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4.3 Momentum Balance Principles 4.3.1 Balance of linear and angular momentum in spatial and material description Consider a continuum body B with a set of particles occupying an arbitrary region Ω with boundary surface ¶Ω at time t . tt= B Ω Ω 0 P¢ P ¶Ω 0 ¶Ω x r X x 0 O¢ O Consider a closed system with a given motion xX= (,)t , spatial mass density rr= (,)x t , and spatial velocity field vvx= (,)t . The total linear momentum L (also called translational momentum) is defined by the vector- valued function, LxvxXVX()t==rr ( ,) t ( ,)d tv ( ) ( ,)d tV. (4.35) ò ò 0 WW0 The total angular momentum J (also called rotational momentum) relative to a fixed point (characterized by the position vector x0 ) is defined by the vector-valued function, JrxvxrXVX()t=´rr ( ,) t ( ,)d tv =´ ( ) ( ,)d tV. (4.36) ò ò 0 WW0 é -1 ù In eqns (4.35) and (4.36) the identity VX( ,)tttt== Vëê (,), xûú vx (,) (eqn (2.8)), conservation of mass in the form rr0 ()dXxVtv= (,)d (eqn (4.6)), and the definition of the position vector r , i.e., rx()=- x x00 = ( X ,)t - x , (4.37) have been used. November 20, 2015 4.3-1 Momentum eqns (4.35) and (4.36) are formulated wrt the current and reference configurations with associated quantities r,,dv v and r0 ,,dV V , respectively. Linear momentum and angular momentum per unit current and reference volume are defined by rrvV, 0 and rvrV´´rr, 0 , respectively. The material time derivatives of linear and angular momentum eqns (4.35.1) and (4.36.1) of the particles which fill an arbitrary region W result in fundamental axioms called momentum balance principles for a continuous body. The balance of linear momentum is postulated as, DD LxvxXVXF ()t==rr (,) t (,)d tv ( ) ( ,)d tV = () t, (4.38) DDttòò0 WW0 and the balance of angular momentum is postulated as, DD JrxvxrXVXM ()tttvtVt=´rr ( ,) ( ,)d =´ ( ) ( ,)d = (), (4.39) DDttòò0 WW0 where F()t is the resultant force and M()t is the resultant moment and both F()t and M()t are vector valued functions. The momentum balance principles are generalizations of Newton’s first and second principles of motion to the context of continuum mechanics, as introduced by Cauchy and Euler. The contributions to linear momentum L and angular momentum J of a body are due to external sources, i.e., F()t and M()t , respectively. If the external sources vanish linear and angular momentum of the body are said to be conserved. From eqns (4.38) and (4.39), using eqn (4.34), LxvxXVXF()t==rr (,) t (,)d tv ( ) ( ,)d tV = () t, (4.40) ò ò 0 WW0 and JrxvxrXVXM()t=´rr ( ,) t ( ,)d tv =´ ( ) ( ,)d tV = () t. (4.41) ò ò 0 WW0 In obtaining eqn (4.41), rv´=´+´=´ rvrv rv since rxv == (see eqns (4.37.1) and (2.28.1)) has been used. The spatial and material acceleration fields are characterized by v and V . The inertia forces per unit current and reference volume are denoted by rv and r0V , respectively. November 20, 2015 4.3-2 t tt= n x ds dv Ω X 33, x ¶Ω O X 22, x X , x 11 Consider a boundary surface ¶Ω of an arbitrary region Ω which is subjected to the Cauchy traction vector ttxn= (,,)t (force measured per unit current surface area of ¶Ω , see Sec. 3.1). The unit vector n is the outward normal to an infinitesimal surface element ds of ¶Ω . The spatial vector field bbx= (,)t is the body force defined per unit current volume of region Ω acting on a particle. (Note: Symbol b should not be confused with the left Cauchy-Green deformation tensor bFF= T .) A body force is, for example, self weight or gravity loading per unit volume, i.e., b = rg with the spatial mass density r and the gravitational acceleration g . Hence, the resultant force F()t and the resultant moment M()t (about a point x0 ) on the body in the current configuration have the additive forms, Ftb()tsv==ò dò d , (4.42) ¶W W and Mrtrb()tsv=´=´ò dò d . (4.43) ¶W W Finally, from eqns (4.38), (4.39), (4.42) and (4.43) the global forms of balance of linear momentum and balance of angular momentum in the spatial description are, D òòòrvtbdddvsv=+ , (4.44) Dt W¶WW and D òòòrv´=´+´r dddvsv rt rb. (4.45) Dt W¶WW Equations (4.44) and (4.45) are fundamental in continuum mechanics. For the balance of angular momentum (4.45), the restriction that distributed resultant couples are neglected is assumed. If resultant couples are considered throughout a body in motion, then the balance of angular momentum eqn (4.45) will be written as, November 20, 2015 4.3-3 D òòò()d()d()drvp´+r vsv = rtm ´+ + rbc ´+ . (4.46) Dt W¶WW In eqn (4.46), m is the distributed assigned coupled traction vector per unit current area acting on the boundary surface ¶Ω while c is the distributed assigned body couple per unit volume acting within the volume of region Ω . The spin angular momentum (or intrinsic angular momentum) per unit current volume is p . A continuum without distributed couples is called non-polar. If any couple acts on parts of the continuum the continuum is called polar. (Polar continua are not considered here.) To express the momentum balance principles in terms of material coordinates, the (pseudo) body force called the reference body force BBX= (,)t is introduced. It acts on the region Ω and is, in contrast to the body force b , referred to the reference position X and measures force per unit reference volume. With volume change ddvJV= and motion xX= (,)t , the transformation of the body force terms of eqns (4.44) and (4.45) is of the form, òòbx(,)dtv== b ( ( X ,),)( ttJ X ,)d tV ò BX ( ,)d tV, (4.47) WW W0 or in the local form as, BX(,)tJt= (,)(,) X bx t or Baa= Jb . (4.48) Using the first Piola-Kirchhoff traction vector TTXN= (,,)t (see eqn (3.1)), eqns (4.38), (4.39), (4.48) and ddvJV= , it can be concluded from eqns (4.44) and (4.45) that, D r VTBdddVSV=+, (4.49) Dt òòò0 W¶WW000 and D r´=´+´r VdddVSV rT rB . (4.50) Dt òòò0 W¶WW000 Equations (4.49) and (4.50) are the global forms of balance of linear momentum and balance of angular momentum, respectively, in the material description. November 20, 2015 4.3-4 4.3.2 Equation of motion in spatial and material description A necessary and sufficient condition that the momentum balance principles eqns (4.44) and (4.45) are satisfied is the existence of a spatial tensor field σ so that tx(,,)tt n = σ (,)xn (see eqn (3.3.1)). By computing the integral form of Cauchy’s stress theorem (3.3.1) and by using divergence theorem (eqn (1.296)), òòòtx(,,)dts n ==σ (,)dxn ts div(,)dσ x tv, (4.51) ¶W ¶W W where σ is the symmetric Cauchy stress tensor. Substituting eqn (4.51) into eqn (4.44) and using eqns (4.38) and (4.40), ò (divσ +-bvr )dv = 0.Cauchy's first equation of motion in global form (4.52) W Equation (4.52) is valid for any volume v . Hence, the integrand should be equal to zero, ¶sab Cauchy's first equation of divσ +=bvr . or +=bvaar , (4.53) ¶xb motion in local form for each pt x of v for all t . Generally, eqn (4.53) is nonlinear in the displacement field u . The nonlinearities are implicitly present due to geometric sources, i.e., the kinematics of motion of the body, and material sources, i.e., the material itself – the Cauchy stress σ may, in general, depend on u . If the acceleration is assumed to be zero for all x ÎW, from eqn (4.53), ¶s Cauchy's equation of divσ +=b0 or ab +=b 0 . (4.54) a equilibrium in elastostatics ¶xb For solid bodies, it is sometimes more convenient to work with the material description. Hence, the equations of motion eqns (4.52) and (4.53) are rearranged in terms of quantities which are referred to the reference configuration. For this purpose, the following important identity is introduced: ¶()JF -1 Div(JF0-T ) = or Aa = 0 .Piola identity (4.58) ¶X A Proof: Choose any region Ω0 of a continuum body with boundary surface ¶Ω0 and apply the divergence theorem twice. With eqn (1.300), Nanson’s formula (2.55) and eqn (1.296), òòòòòDiv(JFFNnInI0--TT )dVSssv===== J d d d div d . (4.59) W¶W¶W¶WW00 With eqn (4.58), Piola transformation (3.8), and identities (1.291) and (2.56.2), the divergence of the first Piola-Kirchoff stress tensor P with respect to material coordinates is, November 20, 2015 4.3-5 DivP == Div(JσF--TT ) Div[σ (JF )] = Gradσ : (JF -- TT ) +σ Div(JF ) , ==JJGradσ : (F--TT ) (Divσ )F . (4.60) From eqns (4.60) and (2.50), ¶¶PJs DivP = J divσ or aB= ab . (4.61) ¶¶X Bbx Combining eqns (4.61), (4.47.2), (4.40) with Cauchy’s first eqn of motion eqn (4.52), obtain after a change of variables and use of ddvJV= , (DivPB+-r V )dV = 0.Global form of the equation of motion (4.62) ò 0 W0 in the reference configuration Since the volume v (and therefore V ) is arbitrary, the associated local form is obtained as, ¶PaA DivPB+=r0 V or +=Baar0V .
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