## TBILISI INTERNATIONAL CENTRE OF

MATHEMATICS AND INFORMATICS

REPORT FOR 2001

The main aim of the project was to further develop the general theory of cusped plates on the basis of I. Vekua's version of the theory of plates with variable thickness, and to develop an analogous theory for elastic bars with variable cross-section [1]. Also to investigate simple elastic cusped plate-fluid interaction problems and simple non-linear models [2].

In practice such plates and beams are encountered in calculating spatial structures with partly fixed edges (e.g., stadium ceiling, aircraft wings etc), in the machine-tool construction (e.g., cutting-machine, planing-machine etc), in the astronautics, and other spheres of practical engineering.

Using the energy projection approach suggested by I.Vekua and developed by Ch. Schwab for plates of constant thickness, the model of elastic plates of variable thickness has now been constructed. In particular, hierarchical models which reduce the original three-dimensional boundary value problem for cusped prismatic shell type elastic bodies to two-dimensional problems have been constructed. It was proved that the model costructed by the energy projection method turns into I. Vekua's model if we change the describing unknown functions. The relation between the unknown functions of these two models has been completely accomplished. As a consequence, both approaches lead finally to the same mathematical model [3,4].

We recall that in the "regular" case (i.e., when the plate thickness does not vanish), the Fourier-Legendre coefficients of the displacement vector u, which solves the original three-dimensional problem in the space H1(W ), automatically belong to the space H1(w ). Moreover, all the moments wNi0,..., wNiN , i=1,2,3 determined by the corresponding two-dimensional hierarchical models belong to the space H1(w ), while the approximation of the displacement vector wN represented by means of these moments belong to the space H1(W ). In the case of cusped plate, the Fourier-Legendre coefficients of the displacement vector uÎ H1(W ) do not belong to the space H1(w ) any more in general. Also, the space of approximation vectors wN represented by the wNi0,..., wNiN , i=1,2,3 do not belong to the space H1(W ). Therefore it is necessary to choose a space for moment functions wNi0,..., wNiN, i=1,2,3 defined on w such that the corresponding linear combinations of these moments with the Legendre polynomials as coefficients, belong to the space H1(W ).In the case when the domain W occupied by the cusped plate is Lipschitzian, the weighted Sobolev space H1N(h(+),h(-),w ,g ) on the plate projection w for the N-th approximation of Vekua's version was constructed and the solvability and uniqueness theorems were proved in these spaces for some classes of admissible boundary value problems [3].

The uniqueness and existence results in the corresponding functional spaces for the obtained two-dimensional variational hierarchical models have been established. We remark here that the well-known approach of previous authors needs some modifications which are connected with the above mentioned pecularities of the appropriate functional spaces where we look for the unknown moments.

The convergence in the space H1(W ) of the approximate solution wN to the exact solution u of the three-dimensional original problem has been obtained. The abstract error estimates with respect to the number of approximation order N and the maximum of the thickness 2h of the plate are given.

Along with the above convergence results we also obtained error estimates in the space H1(W ) with respect to the number N of approximating functions and the maximal thickness of the plate [3].

This leads to a generalization of the mechanical interpretation of Keldysh's effect for degenerate partial differential equatoins previously established for the case of the N=0 approximation and for classical the bending theory (which, actually coincides with the N=1 approximation for cusped plates) to the general N-th approximation.

In the N=1 approximation, the general cusped plate (i.e., a plate considered as a three-dimensional body that may have also cusps) was investigated in detail. Korn's inequality for this case was established and the unique solvability of admissible boundary value problems for weighted displacement moments was proved [5].

The systems of differential equations corresponding to the two-dimensional variational hierarchical models for a general orthogonal system have been also explicitly constructed.

The analogous results for cusped bars with rectangular cross-section are obtained, too. In addition, the dynamical problems in the (0.0) and (1.0) approximations for elastic bars with rectangular cross-sections were investigated in [6].

Admissible static and dynamical problems are investigated for an Euler-Bernoulli cusped beam with a continuously varying cross-section of arbitrary form. The setting of boundary conditions at the beam ends depends on the geometry of the sharpening at the beam ends (sometimes the boundary conditions become weighted ones and sometimes they disappear complitely), while the setting of initial conditions is independent of the sharpening [7].

Also, numerical experiments for some test examples were carried out in [2].

#### References

[1] G. Jaiani. On a mathematical model of bars with variable rectangular cross-sections. ZAMM Z Amgew. Math. Mech. 81, 3, 2001, 147-173

[2] N. Chinchaladze, G. Jaiani. On a cusped elastic solid-fluid interaction problem. Applied Mathematics and Informatics, vol. 6, 2, 2001 (in print)

[3] G. Jaiani, S. Kharibegashvili, D. Natroshvili, W.L. Wendland. Hierarchical models for cusped plates (in print)

[4] G. Jaiani. Application of Vekua's dimension reduction method to cusped plates and bars. Bulletin of TICMI (Tbilisi International Centre of Mathematics and Informatics), vol.5, 2001, 27-34

[5] G. Devdariani, S. Kharibegashvili, D. Natroshvili. The first boundary value problem for the system of cusped prismatic shells in the first approximation. Applied Mathematics and Informatics, vol. 5, 1, 2001

[6] G. Jaiani, S. Kharibegashvili. Dynamical Problems in the (0,0) and (1,0) Aproximations of a Mathematical Model of Bars. Proceedings of the International Graz Workshop (February 12-16, 2001), World Scientific, 2001, 188-248

[7] G. Jaiani. Static and Dynamical Problems for a Cusped Beam. (to appear)

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

Vekua Institute of Applied Mathematics