The analysis of the impact response of a thin plate via fractional derivative standard linear solid model

The impact of a rigid body upon an infinite isotropic plate is investigated for the case when the viscoelastic features of the plate represent themselves only in the place of contact and are governed by the standard linear solid model with fractional derivatives. Thus, the problem concerns the shock interaction of the dropping mass and the target, wherein instead of the Hertz contact law the generalized fractional derivative standard linear solid law is employed as a law of interaction. The part of the plate beyond the contact domain is assumed to be elastic, and its behaviour is described by the equations of motion which take rotary inertia and shear deformations into account. It is assumed that transient waves generate in the plate at the moment of impact, the influence of which on the contact domain is considered using the theory of discontinuities. To determine the desired values behind the transverse shear wave front, one-term ray expansions are used, as well as the equations of motion of the falling mass and the contact region. As a result, we are led to a set of two linear differential equations, the solution of which is found analytically by the Laplace transform and by the Euler substitution method. This allows the contact force to be determined as a function of time.     

Flexible polyurethane foam modelling and identification of viscoelastic parameters for automotive seating applications

A hereditary model and a fractional derivative model for the dynamic properties of flexible polyurethane foams used in automotive seat cushions are presented. Non-linear elastic and linear viscoelastic properties are incorporated into these two models. A polynomial function of compression is used to represent the non-linear elastic behavior. The viscoelastic property is modelled by a hereditary integral with a relaxation kernel consisting of two exponential terms in the hereditary model and by a fractional derivative term in the fractional derivative model. The foam is used as the only viscoelastic component in a foam-mass system undergoing uniaxial compression. One-term harmonic balance solutions are developed to approximate the steady state response of the foam-mass system to the harmonic base excitation. System identification procedures based on the direct non-linear optimization and a sub-optimal method are formulated to estimate the material parameters. The effects of the choice of the cost function, frequency resolution of data and imperfections in experiments are discussed. The system identification procedures are also applied to experimental data from a foam-mass system. The performances of the two models for data at different compression and input excitation levels are compared, and modifications to the structure of the fractional derivative model are briefly explored. The role of the viscous damping term in both types of model is discussed.     

基于分数阶流变模型的铁基块体非晶合金黏弹性行为研究

采用分数阶黏弹单元替代经典模型中的黏壶, 结合非晶合金在外加载荷作用下的微观结构演化, 建立了以分数阶微积分表示的非晶合金黏弹性本构模型. 并根据Hertz弹性理论及分数阶黏弹性本构模型, 推导了块体非晶合金在纳米压痕球形压头下的位移与载荷及时间关系式. 基于推导的解析式, 对铁基块体非晶合金在表观弹性区的纳米压痕位移与载荷及时间曲线进行了非线性拟合分析. 相较于整数阶模型, 分数阶模型不仅具有较高的拟合精度, 其拟合参数能敏锐地反应加载速率对块体非晶合金黏弹性行为的影响, 且参数的变化规律与载荷作用下非晶合金微观结构演化呈现出较强的相关性.     

An analytical model to predict the response of fluid-filled shells to impact—a model for blunt head impacts

An approximate analytical model to predict the response of a fluid-filled shell of arbitrary thickness impacting with a solid elastic sphere is proposed and the limits of applicability of the equations developed are discussed. The model is based on combining the Hertzian contact stiffness and the effective local membrane and bending stiffness to derive implicit expressions for global impact characteristics including the duration of impact, the peak force transmitted, peak global acceleration of shell and sphere, and the resultant pressures induced in the fluid. Closed-form explicit expressions are also derived to predict whether the pressure response in the fluid will be hydrostatic or will exhibit large dynamic transients of pressure (and shear strain). It should be noted that the impact of hollow/empty shells with solid spheres, as well as the impact of shells with an elastic half-space, can be straightforwardly treated as limiting cases. The model is of obvious relevance to head impact modelling and selected parametric studies of the response of fluid-filled shells with geometric and material properties about those typical for the human head are given.     

Comparison of human skull and spherical shell vibrations—implications for head injury modeling

Spherical shell models have been formulated as geometric approximations of the human head. By insuring geometrical and material similarity between the model and human head, impact response and skull fracture studies were expected to yield results in close agreement with model predictions. This, however, was not the case. Model predictions of skull fracture loads were typically twice the observed level. A comparison was made between the resonant frequencies of two dry human skulls and corresponding spherical shell models. Poor agreement was observed. A vibrational analysis of the model revealed that the uniformity of the spherical shell approximmately doubles the effective stiffness and resonant frequencies as compared with the dry human skull. The elastic modulus of the model was adjusted to bring its resonances into closer agreement with those of the skull. The synthesized“effective” modulus was 50% lower than the material property of human cranial bone. Interestingly, this adjustment also brought model predictions of skull fracture into closer agreement with available human cadaver data.     

Size-dependent geometrically nonlinear free vibration analysis of fractional viscoelastic nanobeams based on the nonlocal elasticity theory

In recent decades, mathematical modeling and engineering applications of fractional-order calculus have been extensively utilized to provide efficient simulation tools in the field of solid mechanics. In this paper, a nonlinear fractional nonlocal Euler–Bernoulli beam model is established using the concept of fractional derivative and nonlocal elasticity theory to investigate the size-dependent geometrically nonlinear free vibration of fractional viscoelastic nanobeams. The non-classical fractional integro-differential Euler–Bernoulli beam model contains the nonlocal parameter, viscoelasticity coefficient and order of the fractional derivative to interpret the size effect, viscoelastic material and fractional behavior in the nanoscale fractional viscoelastic structures, respectively. In the solution procedure, the Galerkin method is employed to reduce the fractional integro-partial differential governing equation to a fractional ordinary differential equation in the time domain. Afterwards, the predictor–corrector method is used to solve the nonlinear fractional time-dependent equation. Finally, the influences of nonlocal parameter, order of fractional derivative and viscoelasticity coefficient on the nonlinear time response of fractional viscoelastic nanobeams are discussed in detail. Moreover, comparisons are made between the time responses of linear and nonlinear models.     

Evaluation of the vibrational modes of the human skull as it relates to bone-conducted sound

Analytic and numerical models are used to study bone-conducted sound and how it relates to the vibrational modes of the human skull. The analytic model is based on the solution to the acoustic and elastic wave equations and the constraining boundary conditions for a fluid-filled elastic sphere. Both models predict that most of the acoustic energy of bone-conducted sound exists in the form of surface wave vibrations at the interface between two acoustic media rather than in the bone or cranial chamber. These surface waves have phase speeds much slower than the bulk sound speed for bone. The analytic model, based on spherical elastic shells, predicts a phase speed of 775 m/s and the first resonance frequency at 1500 Hz while the numerical solution yields approximate phase speeds of 450 m/s and provides a visual display of the surface waves and diffraction effects.     

Analysis of the flow of non-Newtonian visco-elastic fluids in fractal reservoir with the fractional derivative

In both the oil reservoir engineering and seepage flow mechanics, heavy oil with relaxation property shows non-Newtonian rheological characteristics. The relationship between shear rate g& and shear stress t is nonlinear. Because of the relaxation phenomena of heavy oil flow in porous media, the equation of motion can be written as[1] 2,rrvpqkppqtrrtll秏骣+=-+琪抖桫 (1) where lv and lp are velocity relaxation and pressure retardation times. For most porous media, the above motion equation (1)…     

Study on vibro-acoustical characteristics of damped composite boxlike shells in water

Based on the structural FEM and the acoustic BEM, a numerical model of coupled elastic layer and viscoelastic layer and outside sound field is established and the vibro-acoustical characteristics of damped composite boxlike shells are studied systematically. It can be concluded that the structural vibration responses and the sound radiation are reduced significantly due to the viscoelastic layer and its effects are dependent on the geometric, physical parameters of the layer and the excitation frequency. It is also shown that compared with the bare elastic shells, the influence of the fluid compressibility on the vibration responses of shells covered with a damping layer is not evident and the effects of the free surface and the rigid plane are weakened.     

Finite element analysis and experimental study on dynamic properties of a composite beam with viscoelastic damping

This paper investigates the frequency dependent viscoelastic dynamics of a multifunctional composite structure from finite element analysis and experimental validation. The frequency-dependent behavior of the stiffness and damping of a viscoelastic material directly affects the system's modal frequencies and damping, and results in complex vibration modes and differences in the relative phase of vibration. A second order three parameter Golla–Hughes–McTavish (GHM) method and a second order three fields Anelastic Displacement Fields (ADF) approach are used to implement the viscoelastic material model, enabling the straightforward development of time domain and frequency domain finite elements, and describing the frequency dependent viscoelastic behavior. Considering the parameter identification a strategy to estimate the fractional order of the time derivative and the relaxation time is outlined. Agreement between the curve fits using both the GHM and ADF and experiment is within 0.001 percent error. Continuing efforts are addressing the material modulus comparison of the GHM and the ADF model. There may be a theoretical difference between viscoelastic degrees of freedom at nodes and elements, but their numerical results are very close to each other in the specific frequency range of interest. With identified model parameters, numerical simulation is carried out to predict the damping behavior in its first two vibration modes. The experimental testing on the layered composite beam validates the numerical predication. Experimental results also show that elastic modulus measured from dynamic response yields more accurate results than static measurement, such as tensile testing, especially for elastomers.     

Calculation of the acoustic radiation force on coated spherical shells in progressive and standing plane waves

In this paper, analytical equations are derived for the time-averaged radiation force induced by progressive and standing acoustic waves incident on elastic spherical shells covered with a layer of viscoelastic and sound-absorbing material. The fluid surrounding the shells is considered compressible and nonviscous. The incident field is assumed to be moderate so that the scattered field from the shells is taken to linear approximation. The analytical results are illustrated by means of a numerical example in which the radiation force function curves are displayed, with particular emphasis on the coating thickness and the content of the hollow region of the shells. The fluid-loading on the radiation force function curves is analysed as well. This study attempts to generalize the various treatments of radiation force due to both progressive and standing waves on spherically-shaped structures immersed in ideal fluids. The results show that various ways can be effectively used for damping resonance peaks, such as by changing the fluid in the interior hollow region of the shells or by changing the coating thickness.     

基于分数阶微分流变模型的非晶合金黏弹性行为及流变本构参数研究

近年来, 基于非晶合金名义弹性区的流变力学行为探索其结构及形变机理是非晶合金领域研究的热点之一. 本文根据非晶合金结构不均匀性的特征, 提出能够比拟树状分形网络结构的分数阶微分流变模型研究非晶合金的黏弹性行为. 通过室温纳米压痕实验, 对三种不同泊松比和玻璃化转变温度的非晶合金的黏弹性变形行为进行了研究. 实验结果表明: 在表观弹性区, 非晶合金的变形表现出与加载速率相关的线性黏弹性性质. 根据Riemann-Liouville分数阶微积分定义, 分别由分数阶微分及整数阶Kelvin模型对实验结果进行了分析. 分析结果表明, 相对于整数阶流变模型, 分数阶微分流变模型能更精细地表征材料的黏弹性变形特征; 在流变模型参数中, 黏性系数η_A和分数阶次α反映出材料的流变特性和流动趋势, 流变参数与玻璃转变温度、泊松比之间具有较好的相关性, 上述相关性有助于从微观结构角度理解材料塑性与泊松比的关联.     

On the durable critic load in creep buckling of viscoelastic laminated plates and circular cylindrical shells

Based on the first order shear deformation theory and classic buckling theory, the paper investigates the creep buckling behavior of viscoelastic laminated plates and laminated circular cylindrical shells. The analysis and elaboration of both instantaneous elastic critic load and durable critic load are emphasized. The buckling load in phase domain is obtained from governing equations by applying Laplace transform, and the instantaneous elastic critic load and durable critic load are determined according to the extreme value theorem for inverse Laplace transform. It is shown that viscoelastic approach and quasi-elastic approach yield identical solutions for these two types of critic load respectively. A transverse disturbance model is developed to give the same mechanics significance of durable critic load as that of elastic critic load. Two types of critic loads of boron/epoxy composite laminated plates and circular cylindrical shells are discussed in detail individually, and the influencing factors to induce creep buckling are revealed by examining the viscoelasticity incorporated in transverse shear deformation and in-plane flexibility.     

Nonlinear analysis of oscillatory indentation in elastic and viscoelastic solids

Determining the mechanical properties at micro- and nanometer length scales using nanoindentation or atomic force microscopy is important to many areas of science and engineering. Here we establish equations for obtaining storage and loss modulus from oscillatory indentations by performing a nonlinear analysis of conical and spherical indentation in elastic and viscoelastic solids. We show that, when the conical indenter is driven by a sinusoidal force, the square of displacement is a sinusoidal function of time, not the displacement itself, which is commonly assumed. Similar conclusions hold for spherical indentations. Well-known difficulties associated with measuring contact area and correcting thermal drift may be circumvented using the newly derived equations. These results may help improve methods of using oscillatory indentation for determining elastic and viscoelastic properties of solids.     

Unsteady MHD flow and heat transfer of fractional Maxwell viscoelastic nanofluid with Cattaneo heat flux and different particle shapes

The paper aims to investigate the unsteady natural convection flow and heat transfer of fractional Maxwell viscoelastic nanofluid in magnetic field over a vertical plate. The effect of nanoparticle shape is first introduced to the study of fractional Maxwell viscoelastic nanofluid. Fractional shear stress and Cattaneo heat flux model are applied to construct the governing boundary layer equations of momentum and energy, which are solved numerically. The quantities of physical interest are graphically presented and discussed in detail. It is found that particle shape and fractional derivative parameters have profound influence on the flow and heat transfer.     

Axisymmetric convection flow of fractional Maxwell fluid past a vertical cylinder with velocity slip and temperature jump

A novel finite volume method is developed to investigate the axisymmetric convection flow and heat transfer of fractional viscoelastic fluid past a vertical cylinder. Fractional cylindrical governing equations are formulated by fractional Maxwell model and generalized Fourier's law. The velocity slip and temperature jump boundary conditions are considered across the fluid-solid interface. Numerical results are validated by exact solutions of special case with source terms. The effects of fractional derivative parameter and boundary condition parameters on flow and heat transfer characteristics are discussed. The viscoelastic fluid performs evident shear thickening property in the fractional Maxwell constitutive relation. Moreover, the boundary condition parameters have remarkable influence on velocity and temperature distributions.     

Rayleigh-Lamb wave propagation on a fractional order viscoelastic plate

A previous study of the authors published in this journal focused on mechanical wave motion in a viscoelastic material representative of biological tissue [Meral et al., J. Acoust. Soc. Am. 126, 3278-3285 (2009)]. Compression, shear and surface wave motion in and on a viscoelastic halfspace excited by surface and sub-surface sources were considered. It was shown that a fractional order Voigt model, where the rate-dependent damping component that is dependent on the first derivative of time is replaced with a component that is dependent on a fractional derivative of time, resulted in closer agreement with experiment as compared with conventional (integer order) models, such as those of Voigt and Zener. In the present study, this analysis is extended to another configuration and wave type: out-of-plane response of a viscoelastic plate to harmonic anti-symmetric Lamb wave excitation. Theoretical solutions are compared with experimental measurements for a polymeric tissue mimicking phantom material. As in the previous configurations the fractional order modeling assumption improves the match between theory and experiment over a wider frequency range. Experimental complexities in the present study and the reliability of the different approaches for quantifying the shear viscoelastic properties of the material are discussed.     

Numerical deflation of beach balls with various Poisson's ratios: from sphere to bowl's shape

We present a numerical study of the shape taken by a spherical elastic surface when the volume it encloses is decreased. For the range of 2D parameters where such a surface may model a thin shell of an isotropic elastic material, the mode of deformation that develops a single depression is investigated in detail. It occurs via buckling from sphere toward an axisymmetric dimple, followed by a second buckling where the depression loses its axisymmetry through folding along portions of meridians. For the thinnest shells, a direct transition from the spherical conformation to the folded one can be observed. We could exhibit unifying master curves for the relative volume variation at which first and second buckling occur, and clarify the role of Poisson's ratio. In the folded conformation, the number of folds and inner pressure are investigated, allowing us to infer shell features from mere observation and/or knowledge of external constraints.     

Creep,relaxation and viscosity properties for basic fractional models in rheology

The purpose of this paper is twofold: from one side we provide a general survey to the viscoelastic models constructed via fractional calculus and from the other side we intend to analyze the basic fractional models as far as their creep, relaxation and viscosity properties are considered. The basic models are those that generalize via derivatives of fractional order the classical mechanical models characterized by two, three and four parameters, that we refer to as Kelvin–Voigt, Maxwell, Zener, anti–Zener and Burgers. For each fractional model we provide plots of the creep compliance, relaxation modulus and effective viscosity in non dimensional form in terms of a suitable time scale for different values of the order of fractional derivative. We also discuss the role of the order of fractional derivative in modifying the properties of the classical models.     

Lyotropic mixture made of potassium laurate/1-undecanol/K2SO4/water presenting high birefringences and large biaxial nematic phase domain: a laser conoscopy study

We present a numerical study of the shape taken by a spherical elastic surface when the volume it encloses is decreased. For the range of 2D parameters where such a surface may model a thin shell of an isotropic elastic material, the mode of deformation that develops a single depression is investigated in detail. It occurs via buckling from sphere toward an axisymmetric dimple, followed by a second buckling where the depression loses its axisymmetry through folding along portions of meridians. For the thinnest shells, a direct transition from the spherical conformation to the folded one can be observed. We could exhibit unifying master curves for the relative volume variation at which first and second buckling occur, and clarify the role of Poisson??s ratio. In the folded conformation, the number of folds and inner pressure are investigated, allowing us to infer shell features from mere observation and/or knowledge of external constraints.