Theory of thin shells as a spatial two-dimensional continuum in an oblique system of coordinates

In contrast to the known approaches in the theory of shells, based on certain assumptions, and the traditional reduction of a three-dimensional problem to a two-dimensional one, in this paper, static and geometric relations are obtained from the position of the interaction of linear forces and moments in a spatially curved two-dimensional continuum. This approach was applied by E. Reissner, but in an orthogonal grid and for physical components. Relations are presented in tensor, vector, and scalar forms.


Introduction
Various methods for constructing versions of the theory of shells based on certain assumptions are known. In this paper, in contrast to the traditional approach of reducing a three-dimensional problem to a two-dimensional one, static and geometric dependences are obtained from the point of view of the interaction of linear forces and moments in a spatially curved two-dimensional continuum. The basic relations of the theory of shells have considerable generality and a compact form. With the help of some transformations and simplifications, these relations are reduced to the equations of the classical theory. The construction of the theory of shells as a two-dimensional continuum was carried out for the general case in a spatial oblique coordinate system, which entailed the use of the apparatus of tensor analysis.

Derivation formulas
Let us present some dependencies related to the differential geometry of surfaces, which are necessary for further constructing the proposed theory of shells. These relations are described in detail in the corresponding courses, as well as in the introductory chapters of fundamental monographs [1-4, 6, 9] on the theory of shells with some differences in notation. Therefore, the formulas in this paragraph are given as a reference without detailed derivation.
So, let some parametrised surface be given by the equation ), , ( The physical significance of 1 A and 2 A is that they are proportionality coefficients in the formulas connecting the differentials of the arc lengths of the coordinate lines with the differentials of the curvilinear coordinates themselves The area of an infinitesimal quadrangular surface element bounded by the coordinate lines , is defined by the equality The second quadratic form of the surface is given by the expression , ) where the coefficients The third quadratic form of the surface is the square of the differential of the unit normal vector to the surface The last form is not independent, it is expressed through the first two , are Gaussian (total) and mean curvature of the surface, respectively.
When deriving the basic relations, we need the Gauss-Weingarten formulas for differentiating the vectors of the basic basis. In surface theory these derivation equalities are analogous to the Serre-Frenet formulas in the theory of spatial curves. Let us represent them in the form Here k ij  are the Christoffel symbols defined by the equalities The coefficients For the vectors of the reciprocal local basis  (7) can be written in the form

Equilibrium equations. Boundary conditions
Let us select an infinitesimal element of the surface, which will be the element of a certain shell, loaded with the distributed load  In this case, vectors of forces and moments can be represented in the form (i, j = 1,2) Each of equations (13), (14) is equivalent to three scalar equations. Expanding equation (13) with the help of (7) and sequentially multiplying by the coordinate vectors of the auxiliary basis n r r , we obtain the known equilibrium equations in scalar form

Equations of moment equilibrium (14), considering the relation
where .
Without taking into account i P and n m , equations (17) are the same as those known from the literature sources [3,9]. Taking these quantities into account, the third moment equation is not algebraic and not satisfied identically. It also implies that for the moment components i P , the component n m of the surface moment is determining. Equilibrium equations (13), (14), which are valid in the interior of the shell, must be supplemented with static boundary conditions, which can be represented in the form ), , cos( ); , cos( are the specified edge forces and moments;  -tangential normal to the surface boundary line.

Geometric relationships
Let us define possible displacements and possible deformations as a system of infinitely small kinematically possible displacements and deformations, which allow us to express possible work in such a form that equations (13), (14), (18) will be equivalent to the following Using the formula for integration by parts, we obtain the equalities The contour integral in (20) can be represented in the form In this expression     From this, passing to actual deformations and displacements, we obtain . ; We represent the vectors of deformations and rotations, translational and rotational displacements in the form Here, in contrast to the classical theory of shells, the vectors i   take into account the components i  directed along the normal to the surface and corresponding to i P in expansions (12). Expanding vector equations (24) using Gauss-Weingarten formulas (7), (10), we obtain the relations between deformations and displacements in scalar form when obtaining geometric relations, we come to the conclusion that these quantities are completely identical to each other. For these quantities, their specific geometric significance is not indicated, although the assumption that "they represent some parameters of deformation" was made in work [8, p.103]. The proposed version of the theory of shells allows us to give auxiliary geometric quantities