Exact solutions of problems of the theory of repeated superposition of large strains for bodies created by successive junction of strained parts. (Russian. English summary) Zbl 1434.74056
Summary: Large strains of composite solids made of incompressible isotropic nonlinear-elastic materials are analyzed for the case in which the parts of these solids are preliminarily strained. The approaches to exact analytical solutions of these problems are given and developed in cooperation with V. An. Levin. He is a professor at the Lomonosov Moscow University. The solution of these problems is useful for stress analysis in members containing preliminarily stressed parts. The results can be used for the verification of industrial software for numerical modeling of additive technologies.
The problems are formulated using the theory of repeated superposition of large strains. Within the framework of this theory these problems can be formulated as follows. Parts of a member, which are initially separated from one another, are subjected to initial strain and passes to the intermediate state. Then these parts are joined with one another. The joint is performed by some surfaces that are common for each pair of connected parts. Then the body, which is composed of some parts, is strained as a whole due to additional loading. The body passes to the final state. It is assumed that the ideal contact conditions are satisfied over the joint surfaces. In other words, the displacement vector in the joined parts is continuous over these surfaces.
The exact solutions for isotropic incompressible materials are obtained using known universal solutions and can be considered as generalizations of these solutions for superimposed large strains.
The problems are formulated using the theory of repeated superposition of large strains. Within the framework of this theory these problems can be formulated as follows. Parts of a member, which are initially separated from one another, are subjected to initial strain and passes to the intermediate state. Then these parts are joined with one another. The joint is performed by some surfaces that are common for each pair of connected parts. Then the body, which is composed of some parts, is strained as a whole due to additional loading. The body passes to the final state. It is assumed that the ideal contact conditions are satisfied over the joint surfaces. In other words, the displacement vector in the joined parts is continuous over these surfaces.
The exact solutions for isotropic incompressible materials are obtained using known universal solutions and can be considered as generalizations of these solutions for superimposed large strains.
MSC:
| 74G05 | Explicit solutions of equilibrium problems in solid mechanics |
| 74B20 | Nonlinear elasticity |
| 74J30 | Nonlinear waves in solid mechanics |
Keywords:
nonlinear theory of elasticity; superposition of large strains; prestrained bodies; exact analytical solutions; additive technologiesReferences:
| [1] | Levin V. A., “Theory of repeated superposition of large deformations. Elastic and viscoelastic bodies”, International Journal of Solids and Structures, 35 (1998), 2585-2600 · Zbl 0918.73026 · doi:10.1016/S0020-7683(98)80032-2 |
| [2] | Levin V. A., Repeated superposition of large deformationsin elastic and viscoelastic bodies, Fizmatlit, M., 1999 |
| [3] | Levin V. A., Zingerman K. M., Plane problems of repeated superposition of large strains. Methods of solution, Fizmatlit, M., 2002 |
| [4] | V.A. Levin, K.M. Zingerman, A.V. Vershinin, E.I. Freiman, A.V. Yangirova, “Numerical analysis of the stress concentration near holes originating in previously loaded viscoelastic bodies at finite strains”, International Journal of Solids and Structures, 50:20-21 (2013), 3119-3135 · doi:10.1016/j.ijsolstr.2013.05.019 |
| [5] | Levin V. A., Vershinin A. V., “Non-stationary plane problem of the successive origination of stress concentrators in a loaded body. Finite deformations and their superposition”, Communications in Numerical Methods in Engineering, 24 (2008), 2229-2239 · Zbl 1154.74016 · doi:10.1002/cnm.1092 |
| [6] | Lurie A. I., Nonlinear theory of elasticity, Nauka, M., 1980 · Zbl 0506.73018 |
| [7] | Truesdell C., A First Cource in Rational Continuum Mechanics, The Johns Hopkins University, Baltimore, Maryland, 1972 |
| [8] | Rivlin R. S., “Large elastic deformations of isotropic materials. VI. Further results in the theory of torsion, shear, and flexure”, Philos. Trans. Roy. Soc. London, A252 (1949), 173-195 · Zbl 0035.41503 · doi:10.1098/rsta.1949.0009 |
| [9] | Bartenev G. M., Khazanovich T. N., “The law of high-elastic strains of mesh polymers”, Vysokomolekulyarnye soedineniya, 2:1 (1960), 20-28 |
| [10] | Gadolin A. V., “Theory of cannons fastened by bands”, Artilleriyskiy zhurnal, 1861, no. 12 |
| [11] | Sedov L. I., Mechanics of Continuous Medium, v. 2, Nauka, M., 1994 |
| [12] | Levin V. A., Zubov L. M., Zingerman K. M., “Torsion of a composite nonlinear elastic cylinder with a prestressed inclusion”, Doklady Physics, 58:12 (2013), 540-543 · doi:10.1134/S102833581310011X |
| [13] | Levin V. A., Zubov L. M., Zingerman K. M., “The torsion of a composite, nonlinear-elastic cylinder with an inclusion having initial large strains”, Int. J. Solids a. Struct., 51:6 (2014), 1403-1409 · doi:10.1016/j.ijsolstr.2013.12.034 |
| [14] | Levin V. A., Morozov E. M., Matvienko Yu. G., Selected Nonlinear Problems of Fracture Mechanics, ed. V.A. Levin, Fizmatlit, M., 2004 |
| [15] | Green A. E., Adkins J. E., Large Elastic Deformations and Non-linear Continuum Mechanics, Clarendon Press, Oxford, 1960 · Zbl 0090.17501 |
| [16] | Rivlin R. S., “Large elastic deformations of isotropic materials”, Philos. Trans. Roy. Soc. London, A240 (1948), 459-508 · Zbl 0029.32605 · doi:10.1098/rsta.1948.0002 |
| [17] | Moony M. A., “Theory of large elastic deformation”, Journal of Applied Physics, 1940, no. 11, 582-592 · JFM 66.1021.04 · doi:10.1063/1.1712836 |
| [18] | Poynting J. H., “On pressure perpendicular to the shear planes in finite pure shears, and on the lengthening of loaded wires when twisted”, Proc. R. Soc. Lond. A, 82 (1909), 546-559 · JFM 40.0875.02 · doi:10.1098/rspa.1909.0059 |
| [19] | Zubov L. M., “Direct and inverse Poynting effects in elastic cylinders”, Doklady Physics, 46 (2001), 675-677 · Zbl 1083.30030 · doi:10.1134/1.1409001 |
| [20] | Mihai L. A., Goriely A., “Positive or negative Poynting effect? The role of adscititious inequalities in hyperelastic materials”, Proc. R. Soc. A, 467 (2011), 3633-3646 · Zbl 1243.74011 · doi:10.1098/rspa.2011.0281 |
| [21] | Moon H., Truesdell C., “Interpretation of adscititious inequalities through the effects pure shear stress produces upon an isotropic elastic solid”, Arch. Ration. Mech. Anal., 55 (1974), 1-17 · Zbl 0298.73003 · doi:10.1007/BF00282431 |
| [22] | Zelenina A. A., Zubov L. M., “The non-linear theory of the pure bending of prismatic elastic solids”, J. Appl. Math. Mech., 64 (2000), 399-406 · Zbl 1103.74334 · doi:10.1016/S0021-8928(00)00062-9 |
| [23] | Levin V. A., Zubov L. M., Zingerman K. M., “Exact solution of the nonlinear bending problem for a composite beam containing a prestressed layer at large strains”, Doklady Physics, 60:1 (2015), 24-27 · doi:10.1134/S102833581501005X |
| [24] | Levin V. A., Zubov L. M., Zingerman K. M., “An exact solution for the problem of flexure of a composite beam with preliminarily strained layers under large strains”, Int. J. Solids a. Struct., 67-68 (2015), 244-249 · doi:10.1016/j.ijsolstr.2015.04.024 |
| [25] | Levin V. A., Zubov L. M., Zingerman K. M., “An exact solution for the problem of flexure of a composite beam with preliminarily strained layers under large strains. Part 2. Solution for different types of incompressible materials”, International Journal of Solids and Structures, 100-101 (2016), 558-565 · doi:10.1016/j.ijsolstr.2016.09.029 |
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.
