Pattern formation in quasistatic fracture


Slow crack propagation gives rise to intriguing patterns that encompass a very wide range of length scales from microns to kilometers. What is the underlying mechanism that is responsible for the diversity of scales and yet the similarity of morphology? What are the characteristics of slow fracture--does it happen in a catastrophe, like a first order phase transition, or gradually and smoothly like a continuous one? We attempt to answer these and similar questions by carrying out analytic and extensive numerical studies of simplified models using concepts and methods of statistical physics, along with simple experiments.
mud cracks
Drying cracks on mud.
Source: Science Photo Lab
To address the pattern formation aspect of quasistatic fracture, we proposed, analyzed, and simulated a model system which describes the typical situation,
namely the quasistatic crack formation in a brittle layer in contact with a substrate[1,2]. We identified the stick-slip action on the frictional substrate and crack propagation on the layer as the two competing, elementary processes, and developed our model based on close experimental observations, including our own.
cracks in paint
Cracks in paint. Summer Palace, Beijing.
The model consists of particles interconnected with springs. Particles sit on a frictional substrate. When the net force on a particle exceeds the force threshold, the particle slips to an equilibrium position. When the tension on a spring exceeds the cracking threshold, it breaks irreversibly. Realistic crack patterns are generated upon iteration.
spring-block model
Basic mechanisms for crack formation. Move pointer over to see animation[5].

We obtained several novel results[1]:

1. The complementary nature of the effects of substrate and thickness, via a scaling variable;
2. Critical exponents and their relationship (scaling relation) which describe and connect dynamic and stationary properties;

3. The growth of correlation in stress field as a selection mechanism of stable fragment size.

These results suggest why morphologically similar patterns are so ubiquitous over a wide range of scales.

In earlier works[2], we also explored the continuous transitions exhibited in quasistatic fracture by means of a coarse-grained spring-block model. On the two-dimensional parameter space spanned by the strain and substrate coupling, a phase transition line was located for the evolution of cracks and fragment-size statistics. The correlation of cracks was shown to crossover from short-range to long-range order across the transition, with power-law decay at the transition, in close analogy to critical phenomena. As similar transitions have been observed in recent fragmentation experiments, the results are not specific to the model we studied. This work served as the basis for later studies.

The effect of disorder and limiting stress amplification were studied analytically using the democratic fiber bundle model[3,4]. We discovered a novel tricritical rupture controlled by disorder, a meeting place between abrupt and continuous behavior. With the model and simulation of a model with local interaction,
we showed that there exist some rupture modes which occur without any noticeable precursor and hence are inherently unpredictable. Our results,
corroborated by extensive simulations, have important implications for failure prediction in industries and especially the feasibility of earthquake predictions.


  1. Pattern formation and selection in quasi-static fracture,
    K.-t. Leung and Z. Neda, Phys. Rev. Lett. 85, 662 (2000).
    Also e-print cond-mat/0006061.
  2. Phase transition in a spring-block model of surface fracture,
    K.-t. Leung and J.V. Andersen, Euro. Phys. Lett. 38, 589 (1997)
  3. Tri-critical behavior in rupture induced by disorder,
    J.V. Andersen, D. Sornette and K.-t. Leung, Phys. Rev. Lett. 78, 2140 (1997)
  4. Reply to comment-Conditions for abrupt failure in the democratic fiber bundle model,
    D. Sornette, K.-t. Leung and J.V. Andersen, Phys. Rev. Lett. 80, 3158 (1998)
  5. A collection of short films (avi and mpeg) from experiments and simulations are available.
  6. There exists a related discussion on curved crack paths.

Last revised October 10, 2002 ©KtL