Discrete homogenization of architectured materials: Implementation of the method in a simulation tool for the systematic prediction of their effective elastic properties

The kinematics and the balance equations for multiphase micro-architectured materials such as foams, textiles, or beam-like structures exhibit a peculiar macroscopic behavior. The topology and mechanical properties of their structural constituents at the microscale induce this behavior. The derivation of the effective mechanical properties of 2D and 3D lattices made of articulated beams is herewith investigated. The asymptotic homogenization technique is used to get closed form expressions of the equivalent properties versus the geometrical and mechanical micro-parameters. The effective behavior of a 2D hexagonal lattice is calculated, and is validated by comparison with FE simulations results. In order to analyze the respective roles of flexion and extension at both the micro and macro scales, a mixed lattice has been conceived, accounting for both extensional and flexional effects in a versatile manner. Its effective moduli are calculated versus geometrical and mechanical parameters of the beams. The scaling law of the effective traction modulus versus density shows a complex nonlinear evolution. This law has a drastic decrease when flexional modes become dominant over extensional ones. The obtained compliance matrix does not exhibit the expected symmetries when shear behavior is considered, which is explained by the too restrictive assumption of rotations being suppressed at the edges. After extending the present methodology towards the 3D case, the effective mechanical behavior of Kelvin foams under compression is obtained with an isotropic continuum behavior which is in good agreement with both the literature and FE simulations. The effective compliance matrix of the equivalent continuum does not exhibit some of the required material symmetries under shear when the edge node rotations are prevented. A classification of lattices with respect to the choice of the equivalent continuum model is proposed, according to the nature of the boundary conditions, considering especially boundary micro-rotations. One of the main results of the present contribution is the need for an extension of the asymptotic homogenization to a micro-polar continuum, by considering lattice micro-rotations as additional degrees of freedom at the microscopic and macroscopic scale

Homogenization à la Piola produces second gradient continuum models for linear pantographic lattices

In the present work, we show that the linearized homogenized model for a pantographic lattice must necessarily be a second gradient continuum, as defined in Germain (1973). Indeed, we compute the effective mechanical properties of pantographic lattices following two routes both based in the heuristic homogenization procedure already used by Piola (see Mindlin, 1965; dell'Isola et al., 2015a): (i) an analytical method based on an evaluation at micro-level of the strain energy density and (ii) the extension of the asymptotic expansion method up to the second order. Both identification procedures lead to the construction of the same second gradient linear continuum. Indeed, its effective mechanical properties can be obtained by means of either (i) the identification of the homogenized macro strain energy density in terms of the corresponding micro-discrete energy or (ii) the homogenization of the equilibrium conditions expressed by means of the principle of virtual power: actually the two methods produce the same results. Some numerical simulations are finally shown, to illustrate some peculiarities of the obtained continuum models especially the occurrence of bounday layers and transition zones. One has to remark that available well-posedness results do not apply immediately to second gradient continua considered here

Flexoelectricity and apparent piezoelectricity of a pantographic micro-bar

We discuss a homogenized model of a pantographic bar considering flexoelectricity. A pantographic bar consists of relatively stiff small bars connected by small soft flexoelectric pivots. As a result, an elongation of the bar relates almost to the torsion of pivots. Taking into account their flexoelectric properties we find the corresponding electric polarization. As a results, the homogenized pantographic bar demonstrates piezoelectric properties inherited from the flexoelectric properties of pivots. The effective stiffness properties of the homogenized bars are determined by the geometry of the structural elements and shear stiffness whereas the piezoelectric properties follow from the flexoelectric moduli of the pivots

Homogenization of magnetoelastic heterogeneous solid bodies based on micropolar magnetoelasticity

A variational-based homogenization method for magnetoelastic composite materials is established in a small strains framework. The existence of a non-symmetrical stress tensor motivates the elaboration of a homogenized Cosserat type magnetoelastic effective medium at the macroscale. Generic expressions of the effective magnetic and elastic properties are derived, showing the existence of couplings between the elastic and magnetic behaviors at the macrolevel. Applications of the developed homogenization methodology are done for periodic heterogeneous media prone to local bending at the scale of a few unit cells. The validation of the homogenized medium is performed by comparing its predictions versus those of fully resolved computations. The influence of the magnetic field intensity and orientation on the strength of micropolar effects is assessed. The proposed formulation opens new possibilities for the efficient design of multifunctional metamaterials via computational modelling.The authors acknowledge support from MCIN/ AEI /10.13039/501100011033 under Grant number PID2020-117894GA-I00. The authors acknowledge support from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 947723, project: 4D-BIOMAP). DGG acknowledges support from the Talent Attraction grant (CM 2018-2018-T2/IND-9992) from the Comunidad de Madrid

Steel and bone: Mesoscale modeling and middle-out strategies in physics and biology

Mesoscale modeling is often considered merely as a practical strategy used when information on lower-scale details is lacking, or when there is a need to make models cognitively or computationally tractable. Without dismissing the importance of practical constraints for modeling choices, we argue that mesoscale models should not just be considered as abbreviations or placeholders for more “complete” models. Because many systems exhibit different behaviors at various spatial and temporal scales, bottom-up approaches are almost always doomed to fail. Mesoscale models capture aspects of multi-scale systems that cannot be parameterized by simple averaging of lower-scale details. To understand the behavior of multi-scale systems, it is essential to identify mesoscale parameters that “code for” lower-scale details in a way that relate phenomena intermediate between microscopic and macroscopic features. We illustrate this point using examples of modeling of multi-scale systems in materials science (steel) and biology (bone), where identification of material parameters such as stiffness or strain is a central step. The examples illustrate important aspects of a so-called “middle-out” modeling strategy. Rather than attempting to model the system bottom-up, one starts at intermediate (mesoscopic) scales where systems exhibit behaviors distinct from those at the atomic and continuum scales. One then seeks to upscale and downscale to gain a more complete understanding of the multi-scale systems. The cases highlight how parameterization of lower-scale details not only enables tractable modeling but is also central to understanding functional and organizational features of multi-scale systems

Thermomechanical couplings in shape memory alloy materials

In this work we address several theoretical and computational issues which are related to the thermomechanical modeling of shape memory alloy materials. More specifically, in this paper we revisit a non-isothermal version of the theory of large deformation generalized plasticity which is suitable for describing the multiple and complex mechanisms occurring in these materials during phase transformations. We also discuss the computational implementation of a generalized plasticity based constitutive model and we demonstrate the ability of the theory in simulating the basic patterns of the experimentally observed behavior by a set of representative numerical examples

A nonlinear Lagrangian particle model for grains assemblies including grain relative rotations

International audienceWe formulate a discrete Lagrangian model for a set of interacting grains, which is purely elastic. The considered degrees of freedom for each grain include placement of barycenter and rotation. Further, we limit the study to the case of planar systems. A representative grain radius is introduced to express the deformation energy to be associated to relative displacements and rotations of interacting grains. We distinguish inter‐grains elongation/compression energy from inter‐grains shear and rotations energies, and we consider an exact finite kinematics in which grain rotations are independent of grain displacements. The equilibrium configurations of the grain assembly are calculated by minimization of deformation energy for selected imposed displacements and rotations at the boundaries. Behaviours of grain assemblies arranged in regular patterns, without and with defects, and similar mechanical properties are simulated. The values of shear, rotation, and compression elastic moduli are varied to investigate the shapes and thicknesses of the layers where deformation energy, relative displacement, and rotations are concentrated. It is found that these concentration bands are close to the boundaries and in correspondence of grain voids. The obtained results question the possibility of introducing a first gradient continuum models for granular media and justify the development of both numerical and theoretical methods for including frictional, plasticity, and damage phenomena in the proposed model