 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
Archive of Issues
| Total articles in the database: | | 12854 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8044
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| In English (Mech. Solids): | | 4810 |
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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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