A mixed boundary problem for a finite internally cracked plate

A displacements and resultant forces model (see eqns 2–5, 13 and 17) for a finite internally cracked plate is proposed. This model satisfies: (a) The equilibrium and compatibility condition in the region occupied by cracked plate; (b) Stress free condition on the surface of crack; (c) Single value condition of displacements around the crack. In this model, some undetermined coefficients are contained, these coefficients are derived from outer boundary condition.

It is proved that, this model is convenient not only for the displacements or resultant forces boundary problem, but also for the mixed boundary problem. Besides this, if the boundary problem is solved, to find the value of displacements of any points in cracked body is also convenient.

Two mixed boundary problems, one for the square cracked plate (see Fig. 3), and another for circular cracked plate (see Fig. 5), are solved. The numerical results obtained are shown in Tables 1 and 2 and Figs. 4 and 6 respectively. These results can explain how the constraint affects the values of the stress intensity factor on the crack tips.     

A mixed heat conduction boundary problem for a semi-infinite plate

An examination is made of a plane stationary heat conduction problem with mixed conditions assigned on the edge of a semi-infinite plate. There is a temperature field in the plate due to the action of a point source of heat located inside the plate.     

A mixed problem of plate bending for a regular octagon weakened with a required full-strength hole

The paper addresses a problem of bending of an isotropic elastic plate, bounded by a regular octagon weakened with a required full-strength hole including the origin of coordinates. Coordinate axes pass the middle points of opposite parallel sides of octagon. Rigid bars are attached to each component of the broken line of the outer boundary of the plate. This plate bends under the action of concentrated moments applied to the middle points of the bars. Unknown part of the boundary is free from external forces. Using the methods of complex analysis, the plate deflection and required full-strength contours are determined. Numerical analysis is performed, and the corresponding graphs are constructed.     

Evaluation of boundary integrals for plate bending

The alternative to quadrature, as a procedure for dealing with the integrations required in the direct boundary element method (DBEM), is to carry out the integration analytically and code the results directly. The potential benefits are efficient computer programs; the avoidance of numerical instability; and generally, better accuracy. The technique is developed in this paper. Serious problems arise when Gauss quadrature is employed for the integration of functions which contain, or are close to singularities. A numerical integration approach may fail at the first stage of the analysis, that is, during the assembly of the discrete equations; or it may fail at the subsequent stage of computing domain points near the boundary. The severity of the problem is dependent both on the strength of the singularity, and on geometry. These points are illustrated with examples.     

A fast multipole boundary element method for solving the thin plate bending problem

A fast multipole boundary element method (BEM) for solving large-scale thin plate bending problems is presented in this paper. The method is based on the Kirchhoff thin plate bending theory and the biharmonic equation governing the deflection of the plate. First, the direct boundary integral equations and the conventional BEM for thin plate bending problems are reviewed. Second, the complex notation of the kernel functions, expansions and translations in the fast multipole BEM are presented. Finally, a few numerical examples are presented to show the accuracy and efficiency of the fast multipole BEM in solving thin plate bending problems. The bending rigidity of a perforated plate is evaluated using the developed code. It is shown that the fast multipole BEM can be applied to solve plate bending problems with good accuracy. Possible improvements in the efficiency of the method are discussed.     

Elastic-plastic problem for a plate with a part-through crack under extension and bending

In this paper a flat plate containing a relatively long and deep part-through crack and subjected to uniform membrane and bending loads is considered. It is assumed that the net ligament and the plate wall in some neighborhood of the crack are fully yielded. In analyzing the problem Reissner's Theory of plate bending is used. The plastic deformations are taken into account by using a plastic strip model. First some elastic results regarding the interaction of two collinear cracks are obtained. The crack opening displacement at various locations in the plate is then calculated for various crack sizes and stress ratios.

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Résumé Dans ce mémoire, on considère une tôle plate comportant une fissure relativement longue et profonde, partiellement tranversante et soumise à une contrainte de membrane uniforme et à des contraintes de flexion. On suppose que le ligament net et les parois de la tôle dans un certain voisinage de la fissure passent complètement en plasticité. On utilise pour l'analyse du problème la théorie de Reissner de la flexion des plaques. Des déformations plastiques sont prises en considération en utilisant un modèle de bande plastique. On obtient en premier lieu certains résultats élastiques concernant l'interaction de deux fissures colinéaires. Ensuite, on calcule le déplacement d'ouverture de la fissure en différents endroits de la plaque et pour diverses dimensions de la fissure et rapports de contrainte.

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This work was supported by the Department of Transportation under the Contract DCT-RC-82007.     

Solution of the mixed boundary value problem of heat conduction in a composite rectangle

The author offers a particular solution of the two-dimensional problem of steady heat conduction for a rectangular region composed of two different materials with mixed boundary conditions.     

A mixed boundary value problem for the unsteady Stokes system in a bounded domain in Rn

A mixed boundary value problem for the unsteady Stokes system is studied from the point of view of the theory of hydrodynamic potentials. Existence and uniqueness results as well as boundary integral representations of the classical solution are given for bounded domains having compact but not connected boundaries of class C^1,α (0<α≤1).     

In-plane bending problem of a double edge cracked plate

In this paper the boundary collocation method is used for evaluating the stress intensity factors (SIF) of a double edge cracked plate under in-plane bending. For the case with a large ratio of the plate height to the width, h/b, the results obtained compare very favorably with existing solutions for an infinite strip. Moreover, this method has been used for different finite plates, and a series of conclusions is provided for application.     

Benchmark computation and performance evaluation for a rhombic plate bending problem

The paper addresses the question of the accuracy and reliability of the computational analysis of the rhombic plate problem. The Kirchhoff model has a large error (measured in the energy norm) in comparison with the three dimensional solution with a soft simple support; also for very thin plates (t = 1/100a) and all angles (up to α = 90°). The Reissner–Mindlin model gives results 3–5 times better and is less sensitive to the change of the plate angle. The Kirchhoff model is a relatively good approximation of three dimensional setting for a hard simple support. The paradoxical (polygon) behaviour of the simply supported Kirchhoff plate extends to the Reissner–Mindlin model with a hard support. The finite element solution of the Kirchhoff model is addressed in detail. It is shown that higher degree methods are clearly preferable and that the skewness of the elements does not influence essentially the accuracy of the method; the singularity of the solution, which strongly depends on the skewness of the plate, is the primary cause of the deterioration of the performance of the FEM.     

Discrete and non-discrete mixed methods for plate bending analysis

For plate bending analysis, both discrete and non-discrete mixed methods are presented by using spline functions in the Hellinger-Reissner variational principle. Piecewise spline interpolation with local supports and overall spline interpolation with boundary knots are applied to each of two mixed functionals. The prescribed boundary conditions can be considered by multiple boundary knots. The convergence behaviour of the mixed methods presented is studied. Numerical examples show that a high degree of accuracy can be obtained due to continuous properties in the high derivatives of the employed shape functions. The proposed mixed methods are found to be very competitive with other numerical methods for plate bending problems.     

One steady-state heat-conduction problem for a polygonal region with mixed boundary conditions

An exact solution is given for one heat-conduction problem involving a polygonal region with mixed boundary conditions. The solution method is based on development of a method presented by the author in previous publications.Translated from Inzhenerno-Fizi-cheskii Zhurnal, Vol. 17, No. 2, pp. 306–312, August, 1969.     

On a mixed finite element method for clamped anisotropic plate bending problems

The paper deals with a new mixed finite element method of solution of the bending problem of clamped anisotropic/orthotropic/isotropic plates with variable/constant thickness. This new mixed method gives simultaneous approximations to displacement u and bending and twisting moment tensor (ψ_ij)_{1 ≤ i, j ≤ 2}. Computer implementation procedures for this mixed method are given along with results of numerical experiments on a good number of interesting problems.     

An inverse boundary problem for one-dimensional heat equation with a multilayer domain

In this paper, we propose a regularization method for determining a moving boundary from Cauchy data in one-dimensional heat equation with a multilayer domain. The numerical scheme is based on the use of the method of fundamental solutions and a discrete Tikhonov regularization technique. The generalized cross validation rule for the choice of a regularization parameter is applied to obtain a stable numerical approximation to the moving boundary. Numerical experiments for five examples show that our proposed method is effective and stable.     

Time domain boundary element formulation for partially saturated poroelasticity

Based on the Mixture theory and the principles of continuum mechanics, a dynamic three-phase model for partially saturated poroelasticity is established as well as the corresponding governing equations in Laplace domain. The three-dimensional fundamental solutions are deduced following Hörmander's method. Based on the weighted residual method, the boundary integral equations are established. The boundary element formulation in time domain for partially saturated media is obtained after regularization by partial integration, spatial discretization, and the time discretization with the Convolution Quadrature Method. The proposed formulation is validated with the semi-analytical one-dimensional solution of a column. Studies with respect to the spatial and temporal discretization show its sensitivity on a fine enough mesh. A half-space example allows to study the wave fronts. Finally, the proposed formulation is used to compute the vibration isolation of an open trench.     

Transformation of domain integrals to boundary integrals in BEM analysis of shear deformable plate bending problems

In this paper two techniques, dual reciprocity method (DRM) and direct integral method (DIM), are developed to transform domain integrals to boundary integrals for shear deformable plate bending formulation. The force term is approximated by a set of radial basis functions. To transform domain integrals to boundary integrals using the dual reciprocity method, particular solutions are employed for three radial basis functions. Direct integral method is also introduced in this paper to evaluate domain integrals. Three examples are presented to demonstrate the accuracy of the two methods. The numerical results obtained by using different particular solutions are compared with exact solutions. Received 27 January 1999     

Winkler plate bending problems by a truly boundary-only boundary particle method

This paper makes the first attempt to use the boundary particle method (BPM) to solve the problems of Winkler plate under lateral loading. In this study, we find that the standard fundamental solution does not work well with the BPM. Instead we construct the modified singular fundamental solution, which satisfies the homogeneous governing equation of Winkler plate and is employed in the BPM to calculate the homogeneous solution. Unlike the other boundary discretization methods, the BPM does not require any inner nodes to evaluate the particular solution of inhomogeneous problems, since the method is a truly boundary-only meshfree technique by using the recursive composite multiple reciprocity technique. Our numerical experiments demonstrate efficiency and high accuracy of the BPM in the solution of Winkler plate bending problems.     

Solution of a drying problem for a colloidal porous material with a moving evaporation boundary

Closed solutions are derived for linear differential equations for one-dimensional drying with an evaporation boundary moving in accordance witht.Translated from Inzhenerno-Fizieheskii Zhurnal, Vol. 25, No. 5, pp. 871–876, November, 1973.     

The problem of a Dugdale crack in a plate under normal bending

The plane problem of a Dugdale crack in a plate under remote normal bending is investigated. By use of complex potentials an exact closed form solution is given and the corresponding crack deformation is discussed.     

Embedded kinematic boundary conditions for thin plate bending by Nitsche's approach

A stabilized variational formulation, based on Nitsche's method for enforcing boundary constraints, leads to an efficient procedure for embedding kinematic boundary conditions in thin plate bending. The absence of kinematic admissibility constraints allows the use of non‐conforming meshes with non‐interpolatory approximations, thereby providing added flexibility in addressing the C¹‐continuity requirements typical of these problems. Work‐conjugate pairs weakly enforce kinematic boundary conditions. The pointwise enforcement of corner deflections is key to good performance in the presence of corners. Stabilization parameters are determined from local generalized eigenvalue problems, guaranteeing coercivity of the discrete bilinear form. The accuracy of the approach is verified by representative computations with bicubic C² B‐splines, exhibiting optimal rates of convergence and robust performance with respect to values of the stabilization parameters. Copyright © 2012 John Wiley & Sons, Ltd.