Abstract
A sequential presentation of the theory of media by conserved dislocations as a variant of the theory of media with a microstructure (according to Mindlin’s definition) is given as well as a rather complete description of the particular variants of the theory, relevant from an applied point of view: Cosserat and Aero–Kuvshinskii media, porous media, media with “twinning”. The correctness of the formulation of models is determined by the use of a “kinematic” variational principle based on a formal description of the kinematics of media, the formulation of kinematic constraints for media of different complexity, and the construction of the corresponding potential energy of deformation using the Lagrange multiplier procedure. A system of defining relations is established and an agreed formulation of the boundary value problem is formulated. In this paper, much attention is paid to the analysis of the physical side of models of the media studied. The interpretation of all physical characteristics responsible for nonclassical effects is proposed, and a description of the spectrum of adhesive mechanical parameters is given. The generalized Aero-Kuvshinskii hypothesis about the proportionality of free and constrained distortions is proposed.
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References
Dislocations in Solids, 1st edn. In: Hirth, J. (eds.) A tribute to F.R.N. Nabarro, vol. 14, p. 650 (2008)
Maugin, G.A.: Material forces: concepts and applications. Appl. Mech. Rev. 48, 213–245 (1995)
Likhachev, V.A., Volkov, A.E., Shudegov, V.E.: Continuum theory of defects. Leningrad State Univ. Publ. 228, Leningrad (in Russian) (1986)
Nabarro, F.R.N.: Theory of Crystal Dislocations. Oxford University Press, Oxford (1967)
Gutkin, M.Y.: Elastic behavior of defects in nanomatarials. I model for infinite and semi-infinite media. Rev. Adv. Mater. Sci. 13, 125–161 (2006)
Fleck, N.A., Hutchinson, J.W.: A phenomenological theory for strain gradient effects in plasticity. J. Mech. Phys. Solids 41, 1825–1857 (1993)
Fleck, N.A., Hutchinson, J.W.: A reformulation of strain gradient plasticity. J. Mech. Phys. Solids 49, 2245–2271 (2001)
Fleck, N.A., Hutchinson, J.W.: Strain gradient plasticity. Adv. Appl. Mech. 33, 295–361 (1997)
Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 1, 51–78 (1964)
Mindlin, R.D., Tiersten, H.F.: Effects of the couple-stress in linear elasticity. Arch. Ration. Mech. Anal. 11, 415–448 (1962)
Kroner, E.: Dislocations and continuum mechanics. Appl. Mech. Rev. 15, 599–606 (1962)
Kroner, E.: Gauge field theories of defects in solids. Stuttgart Max-Plank Inst. (1982)
De Wit, R.: Theory of dislocations: continuous and discrete disclinations in isotropic elasticity. J. Res. Nat. Bur. Stand. 77A(3), 359–368 (1973)
De Wit, R.: The continual theory of the stationary dislocations. Solid State Phys. 10, 249 (1960)
Aifantis, E.C.: Strain gradient interpretation of size effects. Int. J. Fract. 95, 299–314 (1999)
Gao, H., Huang, Y., Nix, W.D., Hutchinson, J.W.: Mechanism-based strain gradient plasticity—I. Theory. J. Mech. Phys. Solids 47, 1239–1263 (1999)
Aero, E.L., Kuvshinskii, E.V.: Fundamental equations of elasticity theory for the media with a rotational interaction of particles. Fiz. Tverd. Tela 2, 1399–1409 (1960)
Lurie, S.A., Belov, P.A., Tuchkova, N.P.: Gradient theory of media with conserved dislocations: application to microstructured materials. In: Maugin, G.A., Metrikine, A.V. (eds.) One hundred years after the Cosserats, advances in mechanics and mathematics, vol. 21, pp. 223–234. Springer, Heidelberg (2010)
Belov, P.A., Lurie, S.A.: Mathematical theory of directness media. Gradient Theories, Formulations, Hierarchy. Analysis. Applications, p. 337. Palmarium Academic Publishing (2014)
Lurie, S.A., Belov, P.A., Volkov-Bogorodsky, D.B., Tuchkova, N.P.: Interphase layer theory and application in the mechanics of composite materials. J. Mat. Sci. 41(20), 6693–6707 (2006)
Belov, P.A., Lurie, S.A.: A continuum model of microheterogeneous media. J. Appl. Math. Mech. 73(5), 599–608 (2009)
Lurie, S., Belov, P., Altenbach, H.: Classification of gradient adhesion theories across length scale. Generalized Continua as Models for Classical and Advanced Materials Series, vol. 42, pp. 261–277 (2016)
Altenbach, H., Eremeyev, V.A.: On the shell theory on the nanoscale with surface stresses. Int. J. Eng. Sci. 49(12), 1294–1301 (2011)
Eremeyev, V.A.: On effective properties of materials at the nano-and microscales considering surface effects. Acta Mech. 227(1), 29–42 (2016)
Cosserat, E., Cosserat, F.: Theore des corps deformables. Hermann, Paris (1909)
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This work was supported particular by the Russian Foundation for Basic Research project No. 15-01-03649 and 16-01-00623, 17-01-00837.
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Lurie, S.A., Belov, P.A., Rabinskiy, L.N. (2018). Model of Media with Conserved Dislocation. Special Cases: Cosserat Model, Aero-Kuvshinskii Media Model, Porous Media Model. In: dell'Isola, F., Eremeyev, V., Porubov, A. (eds) Advances in Mechanics of Microstructured Media and Structures. Advanced Structured Materials, vol 87. Springer, Cham. https://doi.org/10.1007/978-3-319-73694-5_13
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