1,720,991 research outputs found
A variational approach to statics and dynamics of elasto-plastic systems
Full text linkWe prove some existence results for dynamic evolutions in elasto-plasticity and delamination. We study the limit as the data vary very slowly and prove convergence results to quasistatic evolutions.
We model dislocations by mean of currents, we introduce the space of deformations in the presence of dislocations and study the graphs of these maps. We prove existence results for minimum problems. We study the properties of minimizers
On the singular planar Plateau problem
No full textGiven any = γ(S1) ⊂ R2, image of a Lipschitz curve γ : S1 → R2, not necessarily injective, we provide an explicit formula for computing the value of A(γ) := inf | det(∇u)|dx u = γ on S1 , B1(0) where the infimumiscomputedamongall Lipschitz maps u : B1(0) → R2 havingboundary datum γ. This coincides with the area of a minimal disk spanning , i.e., a solution of the Plateau problem of disk type. The novelty of the results relies in the fact that we do not assume the curve γ to be injective and our formula allows arbitrary self-intersections
Existence, Energy Identity, and Higher Time Regularity of Solutions to a Dynamic Viscoelastic Cohesive Interface Model
No full textWe study the dynamics of viscoelastic materials coupled by a common cohesive interface (or, equivalently, two single domains separated by a prescribed cohesive crack) in the antiplane setting. We consider a general class of traction-separation laws featuring an activation threshold on the normal stress, softening, and elastic unloading. In the strong form, the evolution is described by a system of PDEs coupling momentum balance (in the bulk) with transmission and Karush-Kuhn--Tucker conditions (on the interface). We provide a detailed analysis of the system. We first prove the existence of a weak solution, employing a time discrete approach and a regularization of the initial data. Then, we prove our main results: the energy identity and the existence of solutions with acceleration in L-infinity (0, T; L-2). Thanks to the latter we finally prove the existence of strong solutions satisfying the balance of forces in the bulk and on the interface
On the L^1-relaxed area of graphs of BV piecewise constant maps taking three values
No full textGiven a bounded open connected set Omega subset of R-2 with Lipschitz boundary, we consider the class of piecewise constant maps u taking three fixed values alpha , beta , gamma is an element of R-2, vertices of an equilateral triangle; for any u in this class, using a weak notion of Jacobian determinant valid for BV functions, we give a precise description of Det (del u) and show that the relaxed graph area of u is bounded from above by a quantity related to the flat norm of Det (del u) . The provided upper bound allows to show the validity of a De Giorgi conjecture regarding the relaxed area functional when one restricts to this class of piecewise constant functions
A contact problem for viscoelastic bodies with inertial effects and unilateral boundary constraints
Full text linkWe consider a viscoelastic body occupying a smooth bounded domain under the effect of a volumic traction force. Inertial effects are considered; hence the equation for the macroscopic displacement contains a second order term. On a part of the boundary, the body is anchored to a support and no displacement may occur; on a second part, the body can move freely, and on a third part the body is in adhesive contact with a solid support. The boundary forces coming to the action of elastic stresses are responsible for delamination, i.e., progressive failure of adhesive bonds. Following the lines of a new approach based on duality methods in Sobolev-Bochner spaces, we define a suitable concept of weak solution to the resulting PDE system and correspondingly we prove an existence result on finite time intervals of arbitrary length
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
On the viscous Cahn-Hilliard equation with singular potential and inertial term
No full textWe consider a relaxation of the viscous Cahn-Hilliard equation induced by the second-order inertial term u_{tt}. The equation also contains a semilinear term f(u) of "singular" type. Namely, the function f is defined only on a bounded interval of R corresponding to the physically admissible values of the unknown u, and diverges as u approaches the extrema of that interval. In view of its interaction with the inertial term u_{tt}, the term f(u) is difficult to be treated mathematically. Based on an approach originally devised for the strongly damped wave equation, we propose a suitable concept of weak solution based on duality methods and prove an existence result
Regularity and convergence of critical points of an Ambrosio-Tortorelli functional with linear growth and of its Γ-limit
Full text linkIn the one-dimensional setting we consider an Ambrosio-Tortorelli functional Fε(u,v) which has linear growth with respect to u′. We prove that under suitable conditions on the fidelity term, minimizers and critical points of Fε are Sobolev regular, and that the same is true for the Γ-limit F of Fε. As a corollary, we obtain that the functional Aw(u) computing the length of the generalized graph of a function of bounded variation u, under the same conditions on the fidelity term, admits a unique minimizer of class C1. This partially solves a conjecture by De Giorgi [16] in the one-dimensional case
On the relaxed area of the graph of discontinuous maps from the plane to the plane taking three values with no symmetry assumptions
No full textIn this paper, we estimate from above the area of the graph of a singular map u taking a disk to three vectors, the vertices of a triangle, and jumping along three C2 -embedded curves that meet transversely at only one point of the disk. We show that the singular part of the relaxed area can be estimated from above by the solution of a Plateau-type problem involving three entangled nonparametric area-minimizing surfaces. The idea is to “fill the hole” in the graph of the singular map with a sequence of approximating smooth two-codimensional surfaces of graph-type, by imagining three minimal surfaces, placed vertically over the jump of u, coupled together via a triple point in the target triangle. Such a construction depends on the choice of a target triple point, and on a connection passing through it, which dictate the boundary condition for the three minimal surfaces. We show that the singular part of the relaxed area of u cannot be larger than what we obtain by minimizing over all possible target triple points and all corresponding connections
A new approach to topological singularities via a weak notion of Jacobian for functions of bounded variation
No full textWe introduce a weak notion of 2 × 2-minors of gradients for a suitable subclass of BV functions. In the case of maps in BV(R2; R2) such a notion extends the standard definition of Jacobian determinant to non-Sobolev maps. We use this distributional Jacobian to prove a compactness and Γ -convergence result for a new model describing the emergence of topological singularities in two dimensions, in the spirit of Ginzburg-Landau and core-radius approaches. Within our framework, the order parameter is an SBV map u taking values in the unit sphere in R2 and the energy is given by the sum of the squared L2 norm of the approximate gradient ∇u and of the length of (the closure of) the jump set of u multiplied by 1/ε. Here, ε is a length-scale parameter. We show that, in the | log ε| regime, the distributional Jacobians converge, as ε → 0+, to a finite sum μ of Dirac deltas with weights multiple of π, and that the corresponding effective energy is given by the total variation of μ
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