By Morton E. Gurtin
ISBN-10: 0511769806
ISBN-13: 9780511769801
ISBN-10: 052140598X
ISBN-13: 9780521405980
The Mechanics and Thermodynamics of Continua offers a unified remedy of continuum mechanics and thermodynamics that emphasizes the common prestige of the fundamental balances and the entropy imbalance. those legislation are seen as primary development blocks on which to border theories of fabric habit. As a priceless reference resource, this booklet provides an in depth and whole remedy of continuum mechanics and thermodynamics for graduates and complex undergraduates in engineering, physics, and arithmetic. The chapters on plasticity speak about the normal isotropic theories and, additionally, crystal plasticity and gradient plasticity.
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To establish the uniqueness of the transpose, assume that S has a second “transpose” S . 3), S = S . Next, if S has a transpose, then ei · S e j = e j · Sei = S j i , so that (S )i j = S j i . 15) This calculation shows that every tensor S has a transpose: S = S j i ei ⊗ e j . 16) (S + T) = S + T , (S ) = S. 18) S = −S . 20) skw S = 12 (S − S ), and refer to sym S as the symmetric part of S and to skw S as the skew part of S. 21) for any tensor S. 20), this decomposition is unique.
Establish the following identities for any vectors u, v, and w: (u · v)2 + |u × v|2 = |u|2 |v|2 , u × (v × w) + v × (w × u) + w × (u × v) = 0. 1 What Is a Tensor? 1) is also a vector. 1). The linearity of a tensor S is embodied by the requirements: S(u + v) = Su + Sv S(αu) = αSu for all vectors u and v; for all vectors u and scalars α. Tensors S and T are equal if their outputs are the same whenever their inputs are equal; precisely, S = T if and only if Sv = Tv for all vectors v. 2) One way of showing that tensors S and T are equal is a conseqence of the following result: a · Sb = a · Tb for all vectors a and b if and only if S = T.
91) hold for all vectors u and v. This definition yields the representation SC = S2 − (tr S)S + 12 [tr 2 (S) − tr (S2 )]1 . 92). 6. 94) to show that, for any vector u, (u×)C = u ⊗ u, (u ⊗ u)C = 0. 7. 91), show that for any pair of tensors S and T, det (S + T) = det S + tr (S TC ) + tr (S TC ) + det T. 95) 8. 94), show that for any pair of tensors S and T, (ST)C = SC TC and (S + T)C = SC + TC + T S + S T − (tr T)S − (tr S)T + [(tr S)(tr T) − tr (ST)]1. 96) 9. Let S be an invertible tensor and let u and v be vectors.
The Mechanics and Thermodynamics of Continua by Morton E. Gurtin
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