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IssuesArchive of Issues2010-5pp.670-688

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E.L. Aero and A.N. Bulygin, "Nonlinear Theory of Crystalline Solids. Origination of Nano- and Microstructures and Their Stability," Mech. Solids. 45 (5), 670-688 (2010)
Year 2010 Volume 45 Number 5 Pages 670-688
DOI 10.3103/S0025654410050031
Title Nonlinear Theory of Crystalline Solids. Origination of Nano- and Microstructures and Their Stability
Author(s) E.L. Aero (Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, Bol'shoy pr-t 61, St.†Petersburg, 199178 Russia, 16aero@mail.ru)
A.N. Bulygin (Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, Bol'shoy pr-t 61, St.†Petersburg, 199178 Russia, bulygin_an@mail.ru)
Abstract We develop a theory of large deformations of a crystal lattice as a generalization of linear equations of acoustic and optical modes of a complex lattice composed of two interpenetrating sublattices. We suggest a principle of internal translational symmetry with respect to mutual displacements of sublattices by an integer number of periods. Based on this principle, we assume that the force of interaction of the sublattices is a nonlinear periodic function.

We consider some solutions to new nonlinear equations with two characteristics scales and classify all solutions (periodic, increasing, localized). Analysis shows that the local density of the structural reorganization energy nontrivially depends on the dimension (plate thickness H). The fact that the curve has extrema (disappearing in the case of large macroscopic thicknesses) demonstrates conditions for the plate microstructure stability or instability depending on its dimension. This allows one to obtain criteria for the appearance or disappearance of nanoclusters depending on the external stresses and the conditions on the plate boundary.

We also consider the structural transitions at the points of instability, which result in the appearance of defects (interphase boundaries, dislocations, slip lines, small-angle boundaries), domain structures and other inhomogeneities (packing defects, clusters). We study the problem of effective properties of bodies with microstructure associated with the scale level of description.
Keywords nano-structure, defects, domains, plane boundaries, bifurcation transitions, scale effects, effective properties
References
1.  M. Born and K. Huang, Dynamical Theory Of Crystal Lattices (Clarendon, Oxford, 1954; Izd-vo Inostr. Lit., Moscow, 1958).
2.  A. M. Kosevich, Theory of Crystal Lattice. Physical Mechanics of Crystals (Vishcha Shkola, Khar'kov, 1988) [in Russian].
3.  I. A. Kunin, Theory of Elastic Media with Microstructure (Nauka, Moscow, 1975) [in Russian].
4.  E. Cosserat and F. Cosserat, Théorie des Corps Déformables (Hermann, Paris, 1909).
5.  V. I. Erofeev, Wave Processes in Solids with Microstructure (Izd-vo MGU, Moscow, 1999) [in Russian].
6.  S. A. Lurie and P. A. Belov, "Variational Principles for Mathematical Models of Media with Microstructure," in Mathematical Modeling of Systems and Processes, No. 14 (Izd-vo Perm Tekhn. Univ., Perm, 2006), pp. 114-132 [in Russian].
7.  E. L. Aero, "Structural Transitions and Shear Stability of Polyatomic Layers," Izv. Ross. Akad. Nauk. Neorganich. Mater. 35 (8), 1010-1013 (1999) [Inorganic Mater. (Engl. Transl.) 35 (8), 860-862 (1999)].
8.  E. L. Aero, "Essentially Nonlinear Micromechanics of a Media with Variable Periodic Structure," Uspekhi Mekh. 1 (3), 130-176 (2002).
9.  E. L. Aero, "Inhomogeneous Microscopic Shear Strains in a Complex Crystal Lattice Subjected to Large Macroscopic Strains (Exact Solutions)," Fiz. Tverd. Tela 45 (8), 1482-1490 (2003) [Phys. Solid State (Engl. Transl.) 45 (8), 1557-1565 (2003)].
10.  E. L. Aero, "Strongly Nonlinear Theory of Elastic and Nonelastic Strains in Crystal Bodies," in Mathematical Modeling of Systems and Processes, No. 14 (Izd-vo Perm Tekhn. Univ., Perm, 2006), pp. 27-55 [in Russian].
11.  E. L. Aero, "Micromechanics of a Double Continuum in a Model of a Medium with Variable Periodic Structure," J. Engng Mat. 55 (1-4), 81-95 (2006).
12.  E. L. Aero and A. N. Bulygin, "Strongly Nonlinear Theory of Nanostructure Formation Owing to Elastic and Nonelastic Strains in Crystalline Solids," Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 5, 170-187 (2007) [Mech. Solids (Engl. Transl.) 42 (5), 807-822 (2007)].
13.  E. L. Aero, "Plane Boundary Value Problems for the Sine-Helmholtz Equation in the Theory of Elasticity of Liquid Crystals in Non-Uniform Magnetic Fields," Prikl. Mat. Mekh. 60 (1), 79-87 (1996) [J. Appl. Math. Mech. (Engl. Transl.) 60 (1), 75-83 (1996)].
14.  E. Jahnke, F. Emde, and F. Lösch, Tables of Higher Functions (Tourner, Stuttgart, 1960; McGraw-Hill, New York, 1960; Nauka, Moscow, 1977).
15.  V. S. Boiko, R. I. Garber, and A. M. Kosevich, Reversible Plasticity in Crystals (Nauka, Moscow, 1991) [in Russian].
16.  A. L. Korzhenevskii, E. L. Aero, and A. N. Bulygin, "Microscopic Theory of Formation of Martensitic Phases," Izv. Ross. Akad. Nauk. Ser. Fiz. 69, No. 9, 1271-1281 (2005) [Bull. Russ. Acad. Sci. Phys. (Engl. Transl.) 69, No. 9, 1421-1434 (2005)].
17.  P. Hirth and J. Lothe, Theory of Dislocations (Atomizdat, Moscow, 1972) [in Russian].
18.  E. L. Aero and A. N. Bulygin, "The Nonlinear Theory of Localized Waves in a Complex Crystalline Lattice as a Discrete Continual Model," Vych. Mekh. Sploshn. Sred 1, No. 1, 14-30 (2008).
Received 26 January 2010
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