Interesting effects in harmonic generation by plane elastic waves

The harmonics of plane longitudinal and trans-verse waves in nonlinear elastic solids with up to cubic nonlinearity in a one-dimensional setting are investigated in this paper. It is shown that due to quadratic nonlinearity, a transverse wave generates a second longitudinal harmonic. This propagates with the velocity of transverse waves, as well as resonant transverse first and third harmonics due to the cubic and quadratic nonlinearities. A longitudinal wave generates a resonant longitudinal second harmonic, as well as first and third harmonics with amplitudes that increase linearly and quadratically with distance propagated. In a second investigation, incidence from the linear side of a pri-mary wave on an interface between a linear and a nonlinear elastic solid is considered. The incident wave crosses the interface and generates a harmonic with interface conditions that are equilibrated by compensatory waves propagating in two directions away from the interface. The back-propagated compensatory wave provides information on the nonlinear elastic constants of the material behind the interface. It is shown that the amplitudes of the compensatory waves can be increased by mixing two incident longitudinal waves of appropriate frequencies.     

Simulation of multi-cracks in solids using nonlinear elastic wave spectroscopy with a time-reversal process

In non-classical nonlinear media, much characteristic information is contained in their dynamic elastic responses. A method combining nonlinear elastic wave spectroscopy (NEWS) with a time-reversal (TR) process is used in this numerical study, in which the presence of one defect and two defects acting with non-classical nonlinearity in an attenuating medium is simulated. Nonlinear defect behavior is introduced using a modified Preisach–Mayergoyz (PM) model. Two methods are used to determine retrofocal quality: harmonic filtering and modulated wave filtering. In the simulation, the nonlinear signal is filtered from the received continuous wave, then reversed and re-sent; a crack image can be obtained from the nonlinear signal in a lossy solid. By comparison with the actual defect, the image can reflect the distribution of one or two flaws, which show the feasibility and value of the NEWS–TR methodology for microdamage imaging of two defects. These results also show that images obtained with different harmonic and modulated frequencies can reflect the presence of defects. With increasing frequency, the crack positions obtained from the image change due to the influence of solid loss and interaction with sound waves.     

一种土的非线性弹性本构模型参数的反演方法

旨在提出一种土的非线性弹性本构模型参数反演的方法。以现今普遍实行的地基载荷试验为基础,依据遗传算法的组合优化理论,采用正演计算和遗传算法优化相结合的方式,建立了土层非线性弹性本构模型参数反演的方法;并依据某黄土场地地基载荷试验数据,实施了黄土土层非线性弹性本构模型参数反演的全过程。计算结果表明,所建立的方法可以实现土层非线性弹性本构模型中相互关联的多个参数的组合优化,并在对初始值要求较低的情况下,可以获得良好的参数反演精度。从而为土的变形特性分析和土与其中及相邻结构的共同作用分析,提供了较好的土体本构模型参数的确定方法。     

A micromechanical mass sensing method based on amplitude tracking within an ultra-wide broadband resonance

We propose a mass sensing scheme in which amplitude shifts within a nonlinear ultra-wide broadband resonance serve as indicators for mass detection. To achieve the broad resonance bandwidth, we considered a nonlinear design of the resonator comprised of a doubly clamped beam with a concentrated mass at its center. A reduced-order model of the beam system was constructed in the form of a discrete spring-mass system that contains cubic stiffness due to axial stretching of the beam in addition to linear stiffness (Duffing equation). The cubic nonlinearity has a stiffening effect on the frequency response causing nonlinear bending of the frequency response toward higher frequencies. Interestingly, we found that the presence of the concentrated mass broadens the resonant bandwidth significantly, allowing for an ultra-wide operational range of frequencies and response amplitudes in the proposed mass sensing scheme. A secondary effect of the cubic nonlinearity is strong amplification of the third harmonic in the beam’s response. We computationally study the sensitivity of the first and third harmonic amplitudes to mass addition and find that both metrics are more sensitive than the linearized natural frequency and that in particular, the third harmonic amplitude is most sensitive. This type of open-loop mass sensing avoids complex feedback control and time-consuming frequency sweeping. Moreover, the mass resolution is within a functional range, and the design parameters of the resonator are reasonable from a manufacturing perspective.     

Comparative study of different discrete element models and evaluation of equivalent micromechanical parameters

Comparative studies of different discrete element models of a rock-type material are presented. The discrete element formulation employs spherical particles with the cohesive interaction model combining linear elastic behaviour with brittle failure. Numerical studies consisted in simulation of the uniaxial compression test. Two cylindrical specimens with particle size distributions yielding different degree of heterogeneity have been used. Macroscopic response produced by different discrete element models has been compared. The main difference between the compared models consists in the evaluation of micromechanical constitutive parameters. Two approaches are compared. In the first approach, the contact stiffness and strength parameters depend on the local particle size, while in the second approach, global uniform contact parameters are assumed for all the contacting pairs in function of average geometric measures characterizing the particle assembly. The size dependent contact parameters are calculated as functions of geometric parameters characterizing each contacting particle pair. As geometric scaling parameters, the arithmetic and harmonic means, as well as the minimum of the radii of two contacting particles are considered. Two different models with size dependent contact parameters are formulated. The performance of these models is compared with that of the discrete element model with global uniform contact parameters. Equivalence between the models with size dependent and uniform contact parameters has been checked. In search of this equivalence, different methods of evaluation of global uniform parameters have been studied. The contact stiffness has been evaluated in terms of the average radius of the particle assembly or in terms of the averages of the arithmetic and harmonic means of the contact pair radii, the geometric parameters used in the evaluation of the contact stiffness in the size-dependent models. The uniform contact strengths have been determined as functions of the averages of radii squares, squares of arithmetic radii means or squares of minimum radii of the contacting pairs.For the more homogenous specimen, the models with local size dependent parameters and models with global uniform parameters give similar response. The models with uniform parameters evaluated according to the averages of the geometric parameters used in the evaluation of local parameters ensure better agreement with the respective models with size-dependent parameters than the models with uniform parameters evaluated according to the particle radii. Simulations using the more heterogenous specimen reveal differences between the considered models. There are significant differences in stress–strain curves as well as in the failure pattern. The models with local size-dependent parameters are more sensitive to the change of heterogeneity than the model with global uniform parameters.     

Tailoring of pinched hysteresis for nonlinear vibration absorption via asymptotic analysis

The method of multiple scales is adopted to investigate the dynamic response of a nonlinear Vibration Absorber (VA) whose constitutive behavior is governed by hysteresis with pinching. The asymptotic analysis is first devoted to study the response of the absorber to harmonic excitations and to evaluate its sensitivity to the main constitutive parameters. The frequency response obtained in closed form allows to carry out the stability analysis together with a parametric study leading to behavior charts characterizing multi-valued softening/hardening responses or single-valued, quasi-linear responses. A two-degree-of-freedom model of a primary nonlinear structure endowed with the hysteretic vibration absorber is investigated to explore transfers of energy from the structure to the absorber resulting into optimal vibration amplitude reduction. The asymptotic solution is proved to be in good agreement with the numerical solution obtained via continuation. The asymptotic approach is embedded into a differential evolutionary algorithm to obtain a multi-parameter optimization procedure by which the optimal hysteresis parameters are found.     

Nonlinear elastic wave propagation in a phononic material with periodic solid–solid contact interface

Phononic materials enable enhanced dynamic properties, and offer the ability to engineer the material response. In this work we study the wave propagation in such a structure when introduced with nonlinearity. Our system is comprised of pre-compressed material with periodic solid–solid contacts, which exhibit a quadratic nonlinearity for small displacements. We suggest a new approach to modeling this system, where we discretize the unit cell in order to derive an approximate analytical solution using a perturbation method, which we are then able to easily validate numerically. With these methods, we study the band structure in the system and the second harmonic generation originating from the nonlinearity. We qualitatively analyze the second harmonic response of the system in terms of the single-crack response with linear band structure considerations. Significant band structure manipulation by changing system parameters is demonstrated, including possible in-situ tuning. The system also exhibits effective frequency doubling, i.e. the transmitted wave is primarily comprised of the second harmonic wave, for a certain range of frequencies. We demonstrate very high robustness to disorder in the system, in terms of band structure and second harmonic generation. These results have possible applications as frequency-converting devices, tunable engineered materials, and in non-destructive evaluation.     

FORCED WAVE PROPAGATION IN VISCOELASTIC CABLE WITH SMALL SAG

Based on the linear viscoelastic differential constitutive law and cable structural model, the coupled longitudinal-transverse waves that propagate along a viscoelastic cable with small curvature is investigated. A mathematical model is presented that describes the three-dimensional nonlinear response of a viscoelastic cable. An asymptotic form of this model is obtained for the linear response of cables having small equilibrium curvature. The spectral relation governing the propagating waves is derived using transform methods. The spectral relation is employed in deriving a Green's function that is then used to construct solutions for in-plane response under distributed harmonic excitation. Analysis of forced response reveals the existence of two types of periodic waves that propagate through the cable, one characterizing extension-comprehensive deformation and the other characteristic transverse deformation. Project supported by the National Natural Science Foundation of China (No. 59635140).     

Second-harmonic generation of two-dimensional elastic wave propagation in an infinite layered structure with nonlinear spring-type interfaces

The two-dimensional elastic wave propagation in an infinite layered structure with nonlinear interlayer interfaces is analyzed theoretically to investigate the second-harmonic generation due to interfacial nonlinearity. The structure consists of identical isotropic linear elastic layers that are bonded to each other by spring-type interfaces possessing identical linear normal and shear stiffnesses but different quadratic nonlinearity parameters. Explicit analytical expressions are derived for the second-harmonic amplitudes when a single monochromatic Bloch mode propagates in the structure in arbitrary directions by applying the transfer-matrix approach and the Bloch theorem to the governing equations linearized by a perturbation method. The second-harmonic generation by a single nonlinear interface and by multiple consecutive nonlinear interfaces are shown to be profoundly influenced by the band structure of the layered structure, the fundamental Bloch wave mode, and its propagation direction. In particular, the second harmonics generated at multiple consecutive interfaces are found to grow cumulatively with the propagation distance when the phase matching occurs between the Bloch modes at the fundamental and double frequencies.     

大变形下初始斜交异性本构方程

采用材料主轴法,建立了初始斜交异性材料在变形构形(Euler描述)下的斜交异性本构方程,以及在初始构形(Lagrange描述)下的形式.具体给出了斜交异性线弹性材料方程的显式,它在Lagrange描述下形式简洁,可方便地用于有限元计算.文中指出,在变形构形下是线弹性的材料,在Lagrange描述下其本构方程一般已成为非线性,我们称之为本构转换非线性.这种非线性在实际的有限元计算中还未引起重视.为理论简明,本构方程是对二维给出的.     

Seismic base isolation by nonlinear mode localization

Summary In this paper, the performance of a nonlinear base-isolation system, comprised of a nonlinearly sprung subfoundation tuned in a 1∶1 internal resonance to a flexible mode of the linear primary structure to be isolated, is examined. The application of nonlinear localization to seismic isolation distinguishes this study from other base-isolation studies in the literature. Under the condition of third-order smooth stiffness nonlinearity, it is shown that a localized nonlinear normal mode (NNM) is induced in the system, which confines energy to the subfoundation and away from the primary or main structure. This is followed by a numerical analysis wherein the smooth nonlinearity is replaced by clearance nonlinearity, and the system is excited by ground motions representing near-field seismic events. The performance of the nonlinear system is compared with that of the corresponding linear system through simulation, and the sensitivity of the isolation system to several design parameters is analyzed. These simulations confirm the existence of the localized NNM, and show that the introduction of simple clearance nonlinearity significantly reduces the seismic energy transmitted to the main structure, resulting in significant attenuation in the response. This work was supported in part by the National Science Foundation Grant CMS 00-00060. The authors are grateful for this support.     

一种脆性材料动态本构损伤参数的计算反求方法

提出一种利用平板撞击的瞬态速度剖面响应反求陶瓷脆性材料的动态本构损伤参数的方法.通过数值模拟再现平板撞击试验中的正平面波在飞片、样品和窗口材料中的传播情况,分析了"样品/窗口"界面瞬态速度剖面响应与材料参数之间的映射关系,进而建立了用响应反求本构参数的反问题,且通过一些数值试验验证该反问题是适定的.基于平板撞击速度剖面响应反求的方法,为快速地获得高应变率下一些传统方法难以测定出的本构参数提供新途径.     

Pseudo-fault induction in engineering structures: Multi-input multi-output control

In a previous paper, it was demonstrated that any linear system could be made to respond to harmonic excitation as if a static nonlinearity of specified type and position were present, this response being obtained at a single predetermined point on the structure. The method requires the excitation of the linear structure by an additional or auxiliary input. In the present paper, the theory is extended to allow the possibility of producing a specified nonlinear response at more than one point on a linear structure. It is shown that N responses can be obtained by specifying N or less auxiliary inputs. The theory is also extended to provide for polynomial damping in addition to stiffness nonlinearity. The theory is validated using numerical simulation of MDOF lumped-parameter systems.     

Subharmonic and ultra-subharmonic response of nonlinear elastic beams subjected to harmonic excitation

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Thermal Fatigue Damage Assessment in an Isotropic Pipe Using Nonlinear Ultrasonic Guided Waves

The use of nonlinear ultrasonic waves has been accepted as a potential technique to characterize the state of material micro-structure in solids. The typical nonlinear phenomenon is generation of second harmonics. Second harmonic generation of ultrasonic waves propagation has been vigorously studied for tracking material micro-damages in unbounded media and plate-like waveguides. However, there are few studies of launching second harmonic guided wave propagation in tube-like structures. Considering that second harmonics could provide useful information sensitive for material degradation condition, this research aims at developing a procedure for detecting second harmonics of ultrasonic guided wave in an isotropic pipe. The second harmonics generation of guided wave propagation in an isotropic and stress-free elastic pipe is investigated. Flexible polyvinylidene fluoride (PVDF) comb transducers are used to measure fundamental wave and second harmonic one. Experimental results show that nonlinear parameters increase monotonically with propagation distance. This work experimentally verifies that the second harmonics of guided waves in pipe have the cumulative effect with propagation distance. The proposed procedure is applied to assessing thermal fatigue damage indicated by nonlinearity in an aluminum pipe. The experimental observation verifies that nonlinear guided waves can be used to assess damage levels in early thermal fatigue state by correlating them with the acoustic nonlinearity.     

Wave propagation and dispersion in microstructured wool felt

On the basis of the experimental data of the piano hammers study the one-dimensional constitutive equation of the wool felt material is proposed. This relation enables deriving a nonlinear partial differential equation of motion with third order terms, which takes into account the elastic and hereditary properties of a microstructured felt. This equation of motion is used to study pulse evolution and propagation in the one-dimensional case. Thorough analysis both of the linear and nonlinear problems is presented. The physical dimensionless parameters are established and their importance in describing the dispersion effects is discussed. It is shown that both normal and anomalous dispersion types exist in wool felt material. The dispersion analysis shows also that for the certain ranges of physical parameters negative group velocity will appear. The initial value problem is considered and the analysis of the numerical solution describing the strain wave evolution is provided. The influence of the material parameters on the form of a propagating pulse is demonstrated and explained.     

Nonlinear Isolator Dynamics at Finite Deformations: An Effective Hyperelastic,Fractional Derivative,Generalized Friction Model

In presenting a nonlinear dynamic model of a rubber vibrationisolator, the quasistatic and dynamic motion influences on theforce response are investigated within the time and frequencydomain. It is found that the dynamic stiffness at the frequency ofa harmonic displacement excitation, superimposed upon the longterm isolator response, is strongly dependent on staticprecompression, dynamic amplitude and frequency. The problems ofsimultaneously modelling the elastic, viscoelastic and frictionforces are removed by additively splitting them, modelling theelastic force response by a nonlinear, shape factor basedapproach, displaying results that agree with those of aneo-Hookean hyperelastic isolator at a long term precompression.The viscoelastic force is modeled by a fractional derivativeelement, while the friction force governs from a generalizedfriction element displaying a smoothed Coulomb force. A harmonicdisplacement excitation is shown to result in a force responsecontaining the excitation frequency and its every otherhigher-order harmonic, while using a linearized elastic forceresponse model, whereas all higher-order harmonics are present forthe fully nonlinear case. It is furthermore found that the dynamicstiffness magnitude increases with static precompression andfrequency, while decreasing with dynamic excitationamplitude – eventually increasing at the highest amplitudes due tononlinear elastic effects – with its loss angle displaying amaximum at an intermediate amplitude. Finally, the dynamicstiffness at a static precompression, using a linearized elasticforce response model, is shown to agree with the fully nonlinearmodel except at the highest dynamic amplitudes.     

Elasto-plasticity of paper

We develop a constitutive model of paper's in-plane biaxial tensile response accounting for the elastic–plastic hardening behavior, and its orthotropic character. The latter aspect is motivated by machine-made papers, which, in contrast to isotropic laboratory handsheets, are strongly oriented. We focus on modeling paper's response under monotonic loading, this restriction allowing us to treat the elastic-plastic response as a physically nonlinear elastic one. A strain energy function of a hyperbolic tangent form is developed so as to fit the entire range of biaxial and uniaxial experiments on a commercial grade paper. This function may then be introduced as the free energy function into a model based on thermomechanics with internal variables.     

Second-harmonic generation in an infinite layered structure with nonlinear spring-type interfaces

The second-harmonic generation characteristics in the elastic wave propagation across an infinite layered structure consisting of identical linear elastic layers and nonlinear spring-type interlayer interfaces are analyzed theoretically. The interlayer interfaces are assumed to have identical linear interfacial stiffness but can have different quadratic nonlinearity parameters. Using a perturbation approach and the transfer-matrix method, an explicit analytical expression is derived for the second-harmonic amplitude when the layered structure is impinged by a monochromatic fundamental wave. The analysis shows that the second-harmonic generation behavior depends significantly on the fundamental frequency reflecting the band structure of the layered structure. Two special cases are discussed in order to demonstrate such dependence, i.e., the second-harmonic generation by a single nonlinear interface as well as by multiple consecutive nonlinear interfaces. In particular, when the second-harmonic generation occurs at multiple consecutive nonlinear interfaces, the cumulative growth of the second-harmonic amplitude with distance is only expected in certain frequency ranges where the fundamental as well as the double frequencies belong to the pass bands of the layered structure. Furthermore, a remarkable increase of the second-harmonic amplitude is found near the terminating edge of pass bands. Approximate expressions for the low-frequency range are also obtained, which show the cumulative growth of the second-harmonic amplitude with quadratic frequency dependence.     

Enabling reduced-order data-driven nonlinear identification and modeling through naïve elastic net regularization

This work discusses an improved method of reduced-order modeling for existing data-driven nonlinear identification techniques through the incorporation of naïve elastic net regularization. The data-driven methods considered for this study operate using basis functions to represent the observed nonlinearity. Elastic net regularization is used to minimize the number of non-zero coefficients, thus modifying the basis functions and providing a compact representation. The ability of the naïve elastic net to provide reduced-order nonlinear models that can both accurately fit various data sets and computationally simulate new responses is illustrated through studies considering both synthetic data and experimental data. In both cases, the results obtained with the naïve elastic net are shown to match or outperform those from other traditional methods.