Int J Fract (2017) 205:1–21DOI 10.1007/s10704-016-0176-9

ORIGINAL PAPER

Damage in elastomers: nucleation and growth of cavities,micro-cracks, and macro-cracks

X. Poulain · V. Lefèvre · O. Lopez-Pamies ·K. Ravi-Chandar

Received: 12 August 2016 / Accepted: 8 December 2016 / Published online: 3 January 2017© Springer Science+Business Media Dordrecht 2016

Abstract The nucleation of internal cavities andtheir transition to cracks are examined at high spa-tial and temporal resolutionswithin polydimethylsilox-ane (PDMS) elastomers of various cross-link densitiesunder externally applied quasi-static mechanical loads.The focus here is on experiments where the initiationand propagation of internal damage are designed tooccur in between two spherical glass beads that arefirmly embedded within a matrix of the PDMS elas-tomer and are placed close to each other in order togenerate a high triaxial stress state. An optical micro-scope is used tomonitor the various processes of nucle-ation and growth of cavities and cracks at a spatialresolution of about 1µm and a temporal resolution ofabout 66.7ms. In combinationwith corresponding full-field simulations, the experiments show that the nucle-

Electronic supplementary material The online version ofthis article (doi:10.1007/s10704-016-0176-9) containssupplementary material, which is available to authorized users.

X. Poulain · K. Ravi-Chandar (B)Department of Aerospace Engineering and EngineeringMechanics, Center for Mechanics of Solids, Structures,and Materials, University of Texas at Austin, Austin,TX 78712-1221, USAe-mail: ravi@utexas.edu

V. Lefèvre · O. Lopez-PamiesDepartment of Civil and Environmental Engineering,University of Illinois Urbana-Champaign, Urbana,IL 61801-23552, USA

O. Lopez-Pamiese-mail: pamies@illnois.edu

ation of cavities—that is, the onset of cavitation—isan extremely fast process (involving stretch rates inexcess of 100 s−1) that is controlled primarily by thestiffening at large deformations of the underlying elas-tomer and, more critically, by its fracture properties.The experiments and simulations also show that cavi-tation is followed by two distinct events upon furthermacroscopic loading: the transition of the nucleatedcavities to micro-cracks, and the further transition ofsome micro-cracks to macro-cracks. These two dis-tinct events are also controlled primarily by the fractureproperties of the underlying elastomer.

Keywords Cavitation · Neo-Hookean · Fracture ·Polydimethylsiloxane · Triaxiality

1 Introduction

Cavitation in elastomers has occupied the interest ofnumerous investigators in a plurality of fields (rangingfrom materials science, to mechanics, to mathematics)over a very long period of time. This is due in part to itsrole as the initiator of internal damage in the material,and in part to the simplicity of its most basic math-ematical model: the radially symmetric deformationof a single spherical cavity embedded in a homoge-neous elastic medium under remotely applied pressure.Two seminal early works in the literature merit lengthyreview here because one of them has steered much of

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the subsequent research effort and both of them helpplace the present work in context.

The first of these works is that of Gent and Lind-ley (1959a, b) who performed carefully crafted “pokerchip” experiments and provided a first theoreticalexplanation of the phenomenon of cavitation. Specif-ically, they carried out tensile tests on “poker chip”specimens—of 20 mm diameter and thicknesses rang-ing from 0.6 to 9.8 mm—made up of natural rubber,carbon-black filled natural rubber, and a polybutadienesynthetic rubber. At smaller thicknesses, a clear kink inthe nominal macroscopic stress vs extension plot wasobserved. On post-mortem examination, they foundthat below the critical value of the stress, there was novisible internal damage, while above this, cracks werefound within the rubber. Secondary cracks were foundat evenhigher load levels. In someof these experiments,photographs were used to identify volume changes ofthe specimens. It was noted that upon crossing the crit-ical load, there was a large increase in volume sug-gesting that this was due to the nucleation and growthof internal cavities or cracks. In experiments using atransparent rubber, the appearance at first of one andsubsequently (upon increasing the load) of many largeholes was indeed observed. They remarked that, uponunloading, such holes corresponded actually to cracksin the unloaded configuration.

In an attempt to explain their experimental findings,Gent and Lindley (1959a, b) effectively proposed toconsider the nucleation of the observed cavities/cracksas the sudden elastic growth of inherent defects in rub-ber. In particular, leveraging the then recent analysis ofGreen and Zerna (1954) for thick spherical shells, Gentand Lindley (1959a, b) considered the basic problem ofthe radially symmetric elastic deformation of the mate-rial surrounding a single spherical cavity embeddedin an infinite Neo-Hookean medium that is subjectedremotely to hydrostatic tension, and showed that thecavity grows rapidly and unboundedly as a certain crit-ical hydrostatic pressure, namely, Pcr = 5μ/2 with μ

denoting the initial shear modulus of the Neo-Hookeanmedium, is approached. Given that this theoreticalresult agreed reasonably well with some of their exper-imental observations, the pioneering work of Gentand Lindley (1959a, b) prompted a plethora of furtherexperimental, theoretical, and computational investiga-tions (see, e.g., Oberth and Bruenner 1965; Ball 1982;Gent and Park 1984; Cho et al. 1987; Stringfellow andAbeyaratne 1989; Hou and Abeyaratne 1992; Lopez-

Pamies 2009; Cristiano et al. 2010; Lopez-Pamies et al.2011a, b; Xu and Henao 2011; Nakamura and Lopez-Pamies 2012; Lian and Li 2012; Kabaria et al. 2015)clearly following their view of cavitation as an elasticphenomenon.

The second pioneering work is that of Williams andSchapery (1965), who extended the radially symmetriccalculation ofGent andLindley (1959a, b), nowconsid-ering the single spherical cavity to be embedded at thecenter of a Neo-Hookean ball of finite size and allow-ing the cavity surface to deform not only elastically butalso by the creation of new surface (via fracture) asdictated by the energetic approach of Griffith (1921).This extension of the analysis was motivated by therecognition that the stretches at the cavity surface in theanalysis of Gent and Lindley (1959a, b) far exceededrealistic maximum elastic values for rubber. Thus, incontrast to Gent and Lindley (1959a, b), Williams andSchapery (1965), proposed to consider the nucleationof cavities/cracks in rubber as a fracture phenomenon.While no particular geometry of cracks or fracture wasconsidered at the outset, they accounted for the surfaceenergy associatedwith the increase in surface area fromcavity expansion. From their analysis (in which, again,they only allowed for the creation of new radially sym-metric surface), they concluded1 that a small sphericalcavity of initial radiusa0 cangrowby fracturewheneverthe externally applied hydrostatic pressure reaches thecritical value Pcr = μ [5/2−λ−4

a /2−2λ−1a ], where λa

is solution of the nonlinear algebraic equation C(λa) =μ[2λ−1

a − λ2a − 1] + 2γ /a0 = 0 with the materialconstant γ denoting the surface energy per unit unde-formed area that is created. This analysis clearly bringssize dependence to the process of cavity growth—thecritical pressure at which cavity growth occurs dependsnot only on the material properties μ and γ , but alsoon the initial cavity size a0 through the nondimen-sional parameter κ = 2γ / (μa0). Assuming a reason-able range of values for the parameters, 0.1 ≤ μ ≤ 1MPa, 1 ≤ γ ≤ 10 J/m2, and 10−8 ≤ a0 ≤ 10−7m,implies 20 ≤ κ ≤ 20, 000 and yields the range of crit-ical pressures 2.045μ ≤ Pcr ≤ 2.486μ. Remarkably,these critical values are essentially the same as that ofGent and Lindley (1959a, b). Not many investigators,

1 The calculation reported by Williams and Schapery (1965)contains an error, but this did not have any impact in their overallconclusions. The correct calculation was later reported in Gentand Wang (1991b).

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however, have followed up on the study of cavitation asa fracture phenomenon; instead the attention of nearlythe entire community has focused on the view that cavi-tation is the result of the elastic growth of defects (Gent1991a).

With the objective of probing the extent to whichelastic properties are relevant in the onset of cavitationin rubber, Lefèvre et al. (2015) have recently carriedout full-field simulations of the classical experimentsof Gent and Lindley (1959a, b) and those of Gent andPark (1984) under the premise that the rubber utilizedin their specimens was nonlinear elastic and that theunderlying defects at which cavitation could initiatewere vacuous and their spatial distribution was ran-dom and isotropic. Their results have indicated thataccounting just for the nonlinear elasticity of the rub-ber and the presence of defects suffices, to a consid-erable extent, to describe and explain the experimentsin the sense of: (1) where cavitation first occurs, aswell as (2) how cavities continue to grow and interactonce they have been “nucleated”. Their results havealso made it plain that cavitation in rubber is first andforemost a fracture process, essentially because—asalready noted by Williams and Schapery (1965)—thelocal stretches around the defects at which cavitationinitiates far exceed the elastic limit of the rubber, whichtherefore ought to deform by fracturing to accommo-date their growth.Moreover, their results have revealedthat there remains a critical need for more compre-hensive and accurate experimental insight to advancethe understanding of the phenomenon of cavitation andfracture in rubber.

The present paper, which is a continuation of thework ofLefèvre et al. (2015), is devoted to revisiting theexperiments2 of Gent and Park (1984), now performedin a way that allows to investigate the nucleation ofinternal cavities/cracks, their growth, and interactionat high spatial and temporal resolutions. We begin inSect. 2 by describing the techniques used to preparethe specimens—made up of two glass beads placedclose to each other within a PDMS matrix—and the

2 The experimental approach of high spatiotemporal resolutionpresented here can equally be applied to investigate cavitationin rubber confined between other geometries (not just sphericalbeads). Indeed, akin to the work of Cho et al. (1987), we haveemployed the same approach introduced here to study cavita-tion within a PDMS elastomer confined between two cylindricalfibers. The analysis of these experiments will be presented else-where.

experimental schemes used for observation and char-acterization of the nucleation, growth, and interactionof internal cavities/cracks. Specifically, the design ofthe specimen that facilitates the visual observation ofthe region of expected cavitation is discussed. In orderto examine the dependence of the onset of cavitationon the elastic material properties, five different com-positions of PDMS were tested; their uniaxial stress–stretch behavior and the corresponding model calibra-tion are discussed in Sect. 3 and Sect. 4, respectively.The sequence of events beginningwith nonlinear defor-mation, continuing through the nucleation and growthof cavities, followed by their transition tomicro-cracks,and the eventual fracture of the specimens throughmacro-cracks is described in detail in Sect. 5. A numer-ical (finite-element) analysis of the local elastic fieldswithin a representative specimen right up to the pointof the nucleation of cavities is presented in Sect. 6. Aselaborated in our previous analysis (Lefèvre et al. 2015)of the experiments of Gent and Park (1984), these sim-ulations readily allow for the examination of the onsetof cavitation based on an available general elastic cri-terion (Lopez-Pamies et al. 2011b) and its direct com-parison with experiments in order to further probe therelevance of elastic properties on the nucleation of cav-ities in rubber. Numerical (finite-element) analysis ofthe growth of nucleated cavities and micro-cracks arealso presented and confronted to experiments in Sect. 6.Finally, we conclude in Sect. 7 with some remarks onthe governing mechanisms behind cavitation and frac-ture in elastomers.

2 Experimental design for microscopicobservations at high spatial and temporalresolution

2.1 Design of specimens

Because of its high transparency, which allows for thedirect visualization/investigation of the phenomenonof cavitation, and easily controllable cross-link density,which allows for the expedient change of elastic proper-ties, the experiments of this study are on polydimethyl-siloxane (PDMS) elastomers. In particular, the variousgrades of PDMS used were made from Dow CorningSylgard 184 silicone elastomer base with its associatedcuring agent. In this work, we designed two-bead spec-imens similar to those of Gent and Park (1984) with the

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H

D

PDMSMetallic Grip

y

x

Fig. 1 Geometry of the specimen with spherical beads (design 1on top and design 2 in the middle); sections along the mid-planeare shown. The metallic grip into which the beads are gluedalso facilitates holding the specimen in the test machine clamps.The PDMS (shown in gray) is poured using a mold shown inFig. 2. The diameter of the beads D (shown in black) and the gapH between them are varied in order to generate different stressconditions in the constrained PDMS

aim of inducing cavitation within the PDMS confinedbetween the two beads, where the stress field was con-trolled by varying the cross-link density of the PDMS,the diameter of the beads D, and the distance betweenthe beads H . Since the beads were fully embedded intothe PDMS and placed close to each other, there wasenough elastomer in the (short) transverse directionsto neglect any free surface effects on the failure mech-anisms (this was confirmed to be in fact the case bynumerical simulations). The selected geometry for thespecimens also allowed a full examination of the crackpropagation after very large overall deformations andbeyond the domain of interaction between the beads.Sketches of the geometry of the specimens used areshown in Fig. 1.

2.2 Specimen preparation

Two variants of the general specimen design discussedabove were made. The initial specimens (Design 1)were fully composed of PDMS and beads with themetallic end strips shown in Fig. 1. This basic designwas improved in later specimens (Design 2) by substi-tutingmost of thePDMSoneither side of the beadswiththe two extended metallic strips to which beads wereglued as indicated in Fig. 1; in this design, PDMS wasused only to fill the gap between the metallic grips andcover the beads. Thismodification offeredmany advan-

Microscope slab

First layer Second layer

Beads

Fig. 2 Mold arrangement for preparation of the PDMS speci-mens with beads

tages: no failure at grips, smaller overall stretch beforefailure, less PDMS waste, and easier beads clean-upand preparation process for repeatable testing. This sec-tion fully describes the preparation of Design 1 speci-mens and the relevant differences brought to Design 2.

Following themanufacturer’s recommendations, thePDMS base and curing agent were mixed in the ratioof 10:1 in a plastic cup, stirred for 15 min, and thendegassed for about 30 min. Meanwhile, the sphericalbeads were cleaned successively in alcohol and ace-tone, and then coated with a thin layer of Dow Corning92-023 primer. A specimen mold for Design 1, illus-trated in Fig. 2, was made with polymer sheets; a glassmicroscope slide was placed at the bottom in order toachieve the best transparency and surface finish; allsurfaces were coated with a release agent for ease ofremoval of the specimen. A thin layer of the degassedPDMS mixture was poured in the mold to form a firstlayer slightly thicker than the microscope slide. Themold was heated in an oven at 80 ◦C for a few min-utes (typically 10–15 min); this allows partial curingof the PDMS to a semi-solid and sticky state. In thefirst set of specimens, the beads were simply placed atthe desired locations. The beads spacing ranged fromH = 50–900µm. Note that the alignment and spacingof the beads were not controlled, but maintained by thesticky state of the PDMS at this stage. Consequently,the gaps between the beads varied from one specimento another, but weremeasured precisely prior to testing.Finally, more PDMSwas poured over the beads to forma second layer thick enough to cover the beads and pre-vent surface effects. Then, the mold was placed in theoven for final curing at 80 ◦C for about 12 hours beforedemolding and cutting to shape. Note that because ofthe positioning of the microscope slide in the mold,the specimen is thicker at the ends by about 1 mm andthis provides the enlarged region for griping in the testmachine. For Design 2, in the second set of specimens,the beads were embedded in the metal grip parts asshown in Fig. 1, but the distance between the beads

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was still not controlled. The PDMS was poured in onesingle step to build the desired thickness. In all cases,the gap was measured precisely prior to testing.

In our early trials, most of the specimens failed pre-maturely due to debonding at the interface betweenthe beads and the PDMS, despite the use of the rec-ommended Dow Corning 92-023 primer. Several beadmaterials were tested as well to investigate PDMS-beads adhesion properties but the results were incon-clusive; similarly, lightly roughening some beads withsand paper prior to cleaning and priming also did notimprove the adhesion. Since the adhesion properties ofthe interface between the beads and the commerciallygraded PDMS could not be readily improved further,we experimented with the PDMS compositions. Speci-mens were prepared following the procedure describedabove, with the only exception being in the use ofgreater content of the PDMS base. Mixtures of PDMSbase to crosslinking agent in the additional ratios of15:1, 20:1, 25:1, 30:1, and 45:1 were prepared. Thereadily observable increased softness and stickiness ofthese elastomer gradesmayhave resulted in bothmatrixweakening and relative interface strengthening for thePDMS specimens. A number of specimens with thesemixtures were prepared and used to examine the onsetof cavitation and cracking in the modified PDMS elas-tomers.

2.3 Experimental protocol

Prior to testing, high-resolution images of the speci-mens were taken with the help of a Keyence ModelVHX 5000 optical microscope. These images wereused to measure the diameters of the beads and thegaps between them precisely, as well as to monitor theimperfections both in the PDMS matrix (specifically,fabrication induced voids and specimen surface wavi-ness) and on the surfaces of the beads. These imagescorresponding to the undeformed specimens in theiroriginal configurations were critical since they wereused as the reference for both deformation measuresand for identification of cavity nucleation should thetransparency not be optimal. The specimenwas grippedat both ends on a testing machine (MTII Fullam SEMtest frame; see Fig. 3), and the SEM tester placed underthe Keyence Model VHX 5000 microscope. The spec-imen was loaded in tension at a quasi-static constantdisplacement rate of 5 mm/min. A low force capac-ity load-cell with a 50N range was used to measure

loads throughout the experiment but the measurementwas of little use since the force remained quite small(typically <1N)—consistent with the specimen smallcross section (<2 × 5mm2). For all the tests reportedhere, cavitation occurred before the maximum exten-sion limit of the SEM tester was reached. The opticalmicroscope allowed the experiments to be followed athigh magnification and to be fully video-recorded withhigh spatial and temporal resolutions: 1200×1600 pix-els, 15 frames/s, and 10-minmaximumduration.More-over, image focus andmagnification could bemanuallyadjusted. These features of the experimental arrange-ment enabled the capture of the nonlinear deforma-tion process, the first nucleation event, and subsequentdeformation and damage mechanisms leading to finalspecimen failure. At times, for better quality and accu-racy, the SEM tester speed was reduced from ∼5 to∼0.3mm/min. The deformation was observed from thetop view with illumination from the bottom; the viewobserved for the spherical beads would appear as in thesketches of the specimen in Fig. 1. Cavitation could beidentified readily from a local change in the image con-trast of the PDMSelastomer generated by the occlusionof light by the cavity.

3 Elastic constitutive model for PDMS

3.1 Uniaxial tensile response

In order to determine the elastic constitutive behaviorof the various PDMSelastomers—which is an essentialcomponent needed in the analysis presented below inSect. 6 of the above-described experiments—uniaxialtensile tests were performed on specimens with the fivecompositions 10:1, 20:1, 25:1; 30:1, and 45:1. Thesewere performed on rectangular strip specimens, witha self-tightening grip, at a quasi-static constant stretchrate of λ = 10−3 s−1. Digital Image Correlation (DIC)was used to determine the local and global stretchesin the gage section of the specimen. The global nom-inal stress vs stretch response is shown in Fig. 4a bythe discrete symbols for the different PDMS composi-tions. Two clear observations are that with increasingPDMS base to crosslinking agent ratio the initial stiff-ness decreases significantly. Perhaps even more impor-tantly, there is a large increase in the stretch level atwhich the material exhibits a stiffening response. Asexpected, the elastic response of any of these composi-

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Specimen

V

Fig. 3 MTII Fullam SEM test framewith the specimenmountedin the frame located in the view field of a Keyence model VHX5000 Optical microscope. In this case, the specimen has twocylindrical rods that are visible in the image on the monitor

tions cannot be characterized as simply Neo-Hookeanbeyond moderate levels of stretch.

4 Fitting of two constitutive models for the variousPDMS compositions

Wemake use of two different incompressiblemodels todescribe the elastic response of the various PDMS elas-tomers. The first one is a generalized Mooney–Rivlinmodel with polynomial stored-energy function

W (F)

=

⎧⎪⎨

⎪⎩

∑

p,q≥01≤p+q≤2

Cpq (I1 − 3)p (I2 − 3)q if det F = 1

+∞ otherwise

,

(1)

where I1 = Fi j Fi j , I2 = F−1i j F−1

i j stand for the stan-dard principal invariants associated with the deforma-tion gradient tensor F and C10,C01,C20,C11,C02 arematerial constants. A I1-based model introduced byLopez-Pamies (2010) is also used. In this model, thestored-energy function is given by

(a)

(b) (b)

Fig. 4 a Uniaxial nominal stress versus uniaxial stretch forPDMS elastomers with different compositions (the discrete sym-bols represent experimental data). Solid lines indicate the fit (seeTable 1a) to the polynomial Mooney–Rivlin model given by Eq.(1), while the dashed lines indicate the fit (see Table 1b) to theLopez-Pamies model given by Eq. (2). The green dashed line isthe estimated response for PDMS composition 15:1, for whichuniaxial tests were not performed. bComparison between the fitsof the model (2) and the model (4) featuring strong stiffening atlarge deformations (dotted line) to the PDMS composition 45:1.The biaxial response of the models (2) and (4) are also shown inthis figure as solid lines

W (F)

=

⎧⎪⎪⎨

⎪⎪⎩

31−α1

2α1μ1

(I α11 − 3α1

)

+ 31−α2

2α2μ2

(I α21 − 3α2

)if det F = 1

+∞ otherwise

, (2)

where μ1, α1, μ2, α2 are material constants.The material constants in both of these models

are obtained by fitting (via least-squares) the uniaxialexperimental data set in Fig. 4a; the results are givenin Table 1a, b for models (1) and (2), respectively. Fig-ure 4a shows the comparison between the experimen-

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Table 1 (a) PolynomialMooney–Rivlin materialmodel coefficients, (b)Lopez-Pamies materialmodel coefficients

PDMS composition C10 (MPa) C01 (MPa) C20 (MPa) C11 (MPa) C02 (MPa)

(a)

10:1 −2.11697 2.52582 3.46686 −10.3589 8.93568

20:1 −0.09389 0.20666 0.040315 −0.14497 0.19213

25:1 −0.00534 0.038971 0.008982 −0.03421 0.04603

30:1 −0.05564 0.092826 0.00556 −0.0245 0.051858

45:1 −0.038 0.053688 0.001425 −0.00698 0.02249

PDMS composition α1 α2 μ1 (MPa) μ2 (MPa)

(b)

10:1 0.03065 7.24432 0.40625 0.06804

15:1 0.22743 4.53944 0.26915 0.04406

20:1 0.47345 3.16889 0.16198 0.01495

25:1 1.04390 4.48409 0.05825 0.00032

30:1 −1.02103 1.39107 0.01857 0.03192

45:1 −1.10010 1.45673 0.00522 0.00955

tal measurements and the results from the model fit-ting process for both models. Clearly, both of thesemodels are capable of describing the material responseadequately, at least for uniaxial tension and up to thestretch levels presented in Fig. 4a. In this regard, wenote that the generalized Mooney–Rivlin model losesstrong ellipticity beyond certain range of deformationsfor someof thePDMScompositions.On the other hand,the Lopez-Pamies model remains strongly elliptic forall PDMS compositions. We also note that the stiffen-ing response has not been captured completely for the30:1 and 45:1 formulations of the PDMS and we haveto rely on extrapolation of the trend; we will return tothis (and Fig. 4b) in Sect. 6.2. It is also of interest to notethat the stiffening response is even more dramatic forthe case of equibiaxial tension, a state of stress that thecavitywalls aremore likely to experience; this responseis also shown in Fig. 4b.

5 Cavitation, micro-cracks, and macro-cracks inPDMS

We now turn to reporting the observations of nucle-ation and growth of cavities, micro-cracks, and macro-cracks in the experiments described in Sect. 2. Numer-ous experiments were performed on specimens withdifferent compositions of the PDMS elastomer anddifferent gap-to-diameter ratios H/D. In each experi-

ment, while the entire specimens (and hence the beadswithin the PDMS) were pulled apart at the fixed dis-placement rate of ∼ 5mm/min, the region between thetwo beads was monitored using video-capture in theKeyance microscope. Since the force measurementswere not of accurate resolution, we parameterized theloading state with the average stretch across the gapbetween the beads: λ

.= h/H , where h denotes thesize of the gap between the beads in the deformed con-figuration. Two typical movies (captured at 15 framesper second) are included as Supplementary MaterialM1 and M2. The first movie (Test A) correspondsto a PDMS composition of 15:1, an initial gap ofH = 340µm, and a bead diameter of D = 3.175mm,which hence corresponds to a gap-to-diameter ratioof H/D = 0.107. The second movie (Test B) cor-responds to a PDMS composition of 45:1 and the gap-to-diameter ratio H/D = 0.210. All the experimentsfor this system as well as other compositions and sizesexhibited similar response: stretching of the regionbetween the two beads, nucleation of a cavity closeto one of the beads, but within the PDMS, at a criticalpoint (identified here in terms of a critical value of theaverage stretch λ), followed by further nucleation ofadditional cavities in the vicinity of the bead-PDMSinterfaces, transitioning of some (or all) of the cavitiesto cracks, and eventual formation of a single dominantcrack. The details are discussed in the following threesections.

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Fig. 5 Selected sequenceof images from frames 440to 461 of Test A illustratingthe nucleation and growthof multiple cavities in thevicinity of both the left andright interfaces between thebeads (D = 3.178mm) andthe PDMS (15:1composition) for agap-to-diameter ratio ofH/D = 0.107. The redarrows indicate the firstthree cavities formed in thisspecimen

5.1 Onset of cavitation

Selected frames from the video sequence leading upto the nucleation of cavities from Test A are shownin Fig. 5 for a specimen with a PDMS compositionof 15:1, an initial gap of H = 340µm, and a gap-to-diameter ratio of H/D = 0.107. Frame numbers areused to identify individual frames extracted from thevideo. Since the video was recorded at 15 frames persecond, the interframe time interval is 66.7 ms. A cav-ity nucleates sometime between frames 440 and 441.This is indicated in the figure by a red arrow. As alreadyobserved by Oberth and Bruenner (1965) and later byGent and Park (1984) in similar types of experiments,the cavity is within the bulk PDMS and not at the inter-face with the bead. The distance between the center ofthe cavity and the interface is about 15µmin frame441.At this stage, this cavity is already 20µm in diameter,which provides a lower bound on the cavity diametergrowth speed of about 300µm/s (this corresponds toa local stretch rate of at least 100 s−1). The averagestretch over the gap between the two beads is approxi-mately λ = 1.7 when the first nucleation of a cavity isobserved. It is clear that this critical value depends bothon the material behavior and on the gap-to-diameterratio.Wewill examine such a dependence later throughnumerical simulations.

With continueddisplacement of the loadingmachine,the nucleated cavity appears to increase in size toabout 29µm in the next 0.7333 s (frame 451), result-ing in a cavity growth speed of only 12µm/s. Thisimplies the slowing down of the cavity expansion rate,in spite of the fact that the global cross-head displace-

ment rate is maintained constant, due to the compli-ance of the specimen. At this time, a second cavityis nucleated at the right interface and is identified byan arrow in Fig. 5 (frame 451); the stretch incrementbetween the first two nucleation events is almost atthe threshold of measurement resolution. This is fol-lowed by yet another nucleation in frame 453, againnear the interface to the bead at the right, at a dis-tance of about 140µm from the second cavity. Withcontinued loading additional cavities are nucleated inthe vicinity of the bead-PDMS interfaces, presumablybecause these cavities decrease the hydrostatic stressonly locally and other regions of high stress still per-sist near the interfaces, even after the nucleation of thefirst set of cavities. Figure 6 shows selected framesof such secondary nucleated cavities. In other exper-iments, either only one cavity was formed near one ofthe beads or one at each end with one of these cavi-ties eventually governing the final failure of the speci-men.

Results from a number of experiments with differ-ent PDMS compositions (15:1 to 45:1) and with gap-to-diameter ratios varying from H/D = 0.05 to 0.4are compiled in Fig. 7. The critical average stretch λ

between the beads at which the first cavity nucleationwas observed in the experiments is plotted as a func-tion of the gap-to-diameter ratio H/D for the differentmaterials. While there is a large scatter in these criticalstretch levels, there are two major discernible trends.First, the softer materials stretch greater before cavita-tion is first observed. Thus the critical stretch for thePDMS composition 45:1 is much greater than that forthe composition 15:1. Second, within the same mate-

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Damage in elastomers 9

Fig. 6 Selected sequence of images from frames 585 to 1235of Test A illustrating the growth of multiple cavities in thevicinity of both the left and right interfaces between the beads(D = 3.178mm) and the PDMS (15:1 composition) for a gap-

to-diameter ratio of H/D = 0.107. The shape of the cavities haschanged to a conical shape, with the tips of the cone approachingeach other

Fig. 7 Average stretch λ across the gap between the beads atonset of cavitation. Results are shown for different material com-positions as functions of the gap-to-diameter ratio H/D. Thesolid line represents the elastic cavitation criterion (3)

rial, the dependence on the gap-to-diameter ratio H/Dappears to be quite weak.

5.2 Transition to micro-cracks

With continued macroscopic loading, the nucleatedcavities increase in size or “grow”, but there appearsto be a remarkable change in their deformed geometry.Again, this changewas already observed byOberth andBruenner (1965) and later by Gent and Park (1984)in similar experiments, but neither identified the rea-son for such a change, with both groups of authors

apparently considering this to be similar to the shapeobserved when failure occurred by interfacial debond-ing.

Figure 6 shows a selected sequence of images fromframes 585 to 1235 of the same test considered inSect. 5.1. This sequence indicates that most of thenucleated cavities become cone- or tear-drop shaped.As elaborated below, these shapes in the deformedconfiguration correspond to penny-shaped cracks inthe undeformed configuration. We shall refer to suchcracks as “micro-cracks” since they form at the lengthscale of microns and may eventually lead to a “macro-crack” that breaks the specimen. The images in Fig. 6also indicate that the cavities do not grow all the wayinto the bead-PDMS interfaces. This is likely because,once a cavity is formed, the stretches in the two direc-tions in the plane parallel to the bead surface remainat unity due to adhesion with a very stiff medium, andconsequently the free surface of the cavity results in asharp drop of the normal stress at the interface; in otherwords, it is energetically costly to generate debondingalong the interface.

It is emphasized that the deformed micro-cracksin Fig. 6 are roughly axisymmetric and that what isobserved in the projected image is the meridional sec-tion of their three dimensional conical shape. In order tobetter understand the geometry of these micro-cracks,consider the thought experiment illustrated in Fig. 8.Within a compliantmediumoccupying the undeformedconfiguration�0, let a micro-crack be a straight line S0

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10 X. Poulain et al.

Crack tip

S+S-

S0

Fig. 8 Sketch showing the undeformed �0 and the deformed� configurations of a compliant medium that contains a stiffinclusion and an adjacent crack of initial length S0 embedded init. Upon remote stretching (indicated by the arrows), the crackdeforms asymmetrically because of its one-sided proximity tothe stiff inclusion, and opens to a conical shape

that is located at a small distance from a stiff inclusionembedded within the medium. Upon remote stretch-ing of the medium, the opening of the crack will beasymmetric about the crack line because of its one-sided proximity to the stiff inclusion. In particular, theleft surface S− of the crack (closer to the inclusion)will experience a small deformation, while the rightsurface S+ will open by a large amount, resulting inthe triangular shape shown in the deformed configura-tion�.Within the approximately 3D axisymmetric set-ting of the experiments, the complete geometry of thedeformed crack in the corresponding thought experi-ment of Fig. 8would be of conical shape,much like thatobserved in the experiments. We will confirm furtherbelow through numerical simulations that the observedconical shapes correspond indeed to the deformed pro-files of penny-shaped cracks.

Considering the conical geometry of the deformedmicro-cracks carefully, it should be apparent that theircrack tip is roughly along a circle in 3D; in the sketchshown in Fig. 8, it is at the points indicated by the reddots. Again, at this stage of the process, it is clear thatthe nucleated “cavity” is no longer an object that onlygrows elastically, but a crack, with a distinct crack frontand tip. Its expansion occurs through a crack growthprocess in which new surface is created. This processshould be governed not only by the elasticity of thematerial, but also by the fracture properties of themate-rial and by the geometry of the “cavity”. We concludethis section by emphasizing that the spatial and tem-poral resolutions of the present experiments are inad-equate to clearly demarcate the termination of cavitynucleation/growth and the transition to crack growth;

it is not clear when the cavity gives way to the crack,and this aspect remains to be explored.

5.3 Formation of a macro-crack

Selected frames from another representative test (TestB) are shown in Fig. 9 that trace the continued growth ofthe cavities/cracks. In this case, the PDMScompositionis 45:1 and the gap-to-diameter ratio is H/D = 0.210.The magnification in all the frames except the lastimage in this sequence corresponds to 1µm per pixel.It is recalled that displacements are controlled at thespecimen grips, and that due to the compliance of thespecimen (Design 1), the gap between the two beadsis only controlled indirectly. Therefore, in addition tothe frame number, the corresponding average stretch λ

between the beads is also identified in each frame.The sequence of events is as follows. At an average

stretch of about λ = 2.95 (frame 7575), the first cav-ity appears near the interface of the bead on the rightas indicated by a red arrow. Due to material, specimengeometry, and/or loading imperfections, this first cavitydoes not appear exactly on the line of symmetry, but itdoes appear very close to it. With additional displace-ment of the loading grips, this first nucleated cavitytransitions into a conical shape and increases in sizeuntil an average stretch of λ = 3.11 (frame 14,050) atwhich point a second cavity is nucleated near the oppo-site bead on the left (also indicated by a red arrow). Theseparation between the bead-PDMS interface and thesecond nucleated cavity is clearly discernible in thisimage. This second cavity already appears to exhibit aconical shape. Further loading results in the dominantgrowth of the second cavity between frames 14,100and 14,750, with the first cavity remaining unchanged.Beyond this point, the growth of the second cavityresults in a “shrinking” of the first cavity! Betweenframes 14,750 and 15,000, the distance from the apexto the base of the cone increases significantly for thesecond cavity, while it decreases for the first one. Fromframe 15,300 onwards, the global displacement wasdecreased in order to unload the specimen completely.At frame 18,000, it is quite clear that the first cavitynear the right bead has completely disappeared (wewill deal with such “healing” in a future contribution),while the second cavity appears to simply unload elas-tically. The last image shows a view of the completelyunloaded specimen with the two beads and a closed

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Damage in elastomers 11

Fig. 9 Selected sequenceof images from frames 4000to 20,000 of Test Billustrating the nucleation oftwo cavities/cracks, theirtransition to micro-cracks,and the further transition ofone of them to amacro-crack and closing ofthe other for a PDMScomposition of 45:1 andgap-to-diameter ratio ofH/D = 0.210. The lastimage shows the unloadedconfiguration

crack between them, which is located visibly closer tothe left bead. Following the sequence of growth of thesecond cavity, the shrinking of the first cavity, and theeventual unloading and closing of the second cavity, itis clear that these are not cavities, but cracks akin to thatillustrated in Fig. 8. The location of the crack tip of thesecond cavity is marked in frames 14,900, 14,950, and18,000 by a red arrow; again, since we observe only theprojection to a plane, the crack front is actually roughlya circle in 3D. Clearly, the apex of the cone is not thecrack tip!

To conclude this section, it is fitting to note thatin our earlier analysis (Lefèvre et al. 2015) of thetwo sets of sequences of four images of the analo-gous experiments reported by Gent and Park (1984)—Figs. 14 and 15 in their paper—we speculated thatthe crack that appeared in between the two beads wasoriginated from a cavity that was nucleated roughly atthe midpoint between the beads. The above-describedsequence of events captured at higher spatial and tem-poral resolutions has revealed that that is not thecase. Instead, again, it is a cavity that nucleates near

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12 X. Poulain et al.

Fig. 9 continued

a bead-PDMS interface the one that gives way tothe macro-crack that eventually forms in between thebeads.

6 Numerical simulations of the nucleation andgrowth of cavities and micro-cracks

In the sequel, we present a collection of full-field finite-element (FE) simulations of the experiments described

in the previous section with the objective of shed-ding light on the mechanisms that control the observednucleation of cavities, their growth, and their transitionto micro-cracks.

6.1 Simulations of the onset of cavitation

We begin by examining when and where cavitationensues in the experiment shown in Fig. 5 according tothe elastic cavitation criterion of Lopez-Pamies et al.

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Damage in elastomers 13

Fig. 10 FE simulation of the experiment shown in Fig. 5 for aPDMS composition of 15:1 and gap-to-diameter ratio H/D =0.107 at four values of the average stretch λ between the beads.The material points at which the onset of cavitation criterion(3) is progressively satisfied are depicted in red. The availablecorresponding images from the experiment for λ = 1.63 andλ = 1.72 are reproduced here for direct comparison

(2011b). In its more general interpretation, this crite-rion3 states that inside a rubber with initial shear modu-lusμ cavitation can first occur at material points wherethe principal Cauchy stresses ti (i = 1, 2, 3) satisfy thecondition

8t1t2t3 − 12μ (t1t2 + t2t3 + t3t1)

+ 18μ2 (t1 + t2 + t3) − 35μ3 = 0 with ti >3

2μ.

(3)

For convenience, given that free surface effects werechecked to be negligible in the experiment, in the sim-ulation the beads are considered to be located along thecenter of a circular cylindrical block with initial lengthL = 25.75mm and initial radius R = 3.175mm. Asmeasured at the outset of the loading, the diameterof the beads is set at D = 3.175mm, while the gap

3 We recall that the criterion (3) can be thought of (see Section 5in Lopez-Pamies et al. 2011a) as a generalization of the classicalelastic cavitation criterion of Gent and Lindley (1959a, b) for thecasewhen: (1) the rubber contains a random isotropic distributionof vacuous defects (as opposed to just a single spherical vacuousdefect) and (2) the rubber is subjected to arbitrary loading (asopposed to just purely hydrostatic loading).

between them is set at H = 340µm(H/D = 0.107).The PDMS elastomer of composition 15:1 is mod-eled as a nonlinear elastic material characterized bythe Lopez-Pamies stored-energy function (2) with thematerial parameters listed in Table 1b. Since the lin-early elasticmoduli of the glass beads are in the order ofseveral tens of GPa, their constitutive behavior is mod-eled as Neo-Hookean with an initial shear modulus of10 GPa (in the simulation, much like in the experiment,the beads then behave essentially as a rigid solid). Fur-ther, the beads are taken to be perfectly bonded to thePDMS. Due to the constitutive and geometric symme-tries of the problem, it suffices to perform the calcula-tions in just one half of a 2D-axisymmetric configura-tion of the specimen. The geometry of such a region isdiscretized with a structured mesh of a total of about30,000 eight-node hybrid quadratic elements (CAX8Hin ABAQUS) and six-node hybrid quadratic elements(CAX6H in ABAQUS), which proves refined enoughto generate accurate results; the interested reader isreferred to Section 4.1 in Lefèvre et al. (2015) for fur-ther details on the FE analysis.

Figure 10 shows the initial and the computeddeformed configurations of the region between thebeads at the four values of the average stretch betweenthe beads λ = 1, 1.31, 1.63, and 1.72; when avail-able, corresponding images obtained from the exper-iment are also displayed for direct comparison. Mate-rial points at which the cavitation criterion (3) is sat-isfied are depicted in red. Qualitatively, the first pointsto reach the cavitation criterion are those located atthe poles of the particles. The state of stress at thoselocations must be purely hydrostatic. The value ofthe hydrostatic stress happens also to be the largest(within the entire specimen) at those locations. How-ever, these regions are kinematically constrained suchthat all stretches are equal to one, and hence cavita-tion cannot occur at these locations. It is also inter-esting to note that the state of almost purely hydro-static stress near the bead-PDMS interfaces quicklytransitions into an axisymmetric one with significantshear stresses away from the beads; this is illustrated inFig. 11. As the loading increases, the region in whichthe cavitation criterion is satisfied propagates along thebead-PDMS interfaces and also towards the midpointbetween the beads, eventually filling the entire spacebetween the two beads. We should note that this is onlyan admissibility criterion; if a cavity were to developat some point, its further evolution will alter the stress

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14 X. Poulain et al.

(b)

(a)

Fig. 11 FE simulation of the experiment shown in Fig. 5 for aPDMS composition of 15:1 and gap-to-diameter ratio H/D =0.107. a The principal Cauchy stresses t1, t2, t3 along the lineof symmetry at two values of the average stretch λ between the

beads, plotted as functions of the distance (in the undeformedconfiguration) from the poles of the beads up to the midpointbetween them. b Contour plots of t1, t2, t3 in the region betweenthe beads at λ = 1.72

field significantly and hence alter the region in whichthe cavitation criterion is satisfied subsequently. Whiledirect comparison to the experiment indicates that thefirst nucleation of a cavity occurs in the vicinity of abead-PDMS interface, there are significant quantitativedifferences. In particular, according to the simulation,cavitation can first occur at an average stretch betweenthe beads of λ = 1.31. On the other hand, the firstobserved nucleation of a cavity occurs at an averagestretch of λ = 1.72 in the experiment. We next scruti-nize further this discrepancy between the experimentsand the general elastic cavitation criterion (3) of Lopez-Pamies et al. (2011b).

Analogous simulations to that described above wereperformed for all of the different tests that were car-ried out in order to identify the corresponding criti-cal values of the average stretch λ at onset of cavi-tation according to the elastic cavitation criterion (3).The results for the PDMS composition 25:1—again,

modeled as a nonlinear elastic material characterizedby the Lopez-Pamies stored-energy function (2) withthe material parameters listed in Table 1b—are plot-ted (dashed line) in Fig. 7 for direct comparison withthe experimental data (discrete symbols); the resultsfor the other PDMS compositions exhibit little dif-ference and thus, for clarity, are not included in thefigure. As expected from the derivation of the elasticcriterion (3), the theoretical results are seen to boundfrom below all of the experimental data. In particular,as the PDMS composition changes from 10:1 to 15:1,25:1, 30:1, and to 45:1, the theoretical results based onthe elastic criterion (3) are increasingly further awayfrom the experimental results. The reason for this poorquantitative agreement is to be found in the expecta-tion that the strong stiffening of the PDMS elastomersat large deformations must inhibit unrestrained cav-ity expansion, and even more critically, the fact thatthe PDMS elastomers must fail—that is, fracture—at

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Damage in elastomers 15

some large but finite deformation through nucleationand growth of cracks from thewalls of the cavity. Theseare two fundamental features that must be taken intoaccount in the theoretical description of the growth ofthe inherent defects that give rise to cavitation. Therelevance of these two features is explored in the nextsection.

Two additional features that may also play a signif-icant quantitative role are inertia and the viscous dissi-pation innate to the PDMS elastomers. In this regard,we note that the recent analysis by Kumar et al. (2016)of the radially symmetric dynamic growth of defectsin viscoelastic solids has indicated that, while inertiaeffectsmay be safely neglected, viscous dissipation cangreatly increase the critical values of the applied loadsat which defects grow. The extent of the relevance ofviscous dissipation in the nucleation of cavities in elas-tomers will be studied in future work.

6.2 Simulations of the growth of a cavity near abead-PDMS interface

In order to examine the influence of the stiffening atlarge deformations of the PDMS elastomers and thefact that these materials cannot sustain infinitely largeelastic (or otherwise) deformations before fracturingon unrestrained cavity expansion, we turn next to thesimulation of the experiment shown in Fig. 9 (for aPDMS composition of 45:1 and gap-to-diameter ratioof H/D = 0.210), now accounting explicitly for thepresence of a defect near a bead-PDMS interface.Morespecifically, given that the first observable cavity in theexperiment appears roughly along the line of symmetryabout 20µmaway from thepole of oneof the beads (seeframe 7575 in Fig. 9), in the simulation we explicitlyconsider the presence of an initially spherical vacuouscavity of 0.1µm radius at 20µm away from the pole ofone of the beads exactly along the line of symmetry.Weremark that the choice of an initially spherical cavity(as opposed to a non-spherical one) to model the defectin the simulation is consistent with the fact that thefabrication process is not expected to have introduceddefects with strong anisotropic shape in the specimen.Moreover, the choice of 0.1µm for the radius of thecavity is physically reasonable and consistent with thefact that no cavities were observed at the resolution of1µm per pixel in the experiment at the outset of theloading.

Introducing a spherical cavity along the line of sym-metry does not perturb the axisymmetry of the prob-lem. Consequently, similar to the preceding section,the relevant calculations can still be expediently carriedout over a 2D-axisymmetric configuration of a circularcylindrical specimen, in this case, with initial lengthL = 20.00mm and initial radius R = 2.295mm.The geometry of such regions is discretized with four-node hybrid linear elements (CAX4H in ABAQUS)and three-node hybrid linear elements (CAX3H inABAQUS), in a way that small elements are placedaround the cavity. In all, the mesh contains a totalof about 60,000 elements. The smallest finite elementdimension is 5 nm, considerably smaller than the radiusof the defect. Similar to the preceding section, thePDMScomposition of 45:1 ismodeled here as a nonlin-ear elastic material characterized by the Lopez-Pamiesstored-energy function (2)with thematerial parametersindicated in Table 1b. The beads are again modeled asNeo-Hookean solids with an initial shear modulus of10 GPa and are assumed to be perfectly bonded to thePDMS.

Figure 12a shows the initial and the computeddeformed configurations of the region between thebeads at the three values of the average stretch betweenthe beads λ = 1, 1.60, and1.83. Figure 12bpresents thecurrent volume of the cavity v, normalized by its initialvolume V = 4πa30/3 ≈ 4.2× 10−21 m3, as a functionof λ. This latter figure shows that the cavity does notgrow too significantly from λ = 1 up to approximatelyλ = 1.40, at which point (as approximately predictedby the criterion (3)) it starts to grow very rapidly withincreasing stretch. By λ = 1.83, the volume of the cav-ity is more than four orders of magnitude larger thanits initial volume. Throughout the entire loading, thecavity remains roughly spherical in shape. Its radius atλ = 1.83 is∼2.1µm. As illustrated by Fig. 12a, such asize would be observable in the experiment. So, why isit that the first cavity in the experiment is observed at themuch greater stretch of λ = 2.95 and not at λ = 1.83?We posit that the disagreement between the simulationand the experiment is because the constitutive assump-tions made about the behavior of the PDMS need to beaugmented to account for a more accurate descriptionof the stiffening of the PDMS at large deformationsand, more critically, for its fracture properties.

To support our conjecture, we begin by presentingin Fig. 13 the maximum and intermediate principalstretches λmax and λmid (from incompressibility, it fol-

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16 X. Poulain et al.

(b)

1.00=

1.60=

1.83=

(a) initial defect

m radius)

Fig. 12 FE simulation of the experiment shown in Fig. 9 for aPDMS composition of 45:1 and gap-to-diameter ratio H/D =0.210 containing a 0.1µm radius defect 20µm from the bead-PDMS interface. a The initial (λ = 1) and deformed config-urations of the region between the beads at two values of theaverage stretch λ between the beads. b Plot of the volume v ofthe defect (normalized by its initial volume V = 4π/3a30 ≈4.2 × 10−21 m3) as a function of λ

lows that the minimum principal stretch is given byλmin = (λmaxλmid)

−1) at all material points on the sur-face of the cavity for the three illustrative values of theaverage stretch λ = 1.20, 1.40, and 1.60. The resultsare shown as a function of the polar angle � in theundeformed configuration, with � = 0 correspond-ing to the material point on the cavity surface that isclosest to the bead (see the schematic in Fig. 13). For

all average stretches λ considered, the value of λmid

(dashed lines) is similar to the corresponding value ofλmax (solid lines) indicating a state of near equibiaxi-ality. Moreover, the average stretches of λ = 1.20 and1.40, the maximum stretch λmax is fairly homogeneousover the entire cavity surface and attains its highestvalue, λmax = 1.59 and 2.97, respectively, around theequatorial plane (� = π/2). On the other hand, forthe average stretch of λ = 1.60, the maximum stretchλmax exhibits significant spatial variation along the cav-ity surface and attains its highest value, λmax = 7.34,around the poles (� = 0 and π ). Now, the maximum(roughly) biaxial local stretch λmax = 7.34 encoun-tered at λ = 1.60 is admittedly high for a standardelastomer to achieve without fracturing and, moreover,it is exceedingly higher than the values of the stretchesin the uniaxial tension test for the PDMS composition45:1 (see Fig. 4b) to which the nonlinear elastic model(2) was fitted to obtain the material constants listed inTable 1b. Therefore, already at λ = 1.60—when thecavity would still not be visible in the experiment—the simulation may no longer be descriptive of theexperiment because the constitutive model may not bedescriptive of the actual PDMS elastomer response athigh stretch levels.

Indeed, it is possible that at large deformations,the actual PDMS elastomer stiffens more than whatis described by the model used in the simulation tothe point that the growth of the cavity is halted beforebecoming visible. In such a case, further loading ofthe specimen would lead to the buildup of stress, butnot deformation, on the surface of the cavity, whicheventually would develop a crack, as dictated by thefracture properties of the PDMS, and would in turngrow to finally become visible at some larger averagestretch λ. For demonstration purposes, we have carriedout exactly the same simulation described above of theexperiment in Fig. 9, but now taking the nonlinear elas-tic behavior of the PDMS elastomer to be characterizedby the three-termLopez-Pamies stored-energy function

W (F)

=

⎧⎪⎪⎨

⎪⎪⎩

31−α1

2α1μ1

(I α11 − 3α1

) + 31−α2

2α2μ2

(I α21 − 3α2

)

+ 31−α3

2α3μ3

(I α31 − 3α3

)if det F = 1

+∞ otherwise

(4)

with material parameters μ1 = 0.01254 MPa, α1 =0.58119, μ2 = 0.00171 MPa, α2 = 2.45122, μ3 =

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Damage in elastomers 17

bead

defect

Fig. 13 FE simulation of the experiment shown in Fig. 9 for aPDMS composition of 45:1 and gap-to-diameter ratio H/D =0.210 containing a 0.1µm radius defect 20µm from the bead-PDMS interface. Plot of themaximumand intermediate principal

stretches λmax (solid line) and λmid (dashed line) at all materialpoints on the surface of the defect for three values of the averagestretch λ between the beads. The results are shown in terms of thepolar angle � in the undeformed configuration (see schematic)

7.067 × 10−8 MPa, and α3 = 7.51219. As illus-trated by Fig. 4b, this model exhibits essentially iden-tical uniaxial stress–stretch response to that of themodel (2) with the material parameters for the PDMScomposition 45:1 listed in Table 1b up to the uniax-ial stretch at which we have experimental data. Forlarger stretches, however, themodel (4) with the above-indicated material parameters exhibits a much stifferresponse.

Figure 14a, b shows results analogous to thoseshown in Figs. 12 and 13, nowmaking use of the model(4) with strong stiffening at large deformations for thePDMS elastomer. As expected, the results are indis-tinguishable from those in Figs. 12 and 13 up to anaverage stretch of about λ = 1.40. For larger averagestretches, the strong stiffening of the model (4) kicks into limit the local stretches on the surface of the cavityand hence to drastically slow down the growth of thecavity. Rather than the unbounded growth predicted formodel (2), model (4) limits the cavity volume growthdrastically. We were able to compute a solution up toan average stretch of about λ = 2.20. At that point, thevolume of the cavity is only about 80 times larger thanits initial volume. Its radius is ∼ 0.4µm and hence itwould still not be observable in the experiment. More-over, at λ = 2.20, the highest maximum local stretchon the surface of the cavity is only about λmax = 4.27

and remains at � = π/2. These results strongly sug-gest that the nucleation and expansion of cavities in thePDMSelastomers are indeed controlled by the specificsof their stiffening response at large deformations andby their fracture properties, and thus encourage furtherresearch in that direction.

In relation to the above results, it is important to rec-ognize that the influence of the stiffening of the elas-tomer on cavity growth is already apparent from theidealized analysis of Ball (1982) for the radially sym-metric deformation of a single spherical cavity insidean isotropic incompressible elastic solid under hydro-static tension. Indeed, the pressure–stretch responsefor a radially deforming spherical shell with inner andouter radii (a0, b0) and (a, b) in the undeformed anddeformed configurations can be simply written as

P = λ2b

λa∫

λb

1

z3 − 1

dW

dz(z−2, z, z)dz, (5)

where P stands for the externally applied nomi-nal pressure, λa = a/a0 and λb = b/b0 =(1 + (λ3a − 1) a30/b

30

)1/3are the stretches at the inner

and outer surfaces of the shell, and W (λ1, λ2, λ3) =W (F) stands for the stored-energy function character-izing the response of the elastic material making up theshellwritten in terms of the singular valuesλ1, λ2, λ3 of

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18 X. Poulain et al.

(a)

(b)

Fig. 14 FE simulation of the experiment shown in Fig. 9makinguse of the modified material model (4) with strong stiffening atlarge stretches for the PDMS elastomer. a Plot of the volume v

of the defect (normalized by its initial volume V = 4π/3a30 ≈4.2×10−21 m3) as a function of the average stretch λ between thebeads. b Plot of the maximum stretch λmax at all material pointson the surface of the defect for four values of the average stretchλ. The results are shown in terms of the polar angle � in theundeformed configuration (see schematic in Fig. 13). The resultsbased on the material model (2) are included (dashed lines) fordirect comparison. The lock-up of the cavity expansion beyondλ ∼ 1.6 is readily identified

F. The critical pressure at onset of cavitation is obtainedby letting the initial radius of the cavity a0 → 0 andthe local stretch at the cavity wall λa → ∞. In sucha limit, as discussed by Ball (1982), the integral (5)converges to a finite value so long as the stored-energyfunction W satisfy the inequality W (F) < C ‖F‖3 as‖F‖ → ∞, where ‖F‖ = √

Fi jFi j = √I1. For the

Neo-Hookean model, this inequality is satisfied and(5) yields the well-known critical pressure for unstablecavitation: Pcr = 5μ/2. The Lopez-Pamies model (2)fitted to the uniaxial data for the PDMS compositions

10:1, 15:1, 20:1, 25:1 and the Lopez-Pamies model (4)fitted to the PDMS composition 45:1 do not satisfythe above inequality, and consequently do not exhibitunstable cavitation. However, the model (2) fitted tothe uniaxial data for the PDMS compositions 30:1 and45:1 does exhibit unstable cavity growth; this is evidentfrom the numerical simulations presented in Figs. 12and 13.

6.3 Simulations of the deformation of a micro-cracknear a bead-PDMS interface

As discussed in Sect. 5.2, the cavities that are nucleatedin the experiments appear to be roughly spherical atfirst, but then deform into tear-drop- or cone-head-likeshapes (see, e.g., Figs. 6, 9). In this section, we showthrough a direct simulation that the observed shapes areconsistent with the deformed shapes of penny-shapedcracks located near the bead-PDMS interfaces.

For definiteness, as in the preceding section, we con-sider the simulation of the experiment shown in Fig. 9for a PDMS composition of 45:1 and gap-to-diameterratio of H/D = 0.210. More specifically, guided byframe 7650 in Fig. 9, we account explicitly for the pres-ence of a penny-shaped crack of 150µm diameter thatis centered at the line of symmetry and 20µm awayfrom the pole of one of the beads. Since the problemstillpossesses axisymmetry, save for “replacing” the cavitywith the crack, all aspects of the FE analysis discussedin the preceding section remain unchanged. Consistentalso with the preceding section, the PDMS elastomeris modeled here as a nonlinear elastic material charac-terized by the three-term Lopez-Pamies stored-energyfunction (4) featuring strong stiffening at large defor-mations.

Figure 15 shows the initial and the computeddeformed configurations of the region between thebeads at the two values of the average stretch betweenthe beads λ = 1 and 2.92. The corresponding imagesobtained from the experiment are also displayed fordirect comparison. It is clear that the penny-shapedcrack opens roughly to the same profile observed inthe experiment. We emphasize again that, as indicatedby the arrows, the crack tip is not at the apex ofthe conical shape, but near the base. Growth of thecrack is expected to occur from this tip, allowing theapex to deform further and move towards the oppositebead.

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Damage in elastomers 19

1=

2.92=

8.0454.9534.4824.0553.6703.3203.0042.7182.4602.2262.0141.000

crack tip

initial penny-shaped crack (150 µm diameter)

Fig. 15 FE simulation of the experiment shown in Fig. 9 for aPDMS composition of 45:1 and gap-to-diameter ratio H/D =0.210 containing a penny-shaped micro-crack of 150µm diam-eter near a bead-PDMS interface. The initial (λ = 1) and thedeformed configuration at the average stretch λ = 2.92 of the

region between the beads compared to corresponding imagesfrom the experiment. A magnified view of the open crack withan overlay of λmax is shown in order to identify the location ofthe crack tips in this section

7 Summary and conclusions

Experiments have been conducted in the Gent and Park(1984) configuration of two stiff spherical beads ofdiameter D separated by a gap H and embedded ina PDMS matrix of different compositions (and hencemechanical behavior). The onset and growth of dam-age was monitored at high spatial and temporal reso-

lutions. Associated numerical simulations consideredthe elastic analysis of different scenarios that includeda pristine elastomer, a spherical vacuous cavity, and amicro-crack in the vicinity of the bead-PDMS inter-face to examine their potential evolution. These exper-iments and the associated numerical simulations pro-vide a complete physical picture—only partially under-

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20 X. Poulain et al.

stood until now—of how internal damage initiates andpropagates in elastomers.

Indeed, as the first event in a very distinctivesequence, the initiation of internal damage occursthrough the growth of submicron defects into cav-ities/cracks of micron size and roughly sphericaldeformed shape in regions of the elastomer that aresufficiently far away from constraints that may impedetheir large deformation and where the ratio of hydro-static stress to shear stresses is sufficiently high; in theexperiments, by design, these regions occur near thebead-PDMS interfaces. The growth of these defects iscontrolled by the specifics of the nonlinear elastic stiff-eningat large deformations of the underlying elastomerand by its fracture properties. In particular, the defectsdeform elastically up to the point at which the stiffen-ing of the elastomer puts a halt to further deformation.Simulations indicate that a pre-existing cavity mightgrow rapidly, but its growth is inhibited/limited by thestiffening response of the elastomer beyond a certainstretch level; this limitation to sub-micron size makesthese cavities invisible to optical imaging.

The increase of macroscopic loading leads to thebuildup of stress on the surface of the deformed defects,from which cracks may develop at some critical pointdictated by the fracture properties of the elastomer.Beyond that instance, the defects then grow to becomeof micron size. Therefore, as the second distinctiveevent, the nucleated cavities/cracks continue to growsubstantially in size to tens of microns and becomevisible optically. They do so, however, at a signifi-cantly lower rate than that observed during their nucle-ation. The most distinguishing aspect of the growth isthat the nucleated cavities/cracks transition in shapeto be penny-shaped cracks; we refer to these as micro-cracks. In the deformed configuration, these micro-cracks exhibit a distinctive tear-drop-like or conicalshape (due to their one-sided proximity to one of thestiff inclusions). Their growth is controlled by the frac-ture properties of the elastomer.

As the third and last event, one among the vari-ous nucleated micro-cracks—which continue to growat unison during further macroscopic loading—startsto dominate the growth and becomes a macro-crack,while the other micro-cracks stop growing and eventu-ally decrease in size and disappear (i.e., they seem to“heal”). Similar to the micro-cracks, the macro-crackis also of penny shape, but its location is on averagefurther away from the inclusions. The formation and

growth of the macro-crack is controlled as well by thefracture properties of the elastomer.

The above-outlined sequence of events and underly-ing mechanisms provide a clear physical picture of thenucleation and growth of damage in elastomers. Theybring resolution to the competing views of Gent andLindley (1959a, b) and that of Williams and Schapery(1965) that were discussed in the Introduction. Indeed,they indicate that the current lacuna in the quantitativemodeling of internal damage in elastomers and, in alllikelihood, other classes of soft materials can only befilled upon the development of a theory that views cav-itation as a fracture phenomenon—that is, as the com-petition between bulk mechanical energy and surfaceenergy associated with the creation of new surfaces—and is capable of accounting for the transitions fromcavities to micro-cracks, as well as for the posteriordominant growth of somemicro-cracks to formmacro-cracks and the unloading and/or healing of the remain-ing micro-cracks.

Acknowledgements This work was performed during thecourse of a collaborative investigation into cavitation, fractureand damage in soft materials funded by the National ScienceFoundation Grants CMMI-1235352 and CMMI-1235138. Thissupport is gratefully acknowledged.

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