A study of stress concentrations in solids with circular holes by three-dimensional special hybrid stress finite elements

Four kinds of 12-node solid finite elements with two parallel faces and one traction-free cylindrical surface have been developed based on the assumed stress hybrid method. Cylindrical coordinates are used so that the assumed stresses satisfy the equilibrium equations as well as the traction-free condition over the cylindrical boundary. In the limiting case of plane stress the assumed stresses also satisfy the compatibility conditions. Examples have indicated that these present special solid finite elements are far superior in predicting the distributions of the circumferential stresses and the stress concentration factors for solids with circular holes.

Uncoupled Wave Systems and Dispersion in an Infinite Solid Cylinder

In this study, it is shown that there exist uncoupled wave systems for general non-axisymmetric wave propagation in an infinite isotropic cylinder. Two cylindrical surface conditions corresponding to the uncoupled wave systems are discussed. The solutions of the uncoupled wave systems are shown to provide proper bounds of Pochhammer’s equation for a free cylindrical surface. The bounds, which are easy to construct for any Fourier number in the circumferential direction, can be used to trace the branches of Pochhammer’s equation. They also give insight into the modal composition of the branches of Pochhammer’s equation at and between the intersections of the bounds. More refined dispersion relations of Pochhammer’s equation are possible through an asymptotic analysis of the itersections of the branches of Pochhammer’s equation with one family of the bounds. The asymptotic nature of wave motion corresponding to large wave numbers, imaginary or complex, for Pochhammer’s equation is studied. The wave motion is asymptotically equivoluminal for large imaginary wave numbers, and is characterized by coupled dilatation and shear for large complex wave numbers.

Steady and oscillatory thermocapillary convection in liquid columns with free cylindrical surface

In liquid columns (Prandtl number 8·9) with free cylindrical surface heated from above, strong thermocapillary convection (TC) has been observed. Stationary thermocapillary convection exists in the form of a single axially symmetric roll bound to the free surface. For aspect ratios l/a < 1 the radial extension of the roll equals the zone length. The stream velocities and the temperature distribution were measured.The influence of buoyant forces due to horizontal temperature gradients in the experiments was also studied. Buoyant forces become obvious for a contaminated free surface and in bulk regions far from the cylinder surface.The thermocapillary convection shows a transition to time-dependent oscillatory motion when a critical Marangoni number Mac is exceeded. A unique Mac = 7 × 103 has been found for zones with lengths l < 3·5 mm. The oscillatory state of thermocapillary convection has experimentally been proved to be a distortion of the laminar state in form of a wave travelling in the azimuthal direction. A unique non-dimensional wavenumber ≈ 2·2 (near Mac) of the distortion has been found. The non-dimensional frequency of the temperature oscillations is independent of zone size if the aspect ratio is held constant. However, the non-dimensional frequency of temperature oscillations increases linearly with the aspect ratio of the zone. This result is interpreted as a dependence of the phase velocity of the running disturbance on the aspect ratio.

Upper-Bound Solutions of Some Axisymmetric Cold Forging Problems

A new velocity field for the forging of some cylindrical workpieces is presented taking into account the bulging of the free cylindrical surface during deformation. The velocity field is more realistic than that proposed by Avitzur in that the physical boundary and continuity conditions are satisfied. Numerical results indicate that the current analysis improves upon previous solutions published in the literature.