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We report a continuum theory for 2D strain gradient materials accounting for a class of dissipation phenomena. The continuum description is constructed by means of a (reversible) placement function and by (irreversible) damage and plastic functions. Besides, expressions of elastic and dissipation energies have been assumed as well as the postulation of a hemi-variational principle. No flow rules have been assumed and plastic deformation is also compatible, that means it can be derived by a placement function. Strain gradient Partial Differential Equations (PDEs), boundary conditions (BCs) and Karush-Kuhn-Tucker (KKT) type conditions are derived by a hemi variational principle. PDEs and BCs govern the evolution of the placement descriptor and KKT conditions that of damage and plastic variables. Numerical experiments for the investigated homogeneous cases do not need the use of Finite Element simulations and have been performed to show the applicability of the model. In particular, the induced anisotropy of the response has been investigated and the coupling between damage and plasticity evolution has been shown.

Numerical simulations of several planar failure modes of masonry structures are presented, based on the model and solving code from a recent hemivariational block-based model inspired from granular micromechanics (Tran et al. [33]). The numerical tests include a comparison with literature results for a constant shearing load, a parametric study of the influence of mortar thickness, and the simulation of common failures such as bending, shear sliding, and rocking. The results show that the studied model is able to capture common failure modes. Conclusions and insights towards the extension of the studied model are then given.

This paper is devoted to the development of a continuum theory for materials having granular microstructure and accounting for some dissipative phenomena like damage and plasticity. The continuum description is constructed by means of purely mechanical concepts, assuming expressions of elastic and dissipation energies as well as postulating a hemi-variational principle, without incorporating any additional postulate like flow rules. Granular micromechanics is connected kinematically to the continuum scale through Piola’s ansatz. Mechanically meaningful objective kinematic descriptors aimed at accounting for grain–grain relative displacements in finite deformations are proposed. Karush–Kuhn–Tucker (KKT)-type conditions, providing evolution equations for damage and plastic variables associated with grain–grain interactions, are derived solely from the fundamental postulates. Numerical experiments have been performed to investigate the applicability of the model. Cyclic loading–unloading histories have been considered to elucidate the material hysteretic features of the continuum, which emerge from simple grain–grain interactions. We also assess the competition between damage and plasticity, each having an effect on the other. Further, the evolution of the load-free shape is shown not only to assess the plastic behavior, but also to make tangible the point that, in the proposed approach, plastic strain is found to be intrinsically compatible with the existence of a placement function.

Mechanical behavior of materials with granular microstructures is confounded by unique features of their grain-scale mechano-morphology, such as the tension–compression asymmetry of grain interactions and irregular grain structure. Continuum models, necessary for the macro-scale description of these materials, must link to the grain-scale behavior to describe the consequences of this mechano-morphology. Here, we consider the damage behavior of these materials based upon purely mechanical concepts utilizing energy and variational approach. Granular micromechanics is accounted for through Piola’s ansatz and objective kinematic descriptors obtained for grain-pair relative displacement in granular materials undergoing finite deformations. Karush–Kuhn–Tucker (KKT)-type conditions that provide the evolution equations for grain-pair damage and Euler–Lagrange equations for evolution of grain-pair relative displacement are derived based upon a non-standard (hemivariational) variational approach. The model applicability is illustrated for particular form of grain-pair elastic energy and dissipation functionals through numerical examples. Results show interesting damage-induced anisotropy evolution including the emergence of a type of chiral behavior and formation of finite localization zones.

Dieser Beitrag ist mit Zustimmung des Rechteinhabers aufgrund einer (DFG geförderten) Allianz- bzw. Nationallizenz frei zugänglich., This publication is with permission of the rights owner freely accessible due to an Alliance licence and a national licence (funded by the DFG, German Research Foundation) respectively., Strain gradient continuum damage modelling has been applied to quasistatic brittle fracture within an approach based on a maximum energy-release rate principle. The model was implemented numerically, making use of the FEniCS open-source library. The considered model introduces non-locality by taking into account the strain gradient in the deformation energy. This allows for stable computations of crack propagation in differently notched samples. The model can take wedges into account, so that fracture onset can occur at wedges. Owing to the absence of a damage gradient term in the dissipated energy, the normal part of the damage gradient is not constrained on boundaries. Thus, non-orthogonal and non-parallel intersections between cracks and boundaries can be observed.

Granular micromechanics approach (GMA) provides a predictive theory for granular material behavior by connecting the grain-scale interactions to continuum models. Here, we have used GMA to predict the closed-form expressions for elastic constants of macroscale chiral granular metamaterial. It is shown that for macroscale chirality, the grain-pair interactions must include coupling between normal and tangential deformations. We have designed such a grain-pair connection for physical realization and quantified with FE model. The verification of the prediction is then performed using a physical model of 1D bead string obtained by 3D printing. The behavior is also verified using a discrete model of 1D bead string.

This paper aims at modeling pull-out tests of a reinforcement bar from a concrete matrix. The considered system is an elastic cylinder, within which is embedded a rigid bar placed along the central axis. The pull-out test consists in extracting the bar while fixing the outer lateral boundary of the cylinder. Typically, the concrete matrix would be modeled as a Cauchy continuum (i.e. first gradient elastic continuum), while the slender reinforcement bar would be modeled either as an inner elastic cylinder, or – when its radius tends to zero – as a one-dimensional beam. However, we show that such a model yields a null total deformation energy for the concrete cylinder if the bar is modeled as a beam. This result is physically paradoxical, as the extraction of even the slenderest bar requires an energy input, which is transferred to the concrete matrix as deformation energy. In the present work, this paradox is solved by considering a strain-gradient deformation energy for the concrete matrix. It is shown that such a modeling approach allows deformation to occur even when the radius of the inner bar tends to zero. This result is found to be parametrized by a characteristic length which is physically justified through granular micromechanics. Numerical results are also obtained and hint at a possible way to identify the characteristic length experimentally in future works.

This work is dedicated to the study of the effect of stress concentrations in plates induced by notches and holes using a Granular Micromechanics (GM) model. To this end, three structural configurations were analyzed: an intact plate, a plate with a central circular hole, and a plate with two notches. For comparison of the mechanical response of these plates, the radii of the hole and the notches are taken the same. The material behavior of the plates is described through the multiscale GM constitutive law. The equations of this model are formulated based on the mechanical properties of the grains, the grain-pair interactions and the relative movement of the grains. So the identification of the material parameters needs only the determination of the grains’ properties and grain-pair interactions. This micromechanics-based model has been implemented into the finite element code Abaqus/Explicit through a Vectorized User MATerial (VUMAT) subroutine. The model is applied to study the plates under uniaxial and biaxial loadings. This study aims to assess the pertinence of applying the GM model to structural analysis and show the influence of the two geometrical imperfections on the overall behavior of the structure. It has been found that the model can describe the behavior of plates under tension and compression loadings. Moreover, the effects of a hole and notches on the behavior and failure of the granular-made plates are captured.

In this work, we consider a strain gradient elasticity theory with an extended number of field variables: the displacement vector and an additional scalar field defining the internal micro-deformation. The total internal energy of the model depends on the strain, the micro-deformation function, their gradients, and the coupling. The considered model can be treated as gradient/micromorphic. Moreover, the micro-deformation field can be treated as a field of scalar defects distributed along the medium. Based on analytic (one-dimensional) solutions of uniform/non-uniform deformation of the rod, we introduce (i) an apparent stiffness and (ii) an apparent length scale parameter. Subsequently, we provide a variant of continuum-on-continuum homogenization by equating tip displacements for the gradient/micromorphic medium and an equivalent strain gradient one. Elongation of the gradient/micromorphic rod is therefore equated with the corresponding elongation of the equivalent strain gradient rod, whose behavior is characterized by the apparent material constants. Subsequently, the non-dimensional coupling number is identified with a damage parameter. It is shown that, on the one hand, the apparent stiffness of the rod is reduced when such parameter increases. On the other hand, the apparent length scale parameter (i.e. the apparent second gradient elastic coefficient) increases when the damage parameter increases. Therefore, it is shown that the presence of defects in a second gradient linear elastic material may increase its apparent strain gradient behavior.

Resilience is the ability to undergo disruptive events without suffering complete failure and minimizing undesirable consequences thereof. Many attempts have been done in the recent literature in order to define quantitatively synthetic resilience measures of mechanical structures. They have been conceived as to be useful in comparing and classifying the resilience of different structures. In this paper we propose a novel rational and deterministic structural resilience index, based on a worst-case disruptive event, and making use of a minimalistic kinematics-based mechanical description. Particular emphasis is given to the computation of such a new structural resilience index for a generic frame structure, it being possible without the use of automated algorithms and an analytic expression of it has been achieved in terms of the numbers of ports (modules) and floors. First of all, keeping fixed the number of modules, the resilience index decreases as the number of floors increases. Secondly, keeping fixed the number of floors, the resilience index increases as the number modules increases. Finally, we notice that the case with infinite floors has a lower resilience index with respect to case of infinite modules, which is, as expectable, more resilient.

The characterization of the interphase condition between two materials is current in mechanics. In general, its modeling is achieved by considering an interface with only purely elastic properties. In this paper, following previous works, also inertial interface properties are taken into account. For sufficiently low-frequency regime, we investigate two density profiles (affine and quadratic), for the interphase. Moreover, the interface and the interphase are placed between two solids with different characteristics. The first one is non-dispersive, while for the second one three cases are considered: (a) solid without microstructure, i.e., a Cauchy continuum, (b) solid with microstructure characterized by normal dispersion, i.e., a strain gradient continuum, and (c) by anomalous dispersion. The reflection coefficients are plotted for each case. These results are evaluated with respect to a benchmark finite elements simulation of the finite heterogeneous interphase, and the error is discussed. It is shown that the effects of microstructure can be appreciated at higher frequencies and that the proposed model results to be accurate.

Wave propagation in granular materials is known to be dispersive. Micromorphic continuum model based upon granular micromechanics (Misra, A. and P. Poorsolhjouy, Continuum Mech. Thermodyn, 2016. 28(1-2): p. 215-234.) has the ability to describe this dispersion behavior. In this paper we show that the dispersive behavior can be modulated by using electric field when the grains have dielectric properties. To this end, we apply the recently enhanced model that incorporates electro-elastic coupling by connecting microstrain to electric dipole and quadrupole densities due to bound charges in dielectric grains (Romeo, M., Mech. Res. Commun., 2018. 91: p. 33-38.). We particularly investigate the effect of induced polarization that arises due to an imposed electric field. Two cases of dielectric one dimensional infinite rods with the same micromorphic properties have been studied, where case 1 and 2 are in null and nonzero external electric fields, respectively. Parametric studies are performed to understand the contribution of the polarizability (dipole effect), intrinsic quadrupole density, and external electric field on the dispersive behavior of granular media. Results predict an acoustic and an optical branch in the dispersive curve. Polarizability and external electric field are mainly affecting small wavenumber behavior of the wave branches, while quadrupole density alters the behavior of the material at large wavenumbers. A possibility of altering the optical branch to an acoustic branch is also observed, for which instability or attenuation occurs depending upon the direction of the imposed electric field with respect to the wave propagation direction. We find that the location and the width of the frequency band gaps can be altered using external electric field. The possibility of creating or removing frequency band gaps is also shown to exist. The extended theory accounting for electro-elasticity can therefore be utilized as a tool to analyze existing granular media, or to design granular metamaterials, as it systematizes the design process and eliminates ad-hoc manners leading to large data libraries.

Metamaterials response is generally modeled by generalized continuum based theories. Their inherent substructure leads to a necessity for higher-order theories, and especially in damage mechanics, such a generalization is difficult to acquire. We exploit the action formalism in order to obtain the governing equations in generalized damage mechanics for metamaterials. Additionally, by using auxilliary variables, the variational formulation is endowed with the first rate of damage variable that is missing in standard approaches. The presented action formalism with auxilliary variables leads directly to the weak form. We implement a finite element method based approach by using open-source computing platform called FEniCS and solve this weak in order to obtain the deformation and damage numerically. Metamaterials simulations are demonstrated for simple geometries in mixed mode (I and II) as well as in mode III.

We analyze in two dimensions the mechanical behavior of materials with granular microstructures modeled by means of a variationally formulated strain-gradient continuum approach based on micromechanics and show that it can capture microstructural-size-dependent effects. Tension-compression asymmetry of grain-assembly interactions, as well as microscale damage, is taken into account and the continuum scale is linked to the grain-scale mechanisms. Numerical results are provided for finite deformations and substantiate previous research. As expected, results show interesting size-dependent effects that are typical of strain-gradient modeling.

Indexado en Scopus Describing the emerging macro-scale behavior by accounting for the micro-scale phenomena calls for microstructure-informed continuum models accounting properly for the deformation mechanisms identifiable at the micro-scale. Classical continuum theory, in contrast to the micromorphic continuum theory, is unable to take into account the effects of complex kinematics and distribution of elastic energy in internal deformation modes within the continuum material point. In this paper, we derive a geometrically nonlinear micromorphic continuum theory on the basis of granular mechanics, utilizing grain-scale deformation as the fundamental building block. The definition of objective kinematic descriptors for relative motion is followed by Piola’s ansatz for micro–macro-kinematic bridging and, finally, by a limit process leading to the identification of the continuum stiffness parameters in terms of few micro-scale constitutive quantities. A key aspect of the presented approach is the identification of relevant kinematic measures that describe the deformation of the continuum body and link it to the micro-scale deformation. The methodology, therefore, has the ability to reveal the connections between the micro-scale mechanisms that store elastic energy and lead to particular emergent behavior at the macro-scale. info:eu-repo/semantics/publishedVersion Revisión por pares

A hemi-variational formulation for a damaged non-homogeneous Timoshenko beam is proposed here for the purpose of fast simulation of the properties of a dam. The dam is therefore modeled as a damaged non-homogeneous Timoshenko beam embedded in 2D space. The damage evolution and the mechanics of the beam are governed both by the hemi-variational principle and by the assumption on the form of the deformation energy functional, that comprehends the dissipative part owing to damage-aging effects. In the formulation, the Karush–Kuhn–Tucker (KKT) condition is derived and the damage, which locally decreases the stiffness (i.e., axial, bending, and shear stiffness) of the beam, becomes relevant once a measure of the elastic energy, called the normalized undamaged elastic energy, reaches a certain threshold that is constitutively prescribed ab initio. The aging effect is assumed by reducing such a threshold in the neighborhood of the bottom of the beam, where the concentration of the chemical and physical attacks is higher. As a result, we show a method for the calculation of the mechanical failure condition. In addition, we observe that, even though this threshold is assumed to decrease in a large region far from the bottom of the dam, the damage is confined at the bottom of it. Thus, we have proved that the trapezoidal shape that is used in the dam design is useful not only to decrease the state of stress in the elastic regime, but also to confine the effects of damage evolution owing to the aging effects.

We study three different 1D continuous models (extensional rods, Euler and Timoshenko beams) for addressing the dynamic properties of those microstructural materials containing a density of resonators. These models correspond to metamaterials which show interesting properties: In particular, the property that is the objective of this paper is the capacity of eliminating the vibration amplitude in a specific frequency range, which is called hereinafter band gap. The simplicity of these models emphasizes those microstructural properties having a relation with the band gap. We show that the rigidity of the hosting structure does not affect the values of the frequency band gap; it affects only the distance between the load-source of vibration and those points where the amplitude attenuation is visible. We also study, from a numerical point of view and using the Euler beam as the hosting structure, the case of a finite number of resonators. In particular, we study the minimum number of resonators which provides the same band gap as in the case of the presence of a density of resonators. We finally perform a numerical study on a periodic 2D elastic structure, which behaves like the Timoshenko beam model and for which an identification procedure is given.

Due to the latest advancements in 3D printing technology and rapid prototyping techniques, the production of materials with complex geometries has become more affordable than ever. Pantographic structures, because of their attractive features, both in dynamics and statics and both in elastic and inelastic deformation regimes, deserve to be thoroughly investigated with experimental and theoretical tools. Herein, experimental results relative to displacement-controlled large deformation shear loading tests of pantographic structures are reported. In particular, five differently sized samples are analyzed up to first rupture. Results show that the deformation behavior is strongly nonlinear, and the structures are capable of undergoing large elastic deformations without reaching complete failure. Finally, a cutting edge model is validated by means of these experimental results.

In the present contribution, we address the following problem: is it possible to find a microstructure producing, at the macro-level and under loads of the same order of magnitude, a beam which can be both extensible and flexible? Using an asymptotic expansion and rescaling suitably the involved stiffnesses, we prove that a pantographic microstructure does induce, at the macro-level, the aforementioned desired mechanical behavior. Thus, in an analogous fashion to that of variational asymptotic methods, and following a mathematical approach resembling that used by Piola, we have employed asymptotic expansions of kinematic descriptors directly into the postulated energy functional and a heuristic homogenization procedure is presented and applied to the cases of Euler and pantographic beams.

International audience; We consider in this paper analytical solutions for some remarkable cases and for a linear anisotropic D4 second gradient elastic model. The purpose is that of constitutive parameter identification. In general, analytical solutions are considered less important than in the past due to fast numerical tools and to the fact that they are generally very difficult to achieve. However, they still play very important roles such as for numerical comparison and for dealing with pathological mechanical systems.

International audience; In the last decade the exotic properties of pantographic metamaterials have been investigated and different mathematical models (both discreteor continuous) have been introduced. In a previous publication a large part of the already existing literature about pantographic metamaterials has been presented. In this paper we give some details about the next generation of research in this field. We present an organic scheme of the whole process of design, fabrication, experiments, models and image analyses.

International audience; There is a class of planar 1D-continua which can be described exclusively by their placement functions which in turn are curves in a two-dimensional space. In contrast to the Elastica for which the deformation energy depends on the projection of the second gradient to the normal vector of the placement function, i.e. the material curvature, the proposed continuum does also depend on the projection onto the tangent vector, introduced as the stretch gradient. Thus, the deformation energy takes into account the complete second gradient of the placement function. In such a model, non-standard boundary conditions and more generalized forces such as double forces do appear. The deformation energy of the continuum is obtained by applying a heuristic homogenization procedure to a family of slender discrete pantographic structures constituted by extensional and rotational springs. Within the homogenization process, the overall length of the system is kept fixed, the number of the periodically appearing sub-systems, called cells, is increased, and the stiffnesses are appropriately scaled. For two examples, we numerically compare the family of discrete systems with the continuum. The analysis shows that the continuum represents the behaviour of the discrete system already for a relatively moderate number of cells. In particular, the behaviour of the deformation energy error between the discrete and the continuum models when the number of cells tends to infinity is determined by the homogenization process.

International audience; In this paper, we account for the research efforts that have been started, for some among us, already since 2003, and aimed to the design of a class of exotic architectured, optimized (meta) materials. At the first stage of these efforts, as it often happens, the research was based on the results of mathematical investigations. The problem to be solved was stated as follows: determine the material (micro)structure governed by those equations that specify a desired behavior. Addressing this problem has led to the synthesis of second gradient materials. In the second stage, it has been necessary to develop numerical integration schemes and the corresponding codes for solving, in physically relevant cases, the chosen equations. Finally, it has been necessary to physically construct the theoretically synthesized microstructures. This has been possible by means of the recent developments in rapid prototyping technologies, which allow for the fabrication of some complex (micro)structures considered, up to now, to be simply some mathematical dreams. We show here a panorama of the results of our efforts (1) in designing pantographic metamaterials, (2) in exploiting the modern technology of rapid prototyping, and (3) in the mechanical testing of many real prototypes. Among the key findings that have been obtained, there are the following ones: pantographic metamaterials (1) undergo very large deformations while remaining in the elastic regime, (2) are very tough in resisting to damage phenomena, (3) exhibit robust macroscopic mechanical behavior with respect to minor changes in their microstructure and micromechanical properties, (4) have superior strength to weight ratio, (5) have predictable damage behavior, and (6) possess physical properties that are critically dictated by their geometry at the microlevel.

A report on the workshop Computational mechanics of generalized continua and applications to materials with microstructure (Catania 29–31 October 2015) is provided. The constructive atmosphere that was present at the workshop in 2012 and the Euromech Colloquium in 2014, both in Cisterna di Latina, was repeated for this workshop in Catania. The objective of this meeting was to bring together experts within the CNRS International Associate Laboratory (LIA) Francois Cosserat–Tullio Levi Civita Coss & Vita in order to discuss topics of common interest. Particularly, the workshop was dedicated to the following projects of LIA: (i) computational mechanics of generalized continua; and (ii) nonlinearity and stability in continuous media. Also subjects related to the application of generalized continua to multiscale and smart materials were discussed. Approximately 25 Coss & Vita LIA members and other experts (mostly from France and Italy) gathered together, including PhD students and those students from the Scuola Superiore di Catania.

In the present paper, a rational report on Euromech 563, Generalized continua and their applications to the design of composites and metamaterials (Cisterna di Latina 17–21 March 2014), is provided. The frank and constructive spirit which animated the workshop by Dell’Isola et al. ( ZAMM 2014; 94(5): 367–372) also characterized Euromech Colloquium 563. All presentations were video-recorded and are freely available online at the address http://www.memocsevents.eu/euromech563/?page_id=1013 . The topics treated were selected by the organizers in order to allow a comparison of the available experimental evidence with the predictive capability of current theoretical models. The numerical investigations selected and presented aimed to make more effective the aforementioned comparison. The interested reader will find more details about the colloquium at the dedicated webpage http://www.memocsevents.eu/euromech563 . The design and use of advanced materials and structural elements requires an extensive and rigorous process of mathematical modeling. The organizers of Euromech 563, being aware of this need, have chosen as participants renowned and reputed scientists (and some promising young researchers) who all agree on recognizing to mathematical sciences their role of unifying and coordinating tool in the effort for the advance of technology. The Colloquium has indeed surely contributed to the development of more advanced technological possibilities and to the theoretical conception of completely new ones.

A linear elastic second gradient orthotropic two-dimensional solid that is invariant under $$90^{\circ }$$ rotation and for mirror transformation is considered. Such anisotropy is the most general for pantographic structures that are composed of two identical orthogonal families of fibers. It is well known in the literature that the corresponding strain energy depends on nine constitutive parameters: three parameters related to the first gradient part of the strain energy and six parameters related to the second gradient part of the strain energy. In this paper, analytical solutions for simple problems, which are here referred to the heavy sheet, to the nonconventional bending, and to the trapezoidal cases, are developed and presented. On the basis of such analytical solutions, gedanken experiments were developed in such a way that the whole set of the nine constitutive parameters is completely characterized in terms of the materials that the fibers are made of (i.e., of the Young’s modulus of the fiber materials), of their cross sections (i.e., of the area and of the moment of inertia of the fiber cross sections), and of the distance between the nearest pivots. On the basis of these considerations, a remarkable form of the strain energy is derived in terms of the displacement fields that closely resembles the strain energy of simple Euler beams. Numerical simulations confirm the validity of the presented results. Classic bone-shaped deformations are derived in standard bias numerical tests and the presence of a floppy mode is also made numerically evident in the present continuum model. Finally, we also show that the largeness of the boundary layer depends on the moment of inertia of the fibers.

This semi-inverse method is similar to that used in the so-called Saint-Venant problem for cylindrical three-dimensional first-gradient linear homogeneous and isotropic materials. This semi-inverse method is similar to that used by Saint-Venant to solve the omonimus problem for cylindrical three-dimensional first-gradient linear homogeneous and isotropic materials. Two examples are also presented. It is found that wedge forces are necessary to maintain the body in equilibrium and that these are not an artefact of the double application of the divergence theorem in the second-gradient material derivations.

In the literature of granular micromechanics it is standard to achieve, at the macro scale, non-standard enhanced continuum models [1–2]. Among a panoply of exotic behaviors, these models predict that granular materials can show chirality for a specific grain-pair interaction. To verify these predictions, a granular system with a specific grain-pair interaction has been designed and its mechanical behavior under different types of loading has been evaluated via numerical simulations. The resulting granular system, which can be referred to as a granular beam, is a linear array of grains connected via the chosen grain-pair interaction law [1]. The chiral behavior of such mechanical system has been observed experimentally during tensile test. To describe the experimental evidence and numerical results, a continuous one-dimensional beam model has been defined and the four constitutive parameters, which characterize this specific strain energy function, have been identified. The numerical simulations on this granular system, modeled as a 2D deformable-body, have been performed employing the commercial finite element software COMSOL Multiphysics to have a reference data set for the identification process. Comparisons between the results obtained from this 2D FE model and the predictions of the same system analyzed via a specific non standard Timoshenko 1D beam model show an astonishing agreement.

International audience; Wave propagation in architectured materials, or materials with microstructure, is known to be dependent on the ratio between the wavelength and a characteristic size of the microstructure. Indeed, when this ratio decreases (i.e. when the wavelength approaches this characteristic size) important quantities, such as phase and group velocity, deviate considerably from their low frequency/long wavelength values. This well-known phenomenon is called dispersion of waves. Objective of this work is to show that strain-gradient elasticity can be used to quantitatively describe the behaviour of a microstructured solid, and that the validity domain (in terms of frequency and wavelength) of this model is sufficiently large to be useful in practical applications. To this end, the parameters of the overall continuum are identified for a periodic architectured material, and the results of a transient problem are compared to those obtained from a finite element full field computation on the real geometry. The quality of the overall description using a strain-gradient elastic continuum is compared to the classical homogenization procedure that uses Cauchy continuum. The extended model of elasticity is shown to provide a good approximation of the real solution over a wider frequency range.

In this paper, we formulate a linear elastic second gradient isotropic two-dimensional continuum model accounting for irreversible damage. The failure is defined as the condition in which the damage parameter reaches 1, at least in one point of the domain. The quasi-static approximation is done, i.e., the kinetic energy is assumed to be negligible. In order to deal with dissipation, a damage dissipation term is considered in the deformation energy functional. The key goal of this paper is to apply a non-standard variational procedure to exploit the damage irreversibility argument. As a result, we derive not only the equilibrium equations but, notably, also the Karush–Kuhn–Tucker conditions. Finally, numerical simulations for exemplary problems are discussed as some constitutive parameters are varying, with the inclusion of a mesh-independence evidence. Element-free Galerkin method and moving least square shape functions have been employed.

Part of the Advanced Structured Materials book series (STRUCTMAT, volume 87); International audience; In the standard asymptotic micro-macro identification theory, starting from a De Saint-Venant cylinder, it is possible to prove that, in the asymptotic limit, only flexible, inextensible, beams can be obtained at the macro-level. In the present paper we address the following problem: is it possible to find a microstructure producing in the limit, after an asymptotic micro-macro identification procedure, a continuum macro-model of a beam which can be both extensible and flexible? We prove that under certain hypotheses, exploiting the peculiar features of a pantographic microstructure, this is possible. Among the most remarkable features of the resulting model we find that the deformation energy is not of second gradient type only because it depends, like in the Euler beam model, upon the Lagrangian curvature, i.e. the projection of the second gradient of the placement function upon the normal vector to the deformed line, but also because it depends upon the projection of the second gradient of the placement on the tangent vector to the deformed line, which is the elongation gradient. Thus, a richer set of boundary conditions can be prescribed for the pantographic beam model. Phase transition and elastic softening are exhibited as well. Using the resulting planar 1D continuum limit homogenized macro-model, by means of FEM analyses, we show some equilibrium shapes exhibiting highly non-standard features. Finally, we conceive that pantographic beams may be used as basic elements in double scale metamaterials to be designed in future.