Download A First Course in Continuum Mechanics by Professor Oscar Gonzalez, Professor Andrew M. Stuart PDF

By Professor Oscar Gonzalez, Professor Andrew M. Stuart

A concise account of vintage theories of fluids and solids, for graduate and complex undergraduate classes in continuum mechanics.

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Summing all six of the above expressions and making use of index notation we obtain det A = 1 6 ij k pq r Aip Aj q Ak r , which is the desired result. Since the above expression remains unchanged when Am n is replaced by An m we deduce that det A = det AT . 17 Consider any third vector w. Then in any frame we have (F u × F v) · w = det ([F ][u], [F ][v], [w]) = det [F ][u], [F ][v], [F ][F ]−1 [w] = (det[F ]) det [u], [v], [F ]−1 [w] = (det F ) ((u × v) · F −1 w) = (det F ) F −T (u × v) · w. The result follows by the arbitrariness of w.

Furthermore [S]j i ui vj = Su · v = u · S T v = ui [S T ]ij vj . Thus, by the arbitrariness of ui and vj , we have [S T ]ij = [S]j i = [S]Tij as required. 13 Consider any vector v and let u = Sv. Then in the two frames we have the representations [u] = [S][v] and [u] = [S] [v] . Moreover, by definition of A we have [u] = [A][u] and [v] = [A][v] . Using the fact that [A]−1 = [A]T we have [S][v] = [u] = [A][u] = [A][S] [v] = [A][S] [A]T [v]. By the arbitrariness of [v], we obtain [S] = [A][S] [A]T , which implies [S] = [A]T [S][A].

2 Second-Order Tensor Algebra If we let V 2 be the set of second-order tensors, then V 2 has the structure of a real vector space since S + T ∈ V2 ∀S, T ∈ V 2 and αT ∈ V 2 ∀α ∈ IR, T ∈ V 2 . We can also compose second-order tensors in the sense that ST ∈ V 2 ∀S, T ∈ V 2 . In particular, we define the sum S + T via the relation (S + T )v = Sv + T v for all v ∈ V, and we define the composition ST via the relation (ST )v = S(T v) for all v ∈ V. A similar definition holds for αT . 3 Representation in a Coordinate Frame By the components of a second-order tensor S in a coordinate frame {ei } we mean the nine numbers Sij (1 ≤ i, j ≤ 3) defined by Sij = ei · Sej .

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