AN ORTHOGONAL UNIFORM/PARABOLIC SURFACE PRESSURE HALF-SPACE ELEMENT FOR THE ANALYSIS OF NON-HERTZIAN CONTACT
An orthogonal uniform/parabolic surface pressure half-space element is derived for the analysis of non-Hertzian contact. Restriction is made to parabolic variation of the surface gap in the contact in one orthogonal direction. The surface pressure is integrated analytically over the surface element to obtain influence co-efficeints, e.g. for the normal surface displacement. The coefficients are assembled into matrices leading to a linear system of equations. After solving the parabolic pressure variation is transformed to an equivalent semi-elliptic one coinciding with Hertzian line contact. Adjustments are introduced to approximate the finite size of the bodies in contact. The theoretical predictions are compared to experiments involving the non-Hertzian contact between cylinders of finite lengths.
A Mathematical Programming Method for Design of Elastic Bodies in Contact
Equation (6) results from the assumption that transverse normal strain will be produced by the transverse thermal expansion and not be affected by elastic strains. The authors feel that this is a proper assumption for a shell theory consistent with transversely isotropic materials. I t is not unusual for the transverse thermal expansion to be much larger than the in-plane thermal expansion (i.e., acT » aT) of such materials. This holds particularly for pyrolytic graphite where the ratio of thermal expansion of the material in the "c" direction to that in the "a" direction varies approximately between 10 and 30. Therefore, although the transverse normal elastic strain is small, the corresponding thermal strain is not small and is comparable to the in-plane normal strains ex and eg. Equation 11 (a) is developed by a method of asymptotic integration which is adequately described in the literature, e.g., [1] , and done in detail for shells of revolution of pyrolytic graphitetype materials in [2]. Since the derivation is lengthy but straightforward and the procedure adequately documented, only the results were given as equation 11 (a).
Three-dimensional elasticity problems for the prismatic bar
A general solution is given to the three-dimensional linear elastic problem of a prismatic bar subjected to arbitrary tractions on its lateral surfaces, subject only to the restriction that they can be expanded as finite power series in the axial coordinate z. The solution is obtained by repeated differentiation of the tractions with respect to z, establishing a set of sub-problems . A recursive procedure is then developed for generating the solution to from that for . This procedure involves three steps: integration of the stress and displacement fields with respect to z, using an appropriate Papkovich–Neuber (P–N) representation; solution of two-dimensional in-plane and antiplane corrective problems for the tractions in that are independent of z; and expression of these corrective solutions in P–N form. The method is illustrated by an example.