Engineering Transactions, 12, 2, pp. 187-250, 1964

Pewne Zagadnienia Teorii Sprężystości Ciał Anizotropowych

M. Suchar
Politechnika Łódźka

The first section is devoted to the structure of the solution of partial differential equations of the elliptic type with two independent variables, of the fourth or any even order and having constant coefficients. In the theory of elasticity fourth order equations describe the deflection of a thin anisotropic plate and also stress and displacement functions for anisotropic bodies in plane state of stress or strain, assuming that there is a symmetry plane of elasticity at every point of the body. The coefficients of the equation are different in each of the cases considered. For the description of more general states of stress and strainbody of any rectilinear anisotropy leads to sixth of a order equations.
The fourth order equations are analysed by means of the example of the deflection surface of an anisotropic plate. A general form of solution is obtained in cases of simply or multiply connected finite or infinite regions. Also the degree of determinacy of the holomorphic functions occurring in the solution is determined. The analysis is performed for complex parameters of the second kind. Final results are also given for parameters of the first kind. The results are generalized to an elliptic equation of any even order. The proofs contained in the paper constitute a generalization of N.I. MUSKHELISHVILI'S considerations concerning the biharmonic equation. In particular, basic solutions are arrived at for obtaining Green's functions for the problems under consideration.
Making use of the solution for a doubly connected plate bounded by a circle and a point at infinity and loaded in an appropriate manner on the contour, and passing to the limit for the radius of the circle tending to zero, a singular solution is obtained for an infinite anisotropic plate loaded by a concentrated force, previously obtained by J. MOSSAKOWSKI and then for a plate loaded by a concentrated moment and bimoment.
In a similar manner we obtain a singular solution for an infinite anisotropic plate loaded by a concentrated force and moment, obtained by S. G. LETCHNITZKY.
The second section is devoted to the plane distortion problem for anisotropic bodies. The problem is that of finding the state of stress and strain in an anisotropic plate due to initial strain in a portion of the region. This is a generalization to the case of anisotropy of the plane distortion problem solved by W. NOWACKI for isotropic bodies. The method of Green's function is used in the form adapted by W. NOWACKI for the analysis of these problems. Green's functions are, in this case, quantities (stresses displacements, stress or displacement functions) produced by a nucleus of elastic strain. The knowledge of Green's function enables us to obtain, by integration the solution in the case of initial strains distributed in any way over the region. The point of departure is the obtainment of stresses and displacements in an infinite plate produced by a nucleus of, elastic strain. The stresses are determined by means of the Airy function and the displacements - by means of Galerkin functions, the solution being expressed in terms of Fourier integrals which are represented finally in a closed form. They can also be obtained from appropriate singular solutions for an anisotropic plate loaded by a concentrated force, moment or bimoment.
By adding to the singular part of the solution a regular selected in such a manner that their sum satisfies the given boundary conditions, stresses and displacements are obtained in a semi-infinite plate with clamped and free edge conditions and in a plate strip. It is also shown that the computation of the solution for a finite plate reduces to the obtainment of the solution of one of the fundamental plane problems of elasticity of an anisotropic body. A particular case of the above distortion problem are steady-state thermal distortions, to which the last section of the paper is devoted. Making use of the above results, stresses and strains are obtained in an infinite plate and semi-infinite plate produced by a steady-state nucleus of thermo-elastic strain. They can be used for the obtainment of stresses in a plate produced by a discontinuous temperature field. This is illustrated by means of two examples, in which the temperature in an infinite plate and in a semi-infinite plate is constant in the region of the rectangle and is zero outside this region.

Full Text: PDF
Copyright © Polish Academy of Sciences & Institute of Fundamental Technological Research (IPPT PAN).


I. BABUSKA, K. REKTORYS, F. VYÉICHLO, Matematicka theorie rovinné pruZnosti, CSAV, Praga 1955.

H. BATEMAN, Tables of Integral Transforms, New York-Toronto-London 1954.

J. BRILLA, Anizotropické steny, Bratislava SAV, 1958.

J. N. GOODIER, On the integration of the thermo-elastic equation, Phil. Mag., 23, 7 (1937), 1017.

E. GOURSAT, Sur l'équation ΔΔu = 0, Bull. Soc. Math. de France, 26 (1898), 236.

A. E. GREEN, W. ZERNA, Theoretical Elasticity, Oxford 1954.

S. KALISKI, W. NOWACKI, Some problems of structural analysis of plates with mixed boundary conditions, Arch. Mech. Stos., 4, 8 (1956), 413.

M. KRZYZANSKI, Równania różniczkowe cząstkowe rzędu drugiego, Część I, PWN, Warszawa- 1957.

[in Russian]

[in Russian]

[in Russian]

[in Russian]

J. MOSSAKOWSKI. Rozwiązania osobliwe dla płyt anizotropowych, Arch. Mech. Stos., 1, 7 (1955), 97.

J. MOSSAKOWSKI, The state of stress and displacement in a thin anisotropic plate, due to concentrated source of heat, Arch. Mech. Stos., 5, 9 (1957), 565.

J. MOSSAKOWSKI, The Michell problem for anisotropic semi-infinite plate, Arch. Mech. Stos., 4. 8 (1956), 539.

[in Russian]

W. NOWACKI, Zagadnienia termosprężystości, PWN, Warszawa 1960.

W. NOWACKI, A plane distortion problem, Arch. Mech. Stos., 4, 9 (1957), 417.

W. NOWACKI, The stresses in a thin plate due to a nucleus of thermoelastic strain, Arch. Mech. Stos., 1, 9 (1957).

W. NOWACKI, Thermal stresses in orthotropic plates, Bull. Acad. Polon. Sci, Sér. Sci. Techn., 1, 7 (1959), 1.

W. NOWACKI, Naprężenia cieplne w ciałach anizotropowych, Arch. Mech. Stos., 3, 6 (1954) 481.

W. NOWACKI, Ustalone naprężenia w walcu ortotropowym oraz w tarczy ortotropowej, Rozpr. Inz., 3, 8 (1960), 569.

[in Russian]

B. SEN, Note on the stresses produced by nuclei of thermoelastic strain in a semi-infinite solid Quart. Appl. Math., 8, 1951, 365.

M. SOKOŁOWSKI, Some plane problems with boundary conditions in terms of displacements Arch. Mech. Stos., 4, 9 (1957), 439.

C. SOMIGLIANA, Sui sistemi simmetrici di equazioni g derivate parziali, Ann. di Math. Pura e Appl., ser. II, XXI, 1894.

M. SUCHAR, Computation by means of polynomials of influence surfaces. for anisotropic plates with finite dimensions, Arch. Mech, Stos., 5, 10 (1958), 615.

M. SUCHAR, General form of the surface of deflection of a thin anisotropic plate in a multi-connected region, Bull. Acad. Polon. Sci., Sér. Sci. Techn., 2, 8 (1960), 69.

S. TIMOSHENKO, S. WOINOWSKY-KRIEGER, Theory of plates and shells, New York-Toronto- London 1959.

[in Russian]

H. ZORSKI, Plates with discontinuous supports, Arch. Mech. Stos., 3, 10 (1958).