First Term Of Asymptotic Expansion Research Articles

Analysis of the three‐dimensional elasticity theory problem for a radially inhomogeneous cylinder

AbstractIn this paper an axisymmetric problem of elasticity theory for a radially inhomogeneous cylinder of small thickness is studied. It is assumed that homogeneous mixed boundary conditions are specified on lateral surfaces of the cylinder, and boundary conditions that leave the cylinder in equilibrium are specified on the ends of the cylinder. It is considered that elastic moduli are arbitrary continuous functions of a variable along the radius of the cylinder. The formulated boundary value problem is reduced to a spectral problem containing a small parameter characterizing the thinness of the cylinder walls. To determine its solution, the method of asymptotic integration is used, based on three iteration processes. All solutions of the equilibrium equation are constructed, satisfying the specified homogeneous boundary conditions on lateral surfaces. It is obtained that the solution of the problem consists of a penetrating solution and a solution of the nature of the boundary layer. An analysis of stress‐strain states corresponding to different types of solutions is carried out. Based on the constructed asymptotic solutions, a connection is found between the solutions obtained by the equation of elasticity theory and by the applied theory of shells. It is shown that the penetrating solution determines the internal stress‐strain state of a radially inhomogeneous cylinder. The stress state determined by the penetrating solution is equivalent to the principal vector of stress acting in a cross section perpendicular to the axis of the cylinder.The first terms of asymptotic expansions of the penetrating solution in a small parameter coincide with solutions in the applied theory of shells. New classes of solutions are defined as the character of a boundary layer, which are localized at the ends of the cylinder and decrease exponentially with distance from the ends. These solutions are absent in the applied theories of shells. The first terms of the asymptotic expansion of the solution, having the nature of a boundary layer, are equivalent to the Saint‐Venant edge effect in the theory of heterogeneous plates. Asymptotic formulas for displacement and stresses are constructed, allowing one to calculate the three‐dimensional stress‐strain state of a radially heterogeneous cylinder of small thickness with any predetermined accuracy. Based on the obtained asymptotic expansions, one can estimate the applicability areas of applied theories and construct a refined applied theory for radially heterogeneous cylindrical shells.

On the spectrum of a two-dimensional periodic operator with a small localized perturbation

We consider a two-dimensional periodic self-adjoint second-order differential operator on the plane with a small localized perturbation. The perturbation is given by an arbitrary (not necessarily symmetric) operator. It is localized in the sense that it acts on a pair of weighted Sobolev spaces and sends functions of sufficiently rapid growth to functions of sufficiently rapid decay. By studying the spectrum of the perturbed operator, we establish that the essential spectrum is stable, the residual spectrum is absent, and the set of isolated eigenvalues is discrete. We obtain necessary and sufficient conditions for the existence of new eigenvalues arising from the ends of lacunae in the essential spectrum. In the case when such eigenvalues exist, we construct the first terms of asymptotic expansions of these eigenvalues and the corresponding eigenfunctions.

On the 2D models of plates and shells including the transversal shear

AbstractThe 2D Kirchhoff–Love (KL) theory and the Timoshenko–Reissner (TR) theory for thin shells made of the transversally isotropic, homogeneous material are discussed. For the cyclic‐symmetric deformations of shells of revolution the asymptotic analysis of stress–strain states is performed. Two simple linear problems for double‐periodic deformations of plates are studied applying the exact 3D theory and the 2D approximate theories. From these problems it follows that the KL theory is asymptotically correct because it gives the first term of asymptotic expansion of the 3D solution. The TR theory is asymptotically incorrect. It gives correctly the first term only and incorrectly the second term. But if the transversal shear modulus is comparatively small then this theory gives the main part of the second term. The case of the extremely small shear modulus is discussed. As an example the multi‐layered plate with alternating hard and soft isotropic layers is studied.

A note on an asymptotic solution for the contact of two biphasic cartilage layers in a loaded synovial joint at rest

Retaining the first terms of asymptotic expansions and assuming zero gradient of the contact pressure at the contact edge in the perpendicular direction to the edge, Ateshian et al. (1994, J. Biomechanics 27, 1347–1360; 1992, Adv. Bioeng. ASME 22, 191–194) have presented an asymptotic axially symmetric and plane strain solutions to the contact problems of the two thin biphasic cartilage layers in the synovial joint (with the synovial fluid film in between already depleted), subjected to a sudden normal load. Both the immediate and an early time response to a step loading have been analysed. The present note shows that the contact width thus obtained immediately after the load application differs from the numerical values for a dry frictionless contact of incompressible single-phase elastic layers available in the literature. The difference increases with the decreasing contact width-to-layer thickness ratio. It is proposed to improve the above solutions by releasing zero pressure gradient condition at the contact edge and calculating the instantaneous contact widths from the equations proposed for the corresponding case by Matthewson (1981, J. Mech. Phys. Solids 29, 89–113) and Meijers (1967, Appl. Sci. Res. 18, 353–383) that approximate better the numerical values. The instantaneous rate of change in the contact width is then obtained for a step load varying in time.