Eigenvalue Expansions Research Articles

Singular stress field near a crack tip yielding by inclined slip planes.

This paper is concerned with singular stress distribution near a crack tip yielding by two inclined slip planes. Based on the method of eigenvalue expansions of Williams, the mode I near crack tip singular stress field is obtained in the form with an undetermined constant similar to the stress intensity factor. It is found that unbounded stresses occur near the crack tip with the strength of the singularity being somewhat weaker than for the case without slip planes. Finally, in the case of mode III crack, the results from the closed form solution are compared with the present study.

Longitudinal Heat Propagation in Three-Phase Laminated Composites at Low Exciting Frequencies

In order to explore the behavior of matrix-filler bilaminate MFMFM… near the interfaces it is necessary to augment it into a trilaminate MBFBMBF… arrangement (of volume fractions fM, fB, fF), and then consider the condition that the binder layer is infinitely thin, fB → 0. In a previous paper [1] the problem was considered for high frequencies ω of the boundary temperature excitation T(0, y, t) = e−iwtχ(y), using power series representation for the eigenvalues descending in ω1/2 powers. In the first part of the present paper the low frequency regime is determined, using eigenvalue expansions in ascending powers of ω. The second part of the paper is devoted to exhibiting the interface layer as a shock layer across which there occur temperature jump and flux jump, of unexpected nature.

On the Theory of an Impurity Molecule. II. The Born‐Oppenheimer Method

AbstractA nonlinear impurity molecule is considered. The molecule is assumed to show almost free rotations about one axis which can librate. The Born‐Oppenheimer method is applied to solve the Schrödinger equation transformed from cartesian coordinates to coordinates defined by the Eckart conditions and describing the vibrations and rotations. The Hamiltonian, the eigenvalue, and eigenfunction are expanded in power series in the Born‐Oppenheimer parameter and these expansions are inserted in the Schrödinger equation. The eigenfunction and eigenvalue expansions are calculated up to the first and the fifth terms, respectively. Some kinetic corrections caused by the libration are shown to appear in the first order term of the eigenfunction expansion and in the fourth order term of the eigenvalue expansion.