Mixed functionals in the theory of nonlinearly elastic shells

Abstract

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 ...

On the stress distribution in a spherical shell with an off-center curvilinear hole

“…As the rigidity of fixation is increased, the points smear Introduction. The asymmetric stress-strain state (SSS) near curvilinear holes (cutouts) in isotropic [4,6,9,10] and composite [4,5,8,11] plates and shells is of theoretical and practical interest: the results of a stress-strain analysis would allow us to evaluate the stress and strain of various thin-walled structural members under various kinds of loads.Numerical stress-strain analyses were performed mainly for thin-walled spherical caps with a central elliptical hole [4,5,8,10,11].As pointed out in [1,7], in the off-center case, numerical results are mainly available only for a circular hole, with the emphasis being on the stress distribution around the bridge between the outer and inner edges, which does not give an accurate account of the maximum stresses. Most publications employ special orthogonal coordinate frames and address simply connected stress-concentration problems for elastic isotropic shells.…”

Nonlinear two-dimensional static problems for thin shells with reinforced curvilinear holes

“…Methods and results of solving specific classes of physically and geometrically nonlinear problems for some metal shells are discussed in the monographs [20,32,38,42,67,70]. The issue of stress concentration in shells (plates) involving the development of methods of solving nonlinear problems is analyzed in the reviews [24,89,93,94].Note that most studies on nonlinear stress concentration in shells, including problem formulation, development of methods for solving certain classes of nonlinear problems, and analysis of calculated results were carried out by the authors of the present paper and their followers at the S. P. Timoshenko Institute of Mechanics. In solving specific classes of nonlinear…”

“…Methods and results of solving specific classes of physically and geometrically nonlinear problems for some metal shells are discussed in the monographs [20,32,38,42,67,70]. The issue of stress concentration in shells (plates) involving the development of methods of solving nonlinear problems is analyzed in the reviews [24,89,93,94].…”

Stress state of a transversely isotropic spherical shell with a rigid circular inclusion

“…The form of solution depends on the range of the transverse shear compliance parameter. The influence of the relative radius of a rigid inclusion and transverse shear compliance on stress concentration is analyzed Keywords: transverse shear, transversely isotropic shallow spherical shell, rigid circular inclusionThe stress distribution around holes or rigid inclusions in shallow and deep transtropic and orthotropic spherical shells was studied in [1][2][3][4][5][6][7][8][9][10][11]. The results of studies into the influence of the transverse shear stiffness of materials on the stress state of composite shells are reported in [3,7,8].…”

“…The stress distribution around holes or rigid inclusions in shallow and deep transtropic and orthotropic spherical shells was studied in [1][2][3][4][5][6][7][8][9][10][11]. The results of studies into the influence of the transverse shear stiffness of materials on the stress state of composite shells are reported in [3,7,8].…”