9780203009512.ch2

9780203009512.ch2 - 0749_Frame_C02 Page 25 Wednesday,...

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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 ′ = = = = vQ v QAw QAQ Qw QAQ w T T . ()( ) A QAQ QAQ TT T TTT ′ = = T ′′ = = = A B QAQ QBQ QA QQ BQ QABQ T () AB A B QAQ QBQ QA BQ T + ′ = ′ + =+ = = = −− A QAQ QA Q 1T 1 T 1 11 . © 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 Qxx Q TT T ′ = ′′ = = dd d d d d d 2 φ == xHx H xx T T . d d ′′′ = = QxHQx xQHQx , ux u x = , u x T = x u T T = . = u x u x T T T . ′ = 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) ∫= cn c TT dS dV . cA b T = bA n A b Ab T T T dS dV dV dV ∫∫ =∇ () [] . An A 0 TTT dS dV −∇ = . ba ndS b dV ii j j i i = . A 0 b x ad V i j ij i = . A 0 . ∇= = T A i j ij j ji x a x a ∇⋅ A = ∇ AA T . © 2003 by CRC CRC Press LLC
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28 Finite Element Analysis: Thermomechanics of Solids It should be evident that ( ) has different meanings when applied to a tensor as opposed to a vector.
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This note was uploaded on 05/18/2011 for the course MAE 269A taught by Professor Ju during the Spring '11 term at UCLA.

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9780203009512.ch2 - 0749_Frame_C02 Page 25 Wednesday,...

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