so long as care is taken to properly recognize the independent variables. Equation (11) has the advantage of being consistent with Green and Adkins 18, who further discussed the particular functional dependence upon the six E _(m) · (𝒞 − ℐ) E _(n) to be expected for various crystal classes. The approach of Oh and Slattery 2 has the advantage of explicitly recognizing the role of thermodynamic pressure in the description of stress–deformation behavior, in the Euler equation, and in the Gibbs phase rule.

3. This corrects an error in the original derivation. Since  does not depend upon , thermodynamic pressure P will not appear in the extended Gibbs–Duhem equation.