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A Journal of Russian Academy of Sciences | | Founded
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| E. A. Leonova, "Scalar formulation of static problems in elasticity," Mech. Solids. 38 (1), 55-66 (2003) |
| Year |
2003 |
Volume |
38 |
Number |
1 |
Pages |
55-66 |
| Title |
Scalar formulation of static problems in elasticity |
| Author(s) |
E. A. Leonova (Moscow) |
| Abstract |
Static equations of classical elasticity and
uncoupled thermo-elasticity are considered. An invariant representation of the
general solution of these equations is proposed in terms of a minimal number
of scalar harmonic functions.
It is shown that for three-dimensional elasticity equations there are
three such functions, and for the plane problem, there are two functions
and one arbitrary constant. For the basic kinematic characteristics
two vector identities are obtained. On the basis of these,
two types of boundary value problems in domains of a special structure with
given normal (tangential) stresses and given tangential (normal)
displacements on the boundary are reduced to classical problems of
the potential theory. Examples of solutions are given.
Applications of harmonic functions in classical elasticity for
solving particular problems date back to the works of Lam\'e, Betti,
Boussinesq, and Cerruti [1-5], whereas the problem of representing the
general solution in terms of such functions is still the subject matter of
numerous studies [18-26].
The displacement vector is defined by Lam\'e's homogeneous equations [1-4]
and, hence, is a biharmonic vector or can be expressed in terms of
biharmonic vector [4]. Therefore, the displacement vector can be
represented in terms of two harmonic vectors in accordance with the familiar
relations.
It can be shown that all straightforward representations of the general
solution in terms of harmonic functions [7-26] either
contain a harmonic vector [7-20, 22, 24-46] or express the displacement
vector in an orthogonal Cartesian frame in terms of harmonic
functions [21, 23]. When choosing the form of the general
solution of particular problems, one should keep in mind that a
harmonic vector can be split into harmonic components only in a Cartesian
coordinate system or some other specific coordinate systems.
In [7, 16, 26], the displacement vector is expressed in terms of a
harmonic vector and a harmonic scalar-valued function, which may be
either uncoupled [8, 9, 26] or coupled [7, 12, 13] by a single relation.
For three-dimensional equations in terms of components, this fact leads to
four or three independent harmonic functions, respectively. In [21, 23],
four harmonic functions are also linked by a single relation.
The issue of reducing the number of harmonic functions in the general
solutions considered in [8, 9] from four to three was widely
discussed in [8-20, 22, 23, 26]. This reduction has been implemented in
two ways. According to the first approach the displacement vector is
expressed through a single harmonic vector [12, 14-16, 18, 24-26];
while in the second approach, it is expressed through a scalar harmonic
function and two components of a harmonic vector by equating to zero one of
its components [15, 16, 26].
The representation of the general solution in terms of harmonic functions
proposed below does not contain the position vector of a point, in
contrast to [4, 7-9, 14-16, 21, 23]; and it is unrelated to a fixed
coordinate system, in contrast to [21, 23]. In the present work, in
contrast to [1-26], the general solution is expressed in terms of scalar
harmonic functions rather than in terms of a harmonic vector, and the
number of these functions is minimal. |
| References |
| 1. | A. I. Lur'e, Theory of Elasticity [in Russian], Nauka, Moscow, 1970. |
| 2. | W. Nowacki, Theory of Elasticity [Russian translation], Mir, Moscow, 1975. |
| 3. | A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity
[Russian translation], ONTI, Moscow, Leningrad, 1935. |
| 4. | E. Trefftz, Mathematical Theory of Elasticity [Russian translation],
Gostekhizdat, Moscow, 1934. |
| 5. | P. F. Papkovich, "A review of some general solutions of basic
differential equations of rest for an isotropic elastic body," PMM
[Applied Mathematics and Mechanics], Vol. 1, No. 1, pp. 117-132, 1937. |
| 6. | B. G. Galerkin, "To the issue of the stress-strain analysis for an
elastic isotropic body," in Collected Works. Volume 1 [in Russian],
pp. 318-321, Izd-vo AN SSSR, Moscow, 1952. |
| 7. | O. Tedone, "Saggio di una teoria generale delle equazioni dell'
equilibrio elastico peruncorpo isotropo," Ann. di Mat. Pura et Appl.,
Vol. 10, Ser. 3, pp. 13-64, 1904. |
| 8. | P. F. Papkovich, "Expression of the general integral of basic
elasticity equations in terms of harmonic functions," Izv. AN SSSR. Ser.
Mat. i Estestv. Nauk., No. 10, pp. 1425-1435, 1932. |
| 9. | H. Neuber, "Ein neuer Ansatz zur Losung raumlicher Probleme der
Elastizitatstheorie," ZAMM, Bd. 14, H. 4, S. 203-212, 1934. |
| 10. | H. Neuber, Stress Concentration [Russian translation], Gostekhizdat,
Moscow, Leningrad, 1947. |
| 11. | G. D. Grodskii, "Integration of general equations of equilibrium
of an isotropic elastic body by means of Newtonian potentials and harmonic
functions," Izv. AN SSSR. Ser. Mat. i Estestv. Nauk., No. 4,
pp. 587-614, 1935. |
| 12. | R. D. Mindlin, "Note on the Galerkin and Papkovich stress
functions," Bull. Amer. Math. Soc., Vol. 42, No. 6, pp. 373-376, 1936. |
| 13. | L. N. Ter-Mkrtich'yan, "On the general solution of the problem of
elasticity," Trudy Leningrad. Politekh. In-ta, No. 4,
pp. 3-38, 1947. |
| 14. | Yu. A. Krutkov, Stress Function Tensor and General Solutions of
Static Problems in Elasticity [in Russian], Izd-vo AN SSSR, Moscow,
Leningrad, 1949. |
| 15. | I. S. Arzhanykh, "Research in continuum mechanics," in
Transactions of the Institute of Mathematics and Mechanics [in Russian],
Izd-vo AN UzbSSR, Tashkent, No. 9, pp. 60-101, 1952. |
| 16. | M. G. Slobodyanskii,
"General forms of solutions of elasticity equations in simply-connected
and multiply-connected domains in terms of harmonic functions,"
PMM [Applied Mathematics and Mechanics], Vol. 18, No. 1, pp. 55-74, 1954. |
| 17. | M. G. Slobodyanskii, "On general and complete forms of solutions of equations
of elasticity,"
PMM [Applied Mathematics and Mechanics], Vol. 23, No. 3, pp. 468-482, 1959. |
| 18. | R. A. Eubanks and E. Sternberg, "On the completeness of the Boussinesq-Papkovich
stress functions," J. Ration. Mech. Anal., Vol. 5, No. 5, pp. 735-746, 1956. |
| 19. | P. M. Naghdi and S. S. Hsu,
"On a representation of displacements in linear elasticity in terms of
three stress functions," J. Math. Mech., Vol. 10, No. 1, pp. 233-245,
1961. |
| 20. | M. E. Gurtin, "On Helmholtz's theorem and the completeness
of the Papkovich-Neuber stress functions for infinite domains,"
Arch. Ration. Mech. Anal., Vol. 9, No. 3, pp. 225-233, 1962. |
| 21. | M. Stippes, "Completeness of Papkovich potentials," Quart. Appl.
Math., Vol. 26, No. 4, pp. 477-483, 1969. |
| 22. | A. P. Shmakov, "A representation of the general solution
in the theory of elasticity," in
Elasticity and Inelasticity [in Russian], No. 3, pp. 5-12, Izd-vo MGU,
Moscow, 1975. |
| 23. | T. Tran Cong and G. P. Steven, "On the representation of elastic
displacement fields in terms of three harmonic functions," J. Elast.,
Vol. 9, No. 3, pp. 325-333, 1979. |
| 24. | N. I. Ostrosablin, "General representation of solutions of equations of linear
elasticity for an isotropic body," in Dynamics of Continuum:
Non-classical Problems of Solid Mechanics [in
Russian], No. 61, pp. 77-91, Izd-vo In-ta Gidrodinamiki SO AN SSSR,
Novosibirsk, 1983. |
| 25. | M. Z. Wang,
"The Naghdi-Hsu solution and the Naghdi-Hsu transformation,"
J. Elast., Vol. 15, No. 1, pp. 103-108, 1985. |
| 26. | A. I. Borodachev,
"On a generalization of the Naghdi-Hsu transformation and its
application to elasticity problems," PMM [Applied Mathematics and Mechanics], Vol. 51, No. 4, pp. 611-615,
1987. |
|
| Received |
02 December 1999 |
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