Marco Dompé

Mehdi Vahdati

Froukje Van Ligten

Francisco J Cedano-Serrano

Dominique Hourdet

Costantino Creton

Marco Zanetti

Pierangiola Bracco

Jasper Van Der Gucht

Thomas Kodger

Marleen Kamperman

Enhancement of the Adhesive Properties by Optimizing the Water Content in PNIPAM-Functionalized Complex Coacervates

Keywords: complex coacervation, water content, underwater adhesion, extrusion, bioinspired adhesives

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Adhesive technology, despite being known and exploited since the middle Pleistocene, [START_REF] Mazza | A new Palaeolithic discovery: tarhafted stone tools in a European Mid-Pleistocene bone-bearing bed[END_REF] is rarely applied when dealing with adverse environments. In medicine, for instance, surgical tissue closure still relies on conventional techniques, such as suturing and stapling, [START_REF] Lauto | Adhesive biomaterials for tissue reconstruction[END_REF] which have many pitfalls, e.g. inflammatory reactions, scar formation and stress localization. [START_REF] Bhagat | Degradable Adhesives for Surgery and Tissue Engineering[END_REF][START_REF] Bouten | The chemistry of tissue adhesive materials[END_REF][START_REF] Mehdizadeh | Design Strategies and Applications of Tissue Bioadhesives[END_REF] Tissue glues represent a valid alternative to these methods, especially because of the easy handling and minimal tissue damage. [START_REF] Bhagat | Degradable Adhesives for Surgery and Tissue Engineering[END_REF][START_REF] Mehdizadeh | Design Strategies and Applications of Tissue Bioadhesives[END_REF][START_REF] Bré | Taking tissue adhesives to the future: from traditional synthetic to new biomimetic approaches[END_REF] However, most commercially available adhesives fail to offer a proper performance in wet and dynamic environments and do not achieve the required bonding strength. [START_REF] Bhagat | Degradable Adhesives for Surgery and Tissue Engineering[END_REF][START_REF] Rahimnejad | Mussel-inspired hydrogel tissue adhesives for wound closure[END_REF] In nature, many aquatic species (e.g. sandcastle worms, mussels) have managed to solve the challenges related to underwater adhesion by developing protein-based glues that bond strongly to a variety of surfaces even in submerged conditions. [START_REF] Stewart | The tube cement of Phragmatopoma californica: a solid foam[END_REF][START_REF] Waite | Mussel Adhesion: Finding the Tricks Worth Mimicking[END_REF][START_REF] Walker | The histology, histochemistry and ultrastructure of the cement apparatus of three adult sessile barnacles, Elminius modestus, Balanus balanoides and Balanus hameri[END_REF] A phenomenon directly contributing to the processing and delivery of these materials is complex coacervation, [START_REF] Stewart | Complex coacervates as a foundation for synthetic underwater adhesives[END_REF][START_REF] Stewart | The role of coacervation and phase transitions in the sandcastle worm adhesive system[END_REF] an associative phase separation process which requires the presence of two oppositely charged polyelectrolytes: [START_REF] Gucht | Polyelectrolyte complexes: Bulk phases and colloidal systems[END_REF] these, under the proper conditions (pH, ionic strength, mixing ratio), partition into a dilute phase (polymer-poor) and a complex coacervate phase (polymer-rich). [START_REF] Spruijt | Binodal Compositions of Polyelectrolyte Complexes[END_REF] Several research groups have employed these electrostatic interactions to fabricate viscous glues, most of which set underwater due to covalent crosslinking reactions, providing the strength required to oppose detachment. [START_REF] Kaur | Multiphase Adhesive Coacervates Inspired by the Sandcastle Worm[END_REF][START_REF] Shao | Biomimetic Underwater Adhesives with Environmentally Triggered Setting Mechanisms[END_REF][START_REF] Ahn | High-performance mussel-inspired adhesives of reduced complexity[END_REF][START_REF] Seo | Microphase Behavior and Enhanced Wet-Cohesion of Synthetic Copolyampholytes Inspired by a Mussel Foot Protein[END_REF][START_REF] Zhao | Underwater contact adhesion and microarchitecture in polyelectrolyte complexes actuated by solvent exchange[END_REF][START_REF] Zhang | Engineered Bacillus subtilis biofilms as living glues[END_REF] Previously, [START_REF] Dompé | Thermoresponsive Complex Coacervate-Based Underwater Adhesive[END_REF][START_REF] Dompé | Underwater Adhesion of Multiresponsive Complex Coacervates[END_REF][START_REF] Dompé | Tuning the Interactions in Multiresponsive Complex Coacervate-Based Underwater Adhesives[END_REF] we have produced a complex coacervate-based adhesive which sets in response to environmental triggers, forming exclusively physical bonds without the addition of any cross-linking agent. The glue is prepared by mixing oppositely charged polyelectrolytes modified with poly(N-isopropylacrylamide) (PNIPAM) chains. PNIPAM is a water-soluble polymer which phase separates at temperatures above its lower critical solution temperature (LCST, around 32 °C). The adhesive can be reinforced by heating the sample above the LCST (temperature switch), by immersing it in a lower ionic strength medium (salt switch) or by combining the two triggers (temperature + salt switch), mimicking the conditions that the glue would experience in the human body. However, despite optimizing the performance by assessing the effect of different parameters (polymer composition, temperature, ionic strength, environmental trigger and probe surface), we could obtain, at best, a 3-fold enhancement of the adhesive properties compared to the original formulation. [START_REF] Dompé | Thermoresponsive Complex Coacervate-Based Underwater Adhesive[END_REF][START_REF] Dompé | Underwater Adhesion of Multiresponsive Complex Coacervates[END_REF][START_REF] Dompé | Tuning the Interactions in Multiresponsive Complex Coacervate-Based Underwater Adhesives[END_REF] For this adhesive system, and basically for all water-containing adhesives, the adhesive performance and the mechanical properties heavily depend on water content. For instance, commercial poly(ethylene glycol) (PEG) based glues, such as DuraSeal ® and CoSeal ® , bind to tissues with a low adhesive strength, which is mainly attributable to the high water content, ranging from 90w/v% to 99w/v%. [START_REF] Kelmansky | Strong tissue glue with tunable elasticity[END_REF] Another related drawback is the significant swelling in physiological conditions (>700%), [START_REF] Barrett | Mechanically Robust, Negative-Swelling, Mussel-Inspired Tissue Adhesives[END_REF] which can lead to medical complications and further weakening of the adhesive: DuraSeal ® has been reported to cause nerve compression, [START_REF] Lee | DuraSeal-Hematoma: Concealed Hematoma Causing Spinal Cord Compression[END_REF][START_REF] Mulder | Cauda Equina Compression by Hydrogel Dural Sealant After a Laminotomy and Discectomy: Case Report[END_REF] while CoSeal ® has shown a dramatic decrease in both moduli and energy to failure (in compression mode) over a period of three days in physiological conditions due to water sorption. [START_REF] Campbell | Evaluation of absorbable surgical sealants: in vitro testing[END_REF] In order to circumvent issues associated with hydrophilic polymers, relatively hydrophobic surgical glues have been proposed. For example, Karp and co-workers designed a hydrophobic prepolymer (poly(glycerol sebacate acrylate) (PGSA)) which can be UV cross-linked in situ: the glue showed limited swelling in physiological conditions, providing stronger adhesion than standard tissue adhesives in highly dynamic environments. [START_REF] Lang | A Blood-Resistant Surgical Glue for Minimally Invasive Repair of Vessels and Heart Defects[END_REF][START_REF] Nijst | Synthesis and Characterization of Photocurable Elastomers from Poly(glycerol-co-sebacate)[END_REF] However, it is not always desirable to increase the hydrophobicity. For instance, recently Yuk et al. have developed dry double sided tapes whose adhesion mechanism is initiated by the elimination of the interfacial water present at the surface: in this case, water absorption by the adhesive matrix is considered a crucial requirement to promote an intimate contact with the tissue, resulting in the quick establishment of both covalent and non-covalent interactions with the surface. [START_REF] Yuk | Dry double-sided tape for adhesion of wet tissues and devices[END_REF] Additionally, water may also act as plasticizer, which can improve the adhesion performance. [START_REF] Feldstein | Molecular Nature of Pressure-Sensitive Adhesion[END_REF] Feldstein et al. studied how water content affects the adhesive properties of poly(N-vinyl pyrrolidone)-poly(ethylene glycol) (PVP-PEG) blends. [START_REF] Chalykh | Pressuresensitive adhesion in the blends of poly(N-vinyl pyrrolidone) and poly(ethylene glycol) of disparate chain lengths[END_REF][START_REF] Roos | Viscoelasticity and tack of poly(vinyl pyrrolidone)-poly(ethylene glycol) blends[END_REF] By increasing the content of PEG, the water content increased and adhesion was enhanced, enabling the material to sustain higher deformations without considerably affecting the ultimate tensile strength. The optimal performance, in terms of peel force, was detected at a PEG content of 36%, with the mode of failure transitioning from adhesive to cohesive, allowing fibrillation within the material. A further increase in water sorption, however, caused a dilution of the entanglement structure, excessively lowering the modulus and making the material too fluid-like, leading to a decrease in adhesion. [START_REF] Feldstein | Molecular Nature of Pressure-Sensitive Adhesion[END_REF] Analogous to the work of Feldstein, [START_REF] Feldstein | Molecular Nature of Pressure-Sensitive Adhesion[END_REF][START_REF] Chalykh | Pressuresensitive adhesion in the blends of poly(N-vinyl pyrrolidone) and poly(ethylene glycol) of disparate chain lengths[END_REF][START_REF] Roos | Viscoelasticity and tack of poly(vinyl pyrrolidone)-poly(ethylene glycol) blends[END_REF] we believe that the optimization of the water content and, consequently, of the polymer concentration is necessary to promote further enhancement of the adhesion performance.

In the PNIPAM-based adhesive system that we recently developed, [START_REF] Dompé | Thermoresponsive Complex Coacervate-Based Underwater Adhesive[END_REF][START_REF] Dompé | Underwater Adhesion of Multiresponsive Complex Coacervates[END_REF][START_REF] Dompé | Tuning the Interactions in Multiresponsive Complex Coacervate-Based Underwater Adhesives[END_REF] the water content is always above 90%, strongly limiting the adhesive properties. In this work, we test several strategies to reduce and control the water content within the glue, with the goal of enhancing the work of adhesion of at least one order of magnitude. The first strategy is by changing the ionic strength: a lower salt concentration results in a higher polymer concentration in the adhesive. [START_REF] Spruijt | Binodal Compositions of Polyelectrolyte Complexes[END_REF] The second strategy entails raising the polymer concentration of the solutions when preparing the complex coacervates. The third strategy involves the use of mechanical force to squeeze the water out of the complex coacervate phase using an extruder. Earlier research has shown that the high shear forces experienced by polyelectrolyte complexes during extrusion are effective in removing liquids trapped within pores, resulting in the reinforcement of the material. [START_REF] Fu | Extruded Superparamagnetic Saloplastic Polyelectrolyte Nanocomposites[END_REF][START_REF] Shamoun | Extruded Saloplastic Polyelectrolyte Complexes[END_REF][START_REF] Wang | Tough strained fibers of a polyelectrolyte complex: pretensioned polymers[END_REF]

RESULTS AND DISCUSSION

Strategy 1: Effect of Salt Concentration

The first explored strategy to reduce the water content is to decrease the salt concentration at preparation conditions. Previous work has indeed shown that complex coacervates retain less water when prepared at low ionic strength due to the stronger electrostatic interactions. [START_REF] Spruijt | Binodal Compositions of Polyelectrolyte Complexes[END_REF] The characteristics of the polymers (Figures S1-S2-S3 and Table S1) and of the samples (Table S2) used are shown in the Supporting Information. Figure 1 reports the water content of the analysed samples against the added salt concentration at T = 20 °C (no change in water content is observed as function of temperature). [START_REF] Dompé | Thermoresponsive Complex Coacervate-Based Underwater Adhesive[END_REF][START_REF] Dompé | Tuning the Interactions in Multiresponsive Complex Coacervate-Based Underwater Adhesives[END_REF] In line with previous reports, [START_REF] Spruijt | Binodal Compositions of Polyelectrolyte Complexes[END_REF] complex coacervates prepared from homopolymer solutions have a lower water content at lower ionic strength. However, the complex coacervates prepared from graft copolymers show a constant water content over the whole range of salt concentrations analysed. As observed in our previous work, [START_REF] Dompé | Tuning the Interactions in Multiresponsive Complex Coacervate-Based Underwater Adhesives[END_REF] the excess water could be absorbed, at low ionic strength, by the PNIPAM domains and, at high ionic strength (lower PNIPAM hydration), [START_REF] Zhang | Specific Ion Effects on the Water Solubility of Macromolecules: PNIPAM and the Hofmeister Series[END_REF] by the polyelectrolyte matrix. Additionally, while homopolymer complex coacervates are transparent or slightly turbid at every salt concentration, graft copolymer complex coacervates turn white at 0.5 M NaCl and below. At low ionic strength, the formation of a microporous structure, due to the water entrapment within the material, is observed: the higher opacity is attributed to the greater dimensions of the pores, which have sizes big enough to scatter light. [START_REF] Hariri | Mechanical Properties of Osmotically Stressed Polyelectrolyte Complexes and Multilayers: Water as a Plasticizer[END_REF] However, even though the water content does not change significantly upon lowering the salt concentration, the mechanical properties are strongly affected, as shown in Figure S4. At high ionic strength (GS7P1) the material exhibits characteristics of a viscoelastic fluid (Figure S4A), with both storage (G') and loss (G'') moduli crossing each other at a characteristic frequency ωc, from which it is possible to calculate the relaxation time τ (1/ωc): at timescales longer than τ (ω < ωc) the polymer chains can relax the applied stress by sliding along each other, while at shorter timescales (ω > ωc) the chain movement is partially inhibited, leading to a solid-like response. When lowering the salt concentration, a liquid-to-solid transition is detected below the LCST: both moduli display a very weak frequency dependence, with G' always higher than G'', which indicates gel formation. This behaviour is due to the stronger electrostatic interactions at lower ionic strength, which slow down the relaxation of the polyelectrolyte chains. [START_REF] Spruijt | Linear Viscoelasticity of Polyelectrolyte Complex Coacervates[END_REF] This also leads to a high complex viscosity at low ionic strength (Figure S4B), which makes it difficult for the sample to be injected through a small-bore needle during application. Therefore, since the water content is independent on the salt concentration, lowering the ionic strength is not an effective strategy to increase the polymer content within the adhesive matrix.

Strategy 2: Effect of Starting Polymer Concentration

The second strategy consists in increasing the polymer concentration upon mixing. Figure 2 shows a schematic phase diagram of the graft copolymer mixture, plotting the added salt concentration as function of the polymer concentration. In the phase diagram (Figure 2A), two regions can be recognized: a one phase region (white colour) and a two phase region (coloured with a yellow-to-red gradient) in which phase separation into a dilute phase (on the left border of the diagram) and a complex coacervate phase (on the right border) occurs. In Figure 2B we assume that salt partitions in the same way between dilute and complex coacervate phase, in line with other reports in literature: [START_REF] Spruijt | Binodal Compositions of Polyelectrolyte Complexes[END_REF] however, it should be mentioned that other researchers have found the salt concentrations are not the same in the two phases. [START_REF] Li | Phase Behavior and Salt Partitioning in Polyelectrolyte Complex Coacervates[END_REF][START_REF] Perry | PRISM-Based Theory of Complex Coacervation: Excluded Volume versus Chain Correlation[END_REF][START_REF] Qin | Interfacial Tension of Polyelectrolyte Complex Coacervate Phases[END_REF] The dilute phase is always a liquid aqueous solution (as indicated by the yellow colour in Figure 2) at any salt concentration, while the complex coacervate phase transitions from a solid (red) to a liquid (orange) by increasing the added NaCl concentration. When surpassing the critical salt concentration (CSC), which in this system is between 0.8 and 0.85 M NaCl, the electrostatic interactions are completely screened, preventing phase separation. At 0.7 M NaCl, a viscous, phase-separated material with the optimal characteristics of an injectable adhesive is obtained: therefore, this ionic strength is set as standard for all following experiments. When increasing the polymer concentration while maintaining the salt concentration constant, the system (black dots in Figure 2B) always phase separates into the same two phases with the same polymer concentrations (blue and red dots in Figure 2B): the only factor that changes is the volumetric ratio between the two phases, with the complex coacervate phase percentage increasing as the initial polymer concentration increases. When the polymer concentration at mixing reaches the polymer concentration in the complex coacervate phase, the system enters a one phase region: the polymer concentration at mixing is equal to the final one, meaning that an increase in the polymer concentration in the preparation stage leads to a final material with a higher polymer concentration. In order to reach a concentration which is high enough to access this region, several samples, shown in Table S3, were prepared by varying the polymer concentration in the preparation stage. As shown in Figure 3A, over the whole range of polymer concentrations analysed, complex coacervates with nearly the same water content (around 90%), and consequently same polymer concentration, are obtained. Figure 3B indicates that the complex coacervate volume fraction increases linearly when increasing the polymer concentration, as expected.

However, contrary to our expectations, it was not possible to enter the one phase region (and therefore increase the polymer concentration) because of solubility issues, especially concerning the anionic graft copolymer. When mixing the polyelectrolytes at a 1:1 charge ratio, the maximum reachable total polymer content is 4.39% w/v, which, as shown in Figure 3B, is lower than the polymer concentration in the complex coacervate phase (around 5.3% w/v), obtained by extrapolating the regression line in Figure 3B to 100% (border between the two-phase and one-phase region). The value obtained with this method is comparable to the one obtained by performing a thermogravimetric analysis (TGA) on the complex coacervate phase (5.5 % w/v), highlighting the validity of the phase diagram. In conclusion, the devised strategy is not effective in raising the polymer content in the complex coacervate phase.

Strategy 3: Extrusion

As a third strategy, we used a mini-extruder to mechanically force water out of the adhesive (which is prepared at 0.7 M NaCl and at 0.05 M charged monomer concentration). After occurrence of complex coacervation, the dilute phase is removed. The complex coacervate is heated above its LCST and inserted through an inlet on top of the screws in the extrusion chamber. Then, the material is submitted to an extrusion cycle at 50 °C. At first, the material is moved forward in the chamber by the shearing forces of the extruder screws; its progression is then stopped when the whole sample has passed through the screws, with the complex coacervate being solidified in an inner reservoir (Figure S5). After 3 minutes in the chamber, the material is removed and stored in the refrigerator overnight and, if required, the following day another cycle is performed (up to a maximum of 3). After one cycle, expelled water, separated from the material, is visible in the extruded chamber (Figure S5), indicating that this strategy is effective in reducing the water content and, therefore, in increasing the polymer concentration of the material. In order to study the process in more detail, the effect of the extruder frequency and of the number of extrusion cycles are investigated (Figures 4,S6,S7). A drop in water content from 88% to 78% (Figure 4A), giving rise to an increase in viscoelastic moduli G' and G'' (Figure S6), is observed after one extrusion cycle, independently of extruder frequency. Therefore, for the remaining measurements, the extruder frequency was set to 30/min. In contrast, increasing the number of extrusion cycles, after homogenizing the material in the fridge overnight between two successive cycles, significantly influences the complex coacervate phase properties. The water content decreases at a higher number of cycles, reaching 63% after three extrusion processes (Figure 4B).

Differently from the extruder frequency, a higher number of cycles profoundly affects the rheological behaviour: the as-made material shows typical features of a viscous liquid at a temperature below the PNIPAM LCST, with both moduli frequency dependent (Figure S7A). G' overcomes G'' at a crossover frequency ωc of 7.36 rad/s, corresponding to a relaxation time τ = 1/ωc = 0.85 s. After one extrusion cycle, the water content drops to 78%, with the complex coacervates still showing liquid behaviour. However, the moduli become less frequency dependent and increase. The relaxation time increases by an order of magnitude, to τ = 8.33 s. This behaviour becomes more obvious at a higher number of cycles: after three extrusion cycles, the crossover frequency is not detectable in the window of frequencies analysed, meaning that τ ≥ 62.5 s. This is evidence of a higher polymer concentration in the material which slows down chain relaxation, in accordance with the sticky Rouse model, which has been adopted to describe chain dynamics in complex coacervates [START_REF] Spruijt | Linear Viscoelasticity of Polyelectrolyte Complex Coacervates[END_REF][START_REF] Spruijt | Relaxation Dynamics at Different Time Scales in Electrostatic Complexes: Time-Salt Superposition[END_REF] and which predicts a power-law increase of the relaxation time versus the polymer volume fraction, as observed in Figure 4C.

When performing a temperature switch by heating the sample above the LCST of PNIPAM (between 10 °C and 35 °C, Figures S9-S10), the collapse of the thermoresponsive chains is promoted: as a result, all samples acquire characteristics of soft solids, with G' higher than G'', and both frequency independent (Figure S7B). G' shows a power-law increase as function of the polymer concentration (Figure 5A), highlighting that the extrusion process is effective in stiffening the material: this is in accordance with the statistical theory of rubber elasticity, which states that the storage modulus is proportional to the amount of elastically active chains per unit volume. The adhesive can also be reinforced by performing a salt switch, namely by exposing the material to an ionic strength gradient. When soaking the sample in a medium prepared at 0.1 M NaCl, the excess salt ions diffuse out of the adhesive, strengthening the electrostatic interactions between the polyelectrolyte chains. This promotes a reinforcement of the material over time, with the moduli progressively increasing and heading towards a plateau after one hour (Figure 5B). When applying a combined salt and temperature switch, different kinetics are observed (Figure 5B). The rheological properties increase more abruptly during the first stages of the transition, due to the immediate collapse (response) of the thermoresponsive PNIPAM chains when exposed to a medium with a higher temperature than the LCST. This process is much faster than ion diffusion from a confined region. In contrast, for the salt switch, a contact time of one hour is required for the full setting of the material. [START_REF] Dompé | Underwater Adhesion of Multiresponsive Complex Coacervates[END_REF] At the end of the transition, the complex coacervate phase has the characteristics of a soft elastic solid, with higher moduli at a higher number of cycles (Figure S11A). The obtained values (Figure S11B) are similar to the ones obtained after a single temperature switch: this means that the total number of nodes is almost the same. However, in this case, both PNIPAM-PNIPAM and electrostatic interactions contribute to the final moduli, while, when applying a temperature switch, the formed network results mainly from the collapse of the thermoresponsive chains.

Figure 6 shows the rheological properties of the sample obtained after one extrusion cycle and of the homopolymer complex coacervate: despite having a different preparation history, the two materials have the same water content (78% for both samples). At temperatures below PNIPAM LCST, the polymer concentration seems to dictate the rheological properties, with both samples showing a similar behaviour (Figure 6A). However, the presence of PNIPAM is crucial above the LCST: when heated (Figure 6B), the extruded sample turns into a gel because of the collapse of the PNIPAM chains which abruptly slows down the relaxation processes, while, in the homopolymer complex coacervates, no considerable variation is visible due to the absence of any temperature sensitive unit.

However, both samples show a transition in response to a combined temperature-salt switch, mimicking the conditions that the sample would experience in a physiological environment (Figure 6C). Both materials turn into soft gels, with G' overcoming G'' at any analysed frequency. However, the moduli of the homopolymer complex coacervates are more frequency dependent, meaning that the crossover frequency (not detectable because it is out of the frequency range analysed) is anticipated to be higher and the relaxation time will therefore be lower. This indicates that, despite a similar polymer concentration, the presence of the PNIPAM units allows the formation of a soft elastic network which is more effective in slowing down chain relaxation.

The underwater adhesive properties of the extruded graft samples are determined using the setup developed by Sudre at al. [START_REF] Sudre | Reversible adhesion between a hydrogel and a polymer brush[END_REF] In Figure 7A, the effect of a combined temperature-salt switch is reported for samples which were exposed to different extrusion cycles. Compared to the unprocessed sample (0 cycles), the adhesive which has been submitted to one extrusion cycle shows a much higher adhesion energy, with both an increase in the peak stress (60 kPa) and maximum strain (3500%). The mode of failure is the same in both samples, with the material failing cohesively, leaving residues on both retracting surfaces. The higher polymer concentration reinforces the material, enabling a higher resistance to an applied stress: this might be ascribed to a combination of increased stiffness, as observed in Figure 5A, and dissipation, characterized by the damping factor tan δ (G''/G'), which shows a 50% increase after 1 extrusion cycle (Figure 7B). However, a further increase in polymer concentration, obtained when performing additional extrusion cycles, leads to a decrease in the adhesive performance: despite showing a higher peak stress than the unprocessed sample, the strain at break decreases significantly. The mode of failure also changes, with the material now failing adhesively, without residues on the probe. This suggests a much less viscoelastic and more elastic gel-like behaviour. In adhesion science, this is generally quantified by the ratio of the damping factor to the storage modulus, employed as a gauge of expected adhesive performance. [START_REF] Wang | Waterborne, Nanocomposite Pressure-Sensitive Adhesives with High Tack Energy, Optical Transparency, and Electrical Conductivity[END_REF] An excessively high value of G', relative to the dissipative character of the adhesive, will prevent cavities formation in the vertical direction: [START_REF] Wang | Waterborne, Nanocomposite Pressure-Sensitive Adhesives with High Tack Energy, Optical Transparency, and Electrical Conductivity[END_REF] this means that when the material is too stiff and too elastic, i.e. a too low value of tan δ /G' (Figure 7B), resistance to interfacial crack propagation is too low and the material fails adhesively at low strain. [START_REF] Wang | Waterborne, Nanocomposite Pressure-Sensitive Adhesives with High Tack Energy, Optical Transparency, and Electrical Conductivity[END_REF] Therefore an optimal polymer concentration is needed for a balance between cohesive and adhesive properties. [START_REF] Deplace | Fine Tuning the Adhesive Properties of a Soft Nanostructured Adhesive with Rheological Measurements[END_REF] Another requirement that an underwater adhesive should meet is the dimensional stability in submerged conditions. [START_REF] Barrett | Mechanically Robust, Negative-Swelling, Mussel-Inspired Tissue Adhesives[END_REF][START_REF] Jones | Water-Borne Endovascular Embolics Inspired by the Undersea Adhesive of Marine Sandcastle Worms[END_REF] When performing an extrusion process, the complex coacervate phase is brought in an out of equilibrium condition and, if submerged, could likely reabsorb the water which has been previously removed. Swelling experiments were therefore performed on the samples in conditions mimicking the physiological environment (T = 37 °C, 0.1 M [NaCl], pH 7.0) over a period of one hour to check the water uptake/release in the timescale of the experiment (Figure S12A). The unprocessed sample, with the highest water content, shows a negative swelling ratio: this is an indication of shrinking, mainly due to the collapse of the PNIPAM chains, as already observed in other work on thermoresponsive adhesives, [START_REF] Barrett | Mechanically Robust, Negative-Swelling, Mussel-Inspired Tissue Adhesives[END_REF] and to the contraction of the polyelectrolyte matrix at lower ionic strength.

Additionally, the sample turns white (Figure S13), evidence that water also remains trapped within the material, forming a porous structure, as already proposed in our previous work. [START_REF] Dompé | Thermoresponsive Complex Coacervate-Based Underwater Adhesive[END_REF][START_REF] Dompé | Underwater Adhesion of Multiresponsive Complex Coacervates[END_REF][START_REF] Dompé | Tuning the Interactions in Multiresponsive Complex Coacervate-Based Underwater Adhesives[END_REF] The swelling ratio increases as a function of extrusion cycles: the samples submitted to one extrusion cycle shows a higher dimensional stability, exhibiting a swelling ratio close to zero (Figure S12A), similarly to what is observed in hydrophobic tissue adhesives. [START_REF] Lang | A Blood-Resistant Surgical Glue for Minimally Invasive Repair of Vessels and Heart Defects[END_REF][START_REF] Nijst | Synthesis and Characterization of Photocurable Elastomers from Poly(glycerol-co-sebacate)[END_REF] This means that the water removed through the extrusion process is not reabsorbed by the sample (no swelling is observed over a five days period, Figure S12B). However, when the amount of extrusion cycles is further increased, the material swells due to water sorption (Figure S12A), which might also contribute to the decrease in the adhesive performance as seen in Figure 7, as reported for PEG-based tissue adhesives. [START_REF] Campbell | Evaluation of absorbable surgical sealants: in vitro testing[END_REF] When decreasing the amount of water within the material, the adhesive, after setting, becomes less opaque (Figure S13), evidence of a lower amount and of a smaller size of pores containing trapped water within the material. [START_REF] Shamoun | Extruded Saloplastic Polyelectrolyte Complexes[END_REF] Therefore, the optimal balance between adhesive and cohesive properties, together with a proper dimensional stability, leads to a large enhancement of the adhesion performance in the sample submitted to only one extrusion cycle. A poorer adhesive performance is observed when submitting the samples only to a temperature trigger (Figure 8A). Despite obtaining similar moduli when reinforcing the material with a single or a combined trigger (Figures 5A-S11B), the sample fails in an adhesive fashion much earlier when applying only a temperature trigger: this could be correlated to the size of the polymers involved in the reinforcing mechanism. When just a temperature trigger is performed, only the short PNIPAM chains are collapsed, forming domains of a small size. Thereby, when applying a detaching stress, the chains can be stretched only to a small extent. By contrast, by decreasing the ionic strength, stronger interactions between the long backbones are activated. In order to break the sample, also these additional bonds need to be disrupted and the adhesive can be stretched to a higher extent before failure, resulting in a higher toughness. Additionally, this behaviour might also be ascribed to the more viscoelastic character of the material after a combined switch, as confirmed by the slightly higher values of the damping factor and of tan δ/G' as compared to values obtained for a single temperature switch (Figure 9A). Similarly, when probing the homopolymer counterpart with the same water content in physiological conditions, an adhesive failure at low strain is observed (Figure 8B). This behaviour is surprising since, as observed in Figure 9B, the homopolymer complex coacervates have a more dissipative character than the extruded sample. In this case, the linear rheology properties fail to predict the non-linear behaviour observed in the probe-tack test.

The improved adhesive performance shown by the extruded sample might indicate that the presence of two types of interactions in the same material (PNIPAM nodes and electrostatic interactions) favours an increase in toughness (not detectable via linear rheology), which might be related to the variety of bond strength present in the adhesive, as observed for polyampholyte hydrogels and double networks. [START_REF] Sun | Physical hydrogels composed of polyampholytes demonstrate high toughness and viscoelasticity[END_REF][START_REF] Gong | Why are double network hydrogels so tough?[END_REF][START_REF] Sun | Highly stretchable and tough hydrogels[END_REF] Additionally, the higher (in absolute value) swelling ratio (-23.4%) reported in our previous work [START_REF] Dompé | Underwater Adhesion of Multiresponsive Complex Coacervates[END_REF] for homopolymer complex coacervates measured in similar conditions might affect the adhesive properties: a higher amount of released water might weaken the interfacial interactions with the probe, resulting in a lower dimensional stability and a worse adhesive performance compared to the extruded samples, which show a lower swelling ratio (-9.6%). Lastly, in Figure 10 the work of adhesion (Wadh), obtained from the area below the stressstrain curves, is plotted versus the amount of extrusion cycles. After one extrusion cycle, when performing a combined switch, the work of adhesion increases from 3.8 J/m 2 to 60.6 J/m 2 , much higher than the one shown by the homopolymer counterpart (4.7 J/m 2 , not shown in Figure ) and the one obtained after performing a temperature switch only (4.6 J/m 2 ). When further raising the amount of extrusion cycles, the work of adhesion drastically decreases, with a similar trend observed for both triggers. An optimum performance is obtained when preserving a porous structure, when properly balancing the adhesive and cohesive properties and when in-situ swelling is limited.

CONCLUSIONS

Polymer concentration strongly affects the physical properties of an adhesive. PNIPAMfunctionalized complex coacervates exhibit a high water content, which inevitably affects the moduli and the adhesive properties of the material. Several routes have been adopted in this work in order to optimize the water content: among the attempted strategies, extrusion is the most effective in increasing the polymer concentration within the complex coacervate phase. While a higher amount of extrusion cycles does lead to a progressive enhancement of the dynamic moduli, the underwater Wadh initially benefits of the higher polymer concentration within the material, and then drastically drops when the water content is further decreased. By carefully tuning the polymer concentration, promising adhesion data are obtained, reaching an adhesive strength of 60 kPa and work of adhesion of 60 J/m 2 : these values are higher than those reported for commercial adhesives measured underwater, such as fibrin glues, and comparable to the highest values reported for bioinspired adhesives tested in similar conditions. [START_REF] Clancy | Marine Bioinspired Underwater Contact Adhesion[END_REF][START_REF] Rao | Tough Hydrogels with Fast, Strong, and Reversible Underwater Adhesion Based on a Multiscale Design[END_REF] However, differently from those materials, which are already in the solid state before application or need to be solidified in situ with external agents, this glue sets immediately when released in physiological conditions, experiencing an environmentallytriggered phase transition. Further studies are required to systematically address the role of the microstructure on the adhesion properties.

ASSOCIATED CONTENT

Supporting Information.

The following files are available free of charge.

Experimental section, synthesis details of the polymers, additional rheology and swelling data

AUTHOR INFORMATION

Corresponding Author

*E-mail: marleen.kamperman@rug.nl TOC FIGURE

Figure 1

 1 Figure 1 Water content versus added salt concentration for the samples analysed in Strategy 1.

Figure 2 A

 2 Figure 2 A) Complex coacervate (CC) phase diagram. In the two phase region a gradient from yellow to red is used to indicate the transition from a liquid dilute phase (left side) to a liquid CC phase (top right side) and finally to a solid CC phase (bottom right side). B) Effect of polymer concentration: zoom in the liquid CC area, in which the black dots represent the analysed samples, which phase separate into a dilute phase, shown as a light blue dot, and a complex coacervate phase, shown as a red dot.

Figure 3

 3 Figure 3 Effect of the polymer concentration on A) water content and B) complex coacervate volume fraction. The theoretical polymer concentration of the dilute (blue dot) and complex coacervate phase (red dot) are obtained by extrapolating the regression line to y = 100% and y = 0% respectively.

Figure 4

 4 Figure 4 Water content as a function of (A) extruder frequency and (B) number of extrusion cycles. (C) Relaxation time plotted as function of polymer concentration (the last point is not the real value but the relaxation time corresponding to the minimum frequency accessed in the experiment).

45

 45

Figure 5

 5 Figure 5 Setting processes. A) Storage modulus recorded at ω = 1 rad/s after a single temperature switch plotted against the polymer concentration. B) Time sweeps after one extrusion cycle when performing a salt (S) or a combined salt and temperature (S+T) switch;

Figure 6

 6 Figure 6 Rheological properties of extruded and homopolymer complex coacervates A) at 5 °C, B) at 50 °C and C) after a combined temperature and salt switch.

Figure 7 A

 7 Figure7A) Underwater adhesion performance after application of a temperature-salt switch at a nominal strain rate of 0.2 s -1 . B) Tan δ and Tan δ/G' recorded with a rheometer at ω = 1 rad/s (comparable to the strain rate in the tack measurement).

Figure 8

 8 Figure 8Underwater adhesive performance of the sample extruded once after application of a temperature-salt switch compared to A) the application of a single temperature switch and B) the performance of the homopolymer complex coacervate upon a combined trigger.

Figure 9

 9 Figure 9Tan δ and tan δ/G' recorded with a rheometer at ω = 1 rad/s for the sample extruded once after application a combined switch compared to A) the application of a single temperature switch and B) the response of the homopolymer complex coacervate upon a combined trigger.

Figure 10

 10 Figure 10 Work of adhesion versus the amount of extrusion cycles and versus the applied trigger.

ACKNOWLEDGMENTS J.v.d.G. acknowledges the European Research Council for financial support (ERC Consolidator grant Softbreak).

Funding Sources

The project received funding from the European Union's Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement no.642861.

Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.


Aurélien Doitrand

Alberto Sapora

Nonlinear implementation of Finite Fracture Mechanics: A case study on notched Brazilian disk samples

Keywords: Finite Fracture Mechanics, Ramberg-Osgood model, Nonlinearities, Brazilian disk tests, Circular holes

We formalize the coupled criterion (CC) in the framework of Finite Fracture Mechanics (FFM) to take into account the material nonlinear behavior. Once the true stress-strain curve is recorded, we estimate the Ramberg-Osgood (RO) parameters. We then exploit these values in the FFM numerical implementation, considering both an average stress requirement and the energy balance. As a case study, we consider experimental data on Brazilian disks containing a circular hole and involving two materials -PMMA and GPPS. The computational effort of the analysis increases with respect to the linear elastic case, but keeps reasonable. On the other hand, the nonlinear FFM investigation allows a significant improvement of the failure stress predictions, especially those related to smaller holes corresponding to a more pronounced nonlinear behavior detected during tests.

Introduction

Linear elastic fracture mechanics (LEFM) allows studying the propagation of an existing crack but fails to predict its nucleation. Crack initiation in brittle materials can be modeled using several methods such as, e.g., cohesive zone models [START_REF] Barenblatt | The formation of equilibrium cracks during brittle fracture: general ideas and hypotheses[END_REF][START_REF] Cornetti | Short cracks and V-notches: Finite Fracture Mechanics vs[END_REF][START_REF] Cornetti | Finite Fracture Mechanics and cohesive crack model: weight functions vs. cohesive laws[END_REF][START_REF] Doitrand | Comparison between cohesive zone and coupled criterion modeling of crack initiation in rhombus hole specimens under quasi-static compression[END_REF][START_REF] Dugdale | Yielding of steel sheets containing slits[END_REF][START_REF] Martin | Initiation of edge debonding: coupled criterion versus cohesive zone model[END_REF][START_REF] Tvergaard | The relation between crack growth resistance and fracture process parameters in elastic-plastic solids[END_REF] or phase-field models for fracture [START_REF] Bourdin | Numerical experiments in revisited brittle fracture[END_REF][START_REF] Miehe | Thermodynamically consistent phase-field models of fracture: Variational principles and multi-field FE implementations[END_REF][START_REF] Molnar | 2D and 3D Abaqus implementation of a robust staggered phase field solution for modeling brittle fracture[END_REF]]. An approach that has been specifically developed in the framework of Finite Fracture Mechanics (FFM) [START_REF] Hashin | Finite thermoelastic fracture criterion with application to laminate cracking analysis[END_REF][START_REF] Nairn | Exact and variational theorems for fracture mechanics of composites with residual stresses, traction-loaded cracks and imperfect interfaces[END_REF] so as to model crack initiation is the coupled criterion (CC) [START_REF] Leguillon | Strength or toughness? A criterion for crack onset at a notch[END_REF], which allows both the initiation loading and crack length to be determined by simultaneously fulfilling a stress condition and the energy balance. This approach has already been applied to predict crack initiation in a broad range of materials (such as e.g., composites [START_REF] Doitrand | Damage onset modeling in woven composites based on a coupled stress and energy criterion[END_REF][START_REF] Doitrand | Mesoscale analysis of damage growth in woven composites Compos Part A[END_REF][START_REF] García | A numerical study of transverse cracking in cross-ply laminates by 3D finite fracture mechanics[END_REF][START_REF] Martin | A coupled strength and toughness criterion for the prediction of the open hole tensile strength of a composite plate[END_REF], ceramics [START_REF] Leguillon | Application of the coupled stress-energy criterion to predict the fracture behaviour of layered ceramics designed with internal compressive stresses[END_REF][START_REF] Leguillon | What is the tensile strength of a ceramic to be used in numerical models for predicting crack initiation?[END_REF][START_REF] Martin | Understanding the tensile strength of ceramics in the presence of small critical flaws[END_REF]), configurations (interfaces [START_REF] Mantič | Interface crack onset at a circular cylindrical inclusion under a remote transverse tension. Application of a coupled stress and energy criterion[END_REF][START_REF] Mantič | Crack onset and growth at the fibre-matrix interface under a remote biaxial transverse load. Application of a coupled stress and energy criterion[END_REF][START_REF] Martin | Competition between deflection and penetration at an interface in the vicinity of a main crack[END_REF][START_REF] Stein | A model for brittle failure in adhesive lap joints of arbitrary joint configuration[END_REF], V-notch [START_REF] Cornetti | Short cracks and V-notches: Finite Fracture Mechanics vs[END_REF][START_REF] Leguillon | Size effects for crack initiation at blunt notches or cavities[END_REF][START_REF] Leguillon | Failure initiation at V-notch tips in quasi-brittle materials[END_REF]) or loading conditions [START_REF] Doitrand | Numerical modeling of the nucleation of facets ahead of a primary crack under mode I+III loading[END_REF][START_REF] Sapora | Carpinteri V-notched elements under mode II loading conditions[END_REF]. A review of these applications was presented by Weißgraeber et al. [START_REF] Weißgraeber | A review of Finite Fracture Mechanics: crack initiation at singular and non-singular stress raisers[END_REF]. The CC requires only two fracture parameters as input, namely the material strength and fracture toughness. Moreover, compared to other methods cited previously, one advantage of the CC is its numerical efficiency [START_REF] Doitrand | Comparison between cohesive zone and coupled criterion modeling of crack initiation in rhombus hole specimens under quasi-static compression[END_REF]. As a matter of fact, under the assumption of small deformations, only one calculation is needed to compute the stress criterion, and few calculations with varying crack size are needed to compute the energy balance. Besides, since the fracture parameters are only implemented during the post-processing step, no extra calculations are needed to predict crack initiation for another set of material properties. This is particularly useful when these parameters are not known and must be determined by an indirect confrontation with experimental data [START_REF] Doitrand | 3D application of the coupled criterion to crack initiation prediction in epoxy/aluminum specimens under four point bending[END_REF][START_REF] Doitrand | Comparison between 2D and 3D applications of the coupled criterion to crack initiation prediction in scarf adhesive joints[END_REF]. Until now, the CC has been mainly applied in a linear elastic framework and under the assumption of small deformations. There are only few applications involving nonlinearities [START_REF] Leite | Application of finite fracture mechanics to composites: crack onset in a stretched open hole polymer plate with nonlinear behaviour[END_REF][START_REF] Leguillon | Failure initiation at V-notch tips in quasi-brittle materials[END_REF][START_REF] Rosendahl | Crack nucleation in hyperelastic adhesive bonds[END_REF][START_REF] Li | Numerical implementation of the coupled criterion for damaged materials[END_REF]. Leite et al. [START_REF] Leite | Application of finite fracture mechanics to composites: crack onset in a stretched open hole polymer plate with nonlinear behaviour[END_REF] modeled fracture of open hole plate under tension [START_REF] Li | A criterion study for nonsingular stress concentrations in brittle or quasi-brittle materials[END_REF] using either linear elastic or nonlinear material behavior. The expected improved agreement with respect to experimental results using the nonlinear elastic behavior was missing. Leguillon and Yosibash [START_REF] Leguillon | Failure initiation at V-notch tips in quasi-brittle materials[END_REF] studied crack initiation at a V-notch including a small damaged or plastic zone ahead of the notch tip, which was modeled by a decrease in the Young's modulus and varying fracture properties in this zone. The CC was applied using matched asymptotic expansions under the assumption of a small damage area near the V-notch. Two nested loops were implemented in order to both analyze the variation of the material parameters with dam-age and compute crack initiation using the CC. The analysis was however limited to a theoretical framework, lacking a comparison with experimental data. Rosendahl et al. [START_REF] Rosendahl | Crack nucleation in hyperelastic adhesive bonds[END_REF] applied the CC in case of adhesive connections bonded using structural silicone sealants, which undergo large deformations before failure. A stretch criterion was implemented instead of the classical stress criterion to better represent distortional bulk failure surfaces. Recently, Li et al. [START_REF] Li | Numerical implementation of the coupled criterion for damaged materials[END_REF] proposed a numerical implementation of the CC for damaged material. The damaged field in the material was computed using continuum damage mechanics and associated regularization techniques, while the transition from continuous damage field to a discontinuous crack (using element removal strategy) was determined using the CC. The proposed numerical procedure was applied to several examples including single crack initiation and growth, as well as multicracking.

In a previous study, Torabi et al. [START_REF] Torabi | Size effects on brittle fracture of Brazilian disk samples containing a circular hole[END_REF] compared the initiation stress predicted using the CC with experimental data obtained by carrying out Brazilian disk (BD) tests on PMMA and GPPS specimens with circular holes. The application of the CC in a linear elastic framework led to a good matching except for small holes, due to some nonlinear phenomena disregarded by the model [START_REF] Torabi | Size effects on brittle fracture of Brazilian disk samples containing a circular hole[END_REF]. A similar conclusion was drawn by Doitrand and Leguillon [START_REF] Doitrand | 3D application of the coupled criterion to crack initiation prediction in epoxy/aluminum specimens under four point bending[END_REF] in case of aluminum specimens bonded with an epoxy layer, for which neglecting material nonlinearities could explain the bigger discrepancy between numerical predictions and experimental data obtained for the smallest layers.

The objective of this paper is the implementation of the CC in a nonlinear framework. The studied case is that of BD with circular holes mentioned above [START_REF] Torabi | Size effects on brittle fracture of Brazilian disk samples containing a circular hole[END_REF], which is described in Section 3. In Section 2, the CC in a nonlinear framework is presented. The geometry of BD and the Finite Element (FE) model are described in Section 4. In Section 5, crack initiation predictions obtained using both a linear elastic and a nonlinear material behavior are compared to experimental results taken from [START_REF] Torabi | Size effects on brittle fracture of Brazilian disk samples containing a circular hole[END_REF].

FFM Nonlinear implementation

In the framework of FFM, crack initiation under quasi-static loading conditions requires the simultaneous fulfillment of two separate conditions [START_REF] Leguillon | Strength or toughness? A criterion for crack onset at a notch[END_REF]. First, the stress state must be high enough over the whole expected crack path prior to crack nucleation. This condition, originally proposed by Leguillon [START_REF] Leguillon | Strength or toughness? A criterion for crack onset at a notch[END_REF], consists in comparing the stress normal to the crack plane σ nn with the material strength σ c . This condition can also be replaced by a condition involving the average stress over the crack length l, as proposed, e.g., in [START_REF] Cornetti | Finite fracture mechanics: a coupled stress and energy failure criterion[END_REF][START_REF] Sapora | An improved Finite Fracture Mechanics approach to blunt V-notch brittle fracture mechanics: Experimental verification on ceramic, metallic,and plastic materials[END_REF][START_REF] Sapora | Crack onset and propagation stability from a circular hole under biaxial loading[END_REF]:

σ nn (l) = 1 l l 0 σ nn (s)ds ≥ σ c (1)
To be consistent with the previous work about failure predictions in Brazilian disk tests [START_REF] Torabi | Size effects on brittle fracture of Brazilian disk samples containing a circular hole[END_REF], condition 1 will be implemented, the comparison between the approaches going beyond the scope of this paper.

The second condition that must be fulfilled can be obtained through a balance of the potential (δW p ), kinetic (δW k ), and crack surface formation (G c l, G c being the material critical energy release rate) energies between a state before and after crack nucleation. Quasi-static initial conditions lead to a production of kinetic energy (δW k ≥0) allowing the energy condition to be derived, which consists in comparing the incremental energy release rate

G inc with G c . G inc (l) = W p (0) -W p (l) l ≥ G c (2)
It can be noted that the energy dissipated by non-linear mechanisms is not accounted for in the energy balance. As stated in introduction, the application of the CC in a linear elastic framework and under the small deformation assumption only requires few calculations to be performed. In this case, the stress is proportional to the imposed load or displacement U and the potential energy of both the damaged and the undamaged material is proportional to the square of applied load:

σ nn (l, U ) = k(l)U = σ c G inc (l, U ) = A(l)U 2 = G c (3)
where A and k are functions depending only on the material parameters and specimen geometry. Therefore, the stress prior to crack initiation and the incremental energy release rate can be computed once for any load level U . The combination of both criteria leads to the determination of the initiation load U c and crack length l c [START_REF] Doitrand | Comparison between cohesive zone and coupled criterion modeling of crack initiation in rhombus hole specimens under quasi-static compression[END_REF][START_REF] Leguillon | Strength or toughness? A criterion for crack onset at a notch[END_REF] by solving:

k(l c ) 2 A(l c ) = σ 2 c G c . (4)
If nonlinearities are accounted for, originating for instance from the material behavior (e.g., non linear stress-strain curves or large deformations) or from the simulation features (e.g., contact between several parts), the proportionality of the stress and the potential energy to respectively the load and the square load is no longer ensured. In this case, from a rigorous point of view, it is necessary to compute the stress and the energy conditions for several loading levels and determine the minimum loading level U c U 2R 2R 0 x y Fig. 1 Brazilian disk specimen containing a circular hole.

for which both conditions are fulfilled which can be achieved by solving the following minimization problem:

U c = min{U, ∃l, ( G inc (l, U ) G c ≥ 1) ∧ ( σ nn (l, U ) σ c ≥ 1)}
(5) The initiation crack length thus verifies :

min( G inc (l c , U c ) G c , σ nn (l c , U c ) σ c ) = 1 (6)
The practical implementation of the CC solution in a nonlinear framework is discussed more in Section 4.

Experimental tests and data

Let us consider the Brazilian disk tests carried out in [START_REF] Torabi | Size effects on brittle fracture of Brazilian disk samples containing a circular hole[END_REF]. The geometry of the samples is reported in Fig. 1. Some experimental tests aiming at determining fracture initiation were performed by Torabi et al. [START_REF] Torabi | Size effects on brittle fracture of Brazilian disk samples containing a circular hole[END_REF] on t=10mm (PMMA) or t =8mm (GPPS) thick specimens with R 0 =40mm and different hole sizes corresponding to R = 0.25, 0.5, 1, 2 and 4mm. For all the tested configuration, brittle fracture occurred with crack initiation from the hole edge in the loading direction. Torabi et al. [START_REF] Torabi | Size effects on brittle fracture of Brazilian disk samples containing a circular hole[END_REF] determined the failure stress as a function of the hole radius for both PMMA and GPPS specimens, highlighting the size effects of a decreasing initiation stress with an increasing radius. In the following, the experimental results used for the comparison with numerical predictions are taken from [START_REF] Torabi | Size effects on brittle fracture of Brazilian disk samples containing a circular hole[END_REF]. The true stress-strain curves of specimens without holes under uniaxial tensile loading are given in Fig. 2. The material behavior is slightly non linear and can be well reproduced numerically using the RO model [START_REF] Ramberg | Description of stress-strain curves by three parameters[END_REF].

ǫ = σ E + α σ E ( σ σ 0 ) n-1 (7)
PMMA GPPS RO-PMMA RO-GPPS Fig. 2 Experimental (dots and diamonds) and simulated with Ramberg-Osgood (RO) model (dashed and full lines) stress-strain curves for PMMA and GPPS specimen without hole. The corresponding parameters are given in Table 1 together with the strength and fracture toughness of both materials taken from [START_REF] Torabi | Size effects on brittle fracture of Brazilian disk samples containing a circular hole[END_REF]. The Ramberg-Osgood parameters were determined by means of FE calculations on specimens without hole under uniaxial tensile loading so as to fit the experimental stressstrain curves.

Finite element analysis

A 2D plane strain FE model of Brazilian disk test is set-up using Abaqus TM . As demonstrated in [START_REF] Doitrand | Experimental characterization and numerical modeling of crack initiation in rhombus hole PMMA specimens under compression[END_REF], compared to the 3D case, it may be representative of the specimen middle plane. Due to the specimen and test symmetry, only one quarter of the specimen can be modeled. The platen is modeled as infinitely rigid and a displacement of the platen in (Oy) direction is imposed as boundary condition. Contact between the platen and the specimen is taken into account using a normal hard contact and no tangential friction, making the calculations nonlinear. Therefore, even if a linear elastic material behavior is used, the nonlinear solution of the CC as presented in Section 2 must be adopted. The uniaxial compression in (Oy) direction leads to crack initiation in the same direction due to the specimen lateral expansion. Experimentally, it was not possible to check whether symmetric or asymmetric crack initiation occurred [START_REF] Torabi | Size effects on brittle fracture of Brazilian disk samples containing a circular hole[END_REF]. However, the discrepancy between symmetric or asymmetric crack initiation predicted by the CC in such configurations is generally small [START_REF] Sapora | Symmetric vs. asymmetric crack initiation from a circular hole[END_REF][START_REF] Torabi | Size effects on brittle fracture of Brazilian disk samples containing a circular hole[END_REF]. Therefore, symmetric crack initiation is assumed so as to be consistent with [START_REF] Torabi | Size effects on brittle fracture of Brazilian disk samples containing a circular hole[END_REF]. The material parameters of either linear elastic or nonlinear elastic (Ramberg-Osgood model) material behavior are implemented (Table 1).Solving the CC requires a mesh that is fine enough so as to capture the initiation crack length.

It can be shown that this length is a fraction of the material characteristic length L mat = EGc σ 2 c [START_REF] Martin | Understanding the tensile strength of ceramics in the presence of small critical flaws[END_REF]. Fig. 3 shows an example of Brazilian disk specimen mesh consisting of four-nodes linear elements. The minimum mesh size is set to 5×10 -3 mm and a progressive unrefinement of the mesh along the crack path is performed, as explained in [START_REF] Doitrand | Comparison between cohesive zone and coupled criterion modeling of crack initiation in rhombus hole specimens under quasi-static compression[END_REF]. The FE model consists in about 5000 nodes and the typical duration for one calculation is 30s.

Applying the CC to the case of Brazilian disk specimens requires Eq. ( 5) to be solved. An approach similar to that proposed in [START_REF] Leite | Application of finite fracture mechanics to composites: crack onset in a stretched open hole polymer plate with nonlinear behaviour[END_REF] is adopted herein. In practice, a discretization of the imposed displacement {U k } k=1..n can be chosen so as to solve Eq. [START_REF] Cornetti | Finite Fracture Mechanics and cohesive crack model: weight functions vs. cohesive laws[END_REF]. For each loading level U k , one calculation is required so as to compute the stress condition of the CC (Eq. ( 1)) whereas several calculations with varying crack length are needed to compute the energy condition (Eq. ( 2)). For a given loading level, around 100 calculations with varying crack length are performed to compute the potential energy difference

Stress criterion

Energy criterion

U =1.3U c U =0.9U c
Fig. 4 Incremental energy release rate to fracture toughness and stress to strength ratios variation as a function of the crack length l for imposed loading U = 0.9Uc and U = 1.3Uc due to crack initiation and hence the incremental energy release rate. This is done by progressive releasing of the symmetry condition of the nodes lying on the crack path. At each loading level, it is checked whether the CC is fulfilled or not. Then, the initiation loading level is determined using Eq. ( 5) by interpolation of the computed values. Fig 4 shows the variation of (i) stress to strength and (ii) incremental energy release rate to fracture toughness ratios for two imposed displacement levels for which the CC is fulfilled or not. It can be seen that in both cases, a solution to Eq. ( 4) can be found, which allows determining l c . However, a sufficiently high imposed loading must be prescribed so that a solution to Eq. ( 5) can be found. Fig 5 shows the incremental energy release rate to fracture toughness ratios evaluated for l c as a function of the imposed displacement. In the present case, 5 imposed displacement values were used to compute the CC and determine at least one imposed displacement for which the CC is fulfilled. Then, the initiation displacement and crack length are determined by interpolating the values computed on the basis of the discretized imposed displacement. It can be noted that despite the nonlinear CC solution requires more calculations than the usual linear application, they can be executed in parallel. In the present work, 5 calculations were performed in parallel so that the overall analysis for a given hole size specimen finally take around 30 minutes, which is still reasonable compared to the overall analysis time of the classical approach [START_REF] Doitrand | Comparison between cohesive zone and coupled criterion modeling of crack initiation in rhombus hole specimens under quasi-static compression[END_REF]. Note that, in addition, once the calculations are performed, the fracture properties of the material can be easily varied without any extra calculations since they are only implemented during the post-processing step, which is a main advantage of the CC.

Results and discussions

The solution of the CC in nonlinear framework is now applied to the case of PMMA or GPPS Brazilian disk specimens including a central hole. For each case, the critical imposed displacement is computed according to the procedure presented in Section 2. The associated reaction force F c is then estimated so as to obtain the equivalent failure stress σ f given by the following expression [START_REF] Hobbs | An assessment of a technique for determining the tensile strength of rock[END_REF][START_REF] Torabi | Size effects on brittle fracture of Brazilian disk samples containing a circular hole[END_REF]:

σ f = F c πR 0 t (8)
Figs. 6 and7 show the failure stress predicted by the CC using either a linear elastic or a nonlinear material behavior. It can be seen that the hole size effect is very well captured by the CC whatever the material behavior since similar trends as experimentally are obtained numerically. Implementing a linear instead of a nonlinear material behavior results in overestimating of the failure stress. The discrepancy between both cases remains lower than 10% for PMMA and 15% for GPPS. Taking into account the material nonlinear behavior allows reducing the discrepancy with respect to the mean value of experimental data (Tables 2 and3), especially for the smallest holes. For instance, the difference between predictions and mean experimental failure stress is reduced from 27.9% to 9.6% for PMMA and from 18.5% to

CC -linear CC -nonlinear

Experiments-PMMA Fig. 6 Failure stress σ f as a function of hole radius R for PMMA samples. 9.4% for GPPS in the case of the R = 0.25mm hole radius.

CC -linear CC -nonlinear Experiments-GPPS

Conclusions

The CC has been implemented in a nonlinear framework for failure stress prediction in Brazilian disk specimens. Nonlinearities originate from both the contact between the platen and the specimen and the material stress-strain curves. Concerning the FE implementation, taking nonlinearities into account requires more calculations to be performed since there is no longer any proportionality between the stress and the load, and between the potential energy and the square load. Therefore, the dependency of stress and energy to the applied load has to be computed for several load levels, looking for the minimum load for which both criteria are fulfilled.

It is important to note that the fracture parameters G c and σ c are only implemented during the post-processing step. Therefore once the calculations are performed for a given configuration, it is easy to vary these fracture parameters without any additional computational costs, contrary to other fracture approaches such as cohesive zone modeling or phase field models for fracture.

Applying the CC to crack initiation prediction in Brazilian disk specimens with circular holes using linear elastic material behavior results in a reasonable matching except for small holes for which crack initiation occurs at higher loading level and thus with more pronounced nonlinearities. Taking into account the nonlinear material behavior allows to remove this drawback, improving significantly the theoretical FFM predictions.

Fig. 3

 3 Fig. 3 Mesh of one quarter of the Brazilian disk specimen.

Fig. 5

 5 Fig. 5 Incremental energy release rate to fracture toughness ratio evaluated for l = lc as a function of the imposed loading U .

Fig. 7

 7 Fig. 7 Failure stress σ f as a function of the hole radius R for GPPS samples.

Table 1

 1 Material properties for PMMA and GPPS.

		PMMA GPPS
	E (MPa)	2960	3100
	ν	0.38	0.34
	α	2	1.9
	σ0 (MPa)	68	30
	n	1.4	1.2
	σc (MPa)	78	40
	Gc (MPa.mm) 0.88	0.23
	Lmat (mm)	0.428	0.446

Table 2

 2 Deviation between failure stress predictions with the CC using either a linear elastic or a nonlinear material behavior and mean experimental failure stresses for PMMA specimens.

	Hole radius ∆σ f σ f (linear) (%) ∆σ f σ f (nonlinear) (%)
	0.25	27.9	9.6
	0.5	13.2	2.7
	1	5.	3.6
	2	0.01	2.9
	4	14.7	3.97

Table 3

 3 Deviation between failure stress prediction with the CC using either a linear elastic or a nonlinear material behavior and mean experimental failure stresses for GPPS specimens.

	Hole radius ∆σ f σ f (linear) (%) ∆σ f σ f (nonlinear) (%)
	0.25	18.5	9.4
	0.5	8.8	3.2
	1	11.2	4.7
	2	7.7	6.3
	4	13.7	14.8