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Material Geometry
Entity
UAM. Departamento de MatemáticasDate
2019-11-15Subjects
Lie, Grupos de - Tesis doctorados; Distribuciones, Teoría de (Análisis funcional) - Tesis doctorados; MatemáticasNote
Tesis doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de lectura:15-11-2019Esta obra está bajo una licencia de Creative Commons Reconocimiento-NoComercial-SinObraDerivada 4.0 Internacional.
Abstract
In continuum physics the physical properties of a elastic body are
characterized for all the constitutive relations. This measures the
mechanical response produced at each particle by a deformation in a local
neighbourhood of the particle. Di erential geometry provides a rigorous
mathematical framework not only to present the constitutive properties
but to discover and prove results. For applications, it is usual that the
bodies are assumed uniform and homogeneous in the sense of that the
body is made of a unique material and there is a con guration in such a
way that the mechanical response is the same at all the points.
The main purpose of this thesis is to follow the Noll's approach to present a
mathematical framework based on groupoids, algebroids and distributions
to deal with non-uniform and inhomogeneous simple bodies.
For any simple body a unique groupoid, called material groupoid, may
be naturally associated. The unifomity of the body coincides with the
transitivity of the groupoid. If the material groupoid turns out to be
a Lie groupoid the associated Lie algebroid, called material algebroid, is
available. Then, the homogoneneity is characterized by the integrability of
both (material groupoid and material algebroid).
However, the property of being Lie groupoid is not guaranteed. In fact,
smooth uniformity corresponds to that imposition of di erentiability on the
material groupoid. Smooth distributions are now introduced to deal with
this case. In fact, two smooth distributions, called material distributions,
may be canonically de ned generalizing the notion of Lie algebroid. Thus,
it is proved that we can cover the simple body by a material foliation whose leaves are (smoothly) uniform. These new tools are also used to present a
\measure" of uniformity and an homogeneity for non-uniform bodies.
The construction of the material distribution is generalized to a much more
abstract framework in which the case of an arbitrary subgroupoid of a Lie
groupoid is treated. We also study Cosserat media by imposing that the
corresponding material groupoid is a Lie groupoid.
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Google Scholar:Jiménez Morales, Victor Manuel
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