Download e-book for kindle: A First Course in Continuum Mechanics by Professor Oscar Gonzalez, Professor Andrew M. Stuart

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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Extra resources for A First Course in Continuum Mechanics

Example text

11 Show that the transpose of a second-order tensor S is uniquely defined and that [S T ] = [S]T . 12 Prove that a second-order tensor S cannot be both positivedefinite and skew-symmetric. 13 Let A denote the change of basis tensor from a frame {ei } to a frame {ei } with representation [A] in {ei }. Let S be a secondorder tensor with representation [S] and [S] in {ei } and {ei }, respectively. Show that [S] = [A]T [S][A]. 14 Consider a vector a with representation [a] = (1, 1, 1)T in a coordinate frame {ei }.

Then from the definitions of S + T and αT it is straightforward to deduce [S + T ] = [S] + [T ] and [αT ] = α[T ]. To establish a component expression for composition consider any two 16 Tensor Algebra dyadic products a ⊗ b and c ⊗ d. Then for any arbitrary vector v we have ((a ⊗ b)(c ⊗ d)) v = (a ⊗ b) ((c ⊗ d)v) = (a ⊗ b)c(d · v) = a(b · c)(d · v) = (b · c)(a ⊗ d)v, which, by the arbitrariness of v, implies (a ⊗ b)(c ⊗ d) = (b · c)a ⊗ d. With this result we can express tensor composition in terms of components as follows.

Then exp(W ) ∈ V 2 is a rotation in the sense that (exp(W ))T exp(W ) = I and det(exp(W )) = 1. Moreover, (exp(W ))T = exp(W T ). 9 Eigenvalues, Eigenvectors and Principal Invariants By an eigenpair for a second-order tensor S we mean a scalar λ and a unit vector e satisfying Se = λe. Any such λ is called an eigenvalue and any such e an eigenvector of S. From linear algebra we recall that λ is an eigenvalue if and only if it is a root of the characteristic (cubic) polynomial p(λ) = det(S − λI).

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


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