Half-plane Problem Research Articles

On a Contact Problem of an Elastic Half-Plane with a Circular Hole : 2nd Report, The Case of Frictionless Contact

An analytical solution is presented for the stresses produced by the indentation of the straight boundary of a semi-infinite elastic plate with a circular hole by a flat-ended rigid stamp. It is supposed that the stamp is placed on the segment of the straight boundary, just above the hole, and pressed into the plate by a given force, perpendicular to the boundary, and that the coefficient of friction between the stamp and the plate is zero. The complex potentials, which have poles at the center of the hole and disturb the boundary conditions neither along the straight edge nor at the infinity, are added to the solution of the contact problem of a half-plane without hole and the parametric coefficients included in the complex potentials are then determined, so that the conditions at the rim of the hole may be satisfied. Numerical results obtained are shown in graphs and compared with those available.

On a Contact Problem of an Elastic Half-Plane with a Circular Hole

A two-dimensional solution is given for the stresses produced by the indentation of the plane boundary of a semi-infinite elastic medium with a circular hole by a flat-ended rigid stamp. It is assumed that the stamp is placed on the segment of the straight boundary, just above the hole, and pressed into the half-plane by a given force, perpendicular to the boundary, and that the friction between the stamp and the elastic body is so great that no slip can occur. The complex potentials, which have singularities at the center of the hole and do not disturb the boundary conditions along the straight edge as well as at infinity of the half-plane, are added to the solution of the contact problem of a half-plane without hole and the parametric coefficients attached to the potentials are then adjusted, so as to satisfy the conditions at the rim of the hole. Numerical examples are worked out and compared with results available.

Force in Smoothly Joined Elastic Half-Planes

Problem of elastic half-plane in smooth contact with second half-plane and subjected to concentrated force; formulas are given for components of stress in Cartesian coordinates; examination of properties of normal stress transmitted by interface; it is shown that, if elastic plane loaded by concentrated force is separated into 2 half-planes by smooth cut, normal stress across cut is same as in full plane before cutting.

The mixed problem for the elastic anisotropic halfplane

The mixed problem for the elastic anisotropic halfplane

The half-plane diffraction problem for harmonic time dependence

Green’s functions are obtained for the boundary-value problems of mixed type describing the general two-dimensional diffraction problems at a screen in the form of a half-plane (Sommerfeld’s problem), applicable to acoustically rigid or soft screens, and to the full electromagnetic field at a perfectly conducting screen.

On the diffraction field near a plane-screen corner

It is shown that the diffraction field of an incident plane scalar wave in the vicinity of a plane-screen corner of arbitrary angle can be found approximately by solving Laplace's equation. An approximate solution, which satisfies the boundary conditions exactly, is presented as a simple closed expression by generalizing the known solution of the half-plane problem. A special corner condition, in addition to Meixner's edge condition, is not necessary.