Variational estimates for the effective response and field statistics in thermoelastic composites with intra-phase property fluctuations

International audienceIn this work, variational estimates are provided for the macroscopic response, as well as for the first and second moments of the stress and strain fields, in thermoelastic composites with non-uniform distributions of the thermal stress and elastic moduli in the constituent phases. These estimates are obtained in terms of a 'comparison composite' with uniform phase properties depending on the first and second moments of a certain combination of the given intra-phase thermal stresses and modulus field distributions. Under certain hypotheses, these estimates can be shown to lead to upper and lower bounds for the free energy of the composite, which reduce to standard results when the intra-phase fluctuations vanish. An illustrative application is given for rigidly reinforced composites with a non-uniform distribution of the thermal stress in the matrix phase

Discretization of variational regularization in Banach spaces

Consider a nonlinear ill-posed operator equation $F(u)=y$ where $F$ is defined on a Banach space $X$. In general, for solving this equation numerically, a finite dimensional approximation of $X$ and an approximation of $F$ are required. Moreover, in general the given data \yd of $y$ are noisy. In this paper we analyze finite dimensional variational regularization, which takes into account operator approximations and noisy data: We show (semi-)convergence of the regularized solution of the finite dimensional problems and establish convergence rates in terms of Bregman distances under appropriate sourcewise representation of a solution of the equation. The more involved case of regularization in nonseparable Banach spaces is discussed in detail. In particular we consider the space of finite total variation functions, the space of functions of finite bounded deformation, and the $L^\infty$--space

A Computational Method Based on Augmented Lagrangians and Fast Fourier Transforms for Composites with High Contrast

An iterative numerical method based on Fast Fourier Transforms has been proposed by \cite{MOU98} to investigate the effective properties of periodic composites. This iterative method is based on the exact expression of the Green function for a linear elastic, homogeneous reference material. When dealing with linear phases, the number of iterations required to reach convergence is proportional to the contrast between the phases properties, and convergence is therefore not ensured in the case of composites with infinite contrast (those containing voids or rigid inclusions or highly nonlinear materials). It is proposed in this study to overcome this difficulty by using an augmented Lagrangian method. The resulting saddle--point problem involves three steps. The first step consists of solving a linear elastic problem, using the Fourier Transform method. In the second step, a nonlinear problem is solved at each individual point in the volume element. The third step consists of updating the Lagrange multiplier. This method was applied successfully to composites with high or infinite contrast. The first case presented here is that of a linear elastic material containing voids. The second example is that of a two-phase composite with power-law constituents. The third example involves voided rigid-plastic materials

Effect of a nonuniform distribution of voids on the plastic response of voided materials: a computational and statistical analysis

This study investigates the overall and local response of porous media composed of a perfectly plastic matrix weakened by stress-free voids. Attention is focused on the specific role played by porosity fluctuations inside a representative volume element. To this end, numerical simulations using the Fast Fourier Transform (FFT) are performed on different classes of microstructure corresponding to different spatial distributions of voids. Three types of microstructures are investigated: random microstructures with no void clustering, microstructures with a connected cluster of voids and microstructures with disconnected void clusters. These numerical simulations show that the porosity fluctuations can have a strong effect on the overall yield surface of porous materials. Random microstructures without clusters and microstructures with a connected cluster are the hardest and the softest configurations, respectively, whereas microstructures with disconnected clusters lead to intermediate responses. At a more local scale, the salient feature of the fields is the tendency for the strain fields to concentrate in specific bands. Finally, an image analysis tool is proposed for the statistical characterization of the porosity distribution. It relies on the distribution of the ‘distance function’, the width of which increases when clusters are present. An additional connectedness analysis allows us to discriminate between clustered microstructures

Ecological impacts of non-native Pacific oysters (Crassostrea gigas) and management measures for protected areas in Europe

Pacific oysters are now one of the most ‘globalised’ marine invertebrates. They dominate bivalve aquaculture production in many regions and wild populations are increasingly becoming established, with potential to displace native species and modify habitats and ecosystems. While some fishing communities may benefit from wild populations, there is now a tension between the continued production of Pacific oysters and risk to biodiversity, which is of particular concern within protected sites. The issue of the Pacific oyster therefore locates at the intersection between two policy areas: one concerning the conservation of protected habitats, the other relating to livelihoods and the socio-economics of coastal aquaculture and fishing communities. To help provide an informed basis for management decisions, we first summarise evidence for ecological impacts of wild Pacific oysters in representative coastal habitats. At local scales, it is clear that establishment of Pacific oysters can significantly alter diversity, community structure and ecosystem processes, with effects varying among habitats and locations and with the density of oysters. Less evidence is available to evaluate regional-scale impacts. A range of management measures have been applied to mitigate negative impacts of wild Pacific oysters and we develop recommendations which are consistent with the scientific evidence and believe compatible with multiple interests. We conclude that all stakeholders must engage in regional decision making to help minimise negative environmental impacts, and promote sustainable industry development

A self-consistent estimate for linear viscoelastic polycrystals with internal variables inferred from the collocation method

The correspondence principle is customarily used with the Laplace–Carson transform technique to tackle the homogenization of linear viscoelastic heterogeneous media. The main drawback of this method lies in the fact that the whole stress and strain histories have to be considered to compute the mechanical response of the material during a given macroscopic loading. Following a remark of Mandel (1966 Mécanique des Milieux Continus(Paris, France: Gauthier-Villars)), Ricaud and Masson (2009 Int. J. Solids Struct. 46 1599–1606) have shown the equivalence between the collocation method used to invert Laplace–Carson transforms and an internal variables formulation. In this paper, this new method is developed for the case of polycrystalline materials with general anisotropic properties for local and macroscopic behavior. Applications are provided for the case of constitutive relations accounting for glide of dislocations on particular slip systems. It is shown that the method yields accurate results that perfectly match the standard collocation method and reference full-field results obtained with a FFT numerical scheme. The formulation is then extended to the case of time- and strain-dependent viscous properties, leading to the incremental collocation method (ICM) that can be solved efficiently by a step-by-step procedure. Specifically, the introduction of isotropic and kinematic hardening at the slip system scale is considered

Numéro thématique des Comptes Rendus Mécanique en lʼhonneur dʼAndré Zaoui

La Mécanique des Matériaux a connu, en France et dans le monde, un développement spectaculaire au cours des dernières décennies, rendu à la fois nécessaire par les besoins d’innovation et de sûreté de secteurs industriels comme l’énergie et les transports, et possible par les avancées contemporaines en Physique et en Mécanique des Milieux Continus. Tout matériau est, par nature, hétérogène à une et souvent plusieurs échelles. La prise en compte, à une échelle pertinente, de cette hétérogénéité gouvernant les interactions entre mécanismes élémentaires est bien souvent la clef de la compréhension et de la prédiction du comportement mécanique des matériaux à leur échelle macroscopique d’usage. La Micromécanique des Matériaux, à laquelle ce numéro thématique des Comptes Rendus Mécanique est consacré, a précisément pour objet d’aborder ces problèmes de transition d’échelles. Ce numéro thématique est tout naturellement l’occasion d’honorer l’un des acteurs emblématiques du domaine, André Zaoui, qui a contribué de façon essentielle à l’établissement de la démarche micro–macro sur des bases théoriques rigoureuses validées par une approche expérimentale ambitieuse. Par ses travaux personnels, par la création, en avance sur son temps, d’une équipe de recherche dédiée aux expériences micromécaniques, par ses enseignements et ses actions de structuration de la recherche, André Zaoui a initié, puis constamment encouragé,ce domaine en France, l’ancrant solidement dans un dialogue fructueux entre expériences à petite échelle et modélisation

Experimental characterization of the intragranular strain field in columnar ice during transient creep

A digital image correlation (DIC) technique has been adapted to polycrystalline ice specimens in order to characterize the development of strain heterogeneities at an intragranular scale during transient creep deformation (compression tests). Specimens exhibit a columnar microstructure so that plastic deformation is essentially two-dimensional, with few in-depth gradients, and therefore surface DIC analyses are representative of the whole specimen volume. Local misorientations at the intragranular scale were also extracted from microstructure analyses carried out with an automatic texture analyzer before and after deformation. Highly localized strain patterns are evidenced by the DIC technique. Local equivalent strain can reach values as much as an order of magnitude larger than the macroscopic average. The structure of the strain pattern does not evolve with strain in the transient creep regime. Almost no correlation between the measured local strain and the Schmid factor of the slip plane of the underlying grain is observed, highlighting the importance of the mechanical interactions between neighboring grains resulting from the very large viscoplastic anisotropy of ice crystals. Finally, the experimental microstructure was introduced in a full-field fast Fourier transform polycrystal model; simulated strain fields are a good match with experimental ones

Characterization of Generalized Young Measures Generated by Symmetric Gradients

This work establishes a characterization theorem for (generalized) Young measures generated by symmetric derivatives of functions of bounded deformation (BD) in the spirit of the classical Kinderlehrer\ue2\u80\u93Pedregal theorem. Our result places such Young measures in duality with symmetric-quasiconvex functions with linear growth. The \ue2\u80\u9clocal\ue2\u80\u9d proof strategy combines blow-up arguments with the singular structure theorem in BD (the analogue of Alberti\ue2\u80\u99s rank-one theorem in BV), which was recently proved by the authors. As an application of our characterization theorem we show how an atomic part in a BD-Young measure can be split off in generating sequences