By J. N. Reddy
This textbook on continuum mechanics displays the fashionable view that scientists and engineers may be informed to imagine and paintings in multidisciplinary environments. The booklet is perfect for complex undergraduate and starting graduate scholars. The e-book good points: derivations of the fundamental equations of mechanics in invariant (vector and tensor) shape and specializations of the governing equations to numerous coordinate structures; quite a few illustrative examples; chapter-end summaries; and workout difficulties to check and expand the certainty of suggestions provided.
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Extra info for An introduction to continuum mechanics: with applications
Thus, the three-term expression A1 e1 + A2 e2 + A3 e3 can be simply written as A = Ai ei . 33) This notation is called the summation convention. 1 Dummy Index The repeated index is called a dummy index because it can be replaced by any other symbol that has not already been used in that expression. Thus, the expression in Eq. 34) and so on. As a rule, no index must appear more than twice in an expression. For example, Ai Bi Ci is not a valid expression because the index i appears more than twice.
12) The trace of a dyad is defined to be the double-dot product of the dyad with the unit dyad tr = : I. 13) The trace of a tensor is invariant, called the first principal invariant, and it is denoted by I1 ; that is, it is invariant under coordinate transformations (φii = φ¯ ii ). The first, second, and third principal invariants of a dyadic are defined to be I1 = tr , I2 = 1 (tr )2 − tr 2 2 , I3 = det . 14) In terms of the rectangular Cartesian components, the three invariants have the form I1 = φii , I2 = 1 (φii φ j j − φi j φ ji ) , 2 I3 = |φ|.
The products AB and BA are, in general, not equal AB = BA (even if they are of equal size); that is, the matrix multiplication is not commutative. 5. For any real square matrix A, A is said to be normal if AAT = AT A; A is said to be orthogonal if AAT = AT A = I. 6. If A is a square matrix, the powers of A are defined by A2 = AA, A3 = AA2 = A2 A, and so on. 7. Matrix multiplication is associative: (AB)C = A(BC). 8. The product of any square matrix with the identity matrix is the matrix itself. 9.