Soft Matter PAPER. Delayed fracture in gels. Dynamic Article Links C < Cite this: Soft Matter, 2012, 8,

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1 Soft Matter Dynamic Article Links C < Cite this: Soft Matter, 2012, 8, Delayed fracture in gels PAPER Xiao Wang a and Wei Hong* ab Received 9th March 2012, Accepted 21st May 2012 DOI: /c2sm25553g Subject to a subcritical load, a swollen polymeric gel may hesitate for a prolonged period of time without showing any macroscopic symptom, and then break suddenly. Such a phenomenon is usually referred to as the delayed fracture in gels. In this paper, we present a possible mechanism for the delayed fracture in gels from the continuum and fracture mechanics point of view. Using a continuum visco-poroelastic model for polymeric gels, we calculate the evolution of the inhomogeneous stress field around a pre-existing crack, in consequence of the coupled viscoelastic creep and solvent migration. We invoke the instantaneous energy release rate as the local driving force for a crack, and find it to be an increasing function of time. With the dissipation from viscoelastic creep and solvent migration excluded, the criterion for crack advancing is that the instantaneous energy release rate equals the intrinsic fracture energy of the polymer. The fracture delay could thus be attributed to the time needed for viscoelastic creep and solvent migration to bring the instantaneous energy release rate to the level of the intrinsic fracture energy. For most swollen gels, solvent migration is the limiting process, and therefore the delay time depends on the size of a pre-existing crack in a similar way as common diffusion-limited processes. Finally, by assuming a specific size distribution of microcracks, we provide a simple statistical analysis towards the lifetime prediction of a swollen gel. 1 Introduction Because of their biodegradability and mechanical and chemical compatibility with natural tissues, polymeric gels are believed to be the future materials for tissue scaffolding and replacement. 1 However, common synthetic gels suffer from the brittleness which accompanies their natural softness. 2 4 The mechanical strength of regular polymeric gels is insufficient for in vivo loading and manipulation. The increasing interest in developing strong and tough gels 5 has heightened the need for understanding the fundamentals of their failure processes The simplest mechanical test for a gel is to load it with a dead weight. An interest phenomenon known as delayed fracture has been observed when a dead load just below the ultimate strength is applied Instead of instantaneous or continuous crack growth, the gel would hesitate for some time without showing any sign of failure macroscopically, and then undergo a dynamic fracture. The corresponding delay time t f has several distinct characteristics as revealed by experiments: (1) a significant statistical variance exists in the results obtained from samples of identical materials; (2) the average delay time ^t f is highly sensitive to the stress level, i.e. a small decrease of stress s may increase ^t f by orders of magnitude, from seconds to hours; and (3) the shape a Department of Materials Science and Engineering, Iowa State University, Ames, IA 50011, USA b Department of Aerospace Engineering, Iowa State University, Ames, IA 50011, USA. whong@iastate.edu; Fax: ; Tel: of the s ^t f relation depends on material type and swelling condition. The technical importance of understanding the mechanism of delayed fracture could be exemplified with applications such as tissue scaffolding. A gel scaffold must function and keep its structural integrity under specific load; no macroscopic crack initiation should occur during the entire service period for cell growth and differentiation. 1,15 Current understanding of delayed fracture mechanism involves the mechano-chemical damage of stress carrying chains in the gel ,16 18 The applied stress lowers the activation energy barrier of chain breakage, increases the chain dissociation rate, and consequently facilitates nucleation and subcritical growth of microcracks. Since similar dynamic chain breakage and recombination processes also lead to the viscoelastic deformation of a physical gel, the observation that gels with similar viscosity can have drastically different delay time 12 has put such explanations into question. On the other hand, all gels are known to be poroelastic solvent migrates in a gel if the deformation is inhomogeneous. 19,20 In this paper, we will investigate the contribution from the poroelasticity of a gel to the propagation of microcracks. For simplicity in description, we will focus specifically on polymeric gels, while a similar analysis may also be applicable to other types of gels. The concurrent deformation of the polymer network and migration of interstitial solvent driven by the inhomogeneous stress field around a crack will be captured using a recently developed continuum model, 21 as briefed in Section 2. The driving force to advance a microcrack is the instantaneous energy This journal is ª The Royal Society of Chemistry 2012 Soft Matter, 2012, 8,

2 release rate at the crack tip, besides additional energy dissipation through solvent migration and viscoelasticity. Due to the viscoporoelastic deformation around the crack, the instantaneous energy release rate is a function of time as evaluated in Section 3. The classical criterion that a crack advances when the energy release rate exceeds the intrinsic fracture energy G 0 of the polymer network will be used to determine the delay time: G(t f ) ¼ G 0. In the case of a relatively large microcrack, the delay phenomenon is limited by solvent migration, and the time evolution of G follows the typical scaling law of a diffusion limited kinetic process. The size and stress dependency of fracture delay time is reduced to a single master curve in Section 4.1. Finally, the statistical nature of delay fracture events is accounted for by considering the defect size distribution, and a statistical lifetime theory of gels is presented in Section Material model for a visco-poroelastic gel In order to make a comparison between the contributions from viscoelastic deformation and poroelastic solvent migration, a continuum model for the coupled visco-poroelastic behavior of polymeric gels will be used. The theory and the governing equations are briefly summarized as follows, while the detailed formulation and derivation could be found in ref. 21. With the dry state as the reference, the deformation of the polymer network is traced by the current position x(x,t) of the material particle located at X in the reference state. We assume the molecular incompressibility and insist that the deformation gradient F ¼ Vx and the nominal solvent concentration C are related as 19 UC ¼ det F 1, (1) where U is volume occupied by each solvent molecule. The conservation of solvent molecules further relates the time rate of change in C to the nominal diffusion flux J as _ C + V$J ¼ 0. (2) Both solvent migration and viscoelastic deformation are irreversible processes, through which the system free energy is dissipated. Following the usual approach of non-equilibrium thermodynamics, we assume linear kinetics for both dissipative processes. The true flux of solvent migration j is related to its driving force, the gradient of chemical potential m: j ¼ mv x m, (3) with m being the mobility of solvent molecules. Similarly, the viscous shear stress s, as a driving force for viscous flow, is related to the velocity gradient V x v as " # 1 s ¼ h V x v þðv x vþ T ðv x$vþ1 ; (4) with h being the viscosity, the superscript T indicating the tensor transpose, and 1 representing the identity tensor. Eqn (4) assumes that the viscous behavior of the gel is similar to that of an incompressible Newtonian fluid. Here, the differential operator V x, as well as the solvent flux j, is defined with respect to the deformed geometry in the current state. They are related to their nominal counterparts by the simple geometric relations: V ¼ F T $V x and F$J ¼ jdet F. Finally, we prescribe the stress strain relation using a freeenergy function. Adopting the Flory Rehner model, 22 we write the total Helmholtz free energy per unit reference volume, U, into the sum of the elastic energy of stretching U n and the energy of mixing the solvent with polymer chains U m, U ¼ U n + U m. Without specific material information, we take the neo-hookean form for U n, and the Flory Huggins form for U m : 19,22 U n ðfþ ¼ 1 i hf 2 NkT : F 3 2lnðdet FÞ ; (5) U m ðcþ ¼ kt UC ln 1 þ 1 c þ : (6) U UC 1 þ UC Here kt is the temperature in the unit of energy, and c is the Flory Huggins parameter for the enthalpy of mixing. With N being the number of polymer chains per unit volume, the product NkT is the modulus of dry polymer in the reference state. With the contributions from the polymer network, the viscous deformation, and the solvent included, the nominal stress takes the form s ¼ vu n vf þ s$f T det F þ det F dum U dc m F T : (7) The first term on the right-hand side of eqn (7) is the elastic stress from the network, the second term is the viscous stress, and the last term is the osmotic stress from the solvent. Even in the case of fast crack propagation, the velocity and acceleration of a material particle in gel are small. We will thus neglect any inertial effect, and assume the system to be in mechanical equilibrium: V$ s¼ 0. (8) Substituting eqn (1) and (3) into (2), and (7) into (8), we obtain the governing equations of the model, a set of PDEs for the fields x(x,t) and m(x,t). In the case when the energy is mainly dissipated through solvent migration, the viscous stress can be omitted from eqn (7), and the current model reduces to that of a poroelastic gel. 19 The theory described above and in ref. 21 is mostly based on the nominal quantities measured with respect to the geometry in the reference state. However, for the fracture of a polymeric gel, it is often more convenient to consider quantities with respect to the initial state at t ¼ 0. In equilibrium with the solvent of chemical potential m 0, the gel swells isotropically and freely with linear swelling ratio l, given by the nonlinear equation: 23 NkT l 1 l þ kt ln 1 1l l 3 þ 1 þ c Ul 3 l m 0 3 U l3 ¼ 0: (9) In the following discussion, unless otherwise stated, all dimensions refer to those in the initial state, and all physical parameters and fields are measured with respect to this state. For example, the modulus of the gel in the initial state is related to NkT as E ¼ NkT/l Soft Matter, 2012, 8, This journal is ª The Royal Society of Chemistry 2012

3 3 Numerical simulation of delayed fracture The model of visco-poroelastic gel is implemented into a finiteelement method 21 to enable numerical calculations. To exemplify poroelastic effects in gel fracture, let us consider the plane-strain problem of an edge crack of length L in a large sample of size W [ L, as sketched in Fig. 1a. A uniform tensile stress s in the Y-direction is applied remotely to the crack, symmetrically on upper and lower edges of the sample. The edge crack is in pure mode I. A symmetric boundary condition is applied on the right edge of the sample. Both the left edge of the sample and the crack faces are assumed to be traction free. Neglecting solvent evaporation throughout the duration of the processes involved, we assume all boundaries to be impermeable to solvent. Following Lake and Thomas, 24 we divide the fracture energy of a gel into two contributions: the intrinsic fracture energy G 0 needed to cleave polymer chains and form new surfaces, and the energy dissipation from the irreversible deformation in the region near the crack tip. The intrinsic fracture energy could be regarded as a material parameter, while the energy dissipation per unit crack area is highly dependent on the detailed deformation process. Due to the rate dependency of visco-poroelastic deformation, which could be measured using tradition fracture mechanics tests, 2 the traditional definition of energy release rate does not give a unique value. We may still define the instantaneous energy release rate G as the reduction in total potential energy upon advancing of unit area of crack followed by an elastic relaxation. Such a definition excludes the viscous relaxation and the solvent transportation in response to the crack propagation, as if the gel is incompressible and inviscid. Therefore, the instantaneous energy release rate is the direct driving force of polymer chain cleavage. The definition of the instantaneous energy release rate G gives rise to the criterion for crack extension G $ G 0. Such a fracture criterion implies the assumption that the crack opening due to the cleavage of a polymer chain is much faster than viscous relaxation or solvent migration. During irreversible deformation, the states of all material particles surround a crack change, so that the instantaneous energy release rate of a crack would vary with time. As will be shown by our numerical results, G(t) is a monotonically increasing function of time for a static crack under a constant remote stress. We propose the following mechanism for delayed fracture. Immediately after a subcritical load is applied, the instantaneous energy release rate is below the intrinsic fracture energy, G(0) < G 0. After a certain delay time, t f, determined from the criterion G(t f ) ¼ G 0, the crack starts to propagate. Once the propagation starts, G increases with the crack length and the crack becomes unstable. The swollen gel usually fails like a brittle solid. Therefore, rendering the complete fracture process of an advancing crack may be unnecessary, and we will only focus on the stage prior to crack propagation, 0 < t < t f. Dimensional analysis dictates that the instantaneous energy release rate depends on various parameters as: G te EL ¼ g h ; td s L 2; E ; EU kt ; c; m 0 ; (10) kt Fig. 1 (a) Sketch of a mode I edge crack in a gel, remotely loaded by stress s. Crack size L is much smaller than sample size W, and at the same time much larger than the cohesive zone size P. (b) The spring constant of the cohesive zone k plotted as a function of the distance from the crack tip, X. where g is a dimensionless function. The system contains two time scales: the characteristic time for viscoelastic relaxation, t v ¼ h/e, and that of poroelastic solvent migration, t p ¼ L 2 /D. Here D is the diffusion coefficient of solvent in the gel, which relates to the mobility through the Einstein relation D ¼ mktdet F/(det F 1), and is almost independent of the swelling ratio for a highly swollen gel. 19 Using the self-diffusion coefficient of water at room temperature, D ¼ m 2 s 1, 20,25 we estimate the typical time for solvent migration over the heterogeneity created by defects of size 1 mm to 1 mm to be t p ¼ 10 3 to 10 3 s. On the other hand, the characteristic time for viscoelastic deformation strongly depends on composition and environment parameters. In the current paper, we will focus on relatively large cracks with t p [ t v to elucidate the role of poroelasticity. Although t v and t p may overlap in problems concerning very short cracks or highly viscous gels, a complete discussion of a coupled problem is beyond the scope of this paper. For an elastic material, the energy release rate could be calculated by evaluating the J-integral along an arbitrary contour encircling the crack tip. In a visco-poroelastic gel, the J-integral is non-conservative. To evaluate the instantaneous energy release rate, an infinitesimal contour is needed to exclude the visco-poroelastic dissipation. However, such an approach is impractical due to the singularity at the crack tip and the accompanying numerical error. Here this technical difficulty is overcome by invoking a cohesive-zone model. As shown in Fig. 1a, an array of nonlinear springs representing partially This journal is ª The Royal Society of Chemistry 2012 Soft Matter, 2012, 8,

4 damaged chains is placed on the prospective path ahead of a crack, forming a cohesive zone of length P. The cohesive zone physically corresponds to the region where the polymer chains undergo a mechano-chemical process of recombination. It is assumed that such a process is activated by extremely high stresses, and is localized in a domain much smaller than that of typical visco-poroelastic processes. For regular cohesive-zone models, it has been shown that the actual form of the traction separation law is relatively unimportant. 26,27 Here we take the spring constant k to be a smoothed inverse Heaviside function of the horizontal coordinate X, which varies from 0 at the crack tip to infinity at the outer rim of the cohesive zone, X ¼ P, where the material is intact. The cohesive-zone model reduces the singularity at the crack tip and enables the evaluation of the J-integral over a contour just enclosing the nonlinear springs: J ¼ 2 ð P 0 vv þ t y dx; (11) vx where t y is the vertical traction, and v + the vertical displacement of the upper surface of the cohesive zone. This method of calculating the instantaneous energy release rate is validated numerically on a hyperelastic material, of which the instantaneous energy release rate is indifferent from the regular energy release rate. The J-integral is evaluated through three approaches: via the contour over the cohesive zone, via regular contours around the crack tip in the absence of cohesive zone, and via regular contours around the crack tip when a cohesive zone is present. The results plotted in Fig. 2 clearly show that the introduction of the special cohesive-zone model does not affect the energy release rate, and the integral over the cohesive zone is sufficient. To further validate this approach and to ensure accuracy, we run numerical tests on various combinations of crack sizes and cohesive zone sizes relative to the gel specimen. It is found that the relative sizes W/L ¼ 10 and P/L ¼ would already provide repeatable results with acceptable accuracy. 4 Results and discussion 4.1 Poroelastic fracture of an isolated crack In the numerical calculations, we use the stiffness of the gel NkT ¼ 40 kpa, close to that estimated from experiment. 12 In the first example, we take the representative value of the Flory Huggins parameter, c ¼ 0.1. The dependence of the fracture delay on polymer solvent interaction will be discussed later. Initially the gel is in equilibrium with pure solvent and swells with stretch l ¼ 3.4. The elastic modulus of the gel in the initial state can be calculated as E ¼ 12 kpa. Upon application of the remote stress, the homogeneous initial state is perturbed by the highly non-uniform deformation near the crack. Shortly after the load, at t t p, when solvent molecules have hardly migrated, the gel behaves just as an elastic rubber. The hydrostatic tension near the crack tip results in a local decrease of chemical potential (Fig. 3). The inhomogeneous field of chemical potential drives a converging flux of solvent to the crack tip. The solvent transportation is localized at a length scale comparable to the crack size L, while the bulk material away from crack tip is still unaffected (Fig. 4). The redistribution of solvent causes the local swelling near the crack tip. The swelling rate decreases with time and vanishes when a new equilibrium state is reached at t [ t p. A typical curve of the instantaneous energy release rate G(t) is obtained with the method described in Section 3, and plotted in Fig. 5. The material parameters are selected such that the two time scales t v and t p are separated, and the corresponding twostage evolution of G is clearly shown. The monotonous increase Fig. 2 Energy release rate of a static crack in a neo-hookean solid. The value calculated with the current method, i.e. using the J-integral over a cohesive zone, is compared to the J-integrals over regular contours, with and without the presence of a cohesive zone. Four regular contours, different contour sizes L C /L are taken in each case to ensure numerical accuracy. Fig. 3 The inhomogeneous field of chemical potential m/kt at the vicinity of a crack. Snapshots are taken at dimensionless time td/l 2 of 100, 500, 1000, and The arrows illustrate the direction of solvent migration in the upper half specimen Soft Matter, 2012, 8, This journal is ª The Royal Society of Chemistry 2012

5 Fig. 4 The relative volumetric swelling ratios, at material points of different distances away from the crack tip, plotted as functions of the dimensionless time, td/l 2. The numbers in parentheses are the dimensionless coordinates (X/L,Y/L) of the material particles. of the crack, and plot the results in Fig. 6a. Due to the normalization of the time using t p, the size-independent viscoelastic deformation appears to be different in Fig. 6a. With the viscoelastic stage being unimportant here, the readers should direct their focuses to the poroelastic stage. Despite the visco-poroelastic coupling, the shape of G(t) in the poroelastic stage is almost unchanged while plotted in a dimensionless manner. It seems that the evolution of instantaneous energy release rate is captured by a master curve, independent of the coupling with viscoelastic state. The only difference between gels with longer and shorter cracks is the starting point of the poroelastic stage. Furthermore, we have studied the effect of the dimensionless remote load, s ¼ s/e, and plotted the results in Fig. 6a and b. As shown in Fig. 6b, the instantaneous energy release rate scales with the applied stress approximately as s 2 in both the rubbery limit and the long-term limit, just as the scaling relation of a linear elastic material. Following these observations, we may write the expression of instantaneous energy release rate, eqn (10), approximately into GE t s 2 L ¼ G ; EU t p kt ; c; m 0 ; (12) kt in the poroelasticity-dominant stage, t [ t v. The right hand side of eqn (12) is the dimensionless energy-release-rate function, G ¼ GE/s 2 L. More specifically, when constant values of parameters EU/kT, c, and m 0 /kt are chosen, we find the following functional Fig. 5 Time evolution of instantaneous energy release rate and global creep strain after a step load of remote stress. Due to the separation in time scales, the energy-release-rate curve shows a viscoelastic regime and a poroelastic regime, and monotonously increases in both regimes. in G(t) in both stages can be understood as follows. Upon the application of the step load, the viscosity of the gel prevents it from deforming instantaneously. At t ¼ 0, the gel is rigid and thus G(0) ¼ 0. At 0 < t < t v, the gel undergoes a viscoelastic creep. With the concentrated creep stretch at the crack tip, the gel stores locally higher elastic energy which contributes to the instantaneous energy release rate. The contribution from poroelasticity is similar, although the corresponding creep is due to long range solvent migration and thus takes a longer time. It is well known that some dry amorphous elastomers also experience delayed rupture beyond critical elongations, 28 although the main cause there is viscoelastic creep. To this point, we can easily differentiate the poroelastic and viscoelastic effects by examining the global creep stretch, l Y (t), as plotted in Fig. 5. During viscoelastic creep, the entire specimen deforms. In contrast, the solvent migration in the poroelastic stage does not induce significant global deformation on the gel. For ease of description, let us denote the two limits of the instantaneous energy lease rate as G r and G N,namelythe rubbery and the long-term limit, respectively. Although the assumption on the separation of time scales is of theoretical interest, it may not always be the case for an actual material. To explore more general cases, we vary the relative size Fig. 6 (a) Time evolution of instantaneous energy release rate as a function of dimensionless time for various crack sizes and applied stress. FEM results (squares and circles) are compared with the fitting curves using eqn (13). (b) Instantaneous energy release rates at the rubbery and long-term limits plotted as functions of the remote load. FEM results are compared with fitting curves of a quadratic relation. This journal is ª The Royal Society of Chemistry 2012 Soft Matter, 2012, 8,

6 form of G(t) that accurately captures the transition from the rubbery limit to the long-term limit: GðtÞ ¼G r þ G N G r 1 þðal 2 =tdþ ; (13) where G r and G N are the values of G in the rubbery and longterm limits, respectively. The dimensionless parameter a captures the onset of the transition, and is determined by geometry and material properties. By fitting eqn (13) to the numerical data on Fig. 6a, we obtain that a ¼ 700. It was observed in experiments 12,13 that the fracture delay time is often much larger than the characteristic time of viscoelastic deformation, t f [ t v. We therefore hypothesize that the delay is mostly due to the poroelastic deformation of the gels, and in the following discussion we will thus utilize eqn (13), which captures the poroelasticity-dominant stage. We insist the criterion for crack propagation: G $ G 0. If the remote load is relatively high, G r > G 0, the crack will propagate shortly after loading during the viscoelastic stage, when G(t) ¼ G 0. In the other extreme, when the applied load is so low that even the long-term energy release rate is lower than the intrinsic fracture energy, G N < G 0, the crack will never propagate. Only when an intermediate load is applied, such that G r < G 0 < G N, a delay due to poroelastic deformation may be observed. The approximate delay time for a pre-existing edge crack of length L is, from eqn (13), t f D L 2 ¼ a G 0 G r G N G 0 : (14) Containing crack size L, the dimensionless time td/l 2 may not be an objective measure for the fracture delay time. Alternatively, we may normalize the fracture time by G 02 /DE 2, which contains only material parameters, and write eqn (14) into t f DE 2 ¼ a G 0 G r s 2 ; (15) 2 2 G 0 G 0 G N s 2 G 0 where G 0 ¼ G 0 /EL is the dimensionless intrinsic fracture energy. The normalized delay time as in eqn (15) is plotted as a function of the remote stress in Fig. 7. Just as that observed in experiments, the delayed fracture only happens in a very narrow range of subcritical stresses, outside which fracture either happens immediately (after a short time comparable to t v )or never happens. The dependence of the fracture delay time on the crack size L is also shown in Fig. 7, through the dimensionless intrinsic fracture energy G 0. With a longer pre-existing crack, the failure stress takes a lower value, and the delay time is also longer, since solvent will migrate through a larger region during poroelastic deformation. Taking an estimate of the intrinsic fracture energy G 0 ¼ 0.1 Nm 1 and the modulus G 0 ¼ 12 kpa, for a crack of size L ¼ 100 mm, the delay time increases by two orders of magnitude from 10 s to 1000 s, when stress increases slightly from 2 kpa to 2.4 kpa. Moreover, the special scaling law for poroelasticity-dominant processes dictates that the delay time of a specimen with a pre-existing macroscopic crack is much longer, although no experimental result is yet available, perhaps due to the very narrow stress window for delayed fracture of a macroscopic crack. For example, the delay time for a millimetre-size crack may be as long as several days. The reduced functional form of energy release rate in eqn (12) and (13) allows us to study the effect of material parameters simply by looking at their contributions to the two limits, G r and G N. In general, G r and G N are constants depending on material parameters as well as the shape of the specimen. In the case when the viscoelastic and poroelastic processes have separated time scales, there is hardly any solvent migration during the viscoelasticity-dominant stage. As a result, the energy release rate at the rubbery limit, G r, would only be dependent on the stiffness of the gel, instead of the Flory Huggins parameter c or chemical potential m 0. In contrast, the long-term limit, G N, would be influenced more by the swelling behavior. Here we numerically test the effects of material parameters c and initial swelling stretch l and plot the results in Fig. 8. As shown in Fig. 8a, when we gradually change the polymer to be more hydrophobic (c ¼ 0 0.2) while keeping the initial swelling ratio constant, G N shows an increase. If the same type of gel is more swollen, the value of G N is also higher, as shown in Fig. 8b. The more hydrophobic polymers and more swollen gels with higher G N give rise to a more significant delay effect, with a wider stress window and a longer delay time. The dependences on both parameters could be easily verified through experiments, and may be understood as follows. While making a polymer more hydrophobic or adding more solvent has the same effect to a gel by increasing its chemical potential (or osmotic pressure), the regions where the gel is more swollen (e.g. near a crack tip) will have a relatively smaller change. The non-uniform increase in chemical potential results in a convergent flux of solvent towards the crack tip, causing the increase of the instantaneous energy release rate. More numerical results show that in the case when gels of different polymers are all brought to equilibrium with the same solvent (same m 0 /kt), the difference in G N turns out to be smaller than the two cases shown in Fig. 8. This is because a more hydrophobic gel would swell less, so that the two contributions compete against each other and diminish the effect. 4.2 Statistical lifetime theory of gels Fig. 7 Dimensionless fracture delay time as a function of the applied load for various values of intrinsic fracture energy. While the discussion in previous sections relies on the knowledge of the size of a pre-existing crack, such information may not be available in practice. On the other hand, polymeric gels are known to be highly heterogeneous and contain a large amount of defects. The uncertainty in defect size in gels may contribute to the scattering in the experimental data of delayed fracture. In this 8176 Soft Matter, 2012, 8, This journal is ª The Royal Society of Chemistry 2012

7 Fig. 10 Characteristic lifetime ^t f as a function of the remote stress, for gels made of the same polymer at different initial swelling ratios. Fig. 8 Normalized instantaneous energy release rate as a function of dimensionless time during poroelastic deformation regime for gels with (a) the different type of polymer network with fixed initial swelling ratio and (b) the same type of polymer network with a different initial swelling ratio. Fig. 9 Probability of survival of a poroelastic gel as a function of the dimensionless time, t f DE 2 /G 02, under different values of the remote load s/e. section, we propose a lifetime theory to predict the statistics of gel fracture. For simplicity, we regard the pre-existing defects as non-interacting microcracks of various sizes. The results of previous discussion direct our focus to the largest crack in a gel, which is the most vulnerable too. The delay in fracture is limited by the largest crack in a gel, and the statistics of its lifetime could be determined from the probability of finding the largest crack of a certain size. With the lack of detailed information on the statistics of microcracks in a gel, we assume that the size of each crack is a random variable, and the size distributions of individual cracks are identical and independent. Let q(l) be the size distribution of an arbitrary crack in the gel, and QðLÞ ¼ Ð qðlþdl be the cumulative distribution, i.e. the probability of the crack size to be smaller than L. Denote the total number of cracks in a gel as M. Since the crack sizes are independent and identically distributed, the probability of having all cracks in the gel to be smaller than L is Q(L) M. For a crack of known size L, eqn (15) provides a functional relation between the crack size and the expected delay time before propagation, L(t f ), for a specific type of gel under known load. Utilizing this relation, we could further write the probability for a gel to survive after time t as Q m (t) ¼ Q(L(t)) M. With the knowledge of the size distribution of defects in a gel, q(l), its lifetime could be predicted. Here as an example, we assume the exponential distribution of crack sizes: qðlþ ¼ 1 exp LLa ; (16) L a where L a is the mean crack length. Utilizing this specific size distribution and the same material parameters as in Section 4.1, we calculate the survival probability function, Q m (t), and plot the results under various levels of loads in Fig. 9. The survival probability decreases with the normalized applied stress s. Under a large stress, almost all samples are expected to fail after a short time, while almost no sample will fail even after very long time if the load is below some critical value. Only when the level of the remote stress falls within a narrow window, the survival probability Q m (t) starts from a higher value and decreases to a lower value after a delay. If we take the threshold probability to be e 1, a characteristic lifetime ^t f could be obtained from Q m (^t f ) ¼ e 1. The resulting dependence of the dimensionless lifetime ^t f DE 2 2 /G 0 on the applied load is plotted in Fig. 10, and agrees qualitatively with the experimentally measured lifetime of a gel. 12 The dependence of the lifetime on the initial swelling ratio is also shown in Fig. 10, for gel samples made from the same type of polymer (with the same crosslink density and hydrophobicity). A highly swollen gel is expected to have a more significant delay This journal is ª The Royal Society of Chemistry 2012 Soft Matter, 2012, 8,

8 effect in fracture, so that a larger window in terms of the remote stress exists. At the same time, a more swollen gel would have a shorter lifetime, under the same level of remote stress. It should be noted that if the mechano-chemical reaction on polymer chains which causes subcritical crack growth is the dominating mechanism, the fracture delay should not be strongly affected by the swelling ratio, as the solvent molecules usually do not participate in these reactions. The different dependences of delayed fracture on the initial swelling of gels can easily be tested through future experiments, and could be used to identify the possible dominating mechanism and verify the theory. 5 Conclusions From a fracture-mechanics point of view, this paper suggests an alternative mechanism for the delayed fracture phenomena of polymeric gels. The coupled viscoelastic deformation and solvent transportation processes in polymeric gels are modeled using continuum visco-poroelasticity. The local driving force for crack propagation is characterized by an instantaneous energy release rate, to separate the intrinsic fracture energy from the dissipation through viscous deformation and mass transportation. The instantaneous energy release rate is then calculated numerically using a cohesive-zone model. While both the viscoelastic creep and the poroelastic deformation could result in an increase in the instantaneous energy release rate of a crack, the delay in fracture is usually limited by the transportation process, especially in a system with separate time scales of viscoelasticity and poroelasticity (e.g. a system with relatively large cracks and low viscosity). For the poroelasticity-dominant delay in fracture, the delay time is expected to be size dependent a larger crack will have a longer delay before fracture. In this paper, we have limited our discussion to the cases of impermeable crack faces, and excluded liquid-filled cracks. If a crack tip is allowed to exchange solvent directly through the crack faces, the diffusion length and thus the delay time will be greatly reduced. This is consistent with recent observations that crack propagation in gels is facilitated by a small amount of solvent locally supplied at the crack tip, 6,29 which enables fast local swelling, and a sudden increase in energy release rate. It should be noted that a large body of literature is available on the delayed fracture of other heterogeneous or porous materials. 30 In contrast to the delayed fracture of dry materials, in which the origin of uncertainty is dominantly thermal fluctuation and the randomness in structure properties, we propose an alternative mechanism for that in swollen gels, in which the deformation associated with solvent migration may redistribute the stress field and increase the local driving force for crack propagation. Based on the scaling relation developed for the delayed fracture of a single crack, we further develop a statistical theory to study the fracture delay time, i.e. the lifetime of a gel without a macroscopic crack. A simple case of a macroscopically homogenous gel, which contains a large amount of non-interacting microcracks with independently and identically distributed sizes, is used as a model system. Using a Weibull type distribution for crack size as an example, our result qualitatively recovers the experimental findings on the statistical feature and stress sensitivity of the average delay time on similar gels. The model also provides insights towards the dependence of the lifetime of a gel on its material properties as well as the initial swelling ratio. These predictions may be compared with future experiments for verification of the current theory. References 1 J. L. Drury and D. J. Mooney, Biomaterials, 2003, 24, Y. Tanaka, K. Fukao and Y. Miyamoto, Eur. Phys. J. 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