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