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## T. T. C. Ting

Print publication date: 1996

Print ISBN-13: 9780195074475

Published to Oxford Scholarship Online: November 2020

DOI: 10.1093/oso/9780195074475.001.0001

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# Transformation of the Elasticity Matrices and Dual Coordinate Systems

Chapter:
(p.201) Chapter 7 Transformation of the Elasticity Matrices and Dual Coordinate Systems
Source:
Anisotropic Elasticity
Publisher:
Oxford University Press
DOI:10.1093/oso/9780195074475.003.0010

When the elasticity matrices are referred to a rotated coordinate system their elements change and assume different values. We will show in this chapter that, under rotations about the x3-axis, the matrices A and B are tensors of rank one while S, H, L, and M are tensors of rank two. These properties are important in establishing certain invariants that are physically interesting and puzzling. We will also present the amazing Barnett-Lothe integral formalism that allows us to determine S, H, and L without computing the eigenvalues and eigenvectors of elastic constants. New tensors Ni(θ) (i=l,2,3), S(θ), H(θ), L(θ), and Gi(θ) (i=l,3) are introduced, and their properties as well as identities relating them are presented. Also introduced is the idea of dual coordinate systems where the position of a point is referred to one coordinate system while the displacement components are referred to another coordinate system. These will be useful in applications. As in Chapter 6 readers may skip this chapter in the first reading. They can return to this chapter later for specific information.

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