By Y. C. Fung
Revision of a vintage textual content via a wonderful writer. Emphasis is on challenge formula and derivation of governing equations. new version beneficial properties elevated emphasis on functions. New bankruptcy covers long term adjustments in fabrics less than tension.
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Extra info for A first course in continuum mechanics: for physical and biological engineers and scientists
20) then T is said to be symmetric. If then T is said to be skew symmetric (or, simply, skew). 22) skew(T ) ≡ 12 (T − T T ). 23) of its symmetric part and its skew part 20 COMPUTATIONAL METHODS FOR PLASTICITY: THEORY AND APPLICATIONS Basic properties The following basic properties involving the transpose, skew and symmetric parts of a tensor hold: (i) (S + T )T = ST + T T . (ii) (S T )T = T T ST . T (iii) (T T ) = T. (iv) If T is symmetric, then skew(T ) = 0, sym(T ) = T. skew(T ) = T, sym(T ) = 0.
32) 22 COMPUTATIONAL METHODS FOR PLASTICITY: THEORY AND APPLICATIONS Thus, T T has the following Cartesian matrix representation T11 T21 · · · Tn1 T12 T22 · · · Tn2 . [T T ] = . .. .. .. . 4. TRACE, INNER PRODUCT AND EUCLIDEAN NORM For any u, v ∈ U, the trace of the tensor (u ⊗ v) is the linear map deﬁned as tr(u ⊗ v) = u · v. 35) that is, the trace of T is the sum of the diagonal terms of the Cartesian matrix representation [T ]. 36) S : T = Sij Tij . 37) or, in Cartesian component form, The Euclidean norm (or simply norm) of a tensor T is deﬁned as √ 2 + T2 + · · · + T2 .
Ii) T u = Tijkl ul ei ⊗ ej ⊗ ek . (iii) T : S = Tijkl Skl ei ⊗ ej . (vi) S : T = Tklij Skl ei ⊗ ej . (v) T : S = Tijmn Smnkl ei ⊗ ej ⊗ ek ⊗ el . 87) for any second-order tensors S and U . This deﬁnition is analogous to that of symmetric second-order tensors. The Cartesian components of symmetric fourth-order tensors satisfy the major symmetries Tijkl = Tklij . 88) It should be noted that other symmetries are possible in fourth-order tensors. e. 90) for any S. 91) then, T : S = (T : S)T , ‡ Fourth-order S : T = ST : T.