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

Approche probabiliste du comportement mécanique d'interfaces discrètes

La modélisation des interfaces de manière discrète permet une étude fine de leur structure et de leur comportement. L'utilisation d'aspects stochastiques enrichit la modélisation. On s'intéresse ici à la dégradation progressive des liens entre les deux surfaces en contact. Une modélisation récursive est proposée ; les variations des modules d'Young et coefficients de Poisson décrivant le comportement des fibres composant l'interface conduisent à une loi de type Scott Blair caractérisée par un exposant différentiel fractionnaire. Le seuil de rupture potentiel des fibres est décrit par diverses lois de probabilité. Des simulations de fluage et de traction sont réalisées, décrites par des équations aux dérivées fractionnaires stochastiques. En regard de solutions analytiques obtenues dans quelques cas particuliers à l'aide de la transformation de Laplace, une méthode basée sur les polynômes d'Adomian est présentée. La convergence de la méthode est envisagée

Exploiting Viscoelastic Experimental Observations and Numerical Simulations to Infer Biomimetic Artificial Tendon Fiber Designs

Designing biomimetic artificial tendons requires a thorough, data-based understanding of the tendon's inner material properties. The current work exploits viscoelastic experimental observations at the tendon fascicle scale, making use of mechanical and data analysis methods. More specifically, based on reported elastic, volumetric and relaxation fascicle scale properties, we infer most probable, mechanically compatible material attributes at the fiber scale. In particular, the work provides pairs of elastic and viscous fiber-scale moduli, which can reproduce the upper scale tendon mechanics. The computed range of values for the fiber-scale tendon viscosity attest to the substantial stress relaxation capabilities of tendons. More importantly, the reported mechanical parameters constitute a basis for the design of tendon-specific restoration materials, such as fiber-based, engineering scaffolds

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

Investigating the Effect of Aging on the Viscosity of Tendon Fascicles and Fibers

In the current work, we investigate the effect of aging on the viscosity of tendon subunits. To that scope, we make use of experimental relaxation curves of healthy and aged tendon fascicles and fibers, upon which we identify the viscosity parameters characterizing the time-dependent behavior of each tendon subunit. We subsequently combine the obtained results with analytical viscoelastic homogenization analysis methods to extract information on the effective viscous contribution of the embedding matrix substance at the fiber scale. The results suggest that the matrix substance plays a significant role in the relaxation process of the upper tendon subunits both for aged and healthy specimens. What is more, the viscosity coefficients computed for the fibrillar components indicate that aging leads to a viscosity reduction that is statistically significant for both fascicles and fibers. Its impact is more prominent for the lower hierarchical scale of fibers. As such, the reduced stress relaxation capability at the tendon macroscale is to be primarily attributed to the modified viscosity of its inner fibrillar subunits rather than to the matrix substance