Plane Elastic Plate Research Articles

Modelling the inhomogeneous coating of an elastic plate with optimum sound-reflecting properties

The problem of determining the laws governing the inhomogeneity of the coating of a plane elastic plate that ensure the least reflection with a given angle of incidence of a plane sound wave is considered. On the basis of the direct problem solution, a functional representing the intensity of reflection is constructed, and an algorithm for its minimisation is proposed. Analytical expressions describing the mechanical parameters of the inhomogeneous coating are obtained.

Fatigue growth prediction of center slant crack in rectangular plate

ABSTRACTThis paper presents a numerical prediction model of mixed‐mode crack fatigue growth in a plane elastic plate. It involves a formulations of fatigue growth of multiple crack tips under mixed‐mode loading and a displacement discontinuity method with crack‐tip elements (a boundary element method) proposed recently by Yan is extended to analyse the fatigue growth process of multiple crack tips. Due to an intrinsic feature of the boundary element method, a general growth problem of multiple cracks can be solved in a single‐region formulation. In the numerical simulation, for each increment of crack extension, remeshing of existing boundaries is not necessary. Crack extension is conveniently modelled by adding new boundary elements on the incremental crack extension to the previous crack boundaries. At the same time, the element characters of some related elements are adjusted according to the manner in which the boundary element method is implemented. As an example, the present numerical approach is used to analyse the fatigue growth of a centre slant crack in a rectangular plate. The numerical results illustrate the validation of the numerical prediction model and can reveal the effect of the geometry of the cracked plate on the fatigue growth.

A boundary element modeling of fatigue crack growth in a plane elastic plate

This paper presents an extension of a boundary element method to fatigue growth analysis of mixed-mode cracked plane elastic bodies. The method consists of the non-singular displacement discontinuity element presented by Crouch and Starfield and the crack-tip displacement discontinuity element due to the author. In the boundary element implementation the left or the right crack-tip element is placed locally at the corresponding left or right crack tip on top of non-singular displacement discontinuity elements that cover the entire crack surface and the other boundaries. Crack growth is simulated with an incremental crack extension analysis based on the modified maximum strain energy density criterion. In numerical simulation, for each increment of crack extension, remeshing of existing boundaries is not required because of an intrinsic feature of the boundary element method. Crack growth is simulated by adding new boundary elements on the incremental crack extension to the previous crack boundaries. At the same time, the element characters of some related elements are adjusted according to the manner in which the boundary element method is implemented. Some numerical results of fatigue growth in a plane elastic plate with a center-inclined crack under uniaxial cyclic loading are given.

An effective boundary element method for analysis of crack problems in a plane elastic plate

A simple and effective boundary element method for stress intensity factor calculation for crack problems in a plane elastic plate is presented. The boundary element method consists of the constant displacement discontinuity element presented by Crouch and Starfield and the crack-tip displacement discontinuity elements proposed by YAN Xiangqiao. In the boundary element implementation the left or the right crack-tip displacement discontinuity element was placed locally at the corresponding left or right each crack tip on top of the constant displacement discontinuity elements that cover the entire crack surface and the other boundaries. Test examples (i. e., a center crack in an infinite plate under tension, a circular hole and a crack in an infinite plate under tension) are included to illustrate that the numerical approach is very simple and accurate for stress intensity factor calculation of plane elasticity crack problems. In addition, specifically, the stress intensity factors of branching cracks emanating from a square hole in a rectangular plate under biaxial loads were analysed. These numerical results indicate the present numerical approach is very effective for calculating stress intensity factors of complex cracks in a 2-D finite body, and are used to reveal the effect of the biaxial loads and the cracked body geometry on stress intensity factors.

Instability of an Unbounded Elastic Plate in a Gas Flow

The stability of an unbounded plane elastic plate in gas moving on one side of the plate and at rest on the other is analyzed. The gases are inviscid and in general different. The plate is under tension and has flexural stiffness. It is shown that the system is always unstable to plane sinusoidal perturbations with wave vector parallel to the velocity. As limiting cases, a tangential discontinuity between the two gases and unilateral flow past a plate with constant pressure on the opposite side are considered. In these cases, the conditions of stability to plane perturbations are non-trivial and are investigated below.

On the spatial behaviour of the transient and steady-state solutions in thin plates with transverse shear deformation

This paper studies the spatial behaviour of the transient and steady-state solutions in the problem of bending of a plane elastic plate. When the transient solutions are discussed a bounded or unbounded plane plate of an arbitrary regular form is considered. Then an appropriate time-weighted line-integral measure is introduced and, for each fixed t∈[0, T], it is shown that it vanishes at distances from the support of the given data on the time interval [0, T] greater than ct, where c is a characteristic material constant. Moreover, for distances to the support lower than ct, it is established a spatial decay estimate of Saint-Venant-type. For the harmonic vibrations the case of an elastic plate whose middle surface is like a (semi-infinite) strip is considered for which the lateral sides are clamped. Then a cross-sectional line-integral measure is associated with the amplitude of the vibration and the spatial estimate is established for describing the spatial behaviour of the amplitude, provided that the frequency of the harmonic vibration is lower than a critical frequency.

Fluid loading of irregular structures

An analytically and computationally tractable model for the response of complex fluid-loaded structures is that of a plane elastic plate or membrane subject to static fluid loading and single-frequency localized mechanical drive, and reinforced by a number of parallel ‘‘ribs’’ in regular, partially ordered, or random configurations. Coupling along the structure is provided by subsonic surface waves (equivalent to nearest-neighbor mechanical coupling) and by long-range acoustic/hydrodynamic fields. A variety of analytical and computational studies carried out by the Cambridge group will be presented, showing large effects of an isolated irregularity, Anderson localization in extensively irregular configurations under surface wave coupling, and delocalization (short circuiting) by weak long-range fields. [Work supported by ONR, Code 1132SM.]

Long-Range Acoustic Scattering by Surface Inhomogeneities Beneath a Turbulent Boundary Layer

Low-frequency turbulent boundary layer eddies inject a power Ps into the elastic structure over which the boundary layer is formed which generally greatly exceeds the acoustic power Pa directly radiated by the eddies (acoustically equivalent to quadrupoles). The power Ps remains trapped in the surface and an adjacent fluid layer and propagates in subsonic surface wave modes until it encounters a rib, or edge, or other surface inhomogeneity, from which a power Pe is scattered to the far field. While Pe is again small compared with Ps, it may nonetheless greatly exceed Pa, and in that case the dominant acoustic mechanism is associated with the long-range coupling between the quadrupole eddy and the remote inhomogeneity via subsonic surface waves. That interaction, and the acoustic fields produced by it, are examined in detail in this paper for the inhomogeneities represented by a simple line support rib, a simple point support rib, and an edge to a plane elastic plate, either with or without an adjacent rigid baffle, and with a range of edge conditions. Under conditions of “light” fluid loading, the long-range coupling mechanism seems unlikely to be of importance, but at low frequencies and under “heavy” fluid loading it appears that, even for large separations between an eddy and an inhomogeneity, the long-range coupling generates an acoustic field far in excess of that radiated by the same boundary layer turbulence over a homogeneous surface.

An application in hydrodynamics of the Green's function of an elastic plate

Some examples of the bending of a plane elastic plate by transverse forces applied at isolated points are first considered. The plate is infinite and is bounded internally by a circular edge along which it is clamped; simple expressions are found for the displacement. The analogous hydrodynamical problem is that of the steady flow of viscous incompressible liquid past a fixed circular cylinder; the equation for the stream function is in the same form as the equation for the displacement of a plate due to a distributed force of amount Z per unit area. The inertia terms in the hydrodynamical problem correspond to Z . In slow motion there is no stream function with the correct form at infinity, because in this case the inertia terms are ignored so that in the plate problem Z = 0. The effect of a transverse force system can be most simply illustrated by supposing that the forces are concentrated at isolated points; in the corresponding stream functions a simple form of allowance for inertia is therefore made, and they display some of the features of steady flow past a cylinder.

The Green's function of an elastic plate

In this paper a simple expression in finite terms is found for the small transverse displacement of a thin plane elastic plate due to a transverse force applied at an arbitrary point of the plate. The plate is clamped along the semi-infinite straight lines represented by AB, CD in Fig. 1, these lines being the only boundaries of the plate. The transverse displacement w at any point (x, y) of the plate is a biharmonic function of the variables (x, y) which vanishes together with its normal derivative at all points of the boundary. Clearly w is also a function of the coordinates (x0, y0) of the point of application of the force, and it is known ((5), p. 173) that it is a symmetrical function of the coordinate pairs (z, y) and (x0, y0); it is the Green's function associated with the differential equation and the boundary conditions.

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