Structure of weak shock waves in 2-D layered material systems

Shock waves in homogeneous materials in the absence of phase transitions are understood to have a one-wave structure. However, upon loading of a layered heterogeneous material system a two-wave structure is obtained––a leading shock front followed by a complex pattern that varies with time. This dual shock-wave pattern can be attributed to material architecture through which the shock wave propagates, i.e. the impedance (and geometric) mismatch present at various length scales, and nonlinearities arising from material inelasticity and failure.The objective of the present paper is to provide a better understanding of the role of material architecture in determining the structure of weak shock waves in 2-D layered material systems. Normal plate-impact experiments are conducted on 2-D layered material targets to obtain both the precursor decay and the late-time dispersion. The particle velocity at the free surface of the target plate is measured by using a multi-beam VALYN VISAR. In order to understand the effects of layer thickness and the distance of wave propagation on elastic precursor decay and late-time dispersion several different targets with various layer and target thicknesses are employed. Moreover, in order to understand the effects of material inelasticity both elastic–elastic and elastic–viscoelastic bilaminates are utilized.The results of these experiments are interpreted by using asymptotic techniques to analyze propagation of acceleration waves in 2-D layered material systems. The analysis makes use of the Laplace transform and Floquet theory for ODE’s with periodic coefficients [Asymptotic solutions for wave propagation in elastic and viscoelastic bilaminates. In: Developments in Mechanics, Proceedings of the 14th Mid-Eastern Mechanics Conference, vol. 26, no. 8, pp. 399–417]. Both wave-front and late-time solutions for step-pulse loading on layered half-space are compared with the experimental observations. The results of the study indicate that the structure of acceleration waves is strongly influenced by impedance mismatch of the layers constituting the laminates, density of interfaces, distance of wave propagation, and the material inelasticity.     

Dispersion relations and mode shapes for waves in laminated viscoelastic composites by variational methods

The propagation of oscillatory waves through periodic elastic composites has been analysed on the basis of the Floquet theory. This leads to self-adjoint differential equation systems which it was proved convenient to solve by variational methods. Many composites, such as the light-weight high-strength boron-epoxy material, consist of strong reinforcing components in a plastic matrix. The latter can exhibit viscoelastic properties which can have a significant influence on wave propagation characteristics. Replacement of the elastic constant by the viscoelastic complex modulus changes the mathematical structure so that the differential equation system is no longer self-adjoint. However, a modification of the variational principles is suggested which retains formal self-adjointness, and yields variational principles which contain additional boundary terms. These are applied to the determination of wave speeds and mode shapes for a laminated composite made of homogeneous elastic reinforcing plates in a homogeneous viscoelastic matrix for plane waves propagating normally to the reinforcing plates. These results agree well with the exact solution which can be evaluated in this simple case. The variational principles permit solutions for periodic, but otherwise arbitrary variation of material properties.     

On secondary loops in LAOS via self-intersection of Lissajous–Bowditch curves

When the shear stress measured in large amplitude oscillatory shear (LAOS) deformation is represented as a 2-D Lissajous–Bowditch curve, the corresponding trajectory can appear to self-intersect and form secondary loops. This self-intersection is a general consequence of a strongly nonlinear material response to the imposed oscillatory forcing and can be observed for various material systems and constitutive models. We derive the mathematical criteria for the formation of secondary loops, quantify the location of the apparent intersection, and furthermore suggest a qualitative physical understanding for the associated nonlinear material behavior. We show that when secondary loops appear in the viscous projection of the stress response (the 2-D plot of stress vs. strain rate), they are best interpreted by understanding the corresponding elastic response (the 2-D projection of stress vs. strain). The analysis shows clearly that sufficiently strong elastic nonlinearity is required to observe secondary loops on the conjugate viscous projection. Such a strong elastic nonlinearity physically corresponds to a nonlinear viscoelastic shear stress overshoot in which existing stress is unloaded more quickly than new deformation is accumulated. This general understanding of secondary loops in LAOS flows can be applied to various molecular configurations and microstructures such as polymer solutions, polymer melts, soft glassy materials, and other structured fluids.     

瞬态波在层合板中传播的粘弹比拟法

本文发展了粘弹比拟理论,并将之用于求解半无限空间三层复合材料在垂直层合方向传播的瞬态波问题。对于层合板中应力波的传播问题,寻找到了一等效粘弹体,并用一种较好的Laplace变换的数值反演法求得了等效松弛函数和其它一些必要的辅助函数。用特征线法求得了等效粘弹体的应力和速度,进而得到了三层复合材料中心的应力、速度,进一步就得到了层中任意点的应力和速度。对于一个可由精确理论(射线理论)给出计算结果的问题,将粘弹比拟理论的结果和射线理论的结果进行了比较,结果表明,粘弹比拟理论对三层复合材料的瞬态波传播问题是相当成功的。     

TTS in LAOS: validation of time-temperature superposition under large amplitude oscillatory shear

The validation of time-temperature superposition of non-linear parameters obtained from large amplitude oscillatory shear is investigated for a model viscoelastic fluid. Oscillatory time sweeps were performed on a 11?wt.% solution of high molecular weight polyisobutylene in pristane as a function of temperature and frequency and for a broad range of strain amplitudes varying from the linear to the highly non-linear regime. Lissajous curves show that this reference material displays strong non-linear behaviour when the strain amplitude is exceeding a critical value. Elastic and viscous Chebyshev coefficients and alternative non-linear parameters were obtained based on the framework of Ewoldt et al. (J Rheol 52(6):1427?C1458, 2008) as a function of temperature, frequency and strain amplitude. For each strain amplitude, temperature shift factors a _T (T) were calculated for the first order elastic and viscous Chebyshev coefficients simultaneously, so that master curves at a certain reference temperature T _ref were obtained. It is shown that the expected independency of these shift factors on strain amplitude holds even in the non-linear regime. The shift factors a _T (T) can be used to also superpose the higher order elastic and viscous Chebyshev coefficients and the alternative moduli and viscosities onto master curves. It was shown that the Rutgers-Delaware rule also holds for a viscoelastic solution at large strain amplitudes.     

A crack in a piezoelectric layer bonded to a dissimilar elastic layer under transient load

Summary  A piezoelectric layer bonded to the surface of an elastic structure is considered. The piezoelectric and the elastic layers are infinite along the x-axis and have finite thickness in the y-direction. The polarization direction of the piezoelectric material is along the y-axis. By means of the method of singular integral equations, the solution in a Laplace transform plane is demonstrated. Laplace inversion yields the results in the time domain. Numerical values of the crack tip fields under in-plane transient electromechanical loading are obtained. The influence of layers thickness on stress and electric displacement intensity factors is investigated. Received 16 March 2000; accepted for publication 16 August 2000     

Cloaking of a circular cylindrical elastic inclusion from antiplane elastic waves and resonance effects

Cloaking of a circular cylindrical elastic inclusion embedded in a homogeneous linear isotropic elastic medium from antiplane elastic waves is studied. The transformation or change-of-variables method is used to determine the material properties of the cloak and the homogenization theory of composites is used to construct a multilayered cloak consisting of many bi-material cells. The large system of algebraic equations associated with this problem is solved by using the concept of multiple scattering with wave expansion coefficient matrices. Numerical results for cloaking of an elastic inclusion and a rigid inclusion are compared with the case of a cavity. It is found that while the cloaking patterns for the three cases are similar, the major difference is that standing waves are generated in the elastic inclusion and the multilayered cloak cannot prevent the motion inside the elastic inclusion, even though the cloak seems nearly perfect. Waves can penetrate into and cause vibrations inside the elastic inclusion, where the amplitude of standing waves depend on the material properties of the inclusion but are very much reduced when compared to the case when there is no cloak. For a prescribed mass density, the displacements inside the elastic cylinder decrease as the shear modulus increases. Moreover, the cloaking of the elastic inclusion over a range of wavenumbers is also investigated. There is significant low frequency scattering even if the cloak consists of a large number of layers. When the wavenumber increases, the multilayered cloak is not effective if the cloak consists of an insufficient number of layers. Resonance effects that occur in cloaking of elastic inclusions are also discussed.     

Load transfer in fibre-reinforced composites with viscoelastic matrix: an analytical study

In A fibre-reinforced 2D composite material with elastic fibres and viscoelastic, isotropic matrix is studied. Starting from the solution of a reference-problem with elastic matrix material the elastic matrix parameters are substituted by their viscoelastic correspondents in the Laplace domain. For simplification the time-dependent solution is approximated by using limiting value theorems that give information about the time-dependent solution for t → 0 and t → ∞. Then the method of asymptotically equivalent functions is used and illustrated with examples of a steel fibre in a PMMA matrix. The analytical solutions are compared with their numerical counterparts. In summary it can be stated that this paper is a further contribution to the vast literature about the application of the correspondence principle to the solution of special problems of the linear viscoelasticity.     

Time–transient response for ultrasonic guided waves propagating in damped cylinders

Herein, an enhanced spectral finite element (SFE) formulation to calculate the time–transient response in cylindrical waveguides is proposed. The original aspect over SFE-based formulations consists in the possibility to account for the effect of material absorption, i.e. guided waves attenuation, on the calculation of the time–transient response.First, the damped steady-state response is constructed by a weighted superposition of the waveguide modal properties obtained from the spectral decomposition of the governing wave equation. To this purpose an enhanced spectrally formulated finite element is developed, in which material damping is included allowing for complex stress–strain viscoelastic constitutive relations in force of the correspondence principle. Dispersive modal properties for the damped waveguide (phase velocity, energy velocity, attenuation and wavestructures) follow straightforwardly by simple formulae. Next, the frequency response of the problem is calculated by weighting the modal data and the spectrum of the applied time-dependent force via Cauchy residue theorem. Finally, the inverse Fourier transform of the frequency response leads to the time–transient response for propagative damped guided waves.The approach is not restricted to any anisotropy degree, holds for any linear viscoelastic constitutive relation that can be characterized and formulated in the frequency domain and it can be applied to SFE formulations for arbitrary cross-section waveguides. A study on guided waves propagating in a scheduled 4.in-40 ANSI steel pipe is presented, where the steel is considered first as perfectly elastic and then as an hysteretic viscoelastic medium, in order to show the effect of material absorption on the time–transient response.     

Design of armor for protection against blast and impact

The features of blast and impact that can damage a delicate target supported by a structure include both the peak pressure and the impulse delivered to the structure. This study examines how layers of elastic and visco-elastic materials may be assembled to mitigate these features. The impedance mismatch between two elastic layers is known to reduce the pressure, but dissipation is required to mitigate the transmitted impulse in light-weight armor. A novel design concept called impact or blast tuning is introduced in which a multi-layered armor is used to tune the stress waves resulting from an impact or blast to specific frequencies that match the damping frequencies of visco-elastic layers. The material and geometrical parameters controlling the viscous dissipation of the energy within the armor are identified for a simplified one-dimensional system, to provide insight into how the optimal design of multi-use armor might be based on this concept.     

Wave propagation in elastic lattices subjected to a local harmonic loading. I. A quasi-one-dimensional problem

The anti-plane dynamics of infinite (?∞ < x < ∞, ?∞ < y < ∞) material-bond rectangular lattices subjected to a uniform monochromatic excitation of the x = 0 line nodes is studied. A quasi-one-dimensional model is formulated: the original lattice is considered as an infinite waveguide in the x-direction with periodically joined bonds bounded in the y-direction. In such a structure, the wave pattern consists of waves propagated along x-axis and standing waves along y-axis. Steady and unsteady processes are investigated. Dispersion relations are analyzed and resonance points are detected. A combined analytical–numerical approach is used to describe (i) the quasi-steady propagation of waves when the source frequency is within the pass-band, (ii) development of resonance waves, and (iii) percolation of perturbations to the periphery when the excitation frequency is within the stop-band. Long-wave and short-wave components of solutions are compared with those for a simplified 1D mass-spring lattice (MSL) model.     

PREDICTION OF THE VISCOELASTIC PROPERTIES OF THE EQUIVALENT PARTICLE FOR THE INTERCALATED MULTI-LAYER STACK OF NANOPLASTICS

The aim of this paper is to apply the asymptotic homogenization method to deter- mining analytically and numerically the transversely isotropic viscoelastic relaxation moduli of the equivalent particle for the intercalated multi-layer stack of intercalated type nanoplastics.A two-phase multilayered material containing n layers is considered.The matrix is assumed to be an isotropic viscoelastic standard linear body and the reinforcement is assumed to be an isotropic elastic body.Final explicit analytical formulae for the effective elastic moduli of the multilay- ered material are derived first;and then the correspondence principle is employed to obtain the homogenized relaxation moduli of the equivalent intercalated particle.A numerical example is given.Final explicit analytical formulae in the time domain derived here make it convenient to estimate the influence of all the particle parameters of micro-structural details on the effective properties of the equivalent intercalated particle.The results of this paper can also be applied to multi-layer composites.     

On Saint-Venant's principle in linear elastodynamics

We investigate the spatial behaviour of the steady state and transient elastic processes in an anisotropic elastic body subject to nonzero boundary conditions only on a plane end. For the transient elastic processes, it is shown that at distance x ₃ >ct from the loaded end, (c is a positive computable constant and t is the time), all the activity in the body vanishes. For x ₃ , an appropriate measure of the elastic process decays with the distance from the loaded end, the decay rate of end effects being controlled by the factor % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaKazaaiacaGGOaGccaaIXaGaaeiiaiabgkHiTiaabccadaWcaaqa% amXvP5wqonvsaeHbfv3ySLgzaGqbciab-Hha4naaBaaaleaacaqGZa% aabeaaaOqaaiaabogacaqG0baaaKazaakacaGGPaaaaa!4BB0!\[(1{\text{ }} - {\text{ }}\frac{{x_{\text{3}} }}{{{\text{ct}}}})\]. Next, it is shown that for isotropic materials, in the case of a steady state vibration, an analogue of the Phragmén-Lindelöf principle holds for an appropriate cross-sectional measure. One immediate consequence is that in the class of steady state vibrations for which a quasi-energy volume measure is bounded, this measure decays at least algebraically with the distance from the loaded end.     

On a new electromechanical switch using the reversible wavy elastic response of metallic glass ribbons

For the first time, practical applications as an alarm device and automatic filling of an aquarium using an electromechanical switch manufactured from metallic glass (MG) ribbon is proposed. The elastic response of an initial arc-shaped MG ribbon-based Fe_90.65B_3.9Cr_2.75Si_2.7 is studied and exploited. Under the applied load F, the amorphous material exhibits a reversible elastic wavy response. During the elastic deformation and multiplication of harmonic undulations, a perfect linear contact between the waves and support is established. This contact position is the same for the pair waves, and can be employed to ensure the passage of an electric current, since the ribbon is Fe-based. The reversible elastic wavy response of MG ribbon can be used as an electromechanical switch. The lifespan of the ribbon used as a switch is also considered.     

Laminated beams with viscoelastic interlayer

We analytically solve the time-dependent problem of a simply-supported laminated beam, composed of two elastic layers connected by a viscoelastic interlayer, whose response is modeled by a Prony’s series of Maxwell elements. This case applies in particular to laminated glass, a composite made of glass plies bonded together by polymeric films. A practical way to calculate the response of such a package is to consider also the interlayer to be linear elastic, assuming its equivalent elastic moduli to be the relaxed moduli under constant strain, after a time equal to the duration of the design action. The obtained results, that are confirmed by a full 3-D viscoelastic finite-element numerical analysis, emphasize that there is a noteworthy difference between the state of strain and stress calculated in the full-viscoelastic case or in the aforementioned “equivalent” elastic problem.     

黏弹性材料界面裂纹应力场奇性分析

研究两半无限大黏弹性体间Griffith界面裂纹在简谐载荷作用下裂纹尖端动应力场的奇异特性.通过引入裂纹张开位移和裂纹位错密度函数,相应的混合边值问题归结为一组耦合的奇异积分方程.渐近分析表明裂尖动应力场的奇异特征完全包含在奇异积分方程的基本解中.通过对基本解的深入分析发现黏弹性材料界面裂纹裂尖动应力场具有与材料参数和外载荷频率相关的振荡奇异特性.以标准线性固体黏弹材料为例讨论了材料参数和载荷频率对奇性指数和振荡指数的影响.     

On the distance to blow-up of acceleration waves in random media

We study the effects of material spatial randomness on the distance to form shocks from acceleration waves, , in random media. We introduce this randomness by taking the material coefficients and – that represent the dissipation and elastic nonlinearity, respectively, in the governing Bernoulli equation – as a stochastic vector process. The focus of our investigation is the resulting stochastic, rather than deterministic as in classical continuum mechanics studies, competition of dissipation and elastic nonlinearity. Quantitative results for are obtained by the method of moments in special simple cases, and otherwise by the method of maximum entropy. We find that the effect of even very weak random perturbation in and may be very significant on . In particular, the full negative cross-correlation between and $ results in the strongest scatter of , and hence, in the largest probability of shock formation in a given distance x. Received November 6, 2001 / Published online September 4, 2002 Dedicated to Professor Ingo Müller on the occasion of his 65th birthday Communicated by Kolumban Hutter, Darmstadt     

An adaptation of finite linear viscoelasticity theory for rubber-like viscoelasticity by use of a generalized strain measure

In this paper a simplified three-dimensional constitutive equation for viscoelastic rubber-like solids is derived by employing a generalized strain measure and an asymptotic expansion similar to that used by Coleman and Noll (1961) in their derivation of finite linear viscoelasticity (FLV) theory. The first term of the expansion represents exactly the time and strain separability relaxation behavior exhibited by certain soft polymers in the rubbery state and in the transition zone between the glassy and rubbery states. The relaxation spectra of such polymers are said to be deformation independent. Retention of higher order terms of the asymptotic expansion is recommended for treating deformation dependent spectra.Certain assumptions for the solid theory are relaxed in order to obtain a constitutive equation for uncross-linked liquid materials which exhibit large elastic recovery properties.Apart from the strain energyW(I₁,I ₂), which alternatively characterizes the long-time elastic response of solids or the instantaneous elastic response of elastic liquids, only the linear viscoelastic relaxation modulus is required for the first-order theory. Both types of material functions can be obtained, in theory, from simple laboratory testing procedures. The constitutive equations for solids proposed by Chang, Bloch and Tschoegl (1976) and a special form of K-BKZ theory for elastic liquids are shown to be particular cases of the first-order theory.Previously published experimental data on a cross-linked styrene-butadiene rubber (SBR) and an uncross-linked polyisobutylene (PIB) rubber is used to corroborate the theory.     

Harmonic waves in one-, two- and three-dimensional composites: Bounds for eigenfrequencies

Harmonic waves in one-, two- and three-dimensional elastic composites with periodic structure are considered. Based on a new quotient recently proposed by Nemat-Nasser, lower and upper bounds for the eigenfrequencies are developed. For illustration waves propagating normal to the layers in layered composites, and normal to the fibers in fiber-reinforced composites, are considered. These examples show that the new quotient is very effective and yields very accurate results for the considered class of problems. While these results are of interest in their own right, they can be used to check the effectiveness of various approximate theories which recently have been proposed by various authors.     

Determination of the non-linear parameter (mobility factor) of the Giesekus constitutive model using LAOS procedure

The paper introduces a novel procedure to determine the non-linear parameter of the Giesekus model, in relation to the characterization of the non-linear oscillatory shear regime of viscoelastic polymer solutions based on polyacrylamide. Instead of using the shear-thinning viscosity as the representative non-linear effect, the third harmonic in the Fourier spectrum of the shear stress response signal is considered for computing the mobility factor. The fluid is subjected to large amplitude oscillatory shear (LAOS) and its response is recorded. Deviations of this signal from the sinusoidal form are specific to each material and gives both qualitative and quantitative measures of the non-linearity. By fitting the material response with the corresponding numerical solutions of the n-modes Giesekus constitutive relation, one can extract the values of the non-linear α_i-parameters that describe the fluid rheology. It is demonstrated that this procedure, which can be successfully applied to semi-concentrated polymer solutions, provides better results than the classical viscosity-fit method.