arXiv subjects

Reuven Segev

Publications and source records attributed to Reuven Segev.

30 records · Page 2Linked to original sources

arXiv2014

Generalized Electrodynamics as a Special Case of Metric Independent Stress Theory

We use a metric invariant stress theory of continuum mechanics to formulate a simple generalization of the the basic variables of electrodynamics and Maxwell's equations to general differentiable manifolds of any dimension, thus viewing generalized electrodynamics as a special case of continuum mechanics. The basic variable is the potential, or a variation thereof, which is represented as an $r$-form in a $d$-dimensional spacetime. The stress for the case of generalized electrodynamics, is assumed to be represented by an $(d-r-1)$-form, a generalization of the Maxwell $2$-form.

math-ph↗

arXiv2014

Reynolds Transport Theorem for Smooth Deformations of Currents on Manifolds

The Reynolds transport theorem for the rate of change of an integral over an evolving domain is generalized. For a manifold $B$, a differentiable motion $m$ of $B$ in the manifold $\mathcal{S}$, an $r$-current $T$ in $B$, and the sequence of images $m(t)_{\sharp}T$ of the current under the motion, we consider the rate of change of the action of the images on a smooth $r$-form in $\mathcal{S}$. The essence of the resulting computations is that the derivative operator is represented by the dual of the Lie derivative operation on smooth forms.

math-ph↗

arXiv2014

Metric Independent Analysis of Second Order Stresses

A metric independent geometric analysis of second order stresses in continuum mechanics is presented. For a vector bundle $W$ over the $n$-dimensional space manifold, the value of a second order stress at a point $x$ in space is represented mathematically by a linear mapping between the second jet space of $W$ at $x$ and the space of $n$-alternating tensors at $x$. While only limited analysis can be performed on second order stresses as such, they may be represented by non-holonomic stresses, whose values are linear mapping defined on the iterated jet bundle, $J^{1}(J^{1}W)$, and for which an iterated analysis for first order stresses may be performed. As expected, we obtain the surface interactions on the boundaries of regions in space.

math-ph↗

arXiv2014

On the Geometry and Kinematics of Smoothly Distributed and Singular Defects

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 structure. In addition, both continuous distributions of defects as well as singular distributions are encompassed by the theory. In the general case, the material structure is specified by a de Rham current $T$ and the associated defects are given by its boundary. For a motion of defects associated with a family of diffeomorphisms of a material body, it is shown that the rate of change of the distribution of defects is given by the dual of the Lie derivative operator.

math-ph↗

arXiv2013

A unified geometric treatment of material defects

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 by a single differential form or current, whose boundary is identified with the defect itself. The possibility of defining a less restrictive dislocation structure is explored in terms of a plausible weak formulation of the theorem of Frobenius. Several examples are presented and discussed.

math-ph↗

arXiv2013

The configuration space and principle of virtual work for rough bodies

In the setting of an $n$-dimensional Euclidean space, the duality between velocity fields on the class of admissible bodies and Cauchy fluxes is studied using tools from geometric measure theory. A generalized Cauchy flux theory is obtained for sets whose measure theoretic boundaries may be as irregular as flat $(n-1)$-chains. Initially, bodies are modeled as normal $n$-currents induced by sets of finite perimeter. A configuration space comprising Lipschitz embeddings induces virtual velocities given by locally Lipschitz mappings. A Cauchy flux is defined as a real valued function on the Cartesian product of $(n-1)$-currents and locally Lipschitz mappings. A version of Cauchy's postulates implies that a Cauchy flux may be uniquely extended to an $n$-tuple of flat $(n-1)$-cochains. Thus, the class of admissible bodies is extended to include flat $n$-chains and a generalized form of the principle of virtual power is presented. Wolfe's representation theorem for flat cochains enables the identification of stress as an $n$-tuple of flat $(n-1)$-forms.

math-ph↗

arXiv2012

Radiative Transfer and Flux Theory

The fundamental notions of radiative transfer, e.g., Lambert's cosine rule, are studied from the point of view of flux and stress theory of continuum mechanics. For the classical case, where the radiance is distributed regularly over the unit sphere, it is shown that Lambert's rule follows from a balance law for the transfer of radiative power in each direction $\norb$ of the sphere, together with the appropriate Cauchy postulates and the additional assumption that the corresponding flux vector field $\radvec_{\norb}$ be parallel to $\norb$. An analogous theory is presented for the irregular case where the distribution of radiance on the sphere is given as a Borel measure.

math-ph↗

arXiv2012

Geometric Aspects of Singular Dislocations

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 boundary in the sense of currents represents the structure of the dislocations. Frank's rules for dislocations follow naturally from the nilpotency of the boundary operator.

math-ph↗

arXiv2006

Load Capacity of Bodies

For the stress analysis in a plastic body $Ω$, we prove that there exists a maximal positive number $C$, the \emph{load capacity ratio,} such that the body will not collapse under any external traction field $t$ bounded by $Y_{0}C$, where $Y_0$ is the elastic limit. The load capacity ratio depends only on the geometry of the body and is given by $$ \frac{1}{C}=\sup_{w\in LD(Ω)_D} \frac{\int_{\partialΩ}|w|dA} {\int_Ω|ε(w)|dV}=\left\|γ_D\right\|. $$ Here, $LD(Ω)_D$ is the space of isochoric vector fields $w$ for which the corresponding stretchings $ε(w)$ are assumed to be integrable and $γ_D$ is the trace mapping assigning the boundary value $γ_D(w)$ to any $w\in LD(Ω)_D$.

math.AP↗

arXiv2006

Optimal Stresses in Structures

For a given external loading on a structure we consider the optimal stresses. Ignoring the material properties the structure may have, we look for the distribution of internal forces or stresses that is in equilibrium with the external loading and whose maximal component is the least. We present an expression for this optimal value in terms of the external loading and the matrix relating the external degrees of freedom and the internal degrees of freedom. The implementation to finite element models consisting of elements of uniform stress distributions is presented. Finally, we give an example of stress optimization for of a two-element model of a cylinder under external traction.

physics.comp-ph↗

arXiv2005

Bounds on the Trace Mappings of LD-Fields

Bounds on the trace mappings defined on the Sobolev space W^{1,1}(Omega) and the space $LD(Omega)$ of integrable stains are obtained. Such bounds correspond to stress concentration--the ratio between the maximal stress in a body and the maximum of the traction applied to its boundary. The analysis leading to the bounds may be described in the mechanical context of stress theory and stress concentration.

math.AP↗

arXiv2004

Generalized Stress Concentration Factors for Equilibrated Forces and Stresses

As a sequel to a recent work we consider the generalized stress concentration factor, a purely geometric property of a body that for the various external loading fields indicates the worst ratio between the maximum of the optimal stress and maximum of the external loading. The optimal stress concentration factor pertains to a stress field that satisfies the principle of virtual work and for which the stress concentration factor is minimal. Unlike the previous work, we require that the external loading be equilibrated and that the stress field be a symmetric tensor field.

physics.flu-dyn↗