RETRACTED ARTICLE: Crystal’s anisotropic properties and tensor representation: a discussion
Physics Department, Faculty of Science, Urmia University, Urmia, Iran
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Accepted: 9 November 2011
Published online: 23 December 2011
For most applications, crystals need a careful selection to process an appropriate symmetry for a particular application. A crystal is innately symmetrical; hence it presents the same appearance from a number of different directions. Some of the physical properties of crystals which exhibit dependence on symmetry elements are presented. A method of using the inherent symmetry in order to simplify the formulation of the physical properties is needed. The use of tensors is one such tool. The authors review what tensors of different ranks are, and show how such tensors can be used to describe the directional variation of the physical properties within crystals. These arise when a tensor relates a vector to a second-rank tensor. Properties that involve third-rank tensors include the piezoelectric effect. However, to exhibit the reduction of the number of tensor components further, consider the fourth-rank tensors. Both stress and strain can be represented by second-rank tensors, the modulus can be represented by a fourth-rank tensor. Stress is the force acting in any direction divided by the area. The fourth-rank elasticity tensor of an anisotropic and linear elastic material of the material is considered.
© Società Italiana di Fisica and Springer, 2011