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 necessary historical past wisdom in linear algebra and tensor calculus. After introducing the fundamental notions approximately basic kinematics, stability equations, fabric objectivity and constitutive capabilities, the ebook turns to the presentation of rational thermodynamics via 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 workouts around off the cloth provided within the chapters. The publication addresses basically complicated undergraduate scholars in theoretical physics, utilized arithmetic and fabrics sciences.

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1 (Transport Theorem). Let V(t) be a regular region in E and un(x, t) be th e outward normal speed of a surface point x E 8V(t) . Then for any smooth tensor field ,¢(x , t) , we have d dt r'¢ dv = l,r ~'¢ dv + r '¢u n da . 2) J av Proof By definition, d dt r'¢ dv = lim -hI { r Jv h-tO = r + lim -hI {r lim -hI { h -tO JV(t+h) . 3) '¢(x, t) dV} 1 JV(t) ,¢(x, t + h ) dv + '¢(x , t ) dv } 8'¢ --;:}(x,t)dv . V( t) dt The region V(t + h) - V(t) is swept out by 8V in the time int erval (t, t + h) . In other words, a small volume element Llv , as shown in Fig.

We obtain Jump condition. 15) if, in addition, ~~ and a'l/J are bounded in th e neighborhood of x. 15) is also known as the Rankin e-Hugoniot equation. 17) are called the local speeds of propagation of S relative to the motion of the body. If S is a material surface then U± = O. 2 Balance Equations in Material Coordinates Sometimes, for solid bodies, it is mor e convenient to use the material description. 15) in the reference coordinates can be derived in a simil ar manner . 1 Genera l Balance Equation 37 where U,,(X , t) is the outward normal speed of a sur face point X E 8V( t) .

31) states that the total mas s of any part is constant in time. 19) . 35) p + p div e = O. In fluid mechan ics, t his equation is also referr ed to as t he equation of continuity . 35) implies t hat for any t ensor qu an ti ty sb , . p'ljJ fJp'ljJ = 7ft + div(p'ljJ (9 v). 2 Let 'ljJ (x, t ) be an arbitrary function. 37) provid ed t hat t he mass is cons erved . 4) . 3 Laws of Dynamics Let X be a motion of B. 38) x du, Xo of P with respect r p(x-xo) l\xdv. 39) }Pt The angular momentum, also called the m om ent of momentum, is a skewsymmet ric te nsor qu antity, and henc e, more commonly, it is represented as an ax ial vector qu an ti ty.