Crack Blunting

10.1098/ rspa.2002.1057

Crack blunting and the strength of soft elastic solids By C.-Y. H u i1 , A. J ag o ta2 , S. J. B e n n is o n2 a n d J. D. L o n d o n o2 1 Department

of Theoretical and Applied Mechanics, Cornell University, Ithaca, NY 14853, USA ([email protected]) 2 CR&D, The DuPont Company, Experimental Station E356/317A, Wilmington, DE 19880-0356, USA ([email protected])

Received 10 May 2002; accepted 30 July 2002; published online 24 April 2003

When a material is so soft that the cohesive strength (or adhesive strength, in the case of interfacial fracture) exceeds the elastic modulus of the material, we show that a crack will blunt instead of propagating. Large-deformation ­ nite-element model (FEM) simulations of crack initiation, in which the debonding processes are quanti­ ed using a cohesive zone model, are used to support this hypothesis. An approximate analytic solution, which agrees well with the FEM simulation, gives additional insight into the blunting process. The consequence of this result on the strength of soft, rubbery materials is the main topic of this paper. We propose two mechanisms by which crack growth can occur in such blunted regions. We have also performed experiments on two di¬erent elastomers to demonstrate elastic blunting. In one system, we present some details on a void growth mechanism for ultimate failure, post-blunting. Finally, we demonstrate how crack blunting can shed light on some long-standing problems in the area of adhesion and fracture of elastomers. Keywords: blunting; fracture; adhesion; elastomer

1. Introduction The adhesive strength of interfaces and cohesion of solids (called interface fracture toughness or fracture toughness) is often characterized by the amount of energy Gc required to advance an interface crack (or a planar crack in a homogeneous solid) per unit area. For elastomers, Gc varies from 1 to 105 J m¡2 (Gent 1996). It has long been known that Gc depends on both the amount of inelastic deformation (e.g. viscoelasticity) and the separation process (Gent 1996; Gent & Schultz 1972; Andrew & Kinloch 1973; Gent & Lai 1994), although the quantitative coupling between these two mechanisms is still not well understood in soft materials. Several recent theoretical e¬orts have attempted to link microstructural failure mechanisms to the continuum ­ elds governing bulk deformations (Tvergaard & Hutchinson 1992; Xu & Needleman 1994; Rahul Kumar et al . 1999; Knauss 1993; Hui et al . 1992). One of the most powerful theoretical tools used to investigate the coupling between separation and bulk deformation processes is the cohesive zone model, which was ­ rst introduced by Barenblatt (1968) to study fracture. A cohesive zone model describes the interfacial normal and shear forces which resist separation and relative sliding of an interface. The forces, when integrated to complete separation, yield the fracture Proc. R. Soc. Lond. A (2003) 459, 1489{1516

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° c 2003 The Royal Society

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energy, Gc . For sti¬ materials such as metals and ceramics, the material modulus is usually far in excess of the peak cohesive stress or strength of the interface. For soft solids, e.g. elastomers and biological tissues, a physically based value for cohesive strength is often far greater than the modulus of the material. As a consequence, cracks in such materials undergo very large deformations before fracture. The question arises: is it possible that a material will be so soft that it will blunt elastically? Given that the cohesive or adhesive strength of materials and interfaces are so large in comparison with the material modulus, how do soft materials fail? Blunting of a crack in a homogeneous material or at an interface has a profound in®uence on the fracture process, usually resulting in much greater energy dissipation. While the onset of blunting is reasonably well understood in elastic{plastic materials (Tvergaard & Hutchinson 1992), conditions for its occurrence in elastic solids have not been clearly established.y That is, it has not been established under what conditions a crack tip would blunt, simply because the material is soft. We ­ nd that the answer to this question, which is the main subject of this paper, is related to several open and puzzling issues in the failure and adhesion of soft solids. Lake & Thomas (1967) established the relationship between molecular weight between entanglements and the fracture energy of soft elastomers, based on the assumption that all the energy between crosslinks is lost whenever any bond between the two is severed. However, as shown in the following sections, the accompanying cohesive zone is found to be vanishingly small, a paradox. A related unresolved issue emerges when one attempts to explain the rate dependence of peeling force using bulk viscoelastic data for the polymer. It is found that the dissipation region is far too small to account for the increase in dissipation (Gent 1996; Rahul Kumar et al . 2000). In peeling of polymers from a substrate (Gent & Petrich 1969), with increasing rate, the mode of failure changes from cohesive in the elastomer to adhesive at the interface. This is accompanied by a considerable reduction in fracture energy. A quantitative description of the condition that controls such a change in failure mode is lacking. The process of separating soft pressure-sensitive adhesives from a sti¬ substrate includes cavity nucleation (often at the interface), lateral growth of cracks, followed by vertical growth of cracks and stretching of ­ brils (Crosby et al . 2000; Creton & Lakrout 2000; Lakrout et al . 1999). The last process, which implies blunting of the laterally growing cracks, contributes a majority of the dissipated energy (Creton & Lakrout 2000). Our premise is that a crack in a soft material will typically blunt before failure can occur. The region ahead of the blunted crack can be considered as a cohesive zone since its size is small in comparison with typical specimen dimensions. The strength of this cohesive zone, according to our theory, is limited by the elastic modulus of the material. Failure must take place within this cohesive zone, e.g. by cavitation or by the growth of a micro-crack initiating from the original crack. A micro-crack can grow because the material in the blunted zone is highly stretched and therefore has much higher modulus than the material outside the blunted zone. We begin in x 2 with a discussion of some open problems and questions concerning the failure of soft materials. In x 3 we develop the condition for elastic blunting using y See, for example, Atkins & Mai (1988) for a qualitative description of elastic blunting: `In highly extensible materials . . . \cuts" become Inglis \ellipses". . . . relatively harmless notch. . . .’. Atkins & Mai also pose the unanswered question of why cutting of soft materials is often much easier than tearing by remote loading. Proc. R. Soc. Lond. A (2003)

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an approximate analysis of stresses near an ellipse under remote tension. That section also contains results of ­ nite-element computations that support the approximate results and provide a picture of deformation near the crack tip. In x 4 we discuss some mechanisms by which an elastically blunted crack would grow. Section 5 presents experiments on blunting and crack growth in two soft polymers that support the theoretical ­ ndings. For one, plasticized (poly)vinylbutyral, we present some details of the mechanism by which the blunted crack eventually propagates. We conclude in x 6 with further discussion of the implications of our results.

2. Some unresolved problems in the strength of soft solids In this section we discuss some unresolved issues in understanding the strength of soft solids. We begin with the model of Lake & Thomas (1967) for the dependence of elastomer fracture toughness Gc on entanglement molecular weight dependence. This model started with a consideration of the paradoxically high value of the intrinsic fracture toughness of rubbers, Gc ¹= Goc , obtained at low rates or high temperatures where viscoelastic dissipation is minimal. Based on the typical number of chains crossing a unit area of the fracture plane § o (§ o º 108 m¡2 ) and the energy U needed to break a single chemical bond (U º 400 kJ mol¡1 ), Goc should be only ca. 1 J m¡2 , in contrast with experimental values which range from 10 to 100 J m¡2 . Lake & Thomas (1967) resolved this discrepancy in fracture energy by noting that the polymer chains at and in the vicinity of the crack tip are highly stretched. When a bond breaks, the entire chain relaxes to zero load. It was assumed that all the stored elastic energy in the chain from the point of breakage to the nearest crosslink or entanglement is no longer released to the remaining body, but instead is dissipated. Thus, the energy dissipation is proportional to the number of bonds in a polymer chain. This explains why the actual fracture energy is much higher than the energy needed to break one single bond. In addition, since the stored energy in a chain is proportional to the number of bonds, Gc depends on the molecular weight between crosslinks. Implicit in the theory of Lake & Thomas is that there exists a region near the crack tip where the material behaviour is not described by linear elasticity. This idea has recently been explored by Ghatak et al . (2000) using a cohesive zone model. In their model, crack opening is resisted by the stretching of the chains bridging the crack faces. The resisting force f of these chains is directly proportional to the cracking opening displacement ¯ , i.e. f = ks ¯ ; (2.1) where ks º 10¡4 N m¡1 is the spring constant of the polymer chains bridging the crack faces. The energy required to propagate the crack per unit area, under steadystate conditions, is simply the amount of work needed to stretch the chains in a unit area (§ o º 1018 m¡2 ) from their initial state (zero length at the crack tip) to a ­ nal length ¯ m ax , i.e. Goc = 12 § o ks ¯ m2 ax : (2.2) Since 12 ks ¯ m2 ax is approximately equal to nU , where n is the number of bonds in pa chain and U is the energy needed to break the chemical bond, Goc = nU § o / n, which is the classical Lake{Thomas result. The last relation is due to § o / n¡1=2 . Proc. R. Soc. Lond. A (2003)

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So far, the prediction of the cohesive zone model seems to be quite satisfactory, at least from the energy perspective. Let us now estimate the length of the cohesive zone L. Since, outside the cohesive zone, the material is linearly elastic, the stress ­ eld directly ahead of the crack tip at distances large compared with L is K p I ; 2º x

(2.3)

where KI is the stress intensity factor. An estimate of the cohesive zone length, L, can be obtained by equating (2.3) to the characteristic chain fracture stress, ¼ f ² ks ¯ m ax § o . This gives KI2 L= : (2.4) 2º ¼ f2 For an incompressible material, the energy release rate G is related to the stress intensity factor by G = 3KI2 =4E. Using this relation and (2.4), the cohesive zone length L can be expressed in terms of the measured fracture toughness, i.e. L=

2EGoc E = 2 3º ¼ f 3º ks §

;

(2.5)

o

where E is Young’s modulus. For soft materials, using E º 106 N m¡2 , a simple calculation using (2.5) shows that L ¹= 1 nm. Thus, the characteristic length-scale of the zone surrounding the crack tip where energy is dissipated is of the order of the mean square end-to-end distance between crosslinks. This means that the stress and strain in this region must exceed all realistic values if they have to account for most of the energy loss. This apparent paradox is not con­ ned to polymer fracture at low rates. Indeed, this paradox was ­ rst stated in a pioneering paper by Gent & Lai (1994), who reported peel data on thin sheets of elastomers which were adhered together by C{C or S{S interfacial bonds. The goal of their experiments was to elucidate the relationship between the viscoelastic properties of the elastomer and the fracture toughness G of the bonded elastomers strengthened by the interfacial bonding. Their test conditions included a wide range of cracks speeds v (or peel rate), from 4 m m s¡1 to 4 mm s¡1 , and test temperatures, from ¡ 40 to 130 ¯ C. They found that the fracture toughness G increases from its low value Goc at very low crack speeds by several orders of magnitude as the crack speed increases. That viscoelastic losses control the increase in fracture toughness was demonstrated by the success of time{temperature superposition procedures to form log(G=G o ) versus log(vaT ) master curves, where aT is the Williams{Landel{Ferry (WLF) shift factor (Ferry 1980) measured separately. Indeed, these master curves were very similar in shape to curves of log[· 0 (!)=· 0 (0)] versus log(va T ), where · 0 (!) is the storage shear modulus at an angular frequency !. Gent & Lai (1994) estimated the e¬ective size of the dissipative zone by comparing the dependence of the measured fracture toughness with the dependence of · 0 (!) upon oscillation frequency. Speci­ cally, their procedure can be summarized as follows. Let v1 denote the crack speed corresponding to an increment ¢1 of log(G=G o ) in the log(G=G o ) versus log(vaT ) master curves. Let !1 denote the frequency corresponding to the same increment ¢1 of log[· 0 (!)=· 0 (0)] in the log[· 0 (!)=· 0 (0)] versus log(vaT ) curve. A characteristic distance d1 is de­ ned by d1 = v1 =!1 . In this way many values of d can be determined using di¬erent increments of log(G=Go ), and it Proc. R. Soc. Lond. A (2003)

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was argued that these d values represent the size of the dissipative zone near the crack tip. It should be noted that this procedure of estimating d is entirely consistent with the scaling arguments of de Gennes (1996), which are based on linear viscoelastic fracture mechanics. The e¬ective dissipative zone size using this procedure, however, ranges from 0.1 to 10 ¸ A. The size of these dissipative zones was recently computed by Rahul Kumar et al. (2000) using a ­ nite-element model. In these computation models, local failure of the interface is modelled using a rate-independent cohesive zone model. The size of the viscous dissipation zone was found to be of the order of 10¡11 m for peel velocities that showed signi­ cant increase in fracture toughness. Hence the paradox: how could such a small dissipation zone cause a major increase in viscoelastic dissipation? Indeed, theory and simulations of fracture in soft materials based on cohesive zone model often require the use of parameters that are di¯ cult to justify in terms of physical separation processes and length-scales. For example, consider a Johnson{Kendall{ Roberts (JKR) (Johnson et al. 1971) test where two identical poly(dimethylsiloxane) (PDMS) spheres are placed next to each other with no applied external load. It is found that the two spheres will jump into contact. The boundary of the contact region is a circle of radius a, which is proportional to the third power of the surface energy, ® . A detailed analysis of associated contact stresses was carried out by Maugis (1992) using a Dugdale{Barenblatt cohesive zone. In this model, the normal traction ¼ n acting on the PDMS surfaces just outside the contact zone depends only on the separation ¯ between the spheres. In addition, ¼ n vanishes if ¯ > ¯ c . However, if ¯ < ¯ c , then ¼ n = ¼ o , where ¼ o is a positive material constant that is assumed to be independent of ¯ . This simple model of surface interaction implies that the intrinsic fracture toughness is 2Gin = ¼ o ¯ c . Thus, ¯ c can be interpreted as a characteristic decay distance of the surface interaction force and ¼ o the strength of this interaction. The size of the region, d, where these surface forces act is estimated by Maugis (1992) to be º E® d= ; (2.6) 3¼ o2 where the material is assumed to be incompressible. A typical value for surface energy for an elastomer (e.g. PDMS (Chaudhury et al . 1996)) is ® ¹= 25 mJ m¡2 . According to Israelachvili (1992), ¼ o for van der Waals interaction is ca. 7 £ 108 N m¡2 . Since the modulus of PDMS is ca. 4 £ 105 N m¡2 , the size of the cohesive zone estimated using (2.6) is less than 10¡11 m. However, if we do not allow cohesive stress to exceed the modulus, ¼ o º E, then d is tens of micrometres. Another example is the viscoelastic sintering of acrylic beads on a rigid substrate (Lin et al . 2001). As in the case of PDMS, the surface interaction in this case is primarily of van der Waals type. However, the surface interaction strength required to match the experimental data of Mazur & Plazek (1994), ¼ o , was found to be 1:24 £ 106 N m¡2 { about two orders of magnitude lower than the strength of the van der Waals forces. The common source of these paradoxes, we propose, is the stress needed to fail an interface. Near the crack tip, the stress has the form given by (2.3). Crack initiation occurs when the p energy release rate equals the intrinsic fracture toughness Gin , that is, when KI º EGin , where we have ignored constants of order 1. For soft materials with low surface energy or work of adhesion, KI at crack initiation is therefore very small. On the other hand, the stress needed to fail the material or the interface between the material and a substrate is not small since the strength of materials Proc. R. Soc. Lond. A (2003)

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with low surface energy (such as those governed by van der Waals forces) is about two orders of magnitude greater than the elastic modulus. To break the interface at this stress, it is necessary to be extremely close to the crack tip (i.e. x º 10¡12 m). However, attendant blunting due to large deformations is likely to mitigate the ability of remote loading to increase crack tip stress, as explored in the next section.

3. Crack blunting in elastic materials While one may accept intuitively that a crack in a soft elastic material will blunt if the material has su¯ ciently high strength, it is not immediately clear how the condition controlling blunting can be derived. Neither linear elastic fracture mechanics (Lawn 1993) nor analyses of cracks with local large strains (Knowles & Sternberg 1983; Geubelle & Knauss 1994; Gao 1997) yield stress ­ elds that would predict blunting, except in a few degenerate cases. Blunting has been observed in computational analyses of crack propagation in hyperelastic materials where fracture was modelled using a cohesive zone (Jagota et al. 2000; Rahul Kumar et al . 1999) when the maximum cohesive stress and material modulus were of similar magnitude. However, if one poses the question of blunting using a cohesive zone model within the context of linear elasticity (Barenblatt 1968), one again ­ nds that cracks will always propagate at a critical remote loading. The foregoing observations and background point to the conjecture that crack blunting in soft elastic materials has essentially to do with large deformations, and occurs when the cohesive stress exceeds the elastic modulus. This conjecture is explored in this section ­ rstly by deriving an approximate condition for blunting in soft elastic solids, and secondly by numerical simulation of crack growth in an hyperelastic material. (a) Approximate condition for blunting In this section we explore the possibility that blunting in a soft elastic solid has essentially to do with large deformations. In the following, we analyse this conjecture with a simple approximate model, which yields a condition for blunting. Consider a pre-crack of length 2a in a large elastic body subjected to a remote biaxial tension. The interface directly ahead of the crack tip is characterized by its fracture toughness, Gc , and a maximum cohesive stress, ¼ o . This means that the normal stress directly ahead of the crack tip cannot exceed ¼ o . Once this condition is achieved, increase in remote loading will cause the crack to propagate, stably or unstably, depending on whether energy release rate decreases or increases with crack advance. If one disregards the e¬ect of large deformations, increasing the remote loading will always lead to crack growth, eventually. To investigate the e¬ect of ­ nite deformations, we model the deformed crack pro­ le by approximating it as a long, thin, ellipse. Further incremental loading of this ellipse results in a tendency for crack tip stresses to increase due to an increase in remote loading. However, this increase in crack tip stress is mitigated by an increasingly blunted shape. A comparison of the two tendencies yields the condition for blunting. Figure 1 shows that deformed crack modelled as an ellipse with major and minor semi-axes a and b (a ¾ b). Let the ellipse be subjected to a remotely applied biaxial Proc. R. Soc. Lond. A (2003)

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2 b

1

s0

a

Figure 1. Opening crack modelled as an ellipse. Cohesive traction just ahead of the crack tip is equal to or less than the maximum allowed, ¼ o .

tension, ¼ . The maximum tensile stress ¼ Goodier 1970)

m ax

occurs at x = a and is (Timoshenko &

2¼ a : b The radius of curvature, R, at x = a is found to be ¼

m ax

²¼

22;m ax

R=

=

b2 ; a

(3.1)

(3.2)

using (x=a)2 + (y=b)2 = 1. Using (3.2), the maximum tensile stress directly ahead of the ellipse is r a ¼ m ax = 2¼ : (3.3) R For an incremental increase in remote loading, d¼ , the change in stress at x = a is r p a a dR d¼ m ax = 2 d¼ ¡ ¼ d¼ : (3.4) R R 3=2 d¼

Equation (3.4) is an approximation since it assumes that the incremental constitutive response of the material is incompressible and linear elastic. Furthermore, it assumes that Young’s modulus, E, is independent of deformation. The solution to the problem of an ellipse subject to a remotely applied biaxial tension also furnishes a result for the displacement of the ellipse. Let y(x) denote the current deformed shape of the ellipse. The result in Timoshenko & Goodier (1970) (see also Appendix A) implies that near x = a y = yo (1 + ¬ d¼ );

(3.5)

where

2a : (3.6) E¤ b In (3.6), E ¤ = E if the deformation is carried out in plane stress (i.e. the crack is embedded in a large thin sheet of elastic material). If the deformation is carried out under plane strain conditions (i.e. the out-of-plane dimension of the specimen is much greater than the crack length), then E ¤ = 43 E. yo is the deformed shape before application of the incremental remote stress d¼ . Equation (3.6) implies that the rate of change of the radius of curvature R is ¬ =

dR = ¬ R: d¼

(3.7)

The blunting condition, d¼ m ax = 0, can be evaluated by combining (3.4) and (3.7). It is easy to verify that the blunting condition is satis­ ed when ¼ = 2=¬ . Since the Proc. R. Soc. Lond. A (2003)

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maximum tensile stress is related to the remote stress ¼ blunting occurs when ¼ o = 2; E¤

by (3.1) and ¼

m ax

= ¼

o,

(3.8)

that is, blunting is predicted when the maximum cohesive stress exceeds Young’s modulus by about a factor of two. The above condition is approximate since it does not take into account the strain hardening of the elastomer as it deforms. In general, the blunting condition would depend on the deformation behaviour of the material. We note that (3.8) applies also for a uniform tension applied in the y-direction as long as a ¾ b is satis­ ed. Some further insight into this condition for blunting can be extracted by examining the evolution of the maximum stress in this approximate model. The rate of change of the maximum stress as a function of the remote stress can be obtained by substituting (3.7) and (3.6) into (3.4). This results in r µ ¶ d¼ m ax a ¼ a =2 1¡ : (3.9) d¼ R E¤ b Using equations (3.1){(3.3) in (3.9) we get d¼

m ax

d¼

=

¼

m ax

¼

µ

1¡

¼

m ax 2E ¤

¶

:

(3.10)

Equation (3.10) can be readily integrated to give ¼

m ax E¤

=

2¼ =E ¤ ; ¼ =E ¤ + C

(3.11)

where C is a constant of integration. For large ¼ =E ¤ , ¼ m ax =E ¤ ! 2 as predicted by (3.8). The constant C can be determined by noting that, for small ¼ =E ¤ , (3.11) implies that ¼ m ax =E ¤ = 2¼ =C E ¤ . This, together with (2.6), implies that C = (b=a)o ;

(3.12)

where the subscript `o’ denotes the initial aspect ratio of the ellipse. This initial aspect ratio can be estimated by taking a to be the crack length, and b the critical crack opening displacement from the Dugdale{Barenblatt cohesive zone solution (Barenblatt 1968; Lawn 1993) (see also ­ gure 3). Ignoring constants of order 1, b¹

Gc ¼ o

)

C¹ ¼

Gc : oa

(3.13)

A typical value for C for elastomers, based on a = 10 mm, Gc = 50 J m¡2 and ¼ o = 105 N m¡2 would be 0.05. Figure 2 shows the maximum stress as a function of remotely applied stress for several values of C. These ­ gures show that the maximum stress increases rapidly with increasing applied stress. For small C (e.g. long cracks and low fracture toughness) and ¼ o =E ¤ < 2, crack growth occurs at applied stress substantially less than E. If ¼ o =E ¤ > 2, crack blunting due to large geometrical changes mitigates the intensi­ cation of remote stress by the crack. Proc. R. Soc. Lond. A (2003)

Crack blunting and the strength of soft elastic solids

maximum stress smax /E *

2.8

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s0 / E * > 2

2.4 2.0 1.6

s0 / E * < 2

1.2 1.8

C = 0.01 C = 0.05 C = 0.1

1.4 0

0.2

0.4 0.6 remote stress s /E *

0.8

1.0

Figure 2. Maximum stress as a function of remotely applied stress. If the cohesive stress is greater than twice Young’ s modulus, the crack blunts before it propagates. Otherwise, crack tip stress increases with remote stress until local stress equals the cohesive stress. After this point, local stress is limited by cohesive stress and the crack propagates. applied displacement: ‘K’ field

2 1

hyperelastic material

pre-crack

line of cohesive elements

Figure 3. Schematic of boundary conditions for simulation of a crack in a soft elastic material, with cohesive forces ahead of the crack tip holding it shut.

(b) Numerical simulation More realistic material behaviour and accurate change in geometry have been modelled by simulating the propagation of a pre-crack in a hyperelastic material using a commercial ­ nite-element code (ABAQUS® R ). This ­ nite-element code is augmented by cohesive elements that model the fracture process (Jagota et al . 2000; Rahul Kumar et al. 1999). Figure 3 shows the computational domain schematically. A semi-in­ nite plane-strain crack was simulated by applying the small-strain `KI ’ displacement ­ eld at boundary points remote from the crack tip. The material behaviour has been modelled using rate-independent hyperelastic constitutive equations. Let F be the deformation gradient and B the left Cauchy{ Green strain, de­ ned in terms of the position vectors x and X of material points in Proc. R. Soc. Lond. A (2003)

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the deformed and undeformed con­ gurations, respectively: F =

@x ; @X

I1 = Tr(B)

and

B = F ¢ F T:

(3.14)

De­ ne the invariants: I2 = (I12 ¡

Tr(B ¢ B)):

(3.15)

Then, the strain energy density © is represented as a polynomial series in the invariants (Ogden 1984): N X © = Cij (I1 ¡ 3)i (I2 ¡ 3)j : (3.16) i+ j= 1

A material de­ ned by the single constant C10 is termed neo-Hookean (Ogden 1984). A reduced polynomial, in which the elastic energy is a function of I1 only, has been proposed based on data on several elastomers (Yeoh 1993). It proves convenient here to study blunting using a two-parameter model that includes sti¬ening at large strains. That is, we consider n = 2, with corresponding constants C10 and C20 . For the case of uniaxial tension under plane stress conditions, let Lo and L be the initial and stretched lengths, respectively, and " and "l be the nominal and logarithmic (or true) strains 9 L ¶ = = 1 + " is the stretch;= Lo (3.17) ; "l = ln(1 + "): The strain energy density is given by

© = C10 (I1 ¡ 3) + C20 (I1 ¡ 2 I1 = ¶ 2 + ; ¶

9 3)2 ;= ;

(3.18)

and ½ and ¼ , the nominal and Cauchy (or true) stresses, are 9 @© @© > = ; > > > @¶ @" = 6C10 " 4C 20 2 3 4 5 1 2 ½ = (1 + " + 3 " ) + (15" + 18" + 8" + " );> > (1 + ")2 (1 + ")3 > > ; ¡"1 ¡"1 ¡"1 2"1 2"1 2"1 ¼ = 2C10 (e ¡ e ) + 4C20 (e ¡ e )(e + 2e ¡ 3): ½ =

(3.19)

The initial elastic modulus, to be identi­ ed with Young’s modulus E of previous sections, is E = 6C10 : (3.20) Figure 4 shows the Cauchy stress as a function of the logarithmic strain for di¬erent ratios of C20 =C10 . All show considerable increase in tangent modulus, Et = @¼ =@"l , with increasing strain. De­ ne the stretched region as that where the modulus increases over its initial value by a factor of 2. The points in ­ gure 4 show the strain at which this occurs for the di¬erent cases. The interface between the polymer layer and the sti¬ substrate has been modelled using a cohesive zone, implemented as cohesive elements that connect the surfaces. Proc. R. Soc. Lond. A (2003)

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C20 /C10 = 0 C20 /C10 = 0.25 C20 /C10 = 0.50

8

C20 /C10 = 0.75 C20 /C10 = 1.00 linear

stress s /E

6

Et /E = 2

4

2

0

0.2

0.4

0.6 0.8 strain ln (L /L 0 )

1.0

1.2

Figure 4. Cauchy stress as a function of logarithmic strain in hyperelastic materials. In this series, the circle represents the strain at which tangent modulus equals twice the initial, small-strain elastic modulus. We use this to de¯ne a stretched region near the crack tip.

Let T be the tractions across a cohesive zone, with opening and shearing components Tu and Tv , respectively. Let u and v be the jump in opening and sliding displacements across the cohesive zone. We assume that cohesive tractions can be derived from a potential, ª (u; v), as @ª @ª Tu = ; Tv = : (3.21) @u @v The following phenomenological form has been used for the potential, ª (u; v): ª (u; v) = Gin [1 ¡

(1 + u=¯ ¤ )e¡u=¯

¤

¡(v=¯

¤ 2

)

] + (1 ¡

H(u))Au3 :

(3.22)

where ¯ ¤ is a characteristic opening displacement, H(u) is the Heaviside step function, and A is a penalty parameter. The ­ rst term in (3.22) is a simpli­ ed version of the cohesive model of Xu & Needleman (1994). According to (3.21) and (3.22), cohesive tractions vanish for u; v ! 0 and u; v ! 1. For v = 0, the normal tractions increase with increasing u, reaching a maximum value of ¼

o

=

Gin e¯ ¤

(3.23)

and decrease thereafter. The second term in equation (3.22) is introduced to restrict mesh interpenetration under compression, and does not a¬ect the work of separating the interface. A series of simulations was conducted with ­ xed Gin but varying maximum cohesive stress, ¼ o . Remote boundary conditions were increased incrementally until the Proc. R. Soc. Lond. A (2003)

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C.-Y. Hui, A. Jagota, S. J. Bennison and J. D. Londono

s /E = 4/3 C20 /C10 = 0

cohesive traction Ty /E

s /E = 1 C20 /C10 = 0

cohesive traction Ty /E

0.50 0.46 0.42 0.38 0.33 0.29 0.25 0.21 0.17 0.13 0.08 0.04 0

0 -0.4 -0.8 -1.2 -1.6

cohesive traction Ty /E

s /E = 2/3 C20 /C10 = 0

0 -0.4 -0.8 -1.2 -1.6

0 -0.4 -0.8 -1.2 -1.6

cohesive traction Ty /E

s /E = 1/3 C20 /C10 = 0

0 -0.4 -0.8 -1.2 -1.6

Figure 5. Contours of maximum principal logarithmic strain near the crack tip as it begins to propagate for a neo-Hookean material. Fracture toughness is the same in all cases. Remotely applied boundary conditions are increased to just over the fracture toughness to initiate crack propagation. The four cases di® er in the level of cohesive stress. About when cohesive stress exceeds the modulus, the crack blunts. Note that there is little appearance of a stretched region before elastic blunting for this material. Also shown below each contour plot is the distribution of cohesive tractions. Proc. R. Soc. Lond. A (2003)

Crack blunting and the strength of soft elastic solids

1501

s /E = 5/3 C20 /C10 = 1.0

cohesive traction Ty /E

s /E = 4/3 C20 /C10 = 1.0

cohesive traction Ty /E

0.4575 0.1800 0.1650 0.1500 0.1350 0.1200 0.1050 0.0900 0.0750 0.0600 0.0450 0.0300 0.0150 0

0 -0.4 -0.8 -1.2 -1.6

cohesive traction Ty /E

s /E = 1 C20 /C10 = 1.0

-0.4 -0.8 -1.2 -1.6

0 -0.4 -0.8 -1.2 -1.6

cohesive traction Ty /E

s /E = 2/3 C20 /C10 = 1.0

-0.4 -0.8 -1.2 -1.6

Figure 6. Contours of maximum principal logarithmic strain near the crack tip as it begins to propagate for a `hardening’ hyperelastic material. (C 20 =C10 = 1). Fracture toughness is the same in all cases. Remotely applied boundary conditions are increased to just over the fracture toughness to initiate crack propagation. The four cases di® er in the level of cohesive stress. Note that before blunting a considerable stretched region (white) appears near the crack tip. Also shown below each contour plot is the distribution of cohesive tractions.

Proc. R. Soc. Lond. A (2003)

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C.-Y. Hui, A. Jagota, S. J. Bennison and J. D. Londono

Figure 7. Drawing of stretched region as calculated using an isotropic hyperelastic model (in white), compared with the expected Dugdale-like stretched region (uniform grey region) due to developing anisotropy near the crack tip.

applied energy release rate `G’ just exceeded the intrinsic value. Figure 5 shows the results for a neo-Hookean material. It contains contour plots of principal logarithmic strain at the end of the simulation. The range of contours is limited by the value of logarithmic strain at which tangent modulus (in uniaxial plane stress) exceeds the small-strain value by a factor of 2. For each case, we also show the distribution of cohesive tractions ahead of the crack tip. When ¼ o is signi­ cantly smaller than E, a cohesive zone develops from the pre-crack. When the applied stress intensity factor KI reaches its critical value, the crack begins to propagate. The cohesive zone size scales inversely with (¼ o )2 , as predicted by fracture mechanics. Increasing the maximum cohesive stress ¼ o decreases the cohesive zone size and large deformations near the crack tip are evident. At a critical value of ¼ o =E ¹ 1, the crack no longer propagates in the simulations but blunts instead. Note that there is little appearance of a stretched region before elastic blunting for this material. Figure 6 shows the results for a sti¬ening `Yeoh’ material with C20 =C10 = 1. Blunting is delayed to a larger ratio of ¼ o =E. Also, there is a considerable stretched region prior to blunting, shown by the white elliptical patch near the crack tip at higher cohesive stresses. It should be noted that the stretch regions in ­ gure 6 are oriented ca. 90¯ counterclockwise from the cohesive zone. Physically, we expect that these stretch regions should be oriented more or less in the same direction as the cohesive zone (­ gure 7). A possible explanation for this discrepancy is that the constitutive model for the elastomer has been assumed to be isotropic. This assumption is likely to be incorrect near the crack tip since the material there is subjected to a multi-axial state of stress and very large strains. Based on linear fracture mechanics, the cohesive zone scales as K I2 =¼ o2 ; the stretched zone scales as (KI =E)2 . The stretched region becomes much larger than the cohesive zone if E ½ ¼ o . For a neo-Hookean material, a prototype for a soft solid that does not sti¬en much, this does not occur before onset of blunting. For a sti¬ening elastic solid, a stretched region develops along with blunting as shown in ­ gure 6.

4. How do blunted cracks grow? Of course, the elastic blunting condition only captures the fact that the increase in remote loading now causes very large deformations, destroying the incremental increase in stress at the crack tip. Essentially, despite local sti¬ening, there is some Proc. R. Soc. Lond. A (2003)

Crack blunting and the strength of soft elastic solids

1503

region away from the crack tip that will not transmit stress much greater than the elastic modulus to the crack tip. We propose that when the elastic blunting condition is met, one of the following will occur. (1) The blunted region can be viewed as a cohesive zone with a modi­ ed cohesive strength ¼ c º E. It is likely that this modi­ ed cohesive strength varies from material to material. However, it is considerably lower than the actual cohesive strength or intrinsic strength of the material (e.g. ¼ o ). (2) Finite specimen size e¬ects may come into play. For example, if one is cutting with a sharp knife, the elastic blunting e¬ect may be limited by the fact that the crack opening displacement is determined by the knife geometry. For example, if the knife has radius of curvature Rkn ife that is smaller than the cohesive thickness or critical opening displacement, it will prevent blunting. Furthermore, a knife with a very small radius of curvature can give rise to very large stresses near its edge. This may o¬er an explanation for why cutting is energetically easier than peeling, even after accounting for intervening viscoelastic losses. (3) If no failure mechanism is available that operates at a stress of about the modulus, the material will eventually sti¬en enough to exceed the intrinsic strength and the crack will propagate. An important implicit assumption of our argument is that the behaviour of material in the blunted region or the modi­ ed cohesive zone is sharply di¬erent from the material outside the blunted zone. This means that all the stored elastic energy in the blunted region is lost during crack propagation. This will be the case, for example, if the material in the blunted zone is highly anisotropic so that it cannot support lateral stresses. This loss of lateral constraint can be due to cavitation or ­ brillation. Another possibility is that the material in the blunted region undergoes a phase transformation. Clearly, the validity of our hypothesis depends on the nonlinear elastic behaviour of the material at large strains. Thus, although this assumption is entirely consistent with the theory of Lake & Thomas, it clearly cannot be obeyed by all elastomers; counterexamples will be given in the discussion. The most general possibility, (1), explains why, when cohesive zone models are used to simulate experiments, agreement is possible if the cohesive strength is much less than that suggested by physically based mechanisms. Because the blunted zone can be treated as a cohesive zone, the fracture toughness Gc is simply Gc º ¼

c¯ c

º E¯ c ;

(4.1)

where ¯ c is the critical opening of the blunted crack at the onset of crack growth { so far an unknown quantity that must determined by experiment. For example, if the measured toughness of the material is 50 J m¡2 , then ¯ c º 10 m m for a material with E = 5 £ 106 N m¡2 . Since many soft materials do fail by crack growth, the blunted crack in these materials must eventually grow. The questions are as follows. (i) How does a blunted crack provide enough stresses to break bonds or to overcome the high strength of van der Waals forces? (ii) What is the connection between ¯ Proc. R. Soc. Lond. A (2003)

c

and the local failure mechanics?

1504

C.-Y. Hui, A. Jagota, S. J. Bennison and J. D. Londono y

d /2

uniform vertical displacement

x

Figure 8. Geometry of the micromechanics problem: the blunted region is modelled by an in¯nite strip of incompressible elastic material. The e® ective Young’ s modulus of the strip material can be several times that of E, the Young’ s modulus of the elastomer. The strip thickness is ¯ c , the opening of the blunted crack. The strip is loaded by applying a uniform vertical displacement on its boundary so that the normal stress is equal to ¼ c ¼ E at distances su± ciently far away from the crack tip. For the case of fracture by cavitation, the strip material is highly anisotropic.

There are at least two ways in which a blunted crack can grow. The ­ rst is by the nucleation and coalescence of cavities inside the blunted region. The criterion of cavity nucleation has been explored by Gent & Wang (1991), who showed that cavities can form in soft materials when the hydrostatic tension is of the order of the elastic modulus. This is precisely the stress state inside the blunted zone (at least for material points with distances greater than the radius of curvature of the blunted crack tip). The mechanics of cavity growth and failure will not be analysed in this work. However, we expect this process to be similar to the separation of a soft pressure-sensitive adhesive from a rigid substrate (Creton & Lakrout 2000). Typically, this process starts with cavity nucleation at the interface between the adhesive and the substrate, followed by vertical growth of these cavities. The material between these elongated cavities appears as ­ brils that eventually break down leading to the failure. In the following section we show some experiments where cavity growth is the main damage mechanism. The second way a blunted crack can grow is by the initiation and growth of a micro-crack near the blunted crack front. To see this, consider the schematic blunted region as shown in ­ gure 7. For materials with su¯ ciently large strain hardening, the e¬ective modulus of the material close to the blunted crack tip is much sti¬er than the material outside the blunted region as illustrated by our previous numerical results. This being the case, a micro-crack (or micro-cracks) can initiate from the blunted crack tip. Because of strain hardening, this micro-crack can propagate with little or no blunting. The spirit of the analysis below is similar to the theory of craze failure at a crack tip in glassy polymers (Brown 1991). There is, however, an important di¬erence between a craze and the blunted region. In the case of craze breakdown, most the energy dissipation occurs near the craze{bulk interface, where ­ bril drawing occurs. On the other hand, it is unlikely that such an interfacial region exists for elastomers, so that most of the energy loss is due to the elastic unloading of the material in blunted region, as proposed by Lake & Thomas (1967). Let us consider a scenario where we can work out the details of how such blunted cracks fail. Recall that ¯ c in (4.1) is an unknown quantity that, as will be shown below, is determined by the local failure process (e.g. growth and coalescence of cavities, breaking of C{C bonds in a homogeneous elastomer or the adhesive failure of an interface reinforced by van der Waals type surface forces). Since the length-scale of such local processes (say lf ) is much smaller than the length of the cohesive zone (or the blunted zone), we can approximate the cohesive zone as a semi-in­ nite strip of thickness ¯ c occupying the positive x-axis (see ­ gure 8). In this ­ gure, the unloaded Proc. R. Soc. Lond. A (2003)

Crack blunting and the strength of soft elastic solids

1505

pre-crack lies along x < 0 (see ­ gure 8). The micro-crack directly ahead of the precrack is assumed to be much shorter than the pre-crack. Since the chains inside the cohesive zone are highly stretched (especially those that are very close to crack tip), incremental deformations are small and small strain theory is applicable. To simplify analysis, we model this region as a linearly isotropic incompressible elastic material having an e¬ective Young’s elastic modulus, Ee¬ , which is likely to be several times E. The usage of isotropy is not necessary but it simpli­ es the calculations. The normal stress directly ahead of the micro-crack can be estimated using the standard fracture mechanics result Klocal ¼ local º p ; (4.2) 2º x where Klocal is the local stress intensity factor. Assuming plane stress deformation, Klocal can be estimated using p Klocal º Ee¬ (SE1 ); (4.3) where (SE)1 is the elastic strain energy per unit area stored at distances x ¾ ¯ the strip. (SE)1 is approximately given by (SE)1

¹= ¼

2 c¯ c

Ee¬

Therefore, Klocal º ¼

c

p

c

in

:

(4.4)

¯ c:

(4.5)

Equation (4.5) predicts that the local stress in the stretched region near the blunted crack tip is directly proportional to the square root of the blunted crack opening. It shows how crack blunting leads to crack propagation. Indeed, crack growth occurs when the local stress reaches the cohesive strength of the material, that is K p local ¹= ¼ o : (4.6) 2º lf Substituting (4.5) into (4.6) gives the relation p p ¼ c ¯ c = ¼ o 2º lf :

(4.7)

That is, the critical opening of the blunted crack obeys the following scaling law, µ ¶2 µ ¶2 ¯ c ¹ ¼ o ¼ o º ; (4.8) = lf ¼ c E

where we have ignored the numerical p factor of 2º . Another way of interpreting this expression is to multiply (4.7) by ¯ c on both sides, resulting in p p p ¼ c ¯ c = ¼ o 2º lf ¯ c = ¼ o lf 2º ¯ c =lf : (4.9)

The left-hand side of (4.9) can be interpreted as the applied critical energy release rate p (or fracture toughness Gc ). If we take lf to be the characteristic length such that 2º ¼ o lf is the intrinsic fracture toughness Gin , then fracture toughness scales with the intrinsic fracture toughness as r Gc ¯ c ¼ o = º : (4.10) Gin lf E In addition, the critical opening scales as ¯ c º (Gin =E)(¼ o =E). Proc. R. Soc. Lond. A (2003)

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C.-Y. Hui, A. Jagota, S. J. Bennison and J. D. Londono

The above argument can be modi­ ed to account for the case of fracture by cavity growth. Following Brown (1991), we assume that the presence of large cavities inside the cohesive zone results in a material that is highly anisotropic. Speci­ cally, the strip material is strong in tension and weak in shear and cannot support normal stress in the direction of parallel to the cohesive zone. Therefore, the cavitated material can be modelled as a homogeneous linearly orthotropic elastic material. As before, we shall assume the cohesive zone length is long in comparison with its thickness so that it can be approximated as a long thin strip. For an orthotropic material under plane stress deformation, the non-vanishing components of strain and stress are related by ¼

11

¼

22

¼

12

= C11 "11 + C12 "22 ; = C12 "11 + C22 "22 ; = 2C66 "12 ;

(4.11) (4.12) (4.13)

where the Cij are material constants. Our assumption is that the strip material is highly anisotropic so that C66 =C22 ½ 1, C11 =C66 ½ 1 and C12 º C66 . These conditions imply that the strip material is strong in tension and weak in shear (C12 º C66 , C66 =C22 ½ 1) and cannot support normal stress in the direction of parallel to the strip (C11 =C66 ½ 1). For an anisotropic material with this behaviour, the loading on the strip boundary y = §¯ c =2 can be approximated by a uniform normal displacement v = §¼ c ¯ c =2C22 . The normal stress directly ahead of the crack can be estimated using the fracture mechanics result (4.2). The local stress intensity factor, Klocal , can be estimated using a result from Sha et al. (1995) as p Klocal ¹ ¼ c ¬ ¯ c ; (4.14) p where ¬ ² C66 =C22 ½ 1. Equation (4.14) predicts that the local stress near the blunted crack tip is directly proportional to the square root of the blunted crack opening. It shows how crack blunting progresses into crack propagation when damage is by void growth ahead of the crack tip. Indeed, failure of the material occurs when K p local ¹= » ¼ 2º lf

o;

(4.15)

where (1 ¡

» ) is the volume fraction of voids. By (4.15), crack growth occurs when p ¼ c ¬ ¯ c ¹ p (4.16) = » ¼ o; 2º lf p Since the measured toughness Gc ¹= ¼ o ¯ c , we have, using 2º ¼ o lf = Gin , r µ ¶ ¯ c » ¹ p Gc = ¼ c ¯ c = Gin : (4.17) lf ¬ Comparing (4.17) with (4.10), we p note that the fracture toughness due to the void growth is much higher, since » = 2¬ ¾ 1. Equation (4.17) implies the following scaling relation between ¯ c and the intrinsic toughness, i.e. µ 2 ¶µ ¶µ ¶ µ 2 ¶µ ¶µ ¶ » Gin ¼ o » Gin ¼ o ¯ cº º ; (4.18) ¬ ¼ c ¼ c ¬ E E where we have neglected factors of order 1. Proc. R. Soc. Lond. A (2003)

Crack blunting and the strength of soft elastic solids

20 s

1507

60 s

Figure 9. Tensile loading of a crack in two transparent elastomers showing the di® erence between crack propagation and blunting. The experiments have been viewed using transmitted light through cross-polarizers that help visualize stress development. The material to the left is a photoelastic elastomer. With increasing loading, the stress near the crack tip increases monotonically. At a critical point, the crack starts to propagate. The material to the right is plasticized polyvinylbutyral (PVB), an elastomeric material used widely in glass{polymer laminates. It has similar modulus to the ¯rst material, but is tougher. Note that in the case of PVB the pre-crack blunts dramatically. Failure occurs by void formation and coalescence.

5. Experiments on crack blunting In this section we present experiments on crack growth in two soft elastomeric materials. They give a qualitative demonstration of the di¬erence between significant blunting and `normal’ crack growth. For the one that blunts signi­ cantly, we present further details on the void growth mechanism by which the crack eventually grows. Both samples are transparent elastomers with an internal pre-crack created by cutting using a knife. The experiments have been viewed using transmitted light through cross-polarizers to help visualize stress development. A series of Proc. R. Soc. Lond. A (2003)

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C.-Y. Hui, A. Jagota, S. J. Bennison and J. D. Londono

photographs showing stress development are shown in ­ gure 9. In all cases, the white ellipses denoting the cracks have been added to the photographs to indicate the crack pro­ le. The sample to the left is a photoelastic elastomery. With increasing load, the stress near the crack tip increases monotonically. At a critical load, the crack starts to propagate. Presumably, in this case the cohesive stress is small compared with the modulus for such crack propagation, or the material strain hardens considerably so that a micro-crack can grow, as discussed above. The material to the right is plasticized polyvinylbutyralz (PVB), an elastomeric material used widely in glass{polymer laminates. It has a similar modulus to the ­ rst material, but is tougher, meaning higher cohesive stress or less strain hardening. The time interval between pictures on the left is 20 s, and between those on the right is 60 s. Note that in the case of PVB, the pre-crack blunts dramatically. Of course, PVB does eventually fail, but it does so only when a new failure mechanism, operating at a stress of the order of the modulus, takes over. Figure 10 shows an optical micrograph of a propagating crack in PVB. Note the large deformation of the crack ®anks. The whitening ahead of the crack tip, relatively uniform across the intact section, is due to scattering from strain-generated voids. It is interesting to note the clarity of the polymer behind the crack tip, in a region that was voided and whitened when ahead of the crack tip. As mentioned below, we found that void growth in this material is strain dependent, and vanishes upon its removal. Direct evidence of a void growth mechanism in PVB has been obtained by measuring small-angle scattering of X-rays during tensile stretching. Figure 11 shows the measured tensile stress{strain behaviour in plane stress. Details of these measurements are given in Appendix B. The material is nearly incompressible. At these strain rates it shows rate-dependent rubber-like elastic behaviour. Failure strains appear to be approximately constant at a value of about 1.2. Using small-angle Xray scattering (see Appendix B for details), we found that voids nucleate well before failure. Figure 12 shows scattered X-ray intensities at two di¬erent angle ranges. The vertical orientation is the tensile direction. The meridional scattering to the right represents ellipsoidal shaped voids, up to hundreds of nanometres in size. The equatorial scattering to the left represents ­ bril features less than 10 nm in diameter, presumably bridging the larger microvoids. More detailed analyses of these data will be presented elsewhere. Here, approximately, by assuming that the voids are spherical in shape and polydisperse in size, we determine from the scattering data how void size changes with strain (Appendix B). These results are summarized in ­ gure 13. In the context of the crack blunting results developed in this paper, the study of damage development in PVB shows how void formation emerges as a damage and failure mechanism when elastic crack blunting does not permit simple fracture. This leads to the emergence of a cohesive zone that has larger openings and is softer in the sense that it has much lower peak stress. However, the increase in separation distances means that it is usually accompanied by an increase in fracture toughness of the material, as suggested by (4.17). y Photoelastic sheet PS-4C, 1 mm nominal thickness; Measurements Group, Raleigh, NC 27611, USA (www.measurementsgroup.com). z Butacite ® R , DuPont Company. Proc. R. Soc. Lond. A (2003)

Crack blunting and the strength of soft elastic solids

1509

1 mm

Figure 10. Void growth ahead of blunted crack tip in Butacite® R . Note the formation of a cusp at the crack tip due to hardening of the material at large strains. Void damage appears to be strain controlled and independent of rate as measured by SAXS experiments. Voids collapse upon unloading, as can be seen in the unloaded crack ° anks.

120

0.01 mm s-1 0.10 mm s-1

true tensile stress (MPa)

100

1.00 mm s-1 10.0 mm s-1

80 60 40 20

0

0.2

0.4 0.6 0.8 true tensile strain

1.0

1.2

Figure 11. True stress{strain data for plasticized polyvinylbutyral from plane stress tensile tests. The material is nearly incompressible. At these strain rates it shows rate-dependent rubber-like elastic behaviour. Failure strains appear to be approximately constant at a value of about 1.2. Extensional and lateral strains were measured by following ¯ducial marks on the specimen, and were used to determine strain in the gauge section and the true Cauchy stress.

Proc. R. Soc. Lond. A (2003)

1510

C.-Y. Hui, A. Jagota, S. J. Bennison and J. D. Londono SDD = 8500 mm

strain

SDD = 727 mm

log(I ) –0.4

0

0.4 –0.03 q = 4p sin q /l

0

0.03

Figure 12. Scattering intensities of stretched PVB indicating structure at two length-scales. The equatorial scattering to the left represents scattering from ¯brils a few nanometres in dimension aligned with the tensile strain. The meridional scattering shown on the right represents oblong defects 100s of nanometres in size with the long side oriented normal to the tensile axis.

6. Discussion and conclusions We have argued that in soft materials crack blunting due to large deformations often prevents crack tip stresses from reaching a magnitude su¯ cient to decohere the material or interface. Deprived of this direct mode of propagation, failure can now proceed only by another mechanism, for example, void growth or extension in a new stretched region ahead of the crack tip. The blunting condition, approximately that the cohesive strength exceeds the elastic modulus, results in this changed failure mode, which is often accompanied by an increased fracture energy. An example is the transition from cohesive to adhesive failure, which was observed by Gent & Petrich (1969). They found that the toughness of an adhesive bonded to a rigid substrate ­ rst increases rapidly with crack growth rate a, _ reaches a maximum at a_ m ax , then decreases rapidly to a very small fraction of its maximum value. Further increase in crack growth rate results in much smaller peak in toughness. Gent & Petrich noticed that cohesive failure of the adhesive occurs when a_ 6 a_ m ax , whereas interfacial or adhesive failure occurs when a_ > a_ m ax . While it is expected that viscoelastic dissipation is responsible for the large increase in toughness, it is not clear what causes the transition from cohesive to adhesive failure. We propose that this transition is due to crack blunting. At low rates of crack growth, the material near the crack tip is fully relaxed, so that the elastic modulus is approximately Eo , the rubbery modulus or the modulus at zero rate. Since this modulus is ca. 106 Pa, crack blunting is likely. With increasing rate of crack growth the material near the crack tip is loaded at a much higher rate, so that the deformation there is governed by the instantaneous modulus E1 . Since E1 can be several orders of magnitude greater than Eo , it will eventually be su¯ cient to allow the crack to decohere the interface without bluntProc. R. Soc. Lond. A (2003)

140

75

120

50

100

25

80

true stress (MPa)

100

0

0.4 0.8 true logarithmic strain

1.2

1511

effective void diameter (nm)

Crack blunting and the strength of soft elastic solids

60

Figure 13. Evolution of e® ective polydisperse spherical void size with strain.

ing. Since adhesive failure is governed by van der Waals type forces, the adhesive toughness is much less than that of cohesive failure. It is interesting to note that (4.10) or (4.17) predict that the fracture toughness should scale with the work of adhesion. Furthermore, the proportional factor ¼ o =E, a large number, depends only on the elastic constitutive behaviour of the material. This equation is the rate-independent version of the well-known empirical relation Gc = Wad (1 + ¿ (aT a)), _ where aT is the Williams{Landel{Ferry (WLF) shift factor, Wad is the work of adhesion and ¿ is a dimensionless function representing the contribution of crack tip inelastic deformation to the energy dissipation (Gent 1996). Furthermore, (4.10) implies that 1 + ¿ (aT a_ ! 0) ! ¼ o =E for materials capable of large amounts of strain hardening. Since the analysis in this work assumes that the material is elastic, dissipation due to bulk viscoelasticity is not explicitly accounted for. Extension of the results in this work to include viscoelastic material behaviour will be done in a future work. However, we believe the empirical relation Gc = W ad (1 + ¿ (aT a)) _ can be derived from a theoretical model that couples crack blunting and viscoelasticity. Blunting of cracks in a soft elastic material or at the interface between one and a rigid substrate leads to modes of separation that involve signi­ cant increases in energy dissipation. Without crack blunting, it is di¯ cult to rationalize how soft materials can fail since a linear analysis would indicate that the region of energy dissipation has dimensions less than 1 ¸ A. In an elastic material, blunting appears to be contingent upon large changes in geometry, resulting from the modulus being smaller than maximum interfacial cohesive stress. The simple approximate model, and numerical simulations using a neo-Hookean constitutive description are in reasonably good agreement. For elastic materials, most of the energy release to the crack tip is dissipated by the blunting process, only a very small fraction of this energy goes into the actual failure process. Therefore, our analysis is consistent with the theory of Lake & Thomas (1967), that is, most of the elastic stored energy in the blunted region is lost during unloading. Of course, when the elastomer fails by cavity growth, some of the stored energy is used to create new surfaces. For materials capable of large amounts of strain hardening, the measured fracture toughness scales with the Proc. R. Soc. Lond. A (2003)

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C.-Y. Hui, A. Jagota, S. J. Bennison and J. D. Londono

work of adhesion, that is, Gc º (¼ o =Ee¬ )Wad , where E is the small strain elastic modulus and ¼ o is the intrinsic cohesive strength of the material. Thus, crack blunting explains qualitatively the paradox presented by Gent & Lai that there is signi­ cant viscoelastic dissipation even when the dissipative region is only of the order of angstroms in size. That is, crack blunting introduces an additional length-scale that is not in Gent & Lai’s analysis. Indeed, one anticipates that the size of the blunted zone is at least of the order of micrometres for soft materials. Therefore, it is likely that the amount of viscoelastic dissipation is controlled by the size of the blunted zone. It is possible that the length-scale introduced by Gent & Lai is a measure of the size of the actual chain breaking zone at the crack tip. Exactly how a blunted zone interacts with a local failure mechanism when viscoelastic deformation is important is the blunted region is still unclear and is the subject of further research. Finally, we pointed out that some of the scaling relations we have proposed in this work depend on the implicit assumption that all the stored elastic energy in the blunted zone is lost during crack growth. While this assumption is consistent with the theory of Lake & Thomas, it clearly is not valid for all elastomers. In other words, a crack can blunt with smooth changes in material behaviour everywhere. If this were the case, then the stored elastic energy of the material in the blunted region will not be dissipated and the fracture toughness of the material will be close to the intrinsic work of adhesion. This is indeed the case for the fracture of low-modulus gels, which are obviously in the crack blunting regime, yet the adhesion is essentially reversible. The overall energy required to peel these materials from a surface is determined by the thermodynamic work of adhesion (Mowery et al. 1997). We acknowledge the DND-CAT facility, which is supported through E. I. DuPont de Nemours & Company, Northwestern University, the Dow Chemical Company, the State of Illinois through the Department of Commerce and the Board of Higher Education (HECA), the US Department of Energy O± ce of Energy Research, and the US National Science Foundation Division of Materials Research. C.Y.H. acknowledges support from E. I. DuPont de Nemours & Company through a gift to Cornell University. The authors are grateful to C. Creton, E. J. Kramer and K. Shull for valuable suggestions and informative discussions.

Appendix A. Let x and y denote Cartesian coordinates and z be de­ ned as the complex number z = X + iY:

(A 1)

De­ ne elliptic coordinates ¹ and ² by z = c cosh ± ;

± = ¹ + i² :

(A 2)

Lines of constant ¹ are ellipses, and for ¹ = ¹ o , a = c cosh ¹ o ;

b = c cosh ¹ o ;

a2 ¡

b2 = c2 :

(A 3)

The solution we seek, the displacement ­ eld, can be calculated in terms of complex potentials Á(z) and À (z). If u(x; y) and v(x; y) denote the two Cartesian components of displacement, then, for incompressible plane strain, 2G(u + iv) = ª (z) ¡ Proc. R. Soc. Lond. A (2003)

z ª · 0 (· z) ¡

À ·0 (· z );

(A 4)

Crack blunting and the strength of soft elastic solids

1513

where z· denotes the complex conjugate of z, whether of a variable or a function. For an elliptical hole the complex potentials are ) ª (z) = 14 ¼ c[¡ e2¹ o cosh ± + (1 ¡ e2¹ o cos 2(º =2 ¡ ² ))]; (A 5) À (z) = ¡ 14 ¼ c2 [(cosh 2¹ o + 1)± + 12 e2¹ o cosh 2(± ¡ ¹ o ¡ iº =2)]: This results in 2Gv =

3¡ ¸ ¼ a y¡ 1+¸ 2 b

Near the tip of the ellipse, y ¹

¼ (x2 + y2 )ay : 2b((bx=a)2 + (ay=b)2 )

(A 6)

0, this simpli­ es to 3¡ ¸ ¼ a yo ¡ 1+¸ 2 b

a yo ; 2b

(A 7)

y = yo + v = yo (1 + ¬ d¼ );

(A 8)

2Gv ¹ where yo is the initial shape and

which is the result (3.5). The radius of curvature, 1 y (1 + ¬ d¼ )yo = = ; 02 3=2 R (1 + y ) (1 + (1 + ¬ d¼ )2 y22 )3=2

(A 9)

which implies (3.7).

Appendix B. Extensional and lateral strains were measured by following ­ ducial marks on the specimen, and were used to determine strain in the gauge section and the true Cauchy stress. Tensile test specimens, 13 £ 75 mm2 , were prepared from nominally 0.75 mm (30 mil) PVB using a die-punching method. Gripping tabs of PVB itself were cut from the same ­ lm and attached to the specimen by auto-adhesion. Five reference marks were placed by hand on the specimen using a ­ ne-point marker pen. The specimens were conditioned (overnight) before testing by placing in a humidity chamber at 22 § 0:5% relative humidity at 22 ¯ C. Tensile testing was performed using a universal closed-loop control servo-hydraulic testing frame (MTS Corporation) in constant displacement control. The specimens were gripped using hydraulic wedge grips with a SurfAlloy® R ­ nish at a gripping pressure of 5 MPa. A range of displacement rates from 10¡2 to 10 mm s¡1 were studied yielding a range of strain rates from 2:3 £ 10¡4 to 2:3 £ 10¡1 s¡1 for the 25 mm specimen gauge section employed. The relative motions of the markers during loading were monitored by a video camera (30 frames per second) at a magni­ cation of approximately 5£. A video overlay device was used to simultaneously display elapsed time and a voltage corresponding to the applied load. A time-lapse movie of the deformation was made by recording images at speci­ ed time intervals using a digital frame grabber (SCION Corp.) housed in a Macintosh personal computer (Mac IIfx). The movie was analysed, frame by frame, using an image-analysis package (NIH Image). The axial and longitudinal strains were then determined from the relative displacements (motions) of the markers. Small-angle X-ray measurements were performed at the DND-CAT Synchrotron Research Center, Advanced Photon Source, Argonne National Laboratory. The Proc. R. Soc. Lond. A (2003)

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C.-Y. Hui, A. Jagota, S. J. Bennison and J. D. Londono

energy of the X-ray beam from an insertion device (ID) was tunable from 7 to 18 keV. The ID, the double-crystal monochromator, the ­ rst, second and third sets of adjustable slits and the sample were located at 0, 30, 35, 54, 66 and 68 m, respectively, along the X-ray beam path from the synchrotron orbit. The size of the square beam was de­ ned at the ­ rst and second sets of slits, which were both set to 100 m m. A parasitic scattering slit, having the shape of a round pinhole only large enough to circumscribe the square beam, was placed 1{2 mm in front of the sample. The two-dimensional CCD (Mar) detector had 2048 £ 2048 pixels with a 16-bit intensity scale, and a circular active area of 133 mm diameter. In all cases the detector was used in a 4 £ 4 binning mode at a resolution of 512 £ 512, with e¬ective pixel size of 26 m m per pixel. The detector was placed at the end of an evacuated 8 inch pipe ­ tted with Kapton® R windows on both ends. The sample-to-detector distance (SDD) was adjustable from a few centimetres to 8.5 m. PVB samples were stretched using the custom mechanical loadframe (Instron) installed at DND-CAT. Figure 12 shows the observed scattering at two di¬erent angles, showing that the voids have structure in two distinct length-scale ranges. These were studied in separate experiments by choosing detector distance to be 8.5 m for meridional scattering and 0.72 m for equatorial scattering. For meridional scattering, the ID was tuned to an energy of 7.95 keV, corresponding to a wavelength ¶ = 1:560 ¸ A. At this distance the detector covered scattering angles 2³ corresponding to the range in the scattering vector magnitude 8 £ 10¡4 < q (= 4º sin ³ =¶ ) < 3 £ 10¡2 ¸ A¡1 . This regime of scattering angles is termed the ultra-SAXS or USAXS range, which is achieved with di¯ culty in the laboratory. Samples were stretched at the rate of 1 mm s¡1 while load versus strain information in addition to scattering data were collected simultaneously. The bottom actuator moved at an equal rate but in the opposite direction to that of the top actuator. This mode of operation enabled the central portion of the tensile bar to remain in the beam while the sample was stretched. The frame formed by the vertical columns and the crossheads, pivoted on a trundle mount set on a sturdy external support. Remote lateral positioning of the sample, with an accuracy of 250 m m, was achieved by the use of a third actuator, which displaced the frame horizontally in the plane perpendicular to the beam. Scattered intensity (I) and load/strain data were collected simultaneously as the sample was deformed. A video extensiometer was used for local strain measurements. The maximum deformation was 406 mm. The exposure times at the short and long SDD were 1 and 0.2 s. Readout of the pixel array limited the data rate to a minimum of ca. 0.3 Hz. Frames were collected at the rate of 0.43 Hz and the exposure time for each frame was 200 ms. Frames 1{10 were collected on the as-received sample, without the application of strain. The initial pattern was isotropic. Frames 11{58 were collected as the sample was stretched, and anisotropy developed with strain. In this con­ guration the meridional streak was observed. Data were corrected for electronic noise from the detector, for air/instrumental scattering, and for thinning of the sample. For equatorial scattering the ID was tuned to an energy of 8.048 keV, corresponding to a wavelength ¶ = 1:5405 ¸ A. A range 0:02 < q < 0:35 ¸ A¡1 was covered in this con­ guration. This is a typical SAXS range covered by slit and pinhole collimated instruments in the laboratory. The equatorial streak was weaker in comparison with the meridional streak. An interesting observation was that in all experiments, scattering and hence Proc. R. Soc. Lond. A (2003)

Crack blunting and the strength of soft elastic solids

1515

presumably the defect structure depended only on strain, and not on rate of loading nor on time. The meridional component of the scattering increases in intensity with deformation. Kratky plots (Iq 2 versus q) of meridional cuts obtained from the 2D data show a peak that increases in intensity and moves to lower q as elongation proceeds. These data have been ­ tted to a model of polydisperse spherical voids in a dense matrix to obtain an e¬ective void size. More detailed analysis of these data to extract other features of the structure will be presented elsewhere. The shape of the distribution is assumed to be log-normal. Results from the ­ ts, shown in ­ gure 13, are given in terms of the most probable particle diameter 2ro . The stress versus strain data, obtained simultaneously with the scattered intensity, are also shown in ­ gure 13. At a logarithmic strain of about 0.5 the material starts to sti¬en as the stress versus strain curve becomes progressively steeper. As ­ gure 13 indicates, the SAXS intensity is observed concomitantly with the sti¬ening of the material, but not until this point. Subsequently, the change in shape and magnitude of the SAXS intensity is coincident with an increase in the size of the voids.

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