1,721,724 research outputs found
Second-order homogenization of periodic materials based on asymptotic approximation of the strain energy: formulation and validity limits
No full textIn this paper a second-order homogenization approach for periodic material is derived from an appropriate representation of the down-scaling that correlates the micro-displacement field to the macro-displacement field and the macro-strain tensors involving unknown perturbation functions. These functions take into account of the effects of the heterogeneities and are obtained by the solution of properly defined recursive cell problems. Moreover, the perturbation functions and therefore the micro-displacement fields result to be sufficiently regular to guarantee the anti-periodicity of the traction on the periodic unit cell. A generalization of the macro-homogeneity condition is obtained through an asymptotic expansion of the mean strain energy at the micro-scale in terms of the microstructural characteristic size ɛ; the obtained overall elastic moduli result to be not affected by the choice of periodic cell. The coupling between the macro- and micro-stress tensor in the periodic cell is deduced from an application of the generalised macro-homogeneity condition applied to a representative portion of the heterogeneous material (cluster of periodic cell). The correlation between the proposed asymptotic homogenization approach and the computational second-order homogenization methods (which are based on the so called quadratic ansätze) is obtained through an approximation of the macro-displacement field based on a second-order Taylor expansion. The form of the overall elastic moduli obtained through the two homogenization approaches, here proposed, is analyzed and the differences are highlighted. An evaluation of the developed method in comparison with other recently proposed in literature is carried out in the example where a three-phase orthotropic material is considered. The characteristic lengths of the second-order equivalent continuum are obtained by both the asymptotic and the computational procedures here analyzed. The reliability of the proposed approach is evaluated for the case of shear and extensional deformation of the considered two-dimensional infinite elastic medium subjected to periodic body forces; the results from the second-order model are compared with those of the heterogeneous continuum
A multi-scale strain-localization analysis of a layered strip with debonding interfaces
No full textThe paper is focused on the multi-scale modeling of shear banding in a two-phase linear elastic periodically layered strip with damaging interfaces. A two-dimensional layered strip is considered subjected to transverse shear and is assumed to have a finite thickness along the direction of the layers and an infinite extension along the direction perpendicular to layering. The strip is analyzed as a second-gradient continuum resulting from a second-order homogenization procedure developed by the Authors, here specialized to the case of layered materials. This analysis is also aimed to understand the influence on the strain localization and post-peak structural response of the displacement boundary conditions prescribed at the strip edges. To this end, a first model representative of the strip with warping allowed at the edges is analyzed in which the strain localization process is obtained as a results of a bifurcation in analogy to the approach by Chambon et al. (1998). A second model is analyzed in which the warping of the edge is inhibited and the damage propagates from the center of the specimen without exhibiting bifurcation phenomena. For this latter case the effects of a possible interaction between the shear band and the boundary shear layer are considered, which are influenced mainly by the characteristic lengths of the model and the strip length. For realistic values of the relevant parameters it is shown that the boundary conditions have a small effects on the elastic response and on the overall strength of the model. Conversely, the boundary conditions have a significant effect on the shear band location, the post-peak response and the structural brittleness. Since the model parameters directly depend on the material microstructure as a result of the homogenization process, both the extension of the shear band and the occurrence of snap-back in the post-peak phase may be controlled in terms of the constitutive parameters and of the geometry of the phases
Multi-field asymptotic homogenization approach for Bloch wave propagation in periodic thermodiffusive elastic materials
Full text linkMulti-field asymptotic homogenization methods are proposed to describe the behaviour of periodic Cauchy materials subject to several physical phenomena, focusing on thermodiffusion. The resulting homogenized models provide the overall constitutive tensors and overall inertial terms. Moreover, they allow one to investigate the complex band structures associated with damped Bloch waves travelling in periodic materials, avoiding the challenging computations needed by the adoption of micromechanical approaches
Second-gradient homogenized model for wave propagation in heterogeneous periodic media
No full textAbstract The paper is focused on a homogenization procedure for the analysis of wave propagation in materials with periodic microstructure. By a reformulation of the variational-asymptotic homogenization technique recently proposed by Bacigalupo and Gambarotta (2012a), a second-gradient continuum model is derived, which provides a sufficiently accurate approximation of the lowest (acoustic) branch of the dispersion curves obtained by the Floquet–Bloch theory and may be a useful tool for the wave propagation analysis in bounded domains. The multi-scale kinematics is described through micro-fluctuation functions of the displacement field, which are derived by the solution of a recurrent sequence of cell {BVPs} and obtained as the superposition of a static and dynamic contribution. The latters are proportional to the even powers of the phase velocity and consequently the micro-fluctuation functions also depend on the direction of propagation. Therefore, both the higher order elastic moduli and the inertial terms result to depend by the dynamic correctors. This approach is applied to the study of wave propagation in layered bi-materials with orthotropic phases, having an axis of orthotropy parallel to the direction of layering, in which case, the overall elastic and inertial constants can be determined analytically. The reliability of the proposed procedure is analysed by comparing the obtained dispersion functions with those derived by the Floquet–Bloch theory
Homogenization of periodic hexa- and tetrachiral cellular solids
No full textAbstract The homogenization of periodic hexachiral and tetrachiral honeycombs is dealt with two different techniques. The first is based on a micropolar homogenization. The second approach, developed to analyse two-dimensional periodic cells consisting of deformable portions such as the ring, the ligaments and possibly a filling material, is based on a second gradient homogenization developed by the authors. The obtained elastic moduli depend on the parameter of chirality, namely the angle of inclination of the ligaments with respect to the grid of lines connecting the centers of the rings. For hexachiral cells the auxetic property of the lattice together with the elastic coupling modulus between the normal and the asymmetric strains is obtained; a property that has been confirmed here for the tetrachiral lattice. Unlike the hexagonal lattice, the classical constitutive equations of the tetragonal lattice turns out to be characterized by the coupling between the normal and shear strains through an elastic modulus that is an odd function of the parameter of chirality. Moreover, this lattice is found to exhibit a remarkable variability of the Young’s modulus and of the Poisson’s ratio with the direction of the applied uniaxial stress. Finally, a simulation of experimental results is carried out
Metamaterial filter design via surrogate optimization
Full text linkRecently, an increasing research effort has been dedicated to analyse transmission and dispersion properties of periodic metamaterials containing resonators, and to optimize the amplitude of selected acoustic band gaps between consecutive dispersion curves in the Floquet-Bloch spectrum. Potential novel applications of this research are in the design of passive mechanical filters/diodes. The present work proposes a way to interpolate the objective functions in such band gap optimization problems, using Radial Basis Functions. The study is motivated by the high computational effort often needed for an exact evaluation of the original objective functions, when using iterative optimization algorithms. By replacing such functions with surrogate objective functions, well-performing suboptimal solutions can be obtained with a small computational effort. Numerical results demonstrate the feasibility of the approach
Wave propagation properties of one-dimensional acoustic metamaterials with nonlinear diatomic microstructure
No full textAcoustic metamaterials are artificial microstructured media, typically characterized by a periodic locally resonant cell. The cellular microstructure can be functionally customized to govern the propagation of elastic waves. A one-dimensional diatomic lattice with cubic inter-atomic coupling—described by a Lagrangian model—is assumed as minimal mechanical system simulating the essential undamped dynamics of nonlinear acoustic metamaterials. The linear dispersion properties are analytically determined by solving the linearized eigenproblem governing the free wave propagation in the small-amplitude oscillation range. The dispersion spectrum is composed by a low-frequency acoustic branch and a high-frequency optical branch. The two frequency branches are systematically separated by a stop band, whose amplitude is analytically derived. Superharmonic 3:1 internal resonances can occur within a wavenumber-dependent locus defined in the mechanical parameter space. A general asymptotic approach, based on the multiple scale method, is employed to determine the nonlinear dispersion properties. Accordingly, the nonlinear frequencies and waveforms are obtained for the two fundamental cases of non-resonant and superharmonically 3:1 resonant or nearly resonant lattices. Moreover, the invariant manifolds associated with the nonlinear waveforms are parametrically determined in the space of the two principal coordinates. Finally, some examples of non-resonant and resonant lattices are selected to discuss their nonlinear dispersion properties from a qualitative and quantitative viewpoint
Second-order computational homogenization of heterogeneous materials with periodic microstructure
No full textA procedure for second-order computational homogenization of heterogeneous materials is derived from the unit cell homogenization, in which an appropriate representation of the micro-displacement field is assumed as the superposition of a local macroscopic displacement field, expressed in a polynomial form related to the macro-displacement field, and an unknown micro-fluctuation field accounting for the effects of the heterogeneities. This second contribution is represented as the superposition of two unknown functions each of which related to the first-order and to the second-order strain, respectively. This kinematical micro-macro framework guarantees that the micro-displacement field is continuous across the interfaces between adjacent unit cells and implies a computationally efficient procedure that applies in two steps. The first step corresponds to the standard homogenization, while the second step is based on the results of the first step and completes the second-order homogenization. Two multi-phase composites, a three-phase and a laminated composite, are analysed in the examples to assess the reliability of the homogenization techniques. The computational homogenization is carried out by a FE analysis of the unit cell; the overall elastic moduli and the characteristic lengths of the second-order equivalent continuum model are obtained. Finally, the simple shear of a constrained heterogeneous two-dimensional strip made up of the composites considered is analysed by considering a heterogeneous continuum and a homogenized second-order continuum; the corresponding results are compared and discussed in order to identify the validity limits of the proposed technique
Effective elastic properties of planar SOFCs: a non-local dynamic homogenization approach
No full textAbstract The focus of the article is on the analysis of effective elastic properties of planar Solid Oxide Fuel Cell (SOFC) devices. An ideal periodic multi-layered composite (SOFC-like) reproducing the overall properties of multi-layer {SOFC} devices is defined. Adopting a non-local dynamic homogenization method, explicit expressions for overall elastic moduli and inertial terms of this material are derived in terms of micro-fluctuation functions. These micro-fluctuation functions are then obtained solving the cell problems by means of finite element techniques. The effects of the temperature variation on overall elastic and inertial properties of the fuel cells are studied. Dispersion relations for acoustic waves in SOFC-like multilayered materials are derived as functions of the overall constants, and the results obtained by the proposed computational homogenization approach are compared with those provided by rigorous Floquet–Bloch theory. Finally, the influence of the temperature and of the elastic properties variation on the Bloch spectrum is investigated
Auxetic anti-tetrachiral materials: equivalent elastic properties and frequency band-gaps
Full text linkA comprehensive characterization of the novel class of anti-tetrachiral cellular solids, both considering the static and the dynamic response, is provided in the paper. The heterogeneous material is characterized by a periodic microstructure made of equi-spaced rings each interconnected by four ligaments. In the most general case, rings and ligaments are surrounded by a softer matrix and the rings can be filled by a different material. First, the first order linear elastic homogenized constitutive response is estimated resorting to two different microstructural models: a discrete model, in which the ligaments are modeled as beams and the presence of the matrix is neglected and the equivalent elastic properties are evaluated through a simplified analytical approach, and a more detailed continuous model, where the actual properties of matrix, ligaments and rings, varying in the 2D domain, are considered and the first order computational homogenization is adopted. Special attention is given to the dependence of the 2D overall Cauchy-type elastic constants on the mechanical and geometrical parameters characterizing the microstructure. The results, indeed, show the existence of large variations in the linear elastic constants and degree of anisotropy. A comparison with available experimental results confirms the validity of the analytical and numerical approaches adopted. Finally, the rigorous Floquet–Bloch approach is applied to the periodic cell of the cellular solid to evaluate the dispersion of propagation waves along the orthotropic axes in the framework of elasticity and to detect band gaps characterizing the material. A numerical approach, based on the first order computational homogenization, is also adopted and the rigorous and approximate solutions are compared
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