On the boundary conditions of the geometrically nonlinear Kirchhoff–Love shell theory

The present paper addresses the problem of establishing the boundary conditions of a geometrically nonlinear thin shell model, especially the kinematic ones. Our model is consistently derived from general 3D continuum mechanics statements. Generalized cross-sectional strains and stresses are based on the deformation gradient and the first Piola–Kirchhoff stress tensor. Since only the bending deformation is included in this model, no special technique needs to be adopted in order to avoid shear-locking. The theory is derived in such a way that any material model can be considered as a constitutive relation, once the zero transverse normal stress assumption is properly taken into account.     

Refined Analysis of the Influence of the Inhomogeneous Layers on the Stress State of Circular and Noncircular Cylindrical Shells

We study the stress state of laminated inhomogeneous closed cylindrical shells generally with an arbitrary cross section, taking transverse shear into account on the basis of the straight-line element hypothesis. As an example, results are presented for a two-layer cylindrical shell whose cross section is a combination of an oval and a circle.     

Effect of the Boundary Conditions and Influence of the Rotational Inertia on the Vibrational Modes of an Elastic Ring

We present the small-amplitude vibrations of a circular elastic ring with periodic and clamped boundary conditions. We model the rod as an inextensible, isotropic, naturally straight Kirchhoff elastic rod and obtain the vibrational modes of the ring analytically for periodic boundary conditions and numerically for clamped boundary conditions. Of particular interest are the dependence of the vibrational modes on the torsional stress in the ring and the influence of the rotational inertia of the rod on the mode frequencies and amplitudes. In rescaling the Kirchhoff equations, we introduce a parameter inversely proportional to the aspect ratio of the rod. This parameter makes it possible to capture the influence of the rotational inertia of the rod. We find that the rotational inertia has a minor influence on the vibrational modes with the exception of a specific category of modes corresponding to high-frequency twisting deformations in the ring. Moreover, some of the vibrational modes over or undertwist the elastic rod depending on the imposed torsional stress in the ring.     

Stress concentration near a thin elastic inclusion under interaction with harmonic waves in the case of smooth contact

We solve the problem on the interaction of plane harmonic waves with a thin elastic plate-shaped inclusion. The ambient medium is assumed to be in plane strain. The smooth contact conditions are satisfied on both sides of the inclusion. The bending displacements of the inclusion are determined from the corresponding differential equation. In the statement of boundary conditions for this equation, one should take into account the transverse forces and bending moments applied to the lateral edges of the inclusion, while the boundary conditions are posed on the midplane of the inclusion. Using the discontinuous solution method, we reduce the problem to a system of two singular integral equations, which are solved numerically by the mechanical quadrature method. We obtain approximate formulas for the stress intensity coefficients near the ends of the inclusion and for the transverse forces and moments applied to the inclusion.     

Braking of a Body by a Soft Inflatable Shell on Impact on a Surface

The results of mathematical simulation of a solid velocity damping by a soft skeleton fabric shell filled with air on impact on a hard surface are given. The equations of motion of a falling body and of the loading dynamics of membrane shells and the reinforcement rings in the fabric shell are considered together. Themathematical model and the numerical algorithm for solving the spatial problem of the dynamics of inflation of a shell with reinforcement rings are explicitly realized by the finite difference method. The boundary conditions are posed with regard to the contact of the shell elements in compression near the ring belts. The results of numerical experiments considering the interaction of the falling body with the deformable skeleton shell are discussed. The parameters influencing the process of the body braking on impact on a surface are determined.     

Study of failure of layered plates made of composites under impact contact loading

In the present paper, we study the strain and failure of a two-layer plate each of whose layers is made of a composite material. The layers have mutually perpendicular directions of fiber reinforcement. The plate is impacted by a rigid hammer. The layer composite material is modeled by anisotropic elastoviscoplastic damageable media according to two different (one-velocity and two-velocity) models. We propose to use the two-scale theory of fracture of the composite material. The problem is solved numerically by the method of spatial characteristics. This method permits correctly satisfying the boundary and contact conditions and correctly taking into account the material anisotropy in the difference approximation. We show that the one-velocity model increases the degree of plate fracture compared with the two-velocity model, which takes the stress waves dispersion into account. This can be explained by the fact that the stress field in the unloading wave spreads owing to the microinhomogeneity of the composite layers.     

Determining linear vibration frequencies of a ferromagnetic shell

The problems of determining the roots of dispersion equations for free bending vibrations of thin magnetoelastic plates and shells are of both theoretical and practical interest, in particular, in studying vibrations of metallic structures used in controlled thermonuclear reactors. These problems were solved on the basis of the Kirchhoff hypothesis in [1–5]. In [6], an exact spatial approach to determining the vibration frequencies of thin plates was suggested, and it was shown that it completely agrees with the solution obtained according to the Kirchhoff hypothesis. In [7–9], this exact approach was used to solve the problem on vibrations of thin magnetoelastic plates, and it was shown by cumbersome calculations that the solutions obtained according to the exact theory and the Kirchhoff hypothesis differ substantially except in a single case. In [10], the equations of the dynamic theory of elasticity in the axisymmetric problem are given. In [11], the equations for the vibration frequencies of thin ferromagnetic plates with arbitrary conductivity were obtained in the exact statement. In [12], the Kirchhoff hypothesis was used to obtain dispersion relations for a magnetoelastic thin shell. In [5, 13–16], the relations for the Maxwell tensor and the ponderomotive force for magnetics were presented. In [17], the dispersion relations for thin ferromagnetic plates in the transverse field in the spatial statement were studied analytically and numerically.In the present paper, on the basis of the exact approach, we study free bending vibrations of a thin ferromagnetic cylindrical shell. We obtain the exact dispersion equation in the form of a sixth-order determinant, which can be solved numerically in the case of a magnetoelastic thin shell. The numerical results are presented in tables and compared with the results obtained by the Kirchhoff hypothesis. We show a large number of differences in the results, even for the least frequency.     

Stress concentration in a transversely isotropic spherical shell with two circular rigid inclusions

The refined Timoshenko-type theory that takes into account the transverse shear strains is used to find an analytic solution for the stress state of transversely isotropic shallow spherical shell with two circular rigid inclusions. The case of a shell with closely spaced rigid inclusions of unequal radii under internal pressure is analyzed numerically. The stresses in the shell increase considerably with decrease in the distance between the inclusions and increase in the transverse shear parameter     

On the Form of the Contact Reaction in a Solid Circular Plate Simply Supported Along Two Antipodal Edge Arcs and Deflected by a Central Transverse Concentrated Force

A static, purely flexural mechanical analysis is presented for a Kirchhoff solid circular plate, deflected by a transverse central force, and bilaterally supported along two antipodal periphery arcs, the remaining part of the boundary being free. Two kinds of contact reactions are considered, namely the case of distributed reaction force alone, and the situation in which the distributed force is added to a distributed couple of properly selected profile. For both cases this plate problem is formulated in terms of an integral equation of the Prandtl type, coupled with two constraint conditions. The existence of solutions in an appropriate scaled weighted Sobolev space is discussed, and the behaviour of the solution at the endpoints of the support is exhibited. This revised version was published online in June 2006 with corrections to the Cover Date.     

Solid Circular Plate Clamped along Two Antipodal Edge Arcs and Deflected by a Central Transverse Concentrated Force

A static, purely flexural mechanical analysis is presented for a Kirchhoff solid circular plate, deflected by a transverse central force, and clamped along two antipodal arcs, the remaining part of the boundary being free. By adopting an integral formulation, the contact reaction is assumed to be formed by four equal concentrated forces acting at the support extremities, accompanied by two distributed moments with radial and circumferential axis, respectively. This plate problem is rephrased in terms of a complex-valued Hilbert singular integral equation of the second kind, whose solution is obtained in analytical, integral form. A design chart is presented that reports the plate central deflection as a function of the angular width of the plate supports.     

Solvability of boundary value problems in the geometrically and physically nonlinear theory of shallow shells of Timoshenko type

This paper deals with the proof of the existence of solutions of a geometrically and physically nonlinear boundary value problem for shallow Timoshenko shells with the transverse shear strains taken into account. The shell edge is assumed to be partly fixed. It is proposed to study the problem by a variational method based on searching the points of minimum of the total energy functional for the shell-load system in the space of generalized displacements. We show that there exists a generalized solution of the problemon which the total energy functional attains its minimum on a weakly closed subset of the space of generalized displacements.     

Stress state and strength of an inhomogeneous plastic strip with a defect in a stronger part

We consider discontinuous solutions of a boundary value problem for a system of plastic equilibrium equations under plane strain and use these solutions to study the stress state and strength of an inhomogeneous strip with a defect in the form of a transverse cut in a stronger part of the junction.     

Asymptotic Analysis of the Stresses in Thin Elastic Shells

. In this paper, we identify the limiting stress tensor of a thin elastic shell whose thickness approaches zero. In the linear case, we use a convergence result established for the displacement field in order to get the convergence of the contravariant components of the linearised stress tensor. In the nonlinear case, the identification of the first Piola‐Kirchhoff stress tensor is made through a formal asymptotic analysis. In both cases, we show that these limiting stress fields depend on the geometry of the shell and on the boundary conditions. (Accepted June 1, 1998)     

Removal of shallow shell restriction from Touratier kinematic model

A mathematical model to quantify the variation of the displacement field through the thickness of a laminated shell has been proposed previously by Beakou and Touratier (1993). Transverse shear deformations were taken into account and thickness correction factors were not required but the analysis was restricted to shallow shells (i.e. the principal radii of shell curvature were both assumed to be large relative to the shell thickness). In this technical note the later restriction is removed by replacing the Cartesian coordinate form of elasticity tensor with the more general curvilinear form. The modified laminate level model coefficients are derived from the expressions for the curvilinear transverse shear stress components by: applying zero transverse shear stress boundary conditions at the top and bottom of the shell and enforcing interlayer layer continuity at each internal lamina interface. The purpose of this model enhancement is to facilitate the development of a degenerate finite element type that can be used to compute the deformation of a non-shallow laminated shell.     

Reinforcement of a plate with a circular hole by an eccentric circular patch

We study the reinforcement of an infinite elastic plate with a circular hole by a larger eccentric circular patch completely covering the hole and rigidly adjusted to the plate along the entire boundary of itself. We assume that the plate and the patch are in a generalized plane stress state generated by the action of some given loads applied to the plate at infinity and on the boundary of the hole. We use the power series method combined with the conformal mapping method to find the Muskhelishvili complex potentials and study the stress state on the hole boundary and on the adhesion line. We consider several examples, study how the stresses depend on the geometric and elastic parameters, and compare the problem under study with the case of a plate with a circular hole without a patch. In scientific literature, numerous methods for reinforcing plates with holes, in particular, with circular holes, have been studied. In the monographs [1, 2], the problem of reinforcing the hole edges by stiffening ribs is solved. Methods for reinforcing a circular hole by using two-dimensional patches pasted to the entire plate surface are studied in [3, 4]. The case of a plate with a circular cut reinforced by a concentric circular patch adjusted to the plate along the boundary of itself or along some other circle was studied in [5, 6]. The reinforcement of an elliptic hole by a confocal elliptic patch was considered in [7].     

Analysis of right closed thin-walled prismatic shells supported by stringers with geometric and physical nonlinearity taken into account

We consider thin-walled right-angle closed prismatic shells with rigid contour of the transverse cross-section. Such shells underlie the schemes used in the analysis of various thin-walled spatial structures. The use of nonlinear physical and geometric relations in the computations permits numerically obtaining the strength margin of the corresponding structures. In the present paper, we propose methods for obtaining a boundary value problem and analyzing such shells with nonlinear factors taken into account; the problem is presented as a system of linear differential equations with variable coefficients. We show that, within the approach proposed, this boundary value problem has a fixed structure independent of the special form of nonlinearity. The entire variety of problems of static analysis of right-angle prismatic shells with nonlinear factors taken into account can be reduced to solving this boundary value problem. Methods for taking a specific nonlinearity into account are treated as various methods for obtaining expressions for the variable coefficients in the matrices of the boundary value problem. We present methods for solving this boundary value problem numerically; these methods are independent of the specific form of the nonlinearity.     

Analysis of the stress-strain state of an anisotropic plate with an elliptic hole and thin elastic inclusions

We solve the problem of determining the stress-strain state of an anisotropic plate with an elliptic hole and a system of thin rectilinear elastic inclusions. We assume that there is a perfect mechanical contact between the inclusions and the plate. We deal with a more precise junction model with the flexural rigidity of inclusions taken into account. (The tangential and normal stresses, as well as the derivatives of the displacements, experience a jump across the line of contact.) The solution of the problem is constructed in the form of complex potentials automatically satisfying the boundary conditions on the contour of the elliptic hole and at infinity. The problem is reduced to a system of singular integral equations, which is solved numerically. We study the influence of the rigidity and geometry parameters of the elastic inclusions on the stress distribution and value on the contour of the hole in the plate. We also compare the numerical results obtained here with the known data.     

Studies of nonlinear deformation and stability of stiffened oval cylindrical shells under combined loading by bending moment and boundary transverse force

The finite-element statement of stability problems for stiffened oval cylindrical shells is presented with the moments and the nonlinearity of their subcritical stress-strain state taken into account. Explicit expressions for the displacements of elements of noncircular cylindrical shells as solids are obtained by integration of the equations derived by equating the linear deformation components with zero. These expressions are used to construct the shape functions of the effective quadrangular finite element of natural curvature, and an efficient algorithm for studying the shell nonlinear deformation and stability is developed. The stability of stiffened oval cylindrical shells is studied in the case of combined loading by a boundary transverse force and a bending moment. The influence of the shell ovality and the deformation nonlinearity on the shell stability is investigated.     

Optimal design of axisymmetric plastic shallow shells of von Mises material

An optimal design technique developed earlier for axisymmetric plates and circular cylindrical shells is accommodated for shallow spherical shells subjected to uniform transverse pressure. Material of the shells is assumed to be rigid-plastic obeying the von Mises yield condition and the associated deformation law. The post-yield behaviour of the shells is taken into account. The weight minimization is performed under the condition that the maximal deflection of the shell of variable thickness coincides with the deflection of the reference shell of constant thickness. The problem is transformed into a non-linear boundary value problem which is solved numerically.     

Application of the angular superposition method to the contact problem on the compression of an elastic cylinder

The angular superposition method is used to construct an approximate solution of the contact problem on the compression of an elastic cylinder by two rigid plates. The solution thus obtained has a closed-form analytic expression and can be used in the entire domain of the cylinder cross-section. We analyze the absolute error, which takes the largest value near the points of contact between the plates and the cylinder, where the boundary conditions are discontinuous. According to the von Mises criterion, when moving into the depth of the cylinder from the contact site along the symmetry axis, the second invariant J ₂ of the stress deviator tensor first decreases and then, after attaining a minimum, increases and attains the largest value at a small depth, which agrees with Johnson’s photoelastic experiments and Dinnik’s computations. We present the graphs of the displacement and normal stress distributions over the contact site, the dependence of the compressing force on the displacements of rigid plates, and the dependence of the invariant J ₂ on the coordinate along the symmetry axis. If 640 computation points are chosen on the cylinder boundary and the Hertz law for the normal pressure on the contact site is used, then the error in the approximate solution near the endpoint of the contact site is approximately 55%, and if the proposed two-parameter normal law is used, then the error is of the order of 4%. On the free lateral surface of the cylinder boundary, we find the critical pointM*, which separates the cylinder contraction and extension parts.The contact problems are the most difficult problems, and their solution is complicated by the discontinuous boundary conditions [1–5]. In [6], the contact problem is solved by the Fourier method, which can be used only for bodies of classical shapes. In such cases, the problem can be reduced to solving coupled integral equations [7]. The interaction between the bandage and a cylindrical body is considered in [2, 6, 7]. In [8], the possibility of using the finite element method is investigated in the case of contact problems for a differential wheel with roughness of the contacting surfaces taken into account. In [9, 10], the method of homogeneous solutions is used to consider contact problems for a finite-dimensional elastic cylinder loaded on its end surfaces. Note that only error estimates are given in the literature cited above; the absolute error over the entire domain of the elastic body is not studied, although this is one of the important characteristics of the obtained approximate solution. A sufficiently complete survey of the literature in the field of contact interactions of elastic bodies is given in [3–5].In what follows, we propose to solve contact problems by the angular superposition method [11]. This method can be used for bodies of nonclassical shapes, which can be multiply connected, and the friction on the contact site can be taken into account. In the present paper, as a first example of applied character, we show how this method can be used in the simplest case. The multiple connectedness and the curvilinearity of the shape of the body, as well as taking into account the friction on the boundary, do not create new essential difficulties in this method.