A Kinetic Model of Nucleation, Propagation and Fusion of Cracks.

Czechowski, Zbigniew

doi:10.4294/jpe1952.41.127

Cited by 16 publications

(15 citation statements)

References 11 publications

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“…An inspiration to the concept of privilege, which can be treated as a generalization of Simon's (SIMON 1955, 1960 idea of preferential attachment, was an analysis of the coagulation equation in the crack fusion problem (CZECHOWSKI 1993). In our papers (CZECHOWSKI 2001(CZECHOWSKI , 2003(CZECHOWSKI , 2005CZECHOWSKI and RO-ZMARYNOWSKA 2008), it was shown that in many cases (i.e., percolation processes, cellular automata, Cantor set, resource redistribution model, return-to-the-origin problem in random walk, multiplicative processes, and multiplication of probabilities) the hidden privilege can be extracted and explained.…”

Section: Discussionmentioning

confidence: 99%

On Microscopic Mechanisms Which Elongate the Tail of Cluster Size Distributions: An Example of Random Domino Automaton

Czechowski

2014

Pure Appl. Geophys.

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Abstract-On the basis of simple cellular automaton, the microscopic mechanisms, which can be responsible for elongation of tails of cluster size distributions, were analyzed. It was shown that only the appropriate forms of rebound function can lead to inverse power tails if densities of the grid are small or moderate. For big densities, correlations between clusters become significant and lead to elongation of tails and flattening of the distribution to a straight line in log-log scale. The microscopic mechanism, given by the rebound function, included in simple 1D RDA can be projected on the geometric mechanism, which favours larger clusters in 2D RDA.

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Section: Discussionmentioning

confidence: 99%

On Microscopic Mechanisms Which Elongate the Tail of Cluster Size Distributions: An Example of Random Domino Automaton

Czechowski

2014

Pure Appl. Geophys.

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“…The integral (continuous) form of the equation is as follows: where B ( y , y ′) is the probability of fusion of two objects with sizes y and y ′, and f ( t , y ) is the size distribution function of objects in time t . In was shown (Czechowski 1993) that for a constant coagulation coefficient B ( y , y ′) (there is no privilege—small and large objects fuse with the same rate) the solution of has the exponential form On the other hand, for B ( y , y ′) ∼ y + y ′ (then the probability of fusion for larger objects is greater than for smaller objects—larger objects are privileged) we can mention the solution of the coagulation equation for planetesimals evolution (Safronov 1972): for large y . The physical reason for such a form of B ( y , y ′) for gravitational bodies (planetesimals) is presented in Safronov (1972).…”

Section: The Coagulation Equationmentioning

confidence: 98%

“…Another application of the coagulation equation in geophysics was the problem of crack fusion (Czechowski 1991, 1993, 1997). It seems that for brittle materials, such as rocks, the fusion cross‐section may also be given by B ( y , y ′) ∼ y + y ′, because larger cracks induce greater stresses around its tips.…”

Section: The Coagulation Equationmentioning

confidence: 99%

“…where B(y, y ) is the probability of fusion of two objects with sizes y and y , and f (t, y) is the size distribution function of objects in time t. In was shown (Czechowski 1993) that for a constant coagulation coefficient B(y, y ) (there is no privilege-small and large objects fuse with the same rate) the solution of eq. (13) has the exponential form…”

Section: T H E C Oa G U L At I O N E Q Uat I O Nmentioning

confidence: 99%

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The privilege as the cause of power distributions in geophysics

Czechowski

2003

Geophysical Journal International

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SUMMARY Power laws are known to be associated with dynamic systems residing near the critical point in the state space of the system. However, both models, that of phase transitions reached by the tuning parameter (for example, the percolation models) and SOC (self‐organised critical) models, although leading to power‐law relations, still do not explaining causes of their appearance. In this contribution it is assumed that nature is random in its deep structure. The concept of the privilege, meaning the susceptibility of the state on to a change, is introduced. The model describing the privilege is analysed and applied to some known models: the Cantor set, the percolation theory and simple cellular automata. An adequate form of the privilege explains the appearance of the inverse‐power distributions in many phenomena in geophysics. There is a relevance between the privilege concept and the results of a previous paper, in which the influence of the non‐linearity of a model on the output behaviour was investigated.

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“…The kinetic model of evolution of crack population was introduced by Czechowski [5] and developed in [6,7,8,9,10]. It lies at a level intermediate between the purely statistical approach and the fully microscopic treatment.…”

Section: Crack Propagationmentioning

confidence: 99%

Crack nucleation in solid materials under external load - simulations with the Discrete Element Method

Klejment

Dębski

2018

MATEC Web Conf.

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Abstract. Numerical analysis of cracking processes require an appropriate numerical technique. Classical engineering approach to the problem has its roots in the continuum mechanics and is based mainly on the Finite Element Method. This technique allows simulations of both elastic and large deformation processes, so it is very popular in the engineering applications. However, a final effect of cracking -fragmentation of an object at hand can hardly be described by this approach in a numerically efficient way since it requires a solution of a problem of nontrivial evolving in time boundary conditions. We focused our attention on the Discrete Element Method (DEM), which by definition implies "molecular" construction of the matter. The basic idea behind DEM is to represent an investigated body as an assemblage of discrete particles interacting with each other. Breaking interaction bonds between particles induced by external forces imeditelly implies creation/evolution of boundary conditions. In this study we used the DEM approach to simulate cracking process in the three dimensional solid material under external tension. The used numerical model, although higly simplified, can be used to describe behaviour of such materials like thin films, biological tissues, metal coatings, to name a few.

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A Kinetic Model of Nucleation, Propagation and Fusion of Cracks.

Cited by 16 publications

References 11 publications

On Microscopic Mechanisms Which Elongate the Tail of Cluster Size Distributions: An Example of Random Domino Automaton

On Microscopic Mechanisms Which Elongate the Tail of Cluster Size Distributions: An Example of Random Domino Automaton

The privilege as the cause of power distributions in geophysics

Crack nucleation in solid materials under external load - simulations with the Discrete Element Method

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