Modeling Crack Patterns by Modified STIT Tessellations

Roberto León; Werner Nagel; Joachim Ohser; Steve Arscott

doi:10.5566/ias.2245

Authors

Roberto León Universidad Andres Bello, Facultad de Ingeniería, Quillota 980, Vina del Mar, Chile
Werner Nagel University of Jena, Institute of Mathematics, 07737 Jena, Germany
Joachim Ohser University of Applied Sciences, Department of Mathematics and Natural Sciences, Schöfferstr. 3, 64295 Darmstadt, Germany
Steve Arscott Institut d'Electronique, de Microélectronique et de Nanotechnologie (IEMN), CNRS, The University of Lille, Cité Scientifique, 59652 Villeneuve d'Ascq, France

DOI:

https://doi.org/10.5566/ias.2245

Keywords:

fracture pattern, geometry-statistics, Monte Carlo simulation, random tessellation, STIT tessellation

Abstract

Random planar tessellations are presented which are generated by subsequent division of their polygonal cells. The purpose is to develop parametric models for crack patterns appearing at length scales which can change by orders of magnitude in areas such as nanotechnology, materials science, soft matter, and geology. Using the STIT tessellation as a reference model and comparing with phenomena in real crack patterns, three modifications of STIT are suggested. For all these models a simulation tool, which also yields several statistics for the tessellation cells, is provided on the web. The software is freely available via a link given in the bibliography of this article. The present paper contains results of a simulation study indicating some essential features of the models. Finally, an example of a real fracture pattern is considered which is obtained using the deposition of a thin metallic film onto an elastomer material – the results of this are compared to the predictions of the model.

References

Boulogne F, Giorgiutti-Dauphiné F, Pauchard L (2015). Surface patterns in drying films of silica colloidal dispersions. Soft Matter 11:102–8.

Chiu SN, Stoyan D, Kendall WS, Mecke J (2013). Stochastic geometry and its applications. Wiley Series in Probability and Statistics. John Wiley & Sons, Ltd., Chichester, 3rd ed.

Cowan R (2010). New classes of random tessellations arising from iterative division of cells. Adv in Appl Probab 42:26–47.

Graham RL (1972). An efficient algorithm for determining the convex hull of a finite planar set. Information Processing Letters 1:132–3.

Hafver A, Jettestuen E, Kobchenko M, Dysthe1 DK, Meakin P, Malthe-Srenssen A (2014). Classification of fracture patterns by heterogeneity and topology. EPL 105:56004.

Hutchinson J, Suo Z (1991). Mixed mode cracking in layered materials. Advances in applied mechanics 29:63–191.

Iben HN, O’Brien JF (2006). Generating surface crack patterns. In: Eurographics/ ACM SIGGRAPH Symposium on Computer Animation. IEEE.

Kumar A, Pujar R, Gupta N, Tarafdar S, Kulkarni, G.U. (2017). Stress modulation in desiccating crack networks for producing effective templates for patterning metal network based transparent conductors. Appl Phys Lett 111:013502–1 – 4.

León R (2019). Crack pattern simulation. http://crackpatternsimulation.informatica-unab-vm.cl/home/.

Mecke J, Nagel W, Weiß V (2008). A global construction of homogeneous random planar tessellations that are stable under iteration. Stochastics An International Journal of Probability and Stochastic Processes 80:51–67.

Mosser L, Matthäi S(2014). Tessellations stable under iteration.Evaluation of application

as an improved stochastic discrete fracture modeling algorithm. International Discrete Fracture Network Engineering Conference DFNE 2014 - 163.

Nagel W, Mecke J, Ohser J, Weiss V (2008). A tessellation model for crack patterns on surfaces. Image Anal Stereol 27:73–8.

Nagel W, Weiß V (2005). Crack tessellations: characterization of stationary random tessellations stable with respect to iteration. Adv Appl Prob SGSA 37:859–83.

Nandakishore P, Goehring L (2016). Crack patterns over uneven substrates. Soft Matter 12:2233–2492.

Ohser J,Schladitz K (2009). 3d Images of Materials Structures – Processing and Analysis. Weinheim, Berlin: Wiley VCH.

Ohser J, Steinbach B, Lang C (1998). Efficient texture analysis of binary images. J Microsc 192:20–8.

Schneider R, Weil W (2008). Stochastic and Integral Geometry. Berlin Heidelberg: Springer.

Seghir R, Arscott S (2015). Controlled mud-crack patterning and self-organized cracking of polydimethylsiloxane elastomer surfaces. Scientific Reports 5:14787–802.

Serra J(1982). Image Analysis and Mathematical Morphology. London: AcademicPress.

Stoyan D, Mecke J, Pohlmann S (1980). Formulas for stationary planar fibre processes. II. Partially oriented fibre systems. Math Operationsforsch Statist Ser Statist 11:281–6.

Thouless M, Evans A, Ashby M, Hutchinson J (1987). The edge cracking and spalling of brittle plates. Acta Metallurgica 35:1333–41.

Xia Z, Hutchinson J (2000). Crack patterns in thin films. Journal of the Mechanics and Physics of Solids 48:1107–31.

Yu X, Courtat T, Martins P, Decreusefond L, Kelif JM (2014). Crack STIT tessellations for city modeling and impact of terrain topology on wireless propagation. In: The10th International Workshop on Wireless Network Measurements and Experimentation (WiNMeE 2014). IEEE.