Stress analysis of a semi-infinite plate with an oblique edge crack

Summary A semi-infinite plate with an oblique edge crack is analyzed as a thin plate bending problem and a plane elastic problem. The rational mapping function of the sum of fractional expressions and the complex variable method are used. Closed solutions are obtained for these respective problems. Stress distributions, stress intensity factors are investigated for loads causing transverse bending, twisting and uniform tension. The relations between the stress intensity factors and the angle of the oblique edge crack are also investigated.

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Übersicht Eine halbunendliche Platte mit schrägem Anriß wird als Platten- und Scheiben-Problem analysiert. Es werden die Abbildungsfunktionen als die Summe von Partialbruchzerlegungen und die Methode komplexer Veränderlicher zur Analyse verwendet. Damit lassen sich geschlossene Lösungen ausrechnen. Die Lastfälle Querbiegung, Torsion und gleichförmiger Zug werden bezüglich Spannungsverteilung und Spannungsintensitätsfaktor untersucht. Der Zusammenhang zwischen Spannungsintensitätsfaktor und Winkellage des Anrisses wird hergestellt.

     

Velocity and temperature distributions on a semi-infinite flat plate embedded in a saturated porous medium

In this paper the velocity and temperature distributions on a semi-infinite flat plate embedded in a saturated porous medium are obtained for the governing equations (Kaviany [7]) following the technique adopted by Chandrashekara [2] which are concerned with the interesting situations of the existence of transverse, velocity and thermal boundary layers. Here the pressure gradient is just balanced by the first and second order solid matrix resistances for small permeability and observed that by increasing of the flow resistance the asymptotic value for the heat transfer rate increases. Further we concluded that the transverse boundary layers are thicker than that of axial boundary layers. Hence we evaluated the expressions for the boundary layer thickness, the shear stress at the semi-infinite plate and _T (the ratio of the thicknesses of the thermal boundary layer and momentum boundary layer). The variations of these quantities for different values of the porous parameterB and the flow resistanceF have been discussed in detail with the help of tables. The curves for velocity and temperature distributions have been plotted for different values ofB andF.Lastly we have evaluated the heat fluxq(x) and found that it depends entirely upon the Reynolds numberRe, Prandtl numberPr,B andF.     

Stresses in a plate kept in equilibrium by a pin

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 144–148, January–February, 1990.     

Stress distribution about a slowly growing crack determined by photoelastic-coating method

An experimental stress-analysis technique using a birefringent coating is reported for determining the stress distribution about a slowly growing crack. The maximum error of the test method for a large strain gradient is found to be less than 10 percent. For a plate with an internal crack, the experimentally determined stress distribution compares favorably with two numerical solutions. Comparison of stresses about an internal or double-edge crack to those about a single-edge crack indicates that the isochromatics bend over to about 45 deg with the plane of the crack in the former and are inclined at about 60 deg in the latter. Also, the stresses for a single-crack tip vary as the inverse square root of the radius, while the stresses for a double-crack tip follow anr ^?1/4 law more closely.     

Steady temperature field in a semi-infinite composite plate

Summary This paper presents a contribution to generalizing classical problems on heat conduction to the case of composite media. The calculations are comparatively complicated and lead to new results not given in the literature. General deductions are applied to special cases.     

Stress concentration factor due to a functionally graded ring around a hole in an isotropic plate

The aim of this work is to present an analytical solution to reduce the stress concentration factor (SCF) around a circular hole in an isotropic homogeneous plate subjected to far-field uniaxial loading. In this paper the elastic response of an inhomogeneous annular ring made of functionally graded material (FGM), inserted around a hole of a homogeneous plate, is studied. By assuming that Young’s modulus varies in the radial direction with power law and that Poisson’s ratio is constant, the governing differential equations for plane stress conditions are obtained. Using stress function a general solution in explicit closed form is presented and the SCF investigated to highlight the inhomogeneity effects. Furthermore, the explicit solution for an inner homogeneous ring, with different properties with respect to those of the plate, is explicitly obtained and numerical results are compared between homogeneous ring and FGM ring.     

Stress intensity factors for an edge interface crack in a bonded semi-infinite plate for arbitrary material combination

Although a lot of interface crack problems were previously treated, few solutions are available under arbitrary crack lengths and material combinations. In this paper the stress intensity factors of an edge interface crack in a bonded strip are considered under tension with varying the crack length and material combinations systematically. Then, the limiting solutions are provided for an edge interface crack in a bonded semi-infinite plate under arbitrary material combinations. In order to calculate the stress intensity factors accurately, exact solutions in an infinite bonded plate are also considered to produce proportional singular stress fields in the analysis of FEM by superposing specific tensile and shear stresses at infinity. The details of this new numerical solution are described with clarifying the effect of the element size on the stress intensity factor. It is found that for the edge interface crack the normalized stress intensity factors are not always finite depending upon Dunders’ parameters. This behavior can be explained from the condition of the singular stress at the end of bonded strip. Convenient formulas are also given by fitting the computed results.     

Stress concentration factors due to the bending of a thick plate with circular hole

Summary Solutions of a biharmonic equation together with four Helmholtz equations governing a twelfthorder thick plate theory in cylindrical coordinates are derived in a form suitable for finding corrections to previous lower-order thick plate results. As an example, for an infinite plate with a circular hole subjected to bending moments at two parallel outside edges, detailed comparisons of values of a stress couple concentration factor and a stress concentration factor are made between the present theory and previous results due to Alblas, Reissner, and Cheng.

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Spannungskonzentrationsfaktoren für die Biegung dicker Platten mit Kreisloch

Übersicht Für die Grundgleichungen einer Theorie 12. Ordnung für dicke Platten, bestehend aus einer biharmonischen Gleichung und vier Helmholtz-Gleichungen, werden in Polarkoordinaten Lösungen bestimmt, um Korrekturen zu früheren Ergebnissen nach Theorien geringerer Ordnung angeben zu können. Als Beispiel werden für eine unendliche, an zwei parallelen Seiten durch ein Biegemoment belastete Platte mit Kreisloch die Konzentrationsfaktoren für Biegemoment und Randspannung verglichen, die sich nach der hier benutzten Theorie und aus früheren Arbeiten von Alblas, Reissner und Cheng ergeben.

     

Time-dependent interaction of two fluid flows separated by a semi-infinite non-rigid plate

In this paper the following problem is solved in the linear approximation. Let a flat plate separate two uniform inviscid fluid flows with different steady-state densities and velocities. These flows are subject to small time-dependent disturbances due to plate deformation. This problem is solved for arbitrary deformations as well as in the case of the angular harmonic oscillations of a flapping mover. The time-dependent forces acting on the plate are determined, together with the dynamic characteristics of the mover and the position of the fluid-fluid interface. Kazan'. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 55–64, January–February, 1994.     

Impulsive motion of a semi-infinite flat plate in a viscous fluid

The unsteady laminar boundary layer flow is investigated for a semi-infinite flat plate subjected to impulsive motion. An approximate solution is obtained by utilizing Meksyn's method. These results vary smoothly from Rayleigh's unsteady solution to the steady state solution of Blasius. Results are compared to those of Lam and Crocco.Nomenclature A expansion coefficient, see eq. (13) - a expansion coefficient, see eq. (10) - B expansion coefficient, see eq. (14) - b expansion coefficient, see eq. (12) - G function defined by eq. (6) - U free stream velocity - u velocity in x direction - v velocity in y direction - x coordinate along plate - y coordinate normal to plate Greek symbols (l, ) incomplete gamma function - function defined by eq. (15) - y(U/x) ^1/2 - kinematic viscosity - x/Ut - (Uvx)^1/2 f(, )     

The temperature distribution in a semi-infinite insulated cylinder due to an arbitrary heat flux

The temperature distribution in a semi-infinite insulated cylinder with linear temperature dependent heat conductivity and with arbitrary initial temperature subjects at its base to an azimuthal symmetric arbitrary heat flux is found.This is done in two stages: first the problem is solved by assuming a constant initial temperature and constant thermal properties and then the solution is extended to the case in which the heat conductivity varies linearly with temperature provided that the diffusivity is constant and the initial temperature is an arbitrary function. Several particular cases are then checked and found to be in agreement with known solutions.Because of the complexity of the above mentioned solution a more simple solution is developed which corresponds to the case in which the cylinder base can be considered as semi-infinite. Then, the case in which the heat flux has the form (A exp [–r ²/r ₀ ² ]+G) sin t is considered. Two particular cases are considered correspond to narrow and wide beams of heat flux density. In each case the time of maximum temperature and the maximum temperature at the base centre is found.     

Oscillatory free convection heat transfer along a semi-infinite vertical plate

Summary The fluctuating free convection flow along a semi-infinite vertical plate is considered when the plate temperature is of the form T _p –T =(T ₀ –T ) where 0 < 1, denotes the frequency of oscillation and the mean temperature T ₀–T is proportional to ⁿ (0 n < 1). Flow and temperature fields have been obtained by means of two asymptotic expansions. For small values of the frequency parameter , a regular expansion is obtained while for large the method of matched asymptotic expansion is used. It is found that the skin friction and the rate of heat transfer obtained from two expansions overlap satisfactorily for a certain value of . For n=1 the flow governing equations to a semisimilar form, and have been solved by finite difference method. The results obtained from the series and the finite difference methods are in good agreement.

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Oszillierender Wärmeübergang an einer halbunendlichen senkrechten Platte bei freier Konvektion

Übersicht Betrachtet wird die fluktuierende freie Konvektionsströmung längs einer halbunendlichen senkrechten Platte, deren Temperatur dem Gesetz T _p –T =(T ₀–T ) [1+ sin {ie1-03}] folgt, wobei 0 < 1 gelte, {ie1-04} die Frequenz ist und der Temperatur-Mittelwert T ₀–T proportional zu ⁿ (0 n < 1) ist. Mit Hilfe zweier asymptotischer Entwicklungen werden die Strömungs- und Temperaturfelder gewonnen. Für kleine Werte des Frequenzparameters wird eine gewöhnliche Entwicklung benutzt, für große die Methode angepaßter asymptotischer Entwicklungen. Es stellt sich heraus, daß die Oberflächenreibung und die Wärmeübergangsrate aus zwei Entwicklungen für ein bestimmtes zufriedenstellend aufeinander fallen. Für n=1 werden die Grundgleichungen zueinander ähnlich und werden nach der Finite-Differenzen-Methode gelöst. Die Ergebnisse nach den Reihenentwicklungen und der Finite-Differenzen-Methode stimmen gut überein.

     

Effect of friction in a contact problem for a plate with a pin

A solution is obtained for a contact problem concerning the tension of a rectangular elastic plate with a circular hole into which a rigid stationary pin has been inserted. There is a small gap between the hole and the pin, which is of circular cross section. Friction acts in the contact region in accordance with the Coulomb law. The finite-element method and the Boussinesq principle are used to determine the load that realizes a specified contact region. Two variants of boundary conditions on the contour of the hole are examined. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 4, pp. 184–192, July–August, 1998.     

Time-dependent interaction between two inviscid fluid flows separated by a semi-infinite non-rigid plate in a flat-walled channel

The problem of the time-dependent interaction between two inviscid weightless fluids separated by a semi-infinite non-rigid plate in a channel with fixed rigid walls is solved in the linear approximation. The general case of deformation and harmonic oscillations of the plate (flapping mover) are considered. The time-dependent hydrodynamic reaction forces, the position of the interface, and the dynamic characteristics of the mover are determined. Kazan’. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 67–76, January–February, 1997.     

Multiple scattering of flexural waves in a semi-infinite thin plate with a cutout

The multiple scattering of flexural waves and dynamic stress concentration in a semi-infinite thin plate with a cutout are investigated, and the expressions of this problem are obtained. The analytical solutions of wave fields are expressed by employing the wave function expansion method and the expanded mode coefficients are solved by satisfying the boundary condition of the cutout. The image method is used to satisfy the traction free boundary condition of the plate. As an example, the numerical results of dynamic stress concentration factors are graphically presented and discussed. Numerical results show that the analytical results of the scattered waves and dynamic stress in semi-infinite plates are significantly different from those in infinite plates when the ratio of distance b/a is relatively little. In the region of low frequency and long wavelength, the maximum dynamic stress concentration factors occur on the illuminated side of the scattering body with θ = π, but not at the edge of the cutout with θ = π/2. As the incidence frequency increases (the wavelength becomes short), the dynamic stress on the illuminated side of the cutout decreases, however, the dynamic stress on the shadow side increases.