Anders E Boström

The low-frequency reflection and scattering of the S0 Lamb mode from a circular through-thickness hole in a plate: Finite Element, analytical and experimental studies

A study of the interaction of the S0 Lamb wave with a circular through-thickness hole in a plate is presented. The study is limited to the nondispersive frequency range of this wave, in which the distributions of stress and displacement are simple. This allows a Finite Element analysis to be undertaken using a two-dimensional membrane discretization. Predictions of the direct reflection of the S0 mode and the lateral scattering of the SH0 mode are made for a range of diameters of the hole. At the same time, an analytical solution based on modal superposition is developed, and this is also used to predict the reflection and scattering coefficients. Both sets of predictions are validated by experimental measurements. It is found that the trends of the reflection coefficients for different hole diameters, frequencies and distances from the hole satisfy a simple normalization. On a detailed scale, the functions exhibit undulations which are shown to result from the interference of the direct reflection with secondary reflections which arrive slightly later.

Ultrasonic probe modeling and nondestructive crack detection

The mathematical modeling of a typical situation in ultrasonic nondestructive testing for defects is considered. The first objective is the modeling of a reasonably general type of ultrasonic probe. This is performed by prescribing the traction vector on the surface of an elastic half‐space. The effective probe area may be rectangular or elliptic and the traction may or may not include the tangential part (glued or fluid‐coupled probe, respectively). The probe can be of P, SV, or SH type and of any angle. The traction is either constant across the probe (piston‐type source) or it may taper off toward the edges. Numerical results for some representative cases are given showing snapshots of the field beneath the probe. The second objective of the paper is to include the presented probe model into a complete model of the ultrasonic testing situation. To this end the probe field is, via a series of transformations, expressed in spherical vector waves centered at the defect. The influence of the defect is give...

Elastodynamic Scattering From Inclusions Surrounded by Thin Interface Layers

The scattering of elastic waves by elastic inclusions surrounded by interface layers is a problem of interest for nondestructive evaluation of interfaces in composites. In the present paper the scattering by a single elastic inclusion is studied. The scattering problem is solved by means of the null field approach and the properties of the interface layer enters through the boundary conditions on the inclusion

Multiple scattering of elastic waves by bounded obstacles

The transition matrix method for stationary elastic waves is extended to a great class of obstacles characterized by piecewise constant properties. First, the translation properties of the basis functions is used to treat two and, then, several homogeneous obstacles, and thereafter an obstacle with consecutively enclosing layers is considered. It is then indicated how these two basic methods of combination can be applied to treat more complex cases, including an obstacle consisting of several nonenclosing parts. Finally, we give some numerical applications to configurations of spherical and nonspherical obstacles in and below the resonance region.

Elastic wave scattering from an interface crack: antiplane strain

The two-dimensional scalar problem of scattering of elastic waves under antiplane strain from an interface crack between two elastic half-spaces is considered. The method used is a direct integral equation method with the crack-opening displacement as the unknown. Chebyshev polynomials are used as expansion functions and the matrix in the resulting equations is simplified by contour integration techniques. The scattered far field is expressed explicitly in simple functions and the expansion coefficients. The consequences of energy conservation are explored and are used as a check in the numerical implementation. For incoming plane waves numerical results are given for the total scattered energy and the far field amplitude.

Scattering of acoustic waves by a layered elastic obstacle in a fluid—An improved null field approach

Scattering by a layered elastic obstacle submerged in a fluid is considered. The null field approach (T‐matrix method) is used, but the method is modified so that all surface fields can be expanded in spherical harmonics. The resulting method is then compared with the conventional null field approach (which uses expansions in regular wavefunctions), and it is concluded that the present method is the better one in general. It is, however, doubtful if the null field approach is applicable to cases where there is no sphere that circumscribes the core and inscribes the outer surface. Some numerical results are given for a layered spheroid and superspheroid.

On the rational derivation of a hierarchy of dynamic equations for a homogeneous, isotropic, elastic plate

Abstract Flexural equations of motion for a homogeneous, isotropic, elastic plate are derived by an antisymmetric expansion in the thickness coordinate of the displacement components. All but the lowest-order expansion functions are eliminated with the help of the three-dimensional equations of motion, and are plugged into the boundary conditions. Eliminating between these, an equation is obtained for the mean-plane vertical displacement which also includes arbitrary loading on the plate surface. This equation can be truncated to any order in the thickness and it is in particular noted that the corresponding dispersion relation seems to correspond to a power series expansion of the exact Rayleigh–Lamb dispersion relation to all orders. Various truncations of the equation are discussed and are compared numerically with each other, the exact three-dimensional solution and Mindlins plate equation. Both the dispersion relation and the corresponding displacement components as well as an excitation problem are used for the comparisons. The theories are reasonably close to each other and in order to be on the safe side none of them should in fact be used for frequencies above the cutoff of the second flexural mode.

A model of ultrasonic nondestructive testing for internal and subsurface cracks

The scattering of elastic waves in a half-space containing a striplike crack is investigated. As a special case it seems that the crack may be surface breaking. A surface integral equation with the half-space Green tensor is employed. The key point of the method is the expansion of the Green tensor in Fourier representations with the free part of the Green tensor expanded in the crack coordinate system and the half-space part in the half-space coordinate system. The integral equation is discretized by expanding the crack opening displacement in terms of Chebyshew functions having the correct square root behavior along the crack edges. The incident field is emitted from an ultrasonic probe and a recent model for this is employed. The signal response in another (or the same) probe is modeled by a reciprocity argument and the stationary phase approximation is employed to simplify the final answer, which is thus only valid in the far field of the probes (yielding essentially a spherical wave). Numerical resul...

Ultrasonic wave propagation through a cracked solid

The propagation of ultrasonic waves through a perfectly elastic medium containing a random distribution of equally-sized penny-shaped cracks with spring boundary conditions across the crack faces are considered. As limiting cases results for open and fluid-filled cracks are derived also. The medium with the crack distribution is modelled as an effective viscoelastic medium, using the non-interecting scatterer approximation and Foldys thery. For this purpose the scattering by a single crack is solved by an integral equation method. Distributions of both randomly oriented and parallel cracks are considered. Numerical results are presented for the phase velocity and attenuation. For parallel cracks when the effective medium becomes transversely isotropic two further issues are investigated.The first is the extension of a static result due to Kachanov, who showed the transverse isotropy to be of a very special kind. The second is consistency of the wave speeds obtained by using Foldys theory, with the fact that the effective material is transversely isotropic. In particular, the vertically polarized shear wave should have the same wave speeds in directions parallel and normal to the cracks. It is found that the relations established by Kachanov and the consistency requirements are satisfied by the phase velocity for all frequencies considered.

On the boundary conditions for ultrasonic transmission by partially closed cracks

The probability of detecting crack-like defects using ultrasonic techniques can be severely reduced if the crack is closed by a static background pressure. In this paper, we model the contacting faces of a partially closed crack by an array of circular spot-welds randomly distributed over an infinite plane. We give an exact derivation of the reflection and transmission coefficients for a plane elastic wave at such a boundary in terms of the mean interfacial stresses. The latter are estimated in the limit when the contact radius is much smaller than the wavelength and the contacts are sparsely distributed. This calculation is then related to a distributed spring model of the interface. The latter replaces the real interface by an effective homogeneous linear boundary condition which relates the crack opening displacement to the boundary stresses by effective stiffnesses. These unknown parameters are chosen to ensure that the model condition predicts the exact values of the mean interfacial stresses and the reflection and transmission coefficients in the limit already described. Our results are consistent with and complement those of Baik and Thompson(1) who introduced the distributed spring model in this and a number of other contexts. Our analysis provides a systematic assessment of the range of validity of the model and suggests ways in which the present estimates may be improved.