δWint = σ : δd dv

v

™

Jσ : (F ’T δ EF ’1 ) dV

=

V

™

tr(F ’1 JσF ’T δ E) dV

=

V

™

= S : δ E dV (4.43)

V

™

which shows that S is work conjugate to E and enables the material vir-

tual work equation to be alternatively written in terms of the second Piola“

Kirchho¬ tensor as,

™

S : δ E dV = f 0 · δv dV + t0 · δv dA (4.44)

V V ‚V

For completeness the inverse of Equations (4.34a) and (4.41b) are given

as,

σ = J ’1 PF T ; σ = J ’1 F SF T (4.45a,b)

17

4.5 WORK CONJUGACY AND STRESS REPRESENTATIONS

Remark 4.5.2. Applying the pull back and push forward concepts to the

Kirchho¬ and second Piola“Kirchho¬ tensors yields,

S = F ’1 „ F ’T = φ’1 [„ ]; „ = F SF T = φ— [S] (4.46a,b)

—

from which the second Piola“Kirchho¬ and the Cauchy stresses are related

as,

S = Jφ’1 [σ]; σ = J ’1 φ— [S] (4.47a,b)

—

In the above equation S and σ are related by the so-called Piola transfor-

mation which involves a push forward or pull back operation combined with

the volume scaling J.

Remark 4.5.3. A useful interpretation of the second Piola“Kirchho¬

stress can be obtained by observing that in the case of rigid body motion

the polar decomposition given by Equation (3.27) indicates that F = R and

J = 1. Consequently, the second Piola“Kirchho¬ stress tensor becomes,

S = RT σR (4.48)

Comparing this equation with the transformation Equations (2.42) given in

Section 2.2.2, it transpires that the second Piola“Kirchho¬ stress compo-

nents coincide with the components of the Cauchy stress tensor expressed

in the local set of orthogonal axes that results from rotating the global

Cartesian directions according to R.

18 STRESS AND EQUILIBRIUM

EXAMPLE 4.4: Independence of S from Q

A useful property of the second Piola“Kirchho¬ tensor S is its inde-

pendence from possible superimposed rotations Q on the current body

˜

con¬guration. To prove this, note ¬rst that because φ = Qφ, then

˜

˜

F = QF and J = J. Using these equations in conjunction with the

objectivity of σ as given by Equation (4.11) gives,

˜˜ ˜ ˜

˜

S = J F ’1 σ F ’T

= JF ’1 QT QσQT QF ’T

=S

X 3 , x3

Q

dp

n

` ∼

dp

∼

da n

p

N

dP

∼

p

dA P X 2 , x2

X 1 , x1 ∼

`

EXAMPLE 4.5: Biot stress tensor

Alternative stress tensors work conjugate to other strain measures can

be contrived. For instance the material stress tensor T work conjugate

™

to the rate of the stretch tensor U is associated with the name of

Biot. In order to derive a relationship between T and S note ¬rst

that di¬erentiating with respect to time the equations U U = C and

2E = C ’ I gives,

™ ™ ™

1

E = 2 (U U + U U )

(continued)

19

4.5 WORK CONJUGACY AND STRESS REPRESENTATIONS

EXAMPLE 4.5 (cont.)

With the help of this relationship we can express the internal work per

unit of initial volume as,

™ ™ ™

S : E = S : 1 (U U + U U )

2

™ ™

= 1 tr(SU U + S U U )

2

™ ™

1

= 2 tr(SU U + U S U )

™

1

= 2 (SU + U S) : U

and therefore the Biot tensor work conjugate to the stretch tensor is,

1

T = 2 (SU + U S)

Using the polar decomposition and the relationship between S and P ,

namely, P = F S, an alternative equation for T emerges as,

1

T = 2 (RTP + P TR)

4.5.4 DEVIATORIC AND PRESSURE COMPONENTS

In many practical applications such as metal plasticity, soil mechanics, and

biomechanics, it is physically relevant to isolate the hydrostatic pressure

component p from the deviatoric component σ of the Cauchy stress tensor

as,

p = 1 trσ = 1 σ : I

σ = σ + pI; (4.49a,b)

3 3

where the deviatoric Cauchy stress tensor σ satis¬es trσ = 0.

Similar decompositions can be established in terms of the ¬rst and sec-

ond Piola“Kirchho¬ stress tensors. For this purpose, we simply substitute

the above decomposition into Equations (4.34a) for P and (4.41b) for S to

give,

P = P + pJF ’T ; P = Jσ F ’T (4.50a)

S = S + pJC ’1 ; S = JF ’1 σ F ’T (4.50b)

The tensors S and P are often referred to as the true deviatoric components

of S and P . Note that although the trace of σ is zero, it does not follow that

the traces of S and P must also vanish. In fact, the corresponding equa-

tions can be obtained from Equations (4.50a“b) and Properties (2.28,49) of

20 STRESS AND EQUILIBRIUM

the trace and double contractions as,

S :C=0 (4.51a)

P :F =0 (4.51b)

The above equations are important as they enable the hydrostatic pressure

p to be evaluated directly from either S or P as,

p = 1 J ’1 P : F (4.52a)

3

p = 1 J ’1 S : C (4.52b)

3

Proof of the above equations follows rapidly by taking the double contrac-

tions of (4.50a) by F and (4.50b) by C.

EXAMPLE 4.6: Proof of Equation (4.51a)

Equation (4.51a) is easily proved as follows:

S : C = (JF ’1 σ F ’T ) : C

= Jtr(F ’1 σ F ’T C)

= Jtr(σ F ’T F T F F ’1 )

= Jtrσ

=0

A similar procedure can be used for (4.51b).

4.6 STRESS RATES

In Section 3.15 objective tensors were de¬ned by imposing that under rigid

body motions they transform according to Equation (3.135). Unfortunately,

time di¬erentiation of Equation (4.11) shows that the material time deriva-

™

tive of the stress tensor, σ, fails to satisfy this condition as,

™T

™

™

σ = QσQT + QσQT + Qσ Q

™

˜ (4.53)

™

Consequently, σ = QσQT unless the rigid body rotation is not a time-

™

˜

dependent transformation. Many rate-dependent materials, however, must

be described in terms of stress rates and the resulting constitutive models

must be frame-indi¬erent. It is therefore essential to derive stress rate mea-

sures that are objective. This can be achieved in several ways, each leading

to a di¬erent objective stress rate tensor. The simplest of these tensors is

21

4.6 STRESS RATES

due to Truesdell and is based on the fact that the second Piola“Kirchho¬

tensor is intrinsically independent of any possible rigid body motion. The

Truesdell stress rate σ —¦ is thus de¬ned in terms of the Piola transformation

of the time derivative of the second Piola“Kirchho¬ stress as,

d

™

σ —¦ = J ’1 φ— [S] = J ’1 F (JF ’1 σF ’T ) F T (4.54)

dt

The time derivatives of F ’1 in the above equation can be obtained by dif-

ferentiating the expression F F ’1 = I and using Equation (3.93) to give,

d ’1

F = ’F ’1 l (4.55)

dt

™

which combined with Equation (3.127) for J gives the Truesdell rate of stress

as,

σ —¦ = σ ’ lσ ’ σlT + (trl)σ

™ (4.56)

The Truesdell stress rate tensor can be reinterpreted in terms of the Lie

derivative of the Kirchho¬ stresses as,