The geometry of contact, separation, and reformation of continuous bodies
After about two centuries of uninterrupted efforts the theory of deformation of continuous bodies has now reached a satisfactory level of coherence. On the other hand, no such progress has been made in establishing a mathematical theory concerned with the separation and contact of real objects, which may develop cracks or splits, or shatter to pieces. The present article is restricted to the purely geometrical description of the change of coherence for such bodies, needed for subsequent investigations, and analysis of the kinematics and dynamics of these phenomena. Contact and separation involve the disappearing and the creation of boundaries in a continuous body. One may ask, therefore, what it is that retains the identity of a body during such process. The answer is that the subbodies of a given body belong to a materially ordered set of a Boolean algebra and the material points are determined by filters of such subbodies. The separation or contact may be defined according to the set of filters that individuate material points. In terms of filters it is possible to define the ``completion'' \(\overline{\mathcal B}\) of a body \(\mathcal B\) as the set of all minimal filters. If \(\underset\widetilde{}\kappa({\mathcal B})\) is the placement associated with \(B\), it has a continuous extension \(\bar\kappa:\overline{\mathcal B}\to \text{clo Rng }\kappa\), which is a uniform homeomorphism. In classical continuum mechanics, placements are associated with transplacements Tp in order to define the changes of positions of bodies in the Euclidean space. The notion of transplacement is now generalized by introducing the ``extended transplacements'' \(\overline{\text{Tp}}\), which are suitable completions of the transplacements Tp. The mappings in \(\overline{\text{Tp}}/\text{Tp}\) are called ``improper transplacements''. The notion of improper transplacement permits a precise mathematical definition of separation and contact of continuous bodies. The paper is not easy to read because certain axioms are not geometrically evident, but it has the merit of giving for the first time a precise framework to concepts left traditionally vague in continuum mechanics.
scientific article; zbMATH DE number 411743