<<

. 31
( 54 .)



>>


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and (5.92) are identical except for the small strains having been replaced by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

<<

. 31
( 54 .)



>>