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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