35 **scale** models, because of the introduction of some randomness in the **micro**–structures, each macroscopic material point is associated with a local RVE which differs from the other RVEs attached to the other macroscopic positions. In order to take into account this randomness, a library of RVE meshes with a certain number of examples is gen- erated. Each microscopic BVP located at a macroscopic position takes a random RVE mesh from the mesh library. The obtained results show that the proposed DG–based FE 2 scheme can capture the macroscopic localization band for small values of control parameter δ. The size of the RVEs within this scheme is limited by the fact that it may not be larger than the macroscopic length **scale** characterizing the quadratic variation in the displacements at the macroscopic **scale** as mentioned by Kouznetsova et al. (2004a). Using a small RVE size is acceptable for cellular materials as a single cell remains repre- sentative. For random **micro**–structures, choosing a small RVE size potentially leads to large variations of the homogenized properties as shown by Kanit et al. (2003). A com- promise should thus be made. The results obtained with different levels of imperfection ranging from a quasi–perfect **micro**–structure, for which a single cell is representative, to a 30% imperfect **micro**–structure are analyzed. A larger value of the perturbation δ leads to a softer result on the macroscopic force–displacement behavior, which is not necessarily physical because the use of the unit cells with high randomness as the RVEs is no longer valid. Thus this approach cannot be used for structures with high degrees of imperfection, but is shown to predict with accuracy the behavior of structure made of regular **micro**–patterns for which a single cell can be considered as a RVE. In this case, a possible further research towards continuous–discontinuous computational ho- mogenization schemes following the works of Massart et al. (2007), Nguyen et al. (2011) **and** Coenen et al. (2012) should be conducted.

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In this article, we partition the **scale** in mainly two bins, according to Figure 2 . The microscopic **scale** where all orthogonal projections of objects in the scene are bounded by the digital detector size. This allows for acquiring data with traditional holographic setup in orthographic geometries. The macroscopic **scale** encloses geometries where multiple detectors **and** view points or scanning acquisition protocols are needed for data acquisition. Such large **macro** **scale** poses additional challenges in terms of large **scale** complex signal compression **and** transmission. At the microscopic **scale**, the phase of the recorded wavefield is proportional to a depth map of the scene **and** is therefore modeled as a heightfield in the absence of significant **effects** from occlusions **and** speckle noise. At the macroscopic **scale** however, holography presents additional challenges. Different parts of the object are exposed with a large viewing angle, occlusions introduce non-linearities that are essential to handle **and** speckle noise impairs visualization. 1 Nevertheless, the advances in the field at the **micro** **scale** may be translated to the macroscopic imaging scenario, provided that enough computation **and** storage resources are available.

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ABSTRACT. In this paper, a multiscale strategy for the analysis of crack propagation is presented. The purposes of this strategy are, first, to separate the local **effects** from the global **effects** in order to keep a macromesh unchanged during the crack’s propagation **and**, second, to enable one to use a proper fine-**scale** description only where it is required. Two aspects are discussed: the first is the choice of the macroscale in order to include the macroeffect of a crack; the second is the use of a decomposition of the domain into substructures **and** interfaces in order to limit the use of the refined **scale** only around the crack. The integration of the X-FEM as a local enrichment method for the description of a crack is also presented.

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◮ R eq = p8k/φ, comes from analogy with a single pipe. Is it still
valid in complex multi-**scale** porous media?
◮ α: fitting parameter coming from core-flood experiments. Many
hidden physical phenomena (adsorption, time-dependant **effects**, mass transport)...

A number of methods have been developed in the context of SPH. A macroscopic model for the surface tension may be used by introducing a color function [16, 82, 1]. Particles in different phases are assigned a different color **and** the interfaces are located where the gradient of the color function is significantly different from 0. Alternatively, interfaces may be detected at a given level set of the color function seen as an indicator function for the phases. This allows to determine the surface curvature **and** subsequently the expression of a surface force modeling the **effects** of the surface tension at the continuum level. Another approach, which was also extended to SDPD [107], is to introduce additional microscopic attractive potentials that affect all particles [165]. Unlike the previous macroscopic model, it lacks a proper theoretical foundation but is very straightforward to implement. Since surface tension arises at the microscopic **scale** because of inter-particle attractions, this method is able to create surface tension in the SDPD system. The parameters of the additional potential function may be fitted to obtain the desired value. However the addition of extra forces modifies the pressure **and** we must make sure that their influence is negligible.

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Laboratoire de Mecanique des So/ides, Ecole Polytechnique, Palaiseau, France ABSTRACT
This chapter is mainly concerned with the determination of general relationships bet ween microscopic **and** macroscopic mechanical properties for elastoplastic material with or without damage. The overall properties are determined in terms of the unkno wn properties of each constituent phase of the heterogeneous body. At first we must define a representative volume element (RVE) of the heterogeneous material for which the macroscopic mechanical fields are some spatial average of the microscopic one. The determination of local quantities is achieved by solving some particular boundary value problem on the R VE,Jrom which macroscopic quantities are derived. The essential structure of **micro**-**macro** relationships is presented in the case of elastoplastic material. A generalisation is given to take the temperature field into account. We present also a model of damage **and** its averaging process. Finally we discuss the determination of macroscopic temperature through continuum thermodynamics.

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Au-delà de l’intérêt de trouver un découpage optimal qui minimise la variance intra et maximise la variance inter, le jeu des échelles d’analyse est fort utile pour démontrer l’existence de zones urbaines de concentration de la pauvreté de taille bien différente allant de la **macro** à la **micro** zone. Or, une telle connaissance permet d’intégrer l’espace de façon précise lors de l’évaluation d’effets de milieu que ce soit dans les domaines de la santé, du développement des enfants, de la criminalité ou de l’éducation. Ainsi, si effet de milieu il y a, on peut formuler l’hypothèse qui reste à valider, qu’ils sont plus marqués dans les **macro** zones de concentration de la pauvreté que dans les **micro** zones. Par conséquent, pour le vérifier à partir de modèles multiniveaux, il est alors possible d’intégrer plusieurs variables contextuelles au niveau 2 : l’appartenance à une **macro**, puis méso et **micro** zone de concentration de la pauvreté.

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[*] Un pasaje tachado en un texto como este, por ejemplo, que figura en la página 13 de la “note Whitney”, nos enseña que Saussure, en 1984, no había establecido todavía la distinción q[r]

To cite this version : Pasquier, Sylvain **and** Quintard, Michel **and** Davit,
Yohan **Macro**-**scale** modeling of two-phase Flows within structured
packings. (2016) In: Séminaire Fermat franco-allemand à Toulouse : « Gas-
Liquid Flows », 6 June 2016 - 8 June 2016 (Toulouse, France). (Unpublished)

a b s t r a c t
Seismic shaking is an attractive mechanism to explain the destabilisation of regolith slopes **and** the rego- lith migration found on the surfaces of asteroids (Richardson, J.E., Melosh, H.J., Greenberg, R. [2004]. Science 306, 1526–1529. http://dx.doi.org/10.1126/science.1104731; Miyamoto, H., et al., 2007). Here, we use a continuum mechanics method to simulate the seismic wave propagation in an asteroid. Assuming that asteroids can be described by a cohesive core surrounded by a thin non-cohesive regolith layer, our numerical simulations of vibrations induced by **micro**-meteoroids suggest that the surface peak ground accelerations induced by **micro**-meteoroid impacts may have been previously under-estimated. Our lower bound estimate of vertical accelerations induced by seismic waves is about 50 times larger than previous estimates. It suggests that impact events triggering seismic activity are more frequent than previously assumed for asteroids in the kilometric **and** sub-kilometric size range. The regolith lofting is also estimated by a ﬁrst order ballistic approximation. Vertical displacements are small, but lofting times are long compared to the duration of the seismic signals. The regolith movement has a non-linear depen- dence on the distance to the impact source which is induced by the type of seismic wave generating the ﬁrst movement. The implications of regolith concentration in lows of surface acceleration potential are also discussed. We suggest that the resulting surface thermal inertia variations of small fast rotators may induce an increased sensitivity of these objects to the Yarkovsky effect.

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elastic tensile strains at both the axial **and** lateral directions. This is the case for the lateral strain as shown in Fig. 3 . However, one gets a compressive axial strain. This is not verified by the isotropic poroelastic theory. There are several possible reasons to explain such a compressive strain induced by the water injection. Based on some pre- vious studies [ 27 ], the water injection can enhance the pressure solution process in limestone **and** induce a weakening of both elastic modulus **and** mechanical strength of material. Therefore, if such a weakening effect exists, the water injection can induce a decrease in plastic yield stress of limestone **and** then generate additional plastic strains. Due to the applied deviatoric stress, the additional axial plastic strain should be compressive. However, additional investigations are needed to confirm the weakening effect of water injection in limestone. With the increase in pore pressure due to water injection (mov- ing to the left on the diagram), the plastic shearing surface is reached as shown in Fig. 3 . When the peak strength surface is finally reached, there is a material softening **and** a diminution of pore pressure. Moreover, the diminution of pores pressure is directly related to an important volumetric dilatancy. For higher levels of deviatoric stress (42, 47.5 **and** 53 MPa), the injection point is behind the initial pore collapse surface. This means that the material is in the plastic domain when the water injection is started. How- ever, during the water injection phase, due to the increase in pore pressure before peak strength, there is an elastic unloading with respect to the pore collapse surface. How- ever, when the plastic shearing surface is reached, the plastic deformation occurs but only due to the shearing process. Therefore, the plastic deformation during the water injection phase is dominated by the plastic shearing process.

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Il y a donc un vrai besoin de modèle **micro**-**macro** multiaxial prenant en compte le fort couplage ther- momécanique dont nous présentons ici les premiers résultats.
Dans une première partie nous présentons les points clés du comportement que nous tenons à mod- éliser correctement. La deuxième partie reprend le principe la modélisation multiaxiale à l’échelle du grain expliqué précédemment dans [6]. Enfin, une dernière partie montre le passage au polycristal et la simulation d’un essai de traction isotherme. Les pistes envisagées pour passer du VER à la structure sont ensuite évoquées.

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1 Introduction
Models arising from kinetic theory are nowadays a useful technique for the derivation of complex constitutive equations for the numerical simulation of visco-elastic flows. In fact, the majority of constitutive equations used in continuum numerical simulations of such fluids can be derived from kinetic theory [28]. However, the derivation of a constitutive equation from kinetic theory models often involves closure approximations, that can have signifi- cant impact on the final predictions of the model thus constructed [11][27][33]. **Micro**-**macro** techniques aimed at resolving this problem. Although much more computationally expensive, these techniques, that couple coarse-grained molecular **scale** of kinetic theory to the **macro** **scale** of the continuum, have been used increasingly in the last years [29] [20]. In it, advantage is obtained by exploiting the equivalence of the Fokker-Planck equation, that describes the evolution of the probability distribution function governing the config- uration state at the **micro** level, **and** an Itˆo stochastic differential equation. Instead of solving the deterministic Fokker-Planck equation, an equivalent stochastic differential equation is solved by means of a large enough ensem- ble of realizations of the stochastic process. The viscoelastic stress is thus obtained as an ensemble average of some function of the **micro** state at each molecule.

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FACSS 2006 abstract – invited talk at Raman imaging session
**Macro**- **and** **micro**-investigation of arterial tissue by optical coherence tomography **and** Raman spectroscopy.
L.-P. Choo-Smith 1 , M. Hewko 1 , A. Ko 1 , J. Werner 1 , E. Kohlenberg 1 , S. Delorme 2 , R. El-Ayoubi 2 **and** M. Sowa

De plus, il y a quatre températures qui caractérisent les AMF : les températures de début et de fin de transformation inverse, As pour austénite start et Af pour austénite fin[r]

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It is also possible to observe recirculating flows independently of inertia. In the case of complex fluids for example, they result from properties such as viscosity, but also elastic- ity, elongation or relaxation phenomena. Such recirculating areas have been observed for example in flows through non-circular capillaries, typically of square or ellipsoidal section [34] [45] [23] [21]. They have also been observed when complex fluids flow through abrupt or shaped contractions, the fluid being pushed from a large-diameter reservoir, through a contraction **and** then into a capillary of smaller diameter. The high ratio between the capillary **and** reservoir diameters is defined as the contraction ratio. This flow configuration was the subject of thorough numerical as well as experimental studies. The reasons for this are numerous: From an academic point of view, this type of flow is of interest as an appro- priate test problem. They also allow the elongational properties of complex fluids, which are generally difficult to measure, to be evaluated [19] [10]. From an industrial point of view, contraction flows are of great importance in modelling polymer processing involving many contraction **and** expansion sections, such as extrusion. The occurrence, evolution, **and** geometrical characteristics of the recirculating areas developing upstream of the con- traction have a great impact in controlling the end-use properties of the material. Indeed, the recirculating fluid undergoes a thermo-mechanical history which is different from that of the flowing fluid. This can induce heterogeneities in the final properties of the material, thus affecting its mechanical characteristics **and** degradation properties. It is thus essential to understand **and** control the origin **and** development of such recirculating flows, the aim being to predict **and** optimize the kinematics with a view to eliminating secondary flows **and** regions of high stress.

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In this paper, we are interested in studying the macroscopic traffic flow model introduced in [7] from the point of view of measure transport equations in Wasserstein spaces.
Transport equations with non-local velocities have drawn a growing attention in the mathematical community, starting from the Vlasov equation **and** other models in kinetic theory, see e.g. [9, 14, 31]. In this context, non-local means that the velocity at a given point of the space depends not only on the density at that point, but on the density in a whole neighborhood. The first general results of existence **and** uniqueness for such equations are given by Ambrosio-Ganbo [3]. There, the authors show that Wasserstein distances are key tools to deal with these equations, since vector fields resulting from non-local interactions are Lipschitz continuous with respect to such distances. Several extensions have been proposed since then, including definition of gradient flows [5], nu- merical schemes [26, 28], generalizations to domains with boundary [16] **and** to transport equations with sources [25, 27].

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be more favorable for the RD **and** TD orientations than at 45°. These tests show that the mechanical behaviour of unstable stainless steels cannot only be explained in terms of α’ volume fraction. The loading path influence was investigated by testing tensile specimen of both grades following the sketch shown on Fig.11. Large specimens were firstly predeformed along TD by various amounts. Then smaller specimens were cut along 45°/RD **and** tested. In this study all specimens were given a total strain of about 0.35. The martensite amount was then measured **and** the results are presented in Fig.12. When changing the strain path, the extent of martensite amount formed was dependent of the applied pre-strain. For lower pre-strain values (ε = 0.14 **and** ε = 0.23), the new kinetics after an orientation change follows the one obtained without pre-strain for the given orientation (Fig.12b). When all nucleation sites have been exhausted ( ε = 0.28), the kinetics follows the TD orientation curve (Fig.12b).

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(assuming that physical particles of velocity v such that |v| > 10 can be negligible), so
that L x = 4π **and** L v = 20. Concerning the mixture parameters, we take α = δ = 0.5,
γ = 0.1, ν 12 =1.
6.1. Decay rates in the space-homogeneous case. We first propose to val- idate our model in the space-homogeneous case, where we have an estimation of the decay rate of |u 1 (t) −u 2 (t) | 2 **and** of T 1 (t) −T 2 (t) (see section 4 ). Here, we want to check if the behaviour of a gas mixture in the sense of relaxation to a global equilibrium (Maxwell distributions with the same mean velocity **and** temperature) is obtained in a

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1 INTRODUCTION
Analysis of a large system like ship structure, to obtain the fine **scale** results is one of the industrial challenges today. Since there is often only one prototype produced, which is the final product **and** because real-**scale** test of such structures are very expensive **and** difficult, the de- signers now often rely on finite element simulation. Due to the classification society rules **and** regulations, extensive use of numerical simulations started only recently in this domain of en- gineering. With the advent of new design for ships, the old rules **and** regulations were obsolete **and** engineers shifted to the numerical simulation **and** in particular to the finite element method. Due to different specifications, several discretized models are used:

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