Cambridge Introduction To Continuum Mechanics by J. N. Reddy

By J. N. Reddy

This textbook on continuum mechanics displays the fashionable view that scientists and engineers could be expert to imagine and paintings in multidisciplinary environments. The e-book is perfect for complicated undergraduate and starting graduate scholars. The ebook positive factors: derivations of the fundamental equations of mechanics in invariant (vector and tensor) shape and specializations of the governing equations to varied coordinate structures; a variety of illustrative examples; chapter-end summaries; and workout difficulties to check and expand the certainty of ideas awarded.

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2. If a is a constant vector, show that a · dx = 0, C a × dx = 0. 3. Transform the surface integrals I= (n × ∇) · udS, I= S (n × ∇) × udS S into volume integrals and evaluate them. 4. Obtain the differential equations for the vector function φ and the scalar ψ inside an arbitrary volume if the surface integrals n × (r φ)da = 0, S n · (rψ)da = 0, S where r = (xi xi )1/2 and r = x. 5. In the torsion of shafts, the St. Venant warping function φ satisfies the Laplace equation in a simply connected 2D domain D in the x, y plane, representing the cross-section of the shaft and the boundary conditions, n · ∇φ = k · n × r, where n is the unit normal to the boundary, r = xi + y j, and k is the unit vector perpendicular to the plane.

10. 16 25 (c) ∇ · ∇r n = n(n + 1)r (n−2) . (d) If F is any differentiable function, show that ∇ × [F (r )r] = 0. 11. Show that the differential equation (∇ 2 + k2 )ψ = 0, in three dimensions, admits the solution ψ= e±ikr . 12. If A and B are tensors of rank m and n, respectively, in N dimensions, show that (a) the rank of A · B is m + n − 2, (b) the rank of A × B is m + n − 1, and (c) the rank of A B is m + n. 13. If T is a two-dimensional, second-rank, unsymmetrical tensor and n is a twodimensional vector (two-vector), compare n · T · n, (T · n) · n, and n · (n · T).

3 Example: Polar Decomposition Suppose we want to factor the matrix ⎡ 17 1⎢ B = ⎣ 19 5 0 −11 23 0 ⎤ 0 ⎥ 0⎦. 136) V = BRT . The eigenvalues of C are λ1 = 9, λ2 = 16, λ3 = 36. Using the expansion λi = c0 + c1 λi + c2 λi2 , we have the simultaneous equations 3 = c0 + 9c1 + 92 c2 , 4 = c0 + 16c1 + 162 c2 , 6 = c0 + 36c1 + 362 c2 . The solutions of this system are c0 = 52/35, c1 = 23/126, c2 = −1/630. Writing C1/2 = c0 I + c1 C + c2 C2 , we get ⎡ ⎤ 5 1 0 ⎢ ⎥ U = C1/2 = ⎣ 1 5 0 ⎦ , 0 0 3 ⎡ ⎤ 4 3 0 1⎢ ⎥ R = ⎣ −3 4 0 ⎦ , 5 0 0 5 ⎡ ⎤ 101 7 0 1 ⎢ ⎥ V= 149 0 ⎦ .

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