×

Rigorous solution of the problem of the state of a linearly elastic isotropic body under the action of polynomial bulk forces. (Russian. English summary) Zbl 1499.74009

Summary: When solving boundary value problems about the construction of the stress-strain state of an linearly elastic, isotropic body, an important step is finding the internal state generated by the forces, distributed over the area occupied by the body. In the classical version, there is a numerical method for estimating the state at any point of the body based on the singular-integral representation of Cesaro. In the variant of conservative bulk forces, it is possible to construct solutions in an analytical form. With arbitrary regular effects of mechanical and other physical nature the force is not potential and the approaches of Papkovich-Neiber and Arzhanykh-Slobodyansky are powerless. In addition, the solution of nonlinear problems of elastostatics by means of the perturbation method, as well as the use of the Schwarz algorithm in solving problems for the study of multi-cavity solids, lead to the need to solve a sequence of linear problems. At the same time, fictitious bulk forces are necessarily generated, which as a rule have a polynomial nature.
The method proposed by the authors earlier for estimating the stress-strain state of a solid caused by the action of polynomial bulk forces represented in Cartesian coordinates has been improved. The internal state is restored in strict accordance with the forces statically acting on a simply connected bounded linear-elastic body. An effective method for constructing a solution and an algorithm for its computer implementation are proposed and described. Test calculations are demonstrated. The analysis of the state of the ball under the action of a superposition of bulk forces of different nature at different ratios of parameters that emphasize the level of influence of these factors is performed. The results are presented graphically. Conclusions are drawn:
a) the procedure for writing out the stress-strain state on the volume forces represented by polynomials from Cartesian coordinates is justified;
b) the algorithm is implemented in the Mathematica computing system and tested on high-order polynomials;
c) the analysis of the quasi-static state of a linear-elastic isotropic ball exposed to the forces of gravity and inertia at various combinations of parameters corresponding to the variants of slow, fast, compensatory (inertial forces are proportional to the gravitational) rotations is carried out.
The prospects for the development of a new approach to the class of bounded and unbounded bodies containing an arbitrary number of cavities are noted.

MSC:

74A10 Stress
74B05 Classical linear elasticity
PDFBibTeX XMLCite
Full Text: DOI MNR

References:

[1] Truesdell C., A First Course in Rational Continuum Mechanics. Vol. 1: General Concepts, Pure and Applied Mathematics, 71, Academic Press, New York, San Francisco, London, 1977, xxiii+280 pp. · Zbl 0357.73011
[2] Rabotnov Yu. N., Mekhanika deformiruemogo tverdogo tela [Mechanics of a Deformable Rigid Body], Nauka, Moscow, 1988, 712 pp. (In Russian) · Zbl 0648.73002
[3] Lurie A. I., Theory of Elasticity, Foundations of Engineering Mechanics, Springer-Verlag, Berlin, 2005, iv+1050 pp. · Zbl 0269.73013 · doi:10.1007/978-3-540-26455-2
[4] Muskhelishvili N. I., Nekotorye osnovnye zadachi matematicheskoi teorii uprugosti [Some Basic Problems of the Mathematical Theory of Elasticity], Nauka, Moscow, 1966, 707 pp. (In Russian) · Zbl 0151.36201
[5] Green A. E., Zerna W., Theoretical Elasticity, Dover Publications, New York, 1992, xvi+457 pp. · Zbl 0863.73005
[6] Arfken G. B., Weber H. J., Mathematical Methods for Physicists, Elsiver/Academic Press, Amsterdam, 2005, xii+1182 pp. · Zbl 1066.00001
[7] Neuber H., “Ein neuer Ansatz zur Lösung räumlicher Probleme der Elastizitätstheorie. Der Hohlkegel unter Einzellast als Beispiel”, ZAMM, 14:4 (1934), 203-212 · JFM 60.1351.02 · doi:10.1002/zamm.19340140404
[8] Matveenko V. P., Schevelev N. A., “Analytical study of the stress-strain state of rotation bodies under the action of mass forces”, Stress-Strain State of Structures Made of Elastic and Viscoelastic Materials, Sverdlovsk, 1977, 54-60 (In Russian)
[9] Vestyak V. A., Tarlakovsky D. V., “Unsteady axisymmetric deformation of an elastic space with a spherical cavity under the action of bulk forces”, Moscow Univ. Mech. Bull., 71:4 (2016), 87-92 · Zbl 1469.74013 · doi:10.3103/S0027133016040038
[10] Sharafutdinov G. Z., “Functions of a complex variable in problems in the theory of elasticity with mass forces”, J. Appl. Math. Mech., 73:1 (2009), 69-87 · Zbl 1182.74007 · doi:10.1016/j.jappmathmech.2009.03.008
[11] Zaytsev A. V, Fukalov A. A., “Exact analytical solutions of equilibrium problems for elastic anisotropic bodies with central and axial symmetry, which are in the field of gravitational forces, and their applications to the problems of geomechanics”, Matemat. Model. Estestv. Nauk., 1 (2015), 141-144 (In Russian)
[12] Pikul V. V., “To anomalous deformation of solids”, Physical Mesomechanics, 2013, no. 2, 93-100 (In Russian)
[13] Kozlov V. V., “The Lorentz force and its generalizations”, Nelin. Dinam., 7:3 (2011), 627-634 (In Russian)
[14] Satalkina L. V., The method of boundary states in problems of the theory of elasticity of inhomogeneous bodies and thermoelasticity, Thesis of Dissertation (Cand. Phys. & Math. Sci.), Lipetsk, 2010, 108 pp. (In Russian)
[15] Penkov V. B., Novikov E. A., Novikova O. S., Levina L. V., “Combining the method of boundary states and the Lindstedt-Poincaré method in geometrically nonlinear elastostatics”, J. Phys.: Conf. Ser., 1479 (2020), 012135 · doi:10.1088/1742-6596/1479/1/012135
[16] Ivanychev D. A., Novikov E. A., “The solution of physically nonlinear problems for isotropic bodies by the method of boundary states”, Problems of Strength, Plasticity, and Stability in Solid Mechanics, Tver State Techn. Univ., Tver, 2021, 43-47 (In Russian)
[17] Penkov V. B., Ivanychev D. A., Levina L. V., Novikov E. A., “Using the method of boundary states with perturbations to solve physically nonlinear problems of the theory of elasticity”, J. Phys.: Conf. Ser., 1479 (2020), 012134 · doi:10.1088/1742-6596/1479/1/012134
[18] Kuzmenko V. I., Kuzmenko N. V., Levina L. V., Penkov V. B., “A method for solving problems of the isotropic elasticity theory with bulk forces in polynomial representation”, Mech. Solids, 54:5 (2019), 741-749 · Zbl 1459.74018 · doi:10.3103/S0025654419050108
[19] Penkov V. B., Levina L. V., Novikova O. S., “Analytical solution of elastostatic problems of a simply connected body loaded with nonconservative volume forces: theoretical and algorithmic support”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 24:1 (2020), 56-73 (In Russian) · Zbl 1463.74009 · doi:10.14498/vsgtu1711
[20] Ivanychev D. A., “A boundary state method for solving a mixed problem of the theory of anisotropic elasticity with mass forces”, Vestn. Tomsk. Gos. Univ. Mat. Mekh., 2021, no. 71, 63-77 (In Russian) · doi:10.17223/19988621/71/6
[21] Penkov V.B., Rybakova M. R., Satalkina L. V., “Application of the Schwarz algorithm to spatial problems of elasticity theory”, Vesti Vuzov Chernozemya, 2015, no. 2 (40), 23-31 (In Russian)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.