Three-dimensional Contact Problems

It is to be expected that three-dimensional boundary value problems will present greater difficulties than plane problems. In particular, with the far wider choice of boundary regions on which to specify displacement and stress, one rapidly meets problems that are unsolvable - at least analytically. This is true even for elastic materials. In fact, the contact problem with an elliptical contact area is the most general problem that allows an explicit analytic solution - for elastic materials [Galin (1961), Lur e (1964)], in the case of half-space problems. This corresponds to an ellipsoidal indentor, according to classical Hertz theory. The theory can be extended to cover contact between two gently curved bodies. The solution is valid only for quasi-static conditions. 

In the absence of tangential motion, the equivalent viscoelastic problem is solvable, just as in the plane case - discussed in Sect. 3.10 - and by essentially the same methods. However, the problem is worth considering in some detail because, in contrast with the plane case, the indentation is determinate. This enables one to discuss impact problems, which are of considerable interest. These topics are covered in Sects. 5.2 and 5.3. 

Inertial normal contact problems in three dimensions have been considered by Sabin (1975, 1987). By means of an integral transform technique, he reduces the problem to a set of dual Integral equations, which are in turn reduced to a single Volterra integral equation. This is solved numerically. He obtains the interesting result that the contact pressure is not significantly different from that in the non-inertial problem. A similar observation had been made earlier, in connection with the elastic problem, by Tsai (1971). 

The problem of a moving indentor has been considered by Golden (1982) in an approximate manner, by simply assuming that the contact region is elliptical for an ellipsoidal or spherical punch. The difficulty is that the physical significance of this assumption is not totally apparent. It is valid at both low and high veloci- 

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V.M. Alexandrov and D.A. Pozharskii Three-Dimensional Contact Problems. 2001... 

Three-dimensional contact problem solutions are obtained and analyses of impact and hysteretic friction are made. 

The simulation of the dent over-rolling is a complex three-dimensional contact problem, that requires a mesh fine enough to describe the geometry of the dent and large enough to cover the area of contact. To solve this problem, a three dimensional semi-analytical contact code presented by Jacq et al. [10,12] is used. This code can solve 3D ncrmal... 

Primary hepatocytes or liver slices can be used to measure metabolism, but only for a short period of time after the liver sample has been removed from the body however, both models have problems associated with their use. In liver slices, cell-to-cell contact and three-dimensional structure are maintained with a full compliment of cell types (including Kupffer cells) primary hepatocytes have lost the orientation, organization, and nonhepatocyte cells which may contribute to the metabolic activity of the whole liver. Liver slices may suffer from the presence of damaged or dead cells, restricted access of culture media to internal cells, thereby reducing oxygen and nutrient supplies, and from a build-up of toxic products that may result in impaired metabolism. Perfusion of tissue in situ can ameliorate these problems, but of necessity such experiments can normally only be carried out in animals, and these will have different metabolic profiles due to differences in enzyme and transporter expression. 

The three-dimensional properties of a laminate given by Eqns (6.11), (6.12), and (6.32) are needed in situations where out-of-plane stresses develop. Besides the obvious case of out-of-plane loading such as the local indentation and the associated solution of contact stresses in an impact problem, out-of-plane stresses typically arise near free edges of laminates, in the immediate vicinity of plydrops and near matrix cracks or delaminations. Typical examples are shown in Figure 6.4. The red lines indicate regions in the vicinity of which out-of-plane stresses [Pg.132]

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