Fractal Shear Bands at Elastic-Plastic Transitions in Random Mohr-Coulomb Materials
Publication: Journal of Engineering Mechanics
Volume 140, Issue 9
Abstract
This paper studies fractal patterns forming at elastic-plastic transitions in soil- and rock-like materials. Taking either friction or cohesion as nonfractal vector random fields with weak noise-to-signal ratios, it is found that the evolving set of plastic grains (i.e., a shear-band system) is always a monotonically growing fractal under increasing macroscopic load in plane strain. Statistical analysis is used to assess the anisotropy of those shear bands. All the macroscopic responses display smooth transitions, but as the randomness vanishes, they turn into a sharp response of an idealized homogeneous material. Parametric study shows that increasing hardening or friction makes the transition more rapid. In addition, randomness in cohesion has a stronger effect than randomness in friction, whereas dilatation has practically no influence. Adapting the concept of scaling functions, the authors find the elastic-plastic transitions in random Mohr-Coulomb media to be similar to phase transitions in condensed-matter physics: the fully plastic state is a critical point, and with three order parameters (reduced Mohr-Coulomb stress, reduced plastic volume fraction, and reduced fractal dimension), three scaling functions are introduced to unify the responses of different materials. The critical exponents are demonstrated to be universal regardless of the randomness in various constitutive properties and their random noise levels.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
This work was made possible by support from the National Science Foundation (Grant No. CMMI-1030940). Also, partial support of the second author as the Timoshenko Distinguished Visitor in the Division of Mechanics and Computation at Stanford University is gratefully acknowledged.
References
Bakó, B., and Hoffelner, W. (2007). “Cellular dislocation patterning during plastic deformation.” Phys. Rev. B, 76(21), 214108.
Balankin, A. S., Susarrey, O., Mora Santos, C. A., Patiño, J., Yoguez, A., and Garcia, E. J. (2011). “Stress concentration and size effect in fracture of notched heterogeneous material.” Phys. Rev. E, 83(1), 015101.
Berryman, J. G. (1985). “Measurement of spatial correlation functions using image processing techniques.” J. Appl. Phys., 57(7), 2374–2384.
Bigoni, D. (2012). Nonlinear solid mechanics: Bifurcation theory and material instability, Cambridge University Press, Cambridge, UK.
Borodich, F. M. (1997). “Some fractal models of fracture.” J. Mech. Phys. Solids, 45(2), 239–259.
Dassault Systèmes Simulia. (2010). ABAQUS user’s manual version 6.10, Providence, RI.
Falconer, K. (2003). Fractal geometry: Mathematical foundations and applications, Wiley, New York.
Feder, J. (1988). Fractals (physics of solids and liquids), Springer, New York.
Goldenfeld, N. D. (1992). Lectures on phase transitions and the renormalization group, Addison-Wesley, Reading, MA.
Hazanov, S. (1998). “Hill condition and overall properties of composites.” Arch. Appl. Mech., 68(6), 385–394.
Jiang, M., Ostoja-Starzewski, M., and Jasiuk, I. (2001). “Scale-dependent bounds on effective elastoplastic response of random composites.” J. Mech. Phys. Solids, 49(3), 655–673.
Lebedev, V., Didenko, V., and Lapin, A. (2003). “Small-angle neutron scattering investigation of plastically deformed stainless steel.” J. Appl. Cryst., 36(1), 629–631.
Li, J., and Ostoja-Starzewski, M. (2010a). “Fractal pattern formation at elastic-plastic transition in heterogeneous materials.” J. Appl. Mech, 77(2), 021005.
Li, J., and Ostoja-Starzewski, M. (2010b). “Fractals in elastic-hardening plastic materials.” Proc. R. Soc. A, 466(2114), 603–621.
Li, J., and Ostoja-Starzewski, M. (2011). “Fractals in thermoelastoplastic materials.” J. Mech. Mater. Struct., 6(1–4), 351–359.
Li, J., Saharan, A., Koric, S., and Ostoja-Starzewski, M. (2012). “Elastic-plastic transition in 3D random materials: Massively parallel simulations, fractal morphogenesis and scaling functions.” Philos. Mag., 92(22), 2733–2758.
Mandelbrot, B. (1982). The fractal geometry of nature, W. H. Freeman, New York.
MathWorks. (2009). MATLAB user guide, Natick, MA.
Ord, A. (1991). “Deformation of rock: A pressure-sensitive, dilatant material.” Pure Appl. Geophys., 137(4), 337–366.
Ostoja-Starzewski, M. (1990). “Micromechanics model of ice fields. II: Monte Carlo simulation.” Pure Appl. Geophys., 133(2), 229–249.
Ostoja-Starzewski, M. (2008). Microstructural randomness and scaling in mechanics of materials, CRC Press, Boca Raton, FL.
Poliakov, A. N. B., and Herrmann, H. J. (1994). “Self-organized criticality of plastic shear bands in rocks.” Geophys. Res. Lett., 21(19), 2143–2146.
Poliakov, A. N. B., Herrmann, H. J., Podladchikov, Y. Y., and Roux, S. (1994). “Fractal plastic shear bands.” Fractals, 2(4), 567–581.
Ranganathan, S. I., and Ostoja-Starzewski, M. (2008). “Scaling function, anisotropy and the size of RVE in elastic random polycrystals.” J. Mech. Phys. Solids, 56(9), 2773–2791.
Sahimi, M. (2003). Heterogeneous materials, Vols. I and II, Springer, New York.
Sahimi, M., and Arbabi, S. (1993). “Mechanics of disordered solids. III: Fracture properties.” Phys. Rev. B, 47(2), 713–722.
Sornette, D. (2004). Critical phenomena in natural sciences, Springer, New York.
Stinchcombe, R. B. (1989). “Fractals, phase transitions and criticality.” Proc. R. Soc. A, 423(1864), 17–33.
Zaiser, M., Bay, K., and Hahner, P. (1999). “Fractal analysis of deformation-induced dislocation patterns.” Acta Mater., 47(8), 2463–2476.
Information & Authors
Information
Published In
Copyright
© 2014 American Society of Civil Engineers.
History
Received: Mar 6, 2013
Accepted: Nov 15, 2013
Published online: Nov 18, 2013
Discussion open until: Aug 19, 2014
Published in print: Sep 1, 2014
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.