A First Course in Continuum Mechanics - download pdf or read online

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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Then S : D = Sij Dij . Furthermore, if S is symmetric, then S : D = S : sym(D) = sym(S) : sym(D). Proof See Exercise 29. 4 Fourth-Order Tensors In the previous section we defined a second-order tensor as a linear transformation between vectors. In describing the behavior of material bodies we will also need the concept of a linear transformation between second-order tensors. For example, the relationship between stress and strain in a material body is sometimes modeled by such a transformation. Linear transformations between second-order tensors lead to the notion of a fourth-order tensor as defined below.

That is, if we let Sk l be the second-order tensor given by C(ek ⊗ el ), then Cij k l = ei · Sk l ej . These components completely describe the transformation C : V 2 → V 2 . 18) = Cij k l Tk l . Thus the components of C are simply the coefficients in the linear relation between the components of U and T . Example: Let A ∈ V 2 be given and consider the fourth-order tensor defined by C(T ) = AT . Then the components of C are Cij k l = ei · A(ek ⊗ el )ej = ei · Aek δlj = Aik δlj . 4 Fourth-Order Tensors 29 An alternative way to compute the components Cij k l is to use the scalar product for second-order tensors, namely Cij k l = (ei ⊗ ej ) : C(ek ⊗ el ).

Hint: Use the characteristic polynomial and properties of determinants. (b) Show that there is only one independent eigenvector e such that Qe = e. Hint: Use part (c) of Exercise 21 to show that if there were more than one such independent eigenvector, then there must be three, which would imply Q = I. (c) Let n be any unit vector orthogonal to e. Show that Qn is also a unit vector orthogonal to e and that the angle θ ∈ [0, π] between n and Qn satisfies the relation 1 + 2 cos θ = tr Q. Hint: Express Q in the frame {e, n, e × n}.

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


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