On tangent geometry and generalised continuum with defects

This paper introduces tools on fibre geometry towards the framework of mechanics of microstructured continuum. The material is modelled by an appropriate bundle for which the associated connection and metric are induced from the Euclidean space by a smooth transformation represented by a fibre morphism from the bundle to Euclidean space. Furthermore, the general kinematic structure of the theory includes macroscopic and microscopic fields in a multiscaled approach, including large transformation. Defects appear in this geometrical point of view by an induced curvature, torsion and non-metricity tensor in the induced geometry. Special attention is given to transport along a finite path in order to extend the standard infinitesimal analysis of torsion and curvature to a macroscopical point of view. Both theoretical and numerical analysis may be handled without additional difficulties. Accordingly, several examples of transformation involving the distribution of material defects are exhibited and analysed.