It is shown that the bulk defect-deformational (DD) nanostructuring of isotropic solids can be described by a closed three-dimensional (3D) nonlinear DD equation of the Kuramoto-Sivashinsry (KS) type for the nonequilibrium defect concentration, derived here in the framework of the nonlocal elasticity theory (NET). The solution to the linearized DDKS equation describes the threshold appearance of the periodic self-consistent strain modulation accompanied by the simultaneous formation of defect piles at extremes of the strain. The period and growth rate of DD nanostructure are determined. Based on the obtained results, a novel mechanism of nanostructuring of solids under the severe plastic deformation (SPD), stressing the role of defects generation and selforganization, described by the DDKS, is proposed. Theoretical dependencies of nanograin size on temperature and shear strain reproduce well corresponding critical dependencies obtained in experiments on nanostructuring of metals under the SPD, including the effect of saturation of nanofragmentation. The scaling parameter of the NET is estimated and shown to determine the limiting small grain size. 1. Introduction In recent years, the spontaneous formation of periodic nanostructures upon the energy input into solids (laser or particle beams, severe plastic deformation, electrochemical etching, and others) has become the subject of intensive studies. An effective tool for the theoretical investigation of this phenomenon is the nonlinear Kuramoto-Sivashinsky (KS) equation [1, 2]. 2D KS equations with specific variables are used, for example, for the description of nanostructuring of solid surfaces using ion beams [3] and electrochemical etching [4]. Recently, a novel 2D KS equation has been derived for a defect-enriched solid layer formed at the surface by laser irradiation, and its numerical solutions have been used for the interpretation of the formation of surface nano- and microstructures under the action of laser pulses [5]. In [6], a model of the defect-deformational (DD) instability in the system of mobile defects (interstitials and vacancies), interacting in the bulk of an isotropic solid via quasistatic self-consistent strain, has been developed. The mechanism of the bulk DD instability consists in the following. An initial fluctuation of the defect concentration gives rise to self-consistent strain. This is accompanied by the strain-induced defect fluxes enhancing initial defect concentration fluctuation while the diffusion tends to smooth it out. At exceeding of a certain critical defect
References
[1]
Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer, Berlin, Germany, 1984.
[2]
G. I. Sivashinsky, “Instabilities, pattern formation, and turbulence in flames,” Annual Review of Fluid Mechanics, vol. 15, pp. 179–199, 1983.
[3]
J. M. Garcia, L. Vazquez, R. Cuerno, J. A. Garcia, M. Castro, and R. Gago, “Self-organized surface nanopatterning by ion beam sputtering,” in Lecture Notes on Nanoscale Science and Technology 5, Z. Wang, Ed., pp. 323–398, Springer, Heidelberg, Germany, 2009.
[4]
A. G. Limonov, “Numeric simulation of the formation of hexagonal nanoscale structure arrays in Anodic Aluminum Oxide,” Mathematical Models and Computer Simulations, vol. 3, no. 2, pp. 149–157, 2011.
[5]
V. I. Emel’yanov, “Mechanisms of laser-induced self-organization of nano- and microstructures of surface relief in air and in liquid environment,” in Laser Ablation in Liquids, Principles and Applications in the Preparation of Nanomaterials, G. Yang, Ed., chapter 1, pp. 1–109, Pan Stanford, Singapore, 2012.
[6]
V. I. Emel'yanov and I. M. Panin, “Formation of nanometer ordered defect-deformational structures induced by energy fluxes in solids,” Physics of the Solid State, vol. 39, no. 11, pp. 1815–1820, 1997.
[7]
K. Arakawa, K. Ono, M. Isshiki, K. Mimura, M. Uchikoshi, and H. Mori, “Observation of the one-dimensional diffusion of nanometer-sized dislocation loops,” Science, vol. 318, no. 5852, pp. 956–959, 2007.
[8]
I. A. Kunin, Theory of Elastic Media with Microstructure, Springer, New York, NY, USA, 1982.
[9]
A. C. Eringen, Nonlocal Continuum Field Theories, Springer, New York, NY, USA, 2002.
[10]
L. D. Landau and E. M. Lifshitz, Theory of Elasticity, Pergamon, New York, NY, USA, 1986.
[11]
R. Z. Valiev, R. K. Islamgaliev, and I. V. Alexandrov, “Bulk nanostructured materials from severe plastic deformation,” Progress in Materials Science, vol. 45, no. 2, pp. 103–189, 2000.
[12]
R. Pippan, S. Scheriau, A. Taylor, M. Hafok, A. Hohenwarter, and A. Bachmaier, “Saturation of fragmentation during severe plastic deformation,” Annual Review of Materials Research, vol. 40, pp. 319–343, 2010.
[13]
E. Schafler, G. Steiner, E. Korznikova, E. Kerber, and M. J. Zehetbauer, “Lattice defect investigation of ECAP-Cu by means of X-ray line profile analysis, calorimetry and electrical resistometry,” Materials Science and Engineering A, vol. 410411, pp. 169–173, 2005.
[14]
D. Setman, E. Schafler, E. Korznikova, and M. J. Zehetbauer, “The presence and nature of vacancy type defects in nanometals detained by severe plastic deformation,” Materials Science and Engineering A, vol. 493, no. 1-2, pp. 116–122, 2008.
[15]
Y. Y. Zhang, C. M. Wang, W. H. Duan, Y. Xiang, and Z. Zong, “Assessment of continuum mechanics models in predicting buckling strains of single-walled carbon nanotubes,” Nanotechnology, vol. 20, no. 39, Article ID 395707, 2009.
