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

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3 Second-Order Tensors Many physical quantities in mechanics, such as forces, velocities and accelerations, are well-represented by the notion of a vector (first-order tensor). In any Cartesian coordinate frame for IE 3 such quantities are described by three components. In our studies we will encounter other physical quantities, such as stress and strain, which are not wellrepresented by vectors, but rather by linear transformations between vectors, leading to the notion of a second-order tensor as defined below.

13 Consider any vector v and let u = Sv. Then in the two frames we have the representations [u] = [S][v] and [u] = [S] [v] . Moreover, by definition of A we have [u] = [A][u] and [v] = [A][v] . Using the fact that [A]−1 = [A]T we have [S][v] = [u] = [A][u] = [A][S] [v] = [A][S] [A]T [v]. By the arbitrariness of [v], we obtain [S] = [A][S] [A]T , which implies [S] = [A]T [S][A]. 15 By definition of det A we have det A = det[A] = εij k Ai1 Aj 2 Ak 3 . By properties of the permutation symbol and the determinant of 40 Tensor Algebra a matrix under permutations of its columns, we find that det A can be written in six equivalent ways, namely det A = εij k ε123 Ai1 Aj 2 Ak 3 = εij k ε312 Ai3 Aj 1 Ak 2 = εij k ε231 Ai2 Aj 3 Ak 1 = εij k ε321 Ai3 Aj 2 Ak 1 = εij k ε132 Ai1 Aj 3 Ak 2 = εij k ε213 Ai2 Aj 1 Ak 3 .

However, such matrix representations will not be exploited in our developments. 4 Fourth-Order Dyadic Products, Bases The dyadic product of four vectors a, b, c and d is the fourth-order tensor a ⊗ b ⊗ c ⊗ d defined by (a ⊗ b ⊗ c ⊗ d)T = (c · T d)a ⊗ b, ∀T ∈ V 2 . Given any coordinate frame {ei } the eighty-one elementary dyadic products {ei ⊗ ej ⊗ ek ⊗ el } form a basis for V 4 . 19) where Cij k l = ei · C(ek ⊗ el )ej . To see that the above representation is correct consider the expression U = CT , where U = Uij ei ⊗ ej and 30 Tensor Algebra T = Tk l ek ⊗ el .

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

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