5 The Piola-Kirchhoff Stress Tensors, Lagrangian Equations of Motion

where ∆f is the resultant force acting on the material surface, which in the

reference configuration was ∆S°.

The principle of linear momentum can also be written in terms of quantities

which are referred to the referential configuration as



∫



So



ˆ

oN

p ( ) ( X , t)dSo +



∫



Vo



ρ0bo ( X , t)dV o =



∫



Vo



ρ0a o ( X , t)dV o



(5.5-2)



where S°, V°, and ρ0 are the material surface, volume, and density, respectively, referred to the reference configuration. The superscript zero after the

variable is used to emphasize the fact that the function is written in terms

of the reference configuration. For example,

ai(x,t) = ai[␹(X,t),t)] = aio (X,t)

Notice that, since all quantities are in terms of material coordinates, we have

moved the differential operator d/dt of Eq 3.2-4 inside the integral to give

rise to the acceleration a°. In a similar procedure to that carried out in

Section 3.2, we apply Eq 5.5-2 to Portions I and II of the body (as defined in

Figure 3.2a) and to the body as a whole to arrive at the equation

ˆ

ˆ

po(N) + po( − N)  dSo = 0







So 



∫



(5.5-3)



This equation must hold for arbitrary portions of the body surface, and so

ˆ

ˆ

oN

o −N

p ( ) = −p ( )



(5.5-4)



which is the analog of Eq 3.2-6.

ˆ

The stress vector po(N) can be written out in components associated with

the referential coordinate planes as

ˆ

p ( A ) = pio( A )e i

ˆ

o I



ˆ

I



(A = 1,2,3)



(5.5-5)

ˆ



This describes the components of the stress vector po(N) with respect to the

referential coordinate planes; to determine its components with respect to

ˆ

an arbitrary plane defined by the unit vector N , we apply a force balance

to an infinitesimal tetrahedron of the body. As we let the tetrahedron shrink

to the point, we have

ˆ

ˆ

N

I

pio( ) = pio( A ) N A



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(5.5-6)



and defining

ˆ

I

o

PAi ≡ pio( A )



(5.5-7)



ˆ

N

o

pio( ) = PAi N A



(5.5-8)



we obtain



o

where PAi are the components of the first Piola-Kirchhoff stress tensor. These

represent the xi components of the force per unit area of a surface whose

ˆ

referential normal is N .

Using the first Piola-Kirchhoff stress tensor, we can derive the equations

of motion, and hence the equilibrium equations in the referential formulation. Starting with Eq 5.5-2, we introduce Eq 5.5-8 to obtain



∫



So



o

PAi N AdS o +



∫



Vo



ρobio dV o =



∫



Vo



ρoaio dV o



(5.5-9)



and using the divergence theorem on the surface integral, we consolidate

Eq 5.5-9 as



∫ (P

Vo



o

Ai,A



)



+ ρobio − ρoaio dV o = 0



This equation must hold for arbitrary portions of the body so that the integrand is equal to zero, or

o

PAi,A + ρobio = ρoaio



(5.5-10a)



which are the equations of motion in referential form. If the acceleration field

is zero, these equations reduce to the equilibrium equations in referential form

o

PAi,A + ρobio = 0



(5.5-10b)



We note that the partial derivatives of the Piola-Kirchhoff stress components

are with respect to the material coordinates because this stress tensor is

referred to a surface in the reference configuration.

Equilibrium also requires a balance of moments about every point. Summing moments about the origin (Figure 5.1 may be useful in visualizing this

operation) gives us



∫



S



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o



ˆ

oN

ε ijk x j pk( ) dS o +



∫



V



o



o

ε ijk x j ρobk dV o = 0



(5.5-11)



which reduces to



∫



V



o



(



o

ε ijk  x j PAk







)



,A



o

+ x j ρobk  dV o = 0







where we have used Eq 5.5-8 and the divergence theorem. Carrying out the

indicated partial differentiation, we obtain



∫ ε [x

Vo



ijk



)]



(



o

j,A Ak



o

o

P + x j PAk,A + ρobk dV o = 0



and by Eq 5.5-10b this reduces to



∫ (ε

V



o



ijk



)



o

x j,A PAk dV o = 0



(5.5-12)



since the term in parentheses is zero on account of the balance of momentum.

Again, this equation must hold for all portions V° of the body, so the integrand must vanish, giving

o

ε ijk x j,A PAk = 0



(5.5-13)



Following a similar argument to that presented in Section 3.4, we conclude

that Eq 5.5-13 implies

o

o

x j,A PAk = xk,A PAj



(5.5-14)



If we now introduce the definition for sAB

o

PAi = xi,B sBA



(5.5-15)



and substitute into Eq 5.5-14, we observe that

s AB = sBA



(5.5-16)



which is called the second Piola-Kirchhoff stress tensor, or sometimes the

symmetric Piola-Kirchhoff stress tensor.

The Piola-Kirchhoff stresses can be related to the Cauchy stress by considering the differential force exerted on an element of deformed surface dS as

dfi = σjinjdS

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(5.5-17)



This force can also be written in terms of the first Piola-Kirchhoff stress tensor

as

o

dfi = PAi N AdS o



(5.5-18)



Recall from Eq 4.11-4 that the surface element in the deformed configuration

is related to the surface element in the reference configuration by

nq dS = X A,q JN AdS o

Using this, along with Eqs 5.5-17 and 5.5-18, we obtain

o

dfi = σ ji n j dS = σ ji X A,j JN AdS o = PAi N AdS o



(5.5-19)



which can be rewritten as



(σ



ji



)



o

X A,j J − PAi N AdS o = 0



(5.5-20)



From this we see that the Cauchy stress and the first Piola-Kirchhoff stress

are related through

o

Jσ ji = PAi x j,A



(5.5-21)



Also, from Eq 5.5-15 we can write

Jσ ji = x j,A xi,B sAB



(5.5-22)



which relates the Cauchy stress to the second Piola-Kirchhoff stress.

In Chapter Four, we showed that the difference between Eulerian and

Lagrangean strains disappears when linear deformations are considered.

Here, we will show that in linear theories the distinction between Cauchy

and Piola-Kirchhoff stress measures is not necessary.

To show the equivalence of Cauchy and Piola-Kirchhoff stresses in linear

theories, we have to recall some kinematic results from Section 4.7 and also

derive a few more. Introducing a positive number ε that is a measure of

smallness such that the displacement gradients ui,A are of the same order of

magnitude as ε, we may write

ui,A = 0(ε) as ε → 0



(5.5-23)



As we discovered in Section 4.7, the Eulerian and Lagrangean strains are

equivalent as ε → 0, so from Eqs 4.7-1 and 4.7-2 we have

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ΕABδiAδjB = eij = 0(ε)

o

Examination of Eqs 5.5-21 and 5.5-22 relates stress measures σji , PAi , and

sAB. To discuss this relationship in the linear case, we must find an expression

for the Jacobian as we let ε → 0. Starting with the definition of J in the form



J=



1

6



ε ijk ε ABC FiA FjB FkC



we substitute FiA = ui,A + δiA, etc., to get

J=



1

6



(



)(



)(



ε ijk ε ABC ui,A + δ iA u j,B + δ jB uk,C + δ kC



)



Carrying out the algebra and after some manipulation of the indices

J=



1

6



[



( )]



ε ijk ε ABC δ iAδ jBδ kC + 3uk,Cδ iAδ jB + 0 ε 2



where terms on the order of ε2 and higher have not been written out explicitly.

Since εijkεijk = 6 and εijkεijC = 2δkC,

J = 1 + uk,k + 0( ε 2 )



(5.5-24)



Now we can evaluate Eqs 5.5-21 and 5.5-22 as ε → 0, that is, for the case of

a linear theory. Since uk,k is 0(ε).

o

σji + 0(ε ) = PAiδ Aj + 0(ε )



(5.5-25)



With a similar argument for Eq 5.5-22, we find



σji = sABδAiδBj as ε → 0



(5.5-26)



Eqs 5.5-25 and 5.5-26 demonstrate that in linear theory Cauchy, Piola-Kirchhoff, and symmetric Piola-Kirchhoff stress measures are all equivalent.



5.6



Moment of Momentum (Angular Momentum) Principle



Moment of momentum is the phrase used to designate the moment of the

linear momentum with respect to some point. This vector quantity is also

frequently called the angular momentum of the body. The principle of angular

momentum states that the time rate of change of the moment of momentum

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of a body with respect to a given point is equal to the moment of the surface

and body forces with respect to that point. For the body shown in Figure 5.1,

if we take the origin as the point of reference, the angular momentum

principle has the mathematical form

d

dt



∫ε

V



ijk



x j ρ vk dV =



∫ε

S



(

x j tkn ) dS +

ˆ



ijk



∫ε

V



ijk



x j ρ bk dV



(5.6-1)



Making use of Eq 5.3-11 in taking the derivative on the left-hand side of the

equation and applying the divergence theorem to the surface integral after

(ˆ

introducing the identity tkn ) = σ qk nq results in



∫ ε [ x (ρ v˙

V



ijk



j



k



)



]



− σ qk,q − ρ bk − σ jk dV = 0



which reduces to



∫ε

V



ijk



σ kj dV = 0



(5.6-2)



because of Eq 5.4-4 (the equations of motion) and the sign-change property

of the permutation symbol. Again, with V arbitrary, the integrand must

vanish so that



ε ijkσ kj = 0



(5.6-3)



which by direct expansion demonstrates that σ kj = σ jk , and the stress tensor

is symmetric. Note that in formulating the angular momentum principle by

Eq 5.6-1 we have assumed that no body or surface couples act on the body.

If any such concentrated moments do act, the material is said to be a polar

material, and the symmetry property of ␴ no longer holds. But as mentioned

in Chapter Three, this is a rather specialized situation and we shall not

consider it here.



5.7



Law of Conservation of Energy, The Energy Equation



The statement we adopt for the law of conservation of energy is the following: the material time derivative of the kinetic plus internal energies is equal

to the sum of the rate of work of the surface and body forces, plus all other

energies that enter or leave the body per unit time. Other energies may

include, for example, thermal, electrical, magnetic, or chemical energies. In

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this text we consider only mechanical and thermal energies; we also require

the continuum material to be non-polar (free of body or traction couples).

If only mechanical energy is considered, the energy balance can be derived

from the equations of motion (Eq 5.4-4). Here we take a different approach

and proceed as follows. By definition, the kinetic energy of the material occupying an arbitrary volume V of the body in Figure 5.1 is



∫ ρ v ⋅ v dV =∫ ρ v v dV



1

2



K(t) =



V



V



(5.7-1)



i i



Also, the mechanical power, or rate of work of the body and surface forces

shown in the figure is defined by the scalar



∫ t ( )v dS +∫ ρ b v dV

ˆ

n



P(t) =



S



i



i



(5.7-2)



i i



V



Consider now the material derivative of the kinetic energy integral



∫



˙ d

K=

dt

=



V



1

2



ρ vi vi dV =



1

2



∫ ρ(v v ) dV

•



V



∫ ρ(v v˙ )dV = ∫ v (σ

i i



V



i



V



i i



ji,j



)



+ ρ bi dV



where Eq 5.4-4 has been used to obtain the final form of the integrand. But

viσij,j = (viσij),j – vi,jσij, and so

˙

K=



∫









V



( )



ρ bi vi + viσ ij







,j



− vi, jσ ij  dV







which, if we convert the middle term by the divergence theorem and make

use of the decomposition vi,j = Dij + Wij, may be written

˙

K=



∫ ρ b v dV +∫ t ( )v dS − ∫ σ D dV

ˆ

n



V



i i



S



i



i



V



ij



ij



(5.7-3)



By the definition Eq 5.7-2 this may be expressed as

˙

K +S=P



(5.7-4)



where the integral

S=



∫ σ D dV = ∫ tr(σ ⋅ D)dV

V



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ij



ij



V



(5.7-5)