Examinando por Autor "Epstein, Marcelo"

Mostrando 1 - 8 de 8
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    Characteristic distribution: An application to material bodies
    (Elsevier, 2018) León, Manuel de; Epstein, Marcelo; Jiménez Morales, Víctor Manuel
    Associated to each material body B there exists a groupoid (B) consisting of all the material isomorphisms connecting the points of B. The uniformity character of B is reflected in the properties of (B): B is uniform if, and only if, (B) is transitive. Smooth uniformity corresponds to a Lie groupoid and, specifically, to a Lie subgroupoid of the groupoid 1 (B,B) of 1- jets of B. We consider a general situation when (B) is only an algebraic subgroupoid. Even in this case, we can cover B by a material foliation whose leaves are transitive. The same happens with (B) and the corresponding leaves generate transitive Lie groupoids (roughly speaking, the leaves covering B). This result opens the possibility to study the homogeneity of general material bodies using geometric instruments.
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    Characteristic foliations of material evolution: from remodeling to aging
    (SAGE Publications, 2022) León, Manuel de; Epstein, Marcelo; Jiménez Morales, Víctor Manuel
    For any body-time manifold (Formula presented.) there exists a groupoid, called the material groupoid, encoding all the material properties of the material evolution. A smooth distribution, the material distribution, is constructed to deal with the case in which the material groupoid is not a Lie groupoid. This new tool provides a unified framework to deal with general non-uniform material evolution.
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    Lie groupoids and algebroids applied to the study of uniformity and homogeneity of Cosserat media
    (World Scientific, 2018) León, Manuel de; Epstein, Marcelo; Jiménez Morales, Víctor Manuel
    A Lie groupoid, called second-order non-holonomic material Lie groupoid, is associated in a natural way to any Cosserat medium. This groupoid is used to give a new definition of homogeneity which does not depend on a material archetype. The corresponding Lie algebroid, called second-order non-holonomic material Lie algebroid, is used to characterize the homogeneity property of the material. We also relate these results with the previously obtained ones in terms of non-holonomic second-order G¯¯¯¯ -structures.
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    Lie groupoids and algebroids applied to the study of uniformity and homogeneity of material bodies
    (American Institute of Mathematical Sciences, 2018) León, Manuel de; Epstein, Marcelo; Jiménez Morales, Víctor Manuel
    A Lie groupoid, called material Lie groupoid, is associated in a natural way to any elastic material. The corresponding Lie algebroid, called material algebroid, is used to characterize the uniformity and the homogeneity properties of the material. The relation to previous results in terms of G−structures is discussed in detail. An illustrative example is presented as an application of the theory.
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    Material distributions
    (SAGE, 2017) León, Manuel de; Epstein, Marcelo; Jiménez Morales, Víctor Manuel
    The concept of material distribution is introduced as describing the geometric material structure of a general nonuniform 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.
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    Material geometry
    (Springer, 2019) Epstein, Marcelo; León, Manuel de; Jiménez Morales, Víctor Manuel
    Walter Noll's trailblazing constitutive theory of material defects in smoothly uniform bodies is recast in the language of Lie groupoids and their associated Lie algebroids. From this vantage point the theory is extended to non-uniform bodies by introducing the notion of singular material distributions and the physically cognate idea of graded uniformity and homogeneity.
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    Material Geometry: Groupoids in Continuum Mechanics
    (World Scientific Publishing Co. Pte. Ltd., 2021) León, Manuel de; Epstein, Marcelo; Jiménez Morales, Víctor Manuel
    This monograph is the first in which the theory of groupoids and algebroids is applied to the study of the properties of uniformity and homogeneity of continuous media. It is a further step in the application of differential geometry to the mechanics of continua, initiated years ago with the introduction of the theory of G-structures, in which the group G denotes the group of material symmetries, to study smoothly uniform materials. The new approach presented in this book goes much further by being much more general. It is not a generalization per se, but rather a natural way of considering the algebraic-geometric structure induced by the so-called material isomorphisms. This approach has allowed us to encompass non-uniform materials and discover new properties of uniformity and homogeneity that certain material bodies can possess, thus opening a new area in the discipline.
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    On the Homogeneity of Non-uniform Material Bodies
    (Springer, 2020) León, Manuel de; Epstein, Marcelo; Jiménez Morales, Víctor Manuel
    A groupoid (B) called material groupoid is naturally associated to any simple body B (see [11, 9, 10]). The material distribution is introduced due to the (possible) lack of differentiability of the material groupoid (see [13, 15]). Thus, the inclusion of these new objects in the theory of material bodies opens the possibility of studying non-uniform bodies. As an example, the material distribution and its associated singular foliation result in a rigorous and unique subdivision of the material body into strictly smoothly uniform sub-bodies, laminates, filaments and isolated points. Furthermore, the material distribution permits us to present a “measure" of uniformity of a simple body as well as more general definitions of homogeneity for non-uniform bodies.