Isotropic Substrate Research Articles

Toward laser-ultrasonic characterization of an orthotropic–isotropic interface

Unidirectional boron–epoxy composite overlays are used to reinforce highly stressed or damaged regions in aircraft structures. This paper describes work ultimately directed toward an assessment of the adhesive bond strength between the composite overlay and the aircraft substrate. The potential use of leaky interface waves for the adhesive bond assessment is addressed, by considering the existence of such waves along the boundary between an orthotropic composite overlay and an isotropic metallic substrate. Numerical procedures are described to calculate complex leaky interface wave velocities, for propagation parallel or perpendicular to the fibers. It is shown that seven of the nine orthotropic elastic constants are required to calculate these velocities. Next, a new scheme is proposed to determine the seven required orthotropic elastic constants, based on laser-ultrasonic line sources and interferometric detection. Simplifications to the scheme, including the use of piezoelectric detection, are possible for a plate where there is access to both sides. This simplified scheme is demonstrated experimentally for a boron–epoxy plate. For the case where this boron–epoxy plate is well bonded to aluminum, titanium, or steel substrates, leaky interface waves are predicted for propagation perpendicular to the fibers. By contrast, for propagation parallel to the fibers, an interface wave is predicted only for the steel substrate; no wave is predicted for the aluminum substrate, while the titanium substrate is a borderline case.

Coaxial feed in uniaxial substrate between two finite parallel conductors

A coaxial feed located in a uniaxial substrate between two finite parallel conductors is reduced analytically into a finite number of voltage sources over the coaxial probe. The procedure used is the same as originally developed for an isotropic substrate. The resulting expressions for both the new sources and the separated admittance component are interpreted by comparison with the isotropic case.

Computation of the Propagation Characteristics of Finlines

In this paper, first a detailed derivation of the Singular Integral Equation (SIE) method for application to finlines on isotropic substrates is given. The formulation is largely based on the one given by R Mittra et al. [1] for the microstrip line. The derivation is validated by computing the propagation constants of the dominant and higher order modes of bilateral finline. Next, simple perturbation formulas are given to extend the above SIE derivation to obtain the propagation constants of finlines containing anisotropic material. Numerical results are given for the propagation constants of finlines containing either an anisotropic dielectric substrate or a magnetized ferrite.

Stress Driven Patterning of Epitaxial Films

ABSTRACTWe study possible morphologies of epitaxial films atop attractive substrates appearing as a result of competition of misfit stresses, van der Waals forces and surface energy. Corresponding formula for the critical thickness of the dislocation-free Stranski-Krastanov pattern is established for the isotropic deformable films and substrates. If the film thickness exceeds the critical magnitude the layer-by-layer pattern switches to islanding. At the first stage the islands have a shape of striae (i.e. long parallel trenches with periodic spacing). We discuss also i)the circumstances in which surface morphology of the film corresponds to a two-dimensional superlattice of islands rather than a one dimensional lattice of striae and ii)the influence of a buffer inter-layer.

Interface waves along an anisotropic imperfect interface between anisotropic solids

First and second order asymptotic boundary conditions are introduced to model a thin anisotropic layer between two generally anisotropic solids. Such boundary conditions can be used to describe wave interaction with a solid-solid imperfect anisotropic interface. The wave solutions for the second order boundary conditions satisfy energy balance and give zero scattering from a homogeneous substrate/layer/substrate system. They couple the in-plane and out-of-plane stresses and displacements on the interface even for isotropic substrates. Interface imperfections are modeled by an interfacial multiphase orthotropic layer with effective elastic properties. This model determines the transfer matrix which includes interfacial stiffness and inertial and coupling terms. The present results are a generalization of previous work valid for either an isotropic viscoelastic layer or an orthotropic layer with a plane of symmetry coinciding with the wave incident plane. The problem of localization of interface waves is considered. It is shown that the conditions for the existence of such interface waves are less restrictive than those for Stoneley waves. The results are illustrated by calculation of the interface wave velocity as a function of normalized layer thickness and angle of propagation. The applicability of the asymptotic boundary conditions is analyzed by comparison with an exact solution for an interfacial anisotropic layer. It is shown that the asymptotic boundary conditions are applicable not only for small thickness-to-wavelength ratios, but for much broader frequency ranges than one might expect. The existence of symmetric and SH-type interface waves is also discussed.

A combined finite element method/boundary element method technique for V(z) curves of anisotropic-layer/substrate configurations

A numerical technique is presented to calculate V(z) curves for a line-focus acoustic microscope, and for the specimen configuration of an anisotropic layer deposited on an isotropic elastic substrate. In this technique, the finite element method (FEM) and the boundary element method (BEM) are combined to calculate the scattered wave field on the specimen surface. The FEM is used in the layer and the BEM is used in the coupling fluid and the substrate. By combining the FEM with the BEM, the shortcomings of the two methods are avoided and their advantages are maintained. As a result, the present technique can be used for very thin and anisotropic layers. The V(z) curves are obtained by substituting the calculated wave field in an electromechanical reciprocity relation. The attenuation of the water used as the coupling fluid is taken into account by adjusting the calculated V(z) curves. The adjusted curves are compared with measured data, and very good agreement, both in the peak spacing and the amplitudes of the oscillatory curves, is observed.

Evaluation of interfacial properties in isotropic and anisotropic layered substrate

Evaluation of interfacial properties in isotropic and anisotropic layered substrate

Radiation characteristics of Hertzian dipole antennas in a nonreciprocal superstrate-substrate structure

Spectral domain analysis is used in conjunction with matrix algebra to determine the far-field behavior of an arbitrarily oriented Hertzian dipole in a nonreciprocal superstrate-substrate structure. In particular, a gyrotropic superstrate on top of a grounded isotropic substrate is considered. The analysis accounts for completely general 3*3 tensors for the anisotropic superstrate layer. Parameter studies are performed to investigate the effects of changing different variables associated with the geometry as well as the static magnetic field used to bias the gyrotropic material. This work provides a basis for research on the design of microstrip circuits and antennas in nonreciprocal superstrate-substrate geometries.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Microstrip resonators on anisotropic substrates

The spectral domain method is applied to study shielded microstrip resonators printed on anisotropic substrates. A Green's function that takes into account the dielectric anisotropy effects is derived through a fourth-order formulation. Galerkin's method is then applied to form the characteristic equation from which the resonant frequency of the microstrip resonator is numerically obtained. Results for a microstrip situated on an isotropic substrate are used to validate the theory.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The propagation of elastic waves in a certain class of inhomogeneous anisotropic materials. I. The refraction of a horizontally polarized shear wave source

This paper is the first in a series of articles describing the refraction and propagation of infinitesimal disturbances in a 'coarse grained’ inhomogeneous anisotropic material which is fused to an isotropic substrate. Here, the basic constitutive law for the material is motivated by applications to the non-destructive evaluation of austenitic steel welds, although it is clear that the phenomena described and the mathematical analysis used is also of interest in geophysics, the study of composite materials and several other areas of continuum mechanics. This work is concerned with the refraction of a horizontally polarized shear wave source at the fusion interface between a homogeneous isotropic material and transversely isotropic material. The latter is inhomogeneous by virtue of the fact that the zonal axis or axis of symmetry of the crystals varies in direction with the distance from the interface. The mathematical boundary-value problem is solved exactly, and, in the highfrequency limit, a uniform asymptotic expansion for the displacement vector is found. It is shown that in this limit, and for a wide range of material constants, the refracted energy which penetrates certain regions of the ‘weld material’ is totally internally reflected. This conclusion is highly significant in the design of inspection procedures for structurally important welds.

Small-scale yielding interfacial crack-tip fields

B y idealizing the crack surroundings as an isotropic elastic perfectly-plastic material with tensile yield strength σ bonded to a rigid or isotropic elastic substrate, the small-scale yielding (SSY) plane strain asymptotic lields for “elastically” traction-free interfacial cracks are numerically calculated. The resulting fields are described in terms of slip-line theory and elastic potentials. The oscillatory asymptotic elastic field behavior does not persist deep within the plastic zone. However, some loadings produce cusp-like boundaries separating various crack-lip sectors within the near-tip fields, coupling the stress state to the radial distance, r, in non-conventional quasi-constant-slate sectors. When the SSY interfacial load-phase angle, defined as ζ 0 = ∠ K + ɛ ln [K/σ 2 v cosh 2 (πɛ)] (where ∠ K is the phase angle of the complex tractionfree stress intensity factor K, and ɛ is the bi-material constant) is negative, inelastic sectors generally surround the crack tip in the plastically deformable medium, simultaneously producing normal and shear interfacial tractions as large as 3.29σ and σ/√3, respectively. For ζ 0 > 0, the fields usually contain an elastic crack-face sector which typically spans at least 45, a centered-fan sector, as well as various conslantand quasi-constant-state sectors, generally leading to less severe interfacial traction and strain conditions than for ζ 0 < 0. From the individual fields found at various discrete values of ζ 0, the near-tip fields are qualitatively characterised over the entire admissible range of ζ 0. At distances from the crack tip much less than the characteristic plastic zone dimension in the adjacent “yielding” domain, the logarithmically singular asymptotic stress field in the “elastic” domain is accurately described by the solution of a halfspace subject to a uniform stress parallel to its boundary and to step function discontinuities in normal and shear surface tractions at the crack tip.

Numerical modeling of the V(z) curve for a thin-layer/substrate configuration

Abstract A numerical model is presented to calculate V(z) curves for a line-focus acoustic microscope and the specimen configuration of a thin isotropic elastic layer deposited on an isotropic elastic substrate. In this model, a Gaussian beam which is tracked through the lens into the coupling fluid, interacts with the thin-layer/substrate system. The numerical approach is based on the solution of singular integral equations by the boundary element method. The system of singular integral equations follows from the conditions at the interface of the coupling fluid and the thin layer and the interface of the thin layer and the substrate. An electrochemical reciprocity relation is used to express the voltage at the terminals of the microscope's transducer in terms of the calculated incident and back-scattered fields. V(z) curves are presented for various layer thicknesses and various combinations of the elastic constants of the layer and the substrate. The oscillations of the V(z) curves are related to the m...

Thin film stress from nonspherical substrate bending measurements

The usual formula for finding thin film deposition stress from the curvature of substrate wafers assumes small out-of-plane deflections and spherical bending. These are significant limitations because deposition stress often produces large deflections and ellipsoidal surfaces ranging from almost spherical to almost cylindrical depending on the film and substrate materials and on the deposition conditions. The present paper examines a nonlinear bending model proposed by Harper and Wu as well as by Masters, Salamon, and Fahnline and its usefulness for calculating deposition stress for both spherical and nonspherical bending. Algebraic solutions of the model predict spherical bending for small deflections and ellipsoidal bending for large deflections even for isotropic films and substrates. An initial experimental test of the nonlinear theory confirms its usefulness with data obtained by measuring the ellipsoidal and spherical curvatures of thin Ta films direct current magnetron deposited on different thickness and size polycarbonate wafers.

Closed-form expressions for the current distributions on open microstrip lines

The authors systematically clarify the characteristics of current distributions for various cases of substrate relative permittivity, shape ratio, and normalized frequency. These characteristics are clarified, and closed-form expressions are proposed. An open microstrip line with isotropic dielectric substrate is numerically analyzed by the spectral-domain method to clarify current distributions on a strip conductor. The current distributions obtained are illustrated in figures for typical cases. A full view of current distributions on a strip conductor in any microstrip line is provided by showing those close-form expressions. These expressions for the current distribution have been compared with the theoretical results, and good agreement has been obtained. >

Molecular Orientation of Monolayers on Liquid Substrates: Optical Model and FT-IR Methods

The primary emphasis of this work is on the development of an optical model which is descriptive of the reflection-absorption properties from an anisotropic monolayer located on an isotropic liquid substrate. The calculations of electric field intensities, phase changes, and reflection-absorptions are compared for isotropic and anisotropic monolayers in order to explore optical model predictions of chain anisotropy. Relationships are established between the average chain orientation angle, the anisotropic optical constants, and the angle of incident light for s-and p-polarized radiation. The modeling calculations predict that the optimum incident light angles for the extraction of orientation information occur close to the polarizing Brewster angle. A method is proposed whereby the orientation types of isotropy and chain alignment at the isotropic-equivalent angle can be distinguished.

Circular interdigital transducers

The purpose of this article is to establish the focusing properties of a new kind of transducer consisting of circular electrodes deposited on an isotropic substrate and covered by a thin piezoelectric layer. We first express the electromechanical equations of the device and determine its boundary conditions. Then, using the effective dielectric function and an appropriate approximation of the charge density, we obtain various functions (electric potential, displacements) giving a physical description of each medium. Last, we present the numerical results of our study, namely those concerning the displacements and proving the focusing capabilities of our device.

Resonant frequency of a rectangular microstrip patch on several uniaxial substrates

A full-wave analysis for determining the resonant frequency of a rectangular microstrip patch on multiple uniaxial anisotropic layers with or without an anisotropic overlay is presented. Two independent methods are used to derive the immittance matrix for the patch, from which the resonant frequency is determined. They are the Hertz vector potentials and the modified spectral domain immittance approach. Numerical results of the resonant frequency are given for several patch configurations, including cases of a patch on a single anisotropic layer, a patch on a double layer with one layer anisotropic and one layer isotropic, a suspended patch resonator with anisotropic substrate, a patch with anisotropic overlay and a patch on two anisotropic substrates with an anisotropic overlay. Changes in the resonant frequency of up to 58% are reported as n/sub x//n/sub y/ is changed from 1.0 (for isotropic substrates) to 2.0. >

Rayleigh and Love Waves in Cladded Anistropic Medium

Guided waves in coated anisotropic medium are of interest in electronics. Also, in recent years, cladded fiber-reinforced composites are being developed for use as aerospace structures. This paper deals with guided wave propagation in a cladded or coated anisotropic medium. The cladding (coating) is assumed to be a thin isotropic layer, which is bonded to a transversely isotropic substrate with the axis of symmetry parallel to the layer. It is shown that the anisotropy of the substrate affects the dispersion behavior in a manner that is substantially different than in the case of isotropic substrate.

Surface Waves for Material Characterization

Analyses are presented for the propagation of harmonic surface waves on a transversely isotropic layer rigidly bonded to a transversely isotropic substrate of different material. The layer-substrate system is also assumed to be in contact with a liquid and inviscid space. The propagation takes place along an axis of symmetry of both the layer and the substrate. Exact closed-form solutions for the characteristic dispersion relations are presented. Numerical results are presented for material combinations of three classes of centrifugally cast stainless steel material. Results clearly demonstrate the influence of the layer thickness on the propagation speed and, hence, provide a means of material characterization.

Anchoring at the Interface Between a Nematic Liquid Crystal and an Isotropic Substrate

Abstract The anchoring properties of the director at the interface between a nematic LC and an isotropic substrate are discussed. More emphasis is devoted to the special cases of the nematic LC free surface and of the nematic-isotropic interface. Some recent theoretical models for these interfaces are briefly discussed and their predictions are compared with the experimental results.