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This article presents a multifield asymptotic homogenization scheme for the analysis of Bloch wave propagation in non-standard thermoelastic periodic materials, leveraging on the Green-Linsdsay theory that accounts for two relaxation times. The procedure involves several steps. Firstly, an asymptotic expansion of the micro-fields is performed, considering the characteristic size of the microstructure. By utilizing the derived microscale field equations and asymptotic expansions, a series of recursive differential problems are solved within the repetitive unit cell Q. These problems are then expressed in terms of perturbation functions, which incorporate the material's geometric, physical, and mechanical properties, as well as the microstructural heterogeneities. The down-scaling relation, which connects the microscopic and macroscopic fields along with their gradients through the perturbation functions, is then established in a consistent manner. Subsequently, the average field equations of infinite order are obtained by substituting the down-scaling relation into the microscale field equations. To solve these average field equations, an asymptotic expansion of the macroscopic fields is performed based on the microstructural size, resulting in a sequence of macroscopic recursive problems. To illustrate the methodology, a bi-phase layered material is introduced as an example. The dispersion curves obtained from the non-local homogenization scheme are compared with those obtained from the Floquet-Bloch theory. This analysis helps validate the effectiveness and accuracy of the proposed approach in predicting the wave propagation behavior in the considered non-standard thermoelastic periodic materials.

The dynamic behaviour of periodic thermodiffusive multi-layered media excited by harmonic oscillations is studied. In the framework of linear thermodiffusive elasticity, periodic laminates, whose elementary cell is composed by an arbitrary number of layers, are considered. The generalized Floquet-Bloch conditions are imposed, and the universal dispersion relation of the composite is obtained by means of an approach based on the formal solution for a single layer together with the transfer matrix method. The eigenvalue problem associated with the dispersion equation is solved by means of an analytical procedure based on the symplecticity properties of the transfer matrix to which corresponds a palindromic characteristic polynomial, and the frequency band structure associated to wave propagating inside the medium are finally derived. The proposed approach is tested through illustrative examples where thermodiffusive multilayered structures of interest for renewable energy devices fabrication are analyzed. The effects of thermodiffusion coupling on both the propagation and attenuation of Bloch waves in these systems are investigated in detail.

A multiscale asymptotic homogenization method for periodic microstructured materials in presence of thermoelasticity with periodic spatially dependent one relaxation time is introduced. The asymptotic expansions of the micro-displacement and the micro-temperature fields are rewritten on the transformed Laplace space and expressed as power series of the microstructural length scale, leading to a set of recursive differential problems over the periodic unit cell. The solution of such cell problems leads to the perturbation functions. Up-scaling and down-scaling relations are then defined, and the latter allow expressing the microscopic fields in terms of the macroscopic ones and their gradients. Average field equations of infinite order are also derived. The efficiency of the proposed technique was tested in relation to a bi dimensional orthotropic layered body with orthotropy axis parallel to the direction of the layers, where the mechanical and temperature constitutive properties were well stabilised. The dispersion curves of the homogenized medium, truncated at the first order are compared with the dispersion curves of the heterogeneous continuum obtained by the Floquet-Bloch theory. The results obtained with the two different approaches show a very good agreement.

The main purpose of the present paper is to solve the thermodynamic inconsistencies that result when deriving equivalent micropolar models of periodic beam-lattice materials through standard continualization schemes. In fact, this technique identifies higher order micropolar continua characterized by non-positive defined elastic potential energy. Despite this, such models are capable of accurately simulating the optical branches of the discrete Lagrangian model, a property lacking in the thermodynamically consistent standard micropolar continuum. To overcome these thermodynamic inconsistencies while preserving good simulations of the frequency band structure, a dynamic high-frequency consistent continualization is proposed. This continualization scheme is based on a first order regularization approach coupled with a suitable transformation of the difference equation of motion of the discrete Lagrangian system into pseudo-differential equations. A formal Taylor expansion of the pseudo-differential operators allows to obtain differential field equations at various orders according to the continualization order. Thermodynamically consistent higher order micropolar continua with non-local elasticity and inertia are obtained. Finally, the convergence of the frequency band structure of the higher order micropolar models to that of the discrete Lagrangian system is shown as the continualization order increases.

It is well known that the classical energetically consistent micropolar model has limits in simulating the frequency band structure of packed granular materials (see Merkel et al., 2011). It is here shown that if a standard continualization of the difference equation of motion of the discrete model is carried out, an equivalent micropolar continuum is obtained which is able to accurately simulate the optical branches of the discrete model. Nevertheless, this homogenized continuum presents non-positive defined elastic potential energy, a deficiency that limits the reliability of the model and implies instability phenomena (destabilizing effects) in the acoustic branches. This drawback is circumvented here through an high-frequency dynamic homogenization scheme which is based on an enhanced continualization of the discrete governing equations into pseudo-differential equations. Through a formal Taylor expansion of the pseudo-differential operators a higher order differential equation corresponding to the governing equation of a non-local continuum thermodynamically consistent are obtained. The resulting approach allows obtaining an equivalent micropolar continuum characterized by inertial non-locality. Moreover, higher order continua with non-local constitutive and inertial terms may be derived. The proposed continuum models are proved to be able to accurately describe both the static and dynamic behavior of the discrete granular model. Finally, the convergence to the response of the discrete system is shown when increasing the order of the higher order continuum.

In this short paper, a matrix perturbation bound on the eigenvalues found by principal component analysis is investigated, for the case in which the data matrix on which principal component analysis is performed is a convex combination of two data matrices. The application of the theoretical analysis to multi-objective optimization problems (e.g., those arising in the design of acoustic metamaterial filters) is briefly discussed, together with possible extensions.

A promising technique for the spectral design of acoustic metamaterials is based on the formulation of suitable constrained nonlinear optimization problems. Unfortunately, the straightforward application of classical gradient-based iterative optimization algorithms to the numerical solution of such problems is typically highly demanding, due to the complexity of the underlying physical models. Nevertheless, supervised machine learning techniques can reduce such a computational effort, e.g., by replacing the original objective functions of such optimization problems with more-easily computable approximations. In this framework, the present article describes the application of a related unsupervised machine learning technique, namely, principal component analysis, to approximate the gradient of the objective function of a band gap optimization problem for an acoustic metamaterial, with the aim of making the successive application of a gradient-based iterative optimization algorithm faster. Numerical results show the effectiveness of the proposed method.

Two novel chiral block lattice topologies are here conceived having interesting auxetic and acoustic behavior. The architectured chiral material is made up of a periodic repetition of square or hexagonal rigid and heavy blocks connected by linear elastic interfaces, whose chirality results from an equal rotation of the blocks with respect to the line connecting their centroids. The governing equation of the Lagrangian model is derived and a hermitian eigenproblem is formulated to obtain the frequency band structure. An equivalent micropolar continuum is analytically derived through a standard continualization approach in agreement with the procedure proposed by Bacigalupo and Gambarotta (2017) from which an approximation of the frequency spectrum is derived. Moreover, the overall elastic moduli of the equivalent Cauchy continuum are obtained in closed form via a proper condensation procedure. The parametric analysis involving the overall elastic moduli of the Cauchy equivalent continuum model and the frequency band structure is carried out to catch the influence of the chirality angle and of the ratio between the tangential and normal stiffness of the interface. Finally, it is shown how chirality and interface stiffness may affect strong auxeticity and how the equivalent micropolar model provides dispersion curves in excellent agreement with the current ones for a wide range of the wave vector magnitude.

The paper is focused on the dynamic homogenization of lattice-like materials with lumped mass at the nodes to obtain energetically consistent models providing accurate descriptions of the acoustic behavior of the discrete system. The equation of motion of the lattice is transformed according to a unitary approach aimed to identify equivalent non-local continuum models of integral-differential and gradient type, the latter obtained through standard or enhanced continualization. The bilateral Z-transform of the difference equation of motion is matched to the governing integral-differential equation of the equivalent continuum in the transformed Fourier space, which has the same frequency band structure as the Lagrangian one. Firstly, it is shown that the approximation of the kernels via Taylor polynomials leads to the differential field equations of higher order continua endowed with non-local constitutive terms. The field equations derived from such approach corresponds to the ones obtained through the so called standard continualization. However, the differential problem turns out to be ill-posed because the non-positive definiteness of the potential energy density of the higher order continuum. Energetically consistent equivalent continua have been identified through a proper mapping correlating the transformed macro-displacements in the Fourier space with a new auxiliary regularizing continuum macro-displacement field in the same space. Specifically, the mapping here introduced has zeros at the edge of the first Brillouin zone. The integral-differential governing equation and the corresponding differential one has been reformulated through an enhanced continualization that is characterized by energetically consistent differential equations. The constitutive and inertial kernels of the integral-differential equation exhibit polar singularities at the edge of the first Brillouin zone.

A multifield asymptotic homogenization technique for periodic thermo-diffusive elastic materials is provided in the present study. Field equations for the first-order equivalent medium are derived and overall constitutive tensors are obtained in closed form. These lasts depend upon the micro constitutive properties of the different phases composing the composite material and upon periodic perturbation functions, which allow taking into account the effects of microstructural heterogeneities. Perturbation functions are determined as solutions of recursive non homogeneous cell problems emanated from the substitution of asymptotic expansions of the micro fields in powers of the microstructural characteristic size into local balance equations. Average field equations of infinite order are also provided, whose formal solution can be obtained through asymptotic expansions of the macrofields. With the aim of investigating dispersion properties of waves propagating inside the medium, proper integral transforms are applied to governing field equations of the homogenized medium. A quadratic generalized eigenvalue problem is thus obtained, whose solution characterizes the complex valued frequency band structure of the first-order equivalent material. The validity of the proposed technique has been confirmed by the very good matching obtained between dispersion curves of the homogenized medium and the lowest frequency ones relative to the heterogeneous material. These lasts are computed from the resolution of a quadratic generalized eigenvalue problem over the periodic cell subjected to Floquet-Bloch boundary conditions. An illustrative benchmark is conducted referring to a Solid Oxide Fuel Cell (SOFC)-like material, whose microstructure can be modeled through the spatial tessellation of the domain with a periodic cell subjected to thermo-diffusive phenomena.

An innovative special class of tunable periodic metamaterials is designed, suitable for realising high-performance acoustic filters. The metamaterial is made up of a phononic crystal coupled to local resonators. Such local resonators consist of masses enclosed into piezoelectric rings, shunted by either dissipative or non-dissipative electrical circuit. By tuning the impedance/admittance of such electrical circuits, it is possible to fully adjust the constitutive properties of the shunting piezoelectric material. This feature paves the way for unconventional behaviours, well beyond the capabilities achievable with classical materials. It follows that the acoustic properties of the periodic metamaterial can be adaptively modified, in turn, opening new possibilities for the control of pass and stop bands. By exploiting a generalization of the Floquet-Bloch theory, the in-plane free wave propagation in the tunable metamaterial is investigated, by varying a certain tuning parameter, to show the efficiency of the proposed shunting piezoelectric system as a wave propagation control device. Particular attention is devoted to the determination of the in-plane constitutive equations of the shunting piezoelectric phase in the transformed Laplace space. Finally, broad design directions of tunable acoustic filters aiming to a changing performance requirement in real-time, is also provided., Comment: 36 pages, 16 figures

Recently, an increasing research effort has been dedicated to analyse the transmission and dispersion properties of periodic acoustic metamaterials, characterized by the presence of local resonators. Within this context, particular attention has been paid to the optimization of the amplitudes and center frequencies of selected stop and pass bands inside the Floquet-Bloch spectra of the acoustic metamaterials featured by a chiral or antichiral microstructure. Novel functional applications of such research are expected in the optimal parametric design of smart tunable mechanical filters and directional waveguides. The present paper deals with the maximization of the amplitude of low-frequency band gaps, by proposing suitable numerical techniques to solve the associated optimization problems. Specifically, the feasibility and effectiveness of Radial Basis Function networks and Quasi-Monte Carlo methods for the interpolation of the objective functions of such optimization problems are discussed, and their numerical application to a specific acoustic metamaterial with tetrachiral microstructure is presented. The discussion is motivated theoretically by the high computational effort often needed for an exact evaluation of the objective functions arising in band gap optimization problems, when iterative algorithms are used for their approximate solution. By replacing such functions with suitable surrogate objective functions constructed applying machine-learning techniques, well performing suboptimal solutions can be obtained with a smaller computational effort. Numerical results demonstrate the effective potential of the proposed approach. Current directions of research involving the use of additional machine-learning techniques are also presented.

In the present work, the evolution of damage in periodic composite materials is investigated through a novel finite element-based multiscale computational approach. The methodology is developed by means of the original combination of homogenization methods with the phase field approach of fracture. This last is applied at the macroscale level on the equivalent homogeneous continuum, whose constitutive properties are obtained in closed form via a two-scale asymptotic homogenization scheme. The formulation allows considering different assumptions on the evolution of damage at the microscale (e.g., damage in the matrix and not in the inclusion/fiber), as well as the role played by the microstructural topology. Numerical results show that the proposed formulation leads to an apparent tensile strength and a post-peak branch of unnotched and notched specimens dependent not only on the internal length scale of the phase field approach, as for homogeneous materials, but also on the inclusion volumetric content and its shape. Down-scaling relations allow the full reconstruction of the microscopic fields at any point of the macroscopic model, as a simple post-processing operation.

Microstructured honeycomb materials may exhibit exotic, extreme and tailorable mechanical properties, suited for innovative technological applications in a variety of modern engineering fields. The paper is focused on analysing the directional auxeticity of tetrachiral materials, through analytical, numerical and experimental methods. Theoretical predictions about the global elastic properties have been successfully validated by performing tensile laboratory tests on tetrachiral samples, realized with high precision 3D printing technologies. Inspired by the kinematic behaviour of the tetrachiral material, a newly-design bi-layered topology, referred to as bi-tetrachiral material, has been theoretically conceived and mechanically modelled. The novel topology virtuously exploits the mutual collaboration between two tetrachiral layers with opposite chiralities. The bi-tetrachiral material has been verified to outperform the tetrachiral material in terms of global Young modulus and, as major achievement, to exhibit a remarkable auxetic behaviour. Specifically, experimental results, confirmed by parametric analytical and computational analyses, have highlighted the effective possibility to attain strongly negative Poisson ratios, identified as a peculiar global elastic property of the novel bi-layered topology.

In the framework of energy scavenging for applications in flexible/strechable electronics, hybrid piezoelectric nanogenerators, made up with Zinc oxyde nanorods, embedded in a polymeric matrix, and growth on a flexible polymeric support, are investigated. The ZnO nanorods are arranged in clusters, forming nearly regular distributions, so that periodic topologies can be realistically assumed. Focus is on a dynamic multi-field asymptotic homogenization approach, proposed to grasp the overall constitutive behaviour of such complex microstrutcures. A set of applications, both in static and dynamic regime, is proposed to explore different design paradigms, related to nanogenerators based on three working principles. Both extension and bending nanogenerators are, indeed, analysed, considering either extension along the nanorods axis, or orthogonally to it. The study of the wave propagation is, also, exploited to comprehend the main features of such piezoelectric devices in the dynamic regime.

The acoustic dispersion properties of monodimensional waveguide filters can be assessed by means of the simple prototypical mechanical system made of an infinite stack of periodic massive blocks, connected to each other by elastic joints. The linear undamped dynamics of the periodic cell is governed by a two degree-of-freedom Lagrangian model. The eigenproblem governing the free propagation of shear and moment waves is solved analytically and the two dispersion relations are obtained in a suited closed form fashion. Therefore, the pass and stop bandwidths are conveniently determined in the minimal space of the independent mechanical parameters. Stop bands in the ultra-low frequency range are achieved by coupling the stacked material with an elastic half-space modelled as a Winkler support. A convenient fine approximation of the dispersion relations is pursued by formulating homogenised micropolar continuum models. An enhanced continualization approach, employing a proper Maclaurin approximation of pseudo-differential operators, is adopted to successfully approximate the acoustic and optical branches of the dispersion spectrum of the Lagrangian models, both in the absence and in the presence of the elastic support.

This work deals with the micromechanical study of periodic thermo-diffusive elastic multi-layered materials, which are of interest for the fabrication of solid oxide fuel cells (SOFCs). The focus is on the dynamic regime, that is investigating the dispersive wave propagation within the periodic material. In this framework, a generalization of the Floquet-Bloch theory is adopted, able to determine the complex band structure of such materials. The infinite algebraic linear system, obtained by exploiting both bilateral Laplace transform and Fourier transform, is replaced by its finite counterpart, resulting from a proper truncation at a finite number of considered equations. A regularization technique is herein useful to get rid of the Gibbs phenomenon. The solution of the problem is, finally, found in terms of complex angular frequencies, corresponding to a finite sequence of eigenvalue problems for given values of the wave vector. The paper is complemented by numerical examples taking into account thermo-mechanical coupling. The overall behaviour is found to be strongly influenced by the interaction between thermal and mechanical phenomena.

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.