Correspondence principle in plane and axisymmetric mixed boundary-value problems of elasticity
Authors:
Y. A. Antipov and S. M. Mkhitaryan
Journal:
Quart. Appl. Math. 78 (2020), 333-362
MSC (2010):
Primary 44A15, 45E10, 74G05, 74R10
DOI:
https://doi.org/10.1090/qam/1544
Published electronically:
June 20, 2019
MathSciNet review:
4100286
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Abstract: This paper advances the relations by A. Ja. Aleksandrov between axisymmetric and plane strain states to the case of mixed boundary-value problems. We revise two models of a strip-shaped and a circular stamp indented into a half-plane and a half-space, respectively, when the normal and tangential traction components are unknown a priori. Also, two models of a strip-shaped and a penny-shaped interfacial crack are considered. By using the theory of Abelian integral operators and the Riemann-Hilbert problem on a segment we derive the solutions to the model problems and establish the relations between the governing systems of integral equations associated with the four models and their solutions. These relations can be interpreted as mappings between (i) plane and axisymmetric contact problems, (ii) plane and axisymmetric fracture models, (iii) plane contact and fracture problems, and (iv) axisymmetric contact and fracture problems. The mappings enable us to write down the governing systems of integral equations and the solutions to any three models by making use of the governing system and the solution to the fourth problem. The transformations are specified in the scalar cases when there is no friction in the contact zone and when a crack is in a homogeneous elastic medium. By considering the contact frictionless problem of an annulus stamp it is shown that, although an exact solution to the plane strain frictionless contact problem of two stamps is available, a transformation of the plane to the axisymmetric solution in this case is not possible to obtain, and derivation of a closed-form solution to the annulus stamp model is still an open mathematical problem.
References
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References
- V. M. Abramov, Problem on contact of an elastic half-plane with an absolutely rigid foundation when there are friction forces, Dokl. AN SSSR. 17 no.4 (1937), 173–178.
- N. I. Ahiezer and V. A. Ščerbina, On inversion of some singular integrals, Zap. Mat. Otd. Fiz.-Mat. Fak. Kharkov Mat. Obš. 25(4) (1957),191–198.
- A. Ja. Aleksandrov, Solution of axisymmetric problems of the theory of elasticity with the aid of relations between axisymmetric and plane states of stress, J. Appl. Math. Mech. 25 (1961), 1361–1375. MR 0135281, DOI https://doi.org/10.1016/0021-8928%2861%2990013-2
- A. Ja. Aleksandrov and Ju. I. Solov′ev, Three-dimensional problems of the theory of elasticity, Nauka, Moscow, 1978. MR 538075
- Yu. A. Antipov, Exact solution of the problem of pressing an annular stamp into a half-space, Dokl. Akad. Nauk Ukrain. SSR Ser. A 7 (1987), 29–33, 85 (Russian, with English summary). MR 907775
- Yu. A. Antipov, Analytic solution of mixed problems of mathematical physics with a change of boundary conditions over a ring, Mechanics of Solids (MTT). 24(3) (1989), 49–56.
- Y. A. Antipov, Vector Riemann-Hilbert problem with almost periodic and meromorphic coefficients and applications, Proc. A. 471 (2015), no. 2180, 20150262, 18. MR 3394794, DOI https://doi.org/10.1098/rspa.2015.0262
- H. Bateman, Tables of integral transforms, Vol. 1. Bateman Manuscript Project, McGraw-Hill, New York, 1954.
- H. Bateman, Tables of integral transforms, Vol. 2. Bateman Manuscript Project, McGraw-Hill, New York, 1954.
- A. H. England, A crack between dissimilar media, J. Appl. Mech., 32(2) (1965), 400–402.
- F. D. Gakhov, Boundary Value Problems, Third Ed. Nauka, Fiz-Mat. Lit. Moscow, 1977. MR 0486547
- L. A. Galin, Contact problems in the theory of elasticity and viscoelasticity, Nauka, Moscow, 1980. MR 598269
- T. Geike and V. L. Popov, Mapping of three-dimensional contact problems into one dimension, Phys. Review E. 76 (2007), 036710 1–5.
- W. Goldsmith, Review: Non-ideal projectile impact on targets, Int. J. Impact. Engng. 22 (1999), 95–395.
- G. H. Jones and J. A. Zukas, Mechanics of penetration: analysis and experiment, Int. J. Engn. Sci. 16 (1978), 879–903.
- L. M. Keer, Mixed boundary-value problems for an elastic half-space, Proc. Cambridge Philos. Soc. 63(4) (1967), 1379–1386.
- S. M. Mhitarjan, The formulae of N. I. Ahiezer and V. A. Ščerbina for the inversion of certain singular integrals, Mat. Issled. 3 (1968), no. vyp. 1 (7), 61–70 (Russian). MR 0257675
- V. I. Mossakovskiĭ, A fundamental mixed problem of the theory of elasticity for a half-space with a circular curve of separation of the boundary conditions, Prikl. Mat. Meh. 18 (1954), 187–196 (Russian). MR 0066184
- V. I. Mossakovskii and M. T. Rybka, Generalization of the Griffith-Sneddon criterion for the case of a nonhomogeneous body, J. Appl. Math. Mech. 28 (1964), 1277–1286.
- N. I. Muskhelishvili, Some basic problems of the mathematical theory of elasticity. Fundamental equations, plane theory of elasticity, torsion and bending, Translated from the Russian by J. R. M. Radok, P. Noordhoff, Ltd., Groningen, 1963. MR 0176648
- P. F. Papkovich, On the question on analogy between the plane problem of the theory of elasticity and the problem on deformation symmetric with respect to the axis, Prikl. Mat. Mekh. 3(3) (1939), 45–66.
- G. Ia. Popov, Axisymmetric contact problem for an elastic inhomogeneous half-space in the presence of cohesion, J. Appl. Math. Mech., 37 (1973), 1052–1059.
- G. Ja. Popov, The connection between three-dimensional and two-dimensional stressed states in the static and the dynamical elasticity of inhomogeneous bodies, Dokl. Akad. Nauk SSSR 228 (1976), no. 3, 566–569 (Russian). MR 0436724
- Valentin L. Popov and Markus Heß, Method of dimensionality reduction in contact mechanics and friction, Springer, Heidelberg, 2015. English translation by Joshua A. T. Gray. MR 3289592
- J. R. Rice and G. C. Sih, Plane problems of cracks in dissimilar media, J. Appl. Mech., 32(2) (1965), 418–423.
- Ian N. Sneddon, The relation between load and penetration in the axisymmetric Boussinesq problem for a punch of arbitrary profile, Internat. J. Engrg. Sci. 3 (1965), 47–57 (English, with French, German, Italian and Russian summaries). MR 0183169, DOI https://doi.org/10.1016/0020-7225%2865%2990019-4
- I. Ya. Shtaerman, Contact Problem of the Theory of Elasticity, Gos. Izd. Tekh.-Teoret. Lit., Moscow-Leningrad, 1949. MR 0035617
- Ja. S. Ufljand, Integral′nye preobrazovaniya v zadachakh teorii uprugosti, Izdat. Akad. Nauk SSSR, Moscow, 1963 (Russian). MR 0169432
- C. Weber, Achsensymmetrische Deformation von Umdrehungskorpern, Z. Angew. Math. Mech (ZAAM). 25(6) (1925), 466–468.
- J. R. Willis, The penny-shaped crack on an interface, Quart. J. Mech. Appl. Math. 25(3). (1972), 367–385.
- Lothar von Wolfersdorf, Abelsche Integralgleichungen und Randwertprobleme für die verallgemeinerte Tricomi-Gleichung, Math. Nachr. 29 (1965), 161–178 (German). MR 0181877, DOI https://doi.org/10.1002/mana.19650290304
- L. von Wolfersdorf, Zur Lösung der verallgemeinerten Abelschen Integralgleichung mit konstanten Koeffizienten, Z. Angew. Math. Mech. 49 (1969), 759–761 (German). MR 0256104, DOI https://doi.org/10.1002/zamm.19690491212
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Additional Information
Y. A. Antipov
Affiliation:
Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803
MR Author ID:
245270
Email:
yantipov@lsu.edu
S. M. Mkhitaryan
Affiliation:
Department of Mechanics of Elastic and Viscoelastic Bodies, National Academy of Sciences, Yerevan 0019, Armenia
MR Author ID:
194038
Email:
smkhitaryan39@rambler.ru
Keywords:
Abelian operators,
Riemann-Hilbert transform,
plane and axisymmetric problems,
contact and fracture mechanics
Received by editor(s):
March 12, 2019
Received by editor(s) in revised form:
May 21, 2019
Published electronically:
June 20, 2019
Additional Notes:
The research of the first author was sponsored by the Army Research Office and was accomplished under Grant Number W911NF-17-1-0157. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Office or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein.
Article copyright:
© Copyright 2019
Brown University
