Eddy currents in a gradient coil,modeled as circular loops of strips

As a model for the z-coil of an MRI-scanner a set of circular loops of strips, or rings, placed on one cylinder is chosen. The current in this set of thin conducting rings is driven by an external source current. The source, and all excited fields, are time-harmonic. The frequency is low enough to allow for an electro-quasi-static approach. The rings have a thin rectangular cross-section with a thickness so small that the current can be assumed uniformly distributed in the thickness direction. Due to induction, eddy currents occur resulting in an edge-effect. Higher frequencies cause stronger edge-effects. As a consequence, the resistance of the system increases and its self-inductance decreases. The Maxwell equations imply an integral equation for the current distribution in the rings. The Galerkin method with Legendre polynomials as global basis functions is applied. This method shows fast convergence, so only a very restricted number of basis functions is needed. The general method is worked out for N (N ≥ 1) rings, and explicit results are presented for N = 1, N = 2 and N = 24. The derived integral equation and the numerical results of the simulations show that sets of circular rings and plane strips describe the same electromagnetic behavior, thus demonstrating that inductance effects are local.     

加层压电条界面裂纹的稳态扩展

研究当压电条同时与两个不同材料的弹性条粘接在一起,在反平面机械载荷及面内电载荷联合作用下,长度不变的有限Griffith 界面裂纹沿加层压电条界面以常速稳态扩展时裂尖的动态断裂问题。应用Fourier积分变换将问题化为以第二类Fredholm积分方程表示的对偶积分方程,导出了相应的动应力强度因子表达式。给出了动应力强度因子与裂纹传播速度、裂纹长度、压电条及弹性条厚度、电荷载大小及方向的关系曲线。研究结果对结构设计及结构失效的预防具有理论和应用价值。     

Eddy currents in MRI gradient coils configured as rings and patches

The current distribution in a set of parallel rings and patches, called islands, positioned at the surface of a cylinder, is investigated. The current is driven by an externally applied source current. The islands are rectangular pieces of copper (patches) placed in parallel between the rings. The eddy currents in the islands induce currents in the rings that vary in the tangential direction. From the quasi-static Maxwell equations, an integral equation for the current distribution in the strips is derived. The Galerkin method, using global basis functions, is applied to solve this integral equation. It shows fast convergence. The global basis functions are Legendre polynomials in the axial direction and 2π-periodic trigonometric functions in the tangential direction. The Legendre polynomials efficiently cope with the singularity of the kernel function of the integral equation. Explicit numerical results are shown for three configurations. Apart from the current distributions, the resistance and self-inductance of the three systems of rings and islands are computed. The resulting tool can be used to qualitatively understand eddy currents in z-gradient coils, and as such enable the incorporation of eddy currents in the optimization of gradient-coil design.     

Free vibration analysis and shape optimization of prismatic folded plates and shells with circular curved planform

This paper deals with the elastic free vibration analysis and structural shape optimization of prismatic folded plate and shell structures with circular curved planform. The structures are supported on diaphragms at two opposite edges. The basic formulation of a family of curved variable thickness C(0) Mindlin–Reissner finite strips is presented. The accuracy and performance of these newly developed strips are explored through a series of examples including annular plate sectors, a box girder bridge and a cylinder with an interior longitudinal plate. Numerical results obtained are compared with results from other sources. The whole shape optimization process is carried out by integrating finite strip analysis, cubic spline shape and thickness definition, sensitivity analysis and mathematical programming. The objective is either the maximization of the fundamental frequency or the minimization of volume by changing the shape or thickness variation of the cross-section of the structure with constraints on the volume or natural frequencies. Several examples are included to illustrate and highlight various features of the optimization, including annular sector plates, a curved box girder bridge and a cylinder shell segment with curved pianform.     

Static analysis of diaphragm-supported cylindrical shells using a curved finite strip

Two versions are presented of a tranversely-curved finite strip for the static analysis of circular cylindrical shells. The strip is a refined displacement model with sixteen transverse degrees-of-freedom arranged either at two or at four nodal lines. The analysis is specialized for cylinders with diaphragm supports at the ends—via the employment of simple trigonometric series in the longitudinal direction—but extension to other end conditions is straightforward in principle. Detailed results of applications of the curved strip are presented.     

Transverse electric scattering by a dielectric biconvex cylinder loaded with a shallow circular trough in a ground plane

The scattering problem of transverse electric wave from a dielectric biconvex cylinder buried in a shallow circular trough of a ground plane is investigated and a rigorous series solution is also derived. Based on the region-matching method, the analysed region is decomposed into two subregions by introducing a semi-circular auxiliary boundary. The magnetic field of each subregion is expressed in terms of cylindrical wave functions with unknown expansion coefficients. After imposing the matching conditions and the boundary condition on the trough surface with the aid of Graf's addition theorem, the unknown coefficients are determined. Comparisons with published data for a dielectric circular cylinder case show very good agreement. Visible effects of depth-to-half-width ratios of a dielectric biconvex cylinder on echo width, far- field pattern and near-field distribution are illustrated in graphical form.     

A general expression for an area integral of a point-wise J for a curved crack front

The expression for the J-integral at a point on a three-dimensional crack front, obtained from a surface independent integral, is in general a sum of a contour integral and an area integral. In this work a general expression of an area integral for a crack with a curved front is derived in curvilinear coordinates. In certain situations the area integral vanishes and previously known cases are a straight crack front in plane stress or plane strain. The general conditions for a vanishing area integral are studied. It is shown that the area integral is non-zero for cracks with a curved front in the direction of crack extension. Some examples of curved cracks are given, for which the area integral vanishes and that are of interest in practice.     

The plane elastostatic solution for a symmetrically loaded crack in a strip composite

The plane elastostatic problem for a crack in a strip composite loaded with normal or shearing traction is reduced to a single integral equation. The dependence of the solution on the material parameters is exhibited explicitly in the integral equation through two composite parameters. The integral equation is solved numerically and the dependence of the stress intensity factors on the material parameters is displayed graphically for all physically relevant composites for each of several chosen values of the crack length to strip width ratio.     

Current distribution in circular planar coil

The current distribution over the cross section of a planar circular coil is calculated by a Fredholm integral equation technique. An external applied current source is driving the current. The integral equation technique is applied over a two-dimensional cross section of the coil while considering infinitesimally thin windings. The coil windings are divided into equally sized one-dimensional elements. The resulting algebraic system is solved numerically. For low frequencies, the current distribution follows the 1/r behavior. As the frequency increases, the influence of the proximity effect is taken into account. Different cases are studied examining the intensity of these effects on the current distribution as the number of turns, the width of the windings, and the spacing between the turns are varying.     

Numerical solutions of a hypersingular integral equation for antiplane elastic curved crack problems of circular regions

Summary. In this paper, a hypersingular integral equation for the antiplane elasticity curved crack problems of circular regions is suggested. The original complex potential is formulated on a distribution of the density function along a curve, where the density function is the COD (crack opening displacement). The modified complex potential can also be established, provided the circular boundary is traction free or fixed. Using the proposed modified complex potential and the boundary condition, the hypersingular integral equation is obtained. The curve length method is suggested to solve the integral equation numerically. By using this method, the usual integration rule on the real axis can be used to the curved crack problems. In order to prove that the suggested method can be used to solve more complicated cases of the curved cracks, several numerical examples are given.     

Finite element visualization of fatigue crack closure in plane stress and plane strain

Elastic-plastic finite element simulations of growing fatigue cracks in both plane stress and plane strain are used as an aid to visualization and analysis of the crack closure phenomenon. Residual stress and strain fields near the crack tip are depicted by both color fringe plots and x-y graphs. Development of the residual plastic stretch in the wake of a growing plane stress fatigue crack is shown to be associated with the transfer of material from the thickness direction to the axial direction. Finite element analyses indicate that crack closure does occur under pure plane strain conditions. The development of the residual plastic stretch in plane strain is shown to be associated with the transfer of material from the in-plane transverse direction to the axial direction. This in-plane contraction also leads to the generation of complex residual stress fields. The total length of closed crack at minimum load in plane strain is shown to be a small fraction of the total crack length, especially for positive stress ratios. This suggests that experimental measurement of plane strain closure would be extremely difficult, and may explain why some investigators have concluded that closure does not occur in plane strain.     

Current distribution in a parallel set of conducting strips

The current distribution in a parallel set of thin conducting sheets due to an external applied source is investigated. All sheets are placed in one plane. The source, and all excited fields, are time-harmonic. The frequency is low enough to allow for an electro quasi-static approximation (neglecting the displacement current). The conducting sheets are infinitely long and the current is uniform in the longitudinal direction of the sheets. The sheets have a thin rectangular cross-section, so thin that the current can be assumed uniform in the thickness-direction. Hence, the current distribution only depends on the transverse coordinate. Due to the mutual induction between the sheets, the current distribution over the width of the cross-section becomes non-uniform: it accumulates at the edges of the sheets. It is especially this so-called edge-effect, and its dependence on the applied frequency and the distances between the sheets, that is the aim of this investigation. From the Maxwell equations, a set of integral equations for the current distribution in the sheets is derived. These integral equations are solved, as far as possible by analytical means, by writing the current distribution in each sheet as a series of Legendre polynomials. The general method is worked out for N (N 1) sheets, but explicit results are presented for N=1 and 3. It turns out that the edge-effect becomes stronger for increasing frequencies. For this solution, only a very restricted number of Legendre polynomials are needed.     

Mathematical analysis of thermoelastic characteristics in plasma-sprayed thermal barrier coatings

The thermoelastic characteristics of plasma-sprayed thermal barrier coatings (TBCs) have been analyzed using mathematical modeling. Two types of TBC model, cylinder and circular disk which are commercial plasma-sprayed TBCs, subjecting to symmetric temperature distribution to the radial and longitudinal directions, respectively, were taken into consideration. Based on the thermoelastic theories, a second order ordinary differential equation was derived for the cylinder model and a pair of partial differential equations were set up for the circular disk model. The analytic solution was obtained from the ordinary differential equation, while a finite volume method was developed for numerical solutions to the pair of partial differential equations due to the complexity of governing equations. The thermoelastic characteristics of TBC models, such as temperature distributions, displacements, and stresses, were displayed according to the obtained solutions. The rate of heat conduction in the section of the top coat is relatively slow in comparison with the substrate, and no profound difference appears in the temperature distribution between two TBC models. The highest longitudinal tensile stress is expressed at the bond coat of both models, and the substrate is under the compressive stresses to the circumferential direction. While the cylinder expands to the positive longitudinal direction only, the expansion in the circular disk occurs to both the positive and negative longitudinal directions. Relatively large displacement and stresses exhibit in the cylinder as compared with the circular disk. In the circular disk, the stresses to the radial direction undulate at each section, and the displacement profile displays that the width of the circular disk is slightly narrowed. The results demonstrate that the mechanical and thermal properties of the top and bond coats are the crucial factors to be considered in controlling the thermoelastic characteristics of plasma-sprayed TBCs.     

Boundary-value problem of electroelasticity for a piezoceramic elliptic cylinder and a layer with tunnel hollow

We consider a skew-symmetric stressed state of a piezoceramic layer weakened by a through elliptic hollow filled with air. The surfaces of the layer (or a finite elliptic piezoceramic cylinder) operating in contact with air are covered with a diaphragm rigid in its plane and flexible in the perpendicular direction. The stress vector {X ₁, X ₂, X ₃} acts on the lateral surface. The boundary-value problem of electroelasticity is reduced to a system of singular integral equations. This system is numerically solved by the method of mechanical quadratures. The integral representations of the solutions are constructed on the basis of the corresponding matrix of Φ-solutions. The distribution of normal circular stresses over the elliptic contour subjected to the action of normal or tangential stresses is investigated. __________ Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 43, No. 2, pp. 49–54, March–April, 2007.     

Meissner shielding currents and magnetic flux penetration in thin-film superconductors

We have observed the magnetic flux penetration at 4.2 K into superconducting lead films of about 4 µm thickness using a high-resolution magnetooptical technique. The specimens had the shape of small squares or long strips. In increasing magnetic field it was found that magnetic flux starts to penetrate into the bulk of the specimens in the form of flux tubes containing about 60 flux quanta. For the samples with the strip geometry the magnetic fieldH*, at which flux penetration into the bulk sets in, was investigated in the presence of an external electrical currentI along the strip superimposed on the Meissner shielding currents. The fieldH* was found to decrease linearly with increasing transport currentI. The magnetic field near a long superconducting cylinder in the presence of both a transverse applied magnetic field and an applied electrical current parallel to the cylinder axis has been calculated. The calculation is valid for a cylinder of elliptical cross section, which approximates a long, flat strip. The experimental observations for the slope H*/I are in good agreement with the calculations.Work performed in part in the Argonne National Laboratory and in part in the Ames Laboratory of the U.S. Atomic Energy Commission.     

Axisymmetric deformation of a short magneto-strictive current-carrying cylinder

The title-problem has been reduced to that of solving a Fredholm integral equation of the second kind. One end of the cylinder is assumed to be fixed, while the cylinder is deformed by an axial current. The vertical displacement on the upper flat end of the cylinder has been determined from an iterative solution of the Fredholm equation valid for large values of the length. The radial displacement of the curved boundary has also been determined at the middle of the cylinder, by using the iterative solution.     

Ductile penny-shaped crack in a transversely isotropic cylinder

The problem of a penny-shaped crack contained in a transversely isotropic cylinder of elastic perfectly-plastic material is considered for the case when the crack is extended by an axial load. The problem is reduced to solving numerically a Fredholm integral equation of the second kind for the width of the plastic zone. Graphical results are presented showing the effect of transverse isotropy upon the width of the plastic zone and these are compared with the results for isotropic materials.     

Plastic zones in a transversely isotropic solid cylinder containing a ring–shaped crack

In this study, a problem of a ring shaped-crack contained in a infinitely long solid cylinder of elastic perfectly-plastic material is considered.The problem is formulated for a tranversely isotropic matrial by using integral transform technique under uniform load. Due to the geometry of the configuration, Hankel and Fourier integral transform techniques are chosen and the problem is reduced to a singular integral equation. This integral equation is solved numerically by using Gaussian Quadrature Formulae and the values are evaluated for various for discrete points. The plastic zone widths are obtained by using the plastic strip model. They are plotted for various ring-shaped crack sizes and transversely isotropic matrials. It is found that the width of the plastic zone at the inner tip of the crack is greater than the outer one.     

Impact response of a finite crack in an orthotropic strip

Summary The elastodynamic response of a finite crack in an infinite orthotropic strip under normal impact is investigated in this study. The crack is situated symmetrically and oriented in a direction normal to the edges of the strip. Laplace and Hankel transforms are used to reduce the transient problem to the solution of a pair of dual integral equations in the Laplace transform plane. The solution to the dual integral equations is then expressed in terms of a Fredholm integral equation of the second kind. Numerical values on the dynamic stress intensity factor for some fiber-reinforced composite materials are obtained and the results are graphed to display the influence of the material orthotropy.