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We study the effective behavior of heterogeneous thin films with three competing length scales: the film thickness and the
length scales of heterogeneity and material microstructure. We start with three-dimensional nonhomogeneous nonlinear elasticity
enhanced with an interfacial energy of the van der Waals type, and derive the effective energy dens...
Contexts in source publication
Context 1
... a heterogeneous (possibly multilayer) thin film shown in Fig. 1. It occupies a reference domain h = {x ∈ R 3 : (x 1 , x 2 ) ∈ S, 0 < x 3 < ...
Context 2
... at each h > 0, the sequence shown schematically in Fig. 2(a) is clearly a minimizing sequence and it follows that as it contains too much interfacial energy. Now set 1)) converges strongly to f in W 1,4 0 ((0, 1)) as m → ∞. For each fixed m, extend χ (m) periodically to R and consider ...
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Citations
... In this manuscript, we adopt a variational approach to study thin multi-structures, such as a vertical rod upon a horizontal disk, in the context of non-simple grade-two materials, including martensitic materials known for their shape-memory and superelastic characteristics [6,37,39,46,49,60,62,63,64]. We also recall the contribution in [5] where the study of martensitic materials is performed adopting a second-order penalization term expressed in terms of the total variation. ...
... It is also worth mentioning that our analysis differs from previous one, such as [62], because we are targeting energies depending explicitly on the second-order gradient with the aim at emphasizing the presence of bending terms in the limit model. We refer to [27,28,52], where this effect is obtained considering the presence of second-order gradients as a perturbation of the energy. ...
We consider a thin multi-domain of R N , with N ≥ 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 2 U), where W is a continuous function with linear growth at ∞ and D 2 U 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 ≥ 3, and "partially" coupled if N = 2. MSC (2010):
... In this view, in 1999 K. Bhattacharya and R. D. James argued that if one considers a portion of an elastic cylinder between two cross sections at reciprocal finite distance, and compute the Γ-limit of the energy (and related minimizers) as the thickness goes to zero, one may obtain the energy pertinent to a membrane or to a Cosserat's surface, depending on whether the cylinder is made of a simple elastic material or a second-grade one, i.e., one including in the list of energy entries the second gradient of deformation. In the latter case, the limit generates a vector field over the surface on which we shrink the cylinder portion (see [49], [21], [22]). ...
... In the realm of solid-solid phase separation, the zeroth order -limit is a wellstudied problem. Francfort and Müller [19] have studied this problem in a similar framework, which was later extended by Shu [28] to many regimes including dimension reduction. The strategy for the proof of the zeroth order -limit G 0 we use here is, in most aspects, similar to previous work, though in the limsup inequality we employ an argument based on two-scale convergence and measurable selections. ...
A variational model for the interaction between homogenization and phase separation is considered. The focus is on the regime where the latter happens at a smaller scale than the former, and when the wells of the double well potential are allowed to move and to have discontinuities. The zeroth and first order Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document}-limits are identified. The topology considered for the latter is that of two-scale, since it encodes more information on the asymptotic local microstructure. In particular, when the wells are non-constant, the first order Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document}-limit describes the contribution of microscopic phase separation, even in situations where there is no macroscopic phase separation. As a corollary, the minimum of the mass constrained minimization problem is characterized, and it is shown to depend on whether or not the wells are discontinuous. In the process of proving these results, the theory of inhomogeneous Modica Mortola functionals is strengthened.
... In the heterogeneous and bounded case, the problem was solved by Braides, Fonseca and Francfort (see [11]) for γ = 1 and by Shu (see [16]) for γ ̸ = 1. In the present paper we deal with the heterogeneous and unbounded case. ...
... The proof of Theorem 1 is given in Section 4 by using two results: unbounded relaxation (see Corollary 8) and bounded homogenization and 3D-2D dimension reduction (see Theorem 11). These results, proved in [1] and [11,16] respectively, are recalled in Section 3.2. In Section 5 we give applications of Theorem 1 (see Corollary 13). ...
... and we also assume that To establish Theorem 1 we need the following result which was proved by Braides, Fonseca and Francfort (see [11]) for γ = 1 and by Shu (see [16]) for γ ̸ = 1. ...
... This kind of energy blending is studied by [35]. An alternative approach from [41] develops a membrane theory from a 3D model with regularization term ε|D 2 u| 2 and studies the limit ε → 0 as the thickness t → 0. Yet another physically motivated approach would be to consider a pure bending theory, where the nonconvexity lives inside the constraint. A recent example of a bending theory for LCEs is [8]. ...
We design a finite element method (FEM) for a membrane model of liquid crystal polymer networks (LCNs). This model consists of a minimization problem of a non-convex stretching energy. We discuss properties of this energy functional such as lack of rank-1 convexity. We devise a discretization with regularization, propose a novel iterative scheme to solve the non-convex discrete minimization problem, and prove stability of the scheme and convergence of discrete minimizers. We present numerical simulations to illustrate convergence properties of our algorithm and features of the model.
... Typically, the notion is that the constraints on grains in bulk polycrystalline ferroelectrics are so severe that stress-free states are unlikely and complex energy minimising domain patterns form [ 38 , 39 ]. However, in polycrystalline thin films and free-standing lamellae, it is thought that the out-of-plane constraints can be relaxed, giving the material greater freedom to adopt a stress-free configuration [40] , making these ideal candidates to study domain continuity. Firstly, a bi-grain sample, where the grains are of near identical orientation, was used to demonstrate the practical feasibility and provide a proof-of-concept for studying domain compatibility in freestanding thin films. ...
Polycrystalline ferroelectrics constitute the basis of many advanced technologies, including sensors and actuators. Their intricate domain patterns, and switching, drive the macroscopic electrical and mechanical properties of the material, where the domain switching behaviour is largely influenced by the grain-grain interaction of the domain walls. Domain wall continuity across grain boundaries is speculated to affect the domain wall – grain boundary interaction, although the true impact of this phenomenon on the ferroelectric properties, and the conditions under which continuity occurs, are not yet well understood. Whilst there are some theoretical reports, the link to experimental evidence is limited, greatly hindering the applicability and fundamental understanding of current polycrystalline based devices. In this work, we close this gap by studying several grain junctions in free-standing BaTiO3 thin films using microscopy techniques and rationalising the domain configurations with reference to martensite theory. A pleasing agreement of minimal strain and polarisation mismatch for a pair of domain variants were found in cases where domain wall continuity across grain boundaries was observed, confirming that domain continuity is related to the compatibility conditions at the grain boundary. Following this experimental validation, the mismatches for various combinations of Euler angles in bi-grain junctions were theoretically explored, offering valuable insights into specific cases where domain continuity can be expected. These results offer an advancement in the understanding of grain-grain-domain interactions and provides a template for the prediction and control of domain wall continuity in polycrystalline ferroelectrics, appealing to those working in polycrystal design and domain engineering.
... In the realm of solid-solid phase separation, the zeroth order Γ-limit is a well-studied problem. Francfort and Müller in [18] have studied this problem in a similar framework, which was later extended by Shu in [27] to many regimes including dimension reduction. The strategy for the proof of the zeroth order Γ-limit G 0 we use here is, in most aspects, similar to previous work, though in the limsup inequality we employ an argument based on twoscale convergence and measurable selections. ...
A variational model for the interaction between homogenization and phase separation is considered. The focus is on the regime where the latter happens at a smaller scale than the former, and when the wells of the double well potential are allowed to move and to have discontinuities. The zeroth and first order $\Gamma$-limits are identified. The topology considered for the latter is that of two-scale, since it encodes more information on the asymptotic local microstructure. In particular, when the wells are non constant, the first order $\Gamma$-limit describes the contribution of microscopic phase separation, also in situations where there is no macroscopic phase separation. As a corollary, the minimum of the mass constrained minimization problem is characterized, and it is shown to depend on whether or not the wells are discontinuous. In the process of proving these results, the theory of inhomogeneous Modica Mortola functionals is strengthened.
... The second part of the energy contains a quadratic term depending on the tensor of the second derivatives of the deformation. When the quadratic form is the square of the Euclidean norm, we recover the example of the interfacial energy term of Van der Waals type for martensitic materials (see [4,30,31,39,26,27]). ...
We consider a second order thin curved film whose behavior is governed by an energy made up of a first order nonlinear part depending on the gradient of the deformation augmented by a quadratic second order part depending on the tensor of second derivatives of the deformation. We carry out a 3D-2D analysis through an asymptotic expansion in powers of the thickness of the film as it tends to zero.
... This was first carried out in [6,18]. Similar problems in the framework of Γ-convergence have been studied in [32,8,39]. A quantitative analysis of this problem appeared in [27,26,29]. ...
In this work, we study Bloch wave homogenization of periodically heterogeneous media with fourth order singular perturbations. We recover different homogenization regimes depending on the relative strength of the singular perturbation and length scale of the periodic heterogeneity. The homogenized tensor is obtained in terms of the first Bloch eigenvalue. The higher Bloch modes do not contribute to the homogenization limit. The main difficulty is the presence of two parameters which requires us to obtain uniform bounds on the Bloch spectral data in various regimes of the parameter.
... The elliptic operator in (1.2) arises in the study of the formation of the so-called shear bands in elastic materials subject to severe loadings [8]. Variational functionals associated with the related nonlinear operators are also used to model the heterogeneous thin films of martensitic materials [19,7]. Homogenization of the elliptic system (1.1) was first studied by Bensoussan, Lions, and Papanicolaou in [5], where qualitative results were obtained for the case κ = ε. ...
We investigate quantitative estimates in periodic homogenization of second-order elliptic systems of elasticity with singular fourth-order perturbations. The convergence rates, which depend on the scale $\kappa$ that represents the strength of the singular perturbation and on the length scale $\epsilon$ of the heterogeneities, are established. We also obtain the large-scale Lipschitz estimate, down to the scale $\epsilon$ and independent of $\kappa$. This large-scale estimate, when combined with small-scale estimates, yields the classical Lipschitz estimate that is uniform in both $\epsilon$ and $\kappa$.
