FIBER-REINFORCED ELASTOMERS: MACROSCOPIC PROPERTIES,

MICROSTRUCTURE EVOLUTION, AND STABILITY

Oscar Lopez-Pamies

State University of New York, Stony BrookState University of New York, Stony Brook

February 19, 2010 ·

Ames, IADepartment of Aerospace Engineering, Iowa State University

TEM

of Collagen Fibrils in Human Brain Arteries

Rubber reinforced with carbon-black and

fabric

TEM

of a triblock

copolymer with cylindrical morphology

Thermoplastic Elastomers(Self-Assembled Nanodomains)

fPS~ 50 nm

BCC HX-Cyl. Gyroid Lamellar

Soft solids reinforced with fibers

• Complex

initial microstructure

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1 1.2 1.4 1.6 1.8 2l

S bi (

MPa

)

• Nonlinear

constitutive matrix phase and fibers

• Microstructure evolution (geometric nonlinearity)

• Development of instabilities

Issues in constitutive modeling of FREs

Problem setting: Lagrangian

formulationKinematics

N

Deformed WUndeformed

O

X

0W

xF

X¶

=¶

DeformationGradient

Local constitutive behavior

( )W

S X FF

¶

¶,=

( ) ( ) ( )r rr

W WX F X Fc=å2

( ) ( )0

1

, =

where

Right CG Deformation Tensor

TC F F=

( )x X

Stored-energy function of the matrix

( )W F(1)Stored-energy function of the fibers

( )W F(2)

Random variable

characterizing the microstructure

( )rrX

Xcì Îïï= íïïî

( )0

1 if phase

0 otherwise

Problem setting: Lagrangian

formulationKinematics

N

Deformed WUndeformed

O

X

0W

xF

X¶

=¶

DeformationGradient

Local constitutive behavior

( )W

S X FF

¶

¶,=

( ) ( ) ( )r rr

W WX F X Fc=å2

( ) ( )0

1

, =

where

Right CG Deformation Tensor

TC F F=

W I IF = Y(1) (1) 1 2( ) ( , )

W I I I IF = Y(2) (2) 1 2 4 5( ) ( , , , )

Isotropic matrix

Transversely isotropic fibers

I C=1 tr

I N CN= ⋅4 I N C N= ⋅2

5

I C Cé ù= ê úë û2 2

21

(tr ) -tr2

where

( )x X

Problem setting: macroscopic responseDefinition: relation between the volume averages of

the stress and deformation gradient over RVE

S S XW

=W ò 001

d F F XW

=W ò 001

dUndeformed

RVE

l

Lseparation oflength-scales

Hill (1972), Braides

(1985), Müller

(1987)

Variational

Characterization

WS F

F¶

=¶

( )

where

{ }F F F x x FXK ¶= W = W) = 0 0( | in , and on

( )( )

( )K

W WF F

F X F XWÎ

=W ò 001

min , d

and

•

Phenomenological models

( ) ( ) ( )iso aniW F I I G I IF = +1 2 4 5, ,Merodio, Horgan, Ogden,

Pence, Rivlin, Saccomandi,

among others

Formulated on the basis of invariants

Existing analytical approaches

−

“Linear comparison”

variational

estimates for general loading conditions and isotropic constituents (LP & Ponte Castañeda

2006a,b)

•

Homogenization/Micromechanics models

Incorporate direct information from microscopic properties

−

Estimates for special loading conditions, and special matrix and

fiber constituents (He et al., 2006; deBotton

et al., 2006)

fiber failure (local)

Classes of instabilities

Material fiber debonding

(local)matrix cavitation

(local)

Geometricalshort wavelength (local)

long wavelength

(global)

Geymonat, Müller, Triantafyllidis

(1993)

i k j lij kl

WB v v u u

F FF F

= =

ì üï ï¶ï ï= =í ýï ï¶ ¶ï ïî þ||u|| ||v|| 1

2

u,v( ) min ( ) 0

•

The loss of strong ellipticity

of the macroscopic response of the fiber-reinforced elastomer, as characterized by the effective stored-energy function , denotes the onset of long wavelength instabilities

W F( )

Stability and failure

LP (2006)

Microstructure evolution

0W

F

W

Deformed Undeformed

Information on microstructure evolution is important to identify and understand the microscopic mechanisms

that govern the

macroscopic behavior. For that we need information about the local fields ( )F X

New Approach:Iterated Homogenization

Iterated dilute homogenization

Ad-infinitum...

Step 1:

iterated-dilute homogenization

Homogenization

dilute phase

LP, J. App. Mech. (2010)

Wc H W W W Wc

F F¶ é ù- = =ê úë û¶

(1) (2)0

0

, ; 0, ( ,1)

Strategy:

Construct a particulate distribution of fibers ( )

within a hyperelastic

material

for which it is possible to compute exactly

the effective stored-energy function

r Xc( )0 ( )

W

Auxiliary dilute problem: sequential laminates

Rank-1 or simple laminate

( )S

WH W W W W SF F +

Fn

é ù¶é ù ê ú= + ⋅ - Äê úë û ê ú¶ë ûò(1) (1), , max ( ) dx x xw w w

distributional function

related to the two-point statistics of fiber distribution in the undeformed

configuration

When the matrix

phase is dilute:

Rank-2 laminate

... ad-infinitum ...

matrix phase

inclusion phase

Step 2:

sequential laminates

Idiart, JMPS (2008)

( )S

W Wc W W Sc

F +F

né ù¶ ¶ê ú- - ⋅ - Ä =ê ú¶ ¶ë û

ò (1)00

max ( ) d 0x x xw

w w

•

The effective stored-energy function can be finally shown to be given by the following Hamilton-Jacobi

equation

subject to the initial condition

W WF F= (2)( ,1) ( )

W

Iterated homogenization framework

the time

variable is

the spatial

variable is

the Hamiltonian

is

(initial fiber concentration)

( )S

WH W W W W SF F +

Fn

é ù¶é ù ê ú= + ⋅ - Äê úë û ê ú¶ë ûò(1) (1), , max ( ) dx x xw w w

F

t c- 0ln

LP & Idiart, J. Eng. Math. (2010)

( )S

W Wc W W Sc

F +F

t tt n

é ù¶ ¶ê ú- - ⋅ - Ä =ê ú¶ ¶ë ûò (1)0

0

max ( ) d 0x x xw

w w

•

Consider the following perturbed problem

for

subject to the initial condition

W W UF F Ft= +(2)( ,1) ( ) ( )

Wt

Iterated homogenization framework: local fields

LP & Idiart, J. Eng. Math. (2010)

Then, the following identity is true

( ) WUc

F X X t

ttW

=

¶=

¶Wò (2)

0(2)

0 00

1 1( ) d

( )U ⋅ can be any function of interest, e.g., the deformation gradient

•

The computations

amount to solving appropriate Hamilton-Jacobi equations, which are fairly tractable

•

The proposed IH method provides access to local fields, which in turn permits the study of the evolution of microstructure

and the onset of

instabilities

•

In the limit of small deformations as , the IH formulation reduces to the HS lower bound

for fiber-reinforced random media

F I

•

In the further limit of dilute fiber concentration , the IH formulation recovers the

exact result of Eshelby

for a dilute distribution of ellipsodial

fibers

c 0 0

Remarks on the iterated homogenization approach• The proposed IH method provides solutions for in terms of and

and the one-

and

two-point statistics

of the random distribution of fibersW W (1) W (2)

Application toFiber-Reinforced

Neo-Hookean

Solids

Neo-Hookean

matrix

( )W I IF m= Y = -(1)

(1) (1)1 1( ) ( ) 32

Fiber-reinforced Neo-Hookean

solidsrandom, isotropic distribution

N

0W

Stiffer Neo-Hookean

fibers

( )W I IFm

= Y = -(2)

(2) (2)1 1( ) ( ) 32

Macroscopic stored-energy function

W c I I I I cF = Y0 1 2 4 5 0( , ) ( , , , , )

Hamilton-Jacobi

equation

( )c I I Ic II

m mm

æ öæ ö¶Y ¶Y÷ç ÷ç÷ç ÷+ - - Y + - - - =ç÷ ÷ç ç÷ ÷ç¶ ¶ç ÷è øè ø

2(1)(1)

0 1 1 4 (1)0 14

2 1 13 0

2 2

subject to the initial condition I I I I I IY = Y(2)1 2 4 5 1 4( , , , ,1) ( , )

0

1

2

3

4

5

6

1 1.5 2 2.5 3 3.5

l

S

Closed-form solution for

( )f I Im= -1 1( ) 32

where

Overall stress-strain relationW

( ) ( )W I I f I g IF = Y = +1 4 1 4( ) ( , )

( )( )I Ig I

I

m m + --= 4 44

4

2 1( )

2

and

Note I: separable functional form

c cm m m= - + (2)(1)0(1 )c c

c c

m mm m

m m

- + +=

+ + -

(2)(1)(1)0 0

(2)(1)0 0

(1 ) (1 )

(1 ) (1 )

with

and

Note II: no dependence on the second nor fifth invariants I5I2

Overall stress-strain relation

T TW p I pS F F F FN N FF

m m m- - -¶ é ù= - = + - - Ä -ê úë û¶3/2

4( ) ( ) 1

LP & Idiart, J. Eng. Math. (2010)

Macroscopic Instabilities

fiber failure (local)

Classes of instabilities

Material fiber debonding

(local)matrix cavitation

(local)

Geometricalshort wavelength (local)

long wavelength (global)

crI/

mm

æ ö÷ç= - ÷ç ÷÷çè ø

2 3

4 1

Along an arbitrary loading path, the material first becomes unstable at a critical deformation with such that

crFcr

cr crI F N F N= ⋅4

LP & Idiart, J. Eng. Math. (2010)Note: crIm m£ £4 1

Macroscopic

Instabilities

Observations

•

Macroscopic instabilities may only occur when the deformation in the fiber direction, as measured by , reaches a sufficiently large compressive

value

I4crI £4 1

•

The condition states that instabilities may develop whenever the

compressive deformation along the fiber direction reaches a critical value determined by the ratio of hard-to-soft

modes of deformation

0W

m

m

Hard mode

Soft mode

LP (2006), LP & Idiart

(2010)

Microstructure evolution

3e

2e

1e0W

F

W2v

1v

3v

z- = ¥23

z-22

z-21

z- = ¥23

z- =22

1

z- =21

1

Deformed Undeformed

The average shape

and orientation

of the fibers in the deformed configuration are characterized by the Eulerian

ellipsoid

( )( )Z I N N F -= - Ä 1(2)

{ }TE x x Z Zx 1= ⋅ £|where z z z, ,1 2 3

Eigenvalues

of

Eigenvectors

of

TZ Z

TZ Zv , v , v1 2 3

is the average deformation gradient in the fibers. In the IH framework, it is solution of the pde

LP, Idiart, & Li (2010)

Microstructure evolution

F(2)

Sc S

cF F

F¶ ¶

- Ä =¶ ¶ ò

(2) (2)

00

d 0xw

subject to the initial condition F F F=(2)( ,1)

Closed-form solutionT T

I IF F FN N F F N N FN u FN N

n g n gg - -

- -é ùé ù= - Ä - - Ä + Ä + Äê úë û ë û(2) 1 1

14 4

2 2

where , and

( )I I I I II I I I

ng n

+ - -= +

- +

24 1 4 5 4

1

1 4 5 4

2,

2 c c

mnm m

=+ + -

(1)

(1) (2)0 0(1 ) (1 )

Tu I N N F FN= - Ä( )

Sample Results

0

1

2

3

4

5

6

7

1 1.5 2 2.5 3 3.5l

IHFE

c .=0 0 2

c .=0 0 4

t( )m 1

Moraleda

et al. (2009)

l l

IH vs. FEM: in-plane stress-strain response

.l = 2 1

Rigid fibers

S

Axisymmetric

compression at an angle φ0

l

l

1 2/l-1 2/l-

3e

Ν

0j

1e

3e

n

j1e

W0W

F

1 2

1 2

0 00 00 0

/

/F

l

ll

-

-

æ ö÷ç ÷ç ÷ç ÷= ç ÷ç ÷÷ç ÷ç ÷çè ø

Applied Loading Initial fiber orientation

0 1 0 3N e ej j= +cos sin

Axisymmetric

compression at an angle φ0

l

l

1 2/l-1 2/l-

3e

Ν

0j

1e

3e

n

j1e

W0W

F

1 2

1 2

0 00 00 0

/

/F

l

ll

-

-

æ ö÷ç ÷ç ÷ç ÷= ç ÷ç ÷÷ç ÷ç ÷çè ø

Applied Loading Current fiber orientation

1 3n e ej j= +cos sin

0

2 3 20 0

jj

j l j

é ùê ú= ê úê ú+ë û

cos Arcos

cos cos

l

l

1 2/l-1 2/l-

Axisymmetric

compression at an angle φ0

0

10

20

30

40

50

0 20 40 60 80j0

t = 20

c .=0 0 3

( )m =1 1 t = 50

t = 5

S

ll =1

dd

heterogeneity contrastt( )

( )

mm

2

1

Small-deformation response

Large-deformation stress-strain response

0

5

10

15

20

25

30

35

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1l

j = o0 90S

.j = o0 35 3

j = o0 60

t = 20c .=0 0 3

( )m =1 1

j = o0 80

j = o0 10

Axisymmetric

compression at an angle φ0

0

20

40

60

80

100

0 0.2 0.4 0.6 0.8 1l

j

j = o0 90.j = o0 89 9

j = o0 89

j = o0 0

j = o0 80 j = o0 60

.j = o0 35 3

j = o0 10

Evolution of fiber orientation

Note:

Macroscopic instability at

LP, Idiart, & Li (2010)

Axisymmetric

compression at an angle φ0

Effect of fiber orientation

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

30 40 50 60 70 80 90j0

crl

t = 5

t = 20

c .=0 0 3( )m =1 1

t = 50

0.5

0.6

0.7

0.8

0.9

1

20 40 60 80 100

crl

( )m =1 1

j = o0 90

t

c .=0 0 1

c .=0 0 3

c .=0 0 5

Effect of fiber-to matrix contrast

Onset of macroscopic instabilities at crl

Axisymmetric

compression at an angle φ0

Effect of fiber orientation

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

30 40 50 60 70 80 90j0

crl

t = 5

t = 20

c .=0 0 3( )m =1 1

t = 50

Onset of macroscopic instabilities at crl

0

20

40

60

80

100

0 0.2 0.4 0.6 0.8 1l

j

j = o0 90.j = o0 89 9

j = o0 89

j = o0 0

j = o0 80 j = o0 60

.j = o0 35 3

j = o0 10

Evolution of fiber orientation

The rotation of the fibers

— which depends critically on the relative orientation between the loading axes and the fiber direction —

can act as a

dominant geometric softening mechanism.

Remarks

•

It was found that the long axes of the fibers rotate away from the axis of maximum compressive loading

towards the axis of maximum tension.

•

Loadings with predominant compression along the fibers lead to larger rotation of the fibers, which in turn lead to larger geometric softening of the constitutive response, and in some cases —

when the heterogeneity

contrast between the matrix and the fibers is sufficiently high —

also to the loss of macroscopic stability.

The results of this work can help understanding the behavior of many

othersolids with oriented microstructures

Remarks

Polystyrene Ellipsoids2 m

l

S

Reinforced Elastomers

(Wang and Mark 1990)

Liquid crystal elastomers

(Nishikawa and Finkelmann1999)

Transparent Opaque

l

l

1l-1l- »

l

l

1l- 1l-

Rosen (1965); Triantafyllidis

and Maker (1985)

where

•

For the aligned plane-strain loading of a laminate

the onset of macroscopic instabilities occurs at

LLamcr

mlm

æ ö÷ç ÷= -ç ÷ç ÷÷çè ø

1/4

1

L c cmm m

-æ ö- ÷ç ÷= +ç ÷ç ÷çè ø

1

(1) (2)

1c cm m m= - + (2)(1)(1 ) and

?

Contact with earlier work with laminates

0 20 40 60 80 1000.75

0.8

0.85

0.9

0.95

1

crl

0 0.2 0.4 0.6 0.8 10.7

0.75

0.8

0.85

0.9

0.95

1

crl

Effect of fiber concentration c0 Effect of contrast

Laminate

Fibers

0c2 1t ( ) ( )/m m=

Laminate

Fibers

2 1t ( ) ( )/m m=

l

l

1l- 1l-Aligned plane-strain loading conditions

2 1 20t ( ) ( )/m m= =

0 3c .=

Contact with earlier work with laminates

Contact with experiments

Jelf

& Fleck (1992)

Matrix: Silicone —

Young’s Modulus E(1)

= 2.9 MPa

Fibers: Spaghetti — Young’s Modulus E(2)

= 69 MPaVolume fraction

c0

= 31%

Experimental setup

IH prediction

•

An iterated homogenization approach in finite elasticity has been proposed to construct exact (realizable) constitutive models for fiber-reinforced hyperelastic

solids

•

Because the proposed formulation grants access to local fields, it can be used to thoroughly study the onset of failure and the evolution of microstructure in fiber-reinforced soft solids with random microstructures

•

As a first application, closed-form results were derived for fiber-reinforced Neo-Hookean

elastomers

•

These ideas can be generalized to more complex systems of soft heterogeneous media with random microstructures

Final remarks

•

The required analysis reduces to the study of tractable Hamilton-Jacobi equations

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