Deformation of a rectangular plate under impulsive loading

We report the results of the experimental investigation of the behavior of a rectangular plate in the processes of cutting by linear shaped charges according to the axisymmetric and asymmetric modes of unilateral loading and fracture caused by the explosion of a linear charge. It is shown that the transverse velocities and displacements of the same points of the plate subjected to unilateral explosive cutting by shaped charges are much higher (almost by an order of magnitude) than the corresponding parameters for the other modes of fracture. We also observed that the values of longitudinal mass velocities and displacements in the case of bilateral cutting by shaped charges are 1.5–7.0 times higher than the same parameters for unilateral cutting and explosive fracture without using shaped charges. It is also shown that the longitudinal accelerations attain their maximum values in the case where the shaped charge is placed at a focal distance from the plate. Subbotin Institute for Geophysics, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 6, pp. 85–90, November–December, 1998.     

Fourier series solution for a rectangular thick plate with free edges on an elastic foundation

A Fourier series solution is presented for a system of first-order partial differential equations which describe the linear elastic behaviour of a thick rectangular plate resting on an elastic foundation and carrying an arbitrary transverse load. The lateral edges of the plate are unstressed. A central step in the method for solving the system of equations is to combine a complementary function with a particular solution of the system in order to satisfy the boundary conditions. The complementary function is the sum of two series. The terms of the first series are products of a Fourier term in one space variable with the solution of an eigenvalue problem in the other space variable. The second series is similar and comes from reversing the roles of the space variables.     

Twisting of an elastic plate containing a crack

The stress distribution caused by twisting an infinite plate containing a finite crack is analyzed in terms of Reissner's theory for the bending of thin plates. The singular character and the detailed structure of the stresses near the ends of the crack are determined in closed form. Numerical results are given for the magnitudes of the stress couples and stress resultants for a range of plate thicknesses.

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Zusammenfassung Due Spannungsvertcilung, hervorgerufen durch die Torsion einer unendlichen Platte mit einem Ri\ begrenzter Länge, wird mit Hilfe der Reissner-Theorie für die Biegebeanspruchung dünner Platten untersucht. Der singulare Charakter und die genauc Verteilung der Spannungen in Nähe der Ri\enden werden bestimmt. Zahlenmä\ige Ergebenisse für die Gro\e der Spannungsparre und ihrer Resultanten werden für eine Reihe von Plattenstärken angegeben.

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Résumé La distribution des contraintes dans une plague infinie comportant une fisure finic et soumise à torsion est analysée au moyen de la théorie de Reissner pour la flexion des tôles minces.Le caractère singulier, et la structure de détail des contraintes au voisinage des extrémités de la fisure sont explicités.Des résultats numériques sont fournis en ce qui regarde les grandeurs des couples de contraintes et de leurs résultantes, pour une certaine gamme d'éspaisseur de tôles.

     

高速旋转柔性矩形薄板的动力学建模和近似算法

研究了做高速旋转柔性矩形薄板的耦合动力学建模理论和模态截断法的应用.从连续介质力学中关于柔性薄板的变形理论出发,找出了由于在结构动力学中对无大范围运动柔性薄板的动力学性质影响很小而被忽略的变形量.基于Jourdain速度变分原理,先推导出做高速旋转薄板的动力学连续变分方程,再用有限元法对柔性薄板进行离散.因为用有限元离散时,柔性薄板的广义坐标规模较大,故仿真计算需要时间较长,所以模态截断法被用来缩减广义坐标数量,提高计算效率.此外通过对有限元法和模态截断法的计算结果进行比较,揭示了当矩形薄板作高速旋转时,模态截断法截取低阶模态时会引起误差,选取更高的模态可以用来提高计算精度,通过数值对比得到了模态截断的规律.     

Plane-strain deformation of an elastic material compressed in a rough rectangular cavity

Deformation and stress distributions in a linear elastic solid, confined to a rigid cavity with rough walls and subjected to uniform compression from one end, are examined. Wall roughness is modeled by Coulomb friction. At the rigid walls, one boundary condition involves deformation and the other stresses, and this renders the problem non-standard. A Laplace transformation solution is constructed for a semi-infinite cavity, and a computational solution for a cavity with finite length. Agreement between the two solutions is good, and improves with increasing cavity lengths and higher coefficients of friction. There exists a critical value of the coefficient of friction below which the axial displacements decay monotonically with distance from the loaded end and the material points stay in contact with the rough walls. For supercritical values of the coefficient of friction, displacements and stresses on the rough walls exhibit oscillatory behavior in the axial direction. The material loses contact with the walls, and the analytical solution presented here loses validity.     

Crack problems for a rectangular plate and an infinite strip

In this paper, the general plane problem for an infinite strip containing multiple cracks perpendicular to its boundaries is considered. The problem is reduced to a system of singular integral equations. Two specific problems of practical interest are then studied in detail. The first is the investigation of the interaction effect of multiple edge cracks in a plate or beam under tension or bending. The second problem is that of a rectangular plate containing an arbitrarily oriented crack in the plane of symmetry. Particular emphasis is placed on studying the problem of a plate containing an edge crack and subjected to concentrated forces. The plate has the dimensions of a standard compact tension specimen and is intended to simulate the CTS.

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Résumé On considère dans ce mémoire le problème général d'une bande infinie comportant des fissurations multiples perpendiculaires à ses bords. Le problème est réduit à un système d'équations intégrales singulières. Deux problèmes spécifiques d'intérêt pratique sont ensuite étudiés dans le détail. Le premier consiste à étudier l'effet d'interaction de fissures de bord multiples dans une tôle ou dans une poutre sous tension ou en flexion. Le second problème est celui d'une plaque rectangulaire comportant une fissure orientée de manière arbitraire dans son plan de symétrie. Un accent particulier est mis sur l'étude du problème d'une plaque comportant une fissure de bord et soumise à des forces concentrées. La plaque a les dimensions d'une éprouvette standard compacte de traction et est conçue de manière à simuler celle-ci.

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This work was supported by NASA-Langley Research Center under the Grant NGR 39-007-011 and by NSF under the Grant ENG 78-09737.     

Transient SH-waves in an elastic plate with a moving edge

The propagation of transient SH-waves in a plate is discussed. An SH-wave source is placed on a fixed edge of the plate and the other edge moves outward with uniform velocity. Two Laplace transforms in fixed and moving coordinate systems are employed. Developing an exchange formula between two Laplace transforms, a coupled functional equation for two unknown coefficients is derived and solved by iteration. The solution obtained here is exact up to three reflection times of an incident (source) wave, and the time-development of the wave front shape is also discussed.     

Indentation of a laminated composite plate with an interlayer rectangular void

We employ the Eshelby–Stroh formalism to study generalized plane strain infinitesimal deformations caused due to the indentation by a rigid circular cylinder of an elastic laminated plate with a through-the-width rectangular void between two adjoining layers. Assuming that the void does not close during the indentation process, we find the indentation modulus (i.e., the slope of the indentation load vs. the indentation depth curve) as a function of the void size, the void position, elastic moduli of the layers, and boundary conditions at the edges. The change in the indentation modulus caused by an interlayer void parallel to the major surfaces of an anisotropic plate can potentially be used to estimate the void size and location.     

Axisymmetric bending of a heated circular plate on an elastic base with account of its deformability over its thickness

The axisymmetric problem is solved for the bending of a circular plate on a heated half-space under the action of a distributed load and a temperature field.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 48, No. 6, pp. 1004–1008, June, 1985.     

Brittle detachment of a stiffener bonded to an elastic plate

The diffusion of contact stresses between an elastic bar, bonded to an elastic half-plane and loaded longitudinally, requires the integration of a singular integral equation. The solution of this equation is not available in closed form, but only by a series expansion of the contact tangential force mutually transmitted between the stiffener and the plate. Since forty years it has been realized that the expansion of the solution in terms of Chebyshev polynomials is its most convenient method of representation. The procedure can also be extended to treat the brittle detachments of the tips of the stiffener when, according to Griffith's criterion of fracture, a balance can be virtually established between the increase of strain energy due to a propagation of cracks and the surface energy created.     

Interaction of a turbulent boundary layer with an elastic plate

An approximate method is presented for the determination of the efficiency with which an elastic plate reduces turbulent friction, and experimental results are also presented.Deceased.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 56, No. 2, pp. 220–225, February, 1989.     

弹性地基上四边自由矩形薄板的自由振动

将弹性地基用Winkler模型来代替。首先把弹性地基上矩形薄板的动力学方程表示成为Hamilton正则方程，然后采用辛几何方法对全状态相变量进行分离变量，并利用得到的共扼辛正交归一关系，求出弹性地基上四边自由矩形薄板的固有频率和振型的解析解表达式。由于在求解过程中不需要事先人为的选取挠度函数，而是从弹性地基上矩形薄板的动力学基本方程出发，直接利用数学的方法求出可以满足四边自由边界条件的固有频率和振型的解析解表达式，使得问题的求解更加合理化。文中的最后还给出了计算实例来验证本文所采用的方法以及所推导出公式的正确性。     

The instability of a rectangular elastic solid

Summary The paper starts with a critical survey of recent investigations of related problems. The title problem shown in Fig. 2 is then solved whenq is constant directional and whenq is hydrostatic, using the complete formulation given byNovozhilov with the only assumption being that the elongations and shears are small compared to unity. Since the angles of rotation were retained in the formulation, and the obtained expressions did reduce for thin plates to the correspondingEuler-type expressions, it is concluded thatLegenya's suggestion, that the classic stability theory of plates and shells should be reconsidered taking into consideration the angles of rotation, is based on erroneous results. The results of the present paper confirm that whenq=p the solid can become unstable which contradicts the findings of other investigators. The paper concludes with a comparison of buckling pressure versus slenderness ratio curves which are based on results of a number of investigators. These graphs demonstrate that the classical theory of buckling of thin plates (and beams) yields accurate results up toh/L=0.15, thus verifying the validity of a usual assumption of structural mechanics.

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Zusammenfassung Die Arbeit beginnt mit einer kritischen Übersicht über die neueren Untersuchungen verwandter Probleme. Das gestellte Problem, in Fig. 2 erläutert, wird im folgenden für richtungstreue und für hydrostatische Querlastq gelöst, wobei die vonNovozhilov angegebene Formulierung benützt wird und die Dehnungen sowie die Schubverformungen klein gegen eins angenommen werden. Da die Drehwinkel in der Rechnung behalten werden und sich die erhaltenen Ausdrücke für dünne Platten auf entsprechendeEulersche zurückführen lassen, wird gefolgert, daß derLegenyasche Vorschlag, die klassische Stabilitätstheorie sollte durch Hineinnahme des Drehwinkeleinflusses ergänzt werden, auf fehlerhaften Ergebnissen beruht. Das hier gefundene Ergebnis, daß der Quader fürq=p instabil werden kann widerspricht den Ergebnissen anderer Autoren. Die Arbeit schließt mit einem Vergleich von Knicklast-Schlankheitsgrad-Kurven, die aus den Resultaten verschiedener Autoren gewonnen wurden. Die Schaubilder zeigen, daß die klassische Beul-(Knick)-Theorie dünner Platten (und Stäbe) genaue Ergebnisse bis zum Schlankheitsgradh/L=0,15 liefert und bestätigen so die Gültigkeit einer in der Baumechanik üblichen Annahme.

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The results of the present paper were obtained in the course of research conducted under NASA grant NSG-420. Parts of the presented results are also included in a dissertation submitted by the junior author (ST) to New York University.