Continuum Mechanics by Professor I-Shih Liu (auth.)

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By Professor I-Shih Liu (auth.)

This concise textbook develops step-by-step the elemental ideas of continuum mechanics. Emphasis is on mathematical readability, and a longer appendix offers the necessary heritage wisdom in linear algebra and tensor calculus. After introducing the fundamental notions approximately common kinematics, stability equations, fabric objectivity and constitutive features, the booklet turns to the presentation of rational thermodynamics by means of stressing the position of Lagrange multipliers in deriving constitutive funcitions from the underlying entropy precept. a short lecture on prolonged thermodynamics closes the e-book. Many examples and workouts around off the cloth offered within the chapters. The e-book addresses basically complicated undergraduate scholars in theoretical physics, utilized arithmetic and fabrics sciences.

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53) In particular, for n == 1, . 54) Therefore, Al is just twice the strain rate tensor. More generally, the RivlinEri cksen t ensor of order n is a measure for the strain rate of higher order. 29) , the vorticity vector w and the spin t ensor ~ are related by w == (- 2~ ) and not by w == (W), as introduced in Sect. 6. 22 1. 2 Verify that or in t erms of the vorticit y vector w , . ~ av + -1 grad(v · v ) +w x v . 56) o. 29) , w = - (W )). 1. 7 Change of Frame and Objective Tensors A fram e of reference ca n be int erpreted as an observer who observes an event in terms of position and t ime with a rul er and a clock.

I n = Q i ; l . . : M j 1 .. i-. 61) is given by In the physical interpret ation we ca n say that U and U" , T and T* so related, are the same vector and t he same te nsor as obser ved in two different fram es ¢ and ¢*. , So = IR, by Q * : IR -+ IR Q* = 1. 62 ) Note that Q* : Sn -+ Sn dep ends only on the rot at ional part Q(t) of the change of t he fram e *. Ther efore we may also refer to Q * as the induced lin ear m ap on Sn by t he rotati on Q(t) (in the change of fram e) . Let

77) Balance of int ernal energy Bpe {)t + diIV (pex. + q ) - T . grad x. = pr. 75)z. Mor eover , sinc e the balance of kineti c energy is a consequence of t he balance of linear momentum, its field equat ion has not been given above. The balance of internal energy, of cour se, follows from the balanc e of energy and linear momentum. Since it is simpler , the balan ce of internal energy is oft en used in plac e of the balanc e of total energy. 15) can not be applied to the balan ce of int ernal energy nor to the bal an ce of kinetic energy du e to the presence of t he int ernal production (T ·grad x) .

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