Fracture of soft elastic foam

Transcription

1 Frature of soft elasti foam Zhuo Ma Department of Aerospae Engineering, Iowa State University, Ames, IA Xianghao Feng Department of Aerospae Engineering, Iowa State University, Ames, IA Wei Hong 1 Department of Aerospae Engineering, Iowa State University, Ames, IA whong@iastate.edu ASME Membership: Corresponding author. 1

2 Abstrat Consisting of strethable and flexible ell walls or ligaments, soft elasti foams exhibit extremely high frature toughness. Using the analogy between the ellular struture and the network struture of rubbery polymers, this paper proposes a saling law for the frature energy of soft elasti foam. To verify the saling law, a phase-field model for the frature proesses in soft elasti strutures is developed. The numerial simulations in two-dimensional foam strutures of various unit-ell geometries have all ahieved good agreement with the saling law. In addition, the dependenes of the marosopi frature energy on geometri parameters suh as the network onnetivity and spatial orientation have also been revealed by the numerial results. To further enhane the frature toughness, a type of soft foam strutures with non-straight ligaments or folded ell walls has been proposed and its performane studied numerially. Simulations have shown that an effetive frature energy one order of magnitude higher than the base material an be reahed by using the soft foam struture. Keywords: elasti foam, elastomer, toughening Introdution Solid foam, a state of material haraterized by the highly porous ellular struture, is ommonly found in nature and in everyday life. In industrial appliations, solid foams are well known for their superior energy-absorbing apability under ompression [2-4]. Their frature properties have also attrated great interests [4-9]. Saling laws between frature properties and porosity have been proposed and widely aepted [4,5]. However, most existing theories are based on linear elasti frature mehanis and the ell walls of the foams are assumed to be linear elasti prior to rupture. 2

3 While suh theories and preditions an be applied to foams of relatively stiff materials (e.g. eramis and metals), their appliability beomes questionable to those onsisting of soft and highly strethable materials, suh as elastomers. For example, an early experimental study on polyurethane foams found the frature energy to be less dependent on the density, and even exhibiting a slight derease when density inreases [10], while the saling laws of rigid foams all demonstrate linear or power-law dependene of the frature energy over density [4,5]. The major differene between stiff brittle foam and soft elasti foam lies in the porosity and the slenderness of ell walls (or ligaments for an open ell foam). For simpliity, in the following disussion, the two types of foams will be referred to as rigid foam and soft foam, respetively. In ontrast to the ell walls of rigid foam whih partially shares the load after rupture, a fratured ell wall of soft foam merely dangles over the rest of the struture. Suh a strutural differene indues the dramati distintion in energy transmission during frature. Upon rupture, the remaining elasti energy in a ell wall of rigid foam ould be redistributed aording to the rak-tip advanement, while that in a slender omponent ould hardly be transferred to its neighbors. The elasti energy in a ruptured slender omponent is mostly dissipated through loal vibration or visoelasti deformation. As a result, the effetive frature energy would have to inlude the energy of the entire omponent rather than just that at the viinity of the rak faes (i.e. the surfae energy). To better understand this unique mehanism of toughening, one may onsider the frature proess of a rubber, in whih the rosslinked network of long polymer hains ould be regarded as an extreme ase of soft foam when eah slender ligament shrinks down to a moleular sale. In their lassi paper, Lake and Thomas suggest the frature mehanism of rubber frature: the energy needed to rupture a polymer hain is muh larger than that of a single bond as the entire hain is subjet to virtually the same breaking fore [11]. After frature, the broken hains reoil and their entropi elasti 3

4 energy would not be forwarded to neighboring hains. Thus, the intrinsi frature energy of rubber sales approximately as rubber ~ Nn 3 2 l m U, with N being the number of hains per unit volume, n the number of monomers per hain, l the length of eah monomer, and m U the energy needed to rupture eah monomer [11]. This model has been widely used on rubber, but has seldom been related to the frature of soft foam. Just by using the analogy, we may also dedue the saling relation for the frature energy of soft foam as W l (1) Here, we use W to represent the ritial energy density of ell wall at rupture, for the volume fration (i.e. the relative density) of the solid phase, and l for the harateristi size of the foam (e.g. the height of a ell wall). is a dimensionless geometri fator. One may arrive at the same result from a different perspetive. Due to the speial geometry of foam, the sharpness of a rak is always limited by l. The frature energy is thus given by ~ Wl [12], with W W being the effetive strain energy density at the rak tip. In the limiting ase when the material is stiff enough and the frature proess an be modeled by linear elasti frature mehanis, by using the effetive modulus of the struture E eff ~ 2, this saling law (1) redues to the model of stiff foams with the frature toughness given by 3 2 K ~ IC eff E ~ l (for open ells) [4]. However, the appliability of the latter, whih was derived from linear elasti frature mehanis by assuming a square-root singularity in the stress field [4], to soft elasti foams undergoing large deformation remains unknown. It is noteworthy that due to the presene of defets, the rupture strength and thus size dependent, unless the ell walls are thinner than the ritial size for theoretial strength. The energy-dissipation mehanism of the soft foam struture, on the other hand, is never limited to W is usually 4

5 mirosopi sale. One may refer to a two dimensional marosopi analogy of soft foam, a net or a netted struture (e.g. a string bag), ommonly known for its toughness and noth insensitiveness. Even though the saling law (1) seems natural and plausible, it ould be hard to verify it diretly through experiments. In pratie, it is diffiult to ontrol the porosity and ell size independently during polymer proessing, not to mention the size dependeny of W. Alternatively, this paper seeks to verify the saling law through numerial modeling. In the following setions, a phase-field model for rubber frature will be adopted to simulate the damage initiation and evolution in hyperelasti ellular strutures. The phase-field model for frature, whih is apable of alulating the rak growth aording to the energy riterion without a predetermined rak path, is very suitable for strutures with omplex geometries suh as soft elasti foams. The saling law and the speial toughening mehanism will then be demonstrated with the phase-field model. The dependene on the detailed geometry of the foam ells will also be studied. Phase-field model of frature Numerial simulation of frature proesses has the inherited diffiulties in dealing with disontinuities, singularities, and moving boundaries whih auses large geometri and even topologi hanges. To overome some of these diffiulties, phase-field models of brittle frature have been developed [1, 13-18, 22]. Reently, phase-field models have also been applied to the brittle frature of rubbery polymers [23]. Numerial experiments have already shown that these models are apable of apturing both the onset of rak propagation and the damage morphologies of dynami raks [19-21]. Without the need to trak individual rak or to presribe a rak path, the phase field method beomes a promising andidate for modeling the frature of strutures with relatively omplex geometries, suh 5

6 as the soft elasti foam. The model used in this paper losely follows these developments, espeially those by Karma et al [13] and Hakim and Karma [14]. To desribe the state of material damage and to avoid traking the rak front and faes, a phase field X, t ( 0 varying ontinuously between the intat region ( 1 ) and a fully damaged region ) is introdued. The loss of integrity in the solid is modeled by writing the elasti strain energy 0 density as a monotoni inreasing funtion of the damage variable, g F W s, where 0 W is the s strain-energy density of the intat material under the same strain, and g is an interpolation funtion in the interval 0,1 with vanishing derivatives on both ends. In this study, we hoose the interpolant g As a ommon pratie of hyperelastiity, the deformation gradient tensor F is used to represent the state of strain. Following Karma et al. [13], we write the free energy density funtion to inlude three ontributing terms: W 2, F W. (2) s 2 0, g W F 1 g The seond term on the right hand side of Eq. (2) represents the energy assoiated with material damage. When the strain energy at a material partile exeeds the threshold W, the damaged state with 0 beomes energetially favorable. Just as in almost all phase-field models, the gradient energy term is added to regulate a smooth transition between the oexisting states. In this paper, the material onstituting the solid phase of the foam is assumed to be isotropi, so that only a salar oeffiient is needed for the gradient energy term. In equilibrium, the ombination of the seond and third terms on the right hand side of Eq. (2) gives the surfae energy, i.e. half of the intrinsi frature energy. The frature energy of the material modeled by the energy funtion is approximately 6

7 2 W [13]. Here, in a body undergoing finite deformation, all energy densities are measured with respet to the volume in the referene state. Countless number of onstitutive models have been developed for hyperelasti solids. Although speifi stress-strain relations of the solid phase may affet the ultimate frature properties of soft foam, suh dependene is beyond the sope of the urrent paper. Here for simpliity, we will limit the disussion to a neo-hookean material of the strain energy funtion 0 W F F : F 3, (3) s 2 where is the initial shear modulus. In ontrast to linear elasti solids, rubbery polymers are often modelled as inompressible. To enfore volume inompressibility, approahes suh as the appliation of the Lagrange multiplier are often taken, e.g. by adding to the free energy funtion a term p det F 1 with a Lagrange multiplier p representing the pressure field. A physially meaningful model needs to degrade the ompressibility simultaneously with the shear stiffness. Diretly multiplying the Lagrange multiplier term by g obviously does not serve the purpose. Instead, we modify the Lagrange multiplier term slightly by modeling the material as slightly ompressible: 2 1 p pg det F 1. (4) 2 K By taking the variation of (4) with respet to p, one will arrive at an equation of state with a degrading bulk modulus: g K det F 1 p. In the intat state, the large bulk modulus K ensures volume onservation; in the fully damaged state, the added term does not affet the field of deformation, and the ad-ho field p is regulated numerially by the quadrati term in (4). 7

8 With all the aforementioned energy ontributions, the total free energy of the system is simply the volume integral of the energy density, inluding the terms in (4), and the surfae integral of the potential of external trations t : where X, t, x WdV t xda, (5) x symbolizes the urrent oordinates of a material partile loated at X in the referene state. The total free energy is a funtional of the field of damage X, t deformation haraterized by x X, t and the field of. Following Hakim and Karma [14], we neglet inertia and body fores, and assume the system to be in partial mehanial equilibrium, so that x 0 or s 0, (6) in the bulk, and N s t on the surfaes. Here s W F is the nominal stress, and N is the unit normal vetor on a surfae. For the evolution of the phase field, on the other hand, we assume a linear kineti law with isotropi mobility m : 2 0 m m g W p det F W s. (7) To simulate quasi-stati frature proesses, a large enough mobility is taken to ahieve rateindependent results. Further, to model the irreversibility of frature proesses and to prevent the damaged phase from healing, we fore to be a monotonially dereasing funtion of time by taking only the positive part of the driving fore [22]: 0 2 W p det F W m g, (8) s 8

9 where the angular brakets indiate an operation of taking the positive values, 2. Supplemented by proper initial and boundary onditions, Eqs. (6) and (8) onstitute a partial differential system for the oevolution of deformation and damage fields, x X, t and X, t. a stiff a strethed ruptured soft Fig. 1. The rupture and retration proess of a ligament (or ell wall) in a foam struture. Beause of its slenderness, a soft filament will tend to bukle or oil and ould not effetively transdue energy. Energy dissipation and numerial implementation The major differene between soft and stiff foams and the primary means of energy dissipation during frature an be qualitatively understood with the aid of Fig. 1. For ease of desription, we will refer to the solid dividing segments in both open and losed foams as ligaments from now on. Upon rupture from strethed states, a ligament will first be aelerated by its own retrating fores, and the elasti energy stored prior to rupture is mostly onverted to kineti energy. When the ligament retrats further, the differene between stiff and soft ligaments is revealed: while a stiff ligament will remain straight and deelerate and transfer the energy further to the neighboring omponents, a soft ligament will tend to bukle or oil due to its slenderness, and the energy ould not be effetively transferred. The ultimate fator is the stiffness ratio between the surrounding struture and the broken ligament (a bukled ligament has very low stiffness), as elasti wave annot propagate from a ompliant medium to 9

10 a rigid one. As a result, the elasti energy of a soft ligament is mostly damped through subsequent vibration of itself. The detailed proess and the dependene on the strutural geometry, suh as the aspet ratio of eah ligament and the spatial onnetivity at eah node, an be simulated by omputing the full dynami response of a ligament and the surrounding struture. Suh analysis, however, is not of partiular interest to the urrent paper. We will fous on soft foams with very slender ligaments, and hypothesis that most part of the elasti energy stored in the broken ligaments will be dissipated through this proess. The results are thus inappliable to relatively stiff foams. On the other hand, it is also omputationally less feasible to model the full dynami behavior of eah ligament in a foam struture of omplex geometry. Instead, we will neglet the inertia and model the frature proess as quasi-stati. In this limit, the dissipation through rak propagation is negligible, and the frature energy is mainly dissipated through visosity. Without onsidering dynamis, the snap bak of the ligaments are fully damped after eah rupture event. Instead of a proof or evidene, the alulations presented as follows are the onsequene of the proposed energy-dissipation mehanism. Similar as in many other methods for frature and damage simulation (e.g. ohesive element and element deletion), without inertia, the damage-indued softening is intrinsially unstable. To stabilize numerial proedures, and more importantly to dissipate the redundant strain energy in the dangling ligaments after rupture, we introdue a Newtonian-fluid-like damping term to the nominal stress s W F x 2 (9) with being the numerial visosity. For relatively small visosity, the visous stress only has signifiant ontribution at the regions of high deformation rate, whih is expeted to our only in a 10

11 retrating ligament upon rupture. Numerial experiments have shown that when a small value of is taken, the artifiial visosity only hanges the rate of strutural unloading at the wake of a propagating rak, and does not affet the energy onsumption. Substituting Eq. (9) into (6), one may obtain the oevolution of deformation and damage fields by solving the partial differential system (6) and (8) simultaneously. The system has an intrinsi length sale, r W, whih is approximately the thikness of the transition zone from the intat region to a full damaged region. It ould be argued that r physially haraterizes the width of the frature proess zone in the ondensed solid phase. Without losing generality, we rewrite the governing equations into a dimensionless form by normalizing all energy densities and stresses by by r, and time by W, all lengths 1 mw. After normalization, the dimensionless frature energy of the solid phase is approximately 2, and the system has only three dimensionless parameters: the normalized shear modulus W, bulk modulus K W, and visosity m. In the following numerial examples, we will take dimensionless modulus 0. 2, whih orresponds to the representative values of a soft W elastomer: ~ 10 MPa, r ~ 1μm, and ~ 50 N m. The dimensionless bulk modulus is taken to be K W 200, and the visosity m It should also be noted that the ligament thiknesses of most strutures alulated in the urrent paper are omparable to the intrinsi length sale r. In this limit, the alulations ould as well be done by using the regular strength-based material degradation approah. Here, the phase-field approah is taken so that the diret omparison with the frature proess of a bulk material with the same property ould be made when needed. The dimensionless equations are implemented into a finite-element ode through the ommerial software COMSOL Multiphysis 4.3b. For numerial robustness under large deformation, the geometries are disretized by using triangular elements, and both the displaement and damage 11

12 fields are interpolated with linear Lagrange shape funtions. To apture the transition at the interfae between the damaged and intat phases, a maximum mesh size of 0.5r is presribed. The model is integrated over time via a fully oupled impliit sheme, with adaptive step size. To numerially enable damage nuleation, spatial random distributions of the initial shear modulus and the intrinsi frature energy has been introdued to eah model, with standard deviation at 1% of the orresponding magnitudes. Fig. 2. Sketh of the loading onditions for the foam strutures To ompute the frature energy, we load the pre-raked strutures in a similar way as the pureshear test for rubber. As skethed in Fig. 2, the right and bottom edges are onstrained by rollers, and the top edge is loaded by a uniform displaement. For symmetri strutures and if the rak propagates along a symmetry line, only half of the struture is alulated and a symmetry boundary ondition is presribed along the symmetry line. A ramping displaement load is applied within a short time and then held onstant. The rak will start to propagate when the applied displaement exeeds ertain value. In a steady state when the rak tip is far from either ends, the energy release rate is independent of the rak length, G W H, (10) eff 12

13 where H is the undeformed height of the struture. W is the effetive strain-energy density in the eff absene of the rak, and is averaged over the volume inluding the spae of the pores. In ontrast to the standard pure-shear test, the entire struture is under plane-strain ondition, and is allowed to shrink horizontally. The orresponding 2D results are loser to the behavior of 3D losed foam. Although the effetive strain-energy density W an be alulated by integrating W at the region far eff ahead of the rak tip, here we alulate it separately by subjeting a non-raked struture to planestrain uniaxial tension. Results and disussion The simulations are first arried out on hexagonal (honeyomb) strutures, as shown shematially in Fig. 3a. In order to redue boundary effet, the atual omputational domain is muh larger than that shown. Fillets of dimensionless radius 1 have been applied to all orners to redue stress onentration, as the preferential damage of the triple juntions may result in a different saling law. The loal deformation and damage fields of a representative result are shown in Fig. 3b, in whih the preexisting rak has propagated through three ligaments. a b Crak tip 13

14 Fig 3. (a) Part of the 2D honeyomb foam struture being simulated. (b) The alulated deformation and damage fields of a honeyomb struture, during the propagation of a preexsting rak. The shades represent the dimensionless strain energy density W W. The rak profile is indiated shematially by the dash line, whih goes through the the transition zone from the intat to the fully damaged regions in terms of. The deformation is shown to sale, and only part of the struture near the rak tip is shown. The atual omputational domain is muh larger than that shown to irumvent size-effet. In a steady state, a rak is propagating through the struture at a onstant speed, the energy release rate G is given by Eq. (10). However, due to the disrete nature of the struture, the rak propagation appears staggered. To apture the effetive rak veloity, we identify eah event of ligament rupture, and reord the time of the event and the horizontal oordinate of the orresponding ligament in the undeformed state, as shown by Fig. 4a. It is found that when the rak front is far from the edge, the ligament-rupture events are almost equally distributed in time, indiating a steady-state rak propagation. The slope of the linear fit to the rupture events under eah loading ondition is taken to be the nominal rak veloity v. In onsequene of the kineti law, Eq. (7) or (8), the energy release rate G, i.e. the driving fore of the rak, is rate dependent. As shown by Fig. 4b, the effetive frature energy of the struture is a monotoni inreasing funtion of the rak veloity. Here, to ompare between different strutures, we use the threshold value alulated from the vertial interept of the frature-energy-veloity urve, as shown by Fig. 4b. The threshold orresponds to the frature energy of a rak propagating quasi-statially at zero veloity. Following suh a proedure, we ompute the quasi-stati frature energy of various foam strutures, presented as follows. 14

15 a b Fig. 4. (a) Undeformed oordinates of the ligaments as a funtion of the times of rupture. The line is the best linear fit. The slope indiates the dimensionless rak veloity. (b) Dimensionless frature energy (energy-release rate) rw as a funtion of the dimensionless rak veloity v rmw. The line is the best linear fit, and the vertial interept shows the quasi-stati frature energy of the soft foam. The saling relation (1) is verified first through the simulation on the frature proesses of hexagonal soft foams. A set of two-dimensional hexagonal foams of the same volume fration 0.09 but different ligament lengths are modelled, and their quasi-stati frature energies are omputed via the same proedure as desribed above. The resulting frature energies of two different orientations are plotted against the ligament length in Fig. 5a. In both ases, the dimensionless frature energy rw is approximately linear in the ligament length l r. Similarly, we fix the length of eah ligament at l r 4. 2, and vary the solid volume fration from to The resulting frature energies of the two orientations are plotted as funtions of the solid volume fration in Fig. 5b. As expeted, at relatively small volume fration, the frature energy is approximately proportional to the volume fration. Comparing between the two orientations, it is found that the frature energy in an armhair orientation is onsistently higher than that of the same struture in a zigzag orientation. Suh a 15

16 differene ould be attributed to the anisotropy in ligament density. As illustrated by Fig. 6, a horizontal rak mainly goes through the inlined ligaments in the armhair orientation, while a rak through a foam in the zigzag orientation mainly breaks the vertial ligaments. The numbers of ligaments ut by unit rak length in the two orientations differ by a fator of 2 3, whih explains the differene in the effetive frature energies. The same phenomena may also be understood by onsidering the effetive sharpness of a rak. As shown by Fig. 6, the rak path in a zigzag orientation is nearly straight, while that in an armhair orientation is often meandering. With the rak front randomly selets one of two inlined ligaments, whih has almost idential strain energies, the effetive rak tip an be regarded as enompassing the region of both ligaments, and thus the rak is blunter. a b Fig. 5. Calulated frature energy of the hexagonal soft foams versus (a) the normalized ligament length l r at onstant volume fration 9%, and (b) the volume fration of the solid phase at onstant ligament length l r 4.2. Two different orientations are simulated as indiated by the insets (with horizontal raks). 16

17 Fig. 6. Damage patterns of hexagonal soft foams in (a) zigzag and (b) armhair orientations. The olor sale represents the damage variable, plotted in the undeformed geometry. The dash urves show the approximate paths of rak propagation. The simulations on the rak propagation proesses in soft foam strutures of various geometries, inluding those with triangular and square unit ells at different orientations, all exhibit the similar linearity as that observed in the honeyomb foam, whih further supports the saling relation (1). The effet of unit-ell geometry is only refleted in the dimensionless oeffiient, as summarized by Fig. 7. Although the relatively high frature energy of the square-ell foam with vertial/horizontal ligaments may be explained by the higher ligament density than that of the hexagonal foam (one ligament per rak length l versus one ligament per 3 l or 2 l ), the foams of other patterns do not follow the same trend. Despite the higher ligament densities, the effetive frature energies of the triangular foam or the rotated square foam are atually lower. To understand the relatively low frature energies, let us revisit the physial origin of the polymer-network-like toughening mehanism. Two neessary onditions must be met for the mehanism to be effetive: (a) the ligaments must be relatively long and uniform, so that the elasti strain energy everywhere along a ligament is lose to ritial prior to rupture; (b) the network struture must be suffiiently ompliant, so that the remaining strain energy after rupture is not passed to the neighboring ligaments. Even with the same aspet ratio and uniformity in the ligaments, the network onnetivity in either the triangular foam or the square- 17

18 ell foam is higher than that in the hexagon foam. Eah node is onneted to six ligaments in a triangular foam, and four in a square-ell foam, but only three ligaments are onneted to eah node in a hexagonal foam. Therefore, at the tip of a propagating rak in a triangular foam of square-ell foam, two or more ligaments onneted to the same node will be strethed and almost aligned in the diretion perpendiular to the rak. One one of them ruptures, the remaining elasti strain energy may be partially transferred through the ommon node to the other ligaments whih are still standing and arrying the load along the same diretion. The strain energy arried over may ontribute to the further propagation of the rak, and the overall frature energy is thus lower. It should be noted that the square-ell foam with vertial and horizontal ligaments represents a speial ase, in whih the two lateral ligaments are onneted at the diretion almost perpendiular to the load, and thus the marosopi frature energy is less affeted by the relatively high onnetivity. Fig.7. Geometri effet on the frature energy of soft foams. The geometries and orientations are represented by the sketh insets, with raks running horizontally. Despite the apparent linear relation between the effetive frature energy and the solid volume fration (or the ligament length l ), the polymer-network-like toughening mehanism is unlikely to make a soft ellular material with marosopi pores tougher than the bulk solid. By using the same 18

19 method on a bulk solid, we have onfirmed that the dimensionless frature energy of a ondensed struture is approximately 2, just as shown in the literature [23]. The fator whih has not been taken into onsideration here is the size-dependeny of material strength. It is well-known that, due to the presene of defets, larger samples of the same material would exhibit a lower tensile strength. The ritial energy density W whih sales with the squared of the rupture strength, is dependent on the ligament thikness d (and is usually a dereasing funtion). On the other hand, without resolving mirosopi defets, the phase-field model used in the urrent paper will not predit any size effet, and W is taken as a material parameter for normalization. Instead, if we assume the saling relation of brittle solids from linear elasti frature mehanis W ~ d 2 n with the saling index n 1 2 [25], and follow the geometri relation for regular losed-ell foams d ~ l, we will arrive at an effetive frature energy almost independent of or l. For non-brittle materials, the saling index n is usually less than 1 2, and the effetive frature energy will be weakly dependent on and l. Furthermore, in the limiting ase when the ligaments are thin enough that the theoretial strength ould be ahieved, will beome size-independent, and the saling law (1) ould be fully reovered. The size-dependeny of tensile strength, whih has been extensively studied [24], is not a main fous of the urrent paper. W Here we further investigate geometri effets by studying soft elasti foams with non-straight ligaments. The saling relation, d ~ l, represents the geometry of losed-ell foams with relatively straight ligaments (or flat ell walls). In general, if one allows non-straight or folded ligaments, the volume fration an be varied independently from the ligament thikness d. In other words, one may inrease the solid volume fration while keeping the ligament thikness small to ahieve higher frature toughness in the struture. 19

20 a b Fig. 8. (a) Sketh of the unit ell of a soft elasti foam ontaining serpentine ligaments. (b) Simulated deformation and frature proess in the soft elasti foam. The shading shows the dimensionless strain energy density The deformed shape is plotted by downsaling the atual displaement value to 10%. W W. As an illustrative example, we onstrut a numerial model by repeating the unit ell as skethed in Fig. 8 (a). Unlike in the above examples, the initial geometry of a ligament takes a serpentine form. The material is taken to be soft enough so that the ligaments are insensitive to the stress onentration at the folding orners. During deformation, the ligaments will first be straightened and then rupture. As shown by Fig. 8b, the strain-energy distribution in the ligaments at the rak tip is still lose to uniform, with the value lose to W prior to rupture. By varying the width of the serpentine pattern and using the same method as in previous examples, we evaluate the frature energy of several strutures with ligament thikness taken to be r and the unit ell size 18.8r 18. 8r, and plot it as a funtion of the solid volume fration. As shown by Fig. 9., the saling law (1) still holds for the foam strutures with serpentine ligaments. Despite the different geometries of the unit ells, the extrapolation of the urve to lower volume frations will give similar frature energy levels as strutures with straight ligaments. However, due to the muh more ondensed nature of the folding strutures, the frature energy is signifiantly improved. For the struture shown by Fig. 8, the volume fration 20

21 reahes 75 %, and the dimensionless frature energy is rw 25, more than one order of magnitude higher than the foam strutures with straight ligaments or the same material in a bulk form. Fig. 9. Calulated frature energy of soft foam strutures with serpentine ligaments, as a funtion of the solid volume fration. The strutures have idential unit-ell size and ligament thikness. The volume fration is ontrolled by hanging the width of the serpentine pattern, as indiated by the insets. Without realisti material models or optimized design parameters, this numerial example is just an illustration of the toughening mehanism. Nevertheless, it is evident that the frature energy of a material may be signifiantly inreased by adopting similar strutural designs. More interestingly, suh a toughening mehanism is not just limited to soft solids, espeially when the ligament size is small. It should also be noted that the frature energy inrease is obtained at the expense of the initial strutural stiffness. As shown by Fig. 10., the effetive initial modulus is more than two orders of magnitude lower than that of the onstituting solid. Suh a relation between strutural ompliane and frature toughness is similar to that in the mirorak-toughening mehanism, although the latter is usually studied in the ontext of linear elasti frature mehanis [26-28]. 21

22 Fig. 10. The alulated nominal-stress-streth urve of a soft foam with serpentine ligaments as shown by Fig. 8 (without a pre-existing rak). The initial stiffness of the struture is more than two orders of magnitude lower than the solid material. The stress-streth urve exhibits a strain-stiffening behavior, even though the material is taken to be neo-hookean. Moreover, even though the material is taken to be neo-hookean, the stress-streth urve exhibit a lear strain-stiffening segment at relatively large streth, just like the behavior of elastomers at the streth limit of polymer hains. Here, the strain stiffening orresponds to the straightening of the serpentine ligaments. To some extent, suh a struture an be regarded as a marosopi model system for the frature of elastomers. The detailed design and optimization of strutures of the kind, although interesting, is beyond the sope of the urrent paper. We are eagerly awaiting the designs and manufaturing of tough materials by utilizing this mehanism. Conlusion Drawing an analogy between the ompliant ligaments in a soft elasti foam and the polymer hains in an elastomer, this paper proposes a polymer-network-like toughening mehanism and derives 22

23 the saling relation between the marosopi frature energy and the strutural harateristis of soft foam strutures. Different from the energy absorbing mehanism of rigid foams whih is mainly effetive at ompression, the polymer-network-like toughening mehanism allows a soft foam to effetively dissipate energy when the struture is subjet to tension. Through a phase-field model developed speifially for the frature of elastomers, the toughening mehanism as well as the saling relation is then verified on soft foam strutures of various geometries. In addition to the saling law, it is found that the geometri parameters suh as the ligament density and the network onnetivity will also affet the frature energy of soft foams. Finally, to inrease the volume fration of the solid phase without affeting the thikness or slenderness of eah ligament, a type of soft foam strutures of serpentine ligaments is proposed. Numerial study suggests that suh strutures may reah an effetive frature energy muh higher than that of the orresponding bulk material. In other words, one may toughen a soft material just by utting slots or holes in it. Aknowledgement The support from the Bailey Researh Career Development Award is gratefully aknowledged. Referenes [1] Aranson, I. S., Kalatsky, V. A., & Vinokur, V. M., 2000, Continuum field desription of rak propagation, Physial review letters, 85(1), 118. [2] Maiti, S. K., Gibson, L. J., & Ashby, M. F., 1984, Deformation and energy absorption diagrams for ellular solids, Ata metallurgia, 32(11), pp [3] Han, F., Zhu, Z., & Gao, J., 1998, Compressive deformation and energy absorbing harateristi of foamed aluminum, Metallurgial and Materials Transations A, 29(10), pp

24 [4] Gibson, L. J., & Ashby, M. F., 1999, Cellular solids: struture and properties, Cambridge university press. [5] Maiti, S. K., Ashby, M. F., & Gibson, L. J., 1984b, Frature toughness of brittle ellular solids, Sripta Metallurgia, 18(3), pp [6] Gibson, L. J., Ashby, M. F., Zhang, J., & Triantafillou, T. C., 1989, "Failure surfaes for ellular materials under multiaxial loads I. Modelling, International Journal of Mehanial Sienes, 31(9), pp [7] Brezny, R., & Green, D. J., 1991, Fators ontrolling the frature resistane of brittle ellular materials, Journal of the Amerian Cerami Soiety, 74(5), pp [8] Chen, J. Y., Huang, Y., & Ortiz, M., 1998, Frature analysis of ellular materials: a strain gradient model, Journal of the Mehanis and Physis of Solids, 46(5), pp [9] Bažant, Z. P., Zhou, Y., Zi, G., & Daniel, I. M., 2003, Size effet and asymptoti mathing analysis of frature of losed-ell polymeri foam, International journal of solids and strutures, 40(25), pp [10] MIntyre, A., & Anderton, G. E., 1979, Frature properties of a rigid polyurethane foam over a range of densities, Polymer, 20(2), pp [11] Lake, G. J., & Thomas, A. G., 1967, The strength of highly elasti materials, Proeedings of the Royal Soiety of London, Series A, Mathematial and Physial Sienes, 300(1460), pp [12] Thomas, A. G., 1955, Rupture of rubber. II. The strain onentration at an inision, Journal of Polymer siene, 18(88), pp [13] Karma, A., Kessler, D. A., & Levine, H. (2001), Phase-field model of mode III dynami frature, Physial Review Letters, 87(4), p

25 [14] Hakim, V., & Karma, A., 2009, Laws of rak motion and phase-field models of frature, Journal of the Mehanis and Physis of Solids, 57(2), pp [15] Eastgate, L. O., Sethna, J. P., Rausher, M., Cretegny, T., Chen, C. S., & Myers, C. R., 2002, Frature in mode I using a onserved phase-field model, Physial Review E, 65(3), p [16] Wang, Y. U., Jin, Y. M., & Khahaturyan, A. G., 2002, Phase field miroelastiity theory and modeling of elastially and struturally inhomogeneous solid, Journal of Applied Physis, 92(3), pp [17] Maroni, V. I., & Jagla, E. A., 2005, Diffuse interfae approah to brittle frature, Physial Review E, 71(3), p [18] Spatshek, R., Hartmann, M., Brener, E., Müller-Krumbhaar, H., & Kassner, K., 2006, Phase field modeling of fast rak propagation, Physial review letters, 96(1), p [19] Karma, A., & Lobkovsky, A. E., 2004, Unsteady rak motion and branhing in a phase-field model of brittle frature, Physial review letters, 92(24), p [20] Henry, H., & Levine, H., 2004, Dynami instabilities of frature under biaxial strain using a phase field model, Physial review letters, 93(10), p [21] Henry, H., 2008, Study of the branhing instability using a phase field model of inplane rak propagation, EPL Europhysis Letters, 83(1), p [22] Miehe, C., Welshinger, F., & Hofaker, M., 2010, Thermodynamially onsistent phase field models of frature: Variational priniples and multi field FE implementations, International Journal for Numerial Methods in Engineering, 83(10), pp [23] Miehe, C., & Shänzel, L. M., 2014, Phase field modeling of frature in rubbery polymers. Part I: Finite elastiity oupled with brittle failure, Journal of the Mehanis and Physis of Solids, 65, pp

26 [24] Bazant, Zdenek P, Saling of strutural strength, Butterworth-Heinemann, [25] Griffith, A. A. The Phenomena of Rupture and Flow in Solids, Philosophial Transations of the Royal Soiety (London), 221 (1921), pp [26] Hoagland, R. G., Embury, J. D., and Green, D. J.. "On the density of miroraks formed during the frature of eramis." Sripta Metallurgia 9, no. 9 (1975): [27] Evans, A. G. and Fu, Y.. "Some effets of miroraks on the mehanial properties of brittle solids II. Mirorak toughening." Ata Metallurgia 33, no. 8 (1985): [28] Huthinson, J. W.. "Crak tip shielding by miro-raking in brittle solids." Ata metallurgia 35, no. 7 (1987):