2 Material Derivatives of Line, Surface, and Volume Integrals

which upon application of the divergence theorem becomes



˙

PijL (t ) =



∂ Pij*L

dV +

V ∂t



∫



∫vP

S



*

k ijL k



(5.2-3)



n dS



This equation gives the time rate of change of the property Pij as the sum of

the amount created in the volume V, plus the amount entering through the

bounding surface S, and is often spoken of as the transport theorem.

Time derivatives of integrals over material surfaces and material curves

may also be derived in an analogous fashion. First, we consider a tensorial

property Qij of the particles which make up the current surface S, as given by

QijL (t ) =



∫Q



*

ijL



S



(x , t ) dS p = ∫



S



*

QijL (x , t ) n p dS



(5.2-4)



*

where QijL (x , t ) is the distribution of the property over the surface. From

Eq 4.11-7, we have in Eulerian form (again omitting the variables x and t),



˙

QijL (t ) =



=



∫ (Q˙



*

ijL



S



∫ [(Q˙



*

ijL



S



)



∫



*

*

+ vk,k QijL dS p − QijLvq,p dS q

S



]



)



*

*

+ vk,k QijL δ pq − QijLvq,p dS q



(5.2-5)



Similarly, for properties of particles lying on the spatial curve C and

expressed by the line integral



∫R



RijL (t ) =



C



*

ijL



(x , t )dx p



(5.2-6)



we have, using Eq 4.11-1,

˙

RijL (t ) =



=



∫ R˙

C



∫ ( R˙

C



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*

ijL



dx p +



δ



*

ijL pq



∫v

C



p,q



*

RijLdxq



)



*

+ v p,q RijL dxq



(5.2-7)



5.3



Conservation of Mass, Continuity Equation



Every material body, as well as every portion of such a body is endowed

with a non-negative, scalar measure, called the mass of the body or of the

portion under consideration. Physically, the mass is associated with the

inertia property of the body, that is, its tendency to resist a change in motion.

The measure of mass may be a function of the space variables and time. If

∆m is the mass of a small volume ∆V in the current configuration, and if we

assume that ∆m is absolutely continuous, the limit



ρ = lim



∆V →0



∆m

∆V



(5.3-1)



defines the scalar field ρ = ρ(x,t) called the mass density of the body for that

configuration at time t. Therefore, the mass m of the entire body is given by



∫ ρ(x, t ) dV



m=



(5.3-2)



V



In the same way, we define the mass of the body in the referential (initial)

configuration in terms of the density field ρ0 = ρ0(X,t) by the integral



∫



m=



Vo



ρo (X, t ) dV o



(5.3-3)



The law of conservation of mass asserts that the mass of a body, or of any

portion of the body, is invariant under motion, that is, remains constant in

every configuration. Thus, the material derivative of Eq 5.3-2 is zero,

˙

m=



d

dt



∫ ρ (x, t ) dV



=0



(5.3-4)



V



*

which upon application of Eq 5.2-2 with PijL ≡ ρ becomes



˙

m=



∫ (ρ˙ + ρv ) dV = 0

V



i,i



(5.3-5)



and since V is an arbitrary part of the continuum, the integrand here must

vanish, resulting in

˙

ρ + ρvi,i = 0



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(5.3-6)



which is known as the continuity equation in Eulerian form. But the material

derivative of ρ can be written as

˙

ρ=



∂ρ

∂ρ

+ vi

∂t

∂ xi



so that Eq 5.3-6 may be rewritten in the alternative forms



∂ρ

∂ρ

+ vi

+ ρ vi,i = 0

∂t

∂ xi



(5.3-7a)



or



∂ρ

+ ( ρ vi ), i

∂t



=0



(5.3-7b)



˙

If the density of the individual particles is constant so that ρ = 0, the

material is said to be incompressible, and thus it follows from Eq 5.3-6 that

vi,i = 0



or



div v = 0



(5.3-8)



for incompressible media.

Since the law of conservation of mass requires the mass to be the same in

all configurations, we may derive the continuity equation from a comparison

of the expressions for m in the referential and current configurations. Therefore, if we equate Eqs 5.3-2 and 5.3-3,

m=



∫ ρ(x, t )dV = ∫



Vo



V



ρ0 ( X ,t)dV o



(5.3-9)



and, noting that for the motion x = x(X,t), we have



∫ ρ[x(X, t), t]dV = ∫



Vo



V



ρ( X ,t) J dV o



Now if we substitute the right-hand side of this equation for the left-hand

side of Eq 5.3-9 and collect terms,



∫ [ρ(X, t)J − ρ (X, t)]dV

Vo



o



o



=0



But V° is arbitrary, and so in the material description



ρJ = ρo

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(5.3-10a)



˙

and, furthermore, ρo = 0, from which we conclude that



( ρJ )•



=0



(5.3-10b)



Eqs 5.3-10 are called the Lagrangian, or material, form of the continuity

equation.



Example 5.3-1

Show that the spatial form of the continuity equation follows from the

material form.



Solution

Carrying out the indicated differentiation in Eq 5.3-10b,

˙

(ρ J )• = ρ J + ρ J˙



=0



and by Eq 4.11-6,

˙

J = vi,i J

so now

˙

(ρ J )• = J (ρ + ρ vi,i )



=0



˙

But J = det F ≠ 0 (an invertible tensor), which requires ρ + ρ vi,i = 0, the spatial

continuity equation.

As a consequence of the continuity equation, we are able to derive a useful

*

result for the material derivative of the integral in Eq 5.2-1 when PijL is equal

*

*

to the product ρ AijL, where AijL is the distribution of any property per unit

mass. Accordingly, let

d

˙

PijL (t ) =

dt

=



∫



V



*

AijL (x , t )ρ dV =



˙

∫ [A

V



o



*

ijL



d

dt



∫



Vo



*

AijL (X , t )ρ JdV o



(ρ J ) + Aij*L(ρ J )• ]dV o



which because of Eq 5.3-10b reduces to

˙

PijL (t ) =

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∫



V



o



˙*

AijLρ J dV o =



˙

∫A

V



*

ijL



ρ dV



FIGURE 5.1

Material body in motion subjected to body and surface forces.



and so

d

dt



∫A

V



*

ijL



(x , t )ρ dV = ∫



V



˙*

AijL (x , t )ρ dV



(5.3-11)



We shall have numerous occasions to make use of this very important equation.



5.4



Linear Momentum Principle, Equations of Motion



Let a material continuum body having a current volume V and bounding

ˆ

surface S be subjected to surface traction ti( n ) and distributed body forces

ρbi as shown in Figure 5.1. In addition, let the body be in motion under the

velocity field vi = vi(x,t). The linear momentum of the body is defined by the

vector

P (t ) =

i



∫ ρ v dV



(5.4-1)



i



V



and the principle of linear momentum states that the time rate of change of the

linear momentum is equal to the resultant force acting on the body. Therefore,

in global form, with reference to Figure 5.1,

d

dt

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∫ ρ v dV = ∫ t ( )dS + ∫ ρ b dV

ˆ

n



V



i



S



i



V



i



(5.4-2)



and because ti( n ) = σjinj, we can convert the surface integral to a volume

integral having the integrand σji,j. By the use of Eq 5.3-11 on the left-hand

side of Eq 5.4-2 we have, after collecting terms,

ˆ



∫ (ρ v˙ − σ

i



V



ji,j



)



− ρ bi dV = 0



(5.4-3)



˙

where vi is the acceleration field of the body. Again, V is arbitrary and so the

integrand must vanish, and we obtain

˙

σ ji,j + ρ bi = ρ vi



(5.4-4)



which are known as the local equations of motion in Eulerian form.

˙

When the velocity field is zero, or constant so that vi = 0, the equations

of motion reduce to the equilibrium equations



σ ji,j + ρ bi = 0



(5.4-5)



which are important in solid mechanics, especially elastostatics.



5.5



The Piola-Kirchhoff Stress Tensors,

Lagrangian Equations of Motion



As mentioned in the previous section, the equations of motion Eq 5.4-4 are

in Eulerian form. These equations may also be cast in the referential form

based upon the Piola-Kirchhoff tensor, which we now introduce.

Recall that in Section 3.3 we defined the stress components σ ij of the

ˆ

Cauchy stress tensor ␴ as the ith component of the stress vector ti(e j ) acting

ˆ ˆ

on the material surface having the unit normal n = e j . Notice that this unit

normal is defined in the current configuration. It is also possible to define a

stress vector that is referred to a material surface in the reference configuration and from it construct a stress tensor that is associated with that configuration. In doing this, we parallel the development in Section 3.3 for the

Cauchy stress tensor associated with the current configuration.

ˆ

Let the vector po(N) be defined as the stress vector referred to the area

ˆ

ˆ

element ∆S° in the plane perpendicular to the unit normal N = N A I A . Just

as we defined the Cauchy stress vector in Eq 3.2-1, we write

ˆ

oN

∆f

df

= o =p( )

o

dS

∆S →0 ∆S



lim

o



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(5.5-1)