Nonlinear Stress - Strain Behavior
of Nematic Elastomers
using Relative Rotations
Andreas Menzel,1 Harald Pleiner,2 and Helmut R. Brand 1,2
1 Theoretische Physik III, Universität Bayreuth, Germany
2 Max Planck Institute for Polymer Research, Mainz, Germany

ISSP/SOFT 2010
ISSP International Workshop on Soft Matter Physics,
August 9 - 13, 2010,
Tokyo University, Kashiwa Campus, Tokyo, Japan

H.R. Brand (Universität Bayreuth)

Nonlinear Stress - Strain Behavior

Tokyo, August 9, 2010

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Outline

1

Introduction

2

Elasticity Including Nonlinear Relative Rotations
Energetics
Perpendicular Stretching3

Linear Response under Pre-Strain
Effective Linear Shear Modulus
Director Reorientability

4

Conclusions

H.R. Brand (Universität Bayreuth)

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Introduction

Monodomain Side-Chain Nematic Elastomers

Experiment:
linear anisotropic elasticity
nonlinear stress-strain plateau for perpendicular stretchingaccompanied by a complete director reorientation

Description and Interpretation:effective linear modulus and director relaxation under pre-strain?

H.R. Brand (Universität Bayreuth)

Nonlinear Stress - Strain Behavior

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Introduction

Monodomain Side-Chain Nematic Elastomers

Experiment:
linear anisotropic elasticity
nonlinear stress-strain plateau for perpendicular stretchingaccompanied by a complete director reorientation

Description and Interpretation:effective linear modulus and director relaxation under pre-strain?

H.R. Brand (Universität Bayreuth)

Nonlinear Stress - Strain Behavior

Tokyo, August 9, 2010

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Introduction

Results
Stretching a mono-domain nematic elastomer perpendicularly
the resulting elastic plateau at finite strainscomes with a vanishing effective linear modulus and a divergent
director reorientability at its beginning and end (soft mode)
this bifurcation-type behavior is a genuine manifestation of the roleof nonlinear relative rotations
it requires two independent preferred directions and discriminates
nematic LSCEs from simple anisotropic solids
andthis soft mode behavior is not related to the proposed
Nambu-Goldstone mode ("soft-elasticity"), nor is any closeness toan ideal soft-elastic behavior ("semi-soft elasticity") required:the described scenario is found also for cases, where the plateau
starts at very large applied strains
H.R. Brand (Universität Bayreuth)

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Introduction

Results
Stretching a mono-domain nematic elastomer perpendicularly
the resulting elastic plateau at finite strainscomes with a vanishing effective linear modulus and a divergent
director reorientability at its beginning and end (soft mode)
this bifurcation-type behavior is a genuine manifestation of the roleof nonlinear relative rotations
it requires two independent preferred directions and discriminates
nematic LSCEs from simple anisotropic solids
andthis soft mode behavior is not related to the proposed
Nambu-Goldstone mode ("soft-elasticity"), nor is any closeness toan ideal soft-elastic behavior ("semi-soft elasticity") required:the described scenario is found also for cases, where the plateau
starts at very large applied strains
H.R. Brand (Universität Bayreuth)

Nonlinear Stress - Strain Behavior

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Elasticity Including Nonlinear Relative Rotations

Energetics

Elastic and Orientational Degrees of Freedom
Network:daα = Rαj Ξjk dr k
Eulerian strain tensor
εik

=
=
=

Nematic:

12 [δik − Ξij Ξik ]1
2 [δik − (∂aα /∂rk )(∂aα /∂ri )]1
2 [∂ui /∂rk + ∂uk /∂ri − (∂uj /∂ri )(∂uj /∂rk )]Directorn̂ = S · n̂0

H.R. Brand (Universität Bayreuth)

and textures (∇j ni )Nonlinear Stress - Strain Behavior

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Elasticity Including Nonlinear Relative Rotations

Energetics

Relative Rotations
Coupling:
rotations of the anisotropic networkn̂nw = R−1 · n̂0nw(there is no closed expression for R−1 in terms of ∂uj /∂ri )rotations of the nematic directorn̂ = S · n̂0relative rotations (projections)1Ω̃ ≡ n̂ − γ n̂nw
Ω̃nw

≡ −n̂nw + γ n̂with γ ≡ n̂ · n̂nw resulting in Ω̃ · n̂nw = 0 = Ω̃nw · n̂1

A. M. Menzel, H. Pleiner and H. R. Brand, J. Chem. Phys. 126 (2007) 234901.

H.R. Brand (Universität Bayreuth)

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Elasticity Including Nonlinear Relative Rotations

Energetics

Free Energy
Power series expansion in εij , Ω̃i , Ω̃nw
j , and ni and all its couplings up
to some order
here: simplified model (analytical treatment) - elastic nonlinearities
neglected
F= c1 εij εij + 12 c2 εii εjj + . . .
1
(2)
(3)
+ D1 Ω̃i Ω̃i + D1 (Ω̃i Ω̃i )2 + D1 (Ω̃i Ω̃i )32+ D2 ni εij Ω̃j + D2nw ninw εij Ω̃nw
jnw,(2) nw
ni εij εjk Ω̃nwk(2)

+ D2 ni εij εjk Ω̃k + D2
− 21 a (ni Ei )2

reduces in linear order to de Gennes’ expression
H.R. Brand (Universität Bayreuth)

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Elasticity Including Nonlinear Relative Rotations

Perpendicular Stretching

Plateau for Perpendicular Stretch - Eulerian
250

50
200

dF [ kJ ]dA m3

dF [ kJ ]
dA m3

40

150

30

100

20

50

10

0

0
0.0

0.1

0.2

0.3

0.4

0.0

Fig.1: Stress-strain data measured
by Urayama et al.a transferred to therepresentation in terms of the stretch
amplitude A = ∂uz /∂z and dF /dA.

aK. Urayama, R. Mashita, I. Kobayashi, andT. Takigawa, Macromol. 40 (2007) 7665.

H.R. Brand (Universität Bayreuth)

0.1

0.2

0.3

0.4

A

A

Fig.2: Same stress-strain data as in Fig.1
with nonlinear purely elastic contributions
by the network of polymer backbones
subtracted. The line is the result of thetheoretical model aaA. Menzel, H.P., and H.R. Brand, J. Appl. Phys.
105 (2009) 013503.

Nonlinear Stress - Strain Behavior

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Elasticity Including Nonlinear Relative Rotations

Perpendicular Stretching

Plateau for Perpendicular Stretch - LagrangianFig.3: The same stress-strain data points of Urayama et al. and the theoretical line obtained by
the present model (with the nonlinear elastic experimental contributions added) – now in the
representation of the nominal stress as a function of the true strain.

H.R. Brand (Universität Bayreuth)

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Elasticity Including Nonlinear Relative Rotations

Perpendicular Stretching

Director Reorientationϑ [◦ ]

A
Fig.4: Angle ϑ between the director orientation and the x axis under the influence of an
externally imposed strain A for various initial director orientations ϑ0 = ϑ(A = 0), e.g. 0◦ , 0.1◦ ,2◦ , 10◦ , . . . 80◦ , and 89.9◦ , respectively. For ϑ0 = 0◦ (perpendicular stretch) a singularthreshold behavior is found.

H.R. Brand (Universität Bayreuth)

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Elasticity Including Nonlinear Relative Rotations

Perpendicular Stretching

Forward bifurcation

Fig.4a: ϑ = ϑ(A); same as Fig.4
with the area around Ac enlarged

In the vicinity of Ac an amplitude equation can be derived analytically for thecase ϑ0 = 0
0 = ϑ a(Ac − A) + gϑ2 + O(ϑ5 ).
−→ forward bifurcation with exchange of stability between√
ϑ = 0 for A < Ac and ϑ ∼ A − Ac for A > Acfor ϑ0 > 0 an imperfect bifurcation is obtained
H.R. Brand (Universität Bayreuth)

Nonlinear Stress - Strain Behavior

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Elasticity Including Nonlinear Relative Rotations

Perpendicular Stretching

Forward bifurcation

Fig.4a: ϑ = ϑ(A); same as Fig.4
with the area around Ac enlarged

In the vicinity of Ac an amplitude equation can be derived analytically for thecase ϑ0 = 0
0 = ϑ a(Ac − A) + gϑ2 + O(ϑ5 ).
−→ forward bifurcation with exchange of stability between√
ϑ = 0 for A < Ac and ϑ ∼ A − Ac for A > Acfor ϑ0 > 0 an imperfect bifurcation is obtained
H.R. Brand (Universität Bayreuth)

Nonlinear Stress - Strain Behavior

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Elasticity Including Nonlinear Relative Rotations

Perpendicular Stretching

Soft mode
a forward bifurcation is similar to a second order phase transitionan (effective) susceptibility vanishes at the phase transition (atonset)
giving rise to diverging fluctuations (soft mode)in contrast to Nambu-Goldstone modes, where a susceptibility isidentically zero throughout the whole phase due to symmetry
reasons
example: director rotations in a smectic C phase:azimuthal (on the cone) Nambu-Goldstone modetilt angle: soft only at the smectic A to C transition
for imperfect bifurcations no diverging fluctuations

H.R. Brand (Universität Bayreuth)

Nonlinear Stress - Strain Behavior

Tokyo, August 9, 201012 / 27

Elasticity Including Nonlinear Relative Rotations

Perpendicular Stretching

Soft mode
a forward bifurcation is similar to a second order phase transitionan (effective) susceptibility vanishes at the phase transition (atonset)
giving rise to diverging fluctuations (soft mode)in contrast to Nambu-Goldstone modes, where a susceptibility isidentically zero throughout the whole phase due to symmetry
reasons
example: director rotations in a smectic C phase:azimuthal (on the cone) Nambu-Goldstone modetilt angle: soft only at the smectic A to C transition
for imperfect bifurcations no diverging fluctuations

H.R. Brand (Universität Bayreuth)

Nonlinear Stress - Strain Behavior

Tokyo, August 9, 201012 / 27

Elasticity Including Nonlinear Relative Rotations

Perpendicular Stretching

Soft mode
a forward bifurcation is similar to a second order phase transitionan (effective) susceptibility vanishes at the phase transition (atonset)
giving rise to diverging fluctuations (soft mode)in contrast to Nambu-Goldstone modes, where a susceptibility isidentically zero throughout the whole phase due to symmetry
reasons
example: director rotations in a smectic C phase:azimuthal (on the cone) Nambu-Goldstone modetilt angle: soft only at the smectic A to C transition
for imperfect bifurcations no diverging fluctuations

H.R. Brand (Universität Bayreuth)

Nonlinear Stress - Strain Behavior

Tokyo, August 9, 201012 / 27

Elasticity Including Nonlinear Relative Rotations

Perpendicular Stretching

Soft mode
a forward bifurcation is similar to a second order phase transitionan (effective) susceptibility vanishes at the phase transition (atonset)
giving rise to diverging fluctuations (soft mode)in contrast to Nambu-Goldstone modes, where a susceptibility isidentically zero throughout the whole phase due to symmetry
reasons
example: director rotations in a smectic C phase:azimuthal (on the cone) Nambu-Goldstone modetilt angle: soft only at the smectic A to C transition
for imperfect bifurcations no diverging fluctuations

H.R. Brand (Universität Bayreuth)

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Tokyo, August 9, 201012 / 27

Linear Response under Pre-Strain

Effective Linear Shear Modulus

Homeotropic geometry
For a given prestrain A – that results in a given compression B, shear
S, and tilt angle ϑδSBA x̂ Ekx̂

n̂0
ϑ

ŷ

n̂
S

ẑ

Ekẑ

Fig.5: Homeotropic geometry1a small shear δS is added and the effective shear modulus iscalculated2

an external field is applied (k and ⊥ to n̂0 ) and the reorientabilityof the director is calculated

H.R. Brand (Universität Bayreuth)

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Linear Response under Pre-Strain

Effective Linear Shear Modulus

Effective linear shear modulus
60
numerical data

∂ 2F/∂(δS)2 |δS=0 [kPa]

50

40

30

20

10

0
0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

A

Fig.6: Effective shear modulus ∂ 2 F /∂(δS)2 |δS=0 as a function of the prestretching amplitude A.
Here, the system is prestretched in a direction perfectly perpendicular to the initial director
orientation n̂0 . The zeroes of the effective shear modulus at the beginning and end of the plateau
denote diverging fluctuations.
H.R. Brand (Universität Bayreuth)

Nonlinear Stress - Strain Behavior

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Linear Response under Pre-Strain

Effective Linear Shear Modulus

Effective linear shear modulus
60
numerical data

∂ 2F/∂(δS)2 |δS=0 [kPa]

50

40

30

20

10

0
0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

A

Fig.6: Effective shear modulus ∂ 2 F /∂(δS)2 |δS=0 as a function of the prestretching amplitude A.
Here, the system is prestretched in a direction perfectly perpendicular to the initial director
orientation n̂0 . The zeroes of the effective shear modulus at the beginning and end of the plateau
denote diverging fluctuations.
H.R. Brand (Universität Bayreuth)

Nonlinear Stress - Strain Behavior

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Linear Response under Pre-Strain

Director Reorientability

Director reorientability3(1/ǫ0 )(∂ 2 ϑ/∂E 2 )|E=0 [1/kPa]

numerical data
2.5

2

1.5

1

0.5

0
0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

A

Fig.7: Reorientability ∂ 2 ϑ/∂E 2 |E=0 as
a function of the prestretching
amplitude A, where the divergencies
take place at the beginning and endof the plateau (E ⊥ n̂0 )H.R. Brand (Universität Bayreuth)

Nonlinear Stress - Strain Behavior

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Linear Response under Pre-Strain

Director Reorientability

Director reorientability
3

3

2.5

(1/ǫ0 )(∂ 2 ϑ/∂E 2 )|E=0 [1/kPa](1/ǫ0 )(∂ 2 ϑ/∂E 2 )|E=0 [1/kPa]

numerical data

2

1.5

1

0.5

0

numerical data
fitting curve: x = -0.474
analytical result: x = -0.5

2.5

2

1.5

1

0.5

0

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.1

A

2

Fig.7: Reorientability ∂ ϑ/∂E |E=0 as
a function of the prestretching
amplitude A, where the divergencies
take place at the beginning and endof the plateau (E ⊥ n̂0 )H.R. Brand (Universität Bayreuth)

0.11

0.12

0.13

0.14

0.15

0.16

A

2

Fig.8: Same theoretical data fitted in
the region ϑ & 0 by a curve∝ (A − Ac )x with x ≈ −1/2, thus
clearly indicating a soft mode
behavior in mean field description

Nonlinear Stress - Strain Behavior

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Linear Response under Pre-Strain

Director Reorientability

Oblique Pre-Strain
0.3

60

numerical data

numerical data

(1/ǫ0 )(∂ 2 ϑ/∂E 2 )|E=0 [1/kPa]∂ 2F/∂(δS)2 |δS=0 [kPa]

50

40

30

20

10

0

0.25

0.2

0.15

0.1

0.05

0
0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0

0.05

A

0.1

0.15

0.2

0.25

0.3

0.35

0.4

A

Fig.9: Effective shear modulus ∂ 2 F /∂(δS)2 |δS=0 (left) and reorientability ∂ 2 ϑ/∂E 2 |E=0 (right)
as a function of the prestretching amplitude A. Here, the initial director orientation n̂0 slightly
deviates from the perfectly perpendicular orientation by an angle of 0.01 rad (0.57◦ ).

imperfect bifurcation: no divergent fluctuations22
A. Petelin and M. Čopič, Phys. Rev. Lett. 103, 077801 (2009); and presentations at the European Conference on Liquid
Crystals, Colmar, April 2009
H.R. Brand (Universität Bayreuth)

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Conclusions

Remarks
the fluctuations at the bifurcation do not diverge (the effective linear
modulus remains non-zero)for an oblique prestretch
due to boundary induced director inhomogeneities (necking)due to macroscopic material inhomogeneities
if the fluctuations are treated nonlinearly
there is no bifurcation (no diverging fluctuations)in the planar geometry (the small shear added is not in the
director reorientation plane)
for an external field in y direction (perpendicular to the director
reorientation plane)

H.R. Brand (Universität Bayreuth)

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Conclusions

Remarks
the fluctuations at the bifurcation do not diverge (the effective linear
modulus remains non-zero)for an oblique prestretch
due to boundary induced director inhomogeneities (necking)due to macroscopic material inhomogeneities
if the fluctuations are treated nonlinearly
there is no bifurcation (no diverging fluctuations)in the planar geometry (the small shear added is not in the
director reorientation plane)
for an external field in y direction (perpendicular to the director
reorientation plane)

H.R. Brand (Universität Bayreuth)

Nonlinear Stress - Strain Behavior

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Conclusions

Semisoftness
the general scenario – elastic plateau with vanishing effective
linear modulus at its beginning and end – has also been describedby other methods3
often, it is connected to semi-softness, where a small parameter α
describes the (small) deviation from ideal softness;4
the plateau starts at λ1 ≈ 1 + α and the slope of the plateau is
3µα (cf. Chaps. 7.4 and 7.5 of Ref. 4)
however, the smallness of Ac ≈ 0.1 (corresponding to α ≈ 0.1) is
not a necessary condition for the soft mode behavior at the
beginning and end of the plateau

3
4

J. S. Biggins, E. M. Terentjev, and M. Warner, Phys. Rev. E 78 (2008) 041704
and F. F. Ye and T. C. Lubensky, J. Phys. Chem. B 113 (2009) 3853
M. Warner and E.M. Terentjev, Liquid Crystal Elastomers, Clarendon Press, Oxford (2003)

H.R. Brand (Universität Bayreuth)

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Conclusions

Semisoftness
the general scenario – elastic plateau with vanishing effective
linear modulus at its beginning and end – has also been describedby other methods3
often, it is connected to semi-softness, where a small parameter α
describes the (small) deviation from ideal softness;4
the plateau starts at λ1 ≈ 1 + α and the slope of the plateau is
3µα (cf. Chaps. 7.4 and 7.5 of Ref. 4)
however, the smallness of Ac ≈ 0.1 (corresponding to α ≈ 0.1) is
not a necessary condition for the soft mode behavior at the
beginning and end of the plateau

3
4

J. S. Biggins, E. M. Terentjev, and M. Warner, Phys. Rev. E 78 (2008) 041704
and F. F. Ye and T. C. Lubensky, J. Phys. Chem. B 113 (2009) 3853
M. Warner and E.M. Terentjev, Liquid Crystal Elastomers, Clarendon Press, Oxford (2003)

H.R. Brand (Universität Bayreuth)

Nonlinear Stress - Strain Behavior

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Conclusions

High Plateau

Fig.10: In this case the plateau starts at a rather large pre-strain Ac ≈ 0.56
(or λ ≈ 2.3) and ends at A ≈ 0.76 (or λ ≈ 4.2)
– the scenario is the same as for very small Ac .
H.R. Brand (Universität Bayreuth)

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Conclusions

Summary

the scenario of an elastic plateau at finite perpendicular stretching,with a vanishing effective linear modulus and a divergent director
reorientability at its beginning and end (soft mode), is a genuine
manifestation of an instability due to nonlinear relative rotations;
it requires two independent preferred directions and discriminates
these systems from simple anisotropic solids;
there is no need for a small parameter nor for the closeness to an
ideal soft-elastic behavior (Nambu-Goldstone or almost
Nambu-Goldstone mode)– the soft mode scenario can happen, even when the plateau
starts at very high strains

H.R. Brand (Universität Bayreuth)

Nonlinear Stress - Strain Behavior

Tokyo, August 9, 2010

20 / 27

Conclusions

Summary

the scenario of an elastic plateau at finite perpendicular stretching,with a vanishing effective linear modulus and a divergent director
reorientability at its beginning and end (soft mode), is a genuine
manifestation of an instability due to nonlinear relative rotations;
it requires two independent preferred directions and discriminates
these systems from simple anisotropic solids;
there is no need for a small parameter nor for the closeness to an
ideal soft-elastic behavior (Nambu-Goldstone or almost
Nambu-Goldstone mode)– the soft mode scenario can happen, even when the plateau
starts at very high strains

H.R. Brand (Universität Bayreuth)

Nonlinear Stress - Strain Behavior

Tokyo, August 9, 2010

20 / 27

Conclusions

Summary

the scenario of an elastic plateau at finite perpendicular stretching,with a vanishing effective linear modulus and a divergent director
reorientability at its beginning and end (soft mode), is a genuine
manifestation of an instability due to nonlinear relative rotations;
it requires two independent preferred directions and discriminates
these systems from simple anisotropic solids;
there is no need for a small parameter nor for the closeness to an
ideal soft-elastic behavior (Nambu-Goldstone or almost
Nambu-Goldstone mode)– the soft mode scenario can happen, even when the plateau
starts at very high strains

H.R. Brand (Universität Bayreuth)

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Tokyo, August 9, 2010

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Appendix A: Stress-Strain Definition

Appendices

Lagrange description
Comparing the initial dimension l0 to the actual dimension (in the direction of
the external force Fext ), the ratio
λ=

l
l0

(1)

is taken as a measure of the induced strain.
Sometimes, the so called true strain  = ln(λ) is taken as a variable.
Stresses are recorded either as true stress
Fext
lx ly

(2)

Fext
,
lx,0 ly ,0(3)

σext =or as nominal stress
N
σext
=

From the experimental point of view the initial dimension l0 is considered tobe constant and the current sample dimension l is changed.
H.R. Brand (Universität Bayreuth)

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Appendices

Appendix A: Stress-Strain Definition

Eulerian description
In the hydrodynamic (Eulerian) picture the current dimension of the sample lis considered to be constant, and what changes is the initial dimension l0 .
For the displacement field uz = Az (or lz − lz,0 = Alz ) the strain is
λ=

1
.
1−A

(4)

and the stresses are
Fext
lx ly
Fext
N
σext
≡
lx,0 ly ,0σext ≡

=

dF
,
dA

=

(1 − A)

(5)
dF
.
dA

(6)

Here, the expressions on the left of Eqs. (5) and (6) are given as functions of
λ, the expressions on the right as functions of A.
The connection between both follows from Eq. (4).
H.R. Brand (Universität Bayreuth)

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Appendices

Appendix B: ProceduresConstrained equilibrium
As an ansatz we use for the
displacement fields
uz= Az + Sx,

(7)

ux

= Bx

(8)

uy

= Cy .

(9)δS

B

A x̂ Ekx̂n̂0ϑ

and for the director orientation

ŷ

n̂
S

ẑ

Ekẑ

Fig.5: homeotropic geometryn̂ = (cos ϑ, 0, sin ϑ)(10)

For a given initial orientation ϑ0 and external stretch A,
the values S(A), B(A), and ϑ(A) follow from the equilibrium conditions∂F /∂S = 0, ∂F /∂B = 0, and ∂F /∂ϑ = 0(the compression C follows from the incompressibility condition).
H.R. Brand (Universität Bayreuth)

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Appendices

Appendix B: Procedures

Shear and compression

0.2

S

A
0.0

0.15

0.1

0.2

0.3

0.4

0.0
−0.05

0.1
−0.1
−0.15

0.05

B

−0.2
−0.25

0.0
0.0

0.1

0.2

0.3

0.4
−0.3

A

Fig.11: The shear S(A) and the compression B(A) as a function of the pre-strain amplitude A.

H.R. Brand (Universität Bayreuth)

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Appendices

Appendix B: Procedures

Effective linear modulus

For each given pre-strain A a small shear δS is added,
uz = Az + [S(A) + δS]x

(11)

and the free energy (including δS) is again minimized w.r.t. ϑ
and then calculated to lowest order in δS
FA = 12 ceff (A)(δS)2 + O(δS)3(12)

The effective linear modulus ceff (A) = ∂ 2 FA /∂(δS)2 |δS=0 is shown in
Figs. 6 and 9.

H.R. Brand (Universität Bayreuth)

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Appendices

Appendix B: Procedures

Orientability
For given external field E in the x-z plane and a given pre-strain A, thesystem of constrained equilibrium conditions ∂F /∂ϑ = ∂F /∂B = ∂F /∂S = 0
is solved, resulting in F = F (E, A).
For each value of A, there is ∂ϑ/∂E|E=0 = 0 due to stability reasons.
Therefore, we take the second
derivative ∂ 2 ϑ/∂E 2 |E=0 as ameasure for the reorientability ofthe director n̂ in an external field fora given stretching amplitude A. In
Fig.7 this reorientability is shown forE ⊥ n̂0 , while for E k n̂0 the sign of
it is reversed. The case Ex = Ez isshown on the right

H.R. Brand (Universität Bayreuth)

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Appendices

Appendix B: Procedures

Divergence of the orientability
the coefficients of the amplitude equation close to the threshold2(E 2 = Ex,z)
0 = ϑ a(Ac − A) + gϑ2 + O(ϑ5 ).

(13)

2

generally acquire field contributions ∼ E due to the dielectric anisotropy
energyin the limit E → 0 one can write, e.g. Ac (E) = Ac (1 + ζA E 2 )for A & Ac this leads to the field dependence of the tilt angle


q
q
ζA Ac
a
E2
(14)
ϑ = g(E)(A − Ac (E)) ≈ ga (A − Ac ) 1 + ζE 2 +2 Ac − Aand to the orientability∂ 2 ϑ/∂E 2 |E=0 ∼ (A − Ac )−1/2 + O((A − Ac )1/2 )which is observed in Fig.8.
H.R. Brand (Universität Bayreuth)

Nonlinear Stress - Strain Behavior

Tokyo, August 9, 201027 / 27