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Results obtained on the basis of linearized functionals in the theory of nonlinearly elastic composite shells are analyzed and generalized. The Kirchhoff-Love and Timoshenko hypotheses are used. Possible membrane or shear locking is taken into account. New approaches are proposed to improve the convergence of numerical solution for new classes of nonlinear problems for thin and nonthin shells with a curvilinear (circular, elliptical) hole. The stress-strain state of shells is analyzed using different versions of shell theory. The influence of the nonlinear properties and orthotropy of composite materials on the stress distribution in structural members is studied.Introduction. At a certain stage of development of numerical mesh-type methods for shells, so-called locking effects became to manifest themselves as slowed convergence. A similar effect called volumetric or dilatational locking is widely known in relation to weakly compressible bodies [36,38,41]. One of the first locking effects associated with shells was shear locking [61] mainly manifested in attempts to apply shear finite-element models to thin-walled structures [4,41]. Under certain conditions, even the classical model based on the Kirchhoff-Love hypotheses may lead to membrane locking [4,41,61,63]. Further increase in the degrees of freedom of the finite-element method (FEM), beginning with the consideration of reduction and ending with a three-dimensional model, revealed new types of locking: thickness or Poisson's thickness locking [41,62,67] and trapezoidal or curvature thickness locking [67].Originally, locking was overcome using sophisticated mathematical methods such as reduced integration and the like. These methods happened to have a number of shortcomings: they are not general enough to encompass all types of locking, geometrical parameters of shells, and properties of materials, they may produce false solutions, and they are limited to the FEM and involve finite-element models of high order (see [35,36,43] and references therein). These shortcomings are apparently because of the heuristic nature of such methods. This motivated researchers to develop meshfree, or meshless, methods [45,52,54] and mesh methods other than the FEM [2, 6, 7], approaches based on mixed variational principles [42,57,58], and finite-element models of low order [55,60].In the variational difference method (VDM) in the form of the finite-difference method (FDM), locking effects are less significant [2]. The authors of [2] point out that an attempt to satisfy, as accurately as possible, the additional conditions of functionals with the minimum number of varied parameters in the FEM adversely affects the accuracy of the stationarity conditions, while the reasonable inaccuracy of the former, as in the FDM and mixed FEM, produces more accurate trade-off schemes.Different types of locking and modern methods used to overcome them are summarized in Table 1.In the table, the methods are named as is customary in the English-language literature, since their Russian names are ...

The elastoplastic state of thin spherical shells with an elliptic hole is analyzed considering that deflections are finite. The shells are made of an isotropic homogeneous material and subjected to internal pressure of given intensity. Problems are formulated and a numerical method for their solution with regard for physical and geometrical nonlinearities is proposed. The distribution of stresses (strains or displacements) along the hole boundary and in the zone of their concentration is studied. The results obtained are compared with the solutions of problems where only physical nonlinearity (plastic deformations) or geometrical nonlinearity (finite deflections) is taken into account and with the numerical solution of the linearly elastic problem. The stress-strain state in the neighborhood of an elliptic hole in a shell is analyzed with allowance for nonlinear factors Introduction. Stress analysis of simply connected and doubly connected thin and nonthin elastic shells made of metallic and composite materials was performed in [7, 8, 10, 11, 15, 16, 20, etc.]. The basic results have been obtained using analytical, variational, and numerical methods for various shells (plates) with curvilinear holes (notches).The nonaxisymmetric deformation of and stress distribution in isotropic spherical shells with an elliptic hole are studied with allowance for physical or geometrical nonlinearity in [6,7,9,12]. The stress concentration near a circular hole in both spherical and ellipsoidal shells was analyzed in [2, 7, 13] with allowance for physical nonlinearity, in [3,4,7] with allowance for geometrical nonlinearity, and in [1,6,7,14] with allowance for both nonlinear factors (plasticity and large deflections).It is of interest to solve two-dimensional nonlinear static problems for thin shells with curvilinear (noncircular) holes under high surface and contour loads.Using the method developed in [17] and tested against some linear and nonlinear elastic problems, we will numerically analyze the elastoplastic stress-strain state near an elliptic hole in flexible spherical shells. We will also analyze the influence of nonlinear factors on the stress distribution in the stress concentration zones in a shell under surface pressure of given intensity.