r = 0. This is due to the fact that δv is arbitrary, and hence by choosing

δv = [1, 0, 0]T , followed by δv = [0, 1, 0]T and δv = [0, 0, 1]T , the three

components of the equation r = 0 are retrieved. We can now use Equa-

tion (4.16) for the residual vector and integrate over the volume of the body

to give a weak statement of the static equilibrium of the body as,

δW = (·σ + f ) · δv dv = 0 (4.23)

v

A more common and useful expression can be derived by recalling Prop-

erty (2.135e) to give the divergence of the vector σδv as,

·(σδv) = (·σ) · δv + σ : δv (4.24)

Using this equation together with the Gauss theorem enables Equation (4.23)

to be rewritten as,

n · σδv da ’ σ: δv dv + f · δv dv = 0 (4.25)

‚v v v

The gradient of δv is, by de¬nition, the virtual velocity gradient δl. Addi-

tionally, we can use Equation (4.7a) for the traction vector and the symmetry

of σ to rewrite n · σδv as δv · t, and consequently Equation (4.24) becomes,

σ : δl dv = f · δv dv + t · δv da (4.26)

v v ‚v

Finally, expressing the virtual velocity gradient in terms of the symmetric

virtual rate of deformation δd and the antisymmetric virtual spin tensor δw

and taking into account again the symmetry of σ gives the spatial virtual

work equation as,

δW = σ : δd dv ’ f · δv dv ’ t · δv da = 0 (4.27)

v v ‚v

12 STRESS AND EQUILIBRIUM

This fundamental scalar equation states the equilibrium of a deformable

body and will become the basis for the ¬nite element discretization.

4.5 WORK CONJUGACY AND ALTERNATIVE STRESS

REPRESENTATIONS

4.5.1 THE KIRCHHOFF STRESS TENSOR

In Equation (4.27) the internal virtual work done by the stresses is expressed

as,

δW∈ = σ : δd dv (4.28)

v

Pairs such as σ and d in this equation are said to be work conjugate with

respect to the current deformed volume in the sense that their product gives

work per unit current volume. Expressing the virtual work equation in the

material coordinate system, alternative work conjugate pairs of stresses and

strain rates will emerge. To achieve this objective, the spatial virtual work

Equation (4.27) is ¬rst expressed with respect to the initial volume and area

by transforming the integrals using Equation (3.56) for dv to give,

Jσ : δd dV = f 0 · δv dV + t0 · δv dA (4.29)

V V ‚V

where f 0 = Jf is the body force per unit undeformed volume and t0 =

t(da/dA) is the traction vector per unit initial area, where the area ratio

can be obtained after some algebra from Equation (3.68) as,

da J

=√ (4.30)

dA n · bn

The internal virtual work given by the left-hand side of Equation (4.29)

can be expressed in terms of the Kirchho¬ stress tensor „ as,

δW∈ = „ : δd dV ; „ = Jσ (4.31a,b)

V

This equation reveals that the Kirchho¬ stress tensor „ is work conjugate to

the rate of deformation tensor with respect to the initial volume. Note that

the work per unit current volume is not equal to the work per unit initial

volume. However, Equation (4.31b) and the relationship ρ = ρ0 /J ensure

that the work per unit mass is invariant and can be equally written in the

current or initial con¬guration as:

1 1

σ:d= „ :d (4.32)

ρ ρ0

13

4.5 WORK CONJUGACY AND STRESS REPRESENTATIONS

4.5.2 THE FIRST PIOLA“KIRCHHOFF STRESS TENSOR

The crude transformation that resulted in the internal virtual work given

above is not entirely satisfactory because it still relies on the spatial quanti-

ties „ and d. To alleviate this lack of consistency, note that the symmetry

™

of σ together with Equation (3.93) for l in terms of F and the properties of

the trace give,

δW∈ = Jσ : δl dV

V

™

Jσ : (δ F F ’1 ) dV

=

V

™

tr(JF ’1 σδ F ) dV

=

V

™

(JσF ’T ) : δ F dV

= (4.33)

V

We observe from this equality that the stress tensor work conjugate to the

™

rate of the deformation gradient F is the so-called ¬rst Piola“Kirchho¬

stress tensor given as,

P = JσF ’T (4.34a)

Note that like F , the ¬rst Piola“Kirchho¬ tensor is an unsymmetric two-

point tensor with components given as,

3 3

Jσij (F ’1 )Ij

P= PiI ei — E I ; PiI = (4.34b,c)

j=1

i,I=1

We can now rewrite the equation for the principle of virtual work in terms

of the ¬rst Piola“Kirchho¬ tensor as,

™

P : δ F dV = f 0 · δvdV + t0 · δvdA (4.35)

V V ‚V

Additionally, if the procedure employed to obtain the virtual work Equa-

tion (4.27) from the spatial di¬erential equilibrium Equation (4.24) is re-

versed, an equivalent version of the di¬erential equilibrium equation is ob-

tained in terms of the ¬rst Piola“Kirchho¬ stress tensor as,

r 0 = Jr = DIVP + f 0 = 0 (4.36)

where DIVP is the divergence of P with respect to the initial coordinate

system given as,

‚P

DIVP = : I; = (4.37)

0P 0P

‚X

14 STRESS AND EQUILIBRIUM

Remark 4.5.1. It is instructive to re-examine the physical meaning of

the Cauchy stresses and thence the ¬rst Piola“Kirchho¬ stress tensor. An

element of force dp acting on an element of area da = n da in the spatial

con¬guration can be written as,

dp = tda = σda (4.38)

Broadly speaking, the Cauchy stresses give the current force per unit de-

formed area, which is the familiar description of stress. Using Equation (3.68)

for the spatial area vector, dp can be rewritten in terms of the undeformed

area corresponding to da to give an expression involving the ¬rst Piola“

Kirchho¬ stresses as,

dp = JσF ’T dA = P dA (4.39)

This equation reveals that P , like F , is a two-point tensor that relates an

area vector in the initial con¬guration to the corresponding force vector in

the current con¬guration as shown in Figure 4.7. Consequently, the ¬rst

Piola“Kirchho¬ stresses can be loosely interpreted as the current force per

unit of undeformed area.

EXAMPLE 4.3: Rectangular block under self-weight (iii)

Using the physical interpretation for P given in Remark 1 we can ¬nd

the ¬rst Piola“Kirchho¬ tensor corresponding to the state of stresses

described in Example 4.1. For this purpose note ¬rst that dividing

Equation (4.39) by the current area element da gives the traction vector

associated with a unit normal N in the initial con¬guration as,

dA

t(N ) = P N

da

Using this equation with N = E 2 for the case described in Example

4.1 where the lack of lateral deformation implies da = dA gives,

t(E 2 ) = PE 2

2

= PiI (ei — E I )E 2

i,I=1

= P12 e1 + P22 e2

(continued)

15

4.5 WORK CONJUGACY AND STRESS REPRESENTATIONS

X 3, x 3

`

dp

n

N

dP

da

p

dA P

X1 , x1

X 2, x 2

time = t

time = 0

FIGURE 4.7 Interpretation of stress tensors.

EXAMPLE 4.3 (cont.)

Combining the ¬nal equation with the fact that t(E 2 ) = t(e2 ) =

’ρ0 g(H ’ X2 )e2 as explained in Example 4.1, we can identify P12 = 0

and P22 = ρ0 g(X2 ’ H). Using a similar analysis for t(E 1 ) eventually

yields the components of P as,

0 0

[P ] =

0 ρ0 g(X2 ’ H)

which for this particular example coincide with the components of the

Cauchy stress tensor. In order to show that the above tensor P satis¬es

the equilibrium Equation (4.37), we ¬rst need to evaluate the force

vector f 0 per unit initial volume as,

dv

f0 = f

dV

dv

= ’ρ ge2

dV

= ’ρ0 ge2

Combining this expression with the divergence of the above tensor P

immediately leads to the desired result.

16 STRESS AND EQUILIBRIUM

4.5.3 THE SECOND PIOLA“KIRCHHOFF STRESS TENSOR

The ¬rst Piola-Kirchho¬ tensor P is an unsymmetric two-point tensor and

as such is not completely related to the material con¬guration. It is possible

to contrive a totally material symmetric stress tensor, known as the second

Piola“Kirchho¬ stress S, by pulling back the spatial element of force dp

from Equation (4.39) to give a material force vector dP as,

dP = φ’1 [dp] = F ’1 dp (4.40)

—

Substituting from Equation (4.39) for dp gives the transformed force in terms

of the second Piola“Kirchho¬ stress tensor S and the material element of

area dA as,

S = JF ’1 σF ’T

dP = S dA; (4.41a,b)

It is now necessary to derive the strain rate work conjugate to the second

Piola“Kirchho¬ stress in the following manner. From Equation (3.100) it

follows that the material and spatial virtual rates of deformation are related

as,

™

δd = F ’T δ EF ’1 (4.42)

Substituting this relationship into the internal virtual work Equation (4.28)

gives,