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4.3 Balance Principles

4.3.1 Balance of linear and 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 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 xX=  (,)t , spatial density rr= (,)x t , and spatial vvx= (,)t .

The total linear momentum L (also called translational momentum) is defined by the vector- valued ,

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 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 and M()t is the resultant 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 , 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 fields are characterized by v and   V . The 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

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 surface element ds of ¶Ω . The spatial vector field bbx= (,)t is the defined per unit current volume of region Ω acting on a particle. (Note: Symbol b should not be confused with the left Cauchy-Green bFF= T .) A body force is, for example, self weight or 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 .

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 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 σ so that tx(,,)tt n = σ (,)xn (see eqn (3.3.1)).

By computing the integral form of Cauchy’s 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 . 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 field u . The nonlinearities are implicitly present due to geometric sources, i.e., the 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 bodies, it is sometimes more convenient to with the material description. Hence, the 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 . (4.63) ¶X A

4.3.3 Symmetry of the Cauchy tensor Consider the global form of balance of angular momentum eqn (4.45), i.e.,

D òòòrv´=´+´r dddvsv rt rb. (4.45) Dt W¶WW

The first term on the RHS of eqn (4.45), using eqns (3.3.1) and (1.302) is,

òòrt´=dd(div:)dss r ´σnr = ò ´σεσ + T v, (4.64) ¶W ¶W W where ε denotes the permutation tensor (see eqn (1.143)).

The term on the LHS of (4.45),

D òòrv´=´rrddvv rv . Dt WW

Hence, from eqn (4.45),

òòrvb´--(div)d(:)dr  σεσvv = T . (4.65) WW

Using the equation of motion (4.53) and the fact that the current volume v is arbitrary, from eqn (4.65),

T εσ: = 0 or esabc cb = 0 , (4.66)

November 20, 2015 4.3-6 which holds at each point x of the region for all t . The double contraction εσ: T gives a vector with components esabc cb = 0 . From eqn (4.66) can show that,

sscb= bc . (4.67)

This relation eqn (4.67) is satisfied, if and only if the Cauchy stress tensor σ is symmetric, i.e.,

T σσ= or ssab= ba . (4.68)

(Note see Problem 1.3.2 of RWO: Deduce that Tij is symmetric if and only if eijkT jk = 0 .)

The 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) and (3.65.1), the Kirchhoff stress tensor τ and 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 (3.10).

Note that for a polar continuum (resultant couples are not zero) the symmetry property does not hold any longer (σσ¹ T ) and therefore eqn (4368) may also be viewed as a (c.f. Exercise 5, pg. 152, GAH.)

November 20, 2015 4.3-7