Download e-book for iPad: 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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1 Definition By a second-order tensor T on the vector space V we mean a mapping T : V → V which is linear in the sense that: (1) T (u + v) = T u + T v for all u, v ∈ V, (2) T (αu) = αT u for all α ∈ IR and u ∈ V. We denote the set of all second-order tensors on V by the symbol V 2 . Analogous to the zero vector, we define a zero tensor O with the property Ov = 0 for all v ∈ V, and we define an identity tensor I with the property Iv = v for all v ∈ V. Two second-order tensors S and T are said to be equal if and only if Sv = T v for all v ∈ V.

Let S be a secondorder tensor with matrix representations [S] and [S] in the frames {ei } and {ei }, respectively. Then det[S] = det[S] . Thus the numerical value of the determinant is independent of the coordinate frame in which it is computed. Proof See Exercise 20. The quantity | det S| has the geometric interpretation of being the volume of the parallelepiped defined by the three vectors Se1 , Se2 and Se3 . The determinant also arises in the classification of certain types of tensors. For example, a tensor S is invertible if and only if det S = 0.

7) where Sij = ei · Sej . To see that the above representation is correct consider the expression v = Su, where v = vi ei and u = uk ek . Then v = Su = Sij ei ⊗ ej uk ek = Sij δj k uk ei (by dyadic property) = Sij uj ei , which implies vi = Sij uj . 6) and provides a unique representation of S in terms of its components Sij . 5 Second-Order Tensor Algebra in Components Let S = Sij ei ⊗ ej and T = Tij ei ⊗ ej be second-order tensors with matrix representations [S] and [T ]. Then from the definitions of S + T and αT it is straightforward to deduce [S + T ] = [S] + [T ] and [αT ] = α[T ].

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

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