Nonlinear Elasticity

7/18/05 Princeton Elasticity Lectures

Nonlinear Elasticity


Outline

• Some basics of nonlinear elasticity• Nonlinear elasticity of biopolymer

networks• Nematic elastomers


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.


Classical Lagrangian Description

Reference material in D dimensions described by a continuum of mass points x. Neighbors of points do not change under distortion

Material distorted to new positions R(x)

( ) ( )= +R x x ux

ii i i

Rxa a a

a

d h¶

L = = +¶

i iua ah = ¶Cauchy deformation tensor

x

x

( )R x

( )R x


Linear and Nonlinear ElasticityLinear: Small deformations – near 1

Nonlinear: Large deformations – >>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


Deformations and StrainComplete 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

i i i i iua a a a ad d hL = +¶ = +3 3det

det 1:

d R d x= L

L =%

%No volume change

1/ 2

1/ 2

0 0

0 0

0 0

-

-

æ öL ÷ç ÷ç ÷ç ÷ç ÷L = Lç ÷ç ÷ç ÷÷ç L÷÷ççè ø%

Volume preserving stretch along z-axis


Simple shear strain

1

0 1

æ öL÷ç ÷çL = ÷ç ÷÷çè ø%

1 0

1

æ ö÷ç ÷çL = ÷ç ÷L ÷çè ø%

Constant Volume, but note stretching of sides originally along x or y.

Rotate

Not equivalent to

Note: is not symmetric


Pure Shear

2

2

1

1

æ ö+L L ÷ç ÷ç ÷L =ç ÷ç ÷ç ÷L +L ÷çè ø%

Pure shear: symmetric deformation tensor with unit determinant – equivalent to stretch along 45 deg.


Pure shear as stretch1 111 12

x x xUy yy

ji ii

j

Tij j

xRR R

x R x x

U U

2

2

1 0

0 1

TU U


Pure to simple shear

2tan

1

2 2

2 1

2

/

2

2

1 2 2 1 2

0 (1

1cos sin

sin

2

1

)

cos


Cauchy Saint-Venant Strain

( )

1 12 2

12

( ) ( )T T

k k

u

u u u u ua a a aab b b

d h h= L L - » +

= ¶ +¶ +¶ ¶% % % % % %

2 2 2dR dx u dx dxaab b- =

ii i i

Rxa a a

a

d h¶

L = = +¶uis invariant under rotations

in the target space but transforms as a tensor under rotations in the reference space. It contains no information about orientation of object.

R=Reference space

T=Target space

Symmetric!


Elastic energyThe elastic energy should be invariant under rigid rotations in the target space: if is a function of u

12

12

( )

[ ]

D

D

F d xf u

d x K u u u

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


Elastic modulus tensorKis the elastic constant or elastic modulus tensor. It has inherent symmetry and symmetries of the reference space.

K K K K

( )K Isotropic system

1 2

123 4

14 5

( )

( )

( )

T T

T T T T T T

T T T T

K C n n n n C n n n n

C C

C n n n n n n n n

Uniaxial (n = unit vector along uniaxial direction)


Isotropic and Uniaxial Solid

( )

1

1

2 212

23 2

( ) ( )

( ) ( )

Tr Tr Tr

ff f U V

ff u f V uV

C u DBu u uaa m

-

-

= L = L

= =

= ++ - % %%

Invariant under

( ) U (V )®R x R x

Isotropic: free energy density f has two harmonic elastic constants

Uniaxial: five harmonic elastic constants Invariant under

uni( ) U (V )®R x R x2 21 1

2 21 2 3

2 24 5 ;

( , )

zz zz

z

z

f C u C u u C u

C u C u

x

nn nn

nt n

a n

= + +

+ +

=x x

= shear modulus;B = bulk modulus


Force and stress Iext D D

i i i iF d x fu d x u

external force density – vector in target space. The stress tensor i is mixed. This is the engineering or 1st Piola-Kirchhoff stress tensor = force per area of reference space. It is not necessarily symmetric!

i if

( )

( ) ( ) ( )D

i ii i

uF fd x f

u u u

x

x x xab

a aab

dds

d d

¢¶¢- = = = - ¶¢¶ò

12

( )( ) ( )

( ) i ii

u

u

xx x

xab

aa b b

dd

d

¢¢ ¢ ¢= L ¶ +L ¶ -

I Ii i i

fua b b ba

ba

s s¶

= L º L¶

II is the second Piola-Kirchhoff stress tensor - symmetric

Note: In a linearized theory, i = iII


Cauchy stressThe Cauchy stress is the familiar force per unit area in the target space. It is a symmetric tensor in the target space.

d I d Ci i ij j i

d x u d R uaas s¶ = Ñò ò i

iR¶

Ñ º¶

detd dd R d x= L% i

i ii

R

x x Ra aa a

¶¶ ¶¶ = = = L Ñ

¶ ¶ ¶

1 1det det

C I T II Tij i j i ja a a aab

s s s= L = L LL L% %

1det

C II Ts s= L LL% %% %%

Symmetric as required


Coupling to other fieldsWe 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 u 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( ) ( )T T M-L = L L L L L º Q% %% % % % %% 1/ 2( 2 );TM u Md -= L L = + Q = L

% % % % % % %%1/ 2 1/ 2 1/ 2 1/ 2 1( ) ( )T T T T TOO M M M M d- - - - -= L L = L L = L L L L =

%% %% %% %% % % %% % % %M u

O% %

%

is symmetric and depends on only.

is an orthogonal, unimodular rotation matrix


Target-reference conversion

i

O

E Ea

%%

The rotation matrix converts target-space

vectors to reference-space vectors and vice-versa

; Ti i i i

E O E E O Ea aa a= =% %

.i ia a

dL O =%

If is symmetric, 12( )i i i i

i i k k

O u ua a a a

a a

d

d e

» + ¶ - ¶

» - W

To linear order in u, Oi has a term proportional to the antisymmetric part of the strain matrix.


Strain and Rotation

Simple ShearSymmetric shear

Rotation

n

L

% is a reference space vector; it is equal to the

target space vector that is obtained when is

symmetric


Sample couplings

1/ 2 1/ 212

12

( ) ( )( )

( )

Ti i j ij i jj

T T T T T

T

u E E E O u O E v E E

OuO

v

a aab b ab b

d

d

- -

= º

= L L L L L - L L L

= LL - =

% %

%%% %% % % % % % % %

%% % %

Coupling of electric field to strain

( )Ti j ij

ff u gE E v

Free energy no longer depends on the strain u only. The electric field defines a direction in the target space as it should

0i i

i i

xR R

x x xb

a b baa ab

¢¶¶ ¶¢L = = = L L

¢¶ ¶ ¶

i i i Energy depends on both symmetric and anti-symmetric parts of ’


Biopolymer Networks

cortical actin gel neurofilament network


Characteristics of Networks

• Off Lattice• Complex links, semi-flexible rather

than random-walk polymers• Locally randomly inhomogeneous

and anisotropic but globally homogeneous and isotropic

• Complex frequency-dependent rheology

• Striking non-linear elasticity


Goals

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

• Formalisms for treating nonlinear elasticity of random lattices– Affine approximation– Non-affine


Different Networks

10

100

1000

0.01 0.1 1

Strain stiffening of semiflexible biopolymer networksG

or

G' p

lat(P

a)

Strain

NF

Vimentin

Collagen

Actin

Fibrin

polyacrylamide

Max strain ~.25 except for vimentin and NF

Max stretch:L()/L~1.13 at 45 deg to normal


Semi-microscopic models

Random or periodic crosslinked network: Elastic energy resides in bonds (links or strands) connecting nodes

Rb = separation of nodes in bond bVb(| Rb |) = free energy of bond b

0

0

( ) ( )

( )

b bb

b

F V N V

Ff n V

V

R

R

R R

R=f

= =

=

å nb = Number of bonds per unit volume of reference lattice


Affine Transformations

Reference network:Positions R0

Strained target network:Ri=ijR0j

00

( )b

Ff n V

V RR= = L

%

1/ 2 1/ 2

1/ 2

0 1/ 2 0

( ) (1 2 )

( ) :

| | |(1 2 ) |

T

T

b b

O O u

O

uR R

-

L = L L = +

= L L L

L = +

% %% % % %

% %% %

% %

Orthogonal

Depends only on uij


Example: Rubber

2 20

R Nb=

2

2

3( )

2R

V R TNb

=

02 0 0

3 1( ) Tr

2 2T T

b bR R

TF n V Tn

NbR R R= L = L L = L L

% % % % %

20 0

13i j ij

R R Nbd=

Purely entropic force

Average is over the end-to-end separation in a random walk: random direction, Gaussian magnitude

2

2 2

3 3( ) exp

2 2R

P RNb Nbp

é ùê ú= -ê úë û


Rubber : Incompressible Stretch

1 1Tr Tr(1 2 )

2 2T

b bf Tn Tn uf = L L = +

% % %Unstable: nonentropic forces between atoms needed to stabilize; Simply impose incompressibility constraint.

1/ 2

1/ 2

0 0

0 0

0 0

-

-

æ öL ÷ç ÷ç ÷ç ÷ç ÷= Lç ÷ç ÷ç ÷÷ç L÷çè ø

L

21 22 b

f nT=fæ ö÷ç= L + ÷÷çè øL


Rubber: stress -strain

( )( )

R Rz R

R

A L f ff V f A

L L

ff

¶¶ ¶= = =

¶ ¶L ¶L

2

1e z

R

ffnT

Af

sæ ö¶ ÷ç= = = L - ÷÷çè ø¶L L

2 1zff nTA

sæ ö¶ ÷ç= = L = L - ÷÷çè ø¶L L

Engineering stress

Physical Stress

AR= area in reference space

A = AR/ = Area in target space

2 1(1 ) ~3

1nT

Y nTs

gg g g

æ ö÷ç= = + - ÷ç ÷ç +è øY=Young’s modulus


General Case

e ref

1 refdet

ij j ij j

i ji j

dS dS

dS dS

s s-

=

= LL

( )( )

dV RR

dRt =

Engineering stress: not symmetric

Physical Cauchy Stress: Symmetric

Central force

0

e 00 0

0

00 0 0 0

0

( )'( )

| |

( )( ) ( )

| |

iij j

ij

ii j j

fn V R

n R n R

R

RR

R

RR R

R

s

t t

L¶= = L

¶L L

L= L = L

L

%%

%

%% %

%

0

00 0

0

( )

det | |ij ik k jl l

nR R

R

R

R

ts

L= L L

L L%

% %


Semi-flexible Stretchable Link

22 21

| | ( 1)2

dH ds v K v

dsk t^

^

é ùæ öê ú÷ç= + + -÷ç ÷çê úè øë ûò

tt

( ) ( )d

v s sds

=R

t

2| ( ) | 1; ( ) ( ( ), 1 | ( ) | )s s s s^ ^

= = -t t t t

dv

ds=

R

[ ]0

0

0

212

0

,

1 | |

Lz

L

dRR v L ds

ds

dsv^

º =

é ù» -ë û

ò

ò

t

t

t = unit tangentv = stretch


Length-force expressions

0

2 012 2 2

1

02

202

( , ) 1 [1 ( ( , ))];

1 1( ) | | ;

coth( ) 1

( , ) 1 ;

np

p

p

B

L K L g KK

Lg

L n

L

L

LK L

K k T

tt j t

jp j

p j p jp j

t kj t t

kp

¥

^=

æ ö÷ç= + -÷çè ø

= =+

-=

æ ö÷ç= + =÷çè ø

åt

L(,K) = equilibrium length at given and K


Force-extension Curves


Scaling at “Small” StrainG

'/G

' (0

)

Strain/strain8

zero parameter fit to everything

Theoretical curve: calculated from K-1=0


What are Nematic Gels?

• Homogeneous Elastic media with broken rotational symmetry (uniaxial, biaxial)

• Most interesting - systems with broken symmetry that develops spontaneously from a homogeneous, isotropic elastic state


Examples of LC Gels1. Liquid Crystal Elastomers - Weakly crosslinked liquid crystal polymers

4. Glasses with orientational order

2. Tanaka gels with hard-rod dispersion

3. Anisotropic membranes

Nematic Smectic-C


Properties I

• Large thermoelastic effects - Large thermally induced strains - artificial muscles

Courtesy of Eugene Terentjev

300% strain


Properties II

Large strain in smalltemperature range

Terentjev


Properties III

• Soft or “Semi-soft” elasticity

Vanishing xz shear modulus

Soft stress-strain for stressperpendicular to order

Warner Finkelmann


Model for Isotropic-Nematic trans.

( )2 2 23 212 Tr TrTrf Bu u C u D uaa m - += + %% %

13u u uab ab ab ggd= -%

approaches zero signals a transition to a nematic state with a nonvanishing

( )13u S n naab b abd= -%


Spontaneous Symmetry Breaking

Phase transition to anisotropicstate as goes to zero

Direction of n0 isarbitrary

0

0 0 13( )

u u

n n

ab ab

a b abd

=

= Y -

% %

2~uaa Y

( )120 0 0

Tu d= L L -

0 02udL = +

Symmetric-Tracelesspart


Strain of New Phase

u is the deviation ofthe strain relative to the original reference frame R from u0

u’ is the strain relative to the new state at points x’

u is linearly proportionalto u’( )1 1

2 2' ( )T Tu d h h¢ ¢ ¢ ¢= L L - » +

% %

0( ) ( )

( )

i ij j i

i i

R x u

x u

d= L +

¢ ¢ ¢= +

x x

x

( )0

12 0 0

0 0'

T T

T

u u u

u

d = -

= L L - L L

= L L

% % %

% % % %

%%%

0i i k

ij ik kjj jk

xR R

x x x

¢¶¶ ¶¢L = = = L L

¢¶ ¶ ¶


Elasticity of New Phase

Rotation of anisotropy direction costs no energy

C5=0 because ofrotational invariance

This 2nd order expansionis invariant under all U but only infinitesimal V

( )

( )

11 10 0 0 0

1

14 1 1

' ( )

1 cos2 sin2( 1)

sin2 1 cos2

T

r

rr

u V u V u

rq q

q q

-- -= L - L

æ ö- ÷ç ÷ç= - ÷ç ÷- -ç ÷çè ø

( 1)' ~

4xz

ru

rq

-

2

0||20

r^

L=

L

21 12 21 2 3el

54

zz zz

z z

f C u C u u C u u

C u u C u u

nn nn nn

nt nt n n

¢ ¢ ¢ ¢ ¢= + +

¢ ¢ ¢ ¢+ +


Soft Extensional Elasticity

Strain uxx can be converted to a zero energy rotation by developing strains uzz and uxz until uxx =(r-1)/2

( )

1

14 1 1

1 cos2 sin2( 1)

sin2 1 cos2

r

rr

u rq q

q q

æ ö- ÷ç ÷ç= - ÷ç ÷- -ç ÷çè ø

1

1( 1 2 )

2

zz xx

xz xx xx

u ur

u u r ur

= -

= - -


Frozen anisotropy: Semi-soft

( ) ( )hzzf u f u hu= -

( ) ( ) ( 2 )hzz xzf u f u h u uq¢ = - +

System is now uniaxial – why not simply use uniaxial elastic energy? This predicts linear stress-stain curve and misses lowering of energy by reorientation:

2 2 2 21 12 2 51 2 3 4zz zz zf C u C u u C u C u C unn nn nt n= + + + +

Model Uniaxial system:Produces harmonic uniaxial energy for small strain but has nonlinear terms – reduces to isotropic when h=0

f (u) : isotropic

2

2xz xx zz

xx zz xz

u u uu u u

u u uqæ ö- - ÷ç ÷¢® = + ç ÷ç ÷- ÷çè ø

Rotation


Semi-soft stress-strain

2 2 ( ) 2( )

( )0 or

h

xz xz xx zz zz xx xz

xx xzxz xz

xx zzxx

dfhu u u u

dh u

u uu h

s s

ss s

sq

=

= - = - -

-= Þ

-=

+

hfuab

ab

s¶

=¶

Ward Identity

Second Piola-Kirchoff stress tensor.


Semi-soft Extensions

Not perfectly soft because of residual anisotropy arising from crosslinking in the the nematic phase - semi-soft. length of plateau depends on magnitude of spontaneous anisotropy r.

Warner-Terentjev

Stripes form in real systems: semi-soft, BC

Break rotational symmetry

Finkelmann, et al., J. Phys. II 7, 1059 (1997);Warner, J. Mech. Phys. Solids 47, 1355 (1999)

Note: Semi-softness only visible in nonlinear properties


Softness with Director

2 2 21 12 21 2 3 4

2 2125 2 1

2 2 21 12 21 2 3 4

2

21

4 214 2

2 21 12 251 2 1 2 1

[ ( / ) ] [ ( / )]

zz zz

z z z

zz zz

zz

z z

n u

gn

f C u C u u C u C u

C u D n n u D n

C u C u u C u C u

D n D D u C D u

u

D

n

n

n n tt

nn nn nt

n n n n

nn nn nt

n n n

l

l

+

+ +

= + + +

+ + +

= + + +

+ + + -

+

%

% %

%

L%% %

22

5 51

10

2R D

C CD

Soft= - = ÞDirector relaxes to zero

( , )zn n n

Nunit vector along uniaxial direction in reference space; layer normal in a locked SmA phase

2 2 21 ( ) ; , etc.zzn N n c u N u N

Red: SmA-SmC transition

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Nonlinear Elasticity