By J. N. Reddy
ISBN-10: 0511508271
ISBN-13: 9780511508271
This textbook on continuum mechanics displays the trendy view that scientists and engineers could be proficient to imagine and paintings in multidisciplinary environments. The publication is perfect for complex undergraduate and starting graduate scholars. The booklet good points: derivations of the elemental equations of mechanics in invariant (vector and tensor) shape and specializations of the governing equations to numerous coordinate platforms; a number of illustrative examples; chapter-end summaries; and workout difficulties to check and expand the knowledge of techniques provided.
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Example text
Suppose we know the product of A(i, j, k) with an arbitrary tensor behaves as a tensor. Then we can use the quotient rule to establish the nature of A(i, j, k). For example, let vi be components of a vector, and if the product A(i, j, k)vi is known to yield a tensor of rank 2, B jk, then A(i, j, k)vi = B jk. 90) A (i, j, k)vi = B jk. 92) [A (i, j, k) − Q jm Qkn Qil A(l, m, n)]vi = 0. 93) Because vi are arbitrary, it follows that the quantity inside the brackets must be zero for all values of i.
7) From this, the base vector, g i can be related to the Cartesian base vectors ei as gi = ∂x j e j = Li j e j . 8) General Tensors 29 A base vector g i is tangential to the curve ξ i . Here the “curve ξ 1 ” refers to the curve along which ξ 1 changes while ξ 2 and ξ 3 remain constant. Unlike Cartesian coordinates, curvilinear coordinates may not have the dimension of length. As an example, we use angles as coordinates in polar and spherical systems. The term g i dξ i must have the dimension of length, and if dξ i is nondimensional, g i has the dimension of length.
Obtain the factors U, V, and R in the polar decomposition of the matrix ⎡√ ⎤ 3 1 0 ⎢ ⎥ B = ⎣0 2 0⎦ 0 0 1 26 Cartesian Tensors in the form B = RU = V R. 17. For the matrix B= 0 −1 1 , 0 show that e Bθ = I cos θ + B sin θ, where θ is a real number. 18. For C= 5 4 , 0 5 compute C10 and sin C. 19. In the case of an unsymmetric matrix A, show that A and AT have the same eigenvalues, λ(i) , i = 1, 2, 3, and if x (i) are three independent eigenvectors of A and y i , i = 1, 2, 3 are the eigenvectors of AT (these are also called the left eigenvectors of A), show that the two sets of eigenvectors can be selected to have x (i)T y( j) = δi j .
Cambridge Introduction To Continuum Mechanics by J. N. Reddy
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