A note on the divergence problem arising in calculation of Green’s function for a 2D piezoelectric thin film strip with free boundaries and containing straight line defects: Fourier approach for bodies of cubic symmetry

Analytic formulas for the Green’s function and the coupled electro-elastic fields for a 2D piezoelectric strip with free boundaries and containing a distribution of straight line defects have already been found some years ago. These formulas exploit the well-known Stroh formalism and the Fourier approach, so the result is given as the Fourier integral and therefore its numerical implementation should pose no problem. However, in this note we show that for the case of cubic symmetry this form of the Green’s function contains strong divergences, excluding possibilities of direct application of well-known numerical schemes. It is also shown that these divergences translate to divergences of the corresponding electro-elastic fields of a single defect. By means of a rigorous analysis it is demonstrated that imposing physical conditions implied by the nature of the problem all of these divergences cancel and the final, physical result exhibits expected, regular behavior at infinity. Unfortunately, although the nature of this problem is purely mathematical, it leads to irremovable numerical ∞ − ∞ uncertainties which tend to spread over the whole Fourier domain and severely impede engineering applications of the Green’s function. This motivates us to compute the exact form of all divergent terms. These novel formulas will serve as a guide when establishing numerically stable algorithms for engineering computations involving the system in question.     

Deriving the time-dependent dyadic Green’s functions in conductive anisotropic media

A homogeneous anisotropic conductive medium, characterized by symmetric positive definite permeability and conductivity tensors, is considered in the paper. In this anisotropic medium, the electric and magnetic dyadic Green’s functions are defined as electric and magnetic fields arising from impulsive current dipoles and satisfying the time-dependent Maxwell’s equations in quasi-static approximation. A new method of deriving these dyadic Green’s functions is suggested in the paper. This method consists of several steps: equations for electric and magnetic dyadic Green’s functions are written in terms of the Fourier images; explicit formulae for the Fourier images of dyadic Green’s functions are derived using the matrix transformations and solutions of some ordinary differential equations depending on the Fourier parameters; the inverse Fourier transform is applied numerically to obtained formulae to find dyadic Green’s functions values. Using suggested method images of electric and magnetic dyadic Green’s function components are obtained in such conductive anisotropic medium as the white matter of a human brain.     

Analysis of the vibrational mode spectrum of a linear chain with spatially exponential properties

We deduce the dynamic frequency-domain-lattice Green’s function of a linear chain with properties (masses and next-neighbor spring constants) of exponential spatial dependence. We analyze the system as discrete chain as well as the continuous limiting case which represents an elastic 1D exponentially graded material. The discrete model yields closed form expressions for the N×N Green’s function for an arbitrary number N=2,…,∞ of particles of the chain. Utilizing this Green’s function yields an explicit expression for the vibrational mode density. Despite of its simplicity the model reflects some characteristics of the dynamics of a 1D exponentially graded elastic material. As a special case the well-known expressions for the Green’s function and oscillator density of the homogeneous linear chain are contained in the model. The width of the frequency band is determined by the grading parameter which characterizes the exponential spatial dependence of the properties. In the limiting case of large grading parameter, the frequency band is localized around a single finite frequency where the band width tends to zero inversely with the grading parameter. In the continuum limit the discrete Green’s function recovers the Green’s function of the continuous equation of motion which takes in the time domain the form of a Klein-Gordon equation.     

Numerical and experimental analyses of singular electro-elastic fields around a V-shaped notch tip in piezoelectric materials

A super V-shaped notch tip element that simulates electro-elastic behavior near a V-shaped notch tip is first developed based on one-dimensional finite element eigensolutions. The present element is then incorporated with standard four-node hybrid electro-elastic field elements to calculate electro-elastic singularity orders, singular strain distributions and generalized stress intensity factors ahead of V-shaped notch tips in three-point-bending piezoelectric specimens with notch angles α = 0°, 30°, 60° and 90°, respectively. The present numerical solutions are compared with those obtained from the moiré interferometry experiment technique. Finally, the effect of the V-shaped notch angle and electric field on singular electro-elastic fields is also discussed.     

Thermo-electro-elastic fields accompanying plastic deformation

The thermo-electro-elastic coupled fields excited by an arbitrary dynamic dislocation ensemble in a piezoelectric medium are studied in a general statement on the basis of the developed 5D formalism. The found general relations are specified for the case when coupled fields are produced by a single straight dislocation moving at a constant speed v. The limiting behavior of derived expressions is analyzed at v → 0. In particular, the static electro-elastic field of a straight dislocation in a piezoelectric is expressed in a compact form similar to the known Barnett-Swanger formula describing dislocation distortions in a purely elastic medium. It is shown that the thermal component of dislocation fields may be essential only for fast dislocations with the speed v of order of the sound velocity. The temperature distribution around the moving dislocation is also analytically studied in details for the case of non-piezoelectric elastic materials. It is shown that metals and ionic crystals manifest a qualitative difference in described thermal effects. The results of the developed theory are compared with the existing experimental data. It is shown that there is a quantitative agreement between them.     

Probability of cleavage fracture for a crack propagating by fatigue

Standard fracture toughness tests use fatigue pre-cracked specimens loaded monotonically from zero to failure. Scatter in toughness (cleavage) occurs because steel is metallurgically inhomogeneous, and because each specimen has its crack tip in a different local microstructure. A probability of fracture toughness distribution can be obtained by conducting multiple repeat tests on the same steel. This is often used to make probabilistic structural fracture predictions for combinations of crack length and applied load. However, it is likely the true structural situation involves gradual extension of a fatigue crack under a cyclic load. The question then arises as to how often the probability of fracture for the structure needs to be re-calculated. It could be argued that each fatigue load cycle moves the crack tip to a new position and gives a different instantaneous probability of fracture. But if this were the case, the predicted cumulative probability of fracture would quickly tend to unity. This paper describes cold temperature, wide plate fatigue tests designed to investigate this apparent contradiction. The steel is 15 mm thick, grade A, ship plate and the tests involve propagation of a fatigue crack from 300 mm to 650 mm length under a constant amplitude fatigue cycle of 10-100 MPa at −50 °C. The cold temperature fatigue tests do not show an obviously increased probability of fracture compared with the standard monotonic load tests. Nevertheless, in view of uncertainties surrounding the issue, a cumulative probability of fracture determined at 5 mm intervals through the steel is recommended for safe structural predictions.     

Low-frequency scattering from perfectly conducting spheroidal bodies in a conductive medium with magnetic dipole excitation

Inductive electromagnetic means that are currently employed in the exploration of the Earth’s subsurface and embedded voluminous bodies often call for an intensive use, primary at the modeling stage and later on at the inversion stage, of analytically demanding tools of field calculation. Under the aim of modeling implementation, this contribution is concerned with some interesting aspects of the low-frequency interaction of arbitrarily orientated (i.e. three-dimensional) time-harmonic magnetic dipoles, with 3-D perfectly conducting spheroidal bodies embedded in an otherwise homogeneous conductive medium. For many practical applications involving buried obstacles such as Earth’s subsurface electromagnetic probing at low-frequency or any other physical cases (e.g. geoelectromagnetics), non-axisymmetric spheroidal geometry approximates sufficiently such kind of metallic shapes. On the other hand, our analytical approach deals with prolate spheroids, since the corresponding results for the oblate spheroidal geometry can be readily obtained through a simple transformation. The particular physical model concerns a solid impenetrable (metallic) body under a magnetic dipole excitation, where the scattering boundary value problem is attacked via rigorous low-frequency expansions for the incident, scattered and total electric and magnetic fields in terms of positive integral powers of (ik), that is (ik)ⁿ for n ? 0, where k stands for the complex wavenumber of the exterior medium. The purpose of the modeling is to contribute to a simple yet versatile tool to infer information on an unknown body from measurements of the three-component electric and magnetic fields nearby. Our goal is to obtain the most important terms of the low-frequency expansions of the electromagnetic fields, that is the static (for n = 0) and the dynamic (n = 1, 2, 3) terms. In particular, for n = 1 there are no incident fields and thus no scattered ones, while for n = 0 the Rayleigh electromagnetic expression is easily obtained in terms of infinite series. Emphasis is given on the calculation of the next two non-trivial terms (at n = 2 and at n = 3) of the aforementioned fields. Consequently, those are found in closed form from exact solutions of coupled (at n = 2, to the one at n = 0) or uncoupled (at n = 3) Laplace equations and they are given in compact fashion, as infinite series expansions for n = 2 or finite forms for n = 3. Nevertheless, the difficulty of the Poisson’s equation that has to be solved for n = 2 is presented, whereas our analytical approach demands the use of the well-known cut-off method in order to obtain an analytical closed solution. Finally, this research adds useful reference results to the already ample library of scattering by simple shapes using analytical methods.     

Finite element analysis of piezoelectric corner configurations and cracks accounting for different electrical permeabilities

Based on eigenfunctions of asymptotic singular electro-elastic fields obtained from a kind of ad hoc finite element method [Chen MC, Zhu JJ, Sze KY. Finite element analysis of piezoelectric elasticity with singular inplane electroelastic fields. Engng Fract Mech 2006;73(7):855-68], a super corner-tip element model is established from the generalized Hellinger-Reissner variational functional and then incorporated into the regular hybrid-stress finite element to determine the coefficients of asymptotic singular electro-elastic fields near a corner-tip. The focus of this paper is not to discuss the well-known behavior of electrically impermeable and permeable (usually it means fully permeable, hereinafter the same) cracks but analyze the limited permeable crack-like corner configurations embedded in the piezoelectric materials, i.e., study the influence of a dielectric medium inside the corner on the singular electro-elastic fields near the corner-tip. The boundary conditions of the impermeable or permeable corner can be considered as simple approximations representing upper and lower bounds for the electrical energy penetrating the corner. Benchmark examples on the piezoelectric crack problems show that present method yields satisfactory results with fewer elements than existing finite element methods do. As application, a piezoelectric corner configuration accounting for the limited permeable boundary condition is investigated, and it is found that the limited permeable assumption is necessary for corners with very small notch angles.     

Elastic anisotropy effect on indentation-induced thin film interfacial delamination

Effect of elastic anisotropy on indentation-induced thin film interfacial delamination, especially, at the initiation and early growth stage, is examined. The indentation load is modeled as a constant pressure over an expanding semi-spherical cavity. The delamination process is approached by a cohesive zone model. The rest of the problem is formulated within the general anisotropic elasticity theory, and solved numerically by the boundary element method employing a special Green’s function for multilayers. The material system of a Cu(0 0 1) film on a Si(0 0 1) substrate is studied as an example. The interfacial damage initiation and crack development under indentation are captured in the simulation. By comparing the predictions with the materials being modeled as isotropic and as anisotropic (of the cubic symmetry as they are), it is shown that the elastic anisotropy of the copper film plays a significant role in determining the delamination pattern. In the isotropic model, the delamination crack fronts are circular reflecting the problem axisymmetry. In contrast, crack fronts are square with rounded corners in the anisotropic case. This significant difference necessitates a three-dimensional anisotropic stress analysis of the indentation-induced delamination of strongly anisotropic films.     

Piezoelectric study for a coated hole of quasi-polygonal shape in an infinite plate

This paper studies the piezoelectric problems for a coated hole of quasi-polygonal shape embedded in an infinite matrix subjected to electromechanical loadings. The electromechanical loadings considered in this work include a screw dislocation, a point force, a point charge, a far-field anti-plane shear and an in-plane electric field. Each component is assumed to be transversely isotropic medium belonging to a hexagonal crystal class 6 mm and poled in the x₃ direction. Based on the complex variable and analytical continuation method, the general expressions for the complex potentials can be derived in each medium. Numerical results are provided to show the effect of hole shape, the material combinations and the loading condition on the electro-elastic fields due to the presence of the coated film. The image force exerted on a dislocation, which can be used to probe the mobility of the screw dislocation, will be calculated by means of the generalized Peach-Koehler formula.     

Hot wire configuration for depositing device grade nano-crystalline silicon at high deposition rate

The University of Barcelona is developing a pilot-scale hot wire chemical vapor deposition (HW-CVD) set up for the deposition of nano-crystalline silicon (nc-Si:H) on 10 cm × 10 cm glass substrate at high deposition rate. The system manages 12 thin wires of 0.15-0.2 mm diameter in a very dense configuration. This permits depositing very uniform films, with inhomogeneities lower than 2.5%, at high deposition rate (1.5-3 nm/s), and maintaining the substrate temperature relatively low (250 °C). The wire configuration design, based on radicals’ diffusion simulation, is exposed and the predicted homogeneity is validated with optical transmission scanning measurements of the deposited samples. Different deposition series were carried out by varying the substrate temperature, the silane to hydrogen dilution and the deposition pressure. By means of Fourier transform infrared spectroscopy (FTIR), the evolution in time of the nc-Si:H vibrational modes was monitored. Particular importance has been given to the study of the material stability against post-deposition oxidation.     

Ultra-high strength Mg-Li based bulk metallic glasses: Preparation and performance research

In this paper, new Mg-Li based bulk metallic glasses (BMGs) are prepared by conventional copper mold injection casting method. The alloys exhibit excellent mechanical properties, such as ultra-high compressive fracture strength (maximal 729 MPa), high Vickers hardness (>2 GPa) and low elastic modulus (∼35 GPa). Compared with the corresponding crystal alloys, the density of the amorphous alloy samples is reduced by about 1.5% due to their free volume. Thus, it is believed that this new BMGs with these outstanding properties will broaden Mg-Li based alloys’ application fields.     

OM4 bend insensitive multi-mode fibers’ usefulness for MCM integration

For future generations of electronic systems, a severe bottleneck is expected on the interconnection level and the use of optical interconnection is considered as one of the most promising solutions in this matter. Recent progress in fiber development resulted in new generation of optical fibers that are bend insensitive. This makes them ideal for Multi Chip Module (MCM) application. This paper focuses on OM4 bend insensitive multi-mode fibers’ usefulness for MCM integration, particularly the investigation of MM fiber loss is presented, which is influenced by bend diameter and the fiber's mechanical performance under influence of high temperature (400 °C–1000 °C adequate to MCM production process).     

Finite strip with a central crack under tension

In this study, a symmetrical finite strip with a length of 2L and a width of 2h, containing a transverse symmetrical crack of width 2a at the midplane is considered. Two rigid plates are bonded to the ends of the strip through which uniformly distributed axial tensile load of magnitude 2hp₀ is applied. The material of the strip is assumed to be linearly elastic and isotropic. Both edges of the strip are free of stresses. Solution for this finite strip problem is obtained by means of an infinite strip of width 2h which contains a crack of width 2a at y = 0 and two rigid inclusions of width 2c at y = ±L and which is subjected to uniformly distributed axial tensile load of magnitude 2hp₀ at y = ±∞. When the width of the rigid inclusions approach the width of the strip, i.e., when c → h, the portion of the infinite strip between the inclusions becomes identical with the finite strip problem. Fourier transform technique is used to solve the governing equations which are reduced to a system of three singular integral equations. By using the Gauss–Jacobi and the Gauss–Lobatto integration formulas, these integral equations are converted to a system of linear algebraic equations which is solved numerically. Normal and shearing stress distributions and the stress intensity factors at the edges of the crack and at the corners of the finite strip are calculated. Results are presented in graphical and tabular forms.     

Characterization of a low frequency magnetic noise from a two-stage pulse tube cryocooler

Magnetic noise of a two-stage pulse tube cryocooler (PT) was measured by a fundamental mode orthogonal fluxgate magnetometer and by a LTS Double Relaxation Oscillation SQUID (DROS) first-order planar gradiometer. The magnetometer was installed in a dewar made of aluminum at 12 cm distance from a section containing magnetic regenerative materials of the second pulse tube. The magnetic noise spectrum showed a clear peak at 1.8 Hz, which is the fundamental frequency of the He gas pumping rate. The 1.8 Hz magnetic noise registered a peak, during the cooling down process, when the second cold-stage temperature was around 12 K, which is well correlated with the 1.8 Hz variation of the temperature of the second cold stage. Hence, we attributed the main source of this magnetic noise to the temperature variation of the magnetic moments resulting from magnetic regenerative materials, Er₃Ni and HoCu₂, in the presence of background static magnetic fields. We have also pointed out that the superconducting magnetic shield of lead sheets reduced the low frequency magnetic noise generated from the magnetic regenerative materials. With this arrangement, the magnetic noise amplitude measured with the LTS DROS gradiometer, mounted at 7 cm horizontal distance from the magnetic regenerative materials, in the optimum condition, was lower than 500 pT peak-to-peak, whereas the noise level without lead shielding was higher than the dynamic range of DROS instrumentations which was around .     

Transfer matrix solutions for three-dimensional consolidation of a multi-layered soil with compressible constituents

In this paper, transfer matrix solutions for three-dimensional consolidation of a multi-layered soil considering the compressibility of pore fluid are presented. The derivation of the solutions starts with the fundamental differential equations of Biot’s three-dimensional consolidation theory, takes into account the compressibility of pore fluid in the Cartesian coordinate system, and introduces the extended displacement functions. The relationship of displacements, stresses, excess pore water pressure, and flux between the ground surface (z = 0) and an arbitrary depth z is established for Biot’s three-dimensional consolidation problem of a finite soil layer with compressible pore fluid by taking the Laplace transform with respect to t and the double Fourier transform with respect to x and y, respectively. Based on this relationship of the transfer matrix, the continuity between layers, and the boundary conditions, the solutions for Biot’s three-dimensional consolidation problem of a multi-layered soil with compressible constituents in a Laplace-Fourier transform domain is obtained. The final solutions in the physical domain are obtained by inverting the Laplace-Fourier transforms. Numerical analysis is carried out by using a corresponding program based on the solutions developed in this study. This analysis demonstrates that the compressibility of pore fluid has a remarkable effect on the process of consolidation.     

Bleustein-Gulyaev SAW and its associated quasi-particle

Following a previous work that dealt with the case of Rayleigh surface waves in pure elasticity, here it is shown that a quasi-particle with Newtonian point-like mechanics (equation of inertial motion, expression of the kinetic energy) can be associated with the celebrated Bleustein-Gulyaev surface waves of linear piezoelectricity. This association is based on the integration over a vertical band of the sagittal plane of the canonical balance laws that accompany, via Noether’s theorem, the basic field equations. It accounts for the boundary conditions at the limiting surface, the periodicity of the solution in propagation space, and the vanishing of all fields at infinity in the substrate or in the outside vacuum. The proof benefits from the fact that the average (over one wavelength of propagation) of the Lagrangian density at the limiting surface is proportional to the satisfied “dispersion relation”, and hence is zero. The expression found for the “mass” of the said quasi-particle is informative in that it contains information about the frequency, the amplitude of the signal, and the electromechanical coupling.     

An indirectly heated solid cathode for linear electron beam emitter assembly

The paper reports design and development of an indirectly heated solid cathode for a linear electron beam emitter assembly. The solid cathode is made of a 195 mm long 8 mm wide and 2 mm thick tantalum bar with varying cross sections along its length. The emitting surface of the cathode is 110 mm in length and 4 mm in width. Thermionic electron beams emitted from a directly heated strip shaped filament have been used to heat the cathode indirectly. The temperature of the cathode, measured by a two color pyrometer, was raised to 2500 K. The innovative design of the cathode and its holding mechanism accommodates thermal expansion at high temperature, and prevents thermal deformation of the cathode. The cathode is suitable for use in strip electron beam gun and for laboratory experiments on melting and evaporation of refractory materials.     

Oxolane-2,5-dione modified electrospun cellulose nanofibers for heavy metals adsorption

Functionalized cellulose nanofibers have been obtained through electrospinning and modification with oxolane-2,5-dione. The application of the nanofibers for adsorption of cadmium and lead ions from model wastewater samples is presented for the first time. Physical and chemical properties of the nanofibers were characterized. Surface chemistry during preparation and functionalization was monitored using Fourier transform-infrared spectroscopy, scanning electron microscopy, carbon-13 solid state nuclear magnetic resonance spectroscopy and Brunauer Emmett and Teller. Enhanced surface area of 13.68 m² g⁻¹ was recorded for the nanofibers as compared to the cellulose fibers with a surface area of 3.22 m² g⁻¹. Freundlich isotherm was found to describe the interactions better than Langmuir: K_f = 1.0 and 2.91 mmol g⁻¹ (r² = 0.997 and 0.988) for lead and cadmium, respectively. Regenerability of the fiber mats was investigated and the results obtained indicate sustainability in adsorption efficacy of the material.     

Permanent magnet steerers for canted undulators at the ESRF

A number of canted undulators will be installed at the ESRF. Permanent Magnet Steerers will be used for providing canting angles up to 5.4 mrad. These steerers have been optimized in view of the limited space available and the required field quality. The magnetic interaction between the steerers and the undulators may lead to additional optical phase errors: auxiliary poles have been added to obtain a fast decrease of the fringe field. The transverse homogeneity of the magnetic field integral is another strong design constraint. Good homogeneity was reached by optimizing the profiles of the iron poles. Eight steerers have been constructed and magnetic measurements have been performed.