We analyze geometrical structures necessary to represent bulk and surface interactions of standard and substructural nature in complex bodies. Our attention is mainly focused on the influence of diffuse interfaces on sharp discontinuity surfaces. In analyzing this phenomenon, we prove the covariance of surface balances of standard and substructural interactions.
According to [5] we define the * -exponential of a slice-regular function, which can be seen as a generalization of the complex exponential to quaternions. Explicit formulas for exp * (f ) are provided, also in terms of suitable sine and cosine functions. We completely classify under which conditions the * -exponential of a function is either slice-preserving or CJ -preserving for some J ∈ S and show that exp * (f ) is never-vanishing. Sharp necessary and sufficient conditions are given in order that exp *
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