Vectors_Tensors_12_HigherOrderTensors - Section 1.12 Solid...

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Unformatted text preview: Section 1.12 Solid Mechanics Part III Kelly 107 1.12 Higher Order Tensors In this section are discussed some important higher (third and fourth) order tensors. 1.12.1 Fourth Order Tensors After second-order tensors, the most commonly encountered tensors are the fourth order tensors A , which have 81 components. Some properties and relations involving these tensors are listed here. Transpose The transpose of a fourth-order tensor A , denoted by T A , by analogy with the definition for the transpose of a second order tensor 1.10.4, is defined by B C C B : : : : T A A = (1.12.1) for all second-order tensors B and C . It has the property ( ) A A = T T and its components are klij ijkl ) ( ) ( T A A = . It also follows that ( ) A B B A ⊗ = ⊗ T (1.12.2) Identity Tensors There are two fourth-order identity tensors . They are defined as follows: T : : A A A A = = I I (1.12.3) and, from 1.9.7, they have components i j j i l k j i jk il j i j i l k j i jl ik e e e e e e e e e e e e e e e e ⊗...
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This note was uploaded on 01/20/2012 for the course ENGINEERIN 3 taught by Professor Staff during the Fall '11 term at Auckland.

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Vectors_Tensors_12_HigherOrderTensors - Section 1.12 Solid...

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