Vectors_Tensors_10_Special_Tensors

# Vectors_Tensors_10_Special_Tensors - Section 1.10 1.10...

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Section 1.10 Solid Mechanics Part III Kelly 84 1.10 Special Second Order Tensors & Properties of Second Order Tensors In this section will be examined a number of special second order tensors, and special properties of second order tensors, which play important roles in tensor analysis. The following will be discussed: The Identity tensor Transpose of a tensor Trace of a tensor Norm of a tensor Determinant of a tensor Inverse of a tensor Orthogonal tensors Rotation Tensors Change of Basis Tensors Symmetric and Skew-symmetric tensors Axial vectors Spherical and Deviatoric tensors Positive Definite tensors 1.10.1 The Identity Tensor The linear transformation which transforms every tensor into itself is called the identity tensor . This special tensor is denoted by I so that, for example, a Ia = for any vector a In particular, 3 3 2 2 1 1 , , e Ie e Ie e Ie = = = , from which it follows that, for a Cartesian coordinate system, ij ij I δ = . In matrix form, [] = 1 0 0 0 1 0 0 0 1 I (1.10.1) 1.10.2 The Transpose of a Tensor The transpose of a second order tensor A with components ij A is the tensor T A with components ji A ; so the transpose swaps the indices, j i ji j i ij A A e e A e e A = = T , Transpose of a Second-Order Tensor (1.10.2) In matrix notation,

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Section 1.10 Solid Mechanics Part III Kelly 85 [] [] = = 33 23 13 32 22 12 31 21 11 T 33 32 31 23 22 21 13 12 11 , A A A A A A A A A A A A A A A A A A A A Some useful properties and relations involving the transpose are { Problem 2}: ( ) () () () () () () B AC C A B BC A v A u A v u B A B A A B AB u T uT uT Tu u v v u B A B A A A : : : ) ( ) ( : : , T T T T T T T T T T T T T T T T = = = = = = = = + = + = β α (1.10.3) A formal definition of the transpose which does not rely on any particular coordinate system is as follows: the transpose of a second-order tensor is that tensor which satisfies the identity 1 u A v Av u T = (1.10.4) for all vectors u and v . To see that Eqn. 1.10.4 implies 1.10.2, first note that, for the present purposes, a convenient way of writing the components ij A of the second-order tensor A is () ij A . From Eqn. 1.9.4, ( ) j i ij Ae e A = and the components of the transpose can be written as ( ) j i ij e A e A T T = . Then, from 1.10.4, ( ) () ji ji i j j i ij A = = = = A Ae e e A e A T T . 1.10.3 The Trace of a Tensor The trace of a second order tensor A , denoted by A tr , is a scalar equal to the sum of the diagonal elements of its matrix representation. Thus (see Eqn. 1.4.2) ii A = A tr Trace (1.10.5) A more formal definition, again not relying on any particular coordinate system, is A I A : tr = Trace (1.10.6) 1 note that, from the linearity of tensors, Av u v uA = ; for this reason, this expression is usually written simply as uAv
Section 1.10

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Vectors_Tensors_10_Special_Tensors - Section 1.10 1.10...

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