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IssuesArchive of Issues2006-5pp.57-65

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V. M. Mal'kov and Yu. V. Mal'kova, "Study of the nonlinear Flamant problem," Mech. Solids. 41 (5), 57-65 (2006)
Year 2006 Volume 41 Number 5 Pages 57-65
Title Study of the nonlinear Flamant problem
Author(s) V. M. Mal'kov (St. Petersburg)
Yu. V. Mal'kova (St. Petersburg)
Abstract We consider the generalized plane problem of nonlinear elasticity for a half-plane loaded on the boundary by a lumped external force (the nonlinear Flamant problem). For an incompressible material model of neo-Hookean type, we obtain an analytic solution of the boundary value problem without restrictions on the strain magnitude. Comparing the solutions of the nonlinear and linear Flamant problems, we see that they substantially differ both in the character of the stress state and displacements near the point where the force is applied and in other properties. In the nonlinear problem, there are both radial and circumferential stresses, which depend on the material model. Moreover, the radial stresses are compressive, and the circumferential stresses are tensile. The displacements in the nonlinear problem do not have the well-known contradictions of the linear solution; in particular, they are bounded and continuous at the pole. In the nonlinear problem, there are stresses of several types: true Cauchy stresses, conventional stresses, and Piola-Kichhoff stresses. These stresses have different asymptotics in a neighborhood of the pole. In the sense of closeness to the linear stresses in the type of singularity and other parameters, the true Cauchy stresses have priority over stresses of other types.
References
1.  S. P. Timoshenko and J. N. Goodier, Elasticity Theory [Russian translation], Nauka, Moscow, 1975.
2.  D. J. Unger, "Similarity solution of the Flamant problem by means of a one-parameter group transformation," J. Elasticity, Vol. 66, No. 1, pp. 93-97, 2002.
3.  K. F. Chernykh, Nonlinear Singular Elasticity. Part 2 [in Russian], Izd-vo SPbGU, St. Petersburg, 1999.
4.  Y. C. Gao and Z. Zhou, "Large strain contact of a rubber wedge with a rigid notch," Intern. J. Solids and Structures, Vol. 38, No. 48/49, pp. 8921-8928, 2001.
5.  V. M. Malkov, Foundations of Mathematical Nonlinear Elasticity [in Russian], Izd-vo SPbGU, St. Petersburg, 2002.
6.  A. I. Lurie, Elasticity Theory [in Russian], Nauka, Moscow, 1970.
Received 02 July 2004
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