Ferenc Kun
professor, Department of Theoretical Physics, University of Debrecen

Fragmentation phenomena

Fragmentation, i.e. the breaking of particulate materials into a large number of smaller pieces is abundant in nature and underlies several industrial processes. Fragmentation phenomena can be observed on a broad range of length scales from the collisional evolution of asteroids and meteor impacts on the astrophysical scale, through geological phenomena and industrial applications on the intermediate scale down to the break-up of large molecules and heavy nuclei on the atomic scale..

Fragmentation, i.e. the breaking of particulate materials into a large number of smaller pieces is abundant in nature and underlies several industrial processes, which attracted a continuous interest in scientific research over the past decades. Fragmentation phenomena can be observed on a broad range of length scales from the collisional evolution of asteroids and meteor impacts on the astrophysical scale, through geological phenomena and industrial applications on the intermediate scale down to the break-up of large molecules and heavy nuclei on the atomic scale. The most striking observation on fragmentation is that the distribution of fragment sizes or masses shows a power law behavior, independently on the way of imparting energy, relevant microscopic interactions and length scales involved, with an exponent depending only on the dimensionality of the system. The understanding of this observed universality of fragmentation phenomena is till today the main driving force of theoretical studies. Besides one-, two-, and three-dimensional bulk materials, shell-like structures are also widely used in everyday life and industrial applications: they are used in the form of containers, pressure vessels, combustion chambers, airplane pressure chambers. Sudden excessive pressure pulses arising due to an explosion inside closed shells can result in substantial damage and even complete destruction of the structure. From theoretical point of view the fragmentation of shell-like system is a very interesting and challenging problem because locally shells are two-dimensional objects; however, their dynamics occurs in three dimensions giving rise to novel types of breaking mechanisms affecting also the final state distribution of characteristic quantities.
Simulating the collision of two-dimensional disordered solid objects, we have shown that depending on the imparted energy, the collision has two different outcomes: the final state of the solids can be damaged or fragmented with a sharp transition in between at a critical energy. Analyzing the energetics of the collision process and the fragment mass distributions in the final state we found that the transition between the damaged and fragmented states occurs analogously to continuous phase transitions. We concluded that the universal fragment mass distributions observed experimentally are the consequence of the underlying phase transition. Later on experiments confirmed our theoretical findings. We carried out a thorough experimental and theoretical study of the fragmentation of closed shells. Studying the explosion of shells and their impact with a hard wall, we demonstrated that shell fragmentation forms a novel universality class of fragmentation phenomena characterized by the unique exponent of the fragment mass distribution. Increasing the imparted energy the breakup of shells exhibits a transition from the damaged to the fragmented state. Computer simulations revealed that the transition between the two regimes occurs analogous to continuous and abrupt phase transitions for impact and explosion, respectively. We showed experimentally that not only the mass but also the shape of fragments obeys scaling laws. Depending on the primary cracking mechanism, the shape of fragments can be isotropic, or highly anisotropic with self-affine character. In spite of the intensive research of the past decade important questions on the effect of material properties (e.g. ductality), and on the velocity and shape of fragments still remained open.
1. G. Timar, J. Blomer, F. Kun, and H. J. Herrmann, New universality class for the fragmentation of plastic materials, Physical Review Letters 104, 095502 (2010).
2. F. Kun, F. K. Wittel, H. J. Herrmann, B.-H. Kroplin, and K.-J. Maloy, Scaling behaviour of fragment shapes, Physical Review Letters 96, 025504 (2006).
3. F. K. Wittel, F. Kun, H. J. Herrmann, and B.-H. Kroplin, Fragmentation of shells, Physical Review Letters 93, 035504 (2004).

Statistical physics of fracture and related problems

Fragmentation, i.e. the breaking of particulate materials into a large number of smaller pieces is abundant in nature and underlies several industrial processes. Fragmentation phenomena can be observed on a broad range of length scales from the collisional evolution of asteroids and meteor impacts on the astrophysical scale, through geological phenomena and industrial applications on the intermediate scale down to the break-up of large molecules and heavy nuclei on the atomic scale..

The mechanical response and fracture of materials is traditionally the subject of engineering sciences. However, the experimental and theoretical findings on the existence of scaling behavior have demonstrated that the fracture of disordered materials exhibits a high degree of analogy to phase transitions and critical phenomena, so that a comprehensive understanding can only be obtained in the framework of statistical physics. Due to the dominating role of disorder, the theoretical description of the fracture of heterogeneous materials requires the application of novel methods of computer modeling, computational physics and materials science as well. During the last decade at the border line of physics, materials science and engineering sciences a new field of science emerged, namely, the statistical physics of fracture, which have already achieved several novel results with technological importance. Natural catastrophes like earthquakes, snow and rock avalanches, landslides are often caused by crack nucleation under shear load and by the stick-slip motion of compressed crack surfaces. The investigation of fracture phenomena helps to understand the emergence of natural catastrophes revealing relations of observables as well which can be exploited for the forecasting of catastrophic failures.
1. N. Yoshioka, F. Kun, and N. Ito, Size scaling and bursting activity in thermally activated breakdown of fiber bundles, Physical Review Letters 101, 145502 (2008).
2. F. Kun, H. A. Carmona, J. S. Andrade Jr., and H. J. Herrmann, Universality behind Basquin's law of fatigue, Physical Review Letters 100, 094301 (2008).
3. F. Kun, Gy. B. Lenkey, N. Takacs, D. L. Beke, Structure of magnetic noise in dynamic fracture, Physical Review Letters 93, 227204 (2004).

Rheological Fluids

Electro- and magnetorheological fluids, also called smart fluids, are colloidal materials composed of micrometer sized particles suspended in an electromagnetically passive viscous liquid. The most striking feature of ER and MR fluids is that their reological properties can be controlled by an external electric or magnetic field which has major potential in automotive, and aircraft/aerospace applications. They can be used in devices such as dampers, shock absorbers, brakes, clutches, valves, position and speed controllers. Our work is focused on magnetorheological systems where the particles have a permanent magnetic dipole moment. In such colloids in the absence of an external magnetic field the particles aggregate due to the dipolar interaction and biuld up complex structures which gives rise the change of the overall rheological properties of the colloid.

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Socio-economic systems

Recently, the application of statistical physics and of the theory of critical phenomena provided novel insight into the dynamics of socio-economic systems. Various types of models have been developed which capture important aspects of the emergence of communities, opinion spreading or the evolution of financial data. The dynamics of innovation and the spreading of new technological achievements show also interesting analogies to complex physical systems.

The process of innovation has recently been studied by introducing a technology space based on percolation theory \cite{silverberg}. In this model new inventions arise as a result of a random search in the technology space starting from the current best-practice frontier. The model could reproduce the interesting observation that innovations occur in clusters whose sizes are described by the Pareto distribution. Another important aspect of technological development is the spreading of new technological achievements. In a socio-economic system different level technologies may coexist and compete as a result of which certain technologies proliferate while others disappear from the system. One of the key components of the spreading of successful technologies is the copying, i.e. members of the system adopt technologies used by other individuals according to certain decision mechanisms. Decision making is usually based on a cost-benefit balance so that a technology gets adopted by a large number of individuals if the upgrading provides enough benefits. The gradual adaptation of high level technologies leads to spreading of technologies and an overall technological progress of the socio-economic system. In our work we consider a simple agent-based model of the spreading of technological achievements in socio-economic systems. Agents of the model may represent individuals or firms which use certain technologies to collaborate with each other. For simplicity, we assume that costs of the cooperation arise solely due to the incompatibility of technologies used by the agents which then have two origins: on the one hand, difference of technological levels incurs cost, the larger the difference is, the higher the cost gets. On the other hand, technologies used by agents may belong to different providers which induce additional costs. Agents interacting with their social neighborhood can decrease their cost by adopting technologies of their interacting partners. The local rejection-adaptation strategy of agents can lead to interesting changes of the system on the meso- and macro-level, namely, agents can form clusters with identical technological levels, which can also be accompanied by an overall technological progress of the system. We analyze the time evolution of this model socio-economic system starting from a random configuration of technological levels and providers without considering the possibility of innovation. Based on analytic calculations and computer simulations we study how the adaptation of technologies of interacting partners leads to spreading of technological achievements. We characterize the microstructure of communities of agents, and the technological progress of the system on the macro level.
1. G. Kocsis and F. Kun, Competition of information channels in the spreading of innovations, Physical Review E 84 026111 (2011).
2. G. Kocsis and F. Kun, Effect of network topologies on the spreading of technological developments, Journal of Statistical Mechanics: Theory and Experiment P10014 (2008).
3. F. Kun, G. Kocsis, and J. Farkas, Cellular automata for the spreading of technological developments in socio-economic systems, Physica A 383, 660 (2007).

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