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Incompatible deformation field and Riemann curvature tensor

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  • Department of Mechanical Engineering, Cape Peninsula University of Technology, Cape Town 7535, South Africa

Received date: 2016-06-04

  Revised date: 2016-10-08

  Online published: 2017-03-01

Supported by

Project supported by the National Research Foundation of South Africa (NRF) (No. 93918)

Abstract

Compatibility conditions of a deformation field in continuum mechanics have been revisited via two different routes. One is to use the deformation gradient, and the other is a pure geometric one. Variations of the displacement vector and the displacement density tensor are obtained explicitly in terms of the Riemannian curvature tensor. The explicit relations reconfirm that the compatibility condition is equivalent to the vanishing of the Riemann curvature tensor and reveals the non-Euclidean nature of the space in which the dislocated continuum is imbedded. Comparisons with the theory of Kröner and Le-Stumpf are provided.

Cite this article

Bohua SUN . Incompatible deformation field and Riemann curvature tensor[J]. Applied Mathematics and Mechanics, 2017 , 38(3) : 311 -332 . DOI: 10.1007/s10483-017-2176-8

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