I. Vekua Institute of Applied Mathematics

I. Javakhishvili Tbilisi State University

Georgian Academy of Natural Sciences


for 2000



  1. Cusped Plates

1.1. The N-th order Approximation in the Case of a Lipschitz Domain

1.2. The First Order Approximation in the Case of a Non-Lipschitz Domain

  1. Cusped Beams

2.1. The Dynamical Problems in the (0.0) and (1.0) Approximations

2.2. Static and Dynamical Problems for Beams when Barycentric and Beam Axes Coincide



In the fifties of XX century, I.Vekua [1] suggested a new mathematical model of elastic prismatic shells (i.e., of plates of variable thickness) which was based on the expansion of fields of displacement vector, and strain and stress tensors of the three-dimensional theory of linear elasticity into orthogonal Fourier-Legendre series with respect to the variable of plate thickness. Considering only the first N+1 terms of the expansions, he obtained the N-th approximation. Each of the approximations for N=0,1,..., can be considered as an independent mathematical model of plates, e.g., the approximation N=1 actually coincides with the classical plate bending theory. In the sixties, I.Vekua [2] offered the analogous mathematical model for thin shallow shells. All his results concerning plates and shells are collected in the monograph [3]. At the same time he recommended to investigate cusped plates, i.e., plates whose thickness vanishes either on a part of the plate projection boundary or on the whole one (about investigations in this direction see survey [4] and also I.Vekua's comments in [3], p.86). Works of I.Babuska, D.Gordeziani, V.Guliaev, I.Khoma, A.Khvoles, T.Meunargia, C.Schwab, T.Vashakmadze, V.Zgenti, and others (see [5-13] and references therein) are devoted to the analysis of I.Vekua's models (rigorous estimates of an approximation error, i.e., of the modelling error, numerical solutions, etc.) and their generalizations (for non-shallow shells, anisotropic case, etc.).

Here the I.Vekua’s model of prismatic shells is presented and the up-to-date survey of development of I.Vekua’s theory of plates and shells is given.

During the first year the main aim of the project was to work out the general theory of cusped plates on the basis of I. Vekua's [3] version of the theory of plates with variable thickness, and to develop an analogous theory for elastic bars with variable cross-section [14].

Using the energy projection approach, suggested by Ch. Schwab [11] for plates of constant thickness, the model of elastic plates of variable thickness has been constructed.

It has been proved that the Schwab’s model turns into I. Vekua's model if we transform appropriately the unknown functions. The relation between the unknown functions of these two models has been completely accomplished and the corresponding equations have been explicitly written down. As a consequence it has been shown that the both approaches lead finally to the same mathematical model.


1. Cusped Plates

1.1.The N-th order Approximation in the Case of a Lipschitz Domain

Let W1 = W È W denote a bounded domain in R3 occupied by a cusped plate under consideration. We assume the boundary ∂W to be a Lipszhitz surface (i.e., ∂W may have angular singularities but no cusps). Denote by w the plate projection.

For the N-th approximation of Vekua’s version the suitable weighted Sobolev type space VNh(w) is constructed and the solvability and uniqueness theorems are proved for some admissible boundary value problems for the corresponding system of partial differential equations with order degeneration.

Moreover, it is established that the solution of the N-th approximation converges in H1(W ) to the solution of the original three-dimensional problem as N→+¥ .

In the particular case, when the plate thickness vanishes on a part of the boundary ∂w, as a power function of the distance from the reference point to ∂w, the question of existence of traces (on ∂w) of solutions in the space VNh(w) is investigated.

For the N-th approximation this leads to a generalization of the mechanical interpretation of Keldysh's effect for degenerate partial differential equations previously established for the case of the N=0 approximation and for the classical bending theory [4] (which actually coincides with the N=1 approximation of cusped plates).


1.2. The First Order Approximation in the Case of a Non-Lipschitz Domain

In the case of the N=1 approximation, a general cusped plate (i.e., a plate, considered as a three-dimensional body that may have also cusps) is investigated. Korn's inequality for this case is established and the unique solvability of admissible boundary value problems for weighted displacement moments is proved.


2. Cusped Beams

2.1. The Dynamical Problems in the (0.0) and (1.0) Approximations

In [14], expanding the fields of displacements, strains, and stresses of the three-dimensional theory of linear elasticity into double Fourier-Legendre series with respect to the variables of bar thickness and width, an analogous mathematical model of bars with variable rectangular cross-section is constructed. It is assumed that the thickness and width can tend to zero at the ends of the bar, i.e., the case of a cusp is included. More precisely, in Section 1 of [14] some auxiliary formulas are deduced. These formulas give the expressions of the double Fourier-Legendre moments of the first derivatives of a function by means of the double Fourier-Legendre moments of the same function and of the derivatives of these double moments. Section 2 is devoted to the reformulation of the equilibrium equations, Hooke's law, and of the expression of the strain tensor by means of the displacement vector components in terms of the double moments of the displacement vector, strain and stress tensors. In Section 3 the main relations (deduced from the Lamè system) of the (N3,N2) approximation are derived. This is a system of either ordinary differential equations (in the static case) or hyperbolic equations (in the dynamical case), for weighted double moments of displacement vector components. The above equations are degenerate if the area of the variable cross-section (i.e. at least the thickness or the width) equals zero at an endpoint of the bar. If the variable area of the cross-section is always positive, then these equations are nondegenerate. As in the case of plates and shells, all these approximations represent independent mathematical models of bars which go into the three-dimensional model when N2,N3 tend to infinity. In Section 4 the initial and boundary value problems are set up. In the nondegenerate case, i.e., in the case when both the bar thickness and width do not vanish anywhere, the well-posedness of problems follows from the well-known theories (see, e.g., [15], [16]) of nondegenerate systems of ordinary differential equations (in the static case) and nondegenerate systems of hyperbolic equations (in the dynamical case). The existence of the solutions of the static problems are proved in the (0,0) (see [17]) and (1,0) (see [18]) approximations. The relation of the mathematical model of bars under consideration to the three-dimensional theory of elasticity is illustrated in [19].

The present section deals with the dynamical problems in (0,0) and (1,0) approximations. In the first part initial boundary value problems are investigated for the hyperbolic systems with order degeneration containing systems of the (0,0) and (1,0) approximations. In the second part, the results of the first part are applied to the cusped bars.


2.2. Static and Dynamic Problems for Beams when Barycentric and Beam Axes Coincide

Elastic beams with variable cross-section of the general form when barycentric and beam axes coincide are considered on the basis of the classical bending theory. The admissible static and dynamical problems are investigated. Also, numerical experiments for some test examples are carried out.



[1] Vekua, I.N., On a Way of calculating of prismatic shells. Proceedings of A. Razmadze Institute of Mathematics of Georgian Academy of Sciences, 21 (1955), 191-259. (Russian)

[2] Vekua, I.N., The theory of thin shallow shells of variable thickness. Proceedings of A. Razmadze Institute of Mathematics of Georgian Academy of Sciences, 30 (1965), 5-103. (Russian)

[3] Vekua, I.N., Shell theory: General methods of construction. Pitman Advanced Publishing Program, Boston-London-Melbourne 1985.

[4] Jaiani, G.V., Elastic bodies with non-smooth boundaries-cusped plates and shells. ZAMM, 76 (1996) Suppl. 2, 117-120.

[5] Babuska, I., Li, L., Hierarchic modelling of plates. Computers and Structures, 40 (1991), 419-430.

[6] Gordeziani, D.G., To the exactness of one variant of the theory of thin shells. Soviet. Math. Dokl., 215 (1974) 4, 751-754.

[7] Guliaev, V., Baganov, V., Lizunov, P., Nonclassic theory of shells. Vischa Shkola, Lviv 1978. (Russian)

[8] Khoma, I., The generalized theory of anisotropic shells. Naukova Dumka, Kiev 1986. (Russian)

[9] Khvoles, A.R., The general representation for solutions of equilibrium equations of prismatic shell with variable thickness. Seminar of the Institute of Applied Mathematics of Tbilisi State University, Annot. of Reports, 5 (1971), 19-21. (Russian)

[10] Meunargia, T.V., On nonlinear and nonshallow shells. Bulletin of TICMI, 2 (1998) 46-49.(electronic version: TICMI)

[11] Schwab, C., A-posteriori modelling error estimation for hierarchik plate models. Numerische Mathematik, 74 (1996), 221-259.

[12] Vashakmadze, T.S., The theory of anisotropic plates. Kluwer Academic Publishers, Dordrecht-London-Boston 1999.

[13] Zgenti, V.S., To investigation of stress state of isotropic thick-walled shells of nonhomogeneous structure. Applied Mechanics, 27 (1991) 5, 37-44.

[14] Jaiani, G., On a Mathematical Model of a Bar with a Variable Rectangular Cross-section. Preprint 98/21, Institute fur Mathematik, Universitat Potsdam, Potsdam 1998.

[15] Kiguradze, I.T., Initial and boundary value Problems for the systems of ordinary differential equations. V. 1. Linear Theory. "Mezniereba", Tbilisi 1997. (see also, Itogi Naukii Tekhniki, Seria Sovremennje Problemi Matematiki, Noveishje Dostizhenia, V. 30, Moskva 1987, 1-202.) (both in Russian)

[16] Ladyzhenskaya, O.A., Boundary value problems of mathematical physics. "Nauka" Press, Moscow 1973. (Russian)

[17] Jaiani, G., On a model of a bar with variable thikness. Bulletin of TICMI, v.2 (1998), 36-40.

[18] Jaiani G.V., Boundary value problems in (1,0) approximation of a mathematical model of bars. Bulletin of TICMI, v.3 (1997), 7-11. (electronic version:


[19] Jaiani G.V., Relation of a mathematical model of bars to the three-dimensional theory of elasticity. Workshop in partial differential Equations, University of Potsdam, pp. 15 and 16, 1999



For more details contact: jaiani@viam.sci.tsu.ge

Vekua Institute of Applied Mathematics