Marcelo Epstein
The principle of fractal stiffness self-similarity is expanded to encompass structures with several differently-scaled contributors to the total stiffness matrix. The generalized principle is applied to solve the problem of a fractal…
The theory of composite mixtures consisting of $n$ constituents is framed within the schema provided by the notion of $n$-groupoid. The point of departure is the analysis of $n$-dimensional hypercubes and their skeletons, to each of whose…
Although it is often asserted that, in view of their reduced length, axially compressible beams have a higher buckling load than their inextensible counterpart, a detailed analysis demnstrates that this is not necessarily the case. The…
Budiansky's nonlinear shell theory is particularized to a 2D setting, and thereupon generalized to a fully nonlinear, statically and kinematically exact, theory of strain-gradient elasticity of beams. The governing equations are displayed…
The use of double groupoids and their associated double Lie algebroids and characteristic distributions is proposed for the description and analysis of continuous media that carry two different constitutive or geometric structures. Various…
The constitutive characterization of the uniformity and homogeneity of binary elastic composites is presented in terms of a combination of the material groupoids of the individual constituents. The incorporation of these two groupoids…
The 1950's foundational literature on rational mechanics exhibits two somewhat distinct paradigms to the representation of continuous distributions of defects in solids. In one paradigm, the fundamental objects are geometric structures on…
An attempt is made to bring into harmony two of the paradigms commonly used in the theory of continuous distributions of defects. It is shown that the common differential geometric apparatus is provided neatly by the theory of G-structures.…
A hybrid quantum-classical model is proposed whereby a micro-structured (Cosserat-type) continuum is construed as a principal Hilbert bundle
A novel formulation of statics in terms of the exterior algebra of an affine space is shown to be the underlying mathematical structure of Vlasov's thin-walled beam theory in structural mechanics.
Lie groupoids and their associated algebroids arise naturally in the study of the constitutive properties of continuous media. Thus, Continuum Mechanics and Differential Geometry illuminate each other in a mutual entanglement of theory and…
The problem of constant-speed ballistics is studied under the umbrella of non-linear non-holonomic constrained systems. The Newtonian approach is shown to be equivalent to the use of Chetaev's rule to incorporate the constraint within the…
A continuum mechanical framework for the description of the geometry and kinematics of defects in material structure is proposed. The setting applies to a body manifold of any dimension which is devoid of a Riemannian or a parallelism…
The existence of solutions to the boundary tracking of the displacement at one end of a linear Timoshenko beam is discussed on the basis of the Cauchy problem with time and space interchanged.
A unified theory of material defects, incorporating both the smooth and the singular descriptions, is presented based upon the theory of currents of Georges de Rham. The fundamental geometric entity of discourse is assumed to be represented…
The theory of singular dislocations is placed within the framework of the theory of continuous dislocations using de Rham currents. For a general $n$-dimensional manifold, an $(n-1)$-current describes a local layering structure and its…
The notions of uniformity and homogeneity of elastic materials are reviewed in terms of Lie groupoids and frame bundles. This framework is also extended to consider the case Functionally Graded Media, which allows us to obtain some…