Effect of Defects, Inclusions and Inhomogeneities in Elastic Solids
Date
2019-12
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
Abstract
This thesis focusses on the theory of materials with defects introduced by John D. Eshelby in the 50s and the 60s, which today we call Configurational Mechanics or, in his honour, Eshelbian Mechanics. The thesis consists of four interconnected parts. The first part is dedicated to the relation between two of Eshelby’s developments: the energy momentum tensor (or Eshelby stress tensor), describing the net force on a defect, and the Eshelby fourth-order tensor, which relates the strain in an inclusion in an otherwise homogeneous and isotropic matrix to the virtual strain (transformation strain) defining the geometrical misfit between inclusion and matrix, within the theory of small deformations. The second part of the research was prompted by the fact that, although the relation between Eshelby’s inclusion problem (Eshelby, 1951, 1975) and Noether’s theorem has been mentioned in literature, no explicit relation has ever been given, to the best of our knowledge. In a framework based on modern differential geometry, it is shown that the application of Noether’s theorem allows for straightforwardly obtaining the classical results by (Eshelby, 1951, 1975). The third part of the thesis aims at investigating the work of Eshelby (1951, 1975) on configurational forces and of Noll (1967) on material uniformity within a general
framework including thermo-elasticity, volumetric growth inertial effects, in which the divergence of the Eshelby stress is called the Eshelby force. A differential identity is obtained for the modified Eshelby stress, which includes, as a particular case, the identity found by Epstein and Maugin (1990). Moreover, a differential identity is obtained for what is called the modified Eshelby power, representing the time counterpart of the Eshelby force. Then, a relation between the modified Eshelby force and the modified Eshelby power is derived in the dynamical case. Finally, based on the results obtained in the previous parts of the research, a large-deformation counterpart is proposed of the imagined procedure that Eshelby (1957) used to investigate the theory of inclusions in the case of infinitesimal deformations. A mixed multiplicative decomposition of the deformation
gradient, in terms of the Bilby-Kröner-Lee and the Noll-Epstein-Maugin decompositions allows for obtaining the large-deformation fourth-order Eshelby tensor, a novel result.
Description
Keywords
Ellipsoidal inclusion, Eshelby stress, Eshelby fourth-order tensor, Material uniformity, Noether's theorem
Citation
Alhasadi, M. F. (2019). Effect of Defects, Inclusions and Inhomogeneities in Elastic Solids (Doctoral thesis, University of Calgary, Calgary, Canada). Retrieved from https://prism.ucalgary.ca.