Photoelastic and finite element analysis of different size spheres in contact
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The photoelastic stress-freezing technique is applied to observe the stress distribution inside two spheres of different sizes compressed together elastically. After the stress is frozen in, thin slices of the material containing the symmetry axis are prepared for observation through a polariscope. The stress distribution is compared with both the finite element numerical analysis and the Hertz analytical theory which is limited to small deformations. Among the three, the agreement between the experimental results and the finite element analysis is the best. The deviation from the Hertz theory is less in the larger sphere contacting a smaller one than in the smaller sphere contacting a larger one.
I. INTRODUCTION Ever since Hertz1 solved the pressure distribution for two spherical elastic bodies in contact in 1880, many contact problems have been applied to numerous engineering situations. For example, ball bearings in high speed turbomachinery often contact and compress each other. Microhardness indentations are now made between a hard indenter tip and a soft flat surface. The indenter often has a spherical tip.2 Friction and lubrication between rough surfaces are often modeled by spheres of different sizes in passing contact with each other.3'4 With the hot isostatic pressing (HIP) becoming increasingly popular, powder compaction is now an important process for making new materials. The powders are often modeled by spherical particles so that powder compaction is a problem of many spheres in contact.5-6 Upon increasing pressure, the contact is elastic to begin with and the deformation is at first small and then large when it becomes plastic. We are now developing a finite element code which includes elastic deformation, yielding, plastic deformation, creep, and diffusion. It is critical that we compare our numerical results with analytical and experimental findings at every stage of our development before our code becomes too large. Since the Hertz solution is only for small elastic deformation and there is no solution yet available for large elastic deformation, it seems advisable to determine experimentally the stress distribution between two contacting spheres undergoing large elastic deformation. These experimental results can be used on the one hand to determine the limits of the Hertz solution and on the other to compare with the finite element analysis. Photoelastic experiments have been performed for stress analysis in many engineering applications, especially for mechanical parts with complicated geometry. J. Mater. Res., Vol. 7, No. 4, Apr 1992
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The stress-optic law, formulated in 1852 by Maxwell,7 states that the principal refractive indices and the principal stresses are coincident. A fringe pattern can be observed if a thin section is viewed through polarizing filters. Drucker and Woodward8 were the first to show that photoelasticity can be used to solve a general threedimensional problem. Chu and Li9 employed the stressfreezing technique to an
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