On a walk in a strip with inhibitory boundaries

In the paper we have obtained ergodic theorems for walks generated by sums of stationarily connected random variables and two inhibitory boundaries. We have found representations for the steady-state distributions, permitting us to obtain in the case of independent summands a whole series of useful formulas. We discuss the connection of the results established with problems from queueing theory.     

Two problems with mixed boundary conditions for an elastic orthotropic strip

Two problems are considered for an elastic orthotropic strip: the contact problem and the crack problem. Both problems are reduced to integral equations of the first kind with different kernels, containing a singularity: logarithmic for the first problem and singular for the second problem. Regular and singular asymptotic methods are employed to construct approximate solutions of these integral equations. Numerical results are presented.     

Dispersion of normal waves in a linearly orthotropic circular cylinder with a clamped lateral surface

We propose a method of obtaining the dispersion equation for normal waves in an orthotropic cylinder from the boundary conditions on the rigidly clamped boundary using a system of exponential particular solutions of the three-dimensional equations of its stationary wave motions. We compute the real and imaginary branches of the dispersion spectrum for a waveguide made of monocrystalline strontium sulfate. On figure. Bibliography: 7 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 26, 1996, pp. 96–99.     

Nonreflecting distortions of an isotropic strip clamped between rigid punches

The sides of a homogeneous and isotropic strip are considered that are in ideal contact with infinite rigid punches. Smooth distortions of the sides are constructed so that, at one or several given frequencies, an elastic wave propagating through waveguide gives rise to only exponentially decaying perturbations and, possibly, to a small phase shift.     

The vibration and stability of non-homogeneous orthotropic conical shells with clamped edges subjected to uniform external pressures

In this paper an analytical procedure is given to study the free vibration and stability characteristics of homogeneous and non-homogeneous orthotropic truncated and complete conical shells with clamped edges under uniform external pressures. The non-homogeneous orthotropic material properties of conical shells vary continuously in the thickness direction. The governing equations according to the Donnell’s theory are solved by Galerkin’s method and critical hydrostatic and lateral pressures and fundamental natural frequencies have been found analytically. The appropriate formulas for homogeneous orthotropic and isotropic conical shells and for cylindrical shells made of homogeneous and non-homogeneous, orthotropic and isotropic materials are found as a special case. Several examples are presented to show the accuracy and efficiency of the formulation. The closed-form solutions are verified by accurate different solutions. Finally, the influences of the non-homogeneity, orthotropy and the variations of conical shells characteristics on the critical lateral and hydrostatic pressures and natural frequencies are investigated, when Young’s moduli and density vary together and separately. The results obtained for homogeneous cases are compared with their counterparts in the literature.     

Anti-plane analysis of an orthotropic strip weakened by several moving cracks

In this paper several finite cracks with constant length (Yoffe-type crack) propagating in an orthotropic strip were studied. The distributed dislocation technique is used to carry out stress analysis in an orthotropic strip containing moving cracks under anti-plane loading. The solution of a moving screw dislocation is obtained in an orthotropic strip by means of Fourier transform method. The stress components reveal the familiar Cauchy singularity at the location of dislocation. The solution is employed to derive integral equations for a strip weakened by moving cracks. Finally several examples are solved and the numerical results for the stress intensity factor are obtained. The influences of the geometric parameters, the thickness of the orthotropic strip, the crack size and speed have significant effects on the stress intensity factors of crack tips which are displayed graphically.     

The eigenfrequencies of a slightly curved isotropic strip clamped between absolutely rigid profiles

The propagation of elastic waves along an isotropic homogeneous strip, distorted on a finite part and clamped without friction and detachment between two absolutely rigid profiles, is investigated. It is established that, by choosing the shape of the curved sides one can obtain eigenfrequencies, accompanied by trapping of the elastic wave. The result is based on an asymptotic analysis of an auxiliary object – an augmented scattering matrix, which was previously constructed exclusively for scalar problems.     

On the finite integral transform method for exact bending solutions of fully clamped orthotropic rectangular thin plates

Exact bending solutions of fully clamped orthotropic rectangular thin plates subjected to arbitrary loads are derived using the finite integral transform method. In the proposed mathematical method one does not need to predetermine the deformation function because only the basic governing equations of the classical plate theory for orthotropic plates are used in the procedure. Therefore, unlike conventional semi-inverse methods, it serves as a completely rational and accurate model in plate bending analysis. The applicability of the method is extensive, and it can handle plates with different loadings in a uniform procedure, which is simpler than previous methods. Numerical results are presented to demonstrate the validity and accuracy of the approach as compared with those previously reported in the bibliography.     

Interaction between two neighboring circular holes in a prestretched simply supported orthotropic strip under bending

In this work, the influence of initial stretching of a simply supported plate-strip containing two circular holes on the stress concentration around the holes caused by bending of the strip is examined using the finite-element method. The mathematical formulation of the corresponding boundary-value problem is presented within the frame work of the three-dimensional linearized theory of elasticity (TDLTE) under a plane strain state. The material of the plate-strip is linearly elastic, homogeneous, and orthotropic. The numerical results obtained in investigating the influence of the initial stretching and the location of holes on the stress concentration are presented. In particular, it is established that the initial stretching significantly decreases the stress concentration at some characteristic points on the contour of the holes. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 44, No. 6, pp. 827–838, November–December, 2008.     

Elastic buckling and vibration analyses of orthotropic nanoplates using nonlocal continuum mechanics and spline finite strip method

This paper addresses the elastic buckling and vibration characteristics of isotropic and orthotropic nanoplates using finite strip method. In order to consider small scale effect, Eringen’s nonlocal continuum elasticity is employed. The governing nanoplate equations are derived using the principle of virtual work while B3-spline finite strip method is applied to the buckling and vibration analyses. The buckling load and vibration frequency of graphene sheets, which are subjected to biaxial compression and pure shear loading, are determined whilst the effects of different parameters such as sheet size, nonlocal parameter, aspect ratio and boundary conditions are investigated. The interaction curves of the critical biaxial compression loading as well as the interaction curves of the critical uniaxial compression and shear loading are also obtained. It is shown that small scale effect plays considerable role in the analysis of small sizes plates.     

Postbuckling behavior of uniformly compressed circular arches with clamped ends

Summary This communication presents a relatively simple formula for the prediction of the initial postbuckling behavior (in the sense of Koiter) of hingeless elastic circular arches of uniform crosssection subjected to a uniform external gas pressure. Calculations are performed on the basis of an asymptotic method which contains elements of the techniques of analysis used by Budiansky and Hutchinson, and Chien.

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Zusammenfassung Diese Mittelung befaßt sich mit dem nachkritischen Verhalten (im Sinne von Koiter) des unter dem Gasdruck stehenden, eingespannten, elastischen Kreisbogenträgers. Eine verhältnismäßig einfache, genaue Berechnungsformel für die anfängliche nachkritische Belastung wird mittels eines Störungsverfahrens abgeleitet.

     

Cylindrically orthotropic circular plates with large deflections

Zusammenfassung Es werden die Verschiebungs- und Spannungszustände in einer Kreisplatte mit grossen Durchbiegungen und zylindrischer Orthotropie mit Hilfe zweier Methoden untersucht, nämlich der Methode des kleinen Parameters vonPoincaré und einer modifizierten Methode vonGalerkin.Ein numerisches Beispiel für gleichmässige Belastung der Platte, die Poissonsche Zahlv =1/4 und die Beziehungv :v _r =4 wird gegeben.

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This work has been sponsored by the United States Army under contract No. DA-11-002-ORD-2059     

Bounding eigenvalues of clamped plates

Zusammenfassung Es wird ein Verfahren zur Berechnung von oberen und unteren Schranken für die Eigenwerte einer eingespannten Platte beliebiger Form dargestellt. Die Methode besteht darin, exakte Partikulärlösungen der Plattengleichung ( ² u=u) zu konstruieren, welche die Randbedingungen des Plattenproblems approximativ erfüllen. Der Abstand zwischen dem Eigenwert der Partikulärlösung und einem Eigenwert des Plattenproblems kann dann mit Hilfe eines Lemmas vonMoler undPayne abgeschätzt werden.Als Beispiele wurden die ersten Eigenwerte der quadratischen Platte sowie einiger elliptischer Platten berechnet. Für eine spezielle elliptische Platte sind die ersten vier Eigenwerte angegeben.     

L2-flow of elastic curves with clamped boundary conditions

In this paper we investigate the $L^2$-flow of elastic curves with clamped boundary conditions in n-dimensional Euclidean spaces. The $L^2$-flow corresponds to a fourth-order parabolic equation. In the case of closed curves, the long-time existence of solutions of this evolution equation has been derived in the literature. We extend this result to the case of open (i.e., non-closed) curves with clamped boundary conditions.