Nonlinear elastic inclusions in isotropic solids

Type
Article

Authors
Yavari, A.
Goriely, A.

KAUST Grant Number
KUK C1-013-04

Online Publication Date
2013-10-16

Print Publication Date
2013-10-16

Date
2013-10-16

Abstract
We introduce a geometric framework to calculate the residual stress fields and deformations of nonlinear solids with inclusions and eigenstrains. Inclusions are regions in a body with different reference configurations from the body itself and can be described by distributed eigenstrains. Geometrically, the eigenstrains define a Riemannian 3-manifold in which the body is stress-free by construction. The problem of residual stress calculation is then reduced to finding a mapping from the Riemannian material manifold to the ambient Euclidean space. Using this construction, we find the residual stress fields of three model systems with spherical and cylindrical symmetries in both incompressible and compressible isotropic elastic solids. In particular, we consider a finite spherical ball with a spherical inclusion with uniform pure dilatational eigenstrain and we show that the stress in the inclusion is uniform and hydrostatic. We also show how singularities in the stress distribution emerge as a consequence of a mismatch between radial and circumferential eigenstrains at the centre of a sphere or the axis of a cylinder.

Citation
Yavari A, Goriely A (2013) Nonlinear elastic inclusions in isotropic solids. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 469: 20130415–20130415. Available: http://dx.doi.org/10.1098/rspa.2013.0415.

Acknowledgements
This publication was based on work supported in part by award no. KUK C1-013-04, made by King Abdullah University of Science and Technology (KAUST). A.Y. was partially supported by AFOSR (grant no. FA9550-12-1-0290) and NSF (grant nos. CMMI 1042559 and CMMI 1130856). A.G. is a Wolfson/Royal Society Merit Award Holder and acknowledges support from a Reintegration Grant under EC Framework VII.

Publisher
The Royal Society

Journal
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

DOI
10.1098/rspa.2013.0415

PubMed ID
24353470

PubMed Central ID
PMC3857869

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