3

1 ‚2Ψ

c= (F N ± ) — (F N ± ) — (F N β ) — (F N β )

J ‚»2 ‚»2

± β

±,β=1

3

2 S±± ’ Sββ

+ (F N ± ) — (F N β ) — (F N ± ) — (F N β(5.84)

)

J »2 ’ »2

± β

±,β=1

±=β

Noting again that F N ± = »± n± and after some algebraic manipulations us-

ing the standard chain rule we can eventually derive the Eulerian or spatial

elasticity

tensor as,

3 3

‚2Ψ

1

c= 2σ±± n± — n± — n± — n±

n± — n± — nβ — nβ ’

J ‚ ln »± ‚ ln »β

±=1

±,β=1

3

σ±± »2 ’ σββ »2

±

β

+ 2 (5.85)

n± — nβ — n± — nβ

»2 ’ »2

± β

±,β=1

±=β

The evaluation of the Cartesian components of this tensor requires a

similar transformation to that employed in Equation (5.77) for the Cauchy

stresses. Using the same notation, the Cartesian components of the Eulerian

triad T±j are substituted into Equation (5.85) to give after simple algebra

the Cartesian components of c as,

3 3

‚2Ψ

1

c ijkl = T±i T±j Tβk Tβl ’ 2σ±± T±i T±j T±k T±l

J ‚ ln »± ‚ ln »β

±=1

±,β=1

3

σ±± »2 ’ σββ »2

±

β

+ 2 T±i Tβj T±k Tβl (5.86)

»2 ’ »2

± β

±,β=1

±=β

Remark 5.6.2. Again, recalling Remark 4, in the case when »± = »β ,

L™Hospital™s rule yields,

σ±± »2 ’ σββ »2 ‚2Ψ ‚2Ψ

1

±

β

lim 2 = ’ 2σββ

’

»2 ’ »2 J ‚ ln »β ‚ ln »β ‚ ln »± ‚ ln »β

»β ’»± ± β

(5.87)

24 HYPERELASTICITY

5.6.5 A SIMPLE STRETCH-BASED HYPERELASTIC

MATERIAL

A material frequently encountered in the literature is de¬ned by a hypere-

lastic potential in terms of the logarithmic stretches and two material pa-

rameters » and µ as,

»

Ψ(»1 , »2 , »3 ) = µ[(ln »1 )2 + (ln »2 )2 + (ln »3 )2 ] + (ln J)2 (5.88)

2

where, because J = »1 »2 »3 ,

ln J = ln »1 + ln »2 + ln »3 (5.89)

It will be shown that the potential Ψ leads to a generalization of the stress“

strain relationships employed in classical linear elasticity.

Using Equation (5.75) the principal Cauchy stress components emerge

as,

1 ‚Ψ 2µ »

σ±± = = ln »± + ln J (5.90)

J ‚ ln »± J J

Furthermore, the coe¬cients of the elasticity tensor in (5.86) are,

‚2Ψ

1 » 2µ

=+ δ±β (5.91)

J ‚ ln »± ‚ ln »β J J

The similarities between these equations and linear elasticity can be

established if we ¬rst recall the standard small strain elastic equations as,

σ±± = »(µ11 + µ22 + µ33 ) + 2µµ±± (5.92)

Recalling that ln J = ln »1 + ln »2 + ln »3 it transpires that Equations (5.90)

and (5.92) are identical except for the small strains having been replaced by

the logarithmic stretches and » and µ by »/J and µ/J respectively. The

stress“strain equations can be inverted and expressed in terms of the more

familiar material parameters E and ν, the Young™s modulus and Poisson

ratio, as,

J µ(2µ + 3»)

ln »± = [(1 + ν)σ±± ’ ν(σ11 + σ22 + σ33 )]; E= ;

E »+µ

»

ν= (5.93a,b,c)

2» + 2µ

Remark 5.6.3. At the initial unstressed con¬guration, J = »± = 1,

σ±± = 0, and the principal directions coincide with the three spatial di-

rections n± = e± and therefore T±j = δ±j . Substituting these values into

Equations (5.91), (5.87), and (5.86) gives the initial elasticity tensor for this

25

5.6 ISOTROPIC ELASTICITY IN PRINCIPAL DIRECTIONS

type of material as,

c ijkl = »δij δkl + 2µδik δjl (5.94)

which again (see Remark 3) coincides with the standard spatially isotropic

elasticity tensor.

5.6.6 NEARLY INCOMPRESSIBLE MATERIAL IN PRINCIPAL

DIRECTIONS

In view of the importance of nearly incompressible material behavior, cou-

pled with the likelihood that such materials will be described naturally in

terms of principal stretches, it is now logical to elaborate the formulation

in preparation for the case when the material de¬ned by Equation (5.88)

becomes nearly incompressible. Once again, the distortional components of

the kinematic variables being used, namely the stretches »± , must be iden-

ti¬ed ¬rst. This is achieved by recalling Equations (3.43) and (3.61) for F

ˆ

and F to give,

ˆ

F = J ’1/3 F

3

’1/3

=J »± n± — N ±

±=1

3

(J ’1/3 »± ) n± — N ±

= (5.95)

±=1

ˆ

This enables the distortional stretches »± to be identi¬ed as,

ˆ ˆ

»± = J 1/3 »±

»± = J ’1/3 »± ; (5.96a,b)

Substituting (5.96b) into the hyperelastic potential de¬ned in (5.88) yields

after simple algebra a decoupled representation of this material as,

ˆˆˆ

Ψ(»1 , »2 , »3 ) = Ψ(»1 , »2 , »3 ) + U (J) (5.97)

where the distortional and volumetric components are,

ˆˆˆ ˆ ˆ ˆ

Ψ(»1 , »2 , »3 ) = µ[(ln »1 )2 + (ln »2 )2 + (ln »3 )2 ] (5.98a)

U (J) = 1 κ(ln J)2 ; 2

κ = » + 3µ (5.98b)

2

Note that this equation is a particular case of the decoupled Equation (5.56)

with alternative de¬nitions of U (J) and Ψ. The function U (J) will enforce

incompressibility only when the ratio κ to µ is su¬ciently high, typically

103 “104 . Under such conditions the value of J is J ≈ 1 and ln J ≈ 1 ’ J,

26 HYPERELASTICITY

and therefore the value of U will approximately coincide with the function

de¬ned in (5.57).

For the expression U (J), the corresponding value of the hydrostatic pres-

sure p is re-evaluated using Equation (5.59) to give,

dU κ ln J

p= = (5.99)

dJ J

In order to complete the stress description, the additional deviatoric com-

ponent must be evaluated by recalling Equation (5.75) as,

1 ‚Ψ

σ±± =

J ‚ ln »±

1 ‚Ψ 1 ‚U

= +

J ‚ ln »± J ‚ ln »±

1 ‚Ψ κ ln J

= + (5.100)

J ‚ ln »± J

Observing that the second term in this equation is the pressure, the principal

deviatoric stress components are obviously,

1 ‚Ψ

σ±± = (5.101)

J ‚ ln »±

In order to obtain the derivatives of Ψ it is convenient to rewrite this function

with the help of Equation (5.96a) as,

ˆ ˆ ˆ

Ψ = µ[(ln »1 )2 + (ln »2 )2 + (ln »3 )2 ]

= µ[(ln »1 )2 + (ln »2 )2 + (ln »3 )2 ] + 1 µ(ln J)2

3

2

’ 3 µ(ln J)(ln »1 + ln »2 + ln »3 )

= µ[(ln »1 )2 + (ln »2 )2 + (ln »3 )2 ] ’ 1 µ(ln J)2 (5.102)

3

This expression for Ψ is formally identical to Equation (5.88) for the com-

plete hyperelastic potential Ψ with the Lam` coe¬cient » now replaced by

e

’2µ/3. Consequently, Equation (5.90) can now be recycled to give the

deviatoric principal stress component as,

2µ 2µ

σ±± = ln »± ’ ln J (5.103)

J 3J

The ¬nal stage in this development is the evaluation of the volumetric

and deviatoric components of the spatial elasticity tensor c . For a general

decoupled hyperelastic potential this decomposition is embodied in Equa-

tion (5.64), where c is expressed as,

ˆ

c = c +c p +c κ (5.104)

27

5.6 ISOTROPIC ELASTICITY IN PRINCIPAL DIRECTIONS

where the origin of the pressure component c p is the second term in the

general equation for the Lagrangian elasticity tensor (5.61), which is entirely

geometrical, that is, independent of the material being used, and therefore

remains unchanged as given by Equation (5.55b). However, the volumetric

component c κ depends on the particular function U (J) being used and in

the present case becomes,

d2 U

cκ = J I—I

dJ 2

κ(1 ’ pJ)

= (5.105)

I—I

J

ˆ

The deviatoric component of the elasticity tensor c emerges from the push

forward of the ¬rst term in Equation (5.61). Its evaluation is facilitated by

again recalling that Ψ coincides with Ψ when the parameter » is replaced

by ’2µ/3. A reformulation of the spatial elasticity tensor following the

procedure previously described with this substitution and the corresponding

replacement of σ±± by σ±± inevitably leads to the Cartesian components of

ˆ

c as,

3 3

‚2Ψ

1

ˆ = T±i T±j Tβk Tβl ’ 2σ±± T±i T±j T±k T±l

c ijkl

J ‚ ln »± ‚ ln »β

±=1

±,β=1

3

σ±± »2 ’ σββ »2