Plate Of Finite Thickness Research Articles

Short-Wave and Wave Group Scattering by Submerged Porous Plate

An analytical solution for waves propagating through a horizontal porous plate of finite thickness is obtained. The objective of the plate is to reduce the incident short-wave energy and the long-wave energy as well. Consequently, in this study the plate is analyzed in a global perspective [i.e., considering its response to obliquely incident short waves (both regular and irregular) and wave groups (with the consequent generation of free and locked long waves)]. To solve the propagation of regular and irregular waves, an eigenfunction expansion is used and the results are verified with experimental data showing good agreement. The propagation of a wave group past a horizontal porous plate is studied using a multiple-scale perturbation method, and an analytical solution is presented. The results show that the generated long waves are present on both sides of the plate and that maximum short-wave reflection is associated with maximum long-wave transmission.

Heat conduction analysis of bidirectional multipass welding with short bead lengths

A heat conduction analysis is considered for multipass welding conducted in a forward–reverse manner. When the welding is conducted continuously and the bead is short, the interpass temperature sometimes exceeds 500°C. To predict the thermal histories, a heat conduction model is examined using the analytical solution of a moving point heat source in a wide plate of finite thickness. In the model, point heat sources, some of which are moving in one direction and others in the opposite direction, are considered. Comparison of the calculated and experimental results reveals that the proposed model can predict the thermal histories fairly well. This is followed by calculation of the equivalent arc energy, defined as the apparent arc energy which gives either the same cooling time from 800 to 500°C or the same kinetic strength of the heat cycles with no preheating (the concept of kinetic strength has been developed to estimate austenite grain growth in the heat affected zone region), and examination of the effect of maximum interpass temperatures on the equivalent arc energy. When the interpass temperature is greater than 500°C, its effect can be to increase the equivalent arc energy from 2.5 kJ mm-1 to between 6 and 14 kJ mm-1.

Natural convection heat transfer from a finite vertical thick plate at low Grashof numbers

The purpose of this investigation was to discover the natural convection heat transfer characteristics of a finite vertical thick plate on an adiabatic floor, enclosed by a solid wall. One vertical side of the plate was kept cold and the other side hot. Numerical results were obtained for a Grashof number in the range 0.1 to 10 5 , ratio of the temperatures of the hot and cold walls ranging from 0.05 to 1.0, and a Prandtl number of 0.71. The effect of the mutual interferences of the natural convection fields on both walls was investigated. It was found that in the case of a small temperature ratio, when Gr becomes large, convection fields on the hot- and cold-wall sides become fully independent convection fields that do not interfere with each other since a strong natural convection field forms on both wall sides. On the other hand, the fluid flow from the hot-wall side invades the cold-wall side with decreasing Gr. The heat transfer coefficients on the cold-wall side have a negative value at the upper part of the thick plate, resulting in an endothermic phenomenon

Theoretical investigation of eddy-current induction for nondestructive evaluation by superconducting quantum interference devices

An analysis of eddy-current induction is presented for applications in low-frequency electromagnetic nondestructive evaluation (NDE) using superconducting quantum interference devices (SQUID's). Analytical expressions are developed for the induced eddy currents, as well as the resulting magnetic fields produced by circular excitation coil above conducting plates of infinite and finite thicknesses. The in-phase (0/spl deg/) and quadrature-phase (90/spl deg/) components of the eddy-current density are plotted for the low-frequency excitations characteristic of SQUID NDE. The quadrature-phase eddy current has a peak value at the surface of conducting plates, while the in-phase eddy current is maximal at a characteristic depth in the conducting material. Also, both the quadrature- and in-phase eddy currents change signs at characteristic depths that depend on the skin depth. These features can be exploited in determining the depth of material defects. The expression for the magnetic field produced by an excitation coil above a conducting plate of finite thickness is then used to estimate the SQUID's response due to corrosion or variation in material thickness. The expressions derived here can be used to model the magnetic signature of localized defects.

Measurement of Thermal Diffusivity Based on the Photothermal Displacement Technique Using New Phase Method.

A new analysis method of measuring the thermal diffusivity of a solid using photothermal displacement is proposed. To obtain an expression for the photothermal deformation, a homogeneous and isotropic infinite plate of finite thickness with stress-free boundaries is considered. Theoretical study was carried out to investigate the influence of the parameters, such as the radius and modulation frequency of the pump beam and the sample thickness. We obtained the thermal diffusivity from the phase of the signal by two methods. The first method for finding the thermal diffusivity of the theoretical result uses the bi-section method to minimize the standard deviation between experimental and theoretical phase curves. The second method uses the relation between the normalized thermal diffusion length and normalized minimum position. The experimental data obtained by applying these methods for different samples were in good agreement with the literature values.

Surface temperature of a flat plate of finite thickness under conjugate laminar forced convection heat transfer condition

Surface temperature of a flat plate of finite thickness under conjugate laminar forced convection heat transfer condition

A Useful Solution for Estimating Contact Pressure Between a Plate and Foundation

Many manufacturing situations involve a finite thickness plate or layer of material which is pressed against a much thicker foundation of the same or different material. One key example is a blank holder (plate) pressed against a die (foundation) in a sheet metal forming operation. In designing such a plate/foundation system the design objective often involves the contact stress distribution between the plate and foundation and the design variables are typically the thickness and modulus of the plate, the stiffness of the foundation and the applied pressure distribution on the noncontacting side of the plate. In general the problem relating the variables to the contact pressure distribution is three-dimensional and requires a complex finite element or boundary element solution. However, if the applied pressure distribution consists of sufficiently localized patches, which is often the case in applications, then an approximate 3D solution can be constructed by superposition. Specifically, the paper provides a convenient calculation procedure for the contact pressure due to a single circular patch of applied pressure on an infinite, isotropic, elastic layer which rests on a Winkler foundation. The procedure is validated by using known analytical solutions and the finite element method (FEM). Next a sensitivity study is presented for ascertaining the validity of the solution’s use in constructing solutions to practical problems involving multiple patches of loading. This is accomplished through a parametric study of the effects of loading radius, layer thickness, layer elastic properties, foundation stiffness and the form of the applied pressure distribution on the magnitude and extent of the contact pressure distribution. Finally, a procedure for determining an appropriate Winkler stiffness parameter for a foundation is presented. [S1087-1357(00)00603-1]

Prediction of penetration depths during electron beam welding

An analytical penetration model for electron beam welding is presented, based on a combination of the moving line source solution and the solution for a cylindrical cavity moving through a plate of finite thickness. The input parameters are the net beam power, welding speed, keyhole wall temperature, and diameter of the electron beam from which the penetration depth can be calculated for a range of base materials, such as stainless steels, titanium grade 2, and aluminium alloy AA 5052. It is concluded that the penetration model is sufficiently comprehensive and relevant to be used as a predictive tool in ordinary production welding and some possible applications are briefly outlined.

Analytical solution for transient temperature distribution in gas tungsten arc welding with consideration of filler wire

This paper is focused on deriving an analytical solution to predict the transient temperature distribution of the plate during the process of gas tungsten arc (GTA) welding in which a filler wire is fed. An analytical solution is derived from the transient three-dimensional heat conduction equation by applying a convection boundary condition to the top and bottom surface of an infinite plate of finite thickness. The heat input that is applied on the plate is exactly the same as the amount of heat lost from the electric arc while fusing the filler wire. The electric arc is assumed to be a moving heat source with a Gaussian distribution. The point heat sink is located in the spot where the filler wire is fed to the weld pool. However, the feed position of the filler wire is ever-evolving with time because the size of the weld pool changes in accordance with the welding conditions. Nevertheless, the algorithm is capable of determining the feed position. Also, the influence of the filler wire on the temperature distribution of the plate is established by removing the filler wire from the analytical solution and comparing these two different situations. In order to verify the validity of this solution, a GTA welding experiment is conducted for various welding conditions. After this stage, the resulting isotherms of the cross-sections of the plate and the analytical results are compared. The analytical solution, which is in good agreement with experimental results, is the very first to take the filler wire feed into consideration. It can be used as a model to control the GTA welding process with account taken of the filler wire, and it can also be used as an optimization tool for welding process parameters.

Three-dimensional temperature distribution in laser surface hardening processes

This paper introduces a new analytical solution to predict transient temperature distributions in a finite thickness plate during laser surface hardening. This analytical solution was obtained by solving a transient three-dimensional heat conduction equation with convection boundary conditions at the surfaces of the workpiece. To calculate the temperature field numerically in laser surface hardening processes, laser beam absorptivity, one of the most important parameters, should first be determined. It was extremely difficult to find an accurate value for laser beam absorptivity owing to the use of coating material on the workpiece surfaces, essentially to improve the efficiency of laser beam power. Therefore, in this paper, absorptivities were measured experimentally under various hardening conditions, including variations in coating thickness, laser beam power and beam travel velocity. Thereafter, to prove the validity of the model, a series of laser surface hardening experiments was performed on medium-carbon steel under various hardening conditions. The hardened layer of the etched cross-sectional view was examined and compared with the isothermal lines predicted by the proposed analytical model, in which the values of the experimentally measured absorptivity were employed. The results showed that the solution predicted well the transient temperature distribution of finite plates with satisfactory accuracy. Also, owing to the simplicity of the solution method, the analytical model developed may be easily implemented for simulation work for analysis and prediction of laser surface hardening processes under various hardening conditions.

A numerical study of the flow in the wake of a plate of finite thickness

We give a numerical study of the flow near the backside of a plate of finite thickness past which a supersonic flow of viscous compressible heat-conducting gas is occurring, as a function of the Reynolds number, symmetry conditions, and temperature conditions on the surface. Thirteen figures. Bibliography: 7 titles.

Stress intensity factors for corner cracks emanating from fastener holes under tension

In this paper, previous work associated with the stress intensity factor for corner cracks at fastener holes in finite thickness plates is briefly reviewed. The stress intensity factors for two symmetric quarter-elliptical corner cracks subjected to remote tension are evaluated by using both the quarter-point displacement and J-integral methods based on three-dimensional finite element analyses. The geometry ratios analyzed cover a wide range, i.e. depth ratio a/ t: 0.2–0.95, aspect ratio a/ c: 0.2–5, and hole radius ratio r/ t: 0.5–3. Analysis of the J-integral path independence and mutual comparison of the stress intensity factor results between the two methods demonstrate that the present results are of good numerical accuracy. Deviation of the present results from some other solutions found in the literature is also revealed, particularly from Newman and Raju's equations. It is shown that the difference among these results obtained by the different methods is generally within a reasonable bound of error, but Newman and Raju's equations systematically underestimate (up to 15%) the stress intensity factor for cracks of depth ratio larger than 0.8.

A new proposal in gradient plasticity: theory and application in 1‐D quasi‐statics and dynamics

It is well known that, due to the absence of an intrinsic length scale, conventional rate-independent plasticity theories fail to explain the size effects observed in problems such as shear-flow localization, nanoindentation tests and precipitate hardening. Consequently, several new nonlocal/gradient constitutive theories with a preferred-length scale have been proposed recently. One such gradient dependent J-2 flow theory of plasticity that preserves the classical structure of incremental boundary value problem has been proposed in Reference 27. The key feature of this theory that allows for the preservation of the classical structure of the incremental problem is the fact that the gradient measures enter the constitutive relations only through the hardening moduli and no higher-order stresses enter the formulation. This paper deals with a preliminary evaluation of the above theory in the context of two simple problems. The first problem considers quasistatic rate-independent motions of an infinite 1-D bar that is at the point of incipient softening at all points along its length. An analytical solution is provided to this problem and the stabilizing effect of the nonlocal theory is illustrated. The second problem considers 1-D, dynamic, simple-shearing motions of an infinite plate of finite thickness. Numerical results for both the nonclassical theory and the corresponding classical theory are presented. Comparisons show that the non-classical theory predicts a stabilizing trend in the motions and mesh-size insensitivity of the results. Copyright © 1999 John Wiley & Sons, Ltd.

Three-dimensional analyses of plastic constraint for through-thickness cracked bodies

A three-dimensional strip yield model has been proposed to rationalize effects of out-of-plane and in-plane constraints. By use of the model, plastic constraints around a straight-through crack in finite thick plates made of strain hardening materials are analyzed. A global constraint factor α is defined to simulate the three-dimensional effects in two-dimensional analysis. Effects of thickness and stress states on the size of crack-tip plastic zone and α are studied in detail. A unique variation curve of α against normalized thickness is obtained for different combination of materials, load levels and geometry. Influences of the in-plane constraint on the α-thickness curve are analyzed as well. It is shown that the influence of T-stress can be considerable only if the plastic-zone size becomes comparable to the crack length. The difference between the present results and Newman, Bigelow and Shivakumer's three-dimensional finite element results is within 6% over a large range of thickness and stress levels. The three-dimensional shape of the plastic zone is discussed as well. Potential applications of the model are discussed and it is shown by an example that the present model can be used to explain the effects of thickness upon fatigue crack growth.

Modeling of heat transfer in a turbulent flow around a heat-conducting plate with local regions of heating

A solution of the coupled nonstationary boundary-value problem of turbulent flow around a flat heat-conducting plate of finite thickness having local regions with volume heat sources is given. For modeling the heat transfer in the boundary layer, thek-e turbulence model is used. It is shown that the thermal conductivity of the plate material significantly affects the surface distributions of both temperature and local friction.

Flow and mass transfer measurements for a flat plate of finite thickness in pulsating flow

Laboratory measurements were made of flow and mass transfer over a blunt flat plate of finite thickness, which is placed in a pulsating free stream, U x = U 0 (1+ A 0 cos 2 πƒ p t). Low turbulence-intensity wind tunnel experiments were conducted for small and moderate Reynolds numbers, 770 ⩽ Re H ⩽ 8000. Pulsation was generated by means of an acoustic speaker. The majority of experiments were carried out in the ranges of ƒ p = 20.0–80.0 Hz and A 0 ⩽ 0.15. Flow properties were measured by I-type and split-film probes. Mass transfer rates were measured by employing the naphthalene sublimation technique. The present results for non-pulsation flows ( A 0 = 0.0) were shown to be consistent with the published data. For pulsating approach flows, results are provided for the distributions of the wall static pressure, the longitudinal mean velocity and turbulent intensity, and the Sherwood number, Sh, as a function of the stream-wise distance x * measured from the leading-edge separation point. As A 0 or ƒ p increases, the time-mean reattachment length is reduced significantly. This implies that the height and length of the separation bubble shrink simultaneously: the position where C p is recovered moves upstream, and the minimum value of C p decreases; the reverse flow is intensified: and a substantial augmentation of turbulent energy is discernible. In the separation bubble, the effect of pulsation on Sh is conspicuous. Sh decreases monotonically from the separation point to the minimum value near the secondary separation point; and Sh increases appreciably with increasing x *, after passing the secondary separation point to the maximum value at the reattachment point; and afterward, Sh decreases. The secondary separation point and the position where Sh has a maximum move further upstream, as A 0 or ƒ p increases. At large Re H, the relative influence of pulsation on Sh weakens.

Dike propagation through an elastic plate

We study the vertical propagation of a magma‐filled crack which rises due to buoyancy through an elastic plate of finite thickness. The plate rests on a magma source of large volume. Full time‐dependent solutions of the coupled problem with elastic deformation and viscous flow in a thin fracture are obtained numerically. A range of initial conditions corresponding to different regimes of fracture filling, from passive conditions at negligible magma velocity to forceful injection, are investigated. Three phases are distinguished during ascent. In a first phase, as the dike grows out of the initial magma‐filled fracture, the elastic pressure distribution adjusts to a fully developed distribution, with values close to buoyancy in a nose region and smaller values in a conduit region below the nose. Subsequently, the dike develops increasingly wider conduit and nose regions. Close to the plate upper boundary, stress and displacement fields become sensitive to the width of the plate. The initial conditions determine quantitative characteristics of subsequent dike propagation and, in particular, the flux of magma into the dike. As a dike propagates away from source, magma encounters increasingly colder rock but flows through an increasingly wider channel. This provides a self‐shielding mechanism against freezing. At the upper boundary, eruption initially occurs out of the inflated nose region and is driven by the relaxation of elastic stresses.

Wave penetration through slits on stacked thick plates

Wave penetration through slits on single and stacked metal plates of finite thickness is studied by using the Galerkin method. The limiting case of slits on infinitesimally thin plates are also formulated to compare the shielding effectiveness of metal plates with slits against incident plane waves. It is observed that the wave penetrating through slits on stacked plates with a proper separation is much less than that through a single slit on a plate with twice the thickness.

Weight functions and stress intensity factors for corner quarter-elliptical crack in finite thickness plate subjected to in-plane loading

Approximate weight functions for the profile and frontal plane crack front points of a corner quarter-elliptical crack in finite thickness plate, subjected to mode I, in-plane loading, were derived by using the method of universal weight functions by Shen and Glinka ( Engineering Fracture Mechanics, 1991, 40, 1135–1146; Theoretical and Applied Fracture Mechanics, 1991, 15, 247–255.) Closed form expressions were obtained for the coefficients of the weight functions. The coefficients were derived from the reference stress intensity factor solutions obtained by Shiratori and Miyoshi ( Stress Intensity Factors Handbook, Vol. 3, ed. Murakami et al. Pergamon, New York, 1992, pp. 591–597.) using the finite element method. A comparison of the stress intensity factors calculated using the weight functions with the finite element data for various applied stress distributions shows good accuracy of the present results.

Numerical characterization of low Reynolds number flow in the Kenics static mixer

Low Reynolds number flow in a six element Kenics static mixer was modeled using finite element computations. The numerical approach takes into account aspects of the fluid flow within the Kenics mixer which have been neglected in previous studies, including transitions between mixer elements and finite-thickness mixer plates. The pressure drop information obtained from the simulations was compared to several experimental correlations available in the literature for Kenics mixer pressure drop. Analysis of the low Reynolds number velocity field indicates a spatially periodic flow which matches the periodicity of the mixer geometry. Flow transitions at the entrance and exit of each element strongly affect the velocity field for up to ∼25% of the element’s length under creeping flow conditions. Comparison of the velocity field over a range of Reynolds numbers from 0.15 to 100 indicated that Reindependent velocity profiles are obtained up to Re=10, with significant deviations in the velocity field at Reynolds numbers above this limit. The magnitude of the rate-of-strain tensor, which represents an upper bound for mixing efficiency, was computed and profiled within the mixer. The profile for the magnitude of the rate-of-strain tensor was roughly uniform over the central 75% of a single mixer element, but shifted toward higher values in the end regions, indicating that the greatest potential mixing effects take place at the element-to-element transitions.