In fact, this expression de¬nes what is known as the Truesdell rate of the

Kirchho¬ tensor „ —¦ = Jσ —¦ , which can be shown from Equation (4.56) or

Equation (4.57) to be,

„ —¦ = „ ’ l„ ’ „ lT

™ (4.58)

Alternative objective stress rates can be derived in terms of the Lie

derivative of the Cauchy stress tensor to give the Oldroyd stress rate σ • as,

σ • = L [σ]

φ

d

(F ’1 σF ’T ) F T

=F

dt

= σ ’ lσ ’ σlT

™ (4.59)

If the pull back“push forward operations are performed with F T and F ’T

respectively, the resulting objective stress rate tensor is the convective stress

rate σ given as,

d

σ = F ’T (F T σF ) F ’1

dt

= σ + lT σ + σl

™ (4.60)

A simpli¬ed objective stress rate can be obtained by ignoring the stretch

component of F in Equations (4.54), (4.59), or (4.60), thus performing the

22 STRESS AND EQUILIBRIUM

pull back and push forward operations using only the rotation tensor R.

This de¬nes the so-called Green-Naghdi stress rate σ , which with the help

of Equation (3.112) is given as,

d

(RT σR) RT

=R

σ

dt

™ ™

= σ + σ RRT ’ RRT σ

™ (4.61)

™

Finally, if the antisymmetric tensor RRT is approximated by the spin ten-

sor w (see Equation (3.116)) the resulting objective stress rate is known as

Jaumann stress rate,

™

= σ + σw ’ wσ (4.62)

σ

Irrespective of the approximations made to arrive at the above de¬nitions

of σ and σ , they both remain objective even when these approximations

do not apply.

EXAMPLE 4.7: Objectivity of σ —¦

The objectivity of the Truesdell stress rate given by Equation (4.56) can

be proved directly without referring back to the initial con¬guration.

For this purpose recall ¬rst Equations (4.11), (4.53), and (3.137) as,

σ = QσQT

˜

™T

™

™

σ = QσQT + QσQT + Qσ Q

™

˜

™

˜ = QQT + QlQT

l

˜

and note that because J = J then trl = tr˜ With the help of the

l.

above equations, the Truesdell stress rate on a rotated con¬guration

σ —¦ emerges as,

˜

T

σ —¦ = σ ’ ˜˜ ’ σ˜ + (tr˜ σ

™

˜ ˜ lσ ˜ l l) ˜

™T

™ ™

= QσQT + QσQT + Qσ Q ’ (QQT + QlQT )QσQT

™

™

’ QσQT (QQT + QlQT ) + (trl)QσQT

= QσQT ’ QlσQT ’ QσlQT + (trl)QσQT

™

= Qσ —¦ QT

and is therefore objective.

23

4.6 STRESS RATES

Exercises

1. A two-dimensional Cauchy stress tensor is given as,

σ = t — n1 + ± n1 — n2

where t is an arbitrary vector and n1 and n2 are orthogonal unit vectors.

(a) Describe graphically the state of stress. (b) Determine the value of ±

(hint: σ must be symmetric).

2. Using Equation (4.55) and a process similar to that employed in Ex-

ample 4.5, show that, with respect to the initial volume, the stress

™

tensor Π is work conjugate to the tensor H, where H = ’F ’T and

Π = P C = JσF .

Using the time derivative of the equality CC ’1 = I, show that the tensor

3.

™

Σ = CSC = JF T σF is work conjugate to 1 B, where B = ’C ’1 . Find

2

relationships between T , Σ, and Π.

4. Prove Equation (4.51b) P : F = 0 using a procedure similar to Example

4.6.

5. Prove directly that the Jaumann stress tensor, σ is an objective tensor,

using a procedure similar to Example 4.7.

6. Prove that if dx1 and dx2 are two arbitrary elemental vectors moving

with the body (see Figure 3.2) then:

d

(dx1 · σdx2 ) = dx1 · σ dx2

dt

CHAPTER FIVE

HYPERELASTICITY

5.1 INTRODUCTION

The equilibrium equations derived in the previous section are written in

terms of the stresses inside the body. These stresses result from the defor-

mation of the material, and it is now necessary to express them in terms of

some measure of this deformation such as, for instance, the strain. These re-

lationships, known as constitutive equations, obviously depend on the type

of material under consideration and may be dependent on or independent

of time. For example the classical small strain linear elasticity equations

involving Young modulus and Poisson ratio are time-independent, whereas

viscous ¬‚uids are clearly entirely dependent on strain rate.

Generally, constitutive equations must satisfy certain physical principles.

For example, the equations must obviously be objective, that is, frame-

invariant. In this chapter the constitutive equations will be established in

the context of a hyperelastic material, whereby stresses are derived from a

stored elastic energy function. Although there are a number of alternative

material descriptions that could be introduced, hyperelasticity is a partic-

ularly convenient constitutive equation given its simplicity and that it con-

stitutes the basis for more complex material models such as elastoplasticity,

viscoplasticity, and viscoelasticity.

5.2 HYPERELASTICITY

Materials for which the constitutive behavior is only a function of the current

state of deformation are generally known as elastic. Under such conditions,

any stress measure at a particle X is a function of the current deformation

1

2 HYPERELASTICITY

gradient F associated with that particle. Instead of using any of the al-

ternative strain measures given in Chapter 3, the deformation gradient F ,

together with its conjugate ¬rst Piola“Kirchho¬ stress measure P , will be

retained in order to de¬ne the basic material relationships. Consequently,

elasticity can be generally expressed as,

P = P (F (X), X) (5.1)

where the direct dependency upon X allows for the possible inhomogeneity

of the material.

In the special case when the work done by the stresses during a defor-

mation process is dependent only on the initial state at time t0 and the

¬nal con¬guration at time t, the behavior of the material is said to be path-

independent and the material is termed hyperelastic. As a consequence of

the path-independent behavior and recalling from Equation (4.31) that P

™

is work conjugate with the rate of deformation gradient F , a stored strain

energy function or elastic potential Ψ per unit undeformed volume can be

established as the work done by the stresses from the initial to the current

position as,

t

™ ™

™

Ψ(F (X), X) = P (F (X), X) : F dt; Ψ=P :F (5.2a,b)

t0

Presuming that from physical experiments it is possible to construct the

function Ψ(F , X), which de¬nes a given material, then the rate of change

of the potential can be alternatively expressed as,

3

‚Ψ ™

™

Ψ= FiJ (5.3)

‚FiJ

i,J=1

Comparing this with Equation (5.2b) reveals that the components of the

two-point tensor P are,

‚Ψ

PiJ = (5.4)

‚FiJ

For notational convenience this expression is rewritten in a more compact

form as,

‚Ψ(F (X), X)

P (F (X), X) = (5.5)

‚F

Equation (5.5) followed by Equation (5.2) is often used as a de¬nition of a

hyperelastic material.

The general constitutive Equation (5.5) can be further developed by re-

calling the restrictions imposed by objectivity as discussed in Section 3.15.

3

5.3 ELASTICITY TENSOR

To this end, Ψ must remain invariant when the current con¬guration under-

goes a rigid body rotation. This implies that Ψ depends on F only via the

stretch component U and is independent of the rotation component R. For

convenience, however, Ψ is often expressed as a function of C = U 2 = F T F

as,

Ψ(F (X), X) = Ψ(C(X), X) (5.6)

1™ ™

Observing that 2 C = E is work conjugate to the second Piola“Kirchho¬

stress S, enables a totally Lagrangian constitutive equation to be con-

structed in the same manner as Equation (5.5) to give,

‚Ψ ™ 1 ‚Ψ ‚Ψ

™

™

Ψ= : C = S : C; S(C(X), X) = 2 = (5.7a,b)

‚C 2 ‚C ‚E

5.3 ELASTICITY TENSOR

5.3.1 THE MATERIAL OR LAGRANGIAN ELASTICITY

TENSOR

The relationship between S and C or E = 1 (C ’ I), given by Equa-

2

tion (5.7b) will invariably be nonlinear. Within the framework of a po-

tential Newton“Raphson solution process, this relationship will need to be

linearized with respect to an increment u in the current con¬guration. Us-

ing the chain rule, a linear relationship between the directional derivative of

S and the linearized strain DE[u] can be obtained, initially in a component

form, as,

d

DSIJ [u] = SIJ (EKL [φ + u])

d =0

3

‚SIJ d

= EKL [φ + u]

‚EKL d =0

K,L=1

3

‚SIJ

= DEKL [u] (5.8)

‚EKL

K,L=1

This relationship between the directional derivatives of S and E is more

concisely expressed as,

DS[u] = C : DE[u] (5.9)