The elastoplastic shock problem as an example of the resolution of ambiguities in the multiplication of distributions

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Abstract
One-dimensional collisions between two isotropic solids, in which the equations of physics lead to ‘‘multiplications of distributions,’’ are considered. Based on this example, a general physicomathematical method, to be adapted to each particular case, is proposed to resolve the ambiguity inherent in such products. This can be achieved with the aid of a new mathematical theory of generalized functions, which permits dealing with mathematical phenomena of a microscopic nature that govern products of distributions having singularities at the same point. This tool has recently been applied in various situations (in continuum mechanics) in which the equations of physics lead to ‘‘heuristic products of distributions.’’ One obtains new (algebraic) formulas in the simplest cases, and new numerical schemes in more general cases. The key to the resolution of ambiguities lies in more precise statements of the laws of physics than are permitted within distribution theory, and have no analog in classical analysis, so that in general a resolution cannot be obtained from ‘‘formal calculations.’’

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