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 complicated undergraduate classes in continuum mechanics.

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Additional resources for A First Course in Continuum Mechanics

Example text

Use the dot and cross product operations to solve the above equation for v. In particular, show that the unique solution is given by α2 b − α(b × a) + (b · a)a . 5 Let a, b and c be vectors. Find scalars λ and µ such that (a × b) × c = λb − µa. 6 Let v be an arbitrary vector and let n be an arbitrary unit vector. Show that: (a) v = (v · n)n − (v × n) × n, (b) v = (v · n)n + (v ⊗ n)n − (n ⊗ v)n. Remark: The identity in (a) shows that v can always be decomposed into parts parallel and perpendicular to n.

In particular, for each of the basis vectors ei we have S(e1 ) = cos θ e1 + sin θ e2 , S(e2 ) = cos θ e2 − sin θ e1 , S(e3 ) = e3 . S(v) e2 θ o e3 v e1 (3) Reflections. The action of reflecting a vector v with respect to a fixed plane, say with normal e1 , can be expressed in terms of a secondorder tensor T , namely T (v) = v − 2(v · e1 )e1 . 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 .

The above result may be strengthened when, in addition to the properties mentioned above, G is linear and satisfies G(W ) = O for every skew-symmetric W . This case, which we now study, will be of particular importance in Chapters 6 and 7 when discussing linear models for stress in fluid and solid bodies. 16 Isotropic Fourth-Order Tensors. Let C : V 2 → V 2 be an isotropic fourth-order tensor with the properties: (1) C(A) ∈ V 2 is symmetric for every symmetric A ∈ V 2 , (2) C(W ) = O for every skew-symmetric W ∈ V 2 .

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


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