1,720,980 research outputs found
Analytic and geometric properties of dislo- cation singularities
No full textThis paper deals with the analysis of the singularities arising from the solutions of the problem, where F is a 3 × 3 matrix-valued Lp-function ($1les p) and μ a 3 × 3 matrix-valued Radon measure concentrated in a closed loop in 3, or in a network of such loops (as, for instance, dislocation clusters as observed in single crystals). In particular, we study the topological nature of such dislocation singularities. It is shown that, the absolutely continuous part of the distributional gradient Du of a vector-valued function u of special bounded variation. Furthermore, u can also be seen as a multi-valued field, that is, can be redefined with values in the three-dimensional flat torus 3 and hence is Sobolev-regular away from the singular loops. We then analyse the graphs of such maps represented as currents in × 3 and show that their boundaries can be written in term of the measure μ. Readapting some well-known results for Cartesian currents, we recover closure and compactness properties of the class of maps with bounded curl concentrated on dislocation networks. In the spirit of previous work, we finally give some examples of variational problems where such results provide existence of solutions
Variational evolution of dislocations in single crystals
No full textIn this paper, we provide an existence result for the energetic evolution of a set of dislocation lines in a three-dimensional single crystal. The variational problem consists of a polyconvex stored elastic energy plus a dislocation energy and some higher-order terms. The dislocations are modeled by means of integral one-currents. Moreover, we discuss a novel dissipation structure for such currents, namely the flat distance, that will serve to drive the evolution of the dislocation clusters
Currents and dislocations at the continuum scale
No full textA striking geometric property of elastic bodies with dislocations is that the deformation tensor cannot be written as the gradient of a one-to-one immersion, its curl being nonzero and equal to the density of the dislocations, a measure concentrated in the dislocation lines. In this work, we discuss the mathematical properties of such constrained deformations and study a variational problem in finite-strain elasticity, where Cartesian maps allow us to consider deformations in Lp with 1≤p<2, as required for dislocation-induced strain singularities. Firstly, we address the problem of mathematical modeling of dislocations. It is a key purpose of the paper to build a framework where dislocations are described in terms of integral 1-currents and to extract from this theoretical setting a series of notions having a mechanical meaning in the theory of dislocations. In particular, the paper aims at classifying integral 1-currents, with modeling purposes. In the second part of the paper, two variational problems are solved for two classes of dislocations, at the mesoscopic and at the continuum scale. By continuum it is here meant that a countable family of dislocations is considered, allowing for branching and cluster formation, with possible complex geometric patterns. Therefore, modeling assumptions of the defect part of the energy must also be provided, and discussed
Constraint reaction ant the Peach-Koehler force for dislocation networks
No full textIn the presence of dislocations, the elastic deformation tensor F is not a gradient but satisfies the condition Curl F = Lambda(T)(L) (with the dislocation density 3 L a tensor-valued measure concentrated in the dislocation L). Then F is an element of L-p with 1 <= p < 2. This peculiarity is at the origin of the mathematical difficulties encountered by dislocations at the mesoscopic scale, which are here modeled by integral 1-currents free to form complex geometries in the bulk. In this paper, we first consider an energy-minimization problem among the couples (F, L) of strains and dislocations, and then we exhibit a constraint reaction field arising at minimality due to the satisfaction of the condition on the deformation curl, hence providing explicit expressions of the Piola-Kirchhoff stress and PeachKoehler force. Moreover, it is shown that the Peach-Koehler force is balanced by a defect-induced configurational force, a sort of line tension. The functional spaces needed to mathematically represent dislocations and strains are also analyzed and described in a preliminary part of the paper
A variational approach to single crystals with dislocations
Full text linkWe study the graphs of maps u : Omega -> R-3 whose curl is an integral 1-current with coefficients in Z(3). We characterize the graph boundary of such maps under a suitable summability property. We apply these results to study a three-dimensional single crystal with dislocations forming general one-dimensional clusters in the framework of finite elasticity. By virtue of a variational approach, a free energy depending on the deformation field and its gradient is considered. The problem we address is the joint minimization of the free energy with respect to the deformation field and the dislocation lines. We apply closedness results for graphs of torus-valued maps, seen as integral currents and, from the characterization of their graph boundaries, we are able to prove existence of minimizers
A compatible-incompatible decomposition of symmetric tensors in Lp with application to elasticity
No full textA compatible-incompatible decomposition of symmetric tensors in Lp with application to elasticity
No full textIn this paper, we prove the Saint-Venant compatibility conditions in L-p for p is an element of(1, +infinity), in a simply connected domain of any space dimension. As a consequence, alternative, simple, and direct proofs of some classical Korn inequalities in L-p are provided. We also use the Helmholtz decomposition in L-p to show that every symmetric tensor in a smooth domain can be decomposed in a compatible part, which is the symmetric part of a displacement gradient, and in an incompatible part, which is the incompatibility of a certain divergence-free tensor. Moreover, under a suitable Dirichlet boundary condition, this Beltrami-type decomposition is proved to be unique. This decomposition result has several applications, one of which being in dislocation models, where the incompatibility part is related to the dislocation density and where 1 < p < 2. This justifies the need to generalize and prove these rather classical results in the Hilbertian case (p = 2), to the full range p is an element of(1,+infinity)
The Kernel of the Strain Tensor for Solenoidal Vector Fields with Homogeneous Normal Trace
No full text
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
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