Energy dissipation rate and kinetic relations for Eshelby transformations
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2019 Elsevier Ltd Vasoya et al. (J. Appl. Mech., 86, 051005, 2019) gave a general expression for the dissipation associated with a single transforming Eshelby inclusion in a linear elastic solid and showed that requiring the dissipation to be non-negative provides a strong limit on the maximum value of the transformation strain magnitude. Non-negative dissipation is a necessary, but not sufficient, condition for satisfying the ClausiusDuhem inequality which requires the dissipation rate to be non-negative. Here, a general expression is given for the dissipation rate with multiple transforming Eshelby inclusions. When specialized to a single transforming Eshelby inclusion in an infinite solid, the limit on the magnitude of transformation strain for non-negative dissipation rate is one half that for non-negative dissipation. The condition for non-negative dissipation rate is expressed as the product of a configurational force, analogous to the PeachKoehler force for dislocations and to the J-integral for cracks, times the transformation strain rate. This gives a natural framework for specifying kinetic relations that guarantee a non-negative dissipation rate for Eshelby inclusions. Simple kinetic relations are proposed for the mesoscale modeling of the shear transformation zone (STZ) mechanism of plastic deformation in amorphous solids. Numerical examples are presented to illustrate implementation in a computational framework of a kinetic relation that guarantees a non-negative dissipation rate. Possible limitations of a linear elastic Eshelby inclusion model for the STZ plasticity mechanism are discussed in light of results from atomistic analyses.