arXiv subjects
Publications and source records attributed to José Matias.
We extend the theory of structured deformations to the setting of linearized elasticity by providing an integral representation for the underlying energy that features bulk and surface contributions. Our derivation is obtained both via a direct approach by means of a global method for relaxation in BD and via an approximation from nonlinear elastic energies associated to {nonsimple} materials.
The response of many materials to applied forces and boundary constraints depends upon internal geometric changes at multiple submacroscopic levels. Hierarchical structured deformations provide a mathematical setting for the description of such changes and for the variational determination of the corresponding energetic response. The research in this article provides substantial refinements and broadenings of the mathematical setting both for the underlying geometrical structure and for the variational analysis of energetic response. The mathematical tools employed in this research include the global method for relaxation and establish the equivalence of a relaxed energy obtained via relaxation under simultaneous geometrical changes at all levels and a relaxed energy obtained via iterated relaxations proceeding from the deepest submacroscopic level successively to the macroscopic level.
We consider a thin multi-domain of $\mathbb R^N$, with $N\geq 2$, consisting of a vertical rod upon a horizontal disk. In this thin multi-domain, we introduce a bulk energy density of the kind $W(D^2U)$, where $W$ is a continuous function with linear growth at $\infty$ and $D^2U$ denotes the Hessian tensor of a vector-valued function $U$ that represents a deformation of the multi-domain. Considering suitable boundary conditions on the admissible deformations and assuming that the two volumes tend to zero with same rate, we prove that the limit model is well posed in the union of the limit domains, with dimensions $1$ and $N-1$, respectively. Moreover, we show that the limit problem is uncoupled if $N\geq 3$, and ``partially" coupled if $N=2$.
We prove an integral representation result for a class of variational functionals appearing in the framework of hierarchical systems of structured deformations via a global method for relaxation. Some applications to specific relaxation problems are also provided.
Hierarchical (first-order) structured deformations are studied from the variational point of view. The main contributions of the present research are the first steps, at the theoretical level, to establish a variational framework to minimize mechanically relevant energies defined on hierarchical structured deformations. Two results are obtained here: (i) an approximation theorem and (ii) the assignment of an energy to a hierarchical structured deformation by means of an iterative procedure. This has the effect of validating the proposal made in [Deseri & Owen: Elasticity with hierarchical disarrangements: a field theory that admits slips and separations at multiple submacroscopic levels. J.~Elast., 135 (2019), 149--182] to study deformations admitting slips and separations at multiple submacroscopic levels. An explicit example is provided to illustrate the behavior of the proposed iterative procedure and relevant directions for future research are highlighted.
An energy for first-order structured deformations in the context of periodic homogenization is obtained. This energy, defined in principle by relaxation of an initial energy of integral type featuring contributions of bulk and interfacial terms, is proved to possess an integral representation in terms of relaxed bulk and interfacial energy densities. These energy densities, in turn, are obtained via asymptotic cell formulae defined by suitably averaging, over larger and larger cubes, the bulk and surface contributions of the initial energy. The integral representation theorem, the main result of this paper, is obtained by mixing blow-up techniques, typical in the context of structured deformations, with the averaging process proper of the theory of homogenization.
We obtain a measure representation for a functional arising in the context of optimal design problems under linear growth conditions. The functional in question corresponds to the relaxation with respect to a pair $(χ,u)$, where $χ$ is the characteristic function of a set of finite perimeter and $u$ is a function of bounded deformation, of an energy with a bulk term depending on the symmetrised gradient as well as a perimeter term.
We describe multiscale geometrical changes via structured deformations $(g,G)$ and the non-local energetic response at a point $x$ via a function $Ψ$ of the weighted averages of the jumps $[u_{n}](y)$ of microlevel deformations $u_{n}$ at points $y$ within a distance $r$ of $x$. The deformations $u_{n}$ are chosen so that $\lim_{n\to \infty }u_{n}=g$ and $\lim_{n\to \infty }\nabla u_{n}=$ $G$. We provide conditions on $Ψ$ under which the upscaling "$n\to \infty$" results in a macroscale energy that depends through $Ψ$ on (1) the jumps $[g]$ of $g$ and the "disarrangment field" $\nabla g-G$, (2) the "horizon" $r$, and (3) the weighting function $α_{r}$ for microlevel averaging of $[u_{n}](y)$. We also study the upscaling "$n\to \infty$" followed by spatial localization "$r\to 0$" and show that this succession of processes results in a purely local macroscale energy $I(g,G)$ that depends through $Ψ$ upon the jumps $[g]$ of $g$ and the "disarrangment field" $\nabla g-G$, alone. In special settings, such macroscale energies $I(g,G)$ have been shown to support the phenomena of yielding and hysteresis, and our results provide a broader setting for studying such yielding and hysteresis. As an illustration, we apply our results in the context of the plasticity of single crystals.
In this paper we apply both the procedure of dimension reduction and the incorporation of structured deformations to a three-dimensional continuum in the form of a thinning domain. We apply the two processes one after the other, exchanging the order, and so obtain for each order both a relaxed bulk and a relaxed interfacial energy. Our implementation requires some substantial modifications of the two relaxation procedures. For the specific choice of an initial energy including only the surface term, we compute the energy densities explicitly and show that they are the same, independent of the order of the relaxation processes. Moreover, we compare our explicit results with those obtained when the limiting process of dimension reduction and of passage to the structured deformation is carried out at the same time. We finally show that, in a portion of the common domain of the relaxed energy densities, the simultaneous procedure gives an energy strictly lower than that obtained in the two-step relaxations.
Structured deformations provide a multiscale geometry that captures the contributions at the macrolevel of both smooth geometrical changes and non-smooth geometrical changes (disarrangements) at submacroscopic levels. For each (first-order) structured deformation $(g,G)$ of a continuous body, the tensor field $G$ is known to be a measure of deformations without disarrangements, and $M:=\nabla g-G$ is known to be a measure of deformations due to disarrangements. The tensor fields $G$ and $M$ together deliver not only standard notions of plastic deformation, but $M$ and its curl deliver the Burgers vector field associated with closed curves in the body and the dislocation density field used in describing geometrical changes in bodies with defects. Recently, Owen and Paroni [13] evaluated explicitly some relaxed energy densities arising in Choksi and Fonseca's energetics of structured deformations [4] and thereby showed: (1) $(trM)^{+}$, the positive part of $trM$, is a volume density of disarrangements due to submacroscopic separations, (2) $(trM)^{-}$, the negative part of $trM$, is a volume density of disarrangements due to submacroscopic switches and interpenetrations, and (3) $|trM|$, the absolute value of $trM$, is a volume density of all three of these non-tangential disarrangements: separations, switches, and interpenetrations. The main contribution of the present research is to show that a different approach to the energetics of structured deformations, that due to Baía, Matias, and Santos [1], confirms the roles of $(trM)^{+}$, $(trM)^{-}$, and $|trM|$ established by Owen and Paroni. In doing so, we give an alternative, shorter proof of Owen and Paroni's results, and we establish additional explicit formulas for other measures of disarrangements.
In this work we consider an optimal design problem for two-component fractured media for which a macroscopic strain is prescribed. Within the framework of structured deformations, we derive an integral representation for the relaxed energy functional. We start from an energy functional accounting for bulk and surface contributions coming from both constituents of the material; the relaxed energy densities, obtained via a blow-up method, are determined by a delicate interplay between the optimization of sharp interfaces and the diffusion of microcracks. This model has the far-reaching perspective to incorporate elements of plasticity in optimal design of composite media.
Second-order structured deformations of continua provide an extension of the multiscale geometry of first-order structured deformations by taking into account the effects of submacroscopic bending and curving. We derive here an integral representation for a relaxed energy functional in the setting of second-order structured deformations. Our derivation covers inhomogeneous initial energy densities (i.e., with explicit dependence on the position); finally, we provide explicit formulas for bulk relaxed energies as well as anticipated applications.
In this note we present a brief overview of variational methods to solve homogenization problems. The purpose is to give a first insight on the subject by presenting some fundamental theoretical tools, both classical and modern. We conclude by mentioning some open problems.
This work deals with the homogenization of functionals with linear growth in the context of $\mathcal{A}$-quasiconvexity. A representation theorem is proved, where the new integrand function is obtained by solving a cell problem where the coupling between homogenization and the $\mathcal{A}$-free condition plays a crucial role. This result extends some previous work to the linear case, thus allowing for concentration effects.