Wedge Problem Research Articles

The anti-plane shear elastostatic fields near the wedge vertex of an incompressible hyperelastic bimaterial

The anti-plane shear transformation corresponding to a bimaterial wedge problem is studied within the framework of finite elasticity. Two wedges of arbitrary angles are assumed to be incompressible hyperelastic Neo–Hookean materials, and have a common perfect interface and a traction free surface. The resolution of the boundary value problem near the vertex of a bimaterial wedge by means of asymptotic procedure leads to an eigenequation connecting wedges angles and materials properties. Results proved that, contrary to linear elasticity predictions, it is needed to extend development to higher orders because of their contribution in stress singularity.

Exact solution of three-dimensional acoustic field in a wedge with perfectly reflecting boundaries

An exact approach is presented to compute the three-dimensional (3D) acoustic field in a homogeneous wedge-shaped ocean with perfectly reflecting boundaries. This approach applies the Fourier synthesis technique, which reduces a 3D point-source ideal wedge problem into a sequence of two-dimensional (2D) line-source ideal wedge problems, whose analytical solution is well established. A comparison of numerical efficiency is provided between this solution and the solution proposed by Buckingham, which is obtained by a sequence of integral transforms. The details of numerical implementation of these two solutions are also given. To validate the present approach and at the same time compare numerical efficiency between this approach and Buckingham’s analytical solution, two numerical examples are considered. One is the Acoustical Society of America (ASA) benchmark wedge problem and the other is a wide-angle wedge problem. Numerical results indicate that the present approach is efficient and capable of providing accurate 3D acoustic field results for arbitrary receiver locations, and hence can serve as a benchmark model for sound propagation in a homogeneous wedge-shaped ocean.

On three-dimensional asymptotic solution, and applicability of Saint–Venant’s principle to pie-shaped wedge and end face (of a semi-infinite plate) boundary value problems

Three-dimensional asymptotic singular stress fields near the fronts of infinite wedges are presented. This investigation is devoted to establishment of rigorous conditions as to whether the presence of wedge or end face stress singularity, which depends on the prescribed boundary condition, can validate or invalidate the heuristic assumption implicit in Saint–Venant’s principle. This is accomplished by applying a general approach to the solution of canonically singular problems, based on the concept of proper boundary-value problem, the theorem of homogeneous solutions, and classification of boundary value problems (BVP) of three-dimensional elasticity theory into class S (Saint–Venant) or class N (non-Saint–Venant).

Initial stress in the hot forging of a cylindrical blank

A model is proposed for the hot extension of small-diameter cylindrical steel blanks in complex faces to obtain round forgings. Plane deformation is considered. The stress state is calculated on the basis of the Michell problem for an elastic wedge, the Flamant problem for a semiinfinite plate, and a method proposed earlier for assessing the transition of a blank to the plastic state on extension in plane faces. The thermal stress is disregarded; the elastic modulus of the blank depends on the temperature. The distribution of elastic stress tensors in the blank due to the action of three point forces is determined. The total stress field is established; it depends on the vertex angle in the V-shaped lower face. The stress in the plastic-flow zone within the blank’s cross section is estimated, on the basis of the limiting effective elastic stress tensor due to the action of three point forces. As an example, the extension of a steel 45 blank is considered. The limiting effective elastic strain ɛel is assumed to be the elastic limit ɛ0.2 = 0.002. On the assumption that the elastic modulus of steel 45 is 100 GPa at 950°C, the load coefficient when part of the blank passes to the plastic state is determined. The distribution of the components of the total stress tensor is also established. The effective stress corresponding to transition to the plastic state determines the boundary of the plastic zone. The corresponding graphs are plotted. The influence of the vertex angle in the V-shaped lower face on the distribution of the stress-tensor components is established. The optimal value is 120°.

A Benchmark Model for Three-Dimensional Sound Propagation in an Ideal Wedge-Shaped Waveguide

A benchmark model is developed to compute the three-dimensional acoustic field excited by a harmonic point source in a homogeneous wedge-shaped waveguide with impenetrable boundaries. A derivation of the exact solution is presented based on Fourier synthesis, which reduces the three-dimensional ideal wedge problem to a sequence of two-dimensional line-source ideal wedge problems, to which the analytical solution is well established. The details of the numerical implementation of this solution are also provided. Our numerical results indicate that this model is efficient and capable of providing accurate three-dimensional acoustic fields for arbitrary receiver locations, and hence can serve as a benchmark model for sound propagation in a continental shelf environment.

Dynamic Sliding Analysis of a Gravity Dam with Fluid-Structure-Foundation Interaction Using Finite Elements and Newmark’s Sliding Block Analysis

To reduce the natural hazard risks—due to, e.g., earthquake excitation—seismic safety assessments are carried out. Especially under severe loading, due to maximum credible or the so-called safety evaluation earthquake, critical infrastructure, as these are high dams, must not fail. However, under high loading local failure might be allowed as long as the entire structure does not collapse. Hence, for a dam, the loss of sliding stability during a short time period might be acceptable if the cumulative displacements after an event are below an acceptable value. This performance is not only valid for gravity dams but also for rock blocks as sliding is even more imminent in zones with higher seismic activity. Sliding modes cannot only occur in the dam-foundation contact, but also in sliding planes formed due to geological conditions. This work compares the qualitative possible and critical displacements for two methods, the well-known Newmark’s sliding block analysis and a Fluid-Foundation-Structure Interaction simulation with the finite elements method. The results comparison of the maximum displacements at the end of the seismic event of the two methods depicts that for high friction angles, they are fairly close. For low friction angles, the results are differing more. The conclusion is that the commonly used Newmark’s sliding block analysis and the finite elements simulation are only comparable for high friction angles, where this factor dominates the behaviour of the structure. Worth to mention is that the proposed simulation methods are also applicable to dynamic rock wedge problems and not only to dams.

Periodic contact problem for an elastic wedge

A three-dimensional periodic contact problem in the theory of elasticity for a wedge is investigated when an infinite system of identical punches are arranged on one face of the wedge along the edge (at an equal distance from the edge and with equal intervals between neighbouring punches). Identical normal embedding forces are applied to the punches and friction forces are neglected. The other edge of the wedge is rigidly clamped. Without loss of generality of the approaches, the wedge material is assumed to be incompressible. The problem is reduced to an integral equation, from the kernel of which the principal part corresponding to an elastic half-space is separated out. The effect of the edge of a three-dimensional wedge on the contact pressure distribution and the mechanical characteristics of the contact is studied. A regular asymptotic method of solution is used for specified elliptic contact areas. When the contact areas are not known in advance, the method of non-linear boundary value integral equations is used. Calculations were carried out for elliptic and conical punches.

Numerical Simulation of Scattering by a Wedge With Periodic Anisotropic Impedance Faces

Plane wave scattering by an infinite, three-dimensional wedge whose faces are characterized by periodic anisotropic impedances is investigated. An improved hybrid of the moment method (MM) and the physical optics (PO) is proposed to calculate the equivalent electric current on the wedge surface. As for the infinite periodic wedge surface, a periodic representation of the scattering surface is used to prevent nonphysical edge effects in the scattering evaluation. The surface of the wedge is split into an MM region and a PO region. In the implementation of MM, the evaluation of the interaction matrix elements in the MM region is simplified by deriving six universal series. Moreover, the PO-current is modified by a diffracted current to improve its accuracy. The accuracy and effectiveness of the proposed approach is validated and demonstrated by using series solution, the method in [T. B. A. Senior, “Solution of a class of impedance wedge problem for skew incident,” Radio Sci., vol. 21, no. 2, pp. 185-191, 1986], and the two-dimensional improved MM-PO approach under several canonical two-dimensional wedge cases. At last, the proposed approach is applied to analyze the scattering from the wedges with periodic anisotropic impedance faces, and the numerical results are presented.

Fresnel coherent diffractive imaging tomography of whole cells in capillaries

X-ray tomography can be used to study the structure of whole cells in close to their native state. Ptychographic Fresnel coherent diffractive imaging (FCDI) holds particular promise for high-resolution tomographic imaging with quantitative phase sensitivity. To avoid the common missing wedge problem in tomography, cells can be mounted in thin glass capillaries that allow access to the full 180° angular field. However, soft x-rays, which are preferred for cellular imaging, interact strongly with capillaries, sometimes leading to violation of the usual assumptions for coherent diffractive imaging (CDI) and introducing artifacts (i.e., phase wrapping) in the reconstructed images. Here, we describe a method of applying ptychographic FCDI to obtain quantitative x-ray phase images of whole eukaryotic cells mounted in capillaries. The approach eliminates phase-wrapping artifacts due to thick capillaries without the deterioration in image quality that occurs at shallow angles of incidence when using planar mounting schemes. This technique makes it possible to apply CDI tomography to the study of various specimens that can be supported in capillaries and is compatible with established methods of cryogenic preparation.

General solutions of plane elasticity of one-dimensional orthorhombic quasicrystals with piezoelectric effect

By introducing four potential functions, the governing equations of plane problems in 1D orthorhombic quasicrystals with piezoelectric effect are composed of four second-order partial differential equations, in which the quasi-harmonic functions are the essential unknowns. The general solution of these equations is further established, and all expressions are expressed in terms of the potential functions. As an application of the general solution, the closed-form solutions are obtained for wedge problems or half-plane problems of 1D orthorhombic piezoelectric quasicrystals.

Detection of Stator-Slot Magnetic Wedge Failures for Induction Motors Without Disassembly

The recent trend in large ac machines is to employ magnetic stator-slot wedges for improving the motor efficiency, power factor, and power density. The mechanical strength of magnetic wedges is weak compared to the epoxy glass wedges, and many cases of loose and missing wedges have recently been increasingly reported. Magnetic wedge failure deteriorates the performance and reliability of the motor, but there is no method available for testing the wedge quality other than visual inspection after rotor removal. Monitoring of the overall wedge condition without motor disassembly can help reduce the cost of maintenance and risk of degradation in performance. In this paper, a new offline standstill test method for detecting magnetic wedge problems for ac machines without motor disassembly is proposed. An experimental study on 380-V 5.5-kW and 6.6-kV 3.4-MW motors with magnetic wedges is performed to verify the effectiveness of the new test method. It is shown that the new method can provide reliable monitoring of magnetic wedge problems over time, independent of other faults or motor design.

A note on uniform asymptotic wave diffraction by a wedge

Summary New expressions for asymptotically uniform Green’s functions for high-frequency wave diffraction when a plane, cylindrical or point wave field is incident on an ideal wedge are derived. They are useful for deriving a uniform asymptotic expression for the exact solution in terms of the high-frequency diffracted and geometrical optics far field. The present method is simple and consists of differentiating out the singularities of the integral representations and using new representations for trigonometrical sums that arise when the wedge angle is a rational multiple of �. The new results make explicit the continuity of the fields across shadow and reflection boundaries. The geometrical theory of diffraction (GTD) was introduced by Keller (1) as an asymptotic method for the solution of diffraction problems at high frequencies. Keller’s GTD has proved to be very powerful in a wide variety of applications to physical problems. The method uses the saddle point method to derive asymptotic approximations from the exact solution of canonical wedge problems to derive so called ”diffraction coefficients”. The method gives a useful physical representation of the total far-field in terms of the geometrical optics and a diffracted field term. One drawback of Keller’s method is that it breaks down at shadow and reflection boundaries where the diffracted term predicts infinite fields. To overcome this defect, two uniform asymptotic techniques have been developed recently, namely the uniform asymptotic theory (UAT) of edge diffraction and the uniform geometrical theory of diffraction (UTD). A comparison of both these methods is given in the work of Boersma (2) where it is shown that the two different methods of asymptotically approximating integrals by pole subtraction or pole factorisation are equivalent if the complete asymptotic expansions are used. These asymptotic expressions are derived from integrals representing the exact solution. The starting point for our analysis is a particular periodic Green’s function for the solution of of the ideal wedge problem Rawlins(3) . This representation is related to the work of Sommerfeld, Carslaw, Macdonald and others at the beginning of the last century, see Rawlins(4) for a detailed history. The Sommerfeld approach was extended non-trivially to absorbing/impedance wedges by Mayuhzinets and Williams in the 1950’s, see Babich et al (5) for a comprehensive up to date history of general wedge problems solved by the so called Sommerfeld-Malyuzhinets technique. In order to derive useful physical results from these exact solutions the disposition of singularities and saddle points of integrals is crucial for getting accurate uniform results. Oberhettinger (6), (7) used asymptotic methods to derive results for diffraction by ideal

Hamiltonian analysis of a magnetoelectroelastic notch in a mode III singularity

The stress intensity factor (SIF) of a multi-material magnetoelectroelastic wedge in anti-plane deformation is analytically determined by the symplectic method. The Lagrangian equations in configuration variables alone are transformed to Hamiltonian equations in dual variables (configuration and momentum) which allow the use of the method of separation of variables. The solutions of the Hamiltonian equations can be expanded analytically in terms of the symplectic eigenfunctions with coefficients to be determined by the boundary conditions. For the wedge problem, the pairs of anti-plane displacements and shear stresses, electric fields and electric displacements, and magnetic fields and magnetic inductions are proved to be the dual (momentum) variables of the configuration variables. The singularity orders depend directly on the first few eigenvalues whose real parts are less than one but greater than zero. Numerical results for various conditions show the variations of the singularity orders. In particular, special behaviors of the order of the singularity for some special wedge angles are noted.

The Wedge-Type Problem: The Building Brick in High-Frequency Scattering from Complex Objects

This paper presents the problem of high-frequency scattering in a ray-tracing context. It focuses on the wedge problem as the most relevant contribution to the diffracted field from complex objects. The wedge problem's solution is then treated either analytically or numerically. It is treated analytically by resorting to the Malyuzhinets spectral technique and a subsequent approximate solution. It is treated numerically by deriving a parabolic equation from the original elliptic problem, which can then be easily and efficiently solved numerically via finite differences.

Benchmark solutions for sound propagation in an ideal wedge

Sound propagation in a wedge-shaped waveguide with perfectly reflecting boundaries is one of the few range-dependent problems with an analytical solution, and hence provides an ideal benchmark for a full two-way solution to the wave equation. An analytical solution for the sound propagation in an ideal wedge with a pressure-release bottom was presented by Buckingham and Tolstoy [Buckingham and Tolstoy 1990 J. Acoust. Soc. Am. 87 1511]. The ideal wedge problem with a rigid bottom is also of great importance in underwater acoustics. We present an analytical solution to the ideal wedge problem with a perfectly reflecting bottom, either rigid or pressure-release, which may be used to provide a means for investigating the sound field in depth-varying channels, and to establish the accuracy of numerical propagation models. Closed-form expressions for coupling matrices are also provided for the ideal waveguides characterized by a homogeneous water column bounded by perfectly reflecting boundaries. A comparison between the analytical solution and the numerical solution recently proposed by Luo et al. [Luo W Y, Yang C M and Zhang R H 2012 Chin. Phys. Lett. 29 014302] is also presented, through which the accuracy of this numerical model is illustrated.

A three-dimensional parabolic equation model of sound propagation using higher-order operator splitting and Padé approximants

An alternating direction implicit (ADI) three-dimensional fluid parabolic equation solution method with enhanced accuracy is presented. The method uses a square-root Helmholtz operator splitting algorithm that retains cross-multiplied operator terms that have been previously neglected. With these higher-order cross terms, the valid angular range of the parabolic equation solution is improved. The method is tested for accuracy against an image solution in an idealized wedge problem. Computational efficiency improvements resulting from the ADI discretization are also discussed.

Sound Propagation in a Wedge with a Rigid Bottom

Sound propagation in a wedge with perfectly reflecting boundaries is one of the few range-dependent problems with an analytical solution. Since sound propagation towards the wedge apex will be completely backscattered due to the perfectly reflecting boundaries, this test problem is an ideal benchmark for a full two-way solution to the wave equation. An analytical solution for sound propagation in a wedge with a pressure-release sea surface and a pressure-release bottom was presented by Buckingham et al. [J. Acoust. Soc. Am. 87 (1990) 1511]. The ideal wedge problem with a rigid bottom is also of great importance in underwater acoustics. We present an analytical solution to the problem with a wedge bounded above by a pressure-release sea surface and below by a rigid bottom, which may be used to provide informative means of investigating the sound field in depth-varying channels, and to establish the accuracy of numerical propagation models for which it is difficult to treat problems with a pressure-release bottom. A comparison of the analytical solution and the numerical solution recently proposed by Luo et al. [Chin. Phys. Lett. 29(2012) 014302] is also presented, indicating that this numerical propagation model provides high accuracy.

Wedges, cones, cosmic strings and their vacuum energy

One of J Stuart Dowker’s most significant achievements has been to observe that the theory of diffraction by wedges developed a century ago by Sommerfeld and others provided the key to solving two problems of great interest in general-relativistic quantum field theory during the last quarter of the 20th century: the vacuum energy associated with an infinitely thin, straight cosmic string, and (after an interchange of time with a space coordinate) the apparent vacuum energy of empty space as viewed by an accelerating observer. In a sense the string problem is more elementary than the wedge, since Sommerfeld’s technique was to relate the wedge problem to that of a conical manifold by the method of images. Indeed, Minkowski space, as well as all cone and wedge problems, are related by images to an infinitely sheeted master manifold, which we call Dowker space. We review the research in this area and exhibit in detail the vacuum expectation values of the energy density and pressure of a scalar field in Dowker space and the cone and wedge spaces that result from it. We point out that the (vanishing) vacuum energy of Minkowski space results, from the point of view of Dowker space, from the quantization of angular modes, in precisely the way that the Casimir energy of a toroidal closed universe results from the quantization of Fourier modes; we hope that this understanding dispels any lingering doubts about the reality of cosmological vacuum energy.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical in honour of Stuart Dowker’s 75th birthday devoted to ‘Applications of zeta functions and other spectral functions in mathematics and physics’.

A numerically stable coupled-mode formulation for acoustic propagation in range-dependent waveguides

In this paper, a stepwise coupled-mode model with the use of the direct global matrix approach is proposed. This method is capable of handling two-dimensional problems with either a point source in cylindrical geometry or a line source in plane geometry. With the use of the direct global matrix approach, this method is numerically stable. In addition, by introducing appropriately normalized range solutions, thismodel is free from the numerical overflow problem. Furthermore, we put forward source conditions appropriate for the line-source problem in plane geometry. As a result, this method is capable of addressing the scenario with a line source on top of a sloping bottom. Closed-form expressions for coupling matrices are derived and applied in this paper for handling problems with pressure-release boundaries and a homogeneous water column. The numerical simulations indicate that the proposed model is accurate, efficient, and numerically stable. Consequently, this model can serve as a benchmark model in range-dependent propagation modeling. Although this method is verified by an ideal wedge problem in this paper, the formulation applies to realistic problems as well.

Scattering of plane waves by a wedge with different face impedances

The scattering process of plane waves by a wedge with different face impedances is examined in terms of the closed form series solution. A new boundary condition is derived using the solution of the reflection problem of plane waves by an impedance plane. The series solution is obtained for the wedge problem. The results are investigated numerically.