By Professor Oscar Gonzalez, Professor Andrew M. Stuart

ISBN-10: 0511455135

ISBN-13: 9780511455131

ISBN-10: 0521714249

ISBN-13: 9780521714242

ISBN-10: 0521886805

ISBN-13: 9780521886802

ISBN-10: 1282389947

ISBN-13: 9781282389946

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

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4. By the change of basis tensor from {ei } to {ei } we mean the tensor A defined by A = Aij ei ⊗ ej where Aij = ei · ej . 11) We could also define a change of basis tensor B from {ei } to {ei } by B = Bij ei ⊗ ej where Bij = ei · ej . All that we say for A will also apply to B. However, for convenience, we work only with A. Using the components of A we can express the basis vectors of one frame in terms of the other. For example, a basis vector ej may be expressed in the frame {ei } as ej = (e1 · ej )e1 + (e2 · ej )e2 + (e3 · ej )e3 = (ei · ej )ei , 20 Tensor Algebra e3 e 3/ e 2/ e2 o v e1 e1/ Fig.

Then there exists a right-handed, orthonormal basis {ei } for V consisting of eigenvectors of S. The corresponding eigenvalues λi are the same (up to ordering) for any such basis and form the full set of eigenvalues of S. The dyadic representation of S in any such basis is 3 λi ei ⊗ ei , S= i=1 and the matrix representation is  λ1  [S] = 0 0 0 λ2 0  0 0 . λ3 The principal invariants of a second-order tensor S are three scalars defined by I1 (S) = tr S, I2 (S) = 1 2 (tr S)2 − tr(S 2 ) , I3 (S) = det S.

7 (a) For arbitrary v we have S(a ⊗ b)v = S(b · v)a = (b · v)Sa = ((Sa) ⊗ b)v, which implies S(a ⊗ b) = (Sa) ⊗ b. (b) For arbitrary v we have (a ⊗ b)Sv = (b · Sv)a = (S T b · v)a = (a ⊗ S T b)v, which implies (a ⊗ b)S = a ⊗ (S T b). Selected Answers 39 (c) For arbitrary u and v we have u · (a ⊗ b)T v = (a ⊗ b)u · v = (b · u)(a · v) = u · (b ⊗ a)v, which implies (a ⊗ b)T = (b ⊗ a). 9 (a) P T = P and P P = P , thus perpendicular projection. (b) P T = P , thus not perpendicular projection. (c) Perpendicular projection.

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

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