Volume 1, 1997
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
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
sij = 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,
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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