Nonlinear dependence of viscosity in modeling the rate-dependent response of natural and high damping rubbers in compression and shear: Experimental identification and numerical verification

The rate-dependent behavior of filled natural rubber (NR) and high damping rubber (HDR) is investigated in compression and shear regimes. In order to describe the viscosity-induced rate-dependent effects, a constitutive model of finite strain viscoelasticity founded on the basis of the multiplicative decomposition of the deformation gradient tensor into elastic and inelastic parts is proposed. The total stress is decomposed into an equilibrium stress and a viscosity-induced overstress by following the concept of the Zener model. To identify the constitutive equation for the viscosity from direct experimental observations, an analytical scheme that ascertains the fundamental relation between the inelastic strain rate and the overstress tensor of the Mandel type by evaluating simple relaxation test results is proposed. Evaluation of the experimental results using the proposed analytical scheme confirms the necessity of considering both the current overstress and the current deformation as variables to describe the evolution of the rate-dependent phenomena. Based on this experimentally based motivation, an evolution equation using power laws is proposed to represent the effects of internal variables on viscosity phenomena. The proposed evolution equation has been incorporated in the finite strain viscoelasticity model in a thermodynamically consistent way. Simulation results for simple relaxation tests, multi-step relaxation tests and monotonic tests at different strain rates using the developed model show an encouraging correlation with the experiments conducted on HDR and NR in both compression and shear regimes. Finally, an approach to extend the proposed evolution equation for rate-dependent cyclic processes is proposed. The simulation results are critically compared with the experiments.     

基于等效阻尼理论的金属橡胶弹性迟滞力学模型及实验研究

针对弹性多孔金属橡胶非线性迟滞特性力学行为,将迟滞恢复力-位移曲线分解为非线性单值曲线和椭圆,并将等效阻尼理论用于动态力学性能参数识别,从而建立了一种新型的适用于黏弹性阻尼材料的宏观唯象力学模型。采用不同相对密度的环形金属橡胶进行动态实验测试,以验证理论模型的准确性,结果表明该模型可将具有非线性特性的金属橡胶系统进行降阶处理,提高金属橡胶力学模型的预测效率,并能很好地描述金属橡胶的迟滞力学行为。另外,研究了在不同激励频率条件下金属橡胶的阻尼耗能特性。实验结果表明:在高频加载的条件下,黏性阻尼系数对动态加载频率不敏感,阻尼耗能与加载幅值之间呈线性正相关。基于等效阻尼理论的弹性迟滞力学模型具有一定的普适性,可进一步推广应用于类似弹性多孔材料的力学性能表征,为其工程应用提供理论基础。     

Exploring the undrained cyclic behavior of methane-hydrate-bearing sediments using CFD–DEM

Based on the mechanical experimental results of methane hydrate (MH), a bond contact model considering the rate-dependency of MH is proposed. A CFD–DEM scheme considering fluid compressibility is used to simulate a series of undrained cyclic shear tests of numerical methane-hydrate-bearing sediment (MHBS) samples. The dynamic behavior, including stress–strain relationship, dynamic shear modulus, and damping ratio, is investigated. In addition, the force chains, contact fabric and averaged pure rotation rate (APR) are examined to investigate the relationships between micromechanical variables and macromechanical responses in the DEM MH samples. The effects of temperature, confining pressure and MH saturation are also analyzed. Due to the micro-structural strengthening by the MH bonds, no obvious change in microscopic quantities is observed, and the samples remain at the elastic stage under the applied low-shear stress level. When confining pressure and MH saturation increase, the dynamic elastic modulus increases, while the damping ratio decreases. An increasing temperature (leading to weakening of MH bonds) can lower the dynamic elastic modulus, but has almost no impact on the damping ratio. On the contrary, an increasing cyclic shear stress level lowers the damping ratio, but has almost no effect on the dynamic elastic modulus.     

Mechanical properties of isolation bearings identified by a viscoelastic model

The Haringx theory is usually employed to describe the mechanical behavior of rubber bearings subjected to a compressive axial load and a lateral shear deformation, but it does not consider the damping effect. In order to study the behavior of isolation bearings which possess an energy-dissipation capacity, the explicit formulas for the horizontal stiffness of viscoelastic columns and the corresponded height reduction are derived by the method of variable separation. These explicit formulas are then applied to develop an identification procedure to find the shear modulus and loss factor of the rubber using the cyclic shear tests of isolation bearings. Through this identification procedure, the empirical formulas for the shear modulus and the loss factor of rubber are established as functions of the strain amplitude and the excitation frequency.     

Shear relaxation characteristics of rock joints under stepwise loadings

The stress relaxation characteristic of rock mass is an important aspect of rheology and has important practical significance for rock engineering. In order to investigate the relaxation characteristic of rock joints with different slope ratios and normal stresses, a series of shear stress relaxation tests were conducted on artifical rock joints poured by cement mortar. Test results show that the relaxation curves can be divided into three stages, i.e. instantaneous relaxation stage, attenuation relaxation stage, and stable relaxation stage. Furthermore, the nonlinear Maxwell relaxation equation was obtained by using the relation between the viscosity coefficient and time, and the theoretical curves based on the empirical equation agreed well with the test results. Moreover, the change law of the initial viscosity coefficient was investigated. Accordingly, a stress relaxation method, termed as relaxation stress peak method, was proposed to determine the long-term strength of rock joints.     

Influence of excitation amplitude on the characteristics of nonlinear butyl rubber isolators

The purpose of this study is to explore the advantages and characteristics of nonlinear butyl rubber (type IIR) isolators in vibratory shear by comparison with linear isolators. It is known that the mechanical properties of viscoelastic materials exhibit significant frequency and temperature dependence, and in some cases, nonlinear dynamic behavior as well. Nonlinear characteristics in shear deformation are reflected in mechanical properties such as stiffness and damping. Furthermore, even when the excitation amplitude is small the response amplitude may often be large enough that nonlinearities cannot be ignored. The treatment involves developing phenomenological models of the effective storage modulus and effective loss factor of a rubber isolator material as a function of excitation amplitude. The transmissibility of a nonlinear viscoelastic isolator is compared with that of a linear isolator using an equivalent linear damping coefficient. Forced resonance vibration and impedance tests are used to characterize nonlinear parameters and to measure the normalized transmissibility. It is found that as the excitation amplitude of the nonlinear viscoelastic isolator increases, the response amplitude decreases and the transmissibility is improved over that of the linear isolator for excitation frequency that exceeds a particular value governed by the temperature and excitation amplitude. The method of multiple scales and numerical simulations are used to predict the response characteristics of the isolator based on the phenomenological modeling under different values of system parameters.     

Viscoelastic properties of pig kidney in shear, experimental results and modelling

 In this paper, the results from a series of rheological tests of fresh pig kidney have been reported. Using a standard strain-controlled rheometer, the oscillation strain sweep experiment showed a linear viscoelastic strain limit of the order of 0.2% strain. To determine the components of dynamic moduli in terms of frequency, shear oscillation tests were done at strain 0.2% using a stress-controlled rheometer. Shear stress relaxation tests were carried out with a fixed strain of 0.2% and 0.02 s rise time. The model we have developed uses a multi-mode upper convected Maxwell (UCM) model with variable viscosities and time constants, to which we have added a Mooney hyper-elastic response, both multiplied by a damping function. Different forms of damping functions that control the non-linearity of strain-stress profile have been tested. The model was fitted to our experimental data, and matched the entire test data reasonably well with a single set of parameters. Received: 5 May 2000 Accepted: 16 March 2001     

A linear viscoelastic model for solid polyethylene

Stress relaxation tests have been carried out on a blue, pipe grade PE 80 medium density polyethylene (BP Chemicals), to provide thermo-viscoelastic rheology for use in calculating thermal stresses in pipe production. Stresses up to 4 MPa were used, with strains up to about 2%, in tests at temperatures from 23° to 90°C. Within this range a linear viscoelastic model was applicable, provided the initial ramp strain rate was less than 7×10^–5 s^–1. The stress relaxation data was fitted directly by a model incorporating an elastic response to volumetric strains, and a generalised linear solid model, consisting of two Maxwell elements and a purely elastic element in parallel, for deviatoric strains. Arrhenius type temperature dependence of relaxation times and shear moduli is found, and within experimental accuracy the temperature dependence of all these model parameters is the same. As a consequence, and provided that the duration of the strain ramp is sufficiently short relative to relaxation times, the model leads to time-temperature superposition of the relaxation moduli, using the same shift factor on both the response magnitude and time axes.     

循环荷载下岩石阻尼行为的试验研究

为了深入地了解岩石的阻尼特性,利用WDT–1500多功能材料试验机,对砂岩、砾岩和砂砾岩进行了循环荷载试验,研究了岩石的动剪切模量和阻尼参数对应力、应变幅值和应力水平的响应特性,得到了动弹性模量和阻尼参数随应力幅值、应力水平及含水率的变化规律,讨论了其变化机理。证实了双曲线关系能够描述分级循环荷载作用下岩石的应力-应变关系,得到了动剪切模量和阻尼比与动应力、动应变之间的关系,建立了不同应力水平和不同应力幅值条件下岩石的动剪切模量与阻尼比关系模型,结果表明该模型能够描述分级循环荷载过程中阻尼行为。     

A multiscale approach of nonlinear composites under finite deformation: Experimental characterization and numerical modeling

The present paper is devoted to the study of the mechanical behavior of an ethylene propylene diene monomer (EPDM) rubber reinforced by polypropylene (PP) particles, revealed as compressible. The hyperlastic behavior of this blend has been characterized under cyclic uni-axial tensile tests. The experimental results show a significant effect of the fraction of (PP) particles (5%, 10%, 25% and 30% by weight) on the macroscopic behavior of the composite. In order to model this behavior, we first develop and implement a micromechanically-based nonlinear model for hyperelastic composites. The approach is based on the second order homogenization method proposed by Ponte Castaneda and Tiberio (2000) and for which suitable energy densities are adopted for the matrix and the inclusions phases, both assumed as compressible. We then proceed to the model verification by comparison with Finite Element simulations on a unit cell. Finally, we propose an extension of the model in order to take into account damage due to voids growth phenomena. The comparison of the multiscale damage model predictions with the experimental data obtained on the EPDM/PP composite indicates a very good agreement.     

宽γ辐照剂量范围内硅泡沫本构模型研究

基于实验和理论建模研究了白炭黑增强硅泡沫材料在γ辐照剂量范围为0~1000kGy作用后的单轴压缩力学行为。实验结果表明辐照导致硅泡沫出现明显硬化现象，初始杨氏模量和固定应变下应力幅值均随γ辐照剂量近似线性增加。辐照后硅泡沫泡孔结构完整，硅橡胶基体中高分子交联反应占主导，且交联密度随辐照剂量线性增大。基于实验分析结果，实现了Ogden Hyperfoam超弹本构模型参数与辐照剂量的关联。结果表明初始剪切模量参数与辐照剂量成线性关系，硬化指数和泊松比参数与辐照剂量无关。基于应力应变实验数据拟合得到模型参数，并与未参与拟合的实验数据对比，验证了模型的准确性，表明该模型能够表征宽辐照剂量范围内硅泡沫的压缩力学行为。     

宽γ辐照剂量范围内硅泡沫本构模型研究

基于实验和理论建模研究了白炭黑增强硅泡沫材料在γ辐照剂量范围为0~1000kGy作用后的单轴压缩力学行为。实验结果表明辐照导致硅泡沫出现明显硬化现象,初始杨氏模量和固定应变下应力幅值均随γ辐照剂量近似线性增加。辐照后硅泡沫泡孔结构完整,硅橡胶基体中高分子交联反应占主导,且交联密度随辐照剂量线性增大。基于实验分析结果,实现了Ogden Hyperfoam超弹本构模型参数与辐照剂量的关联。结果表明初始剪切模量参数与辐照剂量成线性关系,硬化指数和泊松比参数与辐照剂量无关。基于应力应变实验数据拟合得到模型参数,并与未参与拟合的实验数据对比,验证了模型的准确性,表明该模型能够表征宽辐照剂量范围内硅泡沫的压缩力学行为。     

Thermo-mechanical large deformation response and constitutive modeling of viscoelastic polymers over a wide range of strain rates and temperatures

A phenomenological one-dimensional constitutive model, characterizing the complex and highly nonlinear finite thermo-mechanical behavior of viscoelastic polymers, is developed in this investigation. This simple differential form model is based on a combination of linear and nonlinear springs with dashpots, incorporating typical polymeric behavior such as shear thinning, thermal softening at higher temperatures and nonlinear dependence on deformation and loading rate. Another model, of integral form, namely the modified superposition principle (MSP), is also modified further and used to show the advantage of the newly developed model over MSP. The material parameters for both models are determined for Adiprene-L100, a polyurethane based rubber. The constants once determined are then utilized to predict the behavior under strain rate jump compression, multiple step stress relaxation loading experiment and free end torsion experiments. The new constitutive model shows very good agreement with the experimental data for Adiprene-L100 for the various finite loading paths considered here and provides a flexible framework for a three-dimensional generalization.     

温度对炭黑填充橡胶复合材料力学性能影响的实验研究

采用岛津万能试验机对天然与丁苯共混橡胶(NSBR)试件进行不同温度下多步松弛循环加卸载实验和单向拉伸加卸载循环实验，对比两者的宏观力学响应，并研究温度对炭黑填充复合材料力学性能的影响．结果表明：(1)随着温度的升高，未填充橡胶逐渐变“硬”，而炭黑填充橡胶是“先变软后变硬”．(2)随着温度的升高，橡胶的迟滞损耗呈线性减小．(3)比较单向拉伸和应力松弛平衡点的应力应变曲线，填充橡胶“先变软后变硬”的温度转折点不同，这是由于应力松弛过程基本消除了粘弹性效应．     

弹簧-金属丝网橡胶组合减振器迟滞力学模型及实验研究

金属丝网橡胶材料是一种完全由金属丝编织成的多孔复合材料,与传统螺旋卷制金属橡胶材料相比,其改进了成型工艺,剔除了制备过程中大量的手工工艺干扰,提高机械化程度,重合度更高,拥有更稳定的力学性能.由于金属丝网橡胶材料具有承载能力高、阻尼大、耐高温、耐低温、耐老化、抗油抗腐蚀等优良特性,在很多方面强于传统橡胶,多用于航空航天、船舶、军事武器等军工工业.弹簧$\!$-$\!$-$\!$金属丝网橡胶组合减振器具有可设计刚度和较高承载能力,但因其具有复杂的非线性迟滞特性,目前相关材料的本构模型还难以准确描述其力学特性.本文在弹簧$\!$-$\!$-$\!$金属丝网橡胶组合减振器静态迟滞力学性能实验的基础上,结合其干摩擦阻尼迟滞特性,提出了一种迟滞力学性能理论模型.根据减振器迟滞实验恢复力$\!$-$\!$-$\!$位移曲线特点,利用参数分离的方法将迟滞曲线分解为弹性恢复力和干摩擦阻尼力,分别建模求解等效刚度和干摩擦阻尼系数,以此建立了组合减振器理论模型,并与实验结果进行对比及进行误差分析,验证了理论模型的准确性.      

Generalized Maxwell model for the viscoelastic behavior of a soda-lime-silica glass under low frequency shear loading

The linear viscoelastic behavior of a soda-lime-silica glass under low frequency shear loading is investigated in the glass transition range. Using the time-temperature superposition technique, the master curves of the shear dynamic relaxation moduli are obtained at a reference temperature of 566°C. A method to determine the viscoelastic constants from dynamic relaxation moduli is proposed. However, some viscoelastic constants cannot be directly measured from the experimental curves and others cannot be precisely obtained due to non-linearity effects at very low frequencies. The generalized Maxwell model is investigated from the experimental dynamic moduli without fixing the viscoelastic constants. A set of parameters is shown to be in good agreement with the experimental dynamic relaxation moduli, but does not give the correct values of the viscoelastic constants of the investigated glass. The soda-lime-silica glass exhibits a non-linear viscoelastic behavior at very low stress level which is usually observed for organic glasses. This non-linear behavior is questioned.     

Large-amplitude oscillatory shear flow from the corotational Maxwell model

Using the single relaxation time corotational Maxwell fluid, we derive explicit analytical expressions for the first, third, and fifth harmonics of the alternating shear stress response in large-amplitude oscillatory shear (LAOS). We also derive corresponding expressions for the zeroth, second, and fourth harmonics of both the first and second normal stress differences. These harmonics are found to depend upon just two dimensionless groups: the Deborah and Weissenberg numbers, each of which causes non-Newtonian behavior. The form of the solution for the corotational Maxwell model in LAOS matches the forms of the analytical solutions for two molecular models for dilute solutions and one for concentrated solutions or melts. We also derive an analytical solution for the corotational Maxwell model after startup of LAOS. For this we find that both small and large amplitude cases approach a periodic limit cycle (alternance) at the same rate for both the shear stress response and for the normal stress differences. For molten high density polyethylene that is lightly filled with carbon black, we find good quantitative agreement with measured LAOS behavior when our analytical solution is superposed for multiple relaxation times.