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We present a variational multiscale strategy in the spirit of FE\(^2\) method for simulating the real time evolution of resonant elastic metamaterials with space/time modulation. The implicit time discretization used guarantees the stability of the numerical solution, while the accuracy is quantified by direct comparison with the response obtained...
Similar publications
In this paper, two models of elastic metamaterial containing one and two resonators are proposed to obtain the bandgaps with the aim of providing broadband vibration suppression. The model with one DOF is built by assembling several unite cells in which each unite cell consists of a rectangular frame as the base structure and a rack-and-pinion mech...
Citations
... In contrast to this paper, previous works have successfully established an effective medium theory for the low-frequency regime, in which the operating frequency is significantly smaller than the resonant frequency and the material parameters are static in time [6,7,15]. Furthermore, it is worth emphasising that the time-modulations considered in this paper are very different from the travelling wave-form modulations discussed in [14,16,18,19] and in the references therein. By exploiting the subwavelength resonances of the system, they lead to fundamentally different wave propagation and scattering phenomena [2]. ...
This paper provides a general framework for deriving effective material properties of time-modulated systems of subwavelength resonators. It applies to subwavelength resonator systems with a general form of time-dependent parameters. We show that the resonators can be accurately described by a point-scattering formulation when the width of the resonators is small. In contrast to the static setting, where this point interaction approximation yields a Lippmann-Schwinger equation for the effective material properties, the mode coupling in the time-modulated case instead yields an infinite linear system of Lippmann-Schwinger-type equations. The effective equations can equivalently be written as a system of integro-differential equations. Moreover, we introduce a numerical scheme to approximately solve the system of coupled equations and illustrate the validity of the effective equation.
... Our results can also be generalized to higher space dimensions. Furthermore, it is worth emphasizing that the time-modulations considered in this paper are very different from the travelling wave-form modulations discussed in, for instance, [17][18][19][20][21][22][23][24][25][26]. By exploiting the subwavelength resonant properties of the system, they lead to fundamentally different wave propagation and scattering phenomena. ...
We consider wave scattering from a system of highly contrasting resonators with time-modulated material parameters. In this setting, the wave equation reduces to a system of coupled Helmholtz equations that models the scattering problem. We consider the one-dimensional setting. In order to understand the energy of the system, we prove a novel higher-order discrete, capacitance matrix approximation of the subwavelength resonant quasi-frequencies. Further, we perform numerical experiments to support and illustrate our analytical results and show how periodically time-dependent material parameters affect the scattered wave field.
... The rate-dependence of stiffness, strength, and damage are typically attributed to propagating stress waves that are generated under high strain-rate impact conditions. [19][20][21][22][23] Predicting dynamic deformation and failure in composite structures under high-strain-rate loading requires comprehensive macroscopic constitutive and damage models incorporating lower scale (microstructural) morphological descriptors, along with appropriate damage and deformation mechanisms with strain-rate dependencies. A robust multiscale modeling framework is needed for developing these constitutive damage models. ...
... A common assumption is the periodic boundary conditions (PBC) for the microstructural RVE. 21,22,27 However, the PBC-based homogenization requires non-optimal, large RVEs for microstructures with a high degree of heterogeneity, 28,29 or for microstructures that are subjected to high strain-rates ð≥10 4 s À1 Þ. 19,30 The application of periodic boundary conditions can give unrealistic results for high strain-rate loading conditions, even for periodic microstructures. 19,30 To overcome this limitation, a self-consistent homogenization framework has been developed by the authors 19 to account for stress wave interaction in heterogeneous microstructures under high strain-rate loading conditions. ...
... From Equations (21)- (22), the dynamic part of the PK-2 stress depends on micro-inertia, which is a function of acceleration in the microstructure. As shown in Figure 6, both the quasi-static ðS P, s 11 Þ and dynamic ðS P, d 11 Þ parts of the PK-2 stress increase with increasing strain-rates. ...
This paper develops a Parametrically Upscaled Continuum Damage Mechanics (PUCDM) model for carbon fiber/epoxy matrix composites subjected to high strain-rate loading. The PUCDM model for predicting high strain-rate induced deformation and damage evolution at the macroscopic scale, explicitly incorporates microstructural morphology and micro-inertia in its constitutive coefficients. It is developed using self-consistent homogenization that uses a concurrent multiscale model to account for the effect of micro-inertia and stress wave interaction with the microstructure. The concurrent model embeds a statistically equivalent representative volume element (SERVE) of the microstructure in an exterior domain, whose constitutive and damage response are described by the PUCDM model. Micromechanical modeling for the composite microstructure involves a unified cohesive zone enhanced phase-field damage model for fiber-matrix interface debonding, fiber breakage, and matrix cracking. The PUCDM model parameters for the upscaled domain are determined through micro-macro-scale energy equivalence, along with traction reciprocity and displacement continuity at the SERVE-exterior domain interface that are enforced using a Lagrange multiplier-based approach. Simulations using the concurrent model analyze stress wave propagation and damage evolution in composite microstructures with three different fiber volume fractions at multiple strain rates in the range of 10 ² − 10 ⁵ s ⁻¹ . The fiber volume fraction and spatial distributions affect the overall stiffness and damage evolution in the PUCDM model. Results of the analysis demonstrate the need for representation of micro-inertia, strain rates and microstructure morphology in the PUCDM-based stiffness and damage model parameters for high strain-rates above [Formula: see text].
... Furthermore, wave propagation effects interact with the microstructure and lead to band-gaps. Recent attempts to study these effects can be found in [58], [123], [196], [222], [250]. ...
Nonlinear Mechanics for Composite Heterogeneous Structures applies both classical and multi-scale finite element analysis to the non-linear, failure response of composite structures. These traditional and modern computational approaches are holistically presented, providing insight into a range of non-linear structural analysis problems. The classical methods include geometric and material non-linearity, plasticity, damage and contact mechanics. The cutting-edge formulations include cohesive zone models, the Extended Finite Element Method (XFEM), multi-scale computational homogenization, localization of damage, neural networks and data-driven techniques. This presentation is simple but efficient, enabling the reader to understand, select and apply appropriate methods through programming code or commercial finite element software.
The book is suitable for undergraduate studies as a final year textbook and for MSc and PhD studies in structural, mechanical, aerospace engineering and material science, among others. Professionals in these fields will also be strongly benefited.
An accompanying website provides MATLAB codes for two-dimensional finite element problems with contact, multi-scale (FE2) and non-linear XFEM analysis, data-driven and machine learning simulations.
... Furthermore, wave propagation effects interact with the microstructure and lead to band-gaps. Recent attempts to study these effects can be found in [58], [123], [196], [222], [250]. ...
... Conventional homogenization methods typically account for the averaged motion, generally ignoring local motion in the material (Chen and Fish, 2000;Chen and Ghosh, 2012;Fish et al., 2002Fish et al., , 2012Ghosh, 2019, 2020). For higher loading rates, the inertia associated with the local motion becomes quite significant and cannot be neglected (Liu and Reina, 2019;Nassar et al., 2015;Pham et al., 2011Pham et al., , 2013Shuvalov et al., 2011;Willis, 2009). This example demonstrates the need for more appropriate boundary conditions that account for stress wave propagation in heterogeneous microstructures under high strain-rate loading conditions. ...
This paper proposes a self-consistent homogenization framework for developing finite strain, elastic constitutive models of carbon fiber–epoxy composites for high strain-rate loading. The framework incorporates a concurrent multiscale model, in which an RVE is embedded in a homogenized exterior domain, is developed with self-consistent displacement compatibility and traction reciprocity constraints at the RVE-exterior domain interface. It overcomes limitations of periodic boundary conditions on the RVE for dynamic problems under high strain-rate conditions and allows for the evaluation of material constitutive relations accounting for micro-scale inertia. The proposed self-consistent formulation incorporates the effect of stress waves propagation in the microstructure, along with their interaction with heterogeneities resulting in reflection and transmission at the interfaces. The concurrent model is used to study dynamic composite behavior at strain-rates in the range of 10²-10⁵ s⁻¹ for two RVEs, one containing a single fiber and another with 70 randomly distributed fibers. For strain-rates above 10⁴ s⁻¹, the dynamic part of the stress becomes significant and dominates over the static part of the stress. Results of the analysis show that for high strain-rate dynamic loading, constitutive models for the composites must account for micro-inertia, in addition to strain-rates and microstructural morphology.
... Instead of using Eq. (7) or (21) as the boundary condition, one can link the microscopic matrix domain with the macro scale by using the volume average constraint (VAC) [24] 1ũ f dv = 0 (30) or the point constraint (PC) [15] u f (y 1 = −l/2, y 2 = −l/2) = 0. ...
... Based on the experiment of localized resonant structures by Liu et al. [12], Pham et al. [14] proposed a quasi-2D numerical setup (Fig. 6) satisfying the relaxed scale separa-tion. This example has been used in many other multiscale studies [15,30,31]. The metamaterial consists of three linear isotropic constituent materials, namely, epoxy matrix, rubber coating and lead core. ...
This paper presents an extension of Direct method for the study of dynamic problems in heterogeneous materials. The proposed method can be formulated based on either the Hill–Mandel principle or the extended Hill–Mandel principle, the latter of which enforces the energy contributed by the internal force and the inertial force consistent at two scales. Unlike the traditional method, it is not necessary to conduct two levels of finite element simulations linked by extensive information interchange. Instead, we reformulate the macroscopic variational statement with the microscopic contributions, leading to only a single coupled boundary value problem. The classical microscopic boundary condition used in the traditional method can be employed but it is implemented through kinematical constraints between the macro nodes and the micro nodes in the Direct method. The proposed method is illustrated by two numerical examples including a fiber-reinforced composite and an acoustic metamaterial. The results are verified by direct numerical simulations and it is shown that micro-inertia effects are not important in modeling low-velocity impact behavior of the composite but they are essential in capturing wave attenuation performance of locally resonant acoustic metamaterials.
... This was extended by [25] to account for matrix cracking at the microscale under impact loading. Other, rather FE 2 -type schemes as [32,33,51,52,56,60,61] calculate the full balance of linear momentum at the microscale. In [32] an explicit, periodic, small-strain framework is presented, which was extended to an implicit time integration method for modeling resonant elastic metamaterials in [33]. ...
... Other, rather FE 2 -type schemes as [32,33,51,52,56,60,61] calculate the full balance of linear momentum at the microscale. In [32] an explicit, periodic, small-strain framework is presented, which was extended to an implicit time integration method for modeling resonant elastic metamaterials in [33]. [51,52] use the assumption of linear elasticity to improve the computational performance, by splitting the problem into a purely static and a special dynamic boundary value problem (BVP). ...
In this paper we present a fully-coupled, two-scale homogenization method for dynamic loading in the spirit of FE 2 methods. The framework considers the balance of linear momentum including inertia at the microscale to capture possible dynamic effects arising from micro heterogeneities. A finite-strain formulation is adapted to account for geometrical nonlinearities enabling the study of e.g. plasticity or fiber pullout, which may be associated with large deformations. A consistent kinematic scale link is established as displacement constraint on the whole representative volume element. The consistent macroscopic material tangent moduli are derived including micro inertia in closed form. These can easily be calculated with a loop over all microscopic finite elements, only applying existing assembly and solving procedures. Thus, making it suitable for standard finite element program architectures. Numerical examples of a layered periodic material are presented and compared to direct numerical simulations to demonstrate the capability of the proposed framework. In addition, a simulation of a split Hopkinson tension test showcases the applicability of the framework to engineering problems.
... Several new computational approaches have been recently proposed to describe the transient dynamic behavior of architected materials and probe the topological and material property spaces in search of a better understanding of the structure-dynamic property relationships. These approaches include but are not limited to computational homogenization (Liu and Reina, 2019;Roca et al., 2018;Sridhar et al., 2016), homogenization methods based on Willis' theory (Meng and Guzina, 2018;Nassar et al., 2016;Sridhar et al., 2018), the multiscale finite element method (Casadei et al., 2016), and the method of computational continua (Filonova et al., 2016), among others. Some of these methodologies operate at the scale separation limit, where the microstructure size is much smaller compared to the deformation wavelength. ...
Modeling and simulation have quickly become equivalent pillars of research along with traditional theory and experimentation. The growing realization that most complex phenomena of interest span many orders of spatial and temporal scales has led to an exponential rise in the development and application of multiscale modeling and simulation over the past two decades. In this perspective, the associate editors of the International Journal for Multiscale Computational Engineering and their co-workers illustrate current applications in their respective fields spanning biomolecular structure and dynamics, civil engineering and materials science, computational mechanics, aerospace and mechanical engineering, and more. Such applications are highly tailored, exploit the latest and ever-evolving advances in both computer hardware and software, and contribute significantly to science, technology, and medical challenges in the 21st century.
... In [20], an explicit, periodic, small strain framework is presented for modeling resonant elastic metamaterials. This was extended to an implicit time integration method by Liu and Reina [21]. By splitting the problem into a purely static and a special dynamic boundary value problem (BVP) in [22,23], the assumption of linear elasticity is used to improve the computational performance. ...
This paper presents a numerical two-scale framework for the simulation of fiber reinforced concrete under impact loading. The numerical homogenization framework considers the full balance of linear momentum at the microscale. This allows for the study of microscopic inertia effects affecting the macroscale. After describing the ideas of the dynamic framework and the material models applied at the microscale, the experimental behavior of the fiber and the fiber–matrix bond under varying loading rates are discussed. To capture the most important features, a simplified matrix cracking and a strain rate sensitive fiber pullout model are utilized at the microscale. A split Hopkinson tension bar test is used as an example to present the capabilities of the framework to analyze different sources of dynamic behavior measured at the macroscale. The induced loading wave is studied and the influence of structural inertia on the measured signals within the simulation are verified. Further parameter studies allow the analysis of the macroscopic response resulting from the rate dependent fiber pullout as well as the direct study of the microscale inertia. Even though the material models and the microscale discretization used within this study are simplified, the value of the numerical two-scale framework to study material behavior under impact loading is demonstrated.
