Get Body Tensor Fields in Continuum Mechanics. With Applications PDF

By Arthur S. Lodge

ISBN-10: 0124549500

ISBN-13: 9780124549500

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The other kinds of second-rank tensor may be defined in similar ways. The results are as follows. We include for comparison the contravariant tensors just defined. {9} and {ψ} are respectively the sets of all contravariant and covariant body vectors at P. (7) Second-rank body tensors at P are elements of direct product spaces: (i) (ii) (iii) (iv) Θ G {θ} ® {Θ} Ψ G {ψ} ® {ψ} μ G {Θ} (g) {ψ} μ G {\|/} ® {Θ} (contravariant), (covariant), (right covariant mixed), (left covariant mixed). (8) Second-rank body tensors at P map & onto A3x3 rc (i) 0(P)£ = [0 (£)] = Λρ \Β,_Β)Θλρ \B, B) (ii) Ψ(Ρ)β = \ΨΜ(ξ)] = ΛΡ(Β, 5)ΨΛ Ρ (β, Β) (iii) μ(Ρ)β = \ßrc(0] = Λ Ρ \Β, Β)μΑΫ{Β, Β) (iv) μ(Ρ)β = [μ'Χξ)] = ΛΡ(Β, Β)μλρ \Β, Β) according to the rules (contravariant), (covariant), (right covariant mixed), (left covariant mixed).

5 Wkid-£)UB^)\ β'(£, P) = (3f 73i*)ßk(*, P). The definition of second-rank tensor fields Vectors are also called first-rank tensors', in any coordinate system, a vector is represented by components, having a single dummy index, which, for convenience, we have taken to be elements of a 3 x 1 single-column matrix. 5 SECOND-RANK TENSOR FIELDS 37 second-rank tensor is represented by components having two dummy indexes, which we shall take to be elements of a 3 x 3 (square) matrix. An «th-rank tensor has components, with n dummy indexes, which form an ordered array.

To any given right covariant tensor μ there corre­ sponds a unique left covariant tensor μ whose representative matrix in any B is the transpose of that of μ ; a similar statement with left and right inter­ changed is true. In each case, the transpose of the transpose equals the original tensor. The transpose of an outer product of vectors equals the same vectors in reverse order. A contravariant (or covariant) tensor is said to be symmetric if it equals its transpose and antisymmetric (or skew symmetric) if it equals its transpose times minus one.

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Body Tensor Fields in Continuum Mechanics. With Applications to Polymer Rheology by Arthur S. Lodge


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