Luigi Gambarotta

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.

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

Abstract 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 with each other through homogeneous linear interfaces, have been analyzed. 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 have 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 analyzed: rhombic and hexagonal assemblages and running bond masonry. From the results obtained in the examples, the obtained micropolar model turns out to be particularly accurate to describe dispersive functions for wavelengths greater than 3-4 times the characteristic dimension of the block. Finally, in consideration that the positive definiteness of the second order elastic tensor of the micropolar model is not guaranteed, the hyperbolicity of the equation of motion has been investigated by considering the Legendre–Hadamard ellipticity conditions requiring real values for the wave velocity.

Effects of Layered Accretion on the Mechanics of Masonry Structures

Masonry constructions are built up in successive layers of bricks or blocks that may have considerable effect on the deformation and equilibrium of these structures when they are statically indeterminate and when gravity loads are predominant. This problem is analyzed by referring to a thick arch that reaches its final shape by means of a continuous deposition of heavy brick layers in stress free condition. The brick/block units of the accreting layer are assumed to have a negligible size in comparison to the structural size and the resulting continuous deposition is described taking into account their possible sliding on the current extrados at the instant of deposition. The kinematics of the growing body is described by the superposition of the displacement resulting from the continuing addition of heavy layers to the initial displacement at the considered point when it is attached to the current extrados, i.e., on the accreting layer. The two corresponding strain tensor fields do not satisfy the equations of compatibility, while the total strain field results to be compatible. The stress field is the cumulative effect of the incremental stress induced by the weight of the added layers during the growing process with residual stresses, which are shown to be independent of the initial strain field. Two examples are analyzed to show the effects of the growing process on the stress field and the properties of the strain field are discussed. The first example concerns a masonry wall supported at periodic points while the second example concerns a segmental multileaf thick arch in which the growing process begins from a thin arch resting on its own weight. In both cases a remarkable increase of the stress field is observed in comparison to the solution where the gravity loads are applied on the final domain.

A micropolar model for the analysis of dispersive waves in chiral mass-in-mass lattices

The possibility of obtaining band gap structures in chiral auxetic lattices is here considered and applied to the case of inertial locally resonant structures. These periodic materials are modelled as beam-lattices made up of a periodic array of rigid rings, each one connected to the others through elastic slender ligaments. To obtain low-frequency stop bands, elastic circular resonating inclusions made up of masses located inside the rings and connected to them through an elastic surrounding interface are considered and modeled. The equations of motion are obtained for an equivalent homogenized micropolar continuum and the overall elastic moduli and the inertia terms are given for both the hexachiral and the tetrachiral lattice. The constitutive equation of the beam lattice given by the Authors [15] are then applied and a system of six equations of motion is obtained. The propagation of plane waves travelling along the direction of the lines connecting the ring centres of the lattice is analysed and the secular equation is derived, from which the dispersive functions may be obtained.

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, suitable constrained nonlinear optimization problems on compact sets of admissible geometrical and mechanical parameters are stated. According to functional requirements, sets of parameters which determine the largest low-frequency band gap between selected pairs of consecutive branches of the Floquet-Bloch spectrum are soughted for numerically. The various optimization problems are successfully solved by means of a version of the method of moving asymptotes, combined with a quasi-Monte Carlo multi-start technique.

A simplified evaluation of the influence of the bond pattern on the brickwork limit strength

The influence of the bond pattern on the in-plane limit strength of masonry is analyzed through a simplified procedure based on the application of the safe theorem of limit analysis to the unit cell that generates the whole masonry by periodic repetition. The limit strength domains of running bond, English bond and herringbone bond masonry are obtained with different orientations of the mortar bed joints with respect to the principal directions of the average stress. The effects of different brick geometries are analyzed and comparisons between strength properties of different masonry patterns are made.

History-Based Assessment of the Dome of the Basilica of S. Maria of Carignano in Genoa

This article discusses the assessment of the dome of the Basilica of S. Maria of Carignano in Genoa, designed by Galeazzo Alessi in the 16th century. The access to the private archive of the Sauli Family allowed a detailed history of the Basilica to be reconstructed clarifying several crucial aspects of the structure: the actual depth of the foundation system, the materials of the load bearing structure, mainly of the dome and of the drum, and the actual building sequence. The restoration works (steel hoop) carried out in the past decades, discovered by means of the historical research, are related to the severe crack pattern of the dome, which today necessitates a detailed insight into its mechanical response. It is showed that historical information is the key for understanding several issues that a “technological” engineering approach would leave unsolved. In addition, historical data are used to setup a non-linear incremental finite element model (FEM) procedure, referred to a slice (a ratio of 1/16) of the structure due to symmetry conditions, which is able, in spite of its reduced computational demand, to show that the dome response is strictly related also to the drum features, thus explaining cracking and the role of hooping stresses (and ties) in the global equilibrium. Some issues for the retrofitting of dome-like structures are discussed.

On the Statics of the Dome of the Basilica of S. Maria Assunta in Carignano, Genoa

The paper deals with the dome of the Basilica of S. Maria Assunta in Carignano in Genoa, designed by Galeazzo Alessi and built in the sixteenth century, for which meridian cracking, rather common in masonry domes, requires the assessment of the dome. In order to set a general procedure for the assessment this structures, limit analysis approaches are here discussed and compared. On the basis of classic limit analysis, local (dome only) and global (dome-drum system) collapse mechanisms are considered considering the different behaviour of several structural elements (lantern, shells of the dome, drum, colonnade). A static (safe theorem) and a kinematic approach are applied to the structure by means of equilibrium limit conditions and kinematically admissible collapse mechanisms. Comparisons between the obtained results are carried out so as to: (i) discuss a general approach to the assessment of dome-drum systems based on both numerical tools and standard limit analyses approaches; (ii) provide a first glance in the assessment of the dome.

Micro-polar and second order homogenization of periodic masonry

Micro-polar and second order homogenization procedures for periodic elastic masonry are implemented to include geometric and material length scales in the constitutive equation. By the solution of the RVE equilibrium problems with properly prescribed boundary conditions the orthotropic elastic moduli of the higher order continua are obtained on the basis of an enhanced Hill–Mandel condition. A shear layer problem is analysed and the results from the heterogeneous models are compared with those ones obtained by the homogenization procedures; the second-order homogenization appears to provide better results in comparison to the micro-polar homogenization.