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