Analysis Of Nonlinear Wave Propagation Research Articles

Nonlinear wave propagation in homogenized strain gradient 1D and 2D lattice materials: Applications to hexagonal and triangular networks

AbstractThis work aims to analyze the propagation of fully nonlinear waves, encompassing shear, extension, and bending deformation modes, within homogenized periodic nonlinear hexagonal and triangular networks, successively considering 1D and 2D situations. The wave analysis is conducted from the expression of the effective strain energy density of periodic hexagonal and triangular lattices in the nonlinear regime by a continualization of the discrete lattice equations, considering all forms of energy. We incorporate strain gradient effects into the continuous model to account for the wave‐dispersive nature. The resulting second‐gradient nonlinear continuum exhibits subsonic and supersonic propagation modes. We first examine in a 1D situation the dynamical response of the hexagonal and triangular lattices, considering varying levels of nonlinearity quantified by a single scalar valued parameter. We further evaluate the impact of a fully nonlinear analysis compared to an analysis solely based on the shear energy, regarding both supersonic and subsonic modes. The nonlinear wave propagation analysis is then extended to a 2D situation for the same two lattices. It is shown that the longitudinal mode exhibits a higher frequency at a low degree of nonlinearity; however, as the degree of nonlinearity increases, the shear mode surpasses the longitudinal mode in terms of frequency. As the wavenumber increases, the nonlinearity has a lesser impact on the frequency behavior, and the phase velocity is more influenced by other factors, such as the second gradient contributions of the effective constitutive law. Such a behavior indicates a transition from a highly nonlinear behavior at lower wave numbers to a more linear behavior at higher wave numbers.

Nonlinear wave propagation in homogenized strain gradient 1D and 2D lattice materials: Applications to hexagonal and triangular networks

AbstractThis work aims to analyze the propagation of fully nonlinear waves, encompassing shear, extension, and bending deformation modes, within homogenized periodic nonlinear hexagonal and triangular networks, successively considering 1D and 2D situations. The wave analysis is conducted from the expression of the effective strain energy density of periodic hexagonal and triangular lattices in the nonlinear regime by a continualization of the discrete lattice equations, considering all forms of energy. We incorporate strain gradient effects into the continuous model to account for the wave‐dispersive nature. The resulting second‐gradient nonlinear continuum exhibits subsonic and supersonic propagation modes. We first examine in a 1D situation the dynamical response of the hexagonal and triangular lattices, considering varying levels of nonlinearity quantified by a single scalar valued parameter. We further evaluate the impact of a fully nonlinear analysis compared to an analysis solely based on the shear energy, regarding both supersonic and subsonic modes. The nonlinear wave propagation analysis is then extended to a 2D situation for the same two lattices. It is shown that the longitudinal mode exhibits a higher frequency at a low degree of nonlinearity; however, as the degree of nonlinearity increases, the shear mode surpasses the longitudinal mode in terms of frequency. As the wavenumber increases, the nonlinearity has a lesser impact on the frequency behavior, and the phase velocity is more influenced by other factors, such as the second gradient contributions of the effective constitutive law. Such a behavior indicates a transition from a highly nonlinear behavior at lower wave numbers to a more linear behavior at higher wave numbers.

Analysis of nonlinear wave propagation within architected materials consisting of nonlinear Timoshenko beam structural elements

In the present work, a full nonlinear Timoshenko beam employing nonlinear shape functions is developed. The extended Hamilton principle is employed for deriving the differential equations of motion and the associated boundary conditions. The general form of the boundary conditions is then utilized to determine the static solution of the beam motion. Using this solution for the deformation and rotation of the beam, the nonlinear shape functions of the beam are identified, which leads to the linear and nonlinear mass and stiffness matrices of the Timoshenko beam element. The nonlinear dispersion diagram incorporating the non-linear corrections is obtained using the Linstedt–Poincaré perturbation method. An analysis of the effect of internal transverse shear and bending on the nonlinear dispersion characteristics of wave propagation in two-dimensional periodic network materials made of nonlinear Timoshenko beams is done. The formulated theory shows that the percentage of correction factor of the nonlinear kinematics versus the linear dynamical behavior is inversely proportional to the frequency amplitude. The shear and extension modes are shown to have the higher effect in the non-linear correction term in comparison to the flexural mode.

Damping of three-dimensional waves on coating films dragged by moving substrates

Paints and coatings often feature interfacial defects due to disturbances during the deposition process, which, if they persist until solidification, worsens the product quality. In this article, we investigate the stability of a thin liquid film dragged by a vertical substrate moving against gravity, a fundamental flow configuration in various coating processes. The receptivity of the liquid film to three-dimensional disturbances is analyzed with Direct Numerical Simulations (DNS) and an in-house Integral Boundary Layer (IBL) film model. The latter was used for linear stability analysis and nonlinear wave propagation analysis. The numerical implementation of the IBL film model combines a finite volume formulation with a pseudo-spectral approach for the capillary terms that allows one to investigate non-periodic surface tension-dominated flows. The numerical model was successfully validated with DNS computations. The combination of these numerical tools allows one to describe the mechanisms of capillary and nonlinear damping and identify the instability threshold of the coating processes. The results show that transverse modulations can be beneficial for damping two-dimensional waves within the range of operational conditions considered in this study, which are relevant to air-knife and slot-die coating.

Nonlinear wave propagation analysis in architected materials with consideration of extension, shear and bending effects

The current paper deals with the wave dispersion characteristics of two-dimensional architected materials, considering both the linear and nonlinear regimes. We study the full non-linear coupling between the shear, extension, and bending deformation modes on the inner material deformation of the square, triangular, hexagonal, and re-entrant hexagonal unit cells. The dynamical analysis is done at high and low frequencies and for two different wave amplitudes. Based on the Linstedt–Poincare perturbation method, the non-linear correction to the dispersion diagram is evaluated. The relative contribution of each non-linear energy mode on the nonlinear wave dispersion attributes is quantified. The obtained results show that both the wavenumber and lattice design play an important role in the nonlinear wave corrections of the dispersion characteristics. The shear mode is shown to have the highest effect in the non-linear correction for all considered architected materials, compared to the flexural mode, which has the lowest energy contribution. The eigenwaves vectors are used to compare the deformation of hexagonal and re-entrant hexagonal unit cells at low and high frequencies over the entire Brillouin zone. Overall, the present work provides some guideline as to the proper selection of the inner design of architected materials prone to strong nonlinear dynamical effects.

Phase space analysis of nonlinear wave propagation in a bistable mechanical metamaterial with a defect.

We study the dynamics of solitary waves traveling in a one-dimensional chain of bistable elements in the presence of a local inhomogeneity ("defect"). Numerical simulations reveal that depending upon its initial speed, an incoming solitary wave can get transmitted, captured, or reflected upon interaction with the defect. The dynamics are dominated by energy exchange between the wave and a breather mode localized at the defect. We derive a reduced-order two degree of freedom Hamiltonian model for wave-breather interaction and analyze it using dynamical systems techniques. Lobe dynamics analysis reveals the fine structure of phase space that leads to the complicated dynamics in this system. This work is a step toward developing a rational approach to defect engineering for manipulating nonlinear waves in mechanical metamaterials.

A New Microscopic Traffic Model Using a Spring-Mass-Damper-Clutch System

Microscopic traffic models describe how cars interact with their neighbors in an uninterrupted traffic flow and are frequently used for reference in advanced vehicle control design. In this paper, we propose a novel mechanical system-inspired microscopic traffic model using a mass-spring-damper-clutch system. This model naturally captures the ego vehicle’s resistance to large relative speed and deviation from a (driver- and speed-dependent) desired relative distance when following the lead vehicle. Compared with the existing car-following (CF) models, this model offers physically interpretable insights into the underlying CF dynamics and is able to characterize the impact of the ego vehicle on the lead vehicle, which is neglected in the existing CF models. Thanks to the nonlinear wave propagation analysis techniques for mechanical systems, the proposed model, therefore, has great scalability so that multiple mass-spring-damper-clutch systems can be chained to study the macroscopic traffic flow. We investigate the stability of the proposed model on the system parameters and the time delay using the spectral element method. We also develop a parallel recursive least square with inverse QR decomposition (PRLS-IQR) algorithm to identify the model parameters online. These real-time estimated parameters can be used to predict the driving trajectory that can be incorporated into advanced vehicle longitudinal control systems for improved safety and fuel efficiency. The PRLS-IQR is computationally efficient and numerically stable, and therefore, it is suitable for online implementation. The traffic model and the parameter identification algorithm are validated on both the simulations and naturalistic driving data from multiple drivers. Promising performance is demonstrated.

Nonlinear wave propagation analysis in Timoshenko nano-beams considering nonlocal and strain gradient effects

This article is aimed to investigate the geometrically nonlinear wave propagation of nano-beams on the basis of the most comprehensive size-dependent elasticity theory. To this end, the integral model of nonlocal elasticity theory in the most general form without any simplification in conjunction with the modified strain gradient theory is implemented in the analysis. Also, the Timoshenko beam model is utilized in the presented nonlocal strain gradient elasticity theory. By Hamilton’s principle, the governing integro-partial differential equations of motion are derived. Employing numerical integration and an efficient method called as periodic grid technique, a semi-analytical approach is presented for the solution procedure. To detect the impacts of nonlocality and small scale effects on the nonlinear wave propagation characteristics of beams at nanoscale, adequate numerical examples and comparison studies are presented.

Airy-type evolution equations on star graphs

In the present paper the Airy operator on star graphs is defined and studied. The Airy operator is a third order differential operator arising in different contexts, but our main concern is related to its role as the linear part of the Korteweg-de Vries equation, usually studied on a line or a half-line. The first problem treated and solved is its correct definition, with different characterizations, as a skew-adjoint operator on a star graph, a set of lines connecting at a common vertex representing, for example, a network of branching channels. A necessary condition turns out to be that the graph is balanced, i.e. there is the same number of ingoing and outgoing edges at the vertex. The simplest example is that of the line with a point interaction at the vertex. In these cases the Airy dynamics is given by a unitary or isometric (in the real case) group. In particular the analysis provides the complete classification of boundary conditions giving momentum (i.e., $L^2$-norm of the solution) preserving evolution on the graph. A second more general problem here solved is the characterization of conditions under which the Airy operator generates a contraction semigroup. In this case unbalanced star graphs are allowed. In both unitary and contraction dynamics, restrictions on admissible boundary conditions occur if conservation of mass (i.e., integral of the solution) is further imposed. The above well posedness results can be considered preliminary to the analysis of nonlinear wave propagation on branching structures.

Modeling and Analysis of Nonlinear Wave Propagation in One-Dimensional Phononic Structures

Different from elastic waves in linear periodic structures, those in phononic crystals (PCs) with nonlinear properties can exhibit more interesting phenomena. Linear dispersion relations cannot accurately predict band-gap variations under finite-amplitude wave motions; creating nonlinear PCs remains challenging and few examples have been studied. Recent studies in the literature mainly focus on discrete chain-like systems; most studies only consider weakly nonlinear regimes and cannot accurately obtain some relations between wave propagation characteristics and general nonlinearities. This paper presents propagation characteristics of longitudinal elastic waves in a thin rod and coupled longitudinal and transverse waves in an Euler–Bernoulli beam using their exact Green–Lagrange strain relations. We derive band structure relations for a periodic rod and beam and predict their nonlinear wave propagation characteristics using the B-spline wavelet on the interval (BSWI) finite element method. Influences of nonlinearities on wave propagation characteristics are discussed. Numerical examples show that the proposed method is more effective for nonlinear static and band structure problems than the traditional finite element method and illustrate that nonlinearities can cause band-gap width and location changes, which is similar to results reported in the literature for discrete systems. The proposed methodology is not restricted to weakly nonlinear systems and can be used to accurately predict wave propagation characteristics of nonlinear structures. This study can provide good support for engineering applications, such as sound and vibration control using tunable band gaps of nonlinear PCs.

Nonlinear wave propagation analysis in hyperelastic 1D microstructured materials constructed by homogenization

We analyze the acoustic properties of microstructured beams including a repetitive network material undergoing configuration changes leading to geometrical nonlinearities. The effective constitutive law is evaluated successively as an effective first and second order nonlinear grade 1D continuum, based on a strain driven incremental scheme written over the reference unit cell, taking into account the changes of the lattice geometry. The dynamical equations of motion are next written, leading to specific dispersion relations. The presence of second gradient order term in the nonlinear equation of motion leads to the presence of two different modes: an evanescent subsonic mode for high nonlinearity that vanishes beyond certain values of wave number, and a supersonic mode for a weak nonlinearity. This methodology is applied to analyze wave propagation within different microstructures, including the regular and reentrant hexagons, and plain weave textile pattern.

Comprehensive Seismic Response Analysis for Estimating the Seismic Behavior of Buried Pipelines Enhanced by Three-Dimensional Dynamic Finite Element Analysis of Ground Motion and Soil Amplification

We demonstrate a comprehensive earthquake response analysis method for improving the seismic input force estimation of buried pipelines by combining ground motion and soil amplification analyses. Using this method, the seismic input force of an actual pipeline was estimated and its seismic performance was checked for a largest assumed seismic fault scenario. Three-dimensional inhomogeneity of ground and surface topography is known to greatly affect the results of ground motion and soil amplification analyses. To consider these effects, a linear wave propagation analysis using a 10 × 109 degree-of-freedom three-dimensional finite element model was conducted for the ground motion analysis, and a nonlinear wave propagation analysis using an 80 × 106 degree-of-freedom three-dimensional finite element model was conducted for the soil amplification analysis. The application example showed that three-dimensional inhomogeneity of ground and surface topology caused complex seismic input forces to buried pipelines, and demonstrated the effectiveness of the comprehensive seismic analysis method proposed in this study.

Alternating wavelet-time finite element method: Modeling and analysis of nonlinear wave propagation in one and two-dimensional waveguides

In this paper, a novel method named the alternating wavelet-time finite element method (AWT-FEM) is proposed for studying elastic wave propagation in nonlinear structures. An alternating iterative procedure between the time-domain and a wavelet-domain combined with the spectral finite element method (SFEM) is employed to solve wave equations with general nonlinearities. The advantages of the proposed method are (1) the potential to provide high fidelity results for impacts with high frequency content through the use of the spectral finite element method; (2) nonlinear structures with physically realistic boundary conditions can easily be studied by circumventing wrap-around issues associated with Fourier-based methods; (3) the parallel computing compatible framework and the semi-analytical nature of SFEM make it more computationally efficient for nonlinear systems modeled with structural components. Simulations using the proposed method are conducted to demonstrate its applicability to study nonlinear wave propagation in one-dimensional and two-dimensional systems.

Two-Dimensional Simulation of Nonlinear Acoustic Wave Propagation Using Constrained Interpolation Profile Method

In this study, a numerical simulation of the nonlinear sound wave propagation in the time domain is demonstrated by constrained interpolation profile/cubic-interpolated pseudoparticle (CIP) method. The CIP method is a novel numerical scheme which was proposed by Yabe et al. It is a method of characteristics (MOC) and is a low-dispersion and stable scheme. The CIP method is suitable for analysis of nonlinear wave propagation including weak shock formation because the rapid pressure change, such as that caused by a shock front, easily causes numerical dispersion error in a conventional numerical scheme. Some of the authors have reported application of the CIP method to a one-dimensional nonlinear sound wave propagation in air. Then, we demonstrate a numerical calculation of the nonlinear sound wave propagation for a two-dimensional acoustic field.

Numerical Analysis of Nonlinear Wave Propagation in Elastic Submerged Cables

This paper analyzes the coupled nonlinear tangential-normal waves that propagate along underwater cable suspensions. Taken with the recently developed linear theory governing the in-plane structural waves (Behbahani-Nejad and Perkins, 1996) and an analysis of nonlinear out-of-plane waves for submerged cables (Behbahani-Nejad and Perkins, 1997), this investigation contributes further understanding toward a nonlinear three-dimensional theory for wave propagation along fluid-loaded cables. The nonlinearities present in the in-plane model render the cable/fluid model intractable by exact analytical methods. A numerical solution is pursued in this study using finite difference algorithms. To this end, an infinite cable domain is divided to two subdomains, namely an interior (finite computational) domain and exterior (infinite far-field) domain. Closed-form solutions for the approximate linear theory are employed for the far field in constructing nonreflecting boundary conditions for the computational domain. Numerical results highlight the governing role of nonlinear hydrodynamic drag for underwater cable suspensions. The numerical results further demonstrate that most analyses of cables in air are not useful for underwater cable applications.

Analysis of nonlinear wave propagation and shock wave formation in quasi-brittle material : M. Saka, S. Ohba, N. Aizawa & H. Abe, JSME International Journal, Series A: Mechanics & Material Engineering, 37(4), 1994, pp 421–427

Analysis of nonlinear wave propagation and shock wave formation in quasi-brittle material : M. Saka, S. Ohba, N. Aizawa & H. Abe, JSME International Journal, Series A: Mechanics & Material Engineering, 37(4), 1994, pp 421–427

Analysis of Nonlinear Wave Propagation and Shock Wave Formation in Quasi-Brittle Material

Fracture of quasi brittle materials such as concrete and rocks is known to result from the formation and development of microcracks. Microcracks reduce the macroscopic value of Young's modulus and therefore the corresponding wave velocity, so that shock wave formation is expected. By using of a constitutive law based on experiment, the nonlinar wave propagation was analyzed numerically for an infinite plate. The plate is subjected to an impact compressibe stress at one of its surfaces. The other surface of the plate isassumedto be stress free

Invited paper Characteristics of travelling waves along the non-linear transmission lines for monolithic integrated circuits: a review

Abstract A general analysis of non-linear wave propagation along transmission lines with voltage-dependent capacitance is presented. In particular, slow-wave structures like MIS and Schottky-barrier strip lines are examined. A spatial periodicity is included explicitly. The theoretical treatment is based on suitable equivalent circuits leading to characteristic wave equations. With regard to practical devices, the solutions show a variety of different phenomena as determined by the parameters of the non-linearity, dispersion and dissipation and the boundary conditions. Experimental results performed on a slow-wave model line are included.