Cavitation instabilities in soft solids: A defect-growth theory & applications to elastomers

April 6, 2011Applied Mechanics Colloquium, Harvard University

Oscar Lopez-PamiesState University of New York, Stony Brook

Work supported by the National Science Foundation (DMS)

Force-Deformation Relation

Rubber Disk

Specimen

Metal Plate

Metal Plate

Busse (1938); Yerzley (1939); Gent and Lindley (1959)

Material Right After CavitationUndeformed Material

l

S

Experimental observations

• Practical Relevance/Theoretical Interpretation

• Main Existing Results and Open Problems

• New Strategy: Iterated Homogenization

– Application I: Onset-of-cavitation surfacesfor Neo-Hookean solids

Outline

– Application II: Onset-of-cavitation surfacesfor solids that are not polyconvex

Gent and Lindley (1959), Chen et al. (1995)

Cavitation may lead to material failure, …

it may also be used to toughen hard materials

Rubber-toughened Polycarbonate

Practical relevance

Kundu & Crosby et al (2009), Goriely et al. (2010)

Practical relevance

to indirectly measure mechanical properties

Induced cavitation in a gel

or to induce cavity opening in growing systems like plants

• Sudden growth of initially vanishingly small defects

A theoretical interpretation

Random distributions of flaws, in the order of 0.1 μm in average diameter, are expected to be present in typical elastomers. Physically, they can correspond to:

• Actual voids

• Weak regions of the underlying polymer network

• Particles of dust

• Others…

Gent (1991)

Undeformed State

Infinitesimal defects

Deformed State (I)

Infinitesimal defects

Deformed State (II)

Grown defects

Main Existing Resultsand

Open Problems

• Spherical shell under hydrostatic pressure The classical result for radially symmetric cavitation

Undeformed State

Incompressible, Isotropic Material

Initial Porosity f0

Current porosity:f

fl

l

+ -=

30

3

1

Pressure-stretch relation:

o iR R f= = 1/301,

Deformed State

Po ir r fl l= = + -3

0, 1

• In the limit as The classical result for radially symmetric cavitation

Undeformed State

Incompressible, Isotropic Material

Infinitesimal Cavity(or defect)

o iR R= = +1, 0

f +0 0

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3 3.5 4

20 10f -=

40 10f -=

P

0 0f +

f

Porosity-Pressure relation

Critical Pressure:

Gent and Lindley (1959), Ball (1982)

• General loading conditions

• Compressible and anisotropic materials

• Effect of the geometry and mechanical properties of the defects

• And some others: surface energy, fracture, dynamical effects…

Open problems

Sivaloganathan, Stuart, Spector, Horgan, Abeyaratne, Henao,…

Cavitation: A General Sudden Growth of

Defects Formulation

• Consider a random statistically uniform distribution of nonlinear elastic cavities embedded in an otherwise homogeneous nonlinear elastic solid

( ) ( )W W WX F X F X Fq q+(1) (2)0 0, ( ) ( ) ( ) ( )= 1-

- Local stored-energy function

- The characteristic function is a random variable that describes the initial size, shape, and spatial location of the cavities, and must be characterized in terms of ensemble averages

q0

Problem setting

0W

Undeformed

x FX 0on ¶= W

W

Deformed

W F(2)( )

( )(1)W F

• Total elastic energy

Problem setting

KE W

F FX F X

WÎ=

W ò0( )

0

1min ( , )d

where

{ }K JF F: x = (X) F = x FX= $ > W = ¶W0 0( ) with Grad , 0 in , in c c

• Porosity in the deformed configuration (it measures the size of the cavities in the deformed configuration)

ff F(X) X

F W

W= =

W Wò (2)

0

(2)

0(2)0

det ddet

Problem settingIn the limit as the material under consideration reduces to a nonlinear elastic solid containing a random distribution of zero-volume cavities or defects. When the material is finitely deformed, these defects can suddenly grow to finite sizes signaling the onset of cavitation

0 0f +

1D Schematic of a typical solution

0

0.1

0.2

0.3

0.4

1 1.2 1.4 1.6 1.8

f

60 10f -=

0 0f +

l

20 10f -=

crl

Porosity f

0

0.5

1

1.5

2

2.5

1 1.2 1.4 1.6 1.8

20 10f -=

60 10f -=

l

S

f +0 0

crl

f( )-=0

Defect free solid0

Overall Stress S E l= d /d

Problem setting: Onset-of-cavitation criterionTo put the above observation in a more rigorous setting, we introduce

and postulate:

ff f fF F* +0

00( ) lim ( , )

f

ES fF F

F* +

¶¶0

00( ) lim ( , )and

The onset of cavitation in a nonlinear elastic material with stored-energy function W(1) containing a random distribution of defects with stored-energy function W(2) occurs at critical deformations such thatcrF

cr fF F*é ùÎ ¶ ë û( )Z and crS F*< < +¥0 ( )

where denotes the boundary of the zero-set of . The corresponding 1st PK and critical Cauchy stresses at cavitation are given by

f F*é ù¶ ë û( )Z f F*( )

crS S F*= ( ) and Tcr cr cr crJT S F=

Problem setting: Homogenization• Key advantage of the proposed formulation:

are homogenized or average quantities

ff F(X) X

F W

W= =

W Wò (2)

0

(2)

0(2)0

det ddet

KE W

F FX F X

WÎ=

W ò0( )

0

1min ( , )d

and

These average quantities are much easier to handle than more local quantities, such as for instance , which would likely contain an excess of detail and thus would complicate unnecessarily the analysis of cavitation

F(X)

• Yet, the computation of E and f is, in general, extremely difficult…

An Iterated

Homogenization Method

• Construct a particulate distribution of cavities ( ) within a nonlinear elastic material for which it is possible to compute exactly the total elastic energy E and porosity f.

Xq0 ( )Iterated homogenization

• How? with 2 steps

Lopez-Pamies (2010), Idiart (2008)

1. Derive an iterated homogenization procedure to write an exactsolution for E in terms of an auxiliary dilute problem

2. Formulate and solve the auxiliary dilute problem H by means of sequential laminates

Ef H W E E W

fF F F

¶ é ù- = =ê úë û¶(1) (2)

00

, ; 0, ( ,1) ( )

( )EH E W F +

F¶

= + ⋅ - Ķ

(1)maxw

w x w x

n| |=

= ò1

( )dx

x x Here is an orientational average (2-point statistics)

( )E Ef E W

fF +

F¶ ¶

+ + ⋅ - Ä =¶ ¶

(1)0

0

max 0w

w x w x

• The Total Elastic Energy E in the material can be shown to be given by the following Hamilton-Jacobi equation

subject to the initial condition E WF = (2)( ,1)

• The current porosity f is, in turn, determined by the equation

( )Tf ff f

fF

F-¶ ¶

- + ⋅ Ä + ⋅ Ä =¶ ¶0

0

1 0w x w x

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

Iterated homogenization

Onset-of-cavitation criterion

ff f fF F* +0

00( ) lim ( , )

f

ES fF F

F* +

¶¶0

00( ) lim ( , )and

The onset of cavitation in a nonlinear elastic material with stored-energy function W(1) containing a random distribution of defects with stored-energy function W(2) occurs at critical deformations such thatcrF

cr fF F*é ùÎ ¶ ë û( )Z and crS F*< < +¥0 ( )

Here, and are the functions defined by

where

f fF 0( , ) E fF 0( , )

( )E Ef E W E W

fF + F F

F¶ ¶

+ + ⋅ - Ä = =¶ ¶

(1) (2)0

0

max 0, ( ,1) ( )w

w x w x

and

( )Tf ff f f

fF F

F-¶ ¶

- + ⋅ Ä + ⋅ Ä = =¶ ¶0

0

1 0, ( ,1) 1w x w x

– Arbitrary initial “geometry” of the cavities up to 2-point statistics

– Compressible anisotropic materialsW (1)

– Pressurized cavities (This can be readily accomplished by setting as opposed to just )W g J=(2) ( ) W =(2) 0

• The proposed IH approach is applicable to:

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

Remarks on the IH approach

F– General loading conditions

• A picture of the “microstructure”

point defects randomly distributed

− For isotropic solids, isotropic distribution of vacuous defects

Remarks on the IH approach (continued)

W ( )(F) , ,f l l l=11 2 3( ), W ( )( ) , (F)n

p= =21

04

x

and hydrostatic loading conditions F = Il

the proposed formulation recovers the classical result for radially symmetric cavitation of Ball (1982)

( )E Ef E E

f+

w f l w l l ll

¶ ¶- - + = =

¶ ¶00

, , 0, ( ,1) 03

f ff f f

fw w ll l

æ ö¶ ¶÷ç- + - = =÷ç ÷÷ç¶ ¶è ø00

1 0, ( ,1) 13

Energy:

Porosity:

Identical to the solutions of a shell under hydrostatic loading!!

Why?

Puniform fieldin the cavity!

ª

FEM Approach

FEM Approach for a single defectUndeformed FE model

Some specifics:

f p -

-

= ´

» ´

90

9

/6 10

0.5 10

Cavitation ensues whenever f f= ´5010

Lopez-Pamies, Nakamura, & Idiart (2010)

Mesh near cavity

l3

l1

l2

64,800 8-node brick elements

Initial volume fraction of cavity:

1

5x104

1x105

0 0.5 1 1.5 2 2.5 3

m/s m

ff0

l l l= =1 2 3

FEM Resultst1 = t2 = t3

f f 50/ 10under fixed f f = 5

0/ 10

t1 > t2 > t3

Near Cavity

Application to Neo-Hookean Solids

withIsotropic Distribution of Defects

• For an incompressible NH material with stored-energy

Lopez-Pamies, Nakamura, & Idiart (2010)

• The current porosity is, of course, given by

the solution for E of the Hamilton-Jacobi equation in the limit asis given by

( )G fE Ol l l l lm l l l

l ll

m

l l

æ ö- ÷ç ÷+ +ç ÷ç ÷é ù= + + -ê úë ç øû è

1/301/

2 2 2 1 2 3 1 21 2 3

2 31 2 33

32

3 ( 1),

2( )

f +0 0

where the function G is solution to a pde that needs to be solved numerically

Incompressible Neo-Hookean solid

containing an isotropic distribution of vacuous defects

Onset-of-cavitation criterion for NH solids

S t t t t t t t t t

t t t

: ( )

( )

m y

m y m y m

- + +

+ + + - - =

1 2 3 1 2 1 3 2 3

2 2 3 3 31 2 3

8 12

18 27 8 0

Inside a Neo-Hookean solid cavitation will occur at a material point P whenever along a given loading path the principal Cauchy stresses ti first satisfy the following condition:

where is a known function of ti such that y< £0 1y

NOTE: Cavitation only occurs when it i( , , )> =0 1 2 3

- Axisymmetric loading t t ;=3 2

1t

1t

2t

3 2t t=2t

t

t

t

T =

æ ö÷ç ÷ç ÷ç ÷ç ÷ç ÷÷ç ÷ç ÷çè ø

1

2

2

0 0

0 0

0 0

Onset-of-cavitation : Axisymmetric loading

0

2

4

6

8

10

-12 -8 -4 0 4 8 12

Bound ofHou and Abeyaratne

mt

t t-2 1

IH

FEM

mt t t( ) /= +1 22 3

. m2 5

Onset-of-cavitation surface

m

t t ts

m+ +

= 1 2 3Hydrostatic stress:

Shear stresses:

t t

t t

tm

tm

-=

-=

2 11

3 12

/t m2 /t m1

msm

�50

5�5

05

0

5

10

0 02

46

24

60

2

4

6

8

FE

Theory

/t m- 2 /t m- 1

msm

Loading path2.5

t1

t2

t3

Application to Strongly Elliptic Solids

that are notPolyconvex

• Consider the stored-energy function

Lopez-Pamies (2010)

A general class of I1-based solids

with material parameters

0

0.5

1

1.5

2

2.5

3

3.5

4

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8l

S un (M

Pa)

0

0.5

1

1.5

2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

S ss(M

Pa)

g

Uniaxial Tension Simple Shear

Neo-Hookean

Michelin Elastomer

I1-based Model

0.0

0.5

1.0

1.5

2.0

2.5

-3 -2

-1 0

12

3

-3

-2

-1

0 1

2

Onset-of-cavitation surface: FE results

m

t t ts

m+ +

= 1 2 3Hydrostatic stress:

Shear stresses:t t t t

t tm m- -

= =2 1 3 11 2,

/t m2

/t m1

msm

0

0.5

1

1.5

2

-2 -1 0 1 2

t t t= =1 2

msm

/t m

Axisymmetric Loading

• We have proposed a new strategy (IH) to study the phenomenon of cavitation in nonlinearly elastic solids subjected to general loading conditions.

• The proposed approach is general yet mathematically tractable and thus provides the means to study open problems of cavitation in solids.

• When applied to Neo-Hookean solids containing an isotropic distribution of vacuous defects, the IH method leads to results that significantly improve on the only available bound and are in agreement with FEM calculations for a single defect.

• The results indicate that the onset of cavitation depends very critically on the entire state of stress, not just on the hydrostatic component

Final remarks