1,721,154 research outputs found
Optimal estimates for the triple junction function and other surprising aspects of the area functional
No full textWe consider the relaxed area functional for vector valued maps and its exact value on the triple junction function u : B1(O) → R 2 , a specific function which represents the first example of map whose graph area shows nonlocal effects. This is a map taking only three different values α, β, γ ∈ R 2 in three equal circular sectors of the unit radius ball B1(O). We prove a conjecture due to G. Bellettini and M. Paolini asserting that the recovery sequence provided in [5] (and the corresponding upper bound for the relaxed area functional of the map u) is optimal. At the same time, we show by means of a counterexample that such construction is not optimal if we consider different domains than B1(O), which still contain the same discontinuity set of u in B1(O). Such domains are obtained from B1(O) erasing part of interior of the sectors where u is constant
Limit of dynamic processes in delamination as the viscosity and inertia vanish
Full text linkWe introduce a model of dynamic evolution of a delaminated visco-elastic body with viscous adhesive. We prove the existence of solutions of the corresponding system of PDEs and then study the behavior of such solutions when the data of the problem vary slowly. We prove that a rescaled version of the dynamic evolutions converge to a "local" quasistatic evolution, which is an evolution satisfying an energy inequality and a momentum balance at all times. In the one-dimensional case we give a more detailed description of the limit evolution and we show that it behaves in a very similar way to the limit of the solutions of the dynamic model in [T. Roubicek, SIAM J. Math. Anal. 45 (2013) 101-126], where no viscosity in the adhesive is taken into account
A weak formulation for a rate-independent delamination evolution with inertial and viscosity effects subjected to unilateral constraint
No full textWe consider a system of two viscoelastic bodies attached on one side by an adhesive where a delamination process occurs. We study the dynamic of the system for small strains, subjected to external forces, suitable boundary conditions, and an unilateral constraint on the jump of the displacement at the interface between the bodies. The constraint arises in a graph inclusion, while the delamination coefficient evolves in a rate-independent way. We prove the existence of a weak solution to the corresponding system of PDEs
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
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
Linearization for finite plasticity under dislocation-density tensor regularization
No full textFinite-plasticity theories often feature nonlocal energetic contributions in the plastic variables. By introducing a length-scale for plastic effects in the picture, these nonlocal terms open the way to existence results (Mainik and Mielke in J Nonlinear Sci 19(3):221–248, 2009). We focus here on a reference example in this direction, where a specific energetic contribution in terms of dislocation-density tensor is considered (Mielke and Müller in ZAMM Z Angew Math Mech 86:233–250, 2006). When external forces are small and dissipative terms are suitably rescaled, the finite-strain elastoplastic problem converges toward its linearized counterpart. We prove a Γ -convergence result making this asymptotics rigorous, both at the incremental level and at the level of quasistatic evolution
Dynamics of a viscoelastic membrane with gradient constraint
Full text linkTaking into account inertial and viscosity effects, we consider the dynamics of a two dimensional membrane subjected to an unilateral constraint on its deformation gradient. Specifically, due to the constitutive law, we assume that higher deformations lock the material, leading to the inequality |∇u|≤g, where u denotes the displacement of the membrane and g is a certain positive threshold. We then introduce the concepts of weak and generalised solutions to the associated wave equation, and prove the existence of them for rather general data and homogeneous Dirichlet boundary conditions. The presence of the gradient constraint provides the existence of a Lagrange multiplier λ related to the existence of a reaction term Υ, which corresponds to a strongly nonlinear term in the wave equation. We then extend the existence result to a weak form of the Neumann type boundary condition [Formula presented], for any α≥0, and we show that these solutions tend, as α→∞, in a certain sense to a solution of the homogeneous Dirichlet constrained problem
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
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
A quasi-static evolution generated by local energy minimizers for an elastic material with a cohesive interface
No full textWe consider a model for an elastic material with a cohesive crack along a prescribed fracture set. In the framework of in-plane elasticity we consider a cohesive law with incompenetrability constraint and general loading-unloading regimes. We provide first a time-discrete evolution by means of local minimizers of the energy with respect to the -norm of the crack opening displacement. The choice of this norm is due to technical reasons (the -convexity of the energy) and is in analogy with the classical approach in quasi-static brittle fracture, where the evolution of the system is condensed into the evolution of the crack. In the ``time-continuous" limit we obtain a -evolution, in parametrized form, characterized by Karush-Kuhn-Tuker conditions for the internal variable, equilibrium and energy identity
- …
