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9780203009512.ch2

9780203009512.ch2 - 0749_Frame_C02 Page 25 Wednesday 5:00...

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25 Mathematical Foundations: Tensors 2.1 TENSORS We now consider two n × 1 vectors, v and w , and an n × n matrix, A , such that v = Aw . We now make the important assumption that the underlying information in this relation is preserved under rotation. In particular, simple manipulation furnishes that ) (2.1) The square matrix A is now called a second-order tensor if and only if A = QAQ T . Let A and B be second-order n × n tensors. The manipulations that follow demonstrate that A T , ( A + B ), AB , and A 1 are also tensors. (2.2) (2.3) (2.4) (2.5) 2 ′ = = = = v Qv QAw QAQ Qw QAQ w T T . ( ) ( ) A QAQ Q A Q T T T T T T ′ = = T ′ = = = A B QAQ QBQ QA QQ BQ QABQ T T T T T ( )( ) ( ) ( ) A B A B QAQ QBQ Q A B Q T T T + ′ = ′ + = + = + ( ) = = = A QAQ Q A Q QA Q 1 T 1 T 1 1 1 1 T ( ) . © 2003 by CRC CRC Press LLC
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26 Finite Element Analysis: Thermomechanics of Solids Let x denote an n × 1 vector. The outer product, xx T , is a second-order tensor since (2.6) Next, (2.7) However, (2.8) from which we conclude that the Hessian H is a second-order tensor. Finally, let u be a vector-valued function of x . Then, from which (2.9) and also (2.10) We conclude that (2.11) Furthermore, if d u is a vector generated from d u by rotation in the opposite sense from the coordinate axes, then d u ′ = Q d u and d x = Q d x . Hence, Q is a tensor. Also, since , it is apparent that (2.12) from which we conclude that is a tensor. We can similarly show that I and 0 are tensors. ( ) ( )( ) ( ) xx x x Qx Qx Q xx Q T T T T T ′ = ′ ′ = = d d d d d d d 2 φ φ = = x H x H x x T T . d d d d d d ′ = = x H x Q x H Q x x Q H Q x T T T T ( ) ( ) , d d u x u x = , d d u x u x T T T = d d u x x u T T T T = . = u x u x T T T . d d ′ = u x u x ∂ ′ ∂ ′ = u x Q u x Q T , u x © 2003 by CRC CRC Press LLC
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Mathematical Foundations: Tensors 27 2.2 DIVERGENCE, CURL, AND LAPLACIAN OF A TENSOR Suppose A is a tensor and b is an arbitrary, spatially constant vector of compatible dimension. The divergence and curl of a vector have already been defined. For later purposes, we need to extend the definition of the divergence and the curl to A . 2.2.1 D IVERGENCE Recall the divergence theorem Let , in which b is an arbitrary constant vector. Now (2.13) Consequently, we must define the divergence of A such that * ) (2.14) In tensor-indicial notation, (2.15) Application of the divergence theorem to the vector c j = b i a ij furnishes (2.16) Since b is arbitrary, we conclude that (2.17) Thus, if we are to write as a (column) vector, mixing tensor- and matrix- vector notation, (2.18) = ∫ ∇ c n c T T dS dV . c A b T = b An A b A b b A T T T T T T T T T dS dV dV dV = = = ( ) [ ] . An A 0 T T T dS dV = [ ] . b a n dS b dV i ij j i i [ ] = [ ] .
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