Stress Solutions for an Infinite Plate With Triangular Inlay

1956 ◽  
Vol 23 (3) ◽  
pp. 336-338
Author(s):  
R. M. Evan-Iwanowski
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Unit Circle ◽  
Complex Variable ◽  
Boundary Value ◽  
Infinite Plate ◽  
Mapping Function ◽  
Two Dimensional ◽  
Concentrated Force ◽  
Uniform Tension

Abstract The method of solving two-dimensional problems in elasticity by means of the functions of complex variable, essentially developed by É. Goursat (1), and N. I. Muskhelishvili (2-4), has been applied to the following cases: (a) An infinite plate with a rigid triangular inlay under uniform tension at infinity; (b) A concentrated force; and (c) a moment acting on a triangular inlay in an infinite plate. All these problems are second boundary-value problems; i.e., the displacements are prescribed on the boundary. The first boundary-value problem for a triangular opening in an infinite plate was treated by Hu-Nan Chu (7). The mapping function used in this paper is z = ω ( ζ ) = K ( ζ + n ζ 2 ) , K is real, and 0 < n < 1/2 and real, and it maps an exterior of a triangle with rounded corners, Fig. 1, in the z-plane into an exterior of a unit circle in the ζ-plane [for detailed discussion of this mapping refer to (4)].

scholarly journals Fundamental problems for infinite plate with a curvilinear hole having finite poles

2001 ◽  
Vol 7 (6) ◽  
pp. 485-501 ◽  
Author(s):  
M. A. Abdou ◽  
A. A. El-Bary
Keyword(s):  
Complex Variable ◽  
Arbitrary Shape ◽  
Infinite Plate ◽  
Mapping Function ◽  
Curvilinear Hole ◽  
Two Dimensional ◽  
Rational Mapping ◽  
Special Cases ◽  
Elasticity Problems ◽  
Different Shapes

In the present paper Muskhelishvili's complex variable method of solving two-dimensional elasticity problems has been applied to derive exact expressions for Gaursat's functions for the first and second fundamental problems of the infinite plate weakened by a hole having many poles and arbitrary shape which is conformally mapped on the domain outside a unit circle by means of general rational mapping function. Some applications are investigated. The interesting cases when the shape of the hole takes different shapes are included as special cases.

scholarly journals Parametrization for some boundary value problems of interpolation type

2009 ◽  
Vol 43 (1) ◽  
pp. 229-242
Author(s):  
Miklós Rontó ◽  
Natalia Shchobak
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Analytic Form ◽  
Successive Approximations ◽  
Two Dimensional ◽  
Linear Boundary ◽  
Scalar Parameter ◽  
The Given

Abstract We obtain some results concerning the investigation of two-dimensional non-linear boundary value problems of interpolation type. We show that it is useful to reduce the given boundary value problem, using an appropriate substitution, to a parametrized boundary value problem containing some unknown scalar parameter in the boundary conditions. To study the transformed parametrized problem, we use a method which is based upon special types of successive approximations constructed in an analytic form.

scholarly journals Two-dimensional boundary-value problems of elasticity

1946 ◽  
Vol 187 (1008) ◽  
pp. 87-101 ◽  
Keyword(s):  
Boundary Value Problems ◽  
Stress Function ◽  
Boundary Value ◽  
Infinite Plate ◽  
Thin Disk ◽  
New Method ◽  
Two Dimensional ◽  
Airy’S Stress Function ◽  
Dimensional Boundary

A new method of solving two-dimensional problems of elasticity is developed in this paper. Airy’s stress function is not used; but the stress equations of equilibrium are so transformed with the help of consistency equations as to yield simple solutions when the values of the stresses on the boundary are given. The boundaries discussed in this paper are the straight edges of a semi-infinite plate and the circular boundaries of a thin disk. Action of forces and couples in the interior of the plates is chiefly considered.

THE SECOND BOUNDARY VALUE PROBLEM OF RIEMANN'S TYPE FOR BIANALYTICAL FUNCTIONS WITH DISCONTINUOUS COEFFICIENTS

2004 ◽  
Vol 9 (3) ◽  
pp. 193-200
Author(s):  
L B. Bolottn
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Unit Circle ◽  
Boundary Value ◽  
Discontinuous Coefficients ◽  
Constructive Method ◽  
Analytical Functions ◽  
Basic Boundary

The paper is devoted to the investigation of one of the basic boundary value problems of Riemann's type for bianalytical functions with discontinuous coefficients. In the course of work there was made out a constructive method for solution of the problem in a unit circle. There was also found out that the solution of the problem under consideration consists in consequent solutions of two Riemann's boundary value problems for analytical functions in a unit circle. Besides, the example is constructed.

Approximate Boundary Value Problems of a Deformed Flexible Closed Torso Shell with Excited Edges

2016 ◽  
Vol 08 (04) ◽  
pp. 1650051 ◽  
Author(s):  
I. V. Andrianov ◽  
V. I. Olevskyi ◽  
J. Awrejcewicz
Keyword(s):  
Exact Solution ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Asymptotic Series ◽  
Problem Solution ◽  
Rational Approximations ◽  
Two Dimensional ◽  
Artificial Parameter

A boundary value problem solution is presented to treatment the deformations of a closed flexible elastic torso shell having perturbations at its axial edges. A so-called artificial parameter technique is applied to obtain a solution in the form of a double asymptotic series further summed using two-dimensional fractional rational approximations. Convergence of the approximations to the exact solution is proven.

scholarly journals A note on a method of Milne-Thomson

1963 ◽  
Vol 3 (1) ◽  
pp. 93-98 ◽  
Author(s):  
V. T. Buchwald
Keyword(s):  
Boundary Value Problem ◽  
Analytic Continuation ◽  
Boundary Value Problems ◽  
Complex Variable ◽  
Laurent Series ◽  
Boundary Value ◽  
Series Expansions ◽  
Infinite Strip ◽  
Cauchy Integrals ◽  
Plane Elastostatics

AbstractMilne-Thomson has used the method of analytic continuation to solve boundary value problems of the annulus in plane elastostatics. However, his use of Cauchy integrals is incorrect, and it is shown in this note that the solution is obtained in terms of Laurent Series expansions. The solution is equivalent to that of Muskhelishvili, but is simpler to use in some applications.A similar approach is used to solve the boundary value problem of the infinite strip, the solution being given in terms of functions of a complex variable expressed as Fourier integrals.

An Analysis of Collinear Periodic Crack-Tip Stress Field of an Infinite Plate Subjected to Concentrated Force

2011 ◽  
Vol 121-126 ◽  
pp. 682-686
Author(s):  
Xue Xia Zhang ◽  
Chan Li ◽  
Xiao Chao Cui ◽  
Wen Bin Zhao
Keyword(s):  
Stress Field ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Complex Function ◽  
Infinite Plate ◽  
Crack Tip ◽  
Partial Differential ◽  
Concentrated Force ◽  
Stress Changes ◽  
Periodic Cracks

The Stress field of collinear periodic cracks in an infinite orthotropic composite plate subjected to the concentrated force is studied. The mechanical problem is turned into partial differential boundary value problem. The solution of partial differential equations can be obtained by the complex function methods, which contains unknown coefficients. Using undetermined coefficients method and with the help of boundary conditions, the partial differential boundary value problem is solved. The expressions for stress and displacements near crack tip are derived. Since the distribution of periodic cracks, the stress intensity factor is related to crack shape. Finally, a numerical example shows that the stress changes with the polar angle.

A Solution of the Mixed Boundary Value Problem for an Infinite Plate With a Hole Under Uniform Heat Flux

1991 ◽  
Vol 58 (4) ◽  
pp. 996-1000 ◽  
Author(s):  
Norio Hasebe ◽  
Hideaki Irikura ◽  
Takuji Nakamura
Keyword(s):  
Heat Flux ◽  
Boundary Value Problem ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Infinite Plate ◽  
Mapping Function ◽  
Uniform Heat Flux ◽  
Mixed Boundary Value Problem ◽  
Rational Mapping ◽  
Uniform Heat

A general solution of the mixed boundary value problem with displacements and external forces given on the boundary is obtained for an infinite plate with a hole subjected to uniform heat flux. Complex stress functions, a rational mapping function, and the dislocation method are used for the analysis. The stress function is obtained in a closed form and the first derivative is given by such a form that does not contain the integral term. The mapping function is represented in the form of a sum of fractional expressions. A problem is solved for a crack initiating from a point of a circular hole on which the displacement is rigidly stiffened. Stress distributions and stress intensity factors are calculated.

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