Continuum mechanics, stresses, currents and electrodynamics

Segev, Reuven

doi:10.1098/rsta.2015.0174

Cited by 9 publications

(9 citation statements)

References 32 publications

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“…where K runs through all compact sets in scriptB and C K ∞ ( κ * V Y false) has the topology of uniform convergence of all derivatives [44]. Thus, generalized velocities are represented by sections of κ * V Y having compact supports so that forces may be viewed as tensor-valued currents or generalized sections [37, 45]. An analogous construction can be applied to C 1 -sections.…”

Section: Configurations Velocities and Forcesmentioning

confidence: 99%

“…In the general case, one obtains a generalization of electrodynamics, referred to as p -form electrodynamics, as in Henneaux and Teitelboim [34, 35]. (For further details see Segev [37]. ) A different theory in which form-conjugate forces appear is the theory of dislocations.…”

Section: Some Special Casesmentioning

confidence: 99%

“…We give special attention to the case of smooth stress fields for which we write down the field equation in an explicit di erential form. For example, a weak formulation of p-form, [HT86,HT88], premetric, [HIO06], electrodynamics was shown in [Seg16] to follow from stress theory for fields represented by p-forms in the case where the stress object has a particularly simple form. It should be mentioned that we study here the theory concerning the existence of stresses and the equations it satisfies; we do not study the analytic aspects of the field equations obtained after the constitutive relations are used, in tems of existence and uniqueness of solutions, appropriate function spaces, etc.…”

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confidence: 99%

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Stress theory for classical fields

Kupferman

Olami

Segev

2017

Mathematics and Mechanics of Solids

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A. Classical field theories together with the Lagrangian and Eulerian approaches to continuum mechanics are embraced under a geometric setting of a fiber bundle. The base manifold can be either the body manifold of continuum mechanics, space manifold, or space-time. Di erentiable sections of the fiber bundle represent configurations of the system and the configuration space containing them is given the structure of an infinite dimensional manifold. Elements of the cotangent bundle of the configuration space are interpreted as generalized forces and a representation theorem implies that there exist a stress object representing forces, non-uniquely. The properties of stresses are studies as well as the role of constitutive relations in the present general setting.

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Section: Configurations Velocities and Forcesmentioning

confidence: 99%

Section: Some Special Casesmentioning

confidence: 99%

mentioning

confidence: 99%

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Stress theory for classical fields

Kupferman

Olami

Segev

2017

Mathematics and Mechanics of Solids

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“…The applications of modern differential-geometric approach in continuum mechanics were firstly developed by W. Noll and C.-C. Wang in [38,39]. One can find the state of the art in regard to this approach in [26,28,29,33,35,[40][41][42][43][44][45][46]). It is relevant to note here, that the body manifold does not contain information about the positions of material points in physical space.…”

Section: Introductionmentioning

confidence: 99%

Incompatible Deformations in Additively Fabricated Solids: Discrete and Continuous Approaches

2021

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The present paper is intended to show the close interrelationship between non-linear models of solids, produced with additive manufacturing, and models of solids with distributed defects. The common feature of these models is the incompatibility of local deformations. Meanwhile, in contrast with the conventional statement of the problems for solids with defects, the distribution for incompatible local deformations in additively created deformable body is not known a priori, and can be found from the solution of the specific evolutionary problem. The statement of the problem is related to the mechanical and physical peculiarities of the additive process. The specific character of incompatible deformations, evolved in additive manufactured solids, could be completely characterized within a differential-geometric approach by specific affine connection. This approach results in a global definition of the unstressed reference shape in non-Euclidean space. The paper is focused on such a formalism. One more common factor is the dataset which yields a full description of the response of a hyperelastic solid with distributed defects and a similar dataset for the additively manufactured one. In both cases, one can define a triple: elastic potential, gauged at stress-free state, and reference shape, and some specific field of incompatible relaxing distortion, related to the given stressed shape. Optionally, the last element of the triple may be replaced by some geometrical characteristics of the non-Euclidean reference shape, such as torsion, curvature, or, equivalently, as the density of defects. All the mentioned conformities are illustrated in the paper with a non-linear problem for a hyperelastic hollow ball.

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“…These notes provide an introduction to the fundamentals of global analytic continuum mechanics as developed in [ES80,Seg81,MH94,Seg86b,Seg86a,Seg16]. The terminology "global analytic" is used to imply that the formulation is based on the notion of a configuration space of the mechanical system as in analytic classical mechanics.…”

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confidence: 99%

Notes on Global Stress and Hyper-Stress Theories

Segev¹

2019

Preprint

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A. The fundamental ideas and tools of the global geometric formulation of stress and hyper-stress theory of continuum mechanics are introduced. The proposed framework is the infinite dimensional counterpart of statics of systems having finite number of degrees of freedom, as viewed in the geometric approach to analytical mechanics. For continuum mechanics, the configuration space is the manifold of embeddings of a body manifold into the space manifold. Generalized velocity fields are viewed as elements of the tangent bundle of the configuration space and forces are continuous linear functionals defined on tangent vectors, elements of the cotangent bundle. It is shown, in particular, that a natural choice of topology on the configuration space, implies that force functionals may be represented by objects that generalize the stresses of traditional continuum mechanics.

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Continuum mechanics, stresses, currents and electrodynamics

Cited by 9 publications

References 32 publications

Stress theory for classical fields

Stress theory for classical fields

Incompatible Deformations in Additively Fabricated Solids: Discrete and Continuous Approaches

Notes on Global Stress and Hyper-Stress Theories

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