Volume 1, 1997 16

of straight line Po and s or -Po¹ . Then the solution of the inverse problem is unique.

Theorem 2. Let W 1, W 2 be piece-wise smooth, simply-connected bounded domains in R2, W = W 1 È W 2 . Let us assume, that on W there exists the point of intersection of the curves W 1 and W 2. Then the solution of the inverse problem is unique.

ON A METHOD OF CONSTRUCTION OF A

GEOMETRICAL NONLINEAR THEORY OF NON-SHALLOW SHELLS

T. Meunargia

I. Vekua Institute of Applied Mathematics,

I. Javakhishvili Tbilisi State University

In the present paper the three-dimensional problems of the geometrical nonlinear theory of elasticity are reduced to the two-dimensional problems of non-shallow shells by means of I. N. Vekua method. Under thin and shallow shells I. Vekua meant elastic bodies of shell type, satisfing the conditions

(1)   aa b - x3 ba b » aa b , -h £ x3 £ h, (a , b = 1, 2),

where aa b , ba b are mixed components of the metric tensor and tensor of the curveture of the shell middle surface, x3 is a thickness coordinate, h is a semi-thickness, x1, x2 are curvilinear coordinates of the middle surface.

In our constructions under non-shallow shells we will mean elastic bodies, free from the assumption of the form (1), i. e, in general,

aa b - x3 ba b ¹ aa b , but always | x3 ba b |<1.

There is considered the case, when Hook's law for anisotropic bodies has the following form

s ij = Eijpqe pq,

where s ij are contravariant components of the stress tensor, Eijpq and e pq are contravariant and covariant components of the tensors of elasticity and strain, respectively:

Eijpq = klmn,

 e pq=

here Ri and Ri are co- and contravariant base vectors of the space, U is a vector of desplacement, and klmn are base vectors and elastic constants in the rectangular system

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