Deformations compatibility equations in the general shell theory into the relative coordinate system with projected deformations

This paper presents a new analytical method for obtaining new deformations compatibility equations or, new Saint-Venant’s identities, into the relative coordinate system with projected deformations by applying the hypothesis of the lineal shell theory in general flexion state. The method proposed generalizes the compatibility conditions established by A.L. Goldenveizer for the shell theory. On the other hand, the new equations include the deformations compatibility equations by other authors: Flügge, Saint-Venant, Love-Kirchhoff, Timoshenko, Goldenveizer, and Reissner-Mindlin. The results showed an increase of knowledge in general shell theory, and provide inverse and semi-inverse solutions, whose systems solution correspond to the hyper-statics degrees of the physical model, and not to their degrees of freedom.


Introduction
The "Relative coordinate method" (RCM) was developed for the calculus of shells (Figure 1) [1,2] this method allows projecting the internal forces that appear in a shell with arbitrary geometry ( * ) on top of another one with simpler geometry ( ) only if a scalar function (relative function) ( 1 , 2 ), that relates them as a continuous and bi-univocal form can be defined between them. The RCM, which is a generalization of Pücher's method, provides the possibility of using other reference surfaces with projected solicitations by selecting the most adequate reference surface or projected surface for the calculus of the same shell in each case [1][2][3].
The value of the RCM depends on the generalization of the coordinate system in ℝ 3 [1, 2, 4] since it contains the following orthogonal coordinate systems: (a) Gauss's curvilinear intrinsic coordinate, imposing the condition = 0 on the relative relation, Eq. (3) and (4). (b) Cartesian coordinate, imposing the parametric condition of a Cartesian plane on the relative relation, Eq. (3) and (4). (c) Cylindrical-coordinate, imposing the parametric condition of a Polar plane on the relative relation, Eq. (3) and (4). (d) Spherical coordinate, imposing the parametrical condition of a Sphere on the relative relation, Eq. (3) and (4). (e) Cartesian intrinsic coordinate, imposing the parametrical condition of a coaxial cylinder on the relative relation, Eq. (3) and (4).
It can be said that when Goldenveizer's sufficient conditions [5,6] are not fulfilled, then the membranal model (isostatic problem) plus the simple alteration effect does not provide satisfactory results in the internal equilibrium of the shell modeling [4,6,7]. However, the state equations in the relative coordinate system with projected deformations that provide the mathematical operational models in displacements both for the Love-Kirchhoff's flexion (third order) and the Reissner-Mindlin (fifth order) were obtained [7].
Inverse and semi-inverse models make the solution system coincide with the hyper-statics degree of the physical model [4]. Since the solution systems are expressed in the relative coordinate system the reference surface that generates the most suitable analytical simplification in the shell mathematical modeling of arbitrary geometry ( * ) can be obtained [4]. To get inverse and semiinverse mathematical models in the relative coordinate system with projected deformations, it is necessary to obtain not only Saint-Venant's compatibility equations of the deformations but also the stress functions. As it can be demonstrated [7] the conditions established by A. L. Goldenveizer [5,6] are not sufficient for obtaining the deformations compatibility equations in the relative coordinate system, Eq. (5), as they are only established for Gauss's curvilinear orthogonal coordinate system [7].
The analytical method presented in this paper allows obtaining deformations compatibility equations or Saint-Venant's identities. This method generalizes the condition established by A. L. Goldenveizer, Eq. (5) for the shell lineal theory [5,6].

The new analytical method
Twelve geometric equations, Eq.(6), in components of the shell's general flexion theory were obtained in relative coordinate system with projected deformations considering no dependency between the vectors ⃗ ⃗ (displacement) and ⃗ (rotations) [3,7]. The new method (into the Euclidean vectorial space in ℝ 3 ) [8,9] also permits obtaining the minimal degree of the operator polynomial that generates the sufficient conditions for a symmetric tensor of second order (e.g. the tensor of Green-Lagrange) which corresponds to a deformation tensor and therefore can be integrated so that there is a displacement field from which it originates. This guarantees the continuity of the medium during the deformation process. Appendixes A, B, and C show the application of the method for typical problems of the elasticity theory in two and threedimension Cartesian and Polar coordinate.
By orderly adding the coefficients obtained, Eq. (14) and taking into account the analytical structure of the operator polynomial, Eq. (7), is gets the first deformations compatibility equation, Eq. (15): Where: C ̅ i : Coefficient that accompanies the geometric equation i; D ̅ i : Coefficient that accompanies the partial derivative with respect to coordinate α 1 ; E ̅ i : Coefficient that accompanies the partial derivative with respect to coordinate α 2 .
If Eq. (15) is operate the by

Results and discussion: Particular cases and theoretical results
The new general deformations compatibility equations were obtained by applying the new method in the general shell flexion theory into the relative coordinate system with projected deformations, Eq. (22).
The proposed method has a cyclic character, as each proposal of the operator polynomial of degree N ( ) represents one solution, either trivial (null) or not. One of the purposes of this method is to find the minimal degree of the operator polynomial that generates the nontrivial solution.

Conclusions
The new deformations compatibility equations, Eq. (22), using the mathematical assistant "Wolfram Mathematic 9" can be programmed in the construction of the inverse formulation (type Airy) for the mathematical modeling of different shell problems [19,20,[22][23][24][25][26]. The integration of the non-linear geometrical expressed in the relative coordinate system with projected deformations may be included in the inverse formulation.
The deformations compatibility equations can be obtained at a time, either first or second type Saint-Venant's relationships through the method proposed (appendix case C) [9,16,17,27].
The proposal of the operator polynomial = ( 1 , 2 , 3 , … m ), ∀ , ∈ ℕ provides deformations compatibility equations of different orders. The method can be applied to cases of geometric equations involving on lineal terms.
The general equations, Eq. (22), provide the possibility to obtain new compatibility equations when selecting the reference surface ( ). They also offer the advantage of choosing the reference surface ( ) that generates the least analytical complexity for each problem, so that complex geometric shells can be calculated.
The compatibility conditions of the deformations established by A.L. Goldenveizer, Eq. (5), are only valid when the middle surface of the shell is expressed in Gauss's curvilinear orthogonal coordinate system ( = 2 ⁄ ). The new method allowed to obtain the compatibility equations with projected deformations into the relative coordinate which includes the orthogonal coordinate system ( = 2 ⁄ ) (Figure 4). The principle of the indeterminate coefficient made it possible the formation of the new method since the geometrical equations constitute elements of a Euclidian's vectorial space (E).

Author Contributions:
All authors contributed to the study conception, design, and analysis. All authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.

Acknowledgments:
The authors would like to express their gratitude to Ph.D. Ángel Emilio Castañeda Hevia for his support. The authors wish to thank the organizations that supported the research: The Central University of Las Villas (UCLV).

Conflicts of Interest:
The authors declare no conflict of interest.

Appendix A. Bi-dimensional problem in Cartesian coordinates
In this case, the geometric equations (A.1) are made of differential operators that directly operate on the displacements ( ).   The solution (A.14) expresses that the geometric equations of the bi-dimensional problem in Cartesian coordinates do not require deformations compatibility equations higher than the second order. The expression (A.15) shows a summary of the results of the operator polynomial of different orders, and the convergence to the Saint-Venant's solution (A.11) obtained with an operator polynomial of second degree( (2) ).

Appendix B. Bi-dimensional problems in Polar coordinates
In this case, the geometric equations (B.16) are made up of functions that multiply the differential operators, which function directly on the displacement ( ), and of displacements that are not operated by differential operators.
By applying the proposed procedure to the geometric equations (B.16) using the second-degree operator polynomial ( (2)   Similarly to case A (bi-dimensional problems in Cartesian coordinates), if a first-degree operator polynomial [ (1) ( , ) ] had been used in Polar coordinates, then the solution obtained would have been trivial, which implies the nonexistence of first-degree compatibility equations for this problem. Equally, if an operator polynomial higher than second degree were used, the solution of the lineal combination of the partial derivative equation (B.19) would be obtained.