arXiv subjects

Andrea Bacigalupo

Publications and source records attributed to Andrea Bacigalupo.

33 records · Page 2Linked to original sources

arXiv2018

Dynamic asymptotic homogenization for periodic viscoelastic materials

A non-local dynamic homogenization technique for the analysis of a viscoelastic heterogeneous material which displays a periodic microstructure is herein proposed. The asymptotic expansion of the micro-displacement field in the transformed Laplace domain allows obtaining, from the expression of the micro-scale field equations, a set of recursive differential problems defined over the periodic unit cell. Consequently, the cell problems are derived in terms of perturbation functions depending on the geometrical and physical-mechanical properties of the material and its microstructural heterogeneities. A down-scaling relation is formulated in a consistent form, which correlates the microscopic to the macroscopic transformed displacement field and its gradients through the perturbation functions. Average field equations of infinite order are determined by substituting the down-scale relation into the micro-field equation. Based on a variational approach, the macroscopic field equations of a non-local continuum is delivered and the local and non-local overall constitutive and inertial tensors of the homogenized continuum are determined. The problem of wave propagation in case of a bi-phase layered material with orthotropic phases and axis of orthotropy parallel to the direction of layers is investigated as an example. In such a case, the local and non-local overall constitutive and inertial tensors are determined analytically and the dispersion curves obtained from the non-local homogenized model are analysed.

physics.app-ph↗

arXiv2018

Design of thermo-piezoelectric microstructured bending actuators via multi-field asymptotic homogenization

The use of integrated MicroElectroMechanical systems (MEMS) is recently spread thanks to their improved sensitivity, accuracy and reliability. Accurate preliminary computations born from the need of high precision in the manufacturing process of such devices. Piezoelectric materials are broadly employed in this field as direct converters between mechanical and electrical signals and some of these piezoelectric materials show pyroelectric features, which involve thermo-electrical interactions. Pyroelectric bending actuators are analyzed in the present study in plane conditions. They consists of active PZT layers with in-plane polarization and a microstructured composite layer characterized by a periodic microstructure where PZT fibers with an out of plane polarization are immersed in a polymeric matrix. The constitutive law of the composite layer at the macroscale has been determined by means of a multi-field asymptotic homogenization technique, recently developed for thermo-piezoelectric materials. Overall constitutive equations characterizing the behavior of the microstructured layer at the macroscale have been derived and the closed form of the overall constitutive tensors has been provided for the equivalent first-order (Cauchy) homogenized continuum. Deflection of unimorph and bimorph bender actuators has been investigated in relation to their geometrical features, exploiting the out of plane piezoelectric properties of the composite layer, which modify the stiffness of the entire bender. An accurate description of benders behavior at the structural length scale is of fundamental importance in order to design devices with high performances. In this regard, the influence of the microstructure on the global response of the actuator is investigated in the present study in order to understand how the composite material can be tailored to meet specific design requirements.

physics.app-ph↗

arXiv2018

Bloch wave filtering in tetrachiral materials via mechanical tuning

The periodic cellular topology characterizing the microscale structure of a heterogeneous material may allow the finest functional customization of its acoustic dispersion properties. The paper addresses the free propagation of elastic waves in micro-structured cellular materials. Focus is on the alternative formulations suited to describe the wave propagation in the material, according to the classic canons of solid or structural mechanics. Adopting the centrosymmetric tetrachiral microstructure as prototypical periodic cell, the frequency dispersion spectrum resulting from a synthetic lagrangian beam-lattice formulation is compared with its counterpart derived from different continuous models (high-fidelity first-order heterogeneous and equivalent homogenized micropolar continuum). Asymptotic perturbation-based approximations and numerical spectral solutions are cross-validated. Adopting the low-frequency band gaps of the material band structures as functional targets, parametric analyses are carried out to highlight the descriptive limits of the synthetic models and to explore the enlarged parameter space described by high-fidelity models. The final tuning of the mechanical properties of the cellular microstructure is employed to successfully verify the wave filtering functionality of the tetrachiral material.

physics.app-ph↗

arXiv2018

Multi-parametric sensitivity analysis of the band structure for tetrachiral acoustic metamaterials

Tetrachiral materials are characterized by a cellular microstructure made by a periodic pattern of stiff rings and flexible ligaments. Their mechanical behaviour can be described by a planar lattice of rigid massive bodies and elastic massless beams. The periodic cell dynamics is governed by a monoatomic structural model, conveniently reduced to the only active degrees-of-freedom. The paper presents an explicit parametric description of the band structure governing the free propagation of elastic waves. By virtue of multiparametric perturbation techniques, sensitivity analyses are performed to achieve analytical asymptotic approximation of the dispersion functions. The parametric conditions for the existence of full band gaps in the low-frequency range are established. Furthermore, the band gap amplitude is analytically assessed in the admissible parameter range. In tetrachiral acoustic metamaterials, stop bands can be opened by the introduction of intra-ring resonators. Perturbation methods can efficiently deal with the consequent enlargement of the mechanical parameter space. Indeed high-accuracy parametric approximations are achieved for the band structure, enriched by the new optical branches related to the resonator frequencies. In particular, target stop bands in the metamaterial spectrum are analytically designed through the asymptotic solution of inverse spectral problems.

cond-mat.mtrl-sci↗

arXiv2017

Auxetic behavior and acoustic properties of microstructured piezoelectric strain sensors

The use of multifunctional composite materials adopting piezo-electric periodic cellular lattice structures with auxetic elastic behavior is a recent and promising solution in the design of piezoelectric sensors. In the present work, periodic anti-tetrachiral auxetic lattice structures, characterized by different geometries, are taken into account and the mechanical and piezoelectrical response are investigated. The equivalent piezoelectric properties are obtained adopting a first order computational homogenization approach, generalized to the case of electro-mechanical coupling, and various polarization directions are adopted. Two examples of in-plane and out-of-plane strain sensors are proposed as design concepts. Moreover, a piezo-elasto-dynamic dispersion analysis adopting the Floquet-Bloch decomposition is performed. The acoustic behavior of the periodic piezoelectric material with auxetic topology is studied and possible band gaps are detected.

cond-mat.mtrl-sci↗

arXiv2017

Multi-field asymptotic homogenization of thermo-piezoelectric materials with periodic microstructure

This study proposes a multi-field asymptotic homogenization for the analysis of thermo-piezoelectric materials with periodic microstructures. The effect of the microstructural heterogeneity is taken into account by means of periodic perturbation functions, which derive from the solution of non homogeneous recursive cell problems defined over the unit periodic cell. A strong coupling is present between the micro displacement field and the micro electric potential field, since the mechanical and the electric problems are fully coupled in the asymptotically expanded microscale field equations. The micro displacement, the electric potential, and the relative temperature fields have been related to the macroscopic quantities and to their gradients in the derived down-scaling relations. Average field equations of infinite order have been obtained and the closed form of the overall constitutive tensors has been determined for the equivalent first-order homogenized continuum. A formal solution of such equations has been derived by means of an asymptotic expansion of the macro fields. The accuracy of the proposed formulation is assessed in relation to illustrative examples of a bi-material periodic microstructure subjected to harmonic body forces, free charge densities, and heat sources, whose periodicity is much greater than the characteristic microstructural size. The good agreement obtained between the solution of the homogenized model and the finite element solution of the original heterogeneous material problem confirms the validity of the proposed formulation.

cond-mat.mtrl-sci↗

arXiv2016

Wave propagation in non-centrosymmetric beam-lattices with lumped masses: discrete and micropolar modelling

The in-plane acoustic behavior of non-centrosymmetric lattices having nodes endowed with mass and gyroscopic inertia and connected by massless ligaments with asymmetric elastic properties has been analysed through a discrete model and a continuum micropolar model. In the first case the propagation of harmonic waves and the dispersion functions have been obtained by the discrete Floquet-Bloch approach. It is shown that the optical branch departs from a critical point with vanishing group velocity and for the considered cases this branch is decreasing for increasing the norm of the wave vector from the long wave limit. A micropolar continuum model, useful to approximate the discrete model, has been derived through a continualization method based on a down-scaling law based on a second-order Taylor expansion of the generalized macro-displacement field. It is worth noting that the second order elasticity tensor coupling curvatures and micro-couples turns out to be negative defined also in the general case of non-centrosymmetric lattice. The eigenvalue problem governing the harmonic propagation in the micropolar non-centrosymmetric continuum results in general characterized by a hermitian full matrix that is exact up to the second order in the wave vector. Examples concerning square and equilateral triangular lattices and their acoustic properties have been analysed from both the exact Lagrangian model (within the assumed hypotheses) and the micropolar approximate model. Finally, in consideration of the negative definiteness of the second order elastic tensor of the micropolar model, the loss of strong hyperbolicity of the equation of motion has been investigated.

cond-mat.mtrl-sci↗

arXiv2016

Dispersive wave propagation in two-dimensional rigid periodic blocky materials with elastic interfaces

Dispersive waves in two-dimensional blocky materials with periodic microstructure made up of equal rigid units having polygonal centro-symmetric shape with mass and gyroscopic inertia, connected each other through homogeneous linear interfaces, have been analysed. The acoustic behavior of the resulting discrete Lagrangian model has been obtained through a Floquet-Bloch approach. From the resulting eigenproblem derived by the Euler-Lagrange equations for harmonic wave propagation, two acoustic branches and an optical branch are obtained in the frequency spectrum. A micropolar continuum model to approximate the Lagrangian model has been derived based on a second-order Taylor expansion of the generalized macro-displacement field. The constitutive equations of the equivalent micropolar continuum have been obtained, with the peculiarity that the positive definiteness of the second-order symmetric tensor associated to the curvature vector is not guaranteed and depends both on the ratio between the local tangent and normal stiffness and on the block shape. The same results has been obtained through an extended Hamiltonian derivation of the equations of motion for the equivalent continuum that is related to the Hill-Mandel macro homogeneity condition. Moreover, it is shown that the hermitian matrix governing the eigenproblem of harmonic wave propagation in the micropolar model is exact up to the second order in the norm of the wave vector with respect to the same matrix from the discrete model. To appreciate the acoustic behavior of some relevant blocky materials and to understand the reliability and the validity limits of the micropolar continuum model, some blocky patterns have been analysed: rhombic and hexagonal assemblages and running bond masonry.

cond-mat.mtrl-sci↗

arXiv2016

Optimal design of low-frequency band gaps in anti-tetrachiral lattice meta-materials

The elastic wave propagation is investigated in the beam lattice material characterized by a square periodic cell with anti-tetrachiral microstructure. With reference to the Floquet-Bloch spectrum, focus is made on the band structure enrichments and modifications which can be achieved by equipping the cellular microstructure with tunable local resonators. By virtue of its composite mechanical nature, the so-built inertial meta-material gains enhanced capacities of passive frequency-band filtering. Indeed the number, placement and properties of the inertial resonators can be designed to open, shift and enlarge the band gaps between one or more pairs of consecutive branches in the frequency spectrum. In order to improve the meta-material performance, a nonlinear optimization problem is formulated. The maximum of the largest band gap amplitudes in the low-frequency range is selected as suited objective function. Proper inequality constraints are introduced to restrict the optimal solutions within a compact set of mechanical and geometric parameters, including only physically realistic properties of both the lattice and resonators. The optimization problems related to full and partial band gaps are solved independently, by means of a globally convergent version of the numerical method of moving asymptotes, combined with a quasi-Monte Carlo multi-start technique. The optimal solutions are discussed and compared from the qualitative and quantitative viewpoints, bringing to light the limits and potential of the meta-material performance. The clearest trends emerging from the numerical analyses are pointed out and interpreted from the physical viewpoint. Finally, some specific recommendations about the microstructural design of the meta-material are synthesized.

cond-mat.mtrl-sci↗

arXiv2016

Design of acoustic metamaterials through nonlinear programming

The dispersive wave propagation in a periodic metamaterial with tetrachiral topology and inertial local resonators is investigated. The Floquet-Bloch spectrum of the metamaterial is compared with that of the tetrachiral beam lattice material without resonators. The resonators can be designed to open and shift frequency band gaps, that is, spectrum intervals in which harmonic waves do not propagate. Therefore, an optimal passive control of the frequency band structure can be pursued in the metamaterial. To this aim, a suitable constrained nonlinear optimization problem on a compact set of admissible geometrical and mechanical parameters is stated. According to functional requirements, the particular set of parameters which determines the largest low-frequency band gap between a pair of consecutive branches of the Floquet-Bloch spectrum is obtained. The optimization problem is successfully solved by means of a version of the method of moving asymptotes, combined with a quasi-Monte Carlo multi-start technique.

cond-mat.mtrl-sci↗

arXiv2015

High-frequency parametric approximation of the Floquet-Bloch spectrum for anti-tetrachiral materials

The engineered class of anti-tetrachiral cellular materials is phenomenologically characterized by a strong auxeticity of the elastic macroscopic response. The auxetic behavior, accompanied by a marked anisotropy, is activated by rolling-up deformation mechanisms developed by the periodic pattern of stiff rings and flexible ligaments realizing the material micro-structure. In the absence of a soft matrix, a linear beam lattice model is formulated to describe the free dynamic response of the periodic cell. After a static condensation of the passive degrees-of-freedom, a general procedure is applied to impose the quasi-periodicity conditions of free wave propagation in the low-dimension space of the active degrees-of-freedom. The effects of different mechanical parameters on the band structure are analyzed by comparing the exact dispersion curves with explicit, although approximate, dispersion functions, obtained from asymptotic perturbation solutions of the eigenproblem rising up from the Floquet-Bloch theory. A general scheme for the formulation and solution of the perturbation equations is outlined, for the desired approximation order and depending on the dimension of the perturbation vector. A satisfying approximation accuracy is achieved for low-order asymptotic solutions in wide regions of the parameter space. The explicit dependence of the dispersion functions on the main parameters, including the cell aspect ratio, the ligament slenderness and the ring density, allows the in-depth discussion and - in a design perspective - the fine assessment of the spectrum properties, including specific features like wave velocities and band gap amplitudes.

cond-mat.mtrl-sci↗

arXiv2015

Optimal design of auxetic hexachiral metamaterials with local resonators

A parametric beam lattice model is formulated to analyse the propagation properties of elastic in-plane waves in an auxetic material based on a hexachiral topology of the periodic cell, equipped with inertial local resonators. The Floquet-Bloch boundary conditions are imposed on a reduced order linear model in the only dynamically active degrees-offreedom. Since the resonators can be designed to open and shift band gaps, an optimal design, focused on the largest possible gap in the low-frequency range, is achieved by solving a maximization problem in the bounded space of the significant geometrical and mechanical parameters. A local optimized solution, for a the lowest pair of consecutive dispersion curves, is found by employing the globally convergent version of the Method of Moving asymptotes, combined with Monte Carlo and quasi-Monte Carlo multi-start techniques.

cond-mat.mtrl-sci↗

arXiv2015

Simplified modelling of chiral lattice materials with local resonators

A simplified model of periodic chiral beam-lattices containing local resonators has been formulated to obtain a better understanding of the influence of the chirality and of the dynamic characteristics of the local resonators on the acoustic behavior. The simplified beam-lattices is made up of a periodic array of rigid heavy rings, each one connected to the others through elastic slender massless ligaments and containing an internal resonator made of a rigid disk in a soft elastic annulus. The band structure and the occurrence of low frequency band-gaps are analysed through a discrete Lagrangian model. For both the hexa- and the tetrachiral lattice, two acoustic modes and four optical modes are identified and the influence of the dynamic characteristics of the resonator on those branches is analyzed together with some properties of the band structure. By approximating the generalized displacements of the rings of the discrete Lagrangian model as a continuum field and through an application of the generalized macro-homogeneity condition, a generalized micropolar equivalent continuum has been derived, together with the overall equation of motion and the constitutive equation given in closed form. The validity limits of the micropolar model with respect to the dispersion functions are assessed by comparing the dispersion curves of this model in the irreducible Brillouin domain with those obtained by the discrete model, which are exact within the assumptions of the proposed simplified model.

cond-mat.mtrl-sci↗

arXiv2014

Homogenization of periodic hexa- and tetrachiral cellular solids

The homogenization of auxetic cellular solids having periodic hexachiral and tetrachiral microstructure is dealt with two different techniques. The first approach is based on the representation of the cellular solid as a beam-lattice to be homogenized as a micropolar continuum. The second approach is developed to analyse periodic cells conceived as a two-dimensional domain consisting of deformable portions such as the ring, the ligaments and possibly an embedded matrix internally to these. This approach is based on a second displacement gradient computational homogenization proposed by the Authors (Bacigalupo and Gambarotta, 2010). The elastic moduli obtained by the micropolar homogenization are expressed in analytical form from which it appears explicitly their dependence on the parameter of chirality, which is the angle of inclination of the ligaments with respect to the grid of lines connecting the centers of the rings. For hexachiral cells, the solution of Liu et al., 2012, is found, showing the auxetic property of the lattice together with the elastic coupling modulus between the normal and the asymmetric strains; 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. The properties of the equivalent micropolar continuum are qualitatively detected also in the equivalent second-gradient continuum.

cond-mat.mtrl-sci↗

arXiv2014

Second-order homogenization of periodic materials based on asymptotic approximation of the strain energy: formulation and validity limits

In this paper a second-order homogenization approach for periodic material is derived from an appropriate representation of the down-scaling that correlates the microdisplacement 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 e; the obtained overall elastic moduli result to be not affected by the choice of periodic cell. The coupling between the macro- and microstress tensor in the periodic cell is deduced from an application of the generalised macrohomogeneity 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 is obtained through an approximation of the macrodisplacement 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.

cond-mat.mtrl-sci↗