Application of the second mixed boundary value solution for an interaction of an edge crack and an internal crack of a half plane subjected to uniform traction

2017 ◽  
Vol 97 (6) ◽  
pp. 718-731
Author(s):  
Norio Hasebe ◽  
Atsusi Ueda
Keyword(s):  
Half Plane ◽  
Edge Crack ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Internal Crack

scholarly journals The solution of a mixed boundary value problem in the theory of diffraction by a semi-infinite plane

1975 ◽  
Vol 346 (1647) ◽  
pp. 469-484 ◽  
Keyword(s):  
Boundary Condition ◽  
Boundary Value Problem ◽  
Half Plane ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Mathematical Problem ◽  
Standard Technique ◽  
Infinite Plane ◽  
Mixed Boundary Value Problem ◽  
Rigid Boundary

A solution is obtained for the problem of the diffraction of a plane wave sound source by a semi-infinite half plane. One surface of the half plane has a soft (pressure release) boundary condition, and the other surface a rigid boundary condition. Two unusual features arise in this boundary value problem. The first is the edge field singularity. It is found to be more singular than that associated with either a completely rigid or a completely soft semi-infinite half plane. The second is that the normal Wiener-Hopf method (which is the standard technique to solve half plane problems) has to be modified to give the solution to the present mixed boundary value problem. The mathematical problem which is solved is an approximate model for a rigid noise barrier, one face of which is treated with an absorbing fining. It is shown that the optimum attenuation in the shadow region is obtained when the absorbing lining is on the side of the screen which makes the smallest angle to the source or the receiver from the edge.

On the Application of the Method of Hypersingular Integral Equations to Solving Problems for an Elastic Plane with a Collinear System of Cracks

2020 ◽  
pp. 72-83
Author(s):  
S.M. Mkhitaryan
Keyword(s):  
Integral Equations ◽  
Half Plane ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Plane Deformation ◽  
Elastic Plane ◽  
Antiplane Deformation ◽  
Hypersingular Integral ◽  
Hypersingular Integral Equations ◽  
System Of Cracks

In the present paper, using the method of hypersingular integral equations, based on the formulas of the inversion of the corresponding singular integral equations, the exact quadrature solution of the classical problems of the mechanics of an elastic plane with a collinear system of cracks is constructed. The elastic plane is in a state of antiplane deformation or plane deformation; in case of antiplane deformation, crack edges are symmetrically loaded by tangential forces, while in case of plane deformation, they are again loaded symmetrically but by normal forces. Mixed boundary-value problems for an elastic half-plane equivalent to these problems are formulated. Under plane deformation, the mixed boundary-value problem for an elastic half-plane is discussed as well when the plane boundary is reinforced by two similar and symmetrically located semi-infinite stringers between which a system of an arbitrarily final number of stringers is situated. It is considered that the stringers are absolutely rigid for expansion and compression and absolutely flexible for bending. A particular case of two similar symmetrically located cracks is considered more in detail. In this case, the exact solution to the problem is also constructed by the method of Chebyshev orthogonal polynomials.

The Problem of Several Indentors Moving on a Viscoelastic Half-Plane

1995 ◽  
Vol 62 (2) ◽  
pp. 380-389 ◽  
Author(s):  
H. Z. Fan ◽  
G. A. C. Graham ◽  
J. M. Golden
Keyword(s):  
Steady State ◽  
Boundary Value Problem ◽  
Integral Equations ◽  
Half Plane ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Coupled System ◽  
Mixed Boundary Value Problem ◽  
System Of Integral Equations ◽  
Space And Time

The problem of several indentors moving on a viscoelastic half-plane is considered in the noninertial approximation. The solution of this mixed boundary value problem is formulated in terms of a coupled system of integral equations in space and time. These are solved numerically in the steady-state limit for the case of two indentors. The phenomena of hysteretic friction and interaction between the two indentors are explored.

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