Abstract
Mathematical methods are developed for solving the statics equations of a non-linear model of deforming a crystalline medium with a complex lattice that allows martensitic transformations. In the nonlinear theory, the deformation describes the vector of the acoustic mode U(t, x, y, z) and the vector of the optical mode u(t, x, y, z). They are found from a system of six related nonlinear equations. The vector of the acoustic mode U(t, x, y, z) is sought in the Papkovich–Neuberform. A system of six related nonlinear equations is transformed into a system of individual equations. The equations of the optical mode u(t, x, y, z) are reduced to one sine-Gordon equation with a variable coefficient (amplitude) in front of the sine and two Poisson equations. The definition of the acoustic mode is reduced to solving the scalar and vector Poisson equations. For a constant-amplitude optical mode, particular solutions were found. In the case of plane deformation, a class of doubly-periodic solutions is constructed, which are expressed in terms of Jacobi elliptic functions. The analysis of the solutions found is conducted. It is shown that the nonlinear theory describes the fragmentation of the crystalline medium, the formation of boundaries between fragments, phase transformations, the formation of defects and other deformation features that are realized in the field of high external force effects and which are not described by classical continuum mechanics.
Similar content being viewed by others
References
E. L. Aero, “Microscale Deformations in a Two-Dimensional Lattice: Structural Transitions and Bifurcations under Critical Shear,” Fiz. Tv. Tela 42(6), 1113–1119 (2000) [Phys. Sol. Stat. (Engl. Transl.) 42(6), 1147–1153 (2000)].
E. L. Aero, “Substantially Nonlinear Micromechanics of a Medium with a Variable Periodic Structure,” Usp. Mekh., No. 1, 130–176 (2002)
M. Born and K. Huang, Dynamical Theory of Crystal Lattices (Clarendon Press, Oxford, 1954; Izdvo Inostrannoi Literatury, Moscow, 1958).
D. A. Indeytsev, Yu. I. Meshcheryakov, A. Yu. Kuchmin, and D. S. Vavilov, “A Multiscale Model of Propagation of Steady Elasto-Plastic Waves,” Dokl. Ross. Akad. Nauk 458(2), 490–493 (2014) [Dokl. Phys. (Engl. Transl.) 59 (9), 423–426 (2014)].
V. G. Vaks, Introduction to the Microscopic Theory of Ferroelectrics (Nauka, Moscow, 1973) [in Russian].
A.D. Bruce and R.A. Cowley, Structural Phase Transitions (Taylor & Francis Ltd., London, 1981; Mir, Moscow, 1984).
M. P. Shaskolskaya, Crystallography (Vysshaya Shkola, Moscow, 1984) [in Russian].
T. A. Kontorova and Ya. I. Frenkel, “To the Theory of Plastic Deformation and Twinning,” Zh. Eksp. Teor. Fiz. 8, 89–95 (1938).
O.M. Braun and Y. S. Kivshar, The Frenkel-Kontorova Model. Concepts, Methods, and Applications (Springer, New York, 2004).
A. V. Porubov, E. L. Aero and G. A. Maugin, “Two Approaches to Study Essentially Nonlinear and Dispersive Properties of the Internal Structure of Materials,” Phys. Rev. E. 79, 046608 (2009).
W. Voigt, Lehrbuch der Kristalphysik (Leipzig, 1910).
C. Kittel, Introdution to Solid State Physics (John Willey and Sons, Inc., New York, 1970; GIFML, Moscow, 1978).
E. L. Aero, A.N. Bulygin and Yu.V. Pavlov, “Functionally Invariant Solutions of Nonlinear Klein-Fock- Gordon Equation,” Appl. Math. Comp. 223, 160–166 (2013).
E. L. Aero, A.N. Bulygin and Yu.V. Pavlov, “Mathematical Methods for Solution of Nonlinear Model of Deformation of Crystal Media with Complex Lattice,” in Proc. Int. Conference “Days on Diffraction 2015” (SPbGU, St. Petersburg, 2015), pp. 8–13.
E. L. Aero, A. N. Bulygin and Yu. V Pavlov, “Nonlinear Model of Deformation of Crystal Media with Complex Lattice: Mathematical Methods of Model Implementation,” Math. Mech. Sol. 21, 19–36 (2016).
Acknowledgments
This work was supported by the Russian Foundation for Basic Research (grants No. 16-01-00068-a and No. 17-01-00230-a).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © E.L. Aero, A.N. Bulygin, Yu.V. Pavlov, 2018, published in Izvestiya Akademii Nauk, Mekhanika Tverdogo Tela, 2018, No. 6, pp. 30–40.
About this article
Cite this article
Aero, E.L., Bulygin, A.N. & Pavlov, Y.V. Nonlinear Deformation Model of Crystal Media Allowing Martensite Transformations: Solution of Static Equations. Mech. Solids 53, 623–632 (2018). https://doi.org/10.3103/S0025654418060043
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S0025654418060043