TY - JOUR

T1 - Weyl geometry and the nonlinear mechanics of distributed point defects

AU - Yavari, A.

AU - Goriely, A.

N1 - KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUK C1-013-04
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) and by the National Science Foundation under grant DMS-0907773 (A.G.), CMMI-1130856 (A.Y.) and AFOSR, grant no. FA9550-10-1-0378. A.G. is a Wolfson Royal Society Merit Holder and acknowledges support from a Reintegration Grant under EC Framework VII.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

PY - 2012/9/5

Y1 - 2012/9/5

N2 - The residual stress field of a nonlinear elastic solid with a spherically symmetric distribution of point defects is obtained explicitly using methods from differential geometry. The material manifold of a solid with distributed point defects-where the body is stress-free-is a flat Weyl manifold, i.e. a manifold with an affine connection that has non-metricity with vanishing traceless part, but both its torsion and curvature tensors vanish. Given a spherically symmetric point defect distribution, we construct its Weyl material manifold using the method of Cartan's moving frames. Having the material manifold, the anelasticity problem is transformed to a nonlinear elasticity problem and reduces the problem of computing the residual stresses to finding an embedding into the Euclidean ambient space. In the case of incompressible neo-Hookean solids, we calculate explicitly this residual stress field. We consider the example of a finite ball and a point defect distribution uniform in a smaller ball and vanishing elsewhere. We show that the residual stress field inside the smaller ball is uniform and hydrostatic. We also prove a nonlinear analogue of Eshelby's celebrated inclusion problem for a spherical inclusion in an isotropic incompressible nonlinear solid. © 2012 The Royal Society.

AB - The residual stress field of a nonlinear elastic solid with a spherically symmetric distribution of point defects is obtained explicitly using methods from differential geometry. The material manifold of a solid with distributed point defects-where the body is stress-free-is a flat Weyl manifold, i.e. a manifold with an affine connection that has non-metricity with vanishing traceless part, but both its torsion and curvature tensors vanish. Given a spherically symmetric point defect distribution, we construct its Weyl material manifold using the method of Cartan's moving frames. Having the material manifold, the anelasticity problem is transformed to a nonlinear elasticity problem and reduces the problem of computing the residual stresses to finding an embedding into the Euclidean ambient space. In the case of incompressible neo-Hookean solids, we calculate explicitly this residual stress field. We consider the example of a finite ball and a point defect distribution uniform in a smaller ball and vanishing elsewhere. We show that the residual stress field inside the smaller ball is uniform and hydrostatic. We also prove a nonlinear analogue of Eshelby's celebrated inclusion problem for a spherical inclusion in an isotropic incompressible nonlinear solid. © 2012 The Royal Society.

UR - http://hdl.handle.net/10754/600192

UR - https://royalsocietypublishing.org/doi/10.1098/rspa.2012.0342

UR - http://www.scopus.com/inward/record.url?scp=84872315351&partnerID=8YFLogxK

U2 - 10.1098/rspa.2012.0342

DO - 10.1098/rspa.2012.0342

M3 - Article

SN - 1364-5021

VL - 468

SP - 3902

EP - 3922

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

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

IS - 2148

ER -