# Brittle/quasi-brittle transition in dynamic fracture: An energetic signature

###### Abstract

Dynamic fracture experiments were performed in PMMA over a wide range of velocities and reveal that the fracture energy exhibits an abrupt 3-folds increase from its value at crack initiation at a well-defined critical velocity, below the one associated to the onset of micro-branching instability. This transition is associated with the appearance of conics patterns on fracture surfaces that, in many materials, are the signature of damage spreading through the nucleation and growth of micro-cracks. A simple model allows to relate both the energetic and fractographic measurements. These results suggest that dynamic fracture at low velocities in amorphous materials is controlled by the brittle/quasi-brittle transition studied here.

###### pacs:

46.50.+a, 62.20.M-, 61.43.-jDynamic fracture drives catastrophic material failures. Over the last century, a coherent theoretical framework, the so-called Linear Elastic Fracture Mechanics (LEFM) has developed and provides a quantitative description of the motion of a single smooth crack in a linear elastic material Freund (1990). LEFM assumes that all the mechanical energy released during fracturing is dissipated at the crack tip. Defining the fracture energy as the energy needed to create two crack surfaces of a unit area, the instantaneous crack growth velocity is then selected by the balance between the energy flux and the dissipation rate . This yields Freund (1990):

(1) |

where and are the Rayleigh wave speed and the Young modulus of the material, respectively, and is the Stress Intensity Factor (SIF) for a quasi-static crack of length . depends only on the applied loading and specimen geometry, and characterizes entirely the stress field in the vicinity of the crack front.

Equation (1) describes quantitatively the experimental results for dynamic brittle fracture at slow crack velocities Bergkvist (1974). However, large discrepancies are observed in brittle amorphous materials at high velocities Fineberg and Marder (1999); Ravi-Chandar (2004); Livne et al. (2007, 2008). In particular (i) the measured maximum crack speeds lie in the range , i.e. far smaller than the limiting speed predicted by Eq. (1) and (ii) fracture surfaces become rough at high velocities (see Fineberg and Marder (1999); Ravi-Chandar (2004) for reviews). It has been argued Sharon and Fineberg (1999) that experiments start to depart from theory above a critical associated to the onset of micro-branching instabilities Fineberg et al. (1991): for the crack motion becomes a multi-cracks state. This translates into (i) a dramatic increase of the fracture energy at , due to the increasing number of micro-branches propagating simultaneously and (ii) a non-univocal relation between and Sharon and Fineberg (1999). The micro-branching instability hence yielded many recent theoretical efforts branching (). However, a number of puzzling observations remain at smaller velocities. In particular, even for velocities much lower than , (i) the measured dynamic fracture energy is generally much higher than that at crack initiation Sharon and Fineberg (1999); Kalthoff et al. (1976); Rosakis et al. (1984); Bertram and Kalthoff (2003) and (ii) fracture surfaces roughen over length scales much larger than the microstructure scale ("mist" patterns) Hull (1999), the origin of which remains debated mist (); Lawn (1993).

In this Letter, we report dynamic fracture experiments in polymethylmethacrylate (PMMA), the archetype of brittle amorphous materials, designed to unravel the primary cause of these last discrepancies. We show that dynamic fracture energy exhibits an abrupt 3-folds increase from its value at crack initiation at a well-defined critical velocity well below . This increase coincides with the onset of damage spreading through the nucleation and growth of micro-cracks, the signature of which is the presence of conic patterns on post-mortem fracture surfaces. A simple model for this nominally brittle to quasi-brittle transition is shown to reproduce both the energetic and fractographic measurements.

Dynamic cracks are driven in PMMA with measured Young modulus and Poisson ratio of and , which yields . Its fracture energy at the onset of crack propagation was determined to be , with being the material toughness. Specimen are prepared from parallelepipeds in the (propagation), (loading) and (thickness) directions by cutting a rectangle from the middle of one of the edges and then cutting a groove deeper into the specimen (Fig. 1, bottom inset). Two steel jaws equipped with rollers are placed on both sides of the cut-out rectangle and a steel wedge (semi-angle ) is pushed between them at constant velocity up to crack initiation. In this so-called wedge splitting geometry, the SIF decreases with the crack length . To increase its value at crack initiation, and therefore the initial crack velocity, a circular hole with a radius ranging between and is drilled at the tip of the groove to tune the stored mechanical energy . Dynamic crack growth with instantaneous velocities ranging from to and stable trajectories are obtained. The location of the crack front is measured during each experiment ( and resolutions) using a modified version of the potential drop technique: A series of 90 parallel conductive lines (-thick Cr layer covered with -thick Au layer), -wide with an -period of are deposited on one of the - surfaces of the specimen, connected in parallel and alimented with a voltage source. As the crack propagates, the conductive lines are cut at successive times, these events being detected with an oscilloscope. The instantaneous crack velocity is computed from , and the instantaneous SIF is calculated using 2D finite element calculations (software Castem 2007) on the exact experimental geometry, assuming plane stress conditions and a constant wedge position as boundary condition.

Values for the fracture energy are obtained directly from Eq. (1) by combining the measurements and the calculations. Typical and curves are shown in Fig. 1. The variations of with (Fig. 2) are found to be the same in various experiments performed with various stored mechanical energy at crack initiation. This curve provides evidence for three regimes, separated by two critical velocities. For slow crack velocities, remains of the order of as expected in LEFM. Then, as reaches the first critical velocity , increases abruptly to a value about 3 times larger than . Beyond , increases slowly with up to the second critical velocity, Sharon and Fineberg (1999), above which diverges again with . This second increase corresponds to the onset of the micro-branching instability, widely discussed in the literature Fineberg et al. (1991); Sharon and Fineberg (1999), whereas the first one, at , is reported here for the first time. The high slope of around provides a direct interpretation for the repeated observation of cracks that span a large range of but propagate at a nearly constant velocity of about (see e.g. refs. Ravi-Chandar and Knauss (1984); Ravi-Chandar and Yang (1997)).

The post-mortem fracture surfaces shed light on the nature of the transition at on the curve . Fig. 3 shows the surface morphology for increasing crack velocity. For , the fracture surfaces remain smooth at the optical scale (Fig. 3(a), top). Above conic marks are observed (Figs. 3(b) and 3(c), top). They do not leave any visible print on the sides of the specimens (Fig. 3(b), bottom), contrary to the micro-branches that develop for (Fig. 3(c), bottom).

Similar conic marks were reported in the fracture of many other amorphous brittle materials (see Ravi-Chandar (2004); Hull (1999) and references therein), including polymer glasses, silica glasses and polycrystals. Their formation is thought to arise from inherent toughness fluctuations at the micro-structure scale due to material heterogeneities randomly distributed in the material Smekal (1953); Ravi-Chandar and Yang (1997). The enhanced stress field in the vicinity of the main crack front activates some of the low toughness zones and triggers the initiation of secondary penny-shaped micro-cracks ahead of the crack front. Each micro-crack grows radially under the stress associated with the main crack along a plane different from it. When two cracks intersect in space and time, the ligament separating them breaks up, leaving a visible conic marking on the post-mortem fracture surface.

Figure 4 shows the surface density of conic marks as a function of crack velocity . Below , no conic mark is observed up to magnification, consistently with Sheng and Zhao (1999). Above , increases almost linearly with . The exact correspondence between the critical velocity at which exhibits an abrupt increase and the velocity at which the first conic marks appear on the fracture surfaces strongly suggests that both phenomena are associated with the same transition. The nucleation and growth of micro-cracks can therefore be identified as the new fracture mechanism that starts at . This damage process is generic in brittle materials and is relevant for an even wider range of materials than those that exhibit conic marks, e.g. granite Moore and Lockner (1995).

We now present a simple model reproducing the curve between 0 and . We assume that linear elasticity fails in the material when the local stress reaches a yield stress . It defines a fracture process zone (FPZ) around the crack tip, the size of which is given by where is a dimensionless constant Lawn (1993) and is the dynamic SIF. We consider that all the dissipative phenomena (plastic deformations, crazing or cavitation for instance) occur in the FPZ, with a volumic dissipated energy . The material is then assumed to contain a volume density of discrete "source-sinks" (SS, see e.g. Lawn (1993) for previous uses of this concept). Each SS is assumed to activate into a micro-crack if two conditions are met: (i) the local stress reaches and (ii) the SS is located at a distance from the crack tip larger than note1 (). The nucleation of a micro-crack is assumed to be accompanied by an excluded volume where stress is screened i.e. no SS can acivate anymore. In the following, , , , and are taken as constants throughout the material. Three cases should be considered:

(I) - At the onset of crack propagation, all the volume within contributes to the fracture energy .

(II) - For , no micro-crack nucleates and . The dynamic SIF is then Freund (1990) where is universal and is the dilatational wave speed (here =201060). The volume scanned by the FPZ when the crack surface increases by is . The dissipated energy is given by where is the Griffith surface energy. Since , one finally gets for : {align} Γ(v)=αKd(v)2E+(1-α) Kc2E with α=2 ϵEaσY2. This predicted linear dependence of with for is in agreement with measurements (Fig. 2, inset). A linear fit to the data (correlation coefficient =0.985) gives =1.170.05 and =0.30.2, where stands for 95 confidence interval. The latter value is compatible with the measurements of the fracture energy at crack initiation. By combining Eqs. (1) and (Brittle/quasi-brittle transition in dynamic fracture: An energetic signature), one gets a prediction for the curve unpublished () that reproduces very well the low velocity regime in Fig. 2. Extrapolation of this regime unpublished () exhibits a divergence of the dissipated energy for a finite velocity =2000.23, slightly larger than . In the absence of micro-cracks, this velocity would have therefore set the limiting macroscopic crack velocity.

(III) - For , i.e. micro-cracks start to nucleate. The surface density of micro-cracks is then equal to the number of activated SS beyond per unit of fracture area, i.e. where the third term in the parenthesis stands for the excluded sites around micro-cracks. This yields: {align} ρ(v) = βKd(v)2-Ka2E with β=2 Ea σY2 ρs1+ρsV where =. This linear relationship is in good agreement with the measurements for before the micro-branching onset, beyond which saturates (Fig. 4, inset). A fit to the data (=0.877) between and gives =333. In the micro-cracking regime, the local dynamic SIF is not equal to the macroscopic one anymore, but corresponds to that at the individual micro-crack tips, at which the limiting velocity is expected to be . It is then natural to assume that all micro-cracks propagate at the same velocity , which yields note2 (). The energy dissipated when the crack surface increases by is , yielding: {align} Γ(v) = Γ_a+ χKd(v)2-Ka2E with χ=2 ϵE/a σY21+ρsV where . Eq. (Brittle/quasi-brittle transition in dynamic fracture: An energetic signature) predicts a linear dependence of with , in agreement with the measurements for (Fig. 2, inset). A linear fit to the data between and gives . The corresponding predicted curve unpublished () reproduces very well the intermediate velocity regime (Fig. 2) and exhibits a divergence of the dissipated energy for . This limiting velocity is very close to the observed maximum crack speed in brittle amorphous materials.

This simple scenario allows to illustrate how material defects control the dynamic fracture of amorphous solids before the onset of micro-branching. For , the mechanical energy released at the crack tip is dissipated into both a constant surfacic energy and a volumic energy within the FPZ, the size of which increases with crack speed. With this mechanism alone, the crack speed would be limited to a value slightly larger than . But damage spreading through micro-cracking makes possible to observe much larger velocities: The crack propagates through the nucleation, growth and coalescence of micro-cracks, with a macroscopic effective velocity that can be much larger than the local velocity of each micro-crack tip Ravi-Chandar and Yang (1997); Prades et al. (2005). We suggest that micro-cracks in themselves do not increase dissipation, but rather decrease it by locally screening the stress. At velocities larger than , micro-branches contribute to the dissipated energy proportionally to their surface Sharon et al. (1996). We emphasize that the nominally brittle to quasi-brittle transition occurring at is very likely to be generic for amorphous solids and should therefore be taken into account in future conceptual and mathematical descriptions of dynamic fracture. In this respect, Continuum Damage Mechanics (CDM) Kachanov (1986) initially derived for "real" quasi-brittle materials like ceramics or concrete may be relevant to describe fast crack growth in nominally brittle materials. In particular, a better understanding of the relationship between the dynamics of propagation of both the individual micro-cracks and the macroscopic crack is still needed.

We thank P. Viel and M. Laurent (SPCSI) for gold deposits, T. Bernard (SPCSI) for technical support, K. Ravi-Chandar (University of Texas, Austin) and J. Fineberg (The Hebrew University of Jerusalem) for fruitful discussions and P. Meakin (INL/PGP) for a careful reading of the manuscript. We acknowledge funding from French ANR through Grant No. ANR-05-JCJC-0088 and from Mexican CONACYT through Grant No. 190091.

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