A model for nonlinear viscoelastic torsional response of an elastomeric bushing

Summary An elastomeric bushing is a device used in automotive suspension systems to cushion the loads transmitted from the wheel to the frame of the vehicle. A bushing is essentially an elastomeric hollow cylinder which is bonded to a solid metal shaft at its inner surface and a metal sleeve at its outer surface. The shaft is connected to the suspension and the sleeve is connected to the frame. The elastomeric cylinder provides the cushion when it deforms due to relative motion between the shaft and the sleeve. The relation between the force or moment applied to the shaft or sleeve and the relative displacements or rotations is nonlinear and exhibits features of viscoelasticity.A force(moment)-displacement(rotation) relation for elastomeric bushings is important for multibody dynamics numerical simulations. A boundary value problem for the bushing response leads to a relation which requires extensive computation time to implement and is hence unsuitable. In a separate study, a force(moment)-displacement(rotation) relation for single mode response has been proposed which can be used in multi-body dynamics simulations. The relation is expressed in terms of a force (moment) relaxation function for the bushing, and a method for its determination by experiments on bushings has been presented. The applicability of this relation for torsional mode bushing response is evaluated in the present work.A boundary value problem is formulated for torsional mode bushing response. Numerical solutions of the boundary value problem represent the exact bushing response and act as numerically generated experimental data. The proposed moment-rotation relation is constructed using this data. Numerical solutions of the boundary value problem also allow for comparison between the exact moment-rotation behavior and that predicted by the proposed model. It is shown that the method for determining the bushing relaxation function and the predictions of the proposed moment-rotation relation are in very good agreement with the exact results.     

橡胶材料的非线性黏弹性本构方程

橡胶是一种非线性黏弹性材料,准确描述其非线性黏弹性力学响应的本构方程是橡胶材料及制品设计优化的关键。文中基于超弹性模型和并行流变模型(PRF)描述了橡胶材料的非线性黏弹性响应特征,重点探讨了由实验数据确定PRF本构方程材料参数的方法。首先通过单轴拉伸实验数据拟合得到超弹性模型,将应力松弛实验数据拟合得到表征材料线性黏弹性的模型-Prony级数,再将Prony级数转化为初始的PRF模型,进而不断优化得到PRF模型的准确材料参数,最后进行实验验证。结果表明,PRF模型计算的不同应变下的应力松弛数据与实验数据之间的误差仅为0.067%,PRF能准确地描述橡胶材料非线性响应的应力松弛行为。     

Modeling short- and long-term time-dependent nonlinear behavior of polyethylene

A new methodology for developing macromechanical constitutive formulations for time-dependent materials is presented in this article. In particular, two phenomenological constitutive models for polymer materials are illustrated, describing time-dependent and nonlinear mechanical behavior. In this new approach, short-term creep test data are used for modeling both short-term and long-term responses. The differential form of a model is used to simulate typical nonlinear viscoelastic polymeric behavior using a combination of springs and dashpots. Unified plasticity theory is then used to develop the second model, which is a nonlinear viscoplastic one. Least squares fitting is applied for the determination of material parameters for both models, based on experimental results. Due to practical constraints, experimental data are usually available for short-term time-frames. In the presented proposed formulation, the material parameters determined from short-term testing are used to obtain material parameter relationships for predicting the long-term material response. This is done by extending short-term information for longer time frames. Finally, theoretical and experimental results of tensile tests on polyethylene subjected to various load levels and test times are compared and discussed. Very good agreement of the modeling results with experimental data shows that the developed formulation provides a flexible and reliable framework for predicting load responses of polymers.     

考虑温度效应的沥青混合料参数模型

选取修正Burgers模型作为沥青混合料的粘弹性本构模型,基于最小二乘法原理对单轴压缩蠕变试验数据进行拟合,获得本构模型的粘弹性参数。考虑温度变化对沥青混合料粘弹性的影响,建立修正Burgers模型的粘弹性参数模型,运用实验数据拟合粘弹性参数模型的分项系数,理论计算考虑温度效应的蠕变柔量,与实验结果对比分析表明,参数模型有较好的适用性,拟合精度高。粘弹性参数模型能将有限的实验数据推广到其他温度工况,节约大量实验,具有很好适用性。     

Elastomer bushing response: experiments and finite element modeling

Summary.  Elastomer bushings are essential components in tuning suspension systems since they isolate vibration, reduce noise transmission, accommodate oscillatory motions and accept misalignment of axes. This work presents an experimental study in which bushings are subjected to radial, torsional and coupled radial-torsional modes of deformation. The experimental results show that the relationship between the forces and moments and their corresponding displacements and rotations is nonlinear and viscoelastic due to the nature of the elastomeric material. An interesting feature of the coupling response is that radial force decreases and then increases with torsion. The experimental results were used to assess bushing behavior and to determine the strength of radial-torsional coupling. The experimental results were also compared to finite element simulations of a model bushing. While finite element analysis predicted small displacements at the relaxed state reasonably well, the response to larger radial deformations and coupled deformations was not well captured. Received January 13, 2003 Published online: May 20, 2003 The authors would like to thank Tenneco Automotive and the Great Lakes Truck and Transit Research Center for support of this project.     

损伤粘弹性桩基的非线性动力学行为

在有限变形的假设下，将孔洞作为一种分布损伤建立了粘弹性地基上具有损伤的粘弹性Timoshenko梁的静／动力学行为的初边值问题，并用此来模拟土中具有损伤的桩基非线性动力学行为，其中，本构模型是用具有孔洞的粘弹性固体的卷积方法所表示的，并且地震激励等价于一种横向力的作用。利用Galerkin方法对数学模型进行了简化，得到了一阶和二阶简化系统。非线性动力学中的数值方法被用来求解简化系统，数值上得到了系统的时程曲线、功率谱、相平面图、Poincare截面和分叉图等，讨论了地基和桩基材料参数与载荷参数对桩基动力学行为的影响，也考察了损伤对动力学行为的影响。可以看到，具有损伤的粘弹性桩基构成的动力学系统具有丰富的动力学行为。     

A nonlinear viscoelastic–viscoplastic model for adhesives

We consider the nonlinear viscoelastic–viscoplastic behavior of adhesives. We develop a one-dimensional nonlinear model by combining Schapery’s nonlinear single integral model and Perzyna’s viscoplastic model. The viscoplastic strain was solved iteratively using the von Mises yield criterion and nonlinear kinematic hardening. The combined viscoelastic–viscoplastic model was solved using Newton’s iteration and implemented into a finite element model. The model was calibrated using creep-recovery data from bulk adhesives and verified from the cyclic behavior of the bulk adhesives. The finite element model results agreed with experimental creep and cyclic responses, including recoverable and permanent strain after load removal. Although the contribution of the viscoplastic strain was small, both viscoplastic and viscoelastic components of strain response were required to describe the adhesive creep and cyclic response.

     

基于非线性模型的沥青路面动力响应研究

根据路基变形的非线性及沥青路面具有明显粘弹性的特点,将沥青路面简化为非线性粘弹性地基上的粘弹性无限长梁,建立了移动载荷作用下非线性粘弹性梁系统动力响应数学模型。利用Adomian分解法和小波变换法得到求解稳态响应解析解的新方法。通过实际道路参数对沥青路面动力响应进行了数值仿真,研究了车速、车辆轴载、路面材料及温度对沥青路面动力响应的影响规律。结果表明该文提出的计算方法简便、快捷,是求解该类非线性动力响应问题的一种有效方法;非线性模型更能准确地反映重载及超载时沥青路面实际结构受力状态。     

A variational constitutive framework for the nonlinear viscoelastic response of a dielectric elastomer

We formulate a variational constitutive framework that accounts for nonlinear viscous behavior of electrically sensitive polymers, specifically Dielectric Elastomers (DEs), under large deformation. DEs are highly viscoelastic and their actuation response is greatly affected in dynamic applications. We used the generalized Maxwell model to represent the viscoelastic response of DE allowing the material to relax with multiple mechanisms. The constitutive updates at each load increment are obtained by minimizing an objective function formulated using the free energy and electrostatic energy of the elastomer, in addition to the viscous dissipation potential of the dashpots in each Maxwell branch. The model is then used to predict the electromechanical instability (EMI) of DE. The electro-elastic response of the DE is verified with available analytical solutions in the literature and then the material parameters are calibrated using experimental data. The model is integrated with finite element software to perform a variety of simulations on different types of electrically driven actuators under various electromechanical loadings. The electromechanical response of the DE and the critical conditions at which EMI occurs were found to be greatly affected by the viscoelasticity. Our model predicts that under a dead load EMI can be avoided if the DE operates at a high voltage rate. Subjected to constant, ramp and cyclic voltage, our model qualitatively predicts responses similar to the ones obtained from the analytical solutions and experimental data available in the literature.     

A noninvasive system-level model order reduction scheme for flexible multibody simulation

This paper presents a novel system-level model order reduction scheme for flexible multibody simulation, namely the system-level affine projection (SLAP). Contrary to existing system-level model order reduction approaches for multibody systems simulation, this methodology allows to obtain a constant reduced order basis which can be obtained in a noninvasive fashion with respect to the original flexible multibody model. It is shown that this scheme enables an automatic joint constraint elimination which can be obtained at low computational cost through exploitation of the component level modes typically employed in flexible multibody simulation. The equations of motion are derived such that the computational cost of the resulting SLAP model is independent of the original model size. This approach results in a set of ordinary differential equations with a constant mass matrix and nonlinear internal forces. This structure makes the resulting model suitable for a range of estimation, control, and design applications. The proposed approach is validated numerically on a flexible four-bar mechanism and shows good accuracy for a very low-order SLAP model.     

A thermodynamic study of materials representable by integral expansions

A thermodynamic formulation is developed for nonlinear compressible viscoelastic materials and is used to quantitatively study the thermodynamic states associated with fracture. The Helmholtz free energy is assumed to be approximated by a fourth order multiple integral expansion on the histories of the Lagrangian strain tensor and temperature, and the first and second laws are then utilized to develop a constitutive equation and dissipation function for the material. Simplified expressions are obtained for the special cases of slow motions and long term steady flows. An experimental study is then made on equi-temperature metamorphosed snow, a nonlinear viscoelastic material, to determine the states associated with fracture for a variety of deformation paths. In view of the experimental study, the author is able to conclude that this approach to fracture investigation can produce much insight on the strength properties and may be instrumental in the formulation of a useful fracture criterion of materials for which the fracture condition is history dependent. Additionally, due to the thermodynamic nature of this approach, the information gathered could be useful in investigations of crack growth in nonlinear viscoelastic materials such as snow or even in the formulation of structural constitutive equations.     

On the equivalent static load method for flexible multibody systems described with a nonlinear finite element formalism

The equivalent static load (ESL) method is a powerful approach to solve dynamic response structural optimization problems. The method transforms the dynamic response optimization into a static response optimization under multiple load cases. The ESL cases are defined based on the transient analysis response whereupon all the standard techniques of static response optimization can be used. In the last decade, the ESL method has been applied to perform the structural optimization of flexible components of mechanical systems modeled as multibody systems (MBS). The ESL evaluation strongly depends on the adopted formulation to describe the MBS and has been initially derived based on a floating frame of reference formulation. In this paper, we propose a method to derive the ESL adapted to a nonlinear finite element approach based on a Lie group formalism for two main reasons. Firstly, the finite element approach is completely general to analyze complex MBS and is suitable to perform more advanced optimization problems like topology optimization. Secondly, the selected Lie group formalism leads to a formulation of the equations of motion in the local frame, which turns out to be a strong practical advantage for the ESL evaluation. Examples are provided to validate the proposed method. Copyright © 2016 John Wiley & Sons, Ltd.     

Modeling automotive gas-exchange solenoid valve actuators

We develop a finite-element analysis (FEA) model to describe transient and static operation of gas-exchange valves. Such valves, directly controlled by solenoids, are a promising method for enhancing automotive engine efficiency. The FEA model is validated by experimental testing on an actual automotive prototype valve. We show that a nonlinear lumped-parameter model that uses FEA results also closely matches experimental data. The lumped-parameter model is suitable for optimization of design and can be readily used for closed-loop simulation. We present a simplified lumped-parameter model to facilitate controller design. Finally, we compare a dynamic open-loop simulation with experimental results.     

A formulation to model the nonlinear viscoelastic properties of the vascular tissue

Nearly all soft tissues, including the vascular tissue, present a certain degree of viscoelastic material response, which becomes apparent performing multiple relaxation tests over a wide range of strain levels and plotting the resulting stress relaxation curves, nonlinear viscoelasticity of the tissue. Changes in relaxation rate at each strain may occur at multiple strain levels. A constitutive formulation considering the particular features of the vascular tissue, such as anisotropy, together with these nonlinear viscoelastic phenomena is here presented and used to fit stress?Cstretch curves from experimental relaxation tests. This constitutive model was used to fit several data set of in vitro experimental stress relaxation tests performed on ovine and porcine aorta. The good fitting of the experimental data shows the capability of the model to reproduce the viscoelastic response of the vascular tissue.     

On the safe load limit of columns made of time dependent materials

Summary The safe load limit, a well defined parameter for columns made of linearly viscoelastic solid-type materials with limited creep, is generalized to materially and geometrically nonlinear viscoelastic structures. To illustrate the concepts to be treated, a simple rigid bar-spring/dashpot model simulating the behaviour of imperfect columns is analyzed in detail. Next the imperfect column problem is formulated and solved. Various approximations in the solution technique are introduced which allow relatively simple procedures for the evaluation of the safe load limit. Finally a column made of concrete is analyzed in detail for which the results are compared, where possible, with experimental data, a comparison which shows satisfactory agreement.With 14 FiguresThe results presented here were obtained in the course of research sponsored by the Natural Sciences and Engineering Research Council of Canada, Grant No. A-2736.     

Micro-Macro Analysis of Viscoelastic Unidirectional Laminated Composite Plates Using DR Method

The Dynamic Relaxation (DR) technique together with finite difference discritization is used to study the bending behavior of Mindlin composite plate including geometric nonlinearity. The overall behavior of the unidirectional composite is obtained from a three-dimensional (3D) micromechanical model, in any combination of normal and shear loading conditions, based on the assumptions of Simplified Unit Cell Method (SUCM). The composite system consists of nonlinear viscoelastic matrix reinforced by transversely isotropic elastic fibers. A recursive formulation for the hereditary integral of the Schapery viscoelastic constitutive equation in multiaxial stress state is used to model the nonlinear viscoelastic matrix material in the material level. The creep tests data is used for verification of the predicted response of the current approach. Under uniform lateral pressure, the laminated plate deformation with clamped and hinged edged constraints is predicted for various time steps.     

On the influence of preloading in the nonlinear viscoelastic–viscoplastic response of carbon–epoxy composites

On an ongoing research for the nonlinear viscoelastic response of composites and polymers, a study of the influence of preloading applied to composite laminates subjected to creep–recovery loading is performed. In cases where high stress levels are applied, this response becomes highly nonlinear and has to be taken into account when designing composite parts. A major problem encountered in the experimental investigation of the nonlinear viscoelastic behaviour is the mode of the initial applied loading and its effect in the overall viscoelastic response of the test sample. The damage that occurs due to the instantaneous application of the load leads to an additional viscoelastic/viscoplastic strain component. In order to investigate this effect as well as to compare different preloading modes, as far as viscoelastic/viscoplastic response is concerned, a test program was initiated and the experimental data were investigated in the current study. A preloading mode is applied in each specimen prior to the creep–recovery testing at different applied stress levels. Useful results concerning the effect of preloading in the time dependent response of the material are concluded. Variation of the values of viscoplastic strain in respect to the preloading mode is also of great concern.     

Nonlinear response of a vertically moving viscoelastic beam subjected to a fluctuating contact load

In the present work, the nonlinear response of a vertically moving viscoelastic beam subjected to a periodically varying contact load is investigated. The generalized Galerkin??s method is used to discretize the nonlinear partial differential equation of motion into the temporal equation of motion. The temporal equation of motion contains many nonlinear terms such as cubic geometric and inertial nonlinear terms, nonlinear damping term, and nonlinear parametric excitation terms in addition to forced excitation and parametric excitation terms. The first-order approximate solutions are obtained by using the method of multiple scales, and the stability and bifurcations of the obtained steady-state responses are studied. Extensive numerical simulations are presented to illustrate the influences of various types of system parameters for different resonance conditions. A significant amount of vibration reduction is obtained with the increase in the material loss factor. The results obtained by numerically solving the temporal equation of motion are found to be in good agreement with the results determined by the method of multiple scales. The obtained results are useful for reduction in the vibration of the viscoelastic flexible beam with prismatic joint or single-link viscoelastic Cartesian manipulator with payload subjected to a sinusoidally varying contact load.     

玄武岩纤维混凝土的冲击力学行为及本构模型

采用Φ100mm分离式霍普金森压杆(SHPB)试验装置研究了玄武岩纤维混凝土在不同应变率下的冲击力学行为,并将其与基体混凝土进行对比分析;采用朱-王-唐(ZWT)模型,在试验研究的基础上,建立了考虑纤维三维随机分布效应的玄武岩纤维混凝土非线性粘弹性本构模型,并与SHPB试验结果进行比较。结果表明:玄武岩纤维混凝土的冲击压缩强度与能量吸收能力,较素混凝土有明显提高,具备优异的冲击力学性能;本构模型提供的理论曲线与试验曲线比较接近,改进后的ZWT模型可以较为准确地描述玄武岩纤维混凝土的高应变率力学行为。