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By D. S. Chandrasekharaiah and Lokenath Debnath (Auth.)
An in depth and self-contained textual content written for rookies, Continuum Mechanics deals concise assurance of the fundamental ideas, common ideas, and purposes of continuum mechanics. with no sacrificing rigor, the transparent and straightforward mathematical derivations are made obtainable to plenty of scholars with very little earlier history in sturdy or fluid mechanics. With the inclusion of greater than 250 absolutely worked-out examples and 500 labored workouts, this publication is bound to develop into a typical introductory textual content for college students in addition to an vital reference for execs.
Key Features
* presents a transparent and self-contained remedy of vectors, matrices, and tensors in particular adapted to the desires of continuum mechanics
* Develops the techniques and ideas universal to all parts in stable and fluid mechanics with a standard notation and terminology
* Covers the basics of elasticity idea and fluid mechanics
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Example text
1 8 . Prove that a matrix [α^\ has a multiplicative inverse if and only if det[a^·] ^ 0. 1 9 . Verify that the following matrices are nonsingular. Find their inverses. 1 0 0 0) 0 0 2 0 (ii) 2 3 2 2 0 -1 1 0 0 0-1 2 0 . Verify that the following matrices are orthogonal. Γι o o" (i) 0 0 1 (ii) COS0 -sine? 8 EXERCISES 31 2 1 . Show that (i) ôijôij = 3 (iii) ôikôjmôij (ii) ôijôjkôik = 3 = ôkm 2 2 . Simplify the following (ii) (i) ôu{au - aß) (iii) (âu + au)(âu - ôipôjqapbjCq a^ 2 3 . Write down the following equations in matrix forms: (i) otipOtjp = ou (iii) au = aôubkk (ii) oLpiOLpj = ôu + ßbu 2 4 .
1 5 . Represent the following matrix as a sum of a symmetric matrix and a skewsymmetric matrix: \flu\ = 2 0 4 -6 8 0 8 10 - 8 1 6 . For the matrix given in the previous exercise, compute [α^][α^]τ9 [α^ία^] [ay]2. and 1 7 . 35). 1 8 . Prove that a matrix [α^\ has a multiplicative inverse if and only if det[a^·] ^ 0. 1 9 . Verify that the following matrices are nonsingular. Find their inverses. 1 0 0 0) 0 0 2 0 (ii) 2 3 2 2 0 -1 1 0 0 0-1 2 0 . Verify that the following matrices are orthogonal. Γι o o" (i) 0 0 1 (ii) COS0 -sine?
One may show that if c is a vector with components c, and c\ in the A:, and x\ systems, 42 2 ALGEBRA OF TENSORS respectively, then the entity a (x) b (x) c represented as (tf/fyc*) in the x, system and (a-bjC^) in the x\ system is a Cartesian tensor of order 3. Examples of Cartesian tensors of higher orders may be constructed in an analogous way. 4 SCALAR INVARIANTS Quite often we deal with quantities that may be represented by single numbers not dependent on any coordinate system. , are examples of such quantities; these are referred to as scalar invariants.