Archive for Rational Mechanics and Analysis

, Volume 184, Issue 3, pp 495–551 | Cite as

A New Mathematical Foundation for Contact Interactions in Continuum Physics

  • Friedemann SchurichtEmail author


The investigation of contact interactions, such as traction and heat flux, that are exerted by contiguous bodies across the common boundary is a fundamental issue in continuum physics. However, the traditional theory of stress established by Cauchy and extended by Noll and his successors is insufficient for handling the lack of regularity in continuum physics due to shocks, corner singularities, and fracture. This paper provides a new mathematical foundation for the treatment of contact interactions. Based on mild physically motivated postulates, which differ essentially from those used before, the existence of a corresponding interaction tensor is established. While in earlier treatments contact interactions are basically defined on surfaces, here contact interactions are rigorously considered as maps on pairs of subbodies. This allows the action exerted on a subbody to be defined not only, as usual, for sets with a sufficiently regular boundary, but also for Borel sets (which include all open and all closed sets). In addition to the classical representation of such interactions by means of integrals on smooth surfaces, a general representation using the distributional divergence of the tensor is derived. In the case where concentrations occur, this new approach allows a description of contact phenomena more precise than before.


Contact Interaction Measure Zero Radon Measure Representation Formula Partial Restriction 


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© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität zu KölnKölnGermany

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