Plate Of Finite Thickness Research Articles

Semi-analytical approach for electromagnetic field diffusion using hertz potential

(Paper written in French) This paper contains two parts: In the first part the Hertz vectorial potential is introduced and its possible use in solving electromagnetic diffusion problems common in electrical engineering is considered. Links between this potential and other commonly used potentials are established, and a formulation procedure using three vectorial equations is proposed. This approach has several advantages due to its simplicity and the fact that it requires only a few values for computation. This method is next applied efficiently to the interaction in the static regime between a line conductor crossed by a sine-shaped current and a finite-thickness metallic plate. In the second part, a specific field quadrature method is employed in integral form. This allows the confirmation of the methods accuracy and good numerical stability.

Elastoplastic strain concentration factors in finite thickness plates

The paper presents a comparison of a detailed finite element modelling of elastoplastic strains at a notch root with experimental Moire interferometric data. The three-dimensional nature of the local constraint at a notch root for elastic or elastoplastic material behaviour is confirmed. The elastoplastic analysis shows that the stress concentration factor ratio from the mid-plane and the surface is practically insensitive to the actual σ—ε. relationship when the nominal stress achieves the yield stress.

CTOD for the through-the-thickness crack in a plate of arbitrary thickness

The sectional thickness of plate-like components has a significant influence on the fatigue and fracture properties. This effect is primarily due to the differences in the through-the-thickness stresses prevailing at the tip of a crack in a finite-thickness plate or shell. Characterization of this effect to date has remained largely empirical. The current paper presents new analytical results for CTOD for the through-the-thickness crack in infinite plates with various thicknesses. These results are based on a recently developed solution for an edge dislocation in infinite plate of arbitrary thickness. The analytical predictions of the CTOD and the constraint factor are compared with the three-dimensional FE results. It is shown that the analytical and the numerical results are in good agreement when the numerical calculations are not affected by the size of the FE mesh and by the boundaries of the FE model.

Fluid–Fluid Correlation through a Model Charged Membrane: Analytical Results

We study the properties of electrical double layers separated by a charged plate of finite thickness. The nonlinear Poisson–Boltzmann equation is analytically solved for this system. It is shown that, for an unsymmetrically charged, narrow plate, the charged fluids at both sides of the plate are strongly correlated, and the local electroneutrality condition (LEC) is not satisfied. The LEC is satisfied only for an infinitely thick plate. Analytical expressions for the induced charge and mean electrostatic potential are given and analyzed. These findings could be relevant for the understanding of protein adsorption on membranes.

Elastic T-stress solutions for semi-elliptical surface cracks in finite thickness plates

Three-dimensional finite element analyses have been conducted to calculate the elastic T-stress for semi-elliptical surface cracks in finite thickness plates. Far-field tension and bending loads were considered. The analysis procedures and results were verified using both exact solutions and approximate solutions. The T-stress solutions are presented along the crack front for cracks with a/ t values of 0.2, 0.4, 0.6 or 0.8 and a/ c values of 0.2, 0.4, 0.6 or 1.0. Based on the present finite element calculations for T-stress, empirical equations for the T-stress at three locations: the deepest, the surface and the middle points of the crack front under tension or bending are presented. The numerical results are approximated by empirical formulae fitted with an accuracy of 1% or better. They are valid for 0.2⩽ a/ c⩽1 and 0⩽ a/ t⩽0.8. These T-stress results together with the corresponding K or J values for surface cracks are suitable for the analysis of constraint effects for surface cracked components.

Stress intensity factors and weight functions for deep semi‐elliptical surface cracks in finite‐thickness plates

ABSTRACT Three‐dimensional finite element analyses have been conducted to calculate the stress intensity factors for deep semi‐elliptical cracks in flat plates. The stress intensity factors are presented for the deepest and surface points on semi‐elliptic cracks with a/t‐values of 0.9 and 0.95 and aspect ratios (a/c) from 0.05 to 2. Uniform, linear, parabolic or cubic stress distributions were applied to the crack face. The results for uniform and linear stress distributions were combined with corresponding results for surface cracks with a/t= 0.6 and 0.8 to derive weight functions over the range 0.05 ≤ a/c ≤ 2.0 and 0.6 ≤ a/t ≤ 0.95. The weight functions were then verified against finite element data for parabolic or cubic stress distributions. Excellent agreements are achieved for both the deepest and surface points. The present results complement stress intensity factors and weight functions for surface cracks in finite thickness plate developed previously.

Thermal capacity effect on transient free convection adjacent to a vertical surface in a porous medium

In this paper we analyse how the presence of the thermal capacity of a vertical flat plate of finite thickness, which is embedded in a porous medium affects the transient free convection boundary-layer flow. At the time t = 0, the plate is suddenly loaded internally with a constant heat flux rate q″, so that a transient boundary-layer flow is initiated adjacent to the plate. Initially, the transient effects due to the imposition of the uniform heat flux rate at the plate are confined to a thin fluid region near to the surface and are described by a small time solution. These effects continue to penetrate outwards and eventually evolve into a new steady state flow. Analytical solutions have been derived for these transient (small time) and steady state (large time) flow regimes, which are then matched by a numerical solution of the full boundary-layer equations. It has been found that the non-dimensional fluid temperature (or fluid velocity) profiles are reduced when the thermal capacity effects, described by a parameter Q *, are reduced. For small values of Q *, the approach of these profiles to their steady state values is monotonic. However, for large values of Q *, the temperature profiles are observed to locally exceed (pass through a maximum value) the final steady state values at certain distances from the plate. In general, the maxima in the temperature profiles increase in size as Q * increases and the time taken to approach the steady state solutions increases significantly.

Effect of alignment on the penetration of segmented rods

The purpose of this investigation is to study the effect of alignment on the performance of segmented penetrators. The Eulerian wave propagation code CTH is used for this purpose. A series of calculations using four L/D = 1 tungsten alloy segment trains at varying degrees of misalignment is performed, impacting a single finite-thickness oblique armor steel plate. Obliquity angles of 30° and 60° were considered. This study was performed primarily to investigate the effects of obliquity and is a continuation of a previous study [1] where semi-infinite armor steel plates were examined. It is shown that the obliquity of the plate can have a significant influence on the performance of the segment train. When misalignment is minimal, the performance of the segment train is not adversely affected, particularly if the misalignment positions the train in an orientation aligned with the plate normal. However, for large misalignments, degradation to the performance of the segment train is significant at all orientations.

A unified formalism using effective surface permittivity to study acoustic waves in various anisotropic and piezoelectric multilayers.

A unified formalism is presented that uses the effective surface permittivity (ESP) to study surface acoustic waves (SAW) in layered substrates and guided waves in layered plates. Based on known mathematical tools, such as ordinary differential equation and transfer matrix, a generalized surface impedance (GSI) concept is developed and exploited to investigate the acoustic propagation in various anisotropic and piezoelectric layered structures. The ESP function, originally defined for the surface of a homogeneous and semi-infinite piezoelectric substrate, is extended to both the top surface of and an interface in a layered half space, as well as to either surface of a finite-thickness plate. General ESP expressions for all mentioned configurations are derived in terms of an equivalent GSI matrix. It is shown that, when using the appropriate GSI matrices, the same form of the ESP expressions applies no matter whether the structure is a homogeneous half space alone or coated with a layered plate or a layered plate alone. GSI matrices are explicitly given in terms of the bulk partial mode solutions for a substrate and via the transfer matrix for a plate. Modified GSI matrices for structures consisting of both a plate and a substrate are also specified. Analytical development is fully detailed to suit program implementation. To illustrate its versatility, the formalism is also applied to two-substrate configurations, allowing one to analyze guided waves in a plate sandwiched between and interfacial waves existing along the boundary of two different media. Numerical examples are given to illustrate the spectrum features that the ESP shows for various structures. Deduced ESP expressions allow one to locate directly all piezoelectrically active waves in any structure including at least one piezoelectric layer. Acoustic modes that are not piezoelectrically active and those in non-piezoelectric materials can be also obtained by using the intermediate results, such as derived GSI matrices.

Semi-elliptical surface cracks in finite-thickness plates with built-in ends. I. Stress intensity factor solutions

A crack compliance analysis approach was used to calculate stress intensity factors for surface cracks in flat plates with built-in ends. Stress intensity factor solutions are presented for both constant and linear stress distributions acting on the crack face. The results were verified using finite element analysis and good agreement was obtained. The solutions are suitable for the analysis of surface cracked plates with fixed displacement boundary conditions.

Semi-elliptical surface cracks in finite-thickness plates with built-in ends. II. Weight function solutions

The methods of deriving weight functions from reference stress intensity factors developed for traction type boundary conditions are extended to derive weight functions with displacement boundary conditions. The weight functions for the flat plates with built-in ends were derived from reference stress intensity factors obtained from Part I of the current paper. Validation of the weight functions was conducted by comparing stress intensity factors for several non-linear stress distributions calculated using finite element method. The weight functions are given in closed form, suitable for computer numerical integration. They are suitable for fatigue and fracture analysis of semi-elliptical cracks in complex stress fields with displacement controlled boundary conditions.

Holographic interferometry applied to coupled free convection and radiative transfer in a cavity containing a vertical plate between 290 and 650 K

Holographic interferometry has been applied, in the presence of strong beam deflection effects, to the study of the coupling of gas–gas radiative transfer and free convection in a 0.9 m 3 , isothermal, gastight cavity containing an isothermal plate of finite thickness and filled with N 2 or CO 2. The cavity is heated in the range 290–650 K with the plate at a higher temperature. The focus of this paper is on the validation of an original experimental procedure for the determination of thermal fields in the region close to the plate. Experimental results agree both with independent experimental results using a laser beam deflection technique and, in case of pure free convection, with classical bibliography results related to a vertical wall. It has been shown that gas–gas radiative transfer significantly modifies the temperature field corresponding to free convection and increases the local heat transfer coefficient.

Three-dimensional effects for through-thickness cylindrical inclusions in an elastic plate

A quasi-three-dimensional theoretical closed-form solution for a cylindrical inclusion embedded in an isotropic elastic plate of finite thickness is obtained using the Kane-Mindlin theory. The present solution gives an insight into the transition from plane stress to plane strain. It is shown that strong three-dimensional effects exist near the inclusion even in very thin plates. Moreover, it is shown that two-dimensional plane stress and plane strain solutions are not always the two limits for a three-dimensional problem. The effects of plate thickness and inclusion radius, material constants of the inclusion and the matrix, as well as interference fitting on the stress concentration factor are discussed. The three-dimensional affected zone around the inclusion can be determined from the present solution. The variation of the lateral contraction and shear stresses is discussed. The comparison between the finite element results and the present results shows good agreement.

An experimental study of large-scale vortices over a blunt-faced flat plate in pulsating flow

Laboratory measurements are made of flow over a blunt flat plate of finite thickness, which is placed in a pulsating free stream, U=U o (1+A o cos 2πfpt). Low turbulence-intensity wind tunnel experiments are conducted in the ranges of Stp≤1.23 and A o ≤0.118 at ReH=560. Pulsation is generated by means of a woofer speaker. Variations of the time-mean reattachment length xR as functions of Stp and A o are scrutinized by using the forward-time fraction and surface pressure distributions (Cp). The shedding frequency of large-scale vortices due to pulsation is measured. Flow visualizations depict the behavior of large-scale vortices. The results for non-pulsating flows (A o =0) are consistent with the published data. In the lower range of A o , as Stp increases, xR attains a minimum value at a particular pulsation frequency. For large A o , the results show complicated behaviors of xR. For Stp≥0.80, changes in xR are insignificant as A o increases. The shedding frequency of large-scale vortices is locked-in to the pulsation frequency. A vortex-pairing process takes place between two neighboring large-scale vortices in the separated shear layer.

Turbulent heat transfer around a finite thick plate with incident angle

Turbulent flow and convective heat transfer over an inclined prism is investigated using bound skew hybrid difference scheme. The flow is assumed incompressible two-dimensional recirculating. The k-ε turbulence model with modification to account streamline curvature is used as well as wall function multi-layer turbulent is employed for condition boundary. Numerical computation is carried out for the plate with aspect ratio 4 1 , Re=3×10 4 and incident angels 0, 5, 10, 15 and 20. The fluid flow and heat transfer predictions are compared with experimental measurements and satisfactory agreements is observed. Numerical results provided conclusive evidence that the flow and heat transfer is different substantially on forward and leeward of the plate as the incident angle increases.

Three-dimensional elastic stress fields ahead of blunt V-notches in finite thickness plates

Based on detailed three-dimensional (3D) finite element analyses, elastic fields in front of blunt V-notches in finite thickness plates subjected to uniaxial far-end tensile stress have been investigated. By comparison with the corresponding planar V-notch fields and 3D through-thickness sharp crack fields, various aspects of the 3D fields of the blunt V-notches in finite thickness plates are revealed: (1) The plate thickness and notch angle have obvious effects on the stress concentration factor (SCF) Kt, which is higher in finite thickness plates than in the plane stress and plane strain cases. When the notch angle is smaller than 90°, the SCF is insensitive to the notch angle, but has close relation with the dimensionless plate thickness. With the notch angle increasing further, the SCF decreases and the effect of dimensionless plate thickness on it becomes weaker. (2) For any notch angle considered, the variation of the opening stress σyy normalized by its value σyy0 at the notch-root with the distance x from the root normalized by the root-radius ρ, is insensitive to the plate thickness and coincides well with the two-dimensional (2D) planar solution. (3) The 3D distribution of the out-of-plane constraint factor Tz=σzz/(σyy+σxx) is controlled by the plate thickness (B), the notch-radius (ρ) as well as the notch angle (β), but for deeper V-notches with β≤90°, the distribution of Tz coincides well with that of a U-notch as well as a sharp 3D through-thickness crack and an explicit empirical expression of Tz is presented. (4) The distribution of the in-plane stress ratio Tx=σyy/σxx in front of the 3D V-notch is nearly independent of the plate thickness and coincides well with the corresponding 2D solutions when the opening angle β is smaller than 120°. (5) The gradient of the out-of-plane strain ezz is significant near the free surface in finite thickness plates. On the free surface, the ezz can be 3.5 times the value on the mid-plane, and the through-thickness gradient of the ezz increases with decreasing notch angle. It is of interest to note that most of the field quantities ahead of V-notches are insensitive to the notch angles when the notch angle is smaller than 90°.

Stress Analysis of an undulated plate by the Combined First Order Perturbation and Alternating Method

A new solution method is proposed for the analysis of an undulated plate of finite thickness subjected to in-plane loading conditions. The method treats the boundary conditions on the undulated surfaces by using a first order perturbation method, where the deviation of the undulating surface front a reference plane surface is treated as the small perturbation parameter of the problem. In order to satisfy the boundary conditions on the top and bottom surfaces, solutions of a semi infinite body are alternately applied, where the solution procedure utilizes relatively simple analytical expressions without tedious numerical calculations such as arising in the finite element analyses. It should be noted that an arbitrarily undulated surface of a plate may be approximated by the superpostion of two dimensional sinusoidal waves having arbitrary wave lengths, directions and phases. This means that only certain two-dimensional solutions are necessary to construct a fully three-dimensional solution. The solution accuracy of the proposed method is examined by comparing the present results with finite element solutions, and illustrative examples are also given for the stress analysis of a plate with slightly wavy surfaces in three dimension.

Wave penetration through bent slots

Shielding effectiveness of bent slots carved in a metal plate of finite thickness is analyzed by forming a set of integral equations based on the equivalence principle. Method of moments is used to solve the integral equations for the transmission cross sections. The effects of slot geometry, filling permittivity, and filling conductivity are studied. It is found that the penetrating field is mainly determined by the slot length, filling permittivity, and filling conductivity. Resonances are also observed when the slot length is integer multiples of half wavelengths.

A semi-numerical method for elastic wave scattering calculations

This paper is concerned with the calculation of elastic wavefields in strongly heterogeneous media containing scatterers of arbitrary shape and size. A hybrid method is described in which the elastodynamic field within a finite region enclosing the scatterers is modelled by finite elements while the field away from this region is represented by means of a suitable set of wave functions. Application of a variational principle and the continuity conditions at the interface between the two regions lead to a system of linear equations that is solved by standard techniques. The hybrid method extends the applicability of the finite element method to wave propagation problems in unbounded media by rigorously modelling the radiation field. The method is used to calculate the elastodynamic field in a plate of finite thickness and infinite lateral dimensions containing geometric discontinuities.

Three-dimensional elastic stress fields near notches in finite thickness plates

Based on detailed three-dimensional finite element (3D FE) analyses, elastic notch-root fields in plates with different thicknesses and notch configurations subjected to uniaxial tension have been investigated. By comparing with the planar notch-root fields and crack-tip fields, the following characteristics of the 3D stress–strain fields near the notch front are revealed: (1) The plate thickness and notch configuration have obvious effects on the stress concentration factor (SCF) K t , which is higher in finite thickness plates than in plane stress and plane strain cases. (2) The variation of the opening stress normalized by its value at the notch root with a distance x from the root normalized by the root radius ρ is insensitive to the notch configuration and the plate thickness and coincides well with the two-dimensional (2D) planar solution when x/ ρ<0.75. (3) Strong 3D effects exist within a radius of about three-eighth the plate thickness from the notch root. Further away from the root, the through-thickness variation of field quantities decreases and, at the radial distance of approximately 1.25 times the plate thickness, all of the through-thickness changes disappear completely. At least for notches with opening angles less than 90°, these “2D–3D” transition distances and the variations of the out-of-plane constraint normalized by its value at the notch root with a distance x from the root normalized by the plate thickness B are essentially independent of the notch configuration and plate thickness. (4) In the 3D-effect zone, the gradient of the out-of-plane strain ε zz in the thickness direction is significant near the free surface in thick plates. On the free surface, ε zz can be 3.5 times as high as the value on the mid-plane. It is also found that the in-plane stress ratio in an arbitrary thickness plate coincides very well with the 2D solutions within a distance of about three-tenth of the root radius from the notch root.