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IssuesArchive of Issues2001-4pp.47-53

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B. A. Zhukov, "A modification of the Signorini method for plane strains in an incompressible material," Mech. Solids. 36 (4), 47-53 (2001)
Year 2001 Volume 36 Number 4 Pages 47-53
Title A modification of the Signorini method for plane strains in an incompressible material
Author(s) B. A. Zhukov (Volgograd)
Abstract One of the methods for solving static problems of nonlinear elasticity is the method of expansion of the solution into absolutely convergent series with respect to a small parameter [1], suggested by Signorini for the case where the specific strain energy is represented by a polynomial of the strain tensor invariants. In the case of plane strain, this method allows the application of the powerful means of the complex analysis [2]. If the material is incompressible, the terms in the expansion of the incompressibility condition (beginning from the second one) are nonlinear and cannot be solved by Kolosov-Muskhelishvili potentials. Hence, problems for displacements and mixed boundary-value problems contain an equation to be satisfied in a plane domain. A method permitting one to take into account the incompressibility conditions has been suggested in [3]. This method is based on the original statement of the plane elasticity problem that has been formulated by L. A. Tolokonnikov [4] on the basis of additional hypotheses as regards the relation between stresses and strains. In the present paper, a modification of the Signorini method permitting one to eliminate the incompressibility condition is suggested. Second-order effects are singled out. An example of the solution of a problem for displacements is given.
References
1.  A. E. Green and J. E. Adkins, Large Elastic Deformations and Nonlinear Mechanics of a Continuous Medium [Russian translation], Mir, Moscow, 1965.
2.  N. I. Muskhelishvili, Some Basic Problems of Mathematical Theory of Elasticity [in Russian], Nauka, Moscow, 1966.
3.  V. G. Gromov and L. A. Tolokonnikov, "On the calculation of approximations in the problem of finite plane deformations of an incompressible material," Izv. AN SSSR. OTN. Mekhanika i Mashinostroenie, No. 2, pp. 81-86, 1963.
4.  L. A. Tolokonnikov, "On the stress-strain relation in nonlinear elasticity," PMM [Applied Mathematics and Mechanics], Vol. 20, No. 3, pp. 439-444, 1956.
5.  V. I. Arnold, Mathematical Methods in Classical Mechanics [in Russian], Nauka, Moscow, 1974.
6.  A. I. Lur'e, Nonlinear Theory of Elasticity [in Russian], Nauka, Moscow, 1980.
7.  B. A. Zhukov, "Comparison of Griffith and Sneddon tensile crack models with respect to second-order effects in an incompressible hyperelastic material," in Modern Problems of Continuum Mechanics. Proceedings of the Third International Conference. Volume 1 [in Russian], pp. 154-158, MP Kniga, Rostov-on-Don, 1997.
Received 15 June 1999
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