Persona: Jiménez Morales, Víctor Manuel
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Jiménez Morales
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Víctor Manuel
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Publicación Characteristic distribution: An application to material bodies(Elsevier, 2018) León, Manuel de; Epstein, Marcelo; Jiménez Morales, Víctor ManuelAssociated 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.Publicación Contact Hamiltonian and Lagrangian systems with nonholonomic constraints(American Institute of Mathematical Sciences, 2021) León, Manuel de; Lainz Valcázar, Manuel; Jiménez Morales, Víctor ManuelIn this article we develop a theory of contact systems with nonholonomic constraints. We obtain the dynamics from Herglotz's variational principle, by restricting the variations so that they satisfy the nonholonomic constraints. We prove that the nonholonomic dynamics can be obtained as a projection of the unconstrained Hamiltonian vector field. Finally, we construct the nonholonomic bracket, which is an almost Jacobi bracket on the space of observables and provides the nonholonomic dynamics.Publicación The evolution equation: an application of groupoids to material evolution(CSIC, 2021) León, Manuel de; Jiménez Morales, Víctor ManuelThe aim of this paper is to study the evolution of a material point of a body by itself, and not the body as a whole. To do this, we construct a groupoid encoding all the intrinsic properties of the material point and its characteristic foliations, which permits us to define the evolution equation. We also discuss phenomena like remodeling and aging.Publicación Characteristic foliations of material evolution: from remodeling to aging(SAGE Publications, 2022) León, Manuel de; Epstein, Marcelo; Jiménez Morales, Víctor ManuelFor 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.Publicación A geometric model for non-uniform processes of morphogenesis(Elsevier, 2022-09-23) León, Manuel de; Jiménez Morales, Víctor ManuelIn this paper we present an application of the groupoid theory to the study of relevant case of material evolution phenomena, the process of morphogenesis. Our theory is inspired by Walter Noll’s theories of continuous distributions and provides a unifying and very simple framework of these phenomena. We present the explicit equation, the morphogenesis equation, to calculate the material distributions associated to this phenomenon.Publicación Material Geometry: Groupoids in Continuum Mechanics(World Scientific Publishing Co. Pte. Ltd., 2021) León, Manuel de; Epstein, Marcelo; Jiménez Morales, Víctor ManuelThis 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.Publicación 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 ManuelA 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.Publicación Reduction of a Hamilton — Jacobi Equation for Nonholonomic Systems(Springer, 2019) Esen, Oğul; León, Manuel de; Sardón, Cristina; Jiménez Morales, Víctor ManuelWe discuss, in all generality, the reduction of a Hamilton — Jacobi theory for systems subject to nonholonomic constraints and invariant under the action of a group of symmetries. We consider nonholonomic systems subject to both linear and nonlinear constraints and with different positioning of such constraints with respect to the symmetries.Publicación Material distributions(SAGE, 2017) León, Manuel de; Epstein, Marcelo; Jiménez Morales, Víctor ManuelThe 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.Publicación New notions of uniformity and homogeneity of Cosserat media(AIP Publising, 2023) León, Manuel de; Jiménez Morales, Víctor ManuelIn this paper, we study internal properties of Cosserat media. In fact, by using groupoids and smooth distributions, we obtain three canonical equations. The non-holonomic material equation for Cosserat media characterizes the uniformity of the material. The holonomic material equation for Cosserat media permits us to study when a Cosserat material is a second-grade material. It is remarkable that these two equations also provide us a unique and maximal division of the Cosserat medium into uniform and second-grade parts, respectively. Finally, we present a proper definition of homogeneity of the Cosserat medium, which does not need to assume uniformity. Thus, the homogeneity equation for Cosserat media characterizes this notion of homogeneity.