Far Eastern Mathematical Journal

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Equations of the strain gradient theory in curvilinear coordinates


Guzev M. A., Qi Chengzhi

2013, issue 1, P. 35-42


Abstract
It is shown how to obtain the equilibrium equations of the strain gradient theory in curvilinear coordinates.

Keywords:
gradient theory, curvilinear coordinates

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References

[1] R. A. Toupin, “Elastic materials with couple stresses”, Arch. Ration. Mech. Anal., 11:1 (1962), 385–414.
[2] R. D. Mindlin, “Micro-structure in linear elasticity”, Arch. Ration. Mech. Anal., 16:1 (1964), 51–78.
[3] R. A. Toupin, “Theories of elasticity with couple-stress”, Arch. Ration. Mech. Anal., 17:2 (1964), 85–112.
[4] J. L. Bleustein, “A note on the boundary conditions of Toupin’s strain-gradient theory”, Int. J. Solids Struct., 3 (1967), 1053–1057.
[5] R. D. Mindlin, N. N. Eshel, “On first strain-gradient theories in linear elasticity”, Int. J. Solids Struct., 4 (1968), 109–124.
[6] N. N. Eshel, G. Rosenfeld, “Axi-symmetric problems in elastic materials of grade two”, J. Franklin Inst., 299:1 (1975), 43–51.
[7] P. Germain, “The method of virtual power in continuum mechanics. Part 2: microstructure”, SIAM J. Appl. Math., 25:3 (1973), 556–575.
[8] I. Vardoulakis, E. C. Aifantis, “A gradient flow theory of plasticity for granular materials”, Acta Mech., 87 (1991), 197–217.
[9] N. A. Fleck, J. W. Hutchinson, “A phenomenological theory for strain gradient effects in plasticity”, J. Mech. Phys. Solids, 41:12 (1993), 1825–1857.
[10] N. A. Fleck, J. W. Hutchinson, “A reformulation of strain gradient plasticity”, J. Mech. Phys. Solids, 49 (2001), 2245–2271.
[11] R. Chambon, D. Caillerie, T. Matsuchima, “Plastic continuum with microstructure, local second gradient theories for geomaterials: localization studies”, Int. J. Solids Struct., 38 (2001), 8503–8527.
[12] S. Lurie, P. Belov, N. Tuchkova, “The application of the multiscale models for description of the dispersed composites”, Comput. Mater. Sci., 36:2 (2004), 145–152.
[13] P. A. Belov, A. M. Bodunov, S. A. Lur'e, I. F. Obraztsov, Iu.G. Ianovskii, “O modelirovanii masshtabnykh effektov v tonkikh strukturakh”, Mekhanika kompozitsonnykh materialov i konstruktsii, 8:4 (2002), 585–598.
[14] J. D. Zhao, D. C. Sheng, “Strain gradient plasticity by internal-variable approach with normality structure”, Int. J. Solids Struct., 43 (2006), 5836–5850.
[15] J. D. Zhao, D. C. Sheng, S. W. Sloan, K. Krabbenhoft, “Limit theorems for gradient-dependent elastoplastic geomaterials”, Int. J. Solids Struct., 44 (2007), 480–506.
[16] C. Qi, Q. Qian, M. Wang, and J. Chen, “Derivation of governing equations of strain gradient model of rock mass near deep level tunnels”, Nonlinear geomechanical-geodynamic processes in deep mining, 2nd Russia-China Proceedings, July 2–5, Novosibirsk, 2012, 20–26.
[17] M. A. Guzev, “Sviaz' nelokal'noi i neevklidovoi modelei sploshnoi sredy”, Nelineinye geomekhanicheskie protsessy pri otrabotke mestorozhdenii poleznykh iskopaemykh na bol'shikh glubinakh, 2-ia Rossiisko-Kitaiskaia konferentsiia. Sbornik trudov, IGD SO RAN, Novosibirsk, 2012, 27–31.
[18] M. A. Guzev, “Comparision of non-Euclidean continuum model with strain gradient theory”, Abstract Book of the 23rd International Congress of Theoretical and Applied Mechanics, August 19–24, 2012, Beijing, China, 144.
[19] J. Zhao, D. Pedroso, “Strain gradient theory in orthogonal curvilinear coordinates”, International Journal of Solids and Structures, 45 (2008), 3507–3520.
[20] A. C. Eringen, Mechanics of Continua, John Wiley & Sons, Inc., 1967.

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