Oscar Lopez-Pamies

The nonlinear elastic response of suspensions of rigid inclusions in rubber

Work supported by the National Science Foundation (CMMI)

June 2013SIAM Mathematical Aspects of Materials Science, Philadelphia

Filled elastomers: applications & microstructure

SRI (2009); Kofod et al. (2007); Carpi et al. (2010); Aschwanden & Stemmer (2006)

elastomeric matrix

fillers

Mechanical Reinforcement

Multifunctional (Smart) Applications

muscle-like actuatorsenergy harvesters

Filled elastomers: some milestones• Dilute Solution: Einstein 1906; Smallwood 1944 (one spherical particle)

• Non-spherical, non-dilute: • Eshelby 1957 (one ellipsoidal particle)• Batchelor & Green 1972 (two spherical particles)• Chen & Acrivos 1978 (two spherical particles)

• Finite deformation framework• Hill 1972

difficult (non-convexity) equations to solve!!!

Note: there are not even non-trivial bounds available

Outline

I. Exact solution for Gaussian (Neo-Hookean) rubber filled with an isotropic dilute distribution of rigid particles

II. Extend the dilute solution to Gaussian rubber filled with an isotropic distribution of rigid particles with polydisperse sizes and finite concentration

III. Extend the Gaussian solution to filled non-Gaussian rubber

Via an iterated homogenization technique

Via an iterated dilute homogenization technique

Via a nonlinear comparison medium technique

The problem: local perspectiveDeformed Undeformed

DeformationGradient

Local stored-energy function

Random or periodic indicator function

rigid

Rigid particle behavior

The problem: macroscopic responseDefinition: relation between the volume averages of

the stress and deformation gradient over RVE

l

Lseparation oflength scales

Hill (1972), Ogden (1978)

Variational Characterization

where

and

Undeformed

I. The Dilute Solution for Gaussian Rubber

Strategy: Construct a particulate distribution of rigid particles for which it is possible to compute exactly the total elastic energy

Idea: iterated homogenization

Actual nonlinear material Nonlinear material with

After some work, we arrive at the implicit solution for given by the pde

subject to the initial condition

Idiart (2008); Lopez-Pamies (2010) Lopez-Pamies, Idiart, Nakamura (2011)

Remarks on the IH approach

• By construction, the particles interact in such a manner that their deformation (and stress) is uniform!

– Arbitrary initial “geometry” of the particles up to 2-point correlation functions:

– Compressible anisotropic matrix and particle materials and

• The formulation applies to:– General loading conditions

initial condition

exactly as the ellipsoidal particle in the linear Eshelby problem

with

Application to Neo-Hookean rubber• For Neo-Hookean rubber

the solution for the total elastic energy in the limit as is given by

where

subject to the initial condition

and isotropic distribution of rigid particles

Remarks on H

with

• Upon the change of variables the initial value problem for H can be rewritten as the Eikonal equation (geometrical optics)

• Numerical solution

Remarks on W : asymptotic behavior• In the limit of small deformations as

in terms of the principal invariants this reads as

• In the limit of infinitely large deformations as

agrees with Einstein-Smallwood result for spherical particles!

weak dependence on I2!

independent of I2!

NOT polyconvex!

Remarks on W : comparison with FE

mesh near inclusion

rigid inclusion

FE

Remarks on W : comparison with FE

0

1

2

3

4

5

3 4 5 6 7 8 9 10

H

FEAnalytical

24I =

26I =

1I

0

1

2

3

4

5

6

5 10 15 20 25

H

FEAnalytical

16I =

1 8I =

110I =

2I

14I =

Note: Both solutions are approximately linear in I1

Note: Both solutions are approximately independent of I2

A closed-form approximation

Exact

Approx.

0

5

10

15

20

0 0.5 1 1.5 2 2.5 3 3.5 4

l

H

1 2l l l= =

ExactApprox.

• Based on its asymptotic behavior for small and large deformations, the exact solution for can be approximated by

II. Finite-Concentration Solution for

Gaussian Rubber

Step I: Iterated dilute homogenization

Lopez-Pamies (2010, 2013)

Bruggeman (1935)

[1]W

pW

Step I: Iterated dilute homogenization

In the limit of infinitely many iterations we end up with the IVP

Ad infinitum

Here

Then

[1]W

pW

III. Finite-Concentration Solution for

Non-Gaussian Rubber

III. Finite-concentration solution for non-Gaussian rubber

Talbot & Willis (1985)

Upon introducing the Legendre transformation

Lopez-Pamies, Goudarzi, Danas (2013)

it follows that

upon choosing constant (divergence-free) and

where

III. Finite-concentration solution for non-Gaussian rubber

Upon introducing the Legendre transformation

it follows that

upon choosing constant (divergence-free) and

where

III. Application to rigidly reinforced I1-based materials

• By choosing the comparison medium as a filled Neo-Hookean elastomer with the same microstructure, and setting P = 0 and Q = 0, the variationalapproximation reduces to

• Carrying out the calculations leads to the explicit result

Remarks• In the limit of small deformations as

where

agrees with Brinkman-Roscoe result for polydisperse spherical particles!

• For the case of Neo-Hookean rubber

• If is convex then is strongly elliptic

• is independent of the second invariant I2

Comparison with FE simulationsMonodisperse microstructures Polydisperse microstructures

Typical mesh (undeformed configuration)

~75,000 ten-node tetrahedral hybrid elements

Periodic boundary conditions

1

1.5

2

2.5

3

3.5

4

0 0.1 0.2 0.3 0.4c

Analytical

HS boundFE polydisperseFE monodisperse

mm

Filled Neo-Hookean rubber

-7

-6

-5

-4

-3

-2

-1

0

0.4 0.5 0.6 0.7 0.8 0.9 1l

0.05c =

0.15c =

0.25c =

FEAnalytical

matrix

unS

m

uniaxial compressionlinear elastic response

Note: polydispersity is inconsequential!

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5FEAnalytical

0.05c =

0.15c =

0.25c =

g

matrix

ssS

m

Filled Neo-Hookean rubber

simple shear

0

0.2

0.4

0.6

0.8

1

1.2

1 1.2 1.4 1.6 1.8 2 2.2l

FEAnalytical

0.05c =

0.15c =

0.25c =

matrixS un (M

Pa)

Filled silicone rubber

uniaxial tension

Silicone rubber matrix:

-3

-2.5

-2

-1.5

-1

-0.5

0

0.4 0.5 0.6 0.7 0.8 0.9 1l

FEAnalytical

0.05c =

0.15c =

0.25c =matrix

S un (M

Pa)

uniaxial compression

0

0.1

0.2

0.3

0.4

0.5

3 4 5 6 7FEAnalytical

2I

0.05c =

14.75I =

13.76I =

0.15c =

Y (M

Pa)

Filled silicone rubber

independence of I2

Silicone rubber matrix:

0

0.2

0.4

0.6

0.8

0 0.5 1 1.5FEAnalytical

0.05c =

0.15c =

0.25c =

g

matrixS ss (M

Pa)

simple shear

Main results• Dilute Neo-Hookean (Gaussian) result

• Finite-concentration Neo-Hookean (Gaussian) result

• Finite-concentration non-Gaussian result

Final comments: generalizations

Danas et al. (2012); Qu et al. (2011)

AFM image revealing the presence of “bound’’ rubber

• Interphasial Phenomena

0 1 2 3 4 5 6 7 8 9

10

1 1.5 2 2.5 3 3.5 4 stretch

stre

ss (M

Pa)

elastomeric

matrix

iron

particles• Anisotropic Microstructures