4.3 Momentum Balance Principles

Transcription

1 4.3 Momentum Balance Principles Balance of linear angular momentum in spatial material description Consider a continuum body B with a set of particles occupying an arbitrary region Ω with boundary surface Ω at time t. t = t B Ω Ω P P Ω X x r Ω x O O Consider a closed system with a given motion x= ( X, t), spatial mass density r= r( x, t), spatial velocity field v= v( x, t). he total linear momentum L (also called translational momentum) is defined by the vectorvalued function, L () t = r( x,) t v ( x,)d t v= r ( X ) V ( X,)d t V. (4.35) he total angular momentum J (also called rotational momentum) relative to a fixed point (characterized by the position vector x ) is defined by the vector-valued function, J () t = r r( x,) t v ( x,)d t v= r r ( X ) V ( X,)d t V. (4.36) 1 In eqns (4.35) (4.36) the identity VX - (,) t = V é (,), t tù êë x úû = vx (,) t (eqn (2.8)), conservation of mass in the form r ( X)d V = r ( x, t )d v (eqn (4.6)), the definition of the position vector r, i.e., rx ( ) = x- x = ( X, t) -x, (4.37) have been used. November 2,

2 Momentum eqns (4.35) (4.36) are formulated wrt the current reference configurations with associated quantities r, v,dv r,,dv V, respectively. Linear momentum angular momentum per unit current reference volume are defined by rv, rv r rv, r rv, respectively. he material time derivatives of linear angular momentum eqns (4.35.1) (4.36.1) of the particles which fill an arbitrary region result in fundamental axioms called momentum balance principles for a continuous body. he balance of linear momentum is postulated as, L () t = (,) (,)d ( ) (,)d () t r x t v x t v = r t X V X t V = F t, (4.38) the balance of angular momentum is postulated as, J () t = (,) (,)d ( ) (,)d () t r r x t v x t v = r t r X V X t V = M t, (4.39) where F () t is the resultant force M () t is the resultant moment both F () t M () t are vector valued functions. he momentum balance principles are generalizations of Newton s first second principles of motion to the context of continuum mechanics, as introduced by Cauchy Euler. he contributions to linear momentum L angular momentum J of a body are due to external sources, i.e., F () t M () t, respectively. If the external sources vanish linear angular momentum of the body are said to be conserved. From eqns (4.38) (4.39), using eqn (4.34), L () t = r(,) x t v (,)d x t v= r ( X ) V ( X,)d t V = F () t, (4.4) J () t = r r( x,) t v ( x,)d t v= r r ( X ) V ( X,)d t V = M () t. (4.41) In obtaining eqn (4.41), r v= r v+ r v = r v since r = x = v (see eqns (4.37.1) (2.28.1)) has been used. he spatial material acceleration fields are characterized by v V. he inertia forces per unit current reference volume are denoted by rv rv, respectively. November 2,

3 t = t t n dv x ds Ω X, x 3 3 Ω O X 2, x2 X, x 1 1 Consider a boundary surface Ω of an arbitrary region Ω which is subjected to the Cauchy traction vector t= t( x, t, n ) (force measured per unit current surface area of Ω, see Sec. 3.1). he unit vector n is the outward normal to an infinitesimal surface element ds of Ω. he spatial vector field b= b( x, 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 b= FF.) A body force is, for example, self weight or gravity loading per unit volume, i.e., b = rg with the spatial mass density r the gravitational acceleration g. Hence, the resultant force F () t the resultant moment M () t (about a point x ) on the body in the current configuration have the additive forms, F () t = t ds= b dv, (4.42) M () t = r t ds= r b dv. (4.43) Finally, from eqns (4.38), (4.39), (4.42) (4.43) the global forms of balance of linear momentum balance of angular momentum in the spatial description are, t t rvdv= tds+ b dv, (4.44) r rvdv= r tds+ r b dv. (4.45) Equations (4.44) (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 2,

4 ( r + )d v= ( + )d s+ ( + )d v t r v p r t m r b c. (4.46) 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. he 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.) o express the momentum balance principles in terms of material coordinates, the (pseudo) body force called the reference body force B= B( X, t) is introduced. It acts on the region Ω is, in contrast to the body force b, referred to the reference position X measures force per unit reference volume. ith volume change dv= JdV motion x= ( X, t), the transformation of the body force terms of eqns (4.44) (4.45) is of the form, bx (,)d t v= b( ( X,),) t t J( X,)d t V = BX (,)d t V, (4.47) or in the local form as, BX (, t) = J( X, t) bx (, t) or Ba = Jba. (4.48) Using the first Piola-Kirchhoff traction vector = ( X, t, N ) (see eqn (3.1)), eqns (4.38), (4.39), (4.48) dv= JdV, it can be concluded from eqns (4.44) (4.45) that, t t r = + VdV ds B dv, (4.49) r r VdV = r ds+ r B dv. (4.5) Equations (4.49) (4.5) are the global forms of balance of linear momentum balance of angular momentum, respectively, in the material description. November 2,

5 4.3.2 Equation of motion in spatial material description A necessary sufficient condition that the momentum balance principles eqns (4.44) (4.45) are satisfied is the existence of a spatial tensor field σ so that tx (, t, n) = σ( x, t) n (see eqn (3.3.1)). By computing the integral form of Cauchy s stress theorem (3.3.1) by using divergence theorem (eqn (1.296)), tx (, t, n)d s= σ( x, t) nds= div σ( x, t)dv, (4.51) where σ is the symmetric Cauchy stress tensor. Substituting eqn (4.51) into eqn (4.44) using eqns (4.38) (4.4), (div σ + b- rv )dv =. Cauchy's first equation of motion in global form (4.52) Equation (4.52) is valid for any volume v. Hence, the integr should be equal to zero, for each pt x of v for all t. sab divσ + b= rv. or + ba = rv a, Cauchy's first equation of (4.53) x motion in local form b Generally, eqn (4.53) is nonlinear in the displacement field u. he nonlinearities are implicitly present due to geometric sources, i.e., the kinematics of motion of the body, 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 Î, from eqn (4.53), sab Cauchy's equation of div σ + b= or + ba =. (4.54) x equilibrium in elastostatics b For solid bodies, it is sometimes more convenient to work with the material description. Hence, the equations of motion eqns (4.52) (4.53) are rearranged in terms of quantities which are referred to the reference configuration. For this purpose, the following important identity is introduced: F = or - iv(j ) X -1 ( JFAa ) A =. Piola identity (4.58) Proof: Choose any region Ω of a continuum body with boundary surface Ω apply the divergence theorem twice. ith eqn (1.3), Nanson s formula (2.55) eqn (1.296), - - iv(j )d J d d d div d F V = F N S = n s= In s= I v=. (4.59) ith eqn (4.58), Piola transformation (3.8), identities (1.291) (2.56.2), the divergence of the first Piola-Kirchoff stress tensor P with respect to material coordinates is, November 2,

6 iv = iv(j ) = iv[ (J )] = Grad : (J ) + iv(j ) P σf σ F σ F σ F, - - = JGrad σ : ( F ) = J(iv σ) F. (4.6) From eqns (4.6) (2.5), PaB J sab ivp = Jdivσ or =. (4.61) X x Combining eqns (4.61), (4.47.2), (4.4) with Cauchy s first eqn of motion eqn (4.52), obtain after a change of variables use of dv= JdV, B b (iv P+ B- rv)dv =. Global form of the equation of motion (4.62) in the reference configuration Since the volume v ( therefore V ) is arbitrary, the associated local form is obtained as, ivp+ B= r V PaA or + Ba = rv a. (4.63) X A Symmetry of the Cauchy tensor Consider the global form of balance of angular momentum eqn (4.45), i.e., t r rvdv= r tds+ r b dv. (4.45) he first term on the RHS of eqn (4.45), using eqns (3.3.1) (1.32) is, r tds= r σnd s= ( r div σ+ ε: σ )dv, (4.64) where ε denotes the permutation tensor (see eqn (1.143)). he term on the LHS of (4.45), t Hence, from eqn (4.45), r rvdv= r rv dv. r ( rv -b- div σ)d v= ( ε: σ )dv. (4.65) Using the equation of motion (4.53) the fact that the current volume v is arbitrary, from eqn (4.65), ε: σ = or e s =, (4.66) abc cb November 2,

7 which holds at each point x of the region for all times t. he double contraction vector with components e s =. From eqn (4.66) can show that, abc cb ε: σ gives a s cb = s. (4.67) bc his relation eqn (4.67) is satisfied, if only if the Cauchy stress tensor σ is symmetric, i.e., σ= σ or ab ba s = s. (4.68) (Note see Problem of RO: educe that ij is symmetric if only if e ijkjk =.) he crucial result of eqn (4.68) is a local consequence of the balance of angular momentum eqn (4.45), often referred to as Cauchy s second equation of motion. From eqns (3.62) (3.65.1), the Kirchhoff stress tensor τ the second Piola-Kirchhoff stress tensor S are also symmetric. However, from eqn (3.67), the first Piola-Kirchhoff stress tensor P is, in general, not symmetric as indicated in eqn (3.1). Note that for a polar continuum (resultant couples are not zero) the symmetry property does not hold any longer ( σ¹ σ ) therefore eqn (4368) may also be viewed as a constitutive equation (c.f. Exercise 5, pg. 152, GAH.) November 2,