## Final Report Summary - DYNA META 11 (Controlling elastic waves: structured media and metamaterials in the mechanics of solids and structures.)

As established in the work plan, the emphasis of the research is given to the design and the analytical and numerical modeling of innovative elastic systems capable to control elastic wave propagation in structured media. These include invisibility cloaks, polarizers, filtering and focussing devices, by-pass systems. The project website http://pcwww.liv.ac.uk/~mbrun/index.htm has been constructed and published on the Web.

The main challenges of the research are:

1. Design and modeling of high-contrast semi-discrete structures

2. Modeling of wave propagation in multi-stable high contrast elastic system

3. Design and modeling of invisibility cloaks based on geometrical transforms

4. Optimal design of real life engineering structures

Challenge 1

A type of locally resonant structure controlling the trajectory of in-plane coupled shear and pressure waves has been proposed in [5]. Such an elastic metamaterial, involving arrays of structured coated inclusions, supports tunable low-frequency stop bands associated with localized rotational modes that can be used in the design of filtering, reflecting, and focusing devices. Inertial resonators with inclined ligaments allow for anomalous dispersion (negative group velocity) to occur in the pressure acoustic band and this leads to the physics of negative refraction, whereby a point force located above a finite array of resonators is imaged underneath for a given polarization. For a periodic macrocell with one defect in the middle, an elastic trapped mode exists within a high-frequency stop band.

A chiral elastic metamaterial structure that possesses polarization and stiffening properties is proposed for the full vector, two-dimensional elasticity [7]. The structured media is obtained by embedding into a discrete lattice a system of spinners. Depending on the relative magnitude of the vorticity constants, governed by the system of spinners, the micro-structured medium evidences tunable dynamically induced stiffening of the compressional behavior, anisotropy and shear polarization. Dispersion properties associated to Bloch waves in the discrete vortex-type elastic lattices have been identified analytically.

A new metamaterial microstructured media is proposed having Poisson’s ratio arbitrarily close to the stability limit -1 [12]. Three microstructures are designed with cubic and isotropic behaviors. Effective properties of the lattice structures have been obtained in close analytical form and the Poisson’s ratio has been tested experimentally.

Challenge 2

The advance of a transition of a flexural wave through a beam-like periodically supported discrete structure is analysed in [1]. Computational examples suggest the applicability of the proposed model to real life bridge systems such as the S'Adde bridge and the Millau viaduct. The conditions for the failure wave to exist, to propagate uniformly or to accelerate are found for continuous and discrete-continuous models. A functional equation of the Wiener-Hopf type has led to the expression of the displacement on the interface for discrete models. Remarkably, the solution corresponding to the steady propagation of the interface wave is intersonic.

The transient problem of localization near a fault in a lattice which has substantial impact in assessment of safety and reliability in nuclear industry. The problem of propagation of an edge-crack in a microstructured media is considered in [8]. The propagation is induced by a rapid change of temperature applied on the boundary of the thermoelastic lattice structure. The nonlinear simulations show that the average speed of crack propagation can be estimated from the analysis of the dispersion properties of waves initiated by the crack. It is found that inertia amplifies the elongations of the links, and thus influences the crack advance through the structured solid.

Challenge 3

A comprehensive treatment of a non-singular cloak is obtained in the framework of transformation optics [9]. The quality of the cloaks is assessed numerically using a scattering measure introduced as an L2 norm of the difference between the cloaked field and the ideal unperturbed field. The significant advantage of the continuous square cloak is the straightforward correspondence with a discrete metamaterial lattice structure. Such a connection presents a method through which a physical cloak may be fabricated. The material and geometric properties of the discrete cloak are directly linked to the properties of the formal map. The efficacy of the square push out cloak is confirmed via the quantum mechanical classical Young’s double slit experiment. The interference pattern remains almost entirely unperturbed if the obstacle is cloaked.

The problem of the construction of transformation theory for the dynamic equations of flexural deformations in Kirchhoff–Love plates is addressed in [2,12]. Cloaking design procedures developed for acoustics, vibration of elastic membranes and anti-plane shear problems does not apply to problems of flexural vibrations of elastic plates. For models of square and cylindrical cloaks channeling flexural waves around finite inclusions, it is shown that the transformed equation can be interpreted as the equation of an inhomogeneous locally orthotropic plate including additional terms in the governing equation; these may represent in-plane body forces and pre-stress [12]. The physical nature model is fully explained, with the introduction of pre-stress and in-plane body forces, identified in explicit closed form.

In addition, the transformed equations of three-dimensional vector elasticity are analysed asymptotically for a thin solid [2]. The resulting lower-dimensional model agrees fully with the outcome of the direct application of the radial ‘push-out’ transformation to the equation of motion of a Kirchhoff plate. In contrast to the elastodynamic case and previous work on plates, it is shown that it is possible to construct an invisibility cloak for flexural waves in thin plates without recourse to non-symmetric stresses, tensorial densities, or non-linear theories.

The analytical work is accompanied by illustrative numerical simulations demonstrating the quality of the cloaking effect, via both numerical measures and interferometery.

The exact physical interpretation of the additional terms in the transformed equation may lead to a refinement of the experimental implementations and further developments in the practical implementation of broadband cloaks, lightweight and highly-efficient structured shields and filtering devices for plates.

Challenge 4

Optimal design of slender Civil Engineering Structures has been obtained adapting models of smart interfaces of challenge 1. The appropriate design scheme leads to desired filtering and polarisation properties by mean of a system of ‘‘by-pass” channels for the undesired elastic waves [4,13].

Spectral analysis of Bloch waves in a periodic elastic waveguide determine the dynamic response of a slender but finite structure such as a multispan bridge. Application to the “dancing bridge” in Volgograd, Russia, and the S’Adde bridge in Macomer, Italy, have been considered [4,13].

In addition, the steady-state propagation of a localized damage in a supported bridge model is considered [1]. The analytical model reduces to the analysis of a functional equation of the Wiener-Hopf type, leading to the expression of the displacement on the interface wave to be written as a product of terms evaluated directly from analysis of the kernel function of the Wiener-Hopf equation. The steady propagation of the interface wave is intersonic. Such a model has been applied to study the fire collapse of the San Saba bridge in Texas. The exceptional result is a simple and extremely accurate estimate of the velocity of the failure propagation in the dynamic gravity driven transition fault [3].

Analysis of slender discontinuous bridge structures has been examined considering the propagation of transverse waves in a two-dimensional elastic strip with periodically distributed damage or transverse cracks [6,12]. Numerical simulations concerning infinite periodic strips have shown that band-gaps arise as a consequence of the localized defect present in the structure, and the limits of the band-gaps depend on the depth of the cracked sections. The eigenfrequencies of the finite structures fall within frequency intervals coinciding with the pass-band of the periodic structure. An analytical lower-dimensional periodic beam model has been obtained, in which the cracked sections are represented by elastic junctions with a bending and a shear spring. The effective rotational and translational stiffnesses are derived by means of an asymptotic analysis. A comparison with the numerical findings obtained from the two-dimensional model has shown that the lowest band-gaps of the strip can be determined with a high level of accuracy from the dispersion curves of the periodic beam. The limits of the band-gaps coincide with the eigenfrequencies of elementary beam models. The results of this work can be used to design systems with filtering properties, and to detect and possibly estimate quantitatively the presence of cracks inside structural and mechanical elements by means of non-destructive techniques.

The main challenges of the research are:

1. Design and modeling of high-contrast semi-discrete structures

2. Modeling of wave propagation in multi-stable high contrast elastic system

3. Design and modeling of invisibility cloaks based on geometrical transforms

4. Optimal design of real life engineering structures

Challenge 1

A type of locally resonant structure controlling the trajectory of in-plane coupled shear and pressure waves has been proposed in [5]. Such an elastic metamaterial, involving arrays of structured coated inclusions, supports tunable low-frequency stop bands associated with localized rotational modes that can be used in the design of filtering, reflecting, and focusing devices. Inertial resonators with inclined ligaments allow for anomalous dispersion (negative group velocity) to occur in the pressure acoustic band and this leads to the physics of negative refraction, whereby a point force located above a finite array of resonators is imaged underneath for a given polarization. For a periodic macrocell with one defect in the middle, an elastic trapped mode exists within a high-frequency stop band.

A chiral elastic metamaterial structure that possesses polarization and stiffening properties is proposed for the full vector, two-dimensional elasticity [7]. The structured media is obtained by embedding into a discrete lattice a system of spinners. Depending on the relative magnitude of the vorticity constants, governed by the system of spinners, the micro-structured medium evidences tunable dynamically induced stiffening of the compressional behavior, anisotropy and shear polarization. Dispersion properties associated to Bloch waves in the discrete vortex-type elastic lattices have been identified analytically.

A new metamaterial microstructured media is proposed having Poisson’s ratio arbitrarily close to the stability limit -1 [12]. Three microstructures are designed with cubic and isotropic behaviors. Effective properties of the lattice structures have been obtained in close analytical form and the Poisson’s ratio has been tested experimentally.

Challenge 2

The advance of a transition of a flexural wave through a beam-like periodically supported discrete structure is analysed in [1]. Computational examples suggest the applicability of the proposed model to real life bridge systems such as the S'Adde bridge and the Millau viaduct. The conditions for the failure wave to exist, to propagate uniformly or to accelerate are found for continuous and discrete-continuous models. A functional equation of the Wiener-Hopf type has led to the expression of the displacement on the interface for discrete models. Remarkably, the solution corresponding to the steady propagation of the interface wave is intersonic.

The transient problem of localization near a fault in a lattice which has substantial impact in assessment of safety and reliability in nuclear industry. The problem of propagation of an edge-crack in a microstructured media is considered in [8]. The propagation is induced by a rapid change of temperature applied on the boundary of the thermoelastic lattice structure. The nonlinear simulations show that the average speed of crack propagation can be estimated from the analysis of the dispersion properties of waves initiated by the crack. It is found that inertia amplifies the elongations of the links, and thus influences the crack advance through the structured solid.

Challenge 3

A comprehensive treatment of a non-singular cloak is obtained in the framework of transformation optics [9]. The quality of the cloaks is assessed numerically using a scattering measure introduced as an L2 norm of the difference between the cloaked field and the ideal unperturbed field. The significant advantage of the continuous square cloak is the straightforward correspondence with a discrete metamaterial lattice structure. Such a connection presents a method through which a physical cloak may be fabricated. The material and geometric properties of the discrete cloak are directly linked to the properties of the formal map. The efficacy of the square push out cloak is confirmed via the quantum mechanical classical Young’s double slit experiment. The interference pattern remains almost entirely unperturbed if the obstacle is cloaked.

The problem of the construction of transformation theory for the dynamic equations of flexural deformations in Kirchhoff–Love plates is addressed in [2,12]. Cloaking design procedures developed for acoustics, vibration of elastic membranes and anti-plane shear problems does not apply to problems of flexural vibrations of elastic plates. For models of square and cylindrical cloaks channeling flexural waves around finite inclusions, it is shown that the transformed equation can be interpreted as the equation of an inhomogeneous locally orthotropic plate including additional terms in the governing equation; these may represent in-plane body forces and pre-stress [12]. The physical nature model is fully explained, with the introduction of pre-stress and in-plane body forces, identified in explicit closed form.

In addition, the transformed equations of three-dimensional vector elasticity are analysed asymptotically for a thin solid [2]. The resulting lower-dimensional model agrees fully with the outcome of the direct application of the radial ‘push-out’ transformation to the equation of motion of a Kirchhoff plate. In contrast to the elastodynamic case and previous work on plates, it is shown that it is possible to construct an invisibility cloak for flexural waves in thin plates without recourse to non-symmetric stresses, tensorial densities, or non-linear theories.

The analytical work is accompanied by illustrative numerical simulations demonstrating the quality of the cloaking effect, via both numerical measures and interferometery.

The exact physical interpretation of the additional terms in the transformed equation may lead to a refinement of the experimental implementations and further developments in the practical implementation of broadband cloaks, lightweight and highly-efficient structured shields and filtering devices for plates.

Challenge 4

Optimal design of slender Civil Engineering Structures has been obtained adapting models of smart interfaces of challenge 1. The appropriate design scheme leads to desired filtering and polarisation properties by mean of a system of ‘‘by-pass” channels for the undesired elastic waves [4,13].

Spectral analysis of Bloch waves in a periodic elastic waveguide determine the dynamic response of a slender but finite structure such as a multispan bridge. Application to the “dancing bridge” in Volgograd, Russia, and the S’Adde bridge in Macomer, Italy, have been considered [4,13].

In addition, the steady-state propagation of a localized damage in a supported bridge model is considered [1]. The analytical model reduces to the analysis of a functional equation of the Wiener-Hopf type, leading to the expression of the displacement on the interface wave to be written as a product of terms evaluated directly from analysis of the kernel function of the Wiener-Hopf equation. The steady propagation of the interface wave is intersonic. Such a model has been applied to study the fire collapse of the San Saba bridge in Texas. The exceptional result is a simple and extremely accurate estimate of the velocity of the failure propagation in the dynamic gravity driven transition fault [3].

Analysis of slender discontinuous bridge structures has been examined considering the propagation of transverse waves in a two-dimensional elastic strip with periodically distributed damage or transverse cracks [6,12]. Numerical simulations concerning infinite periodic strips have shown that band-gaps arise as a consequence of the localized defect present in the structure, and the limits of the band-gaps depend on the depth of the cracked sections. The eigenfrequencies of the finite structures fall within frequency intervals coinciding with the pass-band of the periodic structure. An analytical lower-dimensional periodic beam model has been obtained, in which the cracked sections are represented by elastic junctions with a bending and a shear spring. The effective rotational and translational stiffnesses are derived by means of an asymptotic analysis. A comparison with the numerical findings obtained from the two-dimensional model has shown that the lowest band-gaps of the strip can be determined with a high level of accuracy from the dispersion curves of the periodic beam. The limits of the band-gaps coincide with the eigenfrequencies of elementary beam models. The results of this work can be used to design systems with filtering properties, and to detect and possibly estimate quantitatively the presence of cracks inside structural and mechanical elements by means of non-destructive techniques.