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

By Professor I-Shih Liu (auth.)

This concise textbook develops step-by-step the basic rules of continuum mechanics. Emphasis is on mathematical readability, and a longer appendix offers the mandatory historical past wisdom in linear algebra and tensor calculus. After introducing the elemental notions approximately common kinematics, stability equations, fabric objectivity and constitutive features, the publication turns to the presentation of rational thermodynamics through stressing the position of Lagrange multipliers in deriving constitutive funcitions from the underlying entropy precept. a short lecture on prolonged thermodynamics closes the ebook. Many examples and routines around off the cloth awarded within the chapters. The ebook addresses essentially complex undergraduate scholars in theoretical physics, utilized arithmetic and fabrics sciences.

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Show that t he convected time derivatives defined by • Lu , uL:o. = uL:o. T S= S-LS-SL , are objective. 85) with P(T) = Ft(T) , the relative deformation gradi ent from time t to T . 5 Show t hat 1) the relative right stretch t ensor Ut is obj ective, while the relative rotation tensor R; is not obj ective; 2) the Ri vlin-Ericksen tensor An is objective. 2. 1) for any bounded regular subregion of the body, called a part PcB and the vector field n , the outward unit normal t o the boundary of the region P t in the curre nt configurat ion.

21 ) where U" is t he normal speed of the sur face points on S" , while the normal speed of the surface points on 8V" is zero, since a material region is a fixed region in the reference configurat ion. 22) where U"and n " are the norm al speed and the unit normal vect or of the singular sur face. 13) dir ectl y. 22h by t he use of the following relations: ~" = J(~~ + div( 1/J Q9 x)), Div

Then the tractions on the surface planes parallel to e and d , respecti vely are given by (Fig. 7) They are purely shear tractions and, in addit ion, t hey are of t he same magnitude . Indeed , one can easily ver ify t hat t he angle between t he vectors e and e * is equal to t hat between d an d d" , and cos B(e, e") = e · e* lell e *I' d · d* cosB( d, d*) = Idlld*I' which impli es t hat lel le* 1 = Idlld*l. Hence, it follows that Itll = It21. 4) Plane stress If, relative to an orthogonal bas is {e .

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