Free and Forced Vibration of a Kinked Cantilever Beam

In this paper we use the assumed modes method to derive an analytical model of a kinked cantilever beam of unit mass carrying a kink mass (mk) and a tip mass (mt). The model is used to study the free and forced vibration of such a beam. For the free vibration, we obtain the mode shape of the complete beam by solving an eight order polynomial whose coefficients are functions of the kink mass, kink angle and tip mass. A relationship of the form f(mk,mt,δ)=mk+mt(4+103cosδ+23cos2δ)=constant appears to give the same fundamental frequency for a given kink angle, δ, and different combinations of kink mass and tip mass. To derive the dynamic equations of motion, the complete kinked beam mode shape is used in a Lagrangian formulation. The equations of motion are numerically integrated with a torque applied at the base and the tip response for various kink angles are presented. The results match those obtained from a traditional finite element formulation.

Transient Analysis for Antiplane Crack Subjected to Dynamic Loadings

The problem considered here is the antiplane response of an elastic solid containing a half-plane crack subjected to suddenly applied concentrated point forces acting at a finite distance from the crack tip. A fundamental solution for the dynamic dislocation is obtained to construct the dynamic fracture problem containing a characteristic length. Attention is focused on the time-dependent full-field solutions of stresses and stress intensity factor. It is found that at the instant that the first shear wave reaches the crack tip, the stress intensity factor jumps from zero to the appropriate static value. The stresses will take on the appropriate static value instantaneously upon arrival of the shear wave diffracted from the crack tip, and this static value is thereafter maintained. The dynamic stress intensity factor of a kinked crack from this stationary semi-infinite crack after the arrival of shear wave is obtained in an explicit form as a function of the kinked crack velocity, the kink angle, and time. A perturbation method, using the kink angle as the perturbation parameter, is used. If the maximum energy release rate is accepted as the crack propagation criterion, then the crack will propagate straight ahead of the original crack when applying point load at the crack face.

Superlattice Interface and Lattice Strain Study by Ion Channeling

EXTENDED ABSTRACTWe have studied the interface and the lattice strain of superlattices by ion channeling technique. The objective in this work is to verify the existence of alternating tensile and compressive strain in the superlattice and to develop a method for measuring the lattice strain directly. Alternating layers of GaSb/Al Sb were grown epitaxially by MBE with 10 periods. The thickness of each individual layer is 30 nm. Channeling measurements and analysis were made using a 1.76 MeV 4He ion beam. The measurements reveal higher dechanneling along the [110] axis than along the [100] axis. This is consistent with the dechanneling results published earlier. The high dechanneling along the [110] axis has been considered due to the lattice strain that occurs in the layers caused by the slight mismatch between the lattice constants of the two materials. The strain effect make [110] axis slightly bent from layer to layer (“zigzag”), but it does not occur in [100] axis. The axial angular scan analyses were made around the [110] direction at the different depths using a movable energy window setting. We have found that the angular position of the best alignment shifts from layer to layer. The oscillation of those angular positions with depth is of a direct evidence of the existence of alternating tension and compression strain layers in the superlattice. The “kink” angle at the interface is given by the difference of the angular position between the first and second layer. This is found to be 0.17° ± 0.03 %. This is in a good agreement with the result calculated from elasticity. Preliminary result of this experiment is recently published.3We are also investigating the interface and lattice strain by planar angular scan across the (110) plane at a position three degrees from [110] axis.The similar oscillatory results have been found for {110} planar channeling and the “kink” angle measurement is in a good agreement with the results from axial angular scan.We believe that the method of ion beam channeling and angular scan is very effective in strain measurements in multi-layered heteroexpitaxy system.3. W. K. Chu, C. K. Pan and C.-A. Chang, Phys. Rev. Rapid Communication B28, 4033 (1983).

Energy-Release Rate and Crack Kinking Under Combined Loading

Based on the maximum energy-release-rate criterion, kinking from a straight crack is investigated under the plane strain condition. Solutions are obtained by the method that models a kink as a continuous distribution of edge dislocations. The energy-release rate is expressed as a quadratic form of the stress-intensity factors that exist prior to the onset of kinking, and the coefficients of this quadratic form are tabulated for various values of the kink angle. The examination of the results shows that Irwin’s formula for the energy-release rate remains valid for any kink angle provided that the stress-intensity factors in the formula are taken equal to those existing at the tip of a vanishingly small kink.