# Nonlinear Elasticity

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18 Ιουλ 2012 (πριν από 6 χρόνια και 1 μέρα)

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7/18/05Princeton Elasticity Lectures
Nonlinear Elasticity
7/18/05Princeton Elasticity Lectures
Outline

Some basics of nonlinear elasticity

Nonlinear elasticity of biopolymer
networks

Nematic elastomers
7/18/05Princeton Elasticity Lectures
What is Elasticity

Description of distortions of rigid bodies
and the energy, forces, and fluctuations
arising from these distortions.

Describes mechanics of extended
bodies from the macroscopic to the
microscopic, from bridges to the
cytoskeleton.
7/18/05Princeton Elasticity Lectures
Classical Lagrangian
Description
()
()
=+
Rx
x
ux
()
Rx
()

Rx
x

x
Material distorted to new
positions R(x)
Reference material in D
dimensions described by
a continuum of mass
points x. Neighbors of
points do not change
under distortion
i
ii
i
R
x
αα
α
α
δη

Λ
==
+

Cauchy deformation tensor
ii
u
αα
η
=

7/18/05Princeton Elasticity Lectures
Linear and Nonlinear Elasticity
Linear: Small deformations –
Λ
near 1
Nonlinear: Large deformations –
Λ
>>1
Why nonlinear?

Systems can undergo large deformations –
rubbers,
polymer networks , …

Non-linear theory needed to understand properties of
statically strained materials

Non-linearities
can renormalize nature of elasticity

Elegant an complex theory of interest in its own right
Why now:

New interest in biological materials under large strain

Liquid crystal elastomers –
exotic nonlinear behavior

Old subject but difficult to penetrate –
worth a fresh look
7/18/05Princeton Elasticity Lectures
Deformations and Strain
Complete information about shape of body in R(x)= x
+u(x);
u= const. –
translation no energy.
No energy cost unless
u(x) varies in space.
For slow variations, use the Cauchy deformation tensor
ii
i
ii
u
αα
α
αα
δδ
η
Λ
=+

=+
33
det
de
t
1
:

dR
dx
=
Λ
Λ
=

N
o
v
ol
ume

c
hange
1/
2
1/2
00
00
00


Λ

Λ
=
Λ

Λ



Volume preserving stretch along z-axis
Λ−1/2
Λ
7/18/05Princeton Elasticity Lectures
Simple shear strain
1
01

Λ

Λ
=



Constant Volume, but note
stretching of sides
originally along x
or y.
Note: Λ
is not symmetric
10
1


Λ
=

Λ



Rotate
Not equivalent to
7/18/05Princeton Elasticity Lectures
Pure Shear
Pure shear: symmetric deformation tensor with unit
determinant –
equivalent to stretch along 45 deg.
2
2
1
1

+
ΛΛ

Λ
=

Λ
+
Λ



7/18/05Princeton Elasticity Lectures
Pure shear as stretch
11
1
11
2
xx
x
U
yy
y





=≡






′−




j
ii
i
j
T
ij
j
x
R
RR
xR
x
x
UU
β
α
β
α
α
ββ
α

∂∂
Λ=
=
′′

∂∂∂

2
2
10
01
T
UU

Λ=
Λ



=

+
Λ−
Λ



7/18/05Princeton Elasticity Lectures
Pure to simple shear
2
tan
1
θ
Λ
=
+
Λ
22
21
2
/
2
2
12
2
12
0(
1
1
cos
sin
sin
2
1
)
cos
θθ
θθ



Λ


Λ+
Λ

Λ=


Λ


=








7/18/05Princeton Elasticity Lectures
Cauchy Saint-Venant
Strain
(
)
11
22
1
2
()
(
)
TT
kk
u
uu
u
u
u
αα
α
α
αβ
β
β
δη
η
=
ΛΛ

+
=

+

+
∂∂



22
2
dR
dx
u
dx
dx
α
αβ
β

=
i
ii
i
R
x
αα
α
α
δη

Λ
==
+

uαβ
is invariant under rotations
in the target space but
transforms as a tensor under
rotations in the reference
space. It contains no
of object.
Symmetric!
R=Reference space
T=Target space
7/18/05Princeton Elasticity Lectures
Elastic energy
The elastic energy should be invariant under rigid rotations
in the target space: if is a function of uαβ.
1
2
1
2
()
[]
D
D
Fd
x
f
u
dx
K
u
u
u
αβ
α
βγδ
αβ
γδ
αβ
αβ
σ
=
=+

This energy is automatically invariant under rotations in
target space. It must also be invariant under the point-
group operations of the reference space. These place
constraints on the form of the elastic constants.
Note there can be a linear “stress”-like term. This can
be removed (except for transverse random components)
by redefinition of the reference space
7/18/05Princeton Elasticity Lectures
Elastic modulus tensor
K
αβχδ
is the elastic constant or elastic modulus tensor.
It has inherent symmetry and symmetries of the
reference space.
KKK
K
α
βγδ
γδαβ
βαγδ
αβδγ
=
==
Isotropic system
()
K
αγ
α
βγδ
αβ
γ
δ
βδ
αδ
βδ
λ
δδ
µ
δ
δ
δ
δ
=
++
Uniaxial
(
n
= unit vector along uniaxial
direction)
12
1
2
34
1
4
5
()
()
()
TT
TT
T
T
T
T
TTT
T
KC
n
n
n
n
C
n
n
n
n
CC
Cn
n
n
n
n
n
n
n
αγ
α
αβγδ
β
δ
β
γδ
γδ
β
αβ
αγ
αβ
γδ
βδ
αδ
βγ
αγ
α
α
γ
γ
β
δδ
β
γ
β
δ
β
α
δ
δ
δ
δδ
δ
δ
δδ
δ
δδ
δ
=+
+
++
+
+
++
+
+
7/18/05Princeton Elasticity Lectures
Isotropic and Uniaxial
Solid
Isotropic: free energy density f
has
two harmonic elastic constants
(
)
1
1
2
2
1
2
2
32
()
(
)
()
(
)
Tr
Tr
Tr
ff
f
U
V
ff
u
f
V
u
V
Cu
D
Bu
u
u
αα
µ

=
Λ
=
Λ
==
=
+
+



I
nvaria
nt unde
r
()

U
(
V
)

Rx
R
x
µ
= shear modulus;
B = bulk modulus
Uniaxial: five harmonic elastic constants
I
nvaria
nt under
uni
()

U
(
V
)

Rx
R
x
22
11
22
12
3
22
45
;
(,
)
zz
zz
z
z
fC
u
C
u
u
C
u
Cu
C
u
x
νν
νν
ντ
ν
αν
=+
+
++
=
xx
7/18/05Princeton Elasticity Lectures
Force and stress I
ext
DD
ii
i
i
Fd
x
f
u
d
x
u
α
α
σ
=
=−

ii
f
α
α
σ
=∂
external force density –
vector in target space. The
stress tensor
σ
i
α
is mixed. This is the engineering or 1st
Piola-Kirchhoff
stress tensor = force per area of
reference space. It is not necessarily symmetric!
()
()
(
)
()
D
ii
ii
u
Ff
dx
f
uu
u
x
xx
x
αβ
α
α
αβ
δ
δ
σ
δδ

==
=
−∂

1
2
()
()
(
)
()
i
i
i
u
u
x
xx
x
αβ
α
α
ββ
δ
δ
δ

′′

=
Λ∂
+
Λ∂

II
i
ii
f
u
α
ββ
β
α
βα
σσ

=
Λ≡
Λ

σαβII
is the
second Piola-Kirchhoff
stress tensor -
symmetric
Note: In a linearized
theory,
σ
i
α
=
σi
α
II
7/18/05Princeton Elasticity Lectures
Cauchy stress
The Cauchy stress is the familiar force per unit area in the
target space. It is a symmetric tensor in the target space.
dI
d
C
ii
i
j
j
i
dx
u
dR
u
α
α
σσ

=

∫∫
i
i
R

∇≡

de
t
dd
dR
dx
=
Λ

i
ii
i
R
xx
R
α
α
αα

∂∂

==
=
Λ∇
∂∂

11
de
t
det
CI
T
I
I
T
ij
i
j
i
j
αα
α
α
αβ
σσ
σ
=
Λ
=
ΛΛ
ΛΛ

1
de
t
CI
I
T
σσ
=
ΛΛ
Λ


Symmetric as required

7/18/05Princeton Elasticity Lectures
Coupling to other fields
We are often interested in the coupling of target-space vectors
like an electric field or the nematic director to elastic strain.
How is this done? The strain tensor uαβ
is a scalar in the
target space, and it can only couple to target-space scalars,
not vectors.
Answer lies in the polar decomposition theorem
1/2
1/
2
1/
2
()
()
TT
M

Λ
=
ΛΛ
Λ
Λ
Λ

Θ




1/
2
(2
)
;
T
Mu
M
δ

=
ΛΛ
=+
Θ
=
Λ




1/2
1/
2
1/
2
1/
2
1
()
(
)
TT
T
T
T
OO
M
M
M
M
δ
−−

=
ΛΛ
=
ΛΛ
=
ΛΛ
Λ
Λ
=





Mu
O

i
s symm
etri
c
and depends on
only.
is an orthogonal, unimodular
rotation matr
ix

7/18/05Princeton Elasticity Lectures
Target-reference conversion
i
O
EE
α

The r
otation m
atrix
converts
tar
get-space
v
ectors
to re
ference-space v
ectors
and
vice-versa
;
T
ii
i
i
EO
E
E
O
E
αα
αα
==

.
ii
αα
δ
ΛΟ
=

If
is
symm
etric,
1
2
()
ii
i
i
ii
k
k
Ou
u
αα
α
α
αα
δ
δε

+
∂−

≈−

To linear order in u, Oiα
has a term proportional to the
antisymmetric
part of the strain matrix.
7/18/05Princeton Elasticity Lectures
Strain and Rotation
Simple Shear
Rotation
Symmetric
shear
n
Λ

is a reference space v
ector; it is equal to the
target space vec
tor
that is obtained when
is
symm
e
tri
c
7/18/05Princeton Elasticity Lectures
Sample couplings
1/2
1/
2
1
2
1
2
()
(
)
()
()
T
ii
j
i
j
i
j
j
TT
T
T
T
T
uE
E
E
O
u
O
E
v
E
E
OuO
v
α
α
αβ
β
αβ
β
δ
δ
−−
=

=
ΛΛ
Λ
Λ
Λ

Λ
Λ
Λ
=
ΛΛ

=





Coupling of electric field to strain
()
T
ij
i
j
ff
u
g
E
E
v
=−

Free energy no longer depends on the strain u
αβ
only.
The electric field defines a direction in the target
space as it should
0
ii
i
i
x
RR
xx
x
β
α
ββ
α
αα
β

∂∂

Λ
==
=
ΛΛ

∂∂

ii
i
α
αα
δ
η

Λ
=+
Energy depends on
both symmetric and
anti-symmetric parts
of η’
7/18/05Princeton Elasticity Lectures
Biopolymer Networks
cortical actin gel
neurofilament
network
7/18/05Princeton Elasticity Lectures
Characteristics of Networks

Off Lattice

random-walk polymers

Locally randomly inhomogeneous and
anisotropic but globally homogeneous
and isotropic

Complex frequency-dependent rheology

Striking non-linear elasticity
7/18/05Princeton Elasticity Lectures
Goals

Strain Hardening (more resistance to
deformation with increasing strain) –
physiological importance

Formalisms for treating nonlinear
elasticity of random lattices

Affine approximation

Non-affine
7/18/05Princeton Elasticity Lectures
Different Networks
10
100
1000
0.01
0.1
1
Strain stiffening of semiflexible biopolymer networks
G or G'
plat
(Pa)
Strain
NF
Vimentin
Collagen
Actin
Fibrin
polyacrylamide
Max strain
~.25 except for
vimentin
and
NF
Max stretch:
L(Λ)/L~1.13
at 45 deg to
normal
7/18/05Princeton Elasticity Lectures
Semi-microscopic models
Random or periodic
energy resides in bonds
connecting nodes
Rb
= separation of nodes in bond b
Vb(| Rb
|) = free energy of bond b
0
0
()
(
)
()
bb
b
b
FV
N
V
F
fn
V
V
R
R
RR
R
=

==
=

n
b
= Number of
bonds per unit
volume of
reference lattice
7/18/05Princeton Elasticity Lectures
Affine Transformations
Strained target network:
Ri=
Λ
ijR0j
1/2
1/
2
1/2
01
/
2
0
()
(
1
2
)
()
:

||
|
(
1
2
)
|
T
T
bb
OO
u
O
u
RR

Λ
=
ΛΛ
=+
=
ΛΛ
Λ
Λ
=+





Orthogonal

Reference network:
Positions R0
0
0
()
b
F
fn
V
V
R
R
==
Λ

Depends only on uij
7/18/05Princeton Elasticity Lectures
Example: Rubber
2
2
3
()
2
R
T
Nb
=
Purely entropic force
VR
0
2
00
31
()
T
r
22
TT
bb
R
R
T
Fn
V
T
n
Nb
RR
R
=
Λ
=
ΛΛ
=
ΛΛ


2
00
1
3
ij
i
j
RR
N
b
δ
=
2
22
33
()
e
x
p
22
R
PR
Nb
Nb
π

=

22
0
RN
b
=
Average is over the end-to-end separation in a
random walk: random direction, Gaussian magnitude
7/18/05Princeton Elasticity Lectures
Rubber : Incompressible Stretch
11
Tr
Tr
(
1
2
)
22
T
bb
fT
n
T
n
u

=
ΛΛ
=+


Unstable: nonentropic
forces between atoms needed to
stabilize; Simply impose incompressibility constraint.
1/
2
1/
2
00
00
00


Λ

=
Λ

Λ


Λ
2
12
2
b
fn
T



=
Λ
+


Λ
7/18/05Princeton Elasticity Lectures
Rubber: stress -strain
()
()
RR
z
R
R
AL
f
f
fV
f
A
LL

∂∂
==
=
∂∂
Λ

Λ
AR= area in
reference space
2
1
e
z
R
ff
nT
A
σ


==
=
Λ−


∂Λ
Λ
Engineering stress
2
1
z
ff
nT
A
σ


==
Λ
=
Λ−


∂Λ
Λ
Physical Stress
A
= AR/Λ
= Area
in target space
2
1
(1
)
~
3
1
nT
Yn
T
σ
γ
γγ
γ


==
+

+

Y=Young’s modulus
7/18/05Princeton Elasticity Lectures
General Case
Engineering
stress: not
symmetric
σ
er
e
f
1r
e
f
de
t
ij
j
ij
j
ij
i
j
dS
dS
dS
dS
σσ

=
=
Λ
Λ
0
e
0
00
0
0
00
0
0
0
()
'(
)
||
()
()
()
||
i
ij
j
ij
i
ij
j
f
nV
R
nR
n
R
R
R
R
R
R
RR
R
ττ
Λ

==
Λ
∂Λ
Λ
Λ
=
Λ
=
Λ
Λ



()
()
dV
R
R
dR
τ
=
Central force
0
0
00
0
()
de
t
|
|
ij
ik
k
jl
l
n
RR
R
R
R
τ
σ
Λ
=
ΛΛ
ΛΛ


Physical
Cauchy Stress:
Symmetric
7/18/05Princeton Elasticity Lectures
[]
0
0
0
2
1
2
0
,
1|
|
L
z
L
dR
Rv
L
d
s
ds
ds
v

=

≈−

t
t
2
|
()
|
1
;
()
(
()
,
1
|
()
|
)
ss
s
s
⊥⊥
==

tt
t
t
()
()
d
vs
s
ds
=
R
t
d
v
ds
=
R
t
= unit tangent
v
= stretch
2
22
1
||
(
1
)
2
d
Hd
s
v
K
v
ds
κτ





=+
+





t
t
7/18/05Princeton Elasticity Lectures
Length-force expressions
L(
τ
,K) = equilibrium length at given
τ
and K
0
2
0
1
2
22
1
0
2
2
02
(,
)
1
[
1
(
(,
)
)
]
;
11
()
|
|
;
co
th(
)
1
(,
)
1
;
n
p
p
p
B
LK
L
g
K
K
L
g
Ln
L
L
L
KL
Kk
T
τ
τϕ
τ
ϕ
πϕ
πϕ
πϕ
πϕ
τκ
ϕτ
τ
κπ

=


=+


==
+

=


=+
=



t
7/18/05Princeton Elasticity Lectures
Force-extension Curves
7/18/05Princeton Elasticity Lectures
Scaling at “Small” Strain
Theoretical curve:
calculated from
K-1=0
zero parameter fit to everything
G'/G' (0)
Strain/strain8
7/18/05Princeton Elasticity Lectures
What are Nematic Gels?

H
omogeneous Elastic media with
broken rotational symmetry
(uniaxial, biaxial)

Most interesting -
systems with
broken symmetry that develops
spontaneously from a
homogeneous, isotropic elastic
state
7/18/05Princeton Elasticity Lectures
Examples of LC Gels
1. Liquid Crystal Elastomers -
liquid crystal polymers
Nematic
Smectic-C
2. Tanaka gels with hard-rod
dispersion
3. Anisotropic membranes
4. Glasses with orientational order
7/18/05Princeton Elasticity Lectures
Properties I

Large thermoelastic
effects
-
Large
thermally induced strains -
artificial muscles
Courtesy of
Eugene Terentjev
300% strain
7/18/05Princeton Elasticity Lectures
Properties II
Large strain in small
temperature range
Terentjev
7/18/05Princeton Elasticity Lectures
Properties III

Soft or “Semi-soft” elasticity
Vanishing xz
shear modulus
Soft stress-strain for stress
perpendicular to order
Warner Finkelmann
7/18/05Princeton Elasticity Lectures
Model for Isotropic-Nematic trans.
(
)
2
2
2
32
1
2
Tr
Tr
Tr
fB
u
u
Cu
D
u
αα
µ

+
=+

1
3
uu
u
αβ
αβ
αβ
γγ
δ
=

µ
approaches zero signals a transition to a nematic state with a
nonvanishing
(
)
1
3
uS
n
n
α
αβ
β
αβ
δ
=

7/18/05Princeton Elasticity Lectures
Spontaneous Symmetry Breaking
Phase transition to anisotropic
state as
µ
goes to zero
(
)
1
2
00
0
T
u
δ
=
ΛΛ

00
2
u
δ
Λ
=+
Direction of
n0
is
arbitrary
0
00
1
3
()
uu
nn
αβ
αβ
αβ
α
β
δ
=
=
Ψ−

Symmetric-
Traceless
part
2
~
uαα
Ψ
7/18/05Princeton Elasticity Lectures
Strain of New Phase
u

is the strain relative
to the new state at
points
x’
0
()
()
()
ii
j
j
i
ii
Rx
u
xu
δ
=
Λ
+
′′

=+
xx
x
0
ii
k
ij
ik
kj
jj
k
x
RR
xx
x

∂∂

Λ
==
=
ΛΛ

∂∂

δu
is the deviation of
the strain relative to the
original reference frame
R
from
u
0
δu
is linearly proportional
to
u

(
)
11
22
'(
)
T
T
u
δη
η
′′

=
ΛΛ

+

()
0
1
2
00
00
'
TT
T
uu
u
u
δ
=

=
ΛΛ

ΛΛ
=
ΛΛ





7/18/05Princeton Elasticity Lectures
Elasticity of New Phase
(
)
()
1
1
1
00
0
0
1
1
4
11
'(
)
1c
o
s
2
s
i
n
2
(1
)
sin
2
1
cos
2
T
r
r
r
uV
u
V
u
r
θθ
θθ

=
Λ−
Λ


=

−−


2
0||
2
0
r

Λ
=
Λ
Rotation of anisotropy
direction costs no energy
(1
)
'~
4
xz
r
u
r
θ

C5
=0 because of
rotational
invariance
This 2nd order expansion
is invariant under all U
but only infinitesimal V
2
11
22
12
3
el
5
4
zz
zz
zz
fC
u
C
u
u
C
u
u
Cu
u
Cu
u
νν
νν
νν
ντ
ντ
ν
ν
′′

=+
+
′′
′′
++
7/18/05Princeton Elasticity Lectures
Soft Extensional Elasticity
()
1
1
4
11
1c
o
s
2
s
i
n
2
(1
)
sin
2
1
cos
2
r
r
r
ur
θθ
θθ


=

−−


Strain u
xx
can be converted to a
zero energy rotation by
developing strains uzz
and uxz
until uxx
=(r-1)/2
1
1
(1
2
)
2
zz
xx
xz
xx
xx
uu
r
uu
r
u
r
=

=
−−
7/18/05Princeton Elasticity Lectures
Frozen anisotropy: Semi-soft
()
()
h
zz
fu
f
u
h
u
=

System is now uniaxial

why not simply use uniaxial
elastic
energy? This predicts linear stress-stain curve and misses
lowering of energy by reorientation:
22
2
2
11
22
5
12
3
4
zz
zz
z
fC
u
C
u
u
C
u
C
u
C
u
νν
νν
ντ
ν
=+
+
+
+
Model Uniaxial
system:
Produces harmonic uniaxial
energy for small strain but has
nonlinear terms –
reduces to
isotropic when
h=0
f (u)
: isotropic
2
2
xz
xx
zz
xx
z
z
xz
uu
u
uu
u
uu
u
θ

−−

=+


Rotation
()
(
)
(
2
)
h
zz
xz
fu
f
u
h
u
u
θ

=

+
7/18/05Princeton Elasticity Lectures
Semi-soft stress-strain
Ward Identity
22
(
)
2
(
)
()
0 or

h
xz
xz
xx
zz
zz
xx
xz
xx
xz
xz
xz
xx
zz
xx
df
hu
u
u
u
d
hu
uu
uh
σσ
σ
σ
σ
σ
θ
=
=

=
−−

=

=
+
h
f
u
αβ
αβ
σ

=

Second Piola-Kirchoff
stress tensor.
7/18/05Princeton Elasticity Lectures
Semi-soft Extensions
Break rotational symmetry
Stripes form in real
systems: semi-soft, BC
Not perfectly soft because of residual
the the nematic phase -
semi-soft.
length of plateau depends on magnitude
of spontaneous anisotropy
r.
Warner-Terentjev
Note: Semi-softness
only visible in
nonlinear properties
Finkelmann, et al., J. Phys. II 7, 1059 (1997);
Warner, J. Mech. Phys. Solids 47, 1355 (1999)
7/18/05Princeton Elasticity Lectures
Softness with Director
Nα= unit vector along uniaxial
direction in reference space;
layer normal in a locked SmA
phase
(,
)
z
nn
α
ν
=
n


22
2
1
(
)
;
,
etc.
zz
nN
n
c
u
N
u
N
να
α
ν
α
αβ
β
=
−⋅

=

Red: SmA-SmC
transition
22
2
11
22
12
3
4
22
1
2
5
21
22
2
11
22
12
3
4
2
2
1
42
1
4
2
22
11
22
5
12
1
2
1
[(
/
)
]
[
(
/
)
]
zz
zz
zz
z
zz
zz
zz
zz
nu
gn
f
Cu
C
u
u
C
u
C
u
Cu
D
n
n
u
D
n
Cu
C
u
u
C
u
C
u
Dn
D
D
u
C
D
u
u
D
n
ν
νν
τ
τ
νν
νν
ντ
νν
ν
ν
νν
νν
ντ
νν
ν
λ
λ
+
++
=+
+
+
++
+
=+
+
+
++
+

+



"


2
2
55
1
1
0
2
R
D
CC
D
Soft
=

=

Director relaxes to zero