Strain and interface energy of ellipsoidal inclusions subjected to volumetric eigenstrains: shape factors

H J Böhm,G A Zickler,J Svoboda,F D Fischer

doi:10.1007/s00419-021-02066-1

Abstract

Thermodynamic modeling of the development of non-spherical inclusions as precipitates in alloys is an important topic in computational materials science. The precipitates may have markedly different properties compared to the matrix. Both the elastic contrast and the misfit eigenstrain may yield a remarkable generation of elastic strain energy which immediately influences the kinetics of the developing precipitates. The relevant thermodynamic framework has been mostly based on spherical precipitates. However, the shapes of actual particles are often not spherical. The energetics of such precipitates can be met by adapting the spherical energy terms with shape factors. The well-established Eshelby framework is used to evaluate the elastic strain energy of inclusions with ellipsoidal shapes (described by the axes a, b, and c) that are subjected to a volumetric transformation strain. The outcome of the study is two shape factors, one for the elastic strain energy and the other for the interface energy. Both quantities are provided in the form of easy-to-use diagrams. Furthermore, threshold elastic contrasts yielding strain energy shape factors with the value 1.0 for any ellipsoidal shape are studied.

Highlights

Introduction and motivationThe authors outlined in detail [1] how the strain energies of spheroidal inclusions can be obtained from the spherical case by shape factors
Understanding both the positive or negative roles of precipitates is an important topic in materials research; see, e.g., the pertinent chapter in the well-established book on kinetics of materials by Balluffi et al [2] or exact engineering treatments such as [3]
The authors and associated researchers have provided in [4,5,6,7] essentially all kinetic relations, including the mechanical terms, the elastic strain energy contributions, derived for spherical inclusions

Summary

Introduction and motivation

The authors outlined in detail [1] how the strain energies of spheroidal inclusions can be obtained from the spherical case by shape factors. Understanding both the positive or negative roles of precipitates is an important topic in materials research; see, e.g., the pertinent chapter in the well-established book on kinetics of materials by Balluffi et al [2] or exact engineering treatments such as [3]. The authors and associated researchers have provided in [4,5,6,7] essentially all kinetic relations, including the mechanical terms, the elastic strain energy contributions, derived for spherical inclusions. This established concept can be adapted by the so-called shape factors in order to deal with non-spherical shapes of inclusions

Strain energy density and Eshelby tensor

Conclusion