Resolving differential equations of physically nonlinear theory of elasticity in terms of stresses for a plane strain

S. V. Bakushev

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 63 (2020), pp. 72-86

Résumé

The paper is aimed to obtain resolving differential equations of physically nonlinear theory of elasticity in terms of stresses for a plane strain. These equations represent a mathematical model of the continuum whose variable coefficient of the volume expansion (compressibility) is a function of average stress only, and the variable coefficient of the shear is a function of tangential stress intensity only. The resolving differential equations are obtained by inserting the physical relations, in which the strains are expressed in terms of stresses, into Saint-Venant's compatibility condition written for a plane problem. As a result, a physically nonlinear analogue of the Levy equation for linear theory of elasticity is derived. When balance equations are satisfied irrespective of volume forces, stress function introducing yields a physically nonlinear analogue of the Levy equation represented as a physically nonlinear analogue of the biharmonic equation for a plane strain. As opposed to physically linear theory of elasticity, where biharmonic equations are homogeneous, the analogue to the biharmonic equation of physically nonlinear theory of elasticity is inhomogeneous. The form of the right side of the biharmonic equation is governed by the analyzed mathematical model of continuum. The obtained results can be used when solving the problems of physically nonlinear theory of elasticity in terms of stresses.

Détail

Keywords: theory of elasticity, plane strain, physical nonlinearity, resolving differential equation, solution in terms of stresses.

Bibliographie

[1] Dzh. F. Bell, Experimental basis of the deformable solid mechanics, v. 1, Small strains, Nauka, M., 1984

[2] Dzh. F. Bell, Experimental basis of the deformable solid mechanics, v. 2, Finite strains, Nauka, M., 1984

[3] G. Kauderer, Nonlinear Mechanics, Izdatel'stvo inostrannoy literatury, M., 1961

[4] A. I. Lurie, Nonlinear Theory of Elasticity, North-Holland, Amsterdam, 1990 | MR | Zbl

[5] V. V. Novozhilov, Theory of elasticity, Sudpromgiz, Leningrad, 1958

[6] G. A. Geniev, V. S. Leytes, Issues of the mechanics of inelastic bodies, Stroyizdat, M., 1981

[7] S. P. Ivanov, M. N. Akhmetshin, “Solution to the physically nonlinear plane problem of the theory of elasticity and its application in the calculation of the beams interacting with medium”, Stroitel'naya mekhanika inzhenernykh konstruktsiy i sooruzheniy — Structural Mechanics of Engineering Constructions and Buildings, 2012, no. 2, 33–36

[8] S. P. Ivanov, O. G. Ivanov, “Calculation of physically non-linear plate systems, interacting with elastic medium”, Mekhanika kompozitsionnykh materialov i konstruktsiy — Mechanics of Composite Materials and Structures, 11:1 (2005), 146–155

[9] I. V. Dudina, S. A. Zherdeva, “Allowance for physical nonlinearity of materials when estimating the reliability of reinforced concrete structures”, Sistemy. Metody. Tekhnologii — Systems. Methods. Technologies, 2009, no. 2(2), 66–68

[10] S. N. Pichugin, “Calculation of shell constructions in the form of reservoirs with a physically nonlinear strain”, Izvestiya vysshikh uchebnykh zavedeniy. Neft' i gaz Oil and Gas Studies, 2010, no. 3, 64–69 | MR

[11] A. G. Protosenya, V. I. Semenov, I. K. Suprun, “Analysis of load formation on lining of tunnel constructed in soil with nonlinear behavior”, Zapiski Gornogo instituta — Journal of Mining Institute, 199 (2012), 173–175

[12] A. G. Protosenya, “Physically nonlinear plastic problem of the stress distribution around a circular groove”, Izvestiya vysshikh uchebnykh zavedeniy. Gornyy zhurnal — News of the Higher Institutions. Mining Journal, 2014, no. 2, 43–48

[13] A. I. Aleksandrovich, A. V. Gorlova, “Testing of the planar problem of a physically nonlinear elastic body by methods of the theory of functions of a complex variable”, Izvestiya Rossiyskoy akademii nauk. Mekhanika tverdogo tela Mechanics of Solids, 2007, no. 3, 63–72

[14] V. D. Bondar', “Method of complex potentials in nonlinear theory of elasticity”, Prikladnaya mekhanika i tekhnicheskaya fizika — Journal of Applied Mechanics and Technical Physics, 41:1 (2000), 133–143 | MR | Zbl

[15] A. A. Sheina, A. I. Aleksandrovich, “Solution of threedimensional boundary value problems of the nonlinear theory of elasticity using the methods of multi-dimensional complex analysis”, Vestnik Nizhegorodskogo universiteta im. N.I. Lobachevskogo — Vestnik of Lobachevsky University of Nizhni Novgorod, 2011, no. 4–4, 1862–1863

[16] V. A. Nifagin, A. V. Sevrouk, “Common approximate solutions of main problems of spatial non-linear elasticity theory in analytical multi-dimensional functions of matrix variable”, Science and Technique, 2007, no. 1, 60–65 | DOI | DOI

[17] V. M. Mal'kov, Yu. V. Mal'kova, “Analysis of a stress singularity in a non-linear Flamant problem for certain models of a material”, Journal of Applied Mathematics and Mechanics, 72:4 (2008), 468–474 | DOI | MR | Zbl

[18] Lavrov Kirill, Semenov Artem, Benin Andrey, “Modeling of nonlinear multiaxial deformation of concrete on the base of hyperelastic orthotropic model”, MATEC Web of Conferences, 53 (2016), 01043 | DOI

[19] Chenji Wei, Liangang Wang, Baozhu Li, Lihui Xiong, Shuangshuang Liu, Jie Zheng, Suming Hu, Hongqing Song, “A study of nonlinear elasticity effects on permeability of stress sensitive shale rocks using an improved coupled flow and geomechanics model: a case study of the longmaxi shale in China”, Energies, 11:2 (2018), 329 | DOI

[20] N. S. Blokhina, “Calculation of the constructions made of anisotropic material with account for physical nonlinearity”, Stroitel'naya mekhanika i raschet sooruzheniy — Structural Mechanics and Analysis of Constructions, 2012, no. 1, 3–5

[21] V. P. Agapov, K. R. Aydemirov, “Application of finite element method taking into account physical and geometric nonlinearity for the calculation of prestressed reinforced concrete beams”, Vestnik Dagestanskogo gosudarstvennogo tekhnicheskogo universiteta. Tekhnicheskie nauki — Herald of Dagestan State Technical University. Technical Sciences, 44:1 (2017), 127–137 | DOI | DOI

[22] I. N. Serpik, P. Yu. Balabin, R. O. Shkolyarenko, “Calculation of frame constructions in a physically nonlinear formulation with account for the effect of central forces on a bend”, The collected papers of the IV International research to practice conference “Problems of innovative biosphere-compatible socio-economic development in the building, housing and communal, and road complexes”, 2015, 363–366

[23] A. M. Polatov, “Computer modeling of deformed state of physically non-linear transversal-isotropic bodies with hole”, Vychislitel'naya mekhanika sploshnykh sred — Computational Continuum Mechanics, 11:1 (2018), 25–35 | DOI

[24] N. I. Martynov, “Boundary value problems for neo-Hookean material in nonlinear elasticity”, Nauka i mir — Science and World, 2014, no. 9, 25–31

[25] I. A. Brigadnov, V. M. Bukhshtaber, I. A. Antonova, E. G. Sokolova, S. A. Sharov, Mathematical accuracy and problem-solving techniques for boundary problems of nonlinear elasticity, Research report, project No 96-01-00054

[26] K. F. Chernykh, “Variant of nonlinear elasticity theory. Its structure and potential”, Vestnik Sankt-Peterburgskogo universiteta. Prikladnaya matematika. Informatika. Protsessy upravleniya — Vestnik of Saint Petersburg University Applied Mathematics. Computer Science. Control Processes, 2004, no. 3–4, 55–62

[27] S. P. Ivanov, A. S. Ivanova, “The dynamic stability of physically nonlinear plate systems”, Stroitel'naya mekhanika inzhenernykh konstruktsiy i sooruzheniy — Structural Mechanics of Engineering Constructions and Buildings, 2014, no. 4, 9–18

[28] V. A. Shamina, S. Yu. Veselkov, L. V. Slepneva, Basic models in nonlinear mechanics of deformable solids, Research report, project No 95-01-00334

[29] S. V. Bakushev, “Equations of physically nonlinear theory of elasticity in terms of stresses”, Regional'naya arkhitektura i stroitel'stvo — Regional Architecture and Construction, 2011, no. 1(12), 117–123

[30] S. V. Bakushev, “Plane problem of physically nonlinear theory of elasticity–solution in terms of stresses”, Regional'naya arkhitektura i stroitel'stvo — Regional Architecture and Construction, 2014, no. 1(18), 82–88

[31] S. V. Bakushev, “Plane strain in physically nonlinear theory of elasticity — solution in terms of stresses”, Stroitel'naya mekhanika i raschet sooruzheniy — Structural Mechanics and Analysis of Constructions, 2014, no. 2, 2–9