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A computational homogenization framework is presented to study the dynamics of locally resonant acoustic metamaterial structures. Modelling the resonant units at the microscale as representative volume elements and building on well-established scale transition relations, the framework brings as a main novelty a reduced-order macroscopic homogenized continuum whose governing equations involve no additional variables to describe the microscale dynamics unlike micromorphic homogenized continua obtained by alternative computational homogenization approaches. This model-order reduction is obtained by formulating the governing equations of the micro- and macroscale problems in the frequency domain, introducing a finite-element discretization of the two problems and performing an exact dynamic condensation of all the degrees of freedom at the microscale. An appropriate inverse Fourier transform approach is implemented on the frequency-domain equations to capture transient dynamics as well; notably, the implementation involves the Exponential Window Method, here applied for the first time to calculate the time-domain response of undamped locally resonant acoustic metamaterial structures. The framework may handle arbitrary geometries of micro- and macro-structures, any transient excitations and any boundary conditions on the macroscopic domain. [ABSTRACT FROM AUTHOR]

This paper presents a novel locally resonant metamaterial structure and proposes a method to analyze its elastic wave dispersion properties. The structure is conceived as a multiple beam system with parallel beams transversely interconnected by periodic arrays of small resonators, each consisting of a spring-mass-spring subsystem in parallel with a couple of springs. Inserting the resonators between the beams, instead of attaching them as external appendages like in some alternative examples of locally resonant beams in the literature, makes the structure appealing for practical realization and use; furthermore, hosting several arrays of resonators gives the possibility to open multiple band gaps. The proposed method is a homogenization approach for flexural wave dispersion analysis, which removes the equations for the resonators from the set governing the dynamics of the system and reverts the original structure to an equivalent one featuring a tri-diagonal effective mass matrix with frequency dependent terms. The advantage of the homogenization approach is twofold: (1) it demonstrates that the band gaps arise in the frequency ranges where the effective mass matrix is negative definite, generalizing the well-established concept of band gaps attributable to negative mass effects in single locally resonant beams; (2) it provides dispersion curves and band gaps very efficiently, and the band gaps are identified upon calculating the eigenvalues of the tri-diagonal effective mass matrix. Analytical expressions of the band gap edges are obtained for the baseline case of a double beam system, to be readily used for design. Additionally, the exact transfer matrix method in conjunction with the Bloch theorem is formulated as alternative to the homogenization approach for wave dispersion analysis. Finally, the proposed concept of locally resonant structure and pertinent homogenization approach are validated by calculating the transmittance properties of the corresponding finite structure, via the standard finite element method. [ABSTRACT FROM AUTHOR]

Remarkably, the fractional-order homogenization approach proves capable of representing the dynamic behaviour of a bi-material periodic beam within the first few frequency band gaps, which is a strength of the proposed method over classical homogenization techniques whose range of validity is limited to the low-frequency regime. The authors assume the homogenized beam is governed by a fractional-order equation analogue of the Euler-Bernoulli beam equation, where fractional-order kinematic relations involving a space-fractional Riesz-Caputo derivative are defined over a nonlocal horizon. Yang et al. [[13]] propose the fractal scaling-law vector calculus as a connection between fractal geometry and vector calculus. The model involves a new state-dependent stress-dilatancy equation, developed from a fractional differentiation of an elliptic yielding surface; in this context, the dilatancy ratio is determined by the current stress state and the distance from current to critical state, via the fractional order and the shape factor of the yielding surface. [Extracted from the article]

The paper addresses the frequency response of beams in presence of open cracks with interval parameters. On adopting the standard Euler–Bernoulli beam theory, every crack is modelled as a linearly-elastic rotational spring whose stiffness and position are treated as uncertain-but-bounded parameters. A two-step method is proposed to calculate the bounds of all response variables. First, the sensitivity functions of the response are calculated as every uncertain parameter varies within the respective interval. Next, the bounds of the response are computed by either a sensitivity-based method or a global optimization technique, the former if the response is monotonic with respect to all uncertain parameters and the latter if the response is non-monotonic with respect to even one parameter only. The method relies on analytical forms for all response variables and the associated sensitivity functions. The applications focus on the frequency response of multi-cracked beams equipped with tuned mass dampers, showing potential and accuracy of the method. [ABSTRACT FROM AUTHOR]

This work deals with viscoelastic constitutive models involving variable-order fractional operators. There exist two main fractional models in the literature representing the stress-strain relation of viscoelastic materials with time-varying mechanical properties. Here, their features are analyzed and the physical assumptions involved are critically discussed. Specifically, it is shown that only one of these fractional models seems to be actually meaningful in the context of viscoelasticity. Then, a novel formulation is discussed, still in the context of variable-order fractional calculus, to effectively compute the strain response of a viscoelastic material with time-dependent mechanical properties due to any stress input. The proposed formulation exhibits a clear physical meaning and is proved to rely on a consistent application of the Boltzmann superposition principle to a fictitious system that is considered, in some sense, equivalent to the original viscoelastic material under study. The main novelty of the paper is to show that the proposed formulation is strictly related to the meaningful fractional model existing in the literature to which, therefore, a sound mechanical meaning can be given. [ABSTRACT FROM AUTHOR]

In this paper, an improved version of the analog equation method (AEM) is proposed, ideally suitable for non-linear dynamic analysis of time-fractional beams with discontinuities. Various sources of non-linearity will be considered as well as different discontinuities, such as those associated with external point supports (shear force discontinuity) and local flexural flexibility (rotation discontinuity). The main idea of the proposed approach is to reformulate the classical AEM by considering an analog equation with an unknown space-time dependent fictitious load and a spatial generalised operator, which includes the classical 4th-order differential operator and generalised functions modelling the discontinuities along the beam. Consistently, the solution to the analog equation is represented by an integral form involving appropriate Green's functions of the generalised operator. As in the classical AEM formulation, the integral representation of the solution is then approximated dividing the beam in finite elements and considering a constant value of the fictitious load within every beam element. Substituting the so-built approximate solution in the original equation of motion yields a system of fractional differential equations governing the discrete values of the fictitious load, solved by employing a Newmark integration scheme in conjunction with a G-1 algorithm to account for the fractional-derivative memory effects. Computational efficiency and accuracy will be demonstrated against the classical AEM formulation. [ABSTRACT FROM AUTHOR]

The dynamics of beams equipped with tuned mass dampers is of considerable interest in engineering applications. Here, the purpose is to introduce a comprehensive framework to address the stochastic response of the system under stationary and non-stationary loads, considering inertia effects along the spring of every tuned mass damper applied to the beam. For this, the key step is to show that a tuned mass damper with spring inertia effects can be reverted to an equivalent external support, whose reaction force on the beam depends only on the deflection of the attachment point. On this basis, a generalized function approach provides closed analytical expressions for frequency and impulse response functions of the system. The expressions can be used for a straightforward calculation of the stochastic response, for any number of tuned mass dampers. Numerical results show that spring inertia effects may play an important role in applications, affecting considerably the system response. [ABSTRACT FROM AUTHOR]

A novel statistical linearization technique is developed for computing stationary response statistics of randomly excited coupled bending-torsional beams resting on non-linear elastic supports. The key point of the proposed technique consists in representing the non-linear coupled response in terms of constrained linear modes. The resulting set of non-linear equations governing the modal amplitudes is then replaced by an equivalent linear one via a classical statistical error minimization procedure, which provides algebraic non-linear equations for the second-order statistics of the beam response, readily solved by a simple iterative scheme. Data from Monte Carlo simulations, generated by a pertinent boundary integral method in conjunction with a Newmark numerical integration scheme, are used as benchmark solutions to check accuracy and reliability of the proposed statistical linearization technique. [ABSTRACT FROM AUTHOR]

This paper deals with the coupled bending-torsional vibrations of beams carrying an arbitrary number of viscoelastic dampers and attached masses. Exact closed analytical expressions are derived for the frequency response under harmonically varying, arbitrarily placed polynomial loads, making use of coupled bending-torsion theory including warping effects and taking advantage of generalized functions to model response discontinuities at the application points of dampers/masses. In this context, the exact dynamic Green’s functions of the beam are also obtained. The frequency response solutions are the basis to derive the exact dynamic stiffness matrix and load vector of a two-node coupled bending-torsional beam finite element with warping effects, which may include any number of dampers/masses. Remarkably, the size of the dynamic stiffness matrix and load vector is 8×8 and 8×1, respectively, regardless of the number of dampers/masses and loads along the beam finite element. [ABSTRACT FROM AUTHOR]

This paper proposes an original and efficient approach to the moving load problem on Euler-Bernoulli beams, with Kelvin-Voigt viscoelastic translational supports and rotational joints, and in addition, equipped with Kelvin-Voigt viscoelastic tuned mass dampers (TMDs). While supports are taken as representative of external devices such as grounded dampers or in-span supports with flexibility and damping, the rotational joints may model rotational dampers or connections with flexibility and damping arising from imperfections or damage. The theory of generalised functions is used to treat the discontinuities of the response variables, which involves deriving exact complex eigenvalues and eigenfunctions from a characteristic equation built as determinant of a 4 × 4 matrix. Based on built pertinent orthogonality conditions for the deflection eigenfunctions, a closed-form analytical response is established in the time domain. The proposed solution holds for any number of TMDs and along-axis supports/joints. To show its applicability, accuracy and efficiency, in a numerical application a beam with multiple supports/joints is considered, subjected to a moving concentrated force and a series of concentrated forces, respectively. In two different configurations, the beam is equipped with one TMD and three TMDs, respectively. [ABSTRACT FROM AUTHOR]

This paper concerns the vibration response under moving loads of Euler-Bernoulli uniform beams with translational supports and rotational joints, featuring Kelvin-Voigt viscoelastic behaviour. Using the theory of generalized functions to handle the discontinuities of the response variables at the support/joint locations, exact beam modes are obtained from a characteristic equation built as determinant of a $$4\times 4$$ matrix, for any number of supports/joints. Based on pertinent orthogonality conditions for the deflection modes, the response under moving loads is built in the time domain by modal superposition. Remarkably, all response variables are built in a closed analytical form, regardless of the number of supports/joints. [ABSTRACT FROM AUTHOR]

The paper addresses the frequency analysis of bars with an arbitrary number of dampers, subjected to harmonically varying loads. Multiple external/internal dampers occurring at the same position along the bar, modelling external damping devices and internal damping due to damage or imperfect connections, are considered. In this context, the challenge is to handle simultaneous discontinuities of the response variables, i.e. axial force/displacement discontinuities at the location of external/internal dampers. Based on the theory of generalized functions, the paper will present exact closed-form expressions of the frequency response under point/polynomial loads, which hold regardless of the number of dampers. In addition, closed-form expressions will be derived for the exact dynamic stiffness matrix and load vector of the bar, to be used in a standard assemblage procedure for an exact frequency response analysis of 2D truss structures. Changes to consider a single damper at a given position are straightforward. Numerical applications show the advantages of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, a new theoretical framework is presented for modeling non-locality in shear deformable beams. The driving idea is to represent non-local effects as long-range volume forces and moments, exchanged by non-adjacent beam segments as a result of their relative motion described in terms of pure deformation modes of the beam. The use of these generalized measures of relative motion allows constructing an equivalent mechanical model of non-local effects. Specifically, long-range volume forces and moments are associated with three spring-like connections acting in parallel between couples of non-adjacent beam segments, and separately accounting for pure axial, pure bending and pure shear deformation modes. The variational consistency of the proposed non-local beam model is demonstrated by minimization of an appropriate total potential energy functional. Numerical results concerning the static behavior for different boundary and loading conditions are presented. It is shown that the proposed non-local beam model is able to capture experimental data on the static deflection of micro-beams, available in the literature. [ABSTRACT FROM AUTHOR]

Classical foundation models such as the Pasternak and the Reissner models have been recently reformulated within the framework of non-local mechanics, by using the gradient theory of elasticity. To contribute to the research effort in this field, this paper presents a two-dimensional foundation model built by using a mechanically based non-local elasticity theory, recently proposed by the authors. The foundation is thought of as an ensemble of soil column elements resting on an elastic base. It is assumed that each column element is acted upon by a local Winkler-like reaction force exerted by the elastic base, by contact shear forces and volume forces due, respectively, to adjacent and non-adjacent column elements. As in the Pasternak model, the contact shear forces involve the second-order derivative of the column element displacement. The volume forces are non-local forces assumed to depend (1) on the relative displacement between the interacting column elements through power-law distance-decaying attenuation functions and (2) on the product between the volumes of the interacting column elements. As a result, the equilibrium equations are fractional differential equations, for which a numerical solution can be readily found based on the finite difference method. Solutions are built for different foundation shapes and loading conditions. [ABSTRACT FROM AUTHOR]

The theory of generalized functions is used to address the static equilibrium problem of Euler-Bernoulli non-uniform and discontinuous 2-D beams. It is shown that if simple integration rules are applied, the full set of response variables due to end nodal displacements and to in-span loads can be derived, in a closed form, for most common beam profiles and arbitrary discontinuity parameters. On this basis, for finite element analysis purposes, a non-uniform and discontinuous beam element is implemented, for which the exact stiffness matrix and the fixed-end load vector are derived. Upon computing the nodal response, no numerical integration is required to build the response variables along the beam element. [ABSTRACT FROM AUTHOR]

Euler-Bernoulli arbitrary discontinuous beams acted upon by static loads are addressed. Based on appropriate Green's functions here derived in a closed form, the response variables are obtained: (a) for stepped beams with internal springs, as closed-form functions of the beam discontinuity parameters, without enforcing neither internal nor boundary conditions; (b) for stepped beams with internal springs and along-axis supports, as closed-form functions of the unknown reactions of the along-axis supports only, to be computed by enforcing pertinent conditions. A remarkable reduction in computational effort is achieved, in this manner, compared to competing methods in the literature. [ABSTRACT FROM AUTHOR]