Dynamic fracture surface energy values and branching instabilities

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 rubber
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.
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.
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.
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.
metallic coating
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.
Voir aussi / See also
Notes au C.R.A.S., Séries IIb , (2001), 329, p. 195-200.
a (mm)
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
crack branching
no crack branch.
t (µs)
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.
GID (kJ/m²)
t (µs)
100 120
quasi-static analysis
a° = 600 m/s
100 120
Fig. 3. Quasi-static and transient dynamic
finite element analyses: typical results for
crack propagation, without branching.
Fig. 3. Analyses par éléments finis quasistatique et dynamique transitoire : résultats
typiques d'une propagation de fissure en
absence de branchement.
140 160 180 200
a (mm)
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.
< GD > (kJ/m²)
21 24
22 20
21 21
550 560 570 580 590 600 610
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
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
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.
GIc (kJ/m²)
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.
1E-7 1E-6 1E-5 1E-4
a° / c
a° mb
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.
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.