Voir aussi / See also Notes au C.R.A.S., Séries IIb , (2001), 329, p. 195-200.

Dynamic fracture surface energy values and branching instabilities during rapid crack propagation in rub ber toughened PMMA Valeurs de l'énergie de rupture dynamique associées aux instabilités de branchement en fissuration rapide dans un polymère "choc"

C. Fond and R. Schirrer Institut Charles Sadron, 6, rue Boussingault, F67083 Strasbourg. Abstract

An experimental device based on a strip band geometry was designed to explore the brittle behaviour of polymers during rapid crack propagation. The macroscopic crack speed is found to be quasi-constant along any entire RT-PMMA specimen, even in the case of crack branching and until arrest, if any. At the macroscopic branching velocity, the experimental fracture surface energy value and the fracture surface roughness are found to have no single values in RT-PMMA. Indeed, the macroscopic fracture surface energy value increases with the amount of instabilities or aborted cracks branching.

Résumé Un dispositif expérimental basé sur la géométrie de l'éprouvette en bande a été conçu pour étudier le comportement fragile en fissuration rapide des polymères. Pour un PMMA "choc", on observe que la vitesse de la fissure est quasiment constante tout au long de l'essai, qu'il y ait ou non branchement ou arrêt de fissure. A la vitesse de branchement macroscopique, la valeur mesurée de l'énergie de rupture ainsi que la rugosité de la surface de rupture ne sont pas uniques dans le PMMA "choc". En effet, l'énergie de rupture macroscopique croît avec la quantité d'instabilités ou "branchements avortés".

Voir aussi / See also Notes au C.R.A.S., Séries IIb , (2001), 329, p. 195-200.

INTRODUCTION

Most polymeric materials exhibit a brittle fracture mode at high crack propagation speeds. The experimental system shown in Fig. 1(left) was designed to enable a steady state regime of brittle fracture and a simple mechanical analysis of the fracture energy, even for rapid crack propagations (RCP) [1, 2]. The geometry is based on the strip band specimen (SBS), which dimensions are typically L ≈ 200 mm, 25 mm < H < 45 mm, a0 ≈ 3 H, B ≈ 2 mm. The location of the crack tip during the propagation is determined by measuring the resistance of a metallic layer [3]. The loading device, shown in Fig. 1(right), ensures uniform and constant displacement of the strip band boundaries. The symetry of the loading is verified by strain measurments on the sides of the specimen. Crack propagation is started by an impact of low energy on a razor blade placed in contact with one initial blunt notch and a crack propagates symetrically at a macroscopic speed am. Only one of the twin specimens undergoes fracture. Owing to the weight of the grips and the short time to fracture, typically 200 µs, we assume that the boundary conditions are fixed during the crack propagation. Crack branching can be obtained by increasing the mechanical potential energy stored in the specimen. The crack branches are generally symetrical for specimens made of polymethylmethacrylate (PMMA) and rubber toughened-polymethylmethacrylate (RT-PMMA). RT-PMMA is a blend of spherical particles of rubber and PMMA. The volume fraction of particles is approximately 40 % and the diameter of the particles is 200 nm.

+

-a

H

H

y

x

u

u

a

L

metallic coating

B

Fig. 1. (left) Schematic representations of the strip band geometry uniformly loaded and of the conducting layer used to record the crack tip position during propagation. (right) Experimental

device to ensure a symmetrical loading. Fig. 1. (gauche) Schémas de l'éprouvette en bande chargée uniformément et du dépôt métalique

utilisé pour connaître la position du sommet de fissure pendant la propagation. (droite) Dispositif expérimental assurant un chargement symétrique.

DYNAMIC ENERGY RELEASE RATE COMPUTATION AND FRACTURE SURFACE ENERGI ES

Voir aussi / See also Notes au C.R.A.S., Séries IIb , (2001), 329, p. 195-200.

0 50 100 150 20075

100

125

150

175

200crack branchingno crack branch.

branchinglocationa

(mm

)

t (µs)

Fig. 2. Crack lenghts vs. time in a case of crack branching (double branch) and in a case of no macroscopic crack branching. Fig. 2. Longeurs de fissure en fonction du temps dans le cas d'un branchement de fissure (branches doubles) et absence de branchement.

As shown in Fig. 2, the macroscopic crack speed were constant, for a given specimen at a given temperature, whatever branching occured or not. Some experiments revealed effectively that crack propagation in RT-PMMA is unstable between approximately 1 and 0.6 cr, where cr is the Raleigh wave speed. As the fracture surface energy decreases with increasing crack speed, the speed of propagation jumps from 1 to about 550 m/s when an increasing loading initiates the propagation of the crack.

80 100 120 140 160 180 2000.0

0.5

1.0

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2.0

2.5

3.0

°a = 600 m/s

quasi-static analysis

GID

(kJ

/m²)

a (mm)

0 20 40 60 80 100 120t (µs)

Fig. 3. Quasi-static and transient dynamic finite element analyses: typical results for crack propagation, without branching. Fig. 3. Analyses par éléments finis quasi-statique et dynamique transitoire : résultats typiques d'une propagation de fissure en absence de branchement.

The crack tip position during propagation and the stress state at initiation being known, the dynamic energy release rate GID may then be calculated by means of a transient dynamic finite element procedure, based on the software Castem2000©. Owing to the high level of strain rate, a linear elastic behaviour of the fracture mechanics is assumed [4]. It has been shown experimentally that the impact on the razor blade influences the crack propagation only on a few millimetres after initiation. Nevertheless, this crack initiation is simulated by imposing an initial crack tip opening, corresponding to the action of the razor blade at the crack lips. The Fig. 3 shows that the dynamic correction factor is generally of the order of 20 to 30 % for macroscopic crack speeds of about 0.6 cr. Indeed, this specimen geometry is known to induce a low dynamic correction factor [2, 5] and the remote stress field at the crack tip is actually not strongly influenced by the inertial effects in this range of crack speeds. In order to simplify the results, since GID displays relatively small oscillations during crack propagation, the mean value of the fracture energy, < GID>, was calculated for each specimen before branching, if any. Practically, < GID> concerns the steady state regime and does not take into account the first 1.5 mm of propagation after inititation in RT-PMMA. In Fig. 4, it is clear that the experimental mean dynamic fracture surface energy values < GID> for a crack propagating at several hundred meters per second is substantially lower than the fracture energy GIc at the onset of propagation, which is typically close to 10 kJ/m2, at low medium stress intensity loading rates. All speeds, in the range 550-610 m/s for temperatures between 19 °C and 27 °C, are not

Voir aussi / See also Notes au C.R.A.S., Séries IIb , (2001), 329, p. 195-200.

correlated to the values of < GID>. Crack branching occurs at a crack tip speed of nearly 0.6 times cr.

550 560 570 580 590 600 6100,0

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21

°

singlestraight

crack

smallvisible

branches

largecrack

branches

< G

D >

(kJ

/m²)

a (m/s)

Fig. 4. Mean dynamic fracture energy values before branching (if any branching) vs macroscopic crack speed in a RT-PMMA. The numbers are temperature in °C. Fig. 4. Valeurs des énergies de rupture dynamique moyennes avant branchement éventuel en fonction de la vitesse de propagation de la fissure dans un pmma "choc". Les nombres sont les températures en °C.

Unlike in many other polymers like for instance PMMA, am does not change after branching during RCP in RT-PMMA. In RT-PMMA, the macroscopic fracture surface energy decreases with increasing crack speed and at nearly 0.6 cr inertial effects modify the crack tip stress field to induce branching [6]. Subsequently, as the material behaviour tends to accelerate whereas the mechanical - inertial - effects tend to limit the rate of crack propagation, the crack speed stabilizes at approximately amb = 0.6 cr, which is the macroscopic crack branching speed for RT-PMMA.

FRACTURE SURFACE ROUGHNESSES

Photo 1. Optical micrographs showing the roughness change at branching - propagation from

left to right, in a RT-PMMA specimen of thickness 2 mm – (left) 1mm before the crack braching, (right) 1mm after the crack braching .

Photo 1. Micrographies optiques mettant en évidence le changement de rugosité brusque lors d'un branchement de fissure - propagation de gauche à droite dans un échantillon de PMMA

"choc" d'épaisseur 2 mm – (gauche) 1 mm avant le branchement de fissure, (droite) 1 mm après le branchement de fissure.

The fracture surface displays a sharp change in roughness at branching, visible in the comparison of Photo 1(left) and (right). Optical microscopy reveals a relatively coarse surface

Voir aussi / See also Notes au C.R.A.S., Séries IIb , (2001), 329, p. 195-200.

texture a few millimeters before branching and a finer texture a few millimeters after branching. Fig. 5 (left) shows an atomic force microscopy image of a fracture surface a few millimeters before a macroscopic crack branching, corresponding to a high value of the fracture surface energy and Fig. 5 (right) just before a crack arrest, corresponding to the lowest level of the fracture surface energy. It is noticeable that also at the microscpic scale the surface are rougher prior to crack branching than before arrest at a similar crack speed. The fracture surface does not pass through the rubber particles of RT-PMMA.

Fig. 5. Atomic force microscopy images of a fracture surface: (left) rough surface before crack branching, a > 500 m/s, and (right) smooth surface just before crack arrest, a > 500 m/s.

Fig. 5. Images de microscopie à force atomique d'une surface de rupture : (gauche) surface rugueuse avant branchement de fissure, a > 500 m/s, et (droite) juste avant un arrêt de fissure,

a > 500 m/s.

Since the crack forms branches even though the experimental crack speed and the energy released inside the specimen remain quasi-constant, two or sometimes three branches must consume the same energy as a single crack propagating at the same speed. The varying parameter at crack branching is then the roughness of the fracture surface, which means the total surface created (SFt). Indeed, the usual planar crack surface (crack length increase times width, B ∆a) should not be considered. The total surface created may be the approximately the same for two smooth cracks or one rough crack. In the case of a smooth crack, like in Fig. 5 (right), created at speed amb the fracture energy is GIDmin. If the energy release rate is greater than GIDmin the propagating crack may produce microscopic instabilities, since intertial effects are sufficient to render possible crack branching. These small instabilities are themselves smooth and short aborted branches. If the energy release rate exceeds twice GIDmin, then macroscopic smooth branches can appear. SBS fracture tests were also performed using pure PMMA specimens. In these samples, optical microscopy revealed mirror like fracture surfaces, while as expected [7, 8] the measured fracture energy increased with crack speed. Fig. 6 shows that GIDmin is similar for RT-PMMA and pure PMMA at the corresponding crack speed. The fracture energy of RT-PMMA at a crack speed of amb varies from 1 to 3 times GIDmin

Voir aussi / See also Notes au C.R.A.S., Séries IIb , (2001), 329, p. 195-200.

–sometimes we have observed triple branching - and the latter corresponds to smooth crack surfaces. Hence the fracture energy might be expected to be directly correlated with the ratio SFt / (B ∆a) and the fracture energy of pure PMMA.

1E-7 1E-6 1E-5 1E-4 0.1 10.1

1

10

°amb°

PMMA (SBS) RT-PMMA (CT) RT-PMMA (SBS)G

Ic (

kJ/m

²)

a / cr

Fig..6. Fracture surface energy vs. normalized crack speed for PMMA and RT-PMMA. Fig. 6. Fracture surface energy vs. normalized crack speed for PMMA and RT-PMMA.

CONCLUSION

RT-PMMA represents an interesting model material to investigate the micro mechanisms of dynamic fracture processes. It has been experimentally shown that and explained why, at the macroscopic branching velocity, the experimental fracture surface energy have no single value in materials, at least such that the fracture energy decreases in the RCP regime.

REFERENCES

1 Fond, C. and Schirrer, R, Journal de Physique IV, (1997), p. C3-969-C3-974. 2 Nilsson, F, Int. J. Fract. Mech. 8, (1972), p.403-411. 3 Thesken, J. C, The Aeronautical Research Institute of Sweden, FFA-TN, 57, (1994). 4 Ferrer, J. B., Fond, C., Arakawa, K., Takahashi, K., Béguelin, P. and Kausch., H.-H., Letters

in Fract. and Micromech. 87, (1998), L77-L82. 5 Popelar, C. H. et Atkinson, C., J. Mech. Phys. Solids, 28, 14, (1980), p. 77-93. 6 Yoffé, E. H., Phil. Mag. 12, (1951), p 739-750. 7 Willians, J. G., Int. J. Fract. 8, (1972), p. 393-401. 8 Carlsson, J., Dahlberg, L. and Nilsson, F., Proc. Int. Conf. Dyn. Crack Propag., (1972), ed.

1973, p. 165-181.