By J. N. Reddy

ISBN-10: 0521870445

ISBN-13: 9780521870443

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

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Additional resources for An introduction to continuum mechanics: with applications

Example text

Bn be arbitrary vectors. Then we can represent a dyadic as = A1 B1 + A2 B2 + · · · + An Bn . 1) The transpose of a dyadic is defined as the result obtained by the interchange of the two vectors in each of the dyads. For example, the transpose of the dyadic in Eq. 1) is T = B1 A1 + B2 A2 + · · · + Bn An . One of the properties of a dyadic is defined by the dot product with a vector, say V: · V = A1 (B1 · V) + A2 (B2 · V) + · · · + An (Bn · V), V· = (V · A1 )B1 + (V · A2 )B2 + · · · + (V · An )Bn .

A2n    . A =  .. ..  , αA =   ··· ··· ··· ···   . ...  αam1 αam2 . . αamn am1 am2 . . amn Matrix addition has the following properties: 1. Addition is commutative: A + B = B + A. 2. Addition is associative: A + (B + C) = (A + B) + C. 3. There exists a unique matrix 0, such that A + 0 = 0 + A = A. The matrix 0 is called zero matrix; all elements of it are zeros. 4. For each matrix A, there exists a unique matrix −A such that A + (−A) = 0. 5. Addition is distributive with respect to scalar multiplication: α(A + B) = αA + αB.

2 Vector Algebra 21 3. ai = b j ci di . 4. xi xi = r 2 . 5. ai b j c j = 3. SOLUTION: 1. Not a valid expression because the free indices r and s do not match. 2. Valid; both m and s are free indices. There are nine equations (m, s = 1, 2, 3). 3. Not a valid expression because the free index j is not matched on both sides of the equality, and index i is a dummy index in one expression and a free index in the other; i cannot be used both as a free and dummy index in the same equation. The equation would have been valid if i on the left side of the equation is replaced with j; then there will be three equations.