Semi-infinite Elastic Plate Research Articles

Diffraction of a flexural wave by a short joint of semi-infinite elastic plates

The problem of the flexural vibrations of two semi-infinite elastic plates connected along a section of the boundary (the joint) that is short compared with the wavelength of the incident wave, is considered. The problem is reduced to solving integral equations on the section. The use of Green's formula leads to an integral equation with a smooth kernel, the solution of which is a function with singularities of order— 3 2 at the ends of the section. Regularization of this integral equation is carried out. The asymptotic form of the far field over the dimensionless length of the joint is found.

Subsidence history, gravity anomalies, and flexure of the northeast Australian margin in Papua New Guinea

Gravity and well data have been used to model the foreland basin flanking the Papua New Guinea orogenic belt. Backstripping of sediments from well data reveals a former passive margin that has probably experienced at least two major orogenic loading events. The first of these took place at or near the Miocene—Oligocene boundary and was associated with widespread carbonate deposition across the existing shelf and evolving basin. The second event is most likely related to the start of the main phase of uplift in the fold‐and‐thrust belt near the beginning of the Pliocene. Sedimentation in the Papuan Basin altered from a system dominated by carbonate accumulation to one of rapid deposition of terrigenous clastic sediments shed from the emerging orogen. For the purposes of gravity and flexure modeling, the base of the foreland sequence was defined as a horizontal surface that existed at the time of the initial collision. A total of seven gravity, topography, and stratigraphie profiles were constructed and used to model the flexure of the lithosphere supporting the onshore Papuan Basin and New Guinea orogen. The best fit model was a semiinfinite elastic plate in which the plate break was located approximately midway across the orogenic belt. The elastic plate thickness Te was found to be high in the western regions of Papua New Guinea (Te of 70–80 km) with little variation across the first four profiles, but it weakened considerably to the east where Te varied rapidly from 60 km to as little as 8 km. This sudden weakening of the lithosphere occurred as a narrow zone at the foot of the mountain belt. The weakening could be related to a low‐strength zone within the former margin. Alternatively, it reflects the western extent of late Tertiary volcanism that was produced by the subduction of the Solomon Sea Plate beneath Australia.

Incidence of flexural gravitational waves on the line of contact of two floating ice plates of different thickness

We analyze the influence of a rift in an ice field on the propagation of flexural gravitational waves in a basin of finite constant depth. The ice cover is simulated by two floating semiinfinite elastic plates of different thickness. We studied the dependence of the amplitude coefficients of reflection and transmission of waves incident on the rift on the frequency of running waves, the thickness of ice on both sides of the rift, and the type of contact boundary conditions at the rift.

Structural and acoustical wave interaction at a wedge-shaped junction of fluid-loaded plates

Two semi-infinite elastic plates are joined along a line forming a wedge structure with unilateral fluid loading in the outer sector of the two plates. The structure is modeled using thin plate theory, allowing freely propagating flexural and longitudinal waves. The junction is mechanically connected with an applied force and moment acting there to simulate a possible internal connection. The general two-dimensional solution is described for incidence of time harmonic structural or acoustical waves. The total pressure is expressed as a Sommerfeld integral, the integrand comprising Malyuzhinets functions and particular solutions of certain difference equations. The junction conditions reduce to a system of four linear equations. Numerical examples indicate the coupling between the modes for welded steel plates in water. Acoustic plane-wave incidence on the flattest junction considered is converted almost equally, in terms of energy, among diffracted flexural and longitudinal waves. The coupling to flexural energy increases with the angle of the water wedge sector, at the expense of the longitudinal energy which vanishes in the limit as the water occupies the entire domain except for a strip comprising the two plates. An incident longitudinal wave generates relatively little acoustic sound for all wedge angles considered, with most of its energy redistributed among structural modes. The acoustical diffraction is generally greater for flexural incidence. Comparison with the dry structural diffraction coefficients indicates that the fluid loading effects for steel and water configurations can be significant for frequencies less than one-fifth of the coincidence frequency, but are small for higher frequencies.

Analytical solutions for 3-D flexural deformation of semi-infinite elastic plates

SUMMARY The theory of elastic-plate flexure has played a prominent role in isostatic studies and investigations of lithospheric deformation caused by vertical tectonism. Simple analytical, asymmetrical solutions exist when the boundaries of the plate are far removed from the area where the load is applied. When deformation is driven by end loads (i.e. bending moments and shear forces) the problem is usually simplified to yield a 2-D solution that approximates a cross-section of the 3-D solution. We present analytical Green's functions for the point-force response of semi-infinite elastic plates resting on an inviscid substratum and being acted upon by constant in-plane forces and arbitrary end loads. This solution may be used to study deformation and stresses caused by loads close to a free geological boundary (e.g. a trench or weak fracture zone). We also present Green's functions for the case when two semi-infinite plates of different rigidities are mechanically coupled and subject to a point load. This solution may be useful when studying deformation near mechanically strong fracture zones or at ocean-continent transitions. The analytical solutions provide simple alternatives to (and calibration of) complex numerical finite-element or finite-difference solutions, and furthermore give additional physical insight into the geological process to be examined. The analytical methods are used to demonstrate the different stress patterns that may arise near weak and strong geological boundaries. We also illustrate how the observed scatter in 2-D determinations of elastic thickness at subduction zones may partly have its origin in 3-D effects.

The compact Green’s function for a semi-infinite elastic plate, with application to trailing edge noise and blade–vortex interaction noise

Analytic representations are derived for the acoustic Green’s function governing the production of aerodynamic sound by sources near the edge of a thin, semi-infinite elastic plate. Simple closed-form expressions are obtained for the compact Green’s function, which is applicable when the length scale of the source flow near the edge is much smaller than both the acoustic and bending wavelengths. This requires that the characteristic flow Mach number M and frequency f satisfy M2≪f/fc≪1, where fc is the coincidence frequency, above which the bending wave speed in vacuo exceeds the speed of sound in the fluid. At such frequencies the flexural response of the plate can significantly affect the level of sound generation by sources in the neighborhood of the edge. This is illustrated by applications to sound production by (i) turbulent boundary layer flow over an elastic trailing edge (which includes a comparison with more complete predictions applicable for arbitrary values of f/fc≲1), and (ii) the interaction of a rectilinear vortex with the leading and trailing edges of a large plate. Comparisons with analogous predictions for a rigid edge indicate that surface compliance can result in a significant reduction in the intensity of edge-generated sound.

Stress Distribution in an Edge-Stiffened Semi-infinite Elastic Plate Containing a Circular Hole

An edge-stiffened semi-infinite elastic plate containing a circular hole under tension at infinity has been studied using a conformal mapping technique. Melan ’s stiffener condition in Cartesian coordinates has been newly formulated as a resultant shear stress condition in bipolar coordinates. The solution is sought in series form, and by truncating the system of equations, it is numerically solved. Pertinent stress distributions are examined to illustrate the roll of the stiffener under the presence of a circular hole near the straight edge. It has been concluded that the stiffener contributes to indirectly suppressing the stress concentration around the hole.

The propagation of time harmonic Rayleigh–Lamb waves in a bimaterial plate

The time harmonic elastodynamic response of two semi-infinite elastic plates of dissimilar material properties perfectly bonded along their lateral faces is studied. The wave field in either half-plate can be written as a superposition of the so-called Rayleigh–Lamb eigenmodes of an infinite plate. The interaction of a time harmonic incident wave with the interface results in reflected and transmitted fields that contain contributions from all of the real, imaginary, and complex eigenmodes of an infinite plate. Attention is focused on the distribution of energy among the various reflected and transmitted eigenmodes over a range of frequencies. The fundamental symmetric and the fundamental antisymmetric Lamb modes are each used as input excitations. Such excitations can be approximately realized in experiments. It is assumed that the solution of such a canonical problem will facilitate the solution of problems with complicated time-dependent sources.

Linear waves in a fluid flow with constant vorticity located under an ice blanket

The linear dynamics of periodic waves on the surface of a fluid layer of finite depth located under an ice blanket which is simulated by an elastic plate is considered. The fluid particles in the unperturbed state move at a constant horizontal velocity, the profile of which has a linear shift along the vertical. It is shown that several type of waves exist which propagate at the same frequency. The number of waves depends on the frequency, the flow parameters in the fluid and the physico-mechanical parameters of the ice blanket. The problem of the diffraction of waves of fixed frequency on the edge of a semi-infinite elastic plate which floats on the surface of the fluid is considered. The problem is reduced to the solution of Laplace's equation in the strip with specified asymptotic forms at infinity and with boundary conditions on the sides of the strip which have a discontinuity at a point corresponding to the edge of the ice and contact-boundary conditions on the edge of the plate. The solution is constructed using the Wiener-Hopf method. The reflection and transmission coefficients of the waves across the edge of the plate are determined. The results obtained are analysed using the actual parameters of sea ice.

Isochromatic curves for a dynamically loaded elastic plate

Isochromatic curve patterns are calculated for the dynamic stress field in a semi-infinite thin elastic plate whose edge is subject to a point load.

Low-frequency diffraction from a free surface coupled to a semi-infinite elastic surface as modeled by sea ice properties

This paper discusses the solution of a low-frequency plane wave incident upon a semi-infinite elastic plate, such as an Arctic ice lend or free edge, using the Wiener-Hopf method. By low-frequency it is meant that the elastic properties of the plate are adequately described by the thin plate equation. For example, in a floating ice sheet, this translates into frequency-ice thickness products that are ≲ 150. A key issue here is the fluid loading pertaining to sea ice and low-frequency acoustics, which cannot be characterized by simplifying heavy or light fluid loading limits. An approximation to the exact kernel of the Wiener-Hopf functional equation is used here, which is valid in this midrange fluid loading regime. The farfield diffracted pressure is found, which includes a fluid-loaded, sub-sonic (relative to the water) flexural wave in the ice plate. Comparisons are also made with the locally reacting approximation to the input impedance of an ice plate. The combined effects of the ice lead diffraction process represent loss mechanisms that contribute to the transmission loss in long-range Arctic acoustic propagation.

Transient waves in an elastic plate with nonmixed end loadings

The short-time transient response in an elastic semi-infinite plate for nonmixed stress boundary conditions at its end is investigated. The expansions of the end unknowns are constructed in order to complete the boundary conditions. The equations of motion are solved by a double transform technique. The wavefront behavior is derived with the method of steepest descent. The main structure of the wavefronts and the amplitude variation are discussed and compared with experimental results.

The influence of the evolution of the Pyrenees on adjacent basins

Important thicknesses of syn- and post-orogenic sediments are observed in the southern part of the Aquitaine Basin and in the north of the Ebro Basin. Their deposition cannot be explained only as a result of the extensional history of the Early Cretaceous, thus the influence of the compression associated with the Pyrenees formation upon the subsidence of these basins is studied. Europe and Iberia are considered to behave as two semi-infinite elastic plates. The evolution through time of the sedimentary thicknesses is explained by the deflections of these two plates induced not only by the topographic load of the Pyrenees but also by additional forces (10 12 N/m) that may have their origin in the formation of a cold lithospheric root during the shortening. A series of calculations is presented for an approximately north-south profile across the Aquitaine Basin, the Pyrenees and the Ebro Basin; a good fit for the shape of the basins and the gravity anomalies was obtained, with flexural rigidities of 1.5·10 23. Nm for the northern plate and the outer part of the southern plate, and 1.4·10 21 Nm for the inner part, of the southern plate.

On the Kutta condition for the flow along a semi-infinite elastic plate

In this paper we consider a semi-infinite elastic plate, placed in a parallel flow, which performs a waving motion induced by the fluid. We discuss some aspects of the possibility of a smooth flow at the trailing edge. Numerical calculations have been made for a physically realistic part of the parameter space.

The scattering of interval waves by the periphery of an elastic plate

The problem of the scattering of internal waves by the periphery of an elastic semi-infinite plate placed on the surface of a liquid which is exponentially stratified with respect to its density is considered.

The scattering matrix in a waveguide with elastic walls

The spectrum of normal waves is studied and the scattering matrix is determined for a planar waveguide with elastic walls and with an elastic semi-infinite plate situated within it. The mechanical mode of behaviour of elastic plates is described using the general-type differential operators. Problems of this type belong to the class of the boundary contact problems /1, 2/. The unique solvability of these problems requires the formulation of additional boundary contact conditions describing the mechanical behaviour of the edge of the semi-infinite plate situated within the waveguide. The regularization of the integrals appearing when the general-type boundary contact conditions are satisfied is indicated.

On the double cantilever beam technique for studying crack propagation

The problem of a semi-infinite elastic plate loaded along its straight boundary is pursued to develop an analytical and realistic Double Cantilever Beam Model which accounts for the end deflections associated with deformation beyond the crack tip. Deformation of the uncracked portion allows each arm to rotate about the built-in end, producing an end deflection in addition to those caused by bending and shear. Comparisons with the predictions of existing models show reasonable agreement, but comparisons with some available experimental results reveal that the theoretical end deflection is underestimated. This was shown not to be unreasonable when the limitations of the theory were considered.

Sedimentological Effects of a Moving Terrain

To investigate the sedimentologic effects of changing rheology and boundary conditions on lithospheri cflexure due to thrust loading, the following simple linear models for flexure have been examined: the infinite elastic plate to represent thrusting on the craton, the semi-infinite plate to represent thrusting near a plate boundary, and the infinite viscoelastic plate to consider simple time-dependent effects. These models make the following sedimentological predictions: (1) periodic thrust motion on the infinite and semi-infinite elastic plates produces periodic regressions and transgressions on the orogenward and cratonward side of the basin, respectively; (2) periodic thrust motion on a viscoelastic plate with a small relaxation constant produces out of phrase transgressions and regressions on both sides of the basin; and (3) for a given set of flexural parameters the flexure size for these models, in order of decreasing amplitude, are the semi-infinite elastic plate, the infinite viscoelastic plate, and the infinite elastic plate. With cratonward movement of the thrust terrain the deflection of the semi-infinite plate becomes indistinguishable from the infinite plate. In the Western Overthrust belt of Idaho and Wyoming, some sedimentologic indicators are consistent with an infinite viscoelastic plate with a small relaxation time. The sedimentation history in the "Sevier Shale" basin of Tennessee suggests that lithospheric boundary conditions in thrust belts may change with time.

Flexure of Lithosphere Beneath Apennine and Carpathian Foredeep Basins: Evidence for an Insufficient Topographic Load

The Pliocene foredeep basin external to the Apennine thrust belt and the late Miocene foredeep basin external to the outer East Carpathians are considered typical molassic foredeeps that formed contemporaneously with thrusting in the adjacent mountain belts. A simple mechanical model in which the response of the lithosphere to loading was approximately by a semi-infinite (broken) elastic plate overlying a weak fluid was used to analyze the deflection of the foreland lithosphere beneath the Apennine and outer East Carpathian foredeep basins along three transects perpendicular to the thrust belts. Comparison between the calculated and observed depth to basement suggests that the emplacement of thrust sheets onto the foreland lithosphere during mountain building failed to pr duce a sufficient load to create the observed foredeep basins and observed topography regardless of the position of the slab end or the flexural rigidity assumed for the foreland lithosphere. We infer that an additional subsurface load must be present and that its magnitude must be roughly equal to the load created by that part of the thrust belt that is above sea level. For the two Apennine transects, an acceptable fit to the shape and size of the observed foredeep basin was generated by assuming effective elastic plate thicknesses of 8 and 15 km (4.9 and 9.3 mi) and applied vertical forces of 2 × 1015 dynes/cm and 1.3 × 1015 dynes/cm at the slab end. For the Carpathian transect, an effective plate thickness of 30 km (18.6 mi) and an applied force of 1.5 #215; 1015 dynes/cm generated an acceptable fit. This study suggests a method whereby an estimate of the depth to autochthonous basement rocks beneath overthrust belts can be made in some areas where this information is not available from seismic or drilling data. Such an approach may enable exploration geologists to determine if crystalline rocks encountered in drilling overthrust terranes are a part of the autochthonous basement, or if they are part of an overthrust sheet. In the latter situation, oil-bearing sequences of sedimentary rocks might lie below crystalline rocks contained in the allochthonous sheet.

THE DIFFRACTION OF SOUND WAVES BY AN ELASTIC HALF-PLANE ATTACHED TO A VERTICAL RIGID PLANE

The sound field of a line source is investigated when a structure consisting of a semi-infinite thin elastic plate, attached at right angles to a rigid plate of infinite extent, is embedded in a stationary acoustic medium. A formal exact solution is presented and the distant sound field is explicitly derived, with emphasis on the free modes of the coupled fluid-structure system. The significance of the contributions associated with different parts of the structure is exposed by taking two distinct limits of the fluid loading parameter relevant in aerodynamic and underwater problems.