Gradient elasticity and dispersive wave propagation: Model motivation and length scale identification procedures in concrete and composite laminates

Abstract

Nano-scale experimental findings reveal that wave propagation in heterogeneous materials is dispersive. In order to capture such dispersive behavior, in this paper gradient elasticity theory is resorted to. A popular gradient elasticity model arising from Mindlin's theory incorporates two internal length scale parameters, which correspond to one micro-stiffness and one micro-inertia term. As an extension of Mindlin's model, an expanded three-length-scale gradient elasticity formulation with one additional micro-inertia term is used to improve the description of microstructural effects in dynamics. A non-local lattice model is introduced here to give the above micro-stiffness and micro-inertia terms a physical interpretation based on geometrical and mechanical properties of the microstructure. The purpose of this paper is to assess the effectiveness of such a three-length-scale formulation in predicting wave dispersion against experimental and micro-mechanical data from the literature. The dispersive wave propagation through laminated composites with periodic microstructure is investigated first. Length scale identification is carried out based on higher-order homogenization to link the constitutive coefficients of the gradient theory directly to microstructural properties of the layered composite. Secondly, experimental dispersion curves for phonons propagating in aluminum and bismuth crystals are scrutinized, thus highlighting the motivation for including multiple micro-inertia terms. Finally, ultrasonic wave dispersion experimentally observed in concrete specimens with various sand contents and water/cement ratios is analyzed, along with length scale quantification procedures. It is found that the proposed three-length-scale gradient formulation is versatile and effective in capturing a range of wave dispersion characteristics arising from experiments. Advantages over alternative formulations of gradient elasticity from the literature are discussed throughout the paper.

A dynamic high-frequency consistent continualization of beam-lattice materials

“…This technique, which has been also successfully tested in comparative studies by Gómez-Silva et al [17] and Gómez-Silva and Zaera [18], is here applied to a representative case of square beam-lattice to obtain the field equations governing the motion of the equivalent non-local continuum formulated at different orders. Through the presented approach both non-local stiffness and inertia terms are obtained, in agreement with the non-local continuum models proposed by the seminal papers of Mindlin (1964) [34], Eringen (1983) [35], Askes and Aifantis (2011) [36], and more recently by Bacigalupo and Gambarotta (2014) [37], De Domenico and Askes [12], and De Domenico et al [13].…”

Mono-scale and multi-scale formulations of gradient-enriched dynamic piezomagnetics

“…Furthermore, new gradientenriched elasticity models, such as complete anisotropic strain-gradient elasticity [14,15], spatial-temporal nonlocal homogenisation models [16,17] and dispersive gradient elasticity with multiple micro-inertia terms [18,19], have increased the versatility of the gradient elasticity modelling framework. In parallel, procedures for the identification of model coefficients, including higher-order length scales, have been presented in [20,21,22]. An overview of formulations and length scale identification procedures can be found in [23].…”

Wave transmission and reflection analysis through complex media based on the second strain gradient theory

“…Metrikine et Askes developed a stable and dynamic consistent high-order continuum model with two additional material parameters from a 2D lattice by supposing that the displacement vector of a higher-order continuum should represent an average of the particledisplacements of the underlying lattice [27]. In the research of Domenico [28,29], length scale identification is carried out by "continualization" a non-local lattice model with both distributed and lumped mass to a three-lengthscale gradient model by employing a higher-order homogenization. In the present work, one direct map operation, that has been used in research [30], is employed to obtain the continuum equation from a lattice, and the long-range interaction parameters are identified in the SSG theory model.…”