Cagniard-de Hoop Method Research Articles

Temperature fields generated by impulsive interfacial heat sources

Theoretical study of temperature fields generated by an instantaneous line heat source located at the interface between two different media is presented. Green’s functions for the temperature fields in both media are derived using the Cagniard–de Hoop method. It is shown that in a more conductive medium the temperature field has the character of a wave originating at the point source, while in the less conductive medium the temperature field can be decomposed into the “spherical” wave emanating from the two-dimensional point source and the “head” wave arising due to the heat exchange through the interface. The derived analytical expressions are accompanied by the diagrams of the wave surfaces of temperature fields and numerical examples.

Reflection and refraction of a transient temperature field at a plane interface using Cagniard-de Hoop approach.

An instantaneous line heat source located in the medium consisting of two half-spaces with different thermal properties is considered. Green's functions for the temperature field are derived using the Laplace and Fourier transforms in time and space and their inverting by the Cagniard-de Hoop technique known in elastodynamics. The characteristic feature of the proposed approach consists in the application of the Cagniard-de Hoop method to the transient heat conduction problem. The idea is suggested by the fact that the Laplace transform in time reduces the heat conduction equation to a Helmholtz equation, as for the wave propagation. Derived solutions exhibit some wave properties. First, the temperature field is decomposed into the source field and the reflected field in one half-space and the transmitted field in the other. Second, the laws of reflection and refraction can be deduced for the rays of the temperature field. In this connection the ray concept is briefly discussed. It is shown that the rays, introduced in such a way that they are consistent with Snell's law do not represent the directions of heat flux in the medium. Numerical computations of the temperature field as well as diagrams of rays and streamlines of the temperature field are presented.

The stress-intensity factor history for a half plane crack in a transversely isotropic solid due to impact point loading on the crack faces

Three-dimensional analysis is performed for a transversely isotropic solid containing a half plane crack subjected to suddenly applied concentrated point forces acting at a finite distance from the crack edge. The solution of this problem is treated as the superposition of two simpler problems. One considers the transient wave in an elastic half space generated by an impact point loading on the surface, the other problem is that which cancels out the surface displacement ahead of the crack edge induced by problem 1. A half space subjected to a distributed dislocation on the surface is constructed as the fundamental problem and solved by the use of integral transforms, the Wiener–Hopf technique and the Cagniard-de Hoop method. An exact expression is derived for the mode I stress-intensity factor as a function of time and position along the crack edge. Some features of the solution are discussed through numerical results.

Stress Wave Propagation in a Coated Elastic Half-Space due to Water Drop Impact

Stress wave propagation in a coated elastic half-space due to water drop impact is studied by using the Cagniard-de Hoop method. The stresses have singularity at the Rayleigh wavefront whose location and singular behavior are determined from the pressure model and independent of the coating thickness, while reflected waves cause minor changes in amplitudes.

Time-domain Green's function for an infinite sequentially excited periodic line array of dipoles

Green's functions based on truncated periodicity play an important role in the efficient analysis of radiation from, or scattering by, phased-array antennas, frequency selective surfaces and related applications. Such Green's functions exploit the equivalence between summation over the contributions from individual elements in an array and their collective treatment via Poisson summation in terms of an infinite series of Floquet waves (FW). While numerous explorations have been carried out in the frequency domain (FD), much less has been done for transient excitation. In order to gain understanding of the FW critical parameters and phenomenologies governing time-domain (TD) periodicity, we consider the simple canonical problem of radiation from an infinite periodic line array of sequentially pulsed axial dipoles. This problem can be solved in closed form and also by a variety of alternative representations, which include inversion from the FD, spectral decomposition into TD plane waves, the complex space-time analytic signal formulation, and the Cagniard-de Hoop method. These alternatives, some of which apply traditionally only for nondispersive TD events, are shown to still work here because of the special character of the FW dispersion. Particular attention is given to evanescent TB-FWs and their transition through cutoff. Asymptotic techniques grant insight into the TD-FW behavior by identifying their instantaneous frequencies, wavenumbers, and other physics-based parameterizations. A basic question concerns the definition of what constitutes a physically "observable" (causal, etc.) TD-FW. The proposed answer is based on consistency among models arrived at by alternative routes.

Elastodynamic analysis of a half plane crack in a transversely isotropic solid under 3-D transient loading

A three-dimensional analysis of a half plane crack in a transversely isotropic solid is performed. The crack is subjected to a pair of suddenly-applied normal line loadings on its faces. Transform methods are used to reduce the boundary value problem to a single integral equation that can be solved by the Wiener-Hopf technique. The Cagniard-de Hoop method is employed to invert the transforms. An exact expression is derived for the mode I stress intensity factor as a function of time and position along the crack edge. Numerical results are given.

The impact response of a half plane crack in a transversely isotropic solid due to 3-D mixed mode loading

Three-dimensional analysis of a half plane crack in a transversely isotropic solid is performed. The crack is subjected to two opposed pairs of shear line loads on its faces. Transform methods are used to reduce the boundary value problem to a set of coupled integral equations that can be solved by the Wiener-Hopf technique. The Cagniard-de Hoop method is employed to invert the transforms. Exact expressions are derived for the mode II and III stress intensity factors as functions of time and position along the crack edge. Some features of the solutions are discussed through numerical results.

Exact solution for horizontal displacement at center of the surface of an elastic half space under horizontal impulsive punch loading

The exact analytical solution for the horizontal displacement at the center of the surface of an elastic half space under an impulsive loading having the same spatial distribution as the contact stresses that arise underneath a rigid disk when subjected to a static, horizontal load, is obtained using the Cagniard–De Hoop method. The solution can be used to study the dynamical interaction between soil and structures, and can also be used to assess numerical computations with a finite element or a boundary element program.

The 3-D stress intensity factor for a half plane crack in a transversely isotropic solid due to the motion of loads on the crack faces

Three-dimensional analysis of a half plane crack in a transversely isotropic solid is performed. The crack is subjected to a pair of normal point loads moving in a direction perpendicular to the crack edge on its faces. Transform methods are used to reduce the boundary value problem to a single integral equation that can be solved by the Wiener-Hopf technique. The Cagniard-de Hoop method is employed to invert the transforms. An exact expression is derived for the mode I stress intensity factor as a function of time and position along the crack edge. Some features of the solution are discussed through numerical results.

Initial Dirichlet Problem for Half-Plane Diffraction: Global Formulae for its Generalized Eigenfunctions, Explicit Solution by the Cagniard-de Hoop Method

This paper deals with time-dependent plane wave diffraction by a soft/soft Sommerfeld half-plane \sum : x > 0, y = ±0 . The explicit solution is obtained as a time-convolution in two ways: The first is directly applying the Cagniard de Hoop method to the generalized Wiener-Hopf solution of the corresponding stationary problem due to Meister & Speck (1989). The second way makes use of the Laplace integral representation of the generalized eigenfunctions with respect to the spatial (Cartesian) variables derived by Ali Mehmeti in his habilitation thesis (1995) from formulae of Meister (1983). After deforming the path of.integration into the semi-infinte branch cut lines of the characteristic square root \sqrt{\xi^2-k^2} of the Helmholtzian, he obtains representations where real wave numbers may appear. But for convergence of the integrals one has to distinguish the cases x ≥ 0 and x < 0 , where the obstacle is present or not. We set the time Laplace variable s = –ik and recover the time domain functions for the diffracted field from the eigenfunctions of the stationary problem. There follows a global formula representation with polar coordinates having the diffracting edge of \sum as its center. The solution of the initial boundary value problem is seen to coincide with that obtained in the first way, indeed.

Unsteady Dynamic Response of Jointed Dissimilar Materials Subjected to In-Plane Moving Disturbance.

In this study, the problem of jointed dissimilar materials subjected to a suddenly applied dislocation or force which travels constant velocity on the interface is analyzed. The transformed solutions in the Laplace and Fourier domains include the function of Stoneley Wave, and then the inverse transforms are carried out using the Cagniard-deHoop method. The exact transient closedform solutions in the time domain are obtained. The features of Stoneley Wave and the typical stress concentrations near the dislocation tip are showed by graphical representation.

Transient SV-Wave Motion in Plane Strain of an Elastic Plate.

In this study, a plate subjected to a suddenly applied surface force which travels constant velocity on the surface is analyzed. The transformed solutions in the Laplace and Fourier domains include the function of Rayleigh-Lamb frequency equation. Their solutions are expanded into infinite series form, and then the inverse transforms are carried out for the each term of the series using the Cagniard-deHoop method. The exact transient closed-form solutions in the time domain are also expressed in infinite series form and their each term represents physical transient wave. It is shown that the amplitude ratios of reflection of P wave and SV wave are same for the cases of harmonic waves. The numerical results of velocity fields are shown by graphical representation.

Full field analysis of an anti-plane interface crack subjected to dynamic body forces

In this study, the transient full field response of an interface crack between two different media subjected to dynamic body force at one material is investigated. For time t < 0, the bimaterial medium is stress free and at rest. At t = 0, a concentrated anti-plane dynamic point loading is applied at the medium as shown in Fig. 1. The total wave field is due to the effect of this point loading and the scattering of the incident waves by the interface crack. An alternative methodology that is different from the conventional superposition method is used to construct the reflected, refracted and diffracted wave fields. A useful fundamental solution is proposed in this study and the full field solution is determined by superposition of the fundamental solution in the Laplace transform domain. The proposed fundamental problem is the problem of applying an exponentially distributed traction (in the Laplace transform domain) on the interfacial crack faces. The Cagniard–de Hoop method of Laplace inversion is used to obtain the transient solution in time domain. Exact transient closed form solutions for stresses and stress intensity factors are obtained. Numerical results for the time history of stresses and stress intensity factors during the transient process are discussed in detail.

Time-Dependent Plane Wave Diffraction by a Half-Plane: Explicit Solution for Rawlins' Mixed Initial Boundary Value Problem

Eine zeitabhängige, fortschreitende ebene Wellenfront G(t — x cos θ — y sin θ) trifft zum Zeitpunkt t = 0 mit dem Einfallswinkel θ auf die Streukante der Halbebene Σ: x > 0, y = 0. Bei gleichzeitig homogenen Anfangs- und Randbedingungen vom gemischten Typ: Dirichlet an dem oberen und Neumann an dem unteren Ufer von Σ, wird die explizite Lösungsformel für das totale Feld als Zeitfaltung mit G ermittelt. Die auf der Basis von [17] in [16] erhaltene Wiener-Hopf Lösung des stationären Rawlins-Problems [13] mit verallgemeinerter L2-Faktorisierung der stückweise stetigen Fouriersymbol-Matrix bezüglich der reellen Achse bringt hier die Anwendbarkeit der Cagniard-de Hoop-Methode [1]. Dieser Zugang ist auch durch spektraltheoretische Untersuchungen transienter Halbebenenprobleme inspiriert worden ([2,16]). Die Beweismethoden des Grenzabsorptionsprinzips für das Dirichlet-Sommerfeld-Problem in [2] (durch Deformation von Pfadintegralen [9]) hat innere Analogien zur hier benutzten Cagniard-de Hoop-Methode. This paper deals with the diffraction of a time-dependent plane wave field G(t — x cos θ — y sin θ) governed by the two-dimensional wave equation and striking the edge of the half-plane Σ: x > 0, y = 0., at time t = 0 with some incident angle θ. The explicit solution formula for the total wave field is derived as a convolution with respect to time for homogeneous initial data and homogeneous boundary conditions: Dirichlet on the upper, Neumann on the lower bank of Σ. The Cagniard de Hoop method [1] is seen to be applicable due to the Wiener-Hopf solution of the corresponding stationary Rawlins problem [13] obtained in [16] by generalized L2-factorization of its piece-wise continuous Fourier matrix symbol relative to the real line on the basis of [17]. This approach is also inspired by the attempt to solve transient half-plane problems via spectral theory ([2,16]). The method to prove the limiting absorption principle for the pure Dirichlet problem in [2] (by deforming integral paths [9]) has intrinsic anologies to the Cagniard de Hoop method used here.

Dynamical surface response of a semi-infinite anisotropic elastic medium to an impulsive force

The dynamical surface response to an impulsive point source or line source at any location on the free surface of an anisotropic half-space is solved. The calculation of the Green’s functions of the elastic homogeneous medium is based on the Cagniard–de Hoop method. This theoretical study models the generation–detection of transient surface acoustic waves with a point source–point receiver or with a line source–point receiver. An inspection of singular points in the complex plane of the phase slownesses points out the striking phenomena due to anisotropy of surface wave propagation. As a result, the propagation of the Rayleigh wave, of the surface skimming waves, and of the pseudosurface wave are studied as well as the wavefronts focusing. In addition, the diffraction effects by the cusps are emphasized.

Elastodynamic stress intensity factors for a semi-infinite crack due to three-dimensional concentrated shear loadings on the crack faces

The dynamic stress intensity factors for a semi-infinite crack in an otherwise unbounded elastic body is investigated. The crack is subjected to a pair of suddenly-applied shear point loads on its faces at a distance l away from the crack tip. This problem is treated as the superposition of two problems. The first problem considers the disturbance by a concentrated shear force acting on the surface of an elastic half space, while the second problem discusses a half space with its surface subjected to the negative of the tangential surface displacements induced by the first problem in the front of the crack edge. A fundamental problem is proposed and solved by means of integral transforms together with the application of the Wiener-Hopf technique and Cagniard-de Hoop method. Exact expressions are then derived for the mode II and III dynamic stress intensity factors by taking integration over the fundamental solution. Some features of the solutions are discussed.

Transient Analysis of a Subsonic Propagating Interface Crack Subjected to Antiplane Dynamic Loading in Dissimilar Isotropic Materials

In this study, the transient stress fields and the dynamic stress intensity factor of a semi-infinite antiplane crack propagating along the interface between two different media are analyzed in detail. The crack is initially at rest and, at a certain instant, is subjected to an antiplane uniformly distributed loading on the stationary crack faces. After some delay time, the crack begins to move along the interface with a constant velocity, which is less than the smaller of the shear wave speed of these two materials. A new fundamental solution is proposed in this study, and the solution is determined by superposition of the fundamental solution in the Laplace transform domain. The proposed fundamental problem is the problem of applying exponentially distributed traction (in the Laplace transform domain) on the propagating crack faces. The exact full-field solutions and the stress intensity factor are found in the time domain by using the Cagniard-de Hoop method (de Hoop, 1958) of Laplace inversion. The near-tip fields are also obtained from the reduction of the full-field solutions. Numerical results for the dynamically extending crack are evaluated in detail. The region of the stress singular field dominated in the transient process is also discussed.

Reflection of Transient Acoustic Waves by a Continuously Layered Halfspace with Depth-Dependent Attenuation

An integral transformation-type method is presented for the analysis of the space-time domain acoustic wavefield that is reflected by a halfspace consisting of a continuously layered, lossy, isotropic (equivalent) fluid. The application of a vertically varying compliance memory function makes it possible to model a large class of depth-dependent loss properties. The method combines higher-order WKBJ-like asymptotics with the Cagniard-De Hoop method of inverse transformation. The coefficients that occur in the WKBJ asymptotics follow a recurrence scheme that is easy to evaluate by means of a symbolic manipulation program. The form of the transform domain expressions leads to a very fast inversion process. Numerical results are presented that show the reflections from continuously layered lossless and lossy halfspaces.

Dynamic Analysis of a Propagating Antiplane Interface Crack

In this study, the transient problem of a propagating interface crack between two different media is analyzed. For time t< 0, the crack is stress free and at rest. At t= 0, a pair of concentrated antiplane dynamic point loadings are applied at the stationary crack faces. It is assumed that the stationary crack will begin to propagate along the interface with a subsonic speed as the incident wave generated by the point loading in the upper crack face or in the lower crack face arrives at the crack tip. A new fundamental solution is proposed in this study and the transient solution is determined by superposition of the fundamental solution in the Laplace transform domain. The proposed fundamental problem is the problem of applying an exponentially distributed traction (in the Laplace transform domain) on the propagating crack faces. The Cagniard-de Hoop method of Laplace inversion is used to obtain the transient solution in time domain. Theoretical results indicate that the shear stress along the interface ...

Elastodynamic stress intensity factor due to three-dimensional concentrated transient loading on the faces of a crack

The dynamic stress intensity factor for a semi-infinite crack in an otherwise unbounded elastic body is analyzed. The crack is subjected to a pair of suddenly applied point loads on its faces at a distancel away from the crack tip. The solution of the problem is obtained by superposition of the solutions of two simpler problems. The first of these problems is Lamb's problem, while the second problem considers a half space with its surface subjected to the negative of the normal displacement induced by Lamb's problem in the rangex>0. The latter is sovled by means of integral transforms together with the application of Weiner-Hopf technique and Cagniard-de Hoop method. An exact expression is derived for the mode I stress intensity factor as a function of time for any point along the crack edge. Some features of the solution are discussed.