Vectors_Tensors_15_Tensor_Calculus_2

Vectors_Tensors_15_Tensor_Calculus_2 - Section 1.15 1.15...

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Section 1.15 Solid Mechanics Part III Kelly 124 1.15 Tensor Calculus 2: Tensor Functions 1.15.1 Vector-valued functions of a vector Consider a vector-valued function of a vector ) ( ), ( j i i b a a = = b a a This is a function of three independent variables 3 2 1 , , b b b , and there are nine partial derivatives j i b a / . The partial derivative of the vector a with respect to b is defined to be a second-order tensor with these partial derivatives as its components: j i j i b a e e b b a ) ( (1.15.1) It follows from this that 1 = a b b a or ij j m m i a b b a δ = = , I a b b a (1.15.2) To show this, with ) ( ), ( j i i j i i a b b b a a = = , note that the differential can be written as + + = = = 3 1 3 2 1 2 1 1 1 1 1 1 a b b a da a b b a da a b b a da da a b b a db b a da j j j j j j i i j j j j Since 3 2 1 , , da da da are independent, one may set 0 3 2 = = da da , so that 1 1 1 = a b b a j j Similarly, the terms inside the other brackets are zero and, in this way, one finds Eqn. 1.15.2. 1.15.2 Scalar-valued functions of a tensor Consider a scalar valued function of a (second-order) tensor j i ij T e e T T = = ), ( φ . This is a function of nine independent variables, ) ( ij T = , so there are nine different partial derivatives:
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Solid Mechanics Part III Kelly 125 33 32 31 23 22 21 13 12 11 , , , , , , , , T T T T T T T T T φ The partial derivative of with respect to T is defined to be a second-order tensor with these partial derivatives as its components: j i ij T e e T Partial Derivative with respect to a Tensor (1.15.3) The quantity T T / ) ( is also called the gradient of with respect to T . Thus differentiation with respect to a second-order tensor raises the order by 2. This agrees with the idea of the gradient of a scalar field where differentiation with respect to a vector raises the order by 1. Derivatives of the Trace and Invariants Consider now the trace: the derivative of A tr , with respect to A can be evaluated as follows: I e e e e e e e e e e e e A A A A A = + + = + + = + + = 3 3 2 2 1 1 33 22 11 33 22 11 tr j i ij j i ij j i ij A A A A A A A A A (1.15.4) Similarly, one finds that { Problem 1} () ( ) ( ) () () () I A A A I A A A A A A A A A I A A 2 3 2 T 2 3 T 2 ) tr ( 3 ) tr ( ) tr ( 2 ) tr ( 3 tr 2 tr tr = = = I A I A A A A A A A A Derivatives of the Invariants (1.15.6)
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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_15_Tensor_Calculus_2 - Section 1.15 1.15...

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