By J. N. Reddy
This textbook on continuum mechanics displays the fashionable view that scientists and engineers may be knowledgeable to imagine and paintings in multidisciplinary environments. The booklet is perfect for complex undergraduate and starting graduate scholars. The ebook positive aspects: derivations of the elemental equations of mechanics in invariant (vector and tensor) shape and specializations of the governing equations to numerous coordinate platforms; various illustrative examples; chapter-end summaries; and workout difficulties to check and expand the certainty of options offered.
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Additional resources for Cambridge Introduction To Continuum Mechanics
We refer to these two configurations as undeformed and deformed configurations, respectively. We use a fixed Cartesian system to describe the geometry of the two configurations. As shown in Fig. 1, a generic particle in the undeformed configuration has the position X, and the same particle in the deformed configuration has the position x. We simultaneously develop descriptions of the geometry of this mapping from the state B0 to B, using X1 , X2 , and X3 as independent variables and using x1 , x2 , and x3 as independent variables.
43) g √ v = v 1 g11 √ 1 + · · · + . 44) can be written as From this we can extract the physical component: √ v (1) = v 1 g11 . 46) the physical components are of the form v(1) = v1 g 11 . 3 Tensor Calculus Tensor calculus deals with differentiating and integrating tensors in curvilinear coordinates. In the Cartesian system the base vectors are constants, and ∂u = ei ∂kui . 48) In the case of general tensors in a curvilinear system, ξ , the base vectors are not constants, and ∂u = g i ∂kui + ui ∂k g i .
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. That is, A (i, j, k) = Q jm Qkn Qil A(l, m, n). Thus A is a third-rank tensor. This is known as the quotient rule.
Cambridge Introduction To Continuum Mechanics by J. N. Reddy