Stress-strain state of spherical shells with two unequal holes

This article presents the results of a numerical study of the stress concentration around two equal and unequal holes in an orthotropic spherical shell made of composite materials under the action of internal pressure. The influence of geometric (hole radii, shell thickness, distance between holes) characteristics, as well as material orthotropy and shear stiffness, on the stress state of spherical shells made of composite materials is studied. A numerical algorithm based on the finite element method has been developed and a software package has been implemented on a computer that allows solving the problem of stress concentration near two unequal holes in spherical shells made of composite materials.


Introduction
A fairly large number of works are devoted to the methods of calculating shell structures made of composite materials and the study of stress concentration near holes. However, most of the studies were carried out within the framework of the classical Kirchhoff-Love hypothesis, which does not take into account interlayer and transverse shears characteristic of composite materials [1,2].
Within the framework of the improved theory of the Timoshenko type, the application of the finite element method is proposed for solving problems of orthotropic shells made of composite materials, weakened by several holes [3,4].
Multiply connected shells made of composite materials have been considered by a few authors. The main results are given in [2,[7][8][9]. For an orthotropic cylindrical shell with several holes, a solution was constructed in the work of K I Shnerenko, N Kh Noraliev [10][11][12][13][14]. Moreover, the distribution of stresses around two equal circular holes in an orthotropic spherical shell made of a composite material was investigated taking into account shear stiffness [5,6]. As follows from the above review of works, at present, the study of the stress-strain state of shells made of composite materials with two unequal holes is insufficient.
Therefore, in this paper, we study the stress distribution around two unequal circular holes in an orthotropic spherical shell made of a composite material. Using the refined theory of Timoshenko-type shells, we will take into account the effect of transverse shear deformations for the entire package of the shell as a whole. In this case, the composite material will be considered as homogeneous with the given characteristics.

Materials and methods
A spherical shell with two unequal circular holes of radii and (Fig. 1) and intensity under the influence of internal pressure e is considered. The study area is divided into finite elements using the finite element method [5,6]. In the middle surface of the shell, choose an orthogonal curvilinear coordinate system the third coordinate is reported in the direction of the outer normal to the middle surface of the shell.
We will proceed from the variational equation [1]: where Г -outlines of holes; Ω -area, the boundary of which is far enough from the contours of the holes, and on it the damping conditions are satisfied; components of the disturbed stress state; -components of the main stress state. As can be seen from the table, the stresses on the contour of the larger hole at the point θ = 0 up to the distance hardly change. With further approach of the holes to at the points of the contour of the larger hole, the coefficient increases, and the coefficient p at the points , π/2 decreases, and at the point , it increases. Of interest is the influence of the radius of the larger holes and the value of the jumper d on the stresses at the point of the contour of the small holes. This can be seen from the first column of the table. When the holes approach each other at the point on the contour of the larger hole, the stress concentration and increase by 131% and 102%, respectively. Note that when the holes approach each other in the considered example, a non-monotonic character of the change in the stress concentration coefficients on the hole contours is also observed. A similar phenomenon was previously found for an isotropic shell with equal holes [7,8]. In Fig. 2 and 3 show the results of the study of stress concentration and at different values of the shear parameter .. The shell parameters were chosen as in the case of Table 1. Curves 1, 2, and 3 respectively refer to the points , π/2, π. It can be seen that the influence of on the stress concentration is most pronounced at the points θ = 0 and θ = π. The effect of orthotropy of the material on the stress concentration (top line) and (bottom line) at various points of the hole contour is given in Table 2. The radius of one hole was fixed p , and the second hole g g g g g g . It is seen that the orthotropy of the material significantly affects the stress concentration along both contours of the holes. With a change in the orthotropy of the material from 2.5 to 0.8 in points θ = 0 and θ = π on the contour of the larger hole, the coefficient increases by 41% and 25%, respectively, and by 61% and 40%.  This increase on the contour of the small hole at the points θ = 0 and θ = π for the coefficient is 35% and 15%, and for 140% and 32%. At the point with a change in , the coefficients and decrease. Table 3 shows the results of studying the stress concentration factors and depending on the change in the radii of the holes . In this case, the radius of one hole was fixed g , and the radius of the other hole was changed . As the radius of the second hole increases, the stress concentration increases on the contour of both holes. When the parameter changes at the points and θ = π on the contour of the first hole , the coefficient g increases by 6% and 21%, respectively, and at the point θ = 0 decreases by 10%.  The coefficient on the contour of the first hole at the points and θ = π decreases, and at the point θ = 0 it increases. At the points p , π on the contour of the second hole with an increase in the radius of the holes, the coefficient increases by 200%, 170%, 105%, respectively, and the coefficient by 60%, 300% , and 100%. With a change in the shear stiffness from 1.0 to 0.01 at the points , π on the contour , the coefficient increases by 7%, 40%, 3%, respectively, and the coefficient -by 8%, 4%, 30%. At points y on the contour with a change in , the coefficient increases, respectively, by 6%, 10%, 4%, and the coefficient by 16%, 31%, 10%.   For the same shell, Table 5 shows the nature of changes in the coefficients and from the shell radius. It is seen that with an increase in the shell radius , the stress concentration factor decreases along both contours of the holes. With an increase in the shell radius from 5 to 20 at the points on the contour of the larger hole , the coefficient decreases by 16%, 36%, 84%, respectively, and the coefficient -by 27%, 26%, 14%. With an increase in the radius of the shell at the indicated points on the contour of a small hole , the coefficient decreases by 26%, 36%, 10%, and the coefficient decreases by 47%, 43%, 7%.

Conclusions
In this paper, we studied the stress distribution around two unequal circular holes in an orthotropic spherical shell made of a composite material. Using the refined theory of Timoshenko-type shells, we considered the effect of transverse shear deformations for the entire package of the shell as a whole. In this case, the composite material is considered as homogeneous with the given characteristics. Thus, the developed technique can be used to calculate the elements of shell structures made of composite materials with several holes.