Material distributions

Abstract

The concept of material distribution is introduced as describing the geometric material structure of a general non-uniform body. Any smooth constitutive law is shown to give rise to a unique smooth integrable singular distribution. Ultimately, the material distribution and its associated singular foliation result in a rigorous and unique subdivision of the material body into strictly smoothly uniform components. Thus, the constitutive law induces a unique partition of the body into smoothly uniform sub-bodies, laminates, filaments and isolated points.

Characteristic foliations of material evolution: from remodeling to aging

“…We now present a formulation of the time evolution of a material body, following mainly [6][7][8]. In our geometrical description of the theory of simple bodies, the time variable has not played a role.…”

“…The collection of all the material isomorphisms of a body constitutes always a groupoid, called the material groupoid. This point of view has been developed extensively in the book [5] (see also [6][7][8]) even for more general bodies where the material groupoid is not a Lie groupoid.…”

An Intrinsic Geometric Formulation of Hyper-Elasticity, Pressure Potential and Non-Holonomic Constraints

“…This point was emphasized by Noll [42,43], who called a formulation of continuum mechanics on the body B as intrinsic 1 and by Rougée [49,50] who refers to it as an intrinsic Lagrangian framework, whereas a formulation on a reference configuration is denominated as a standard Lagrangian approach. Intrinsic formulations nowadays use modern tools in differential geometry [35,32,15,33,55]. In this direction, we insist on the fundamental role played by the manifold of all the Riemannian metrics on the body -introduced in solids mechanics by Rougée [49,51,52] and Fiala [18,19,20] -in the formulation of hyper-elasticity (see section 3).…”

Material Geometry of Binary Composites

“…The Lie algebroid associated with a Lie groupoid is a kind of infinitesimal version of the latter and carries information pertinent to the determination of the presence of inhomogeneities [10]. Somewhat surprisingly, even when the material groupoid is only an algebraic (rather than a Lie) subgroupoid of the 1-jet groupoid of the body, a singular material distribution can be defined [11,12] and used to obtain a material foliation, with transitive leaves representing smoothly uniform sub-bodies, layers, filaments, and isolated points.…”