Modelling of a Cellular Rubber with Nonlinear Viscosity Functions

In this paper a porous carbon black-filled rubber is investigated under uniaxial tension. On the experimental site the main focus of attention lies on the Mullins effect, the thixotropic and the viscoelastic behaviour. Because of the two phase character of cellular rubber, the Theory of Porous Media is taken into account. Performing a proper preconditioning, the Mullins effect can be eliminated. Hence, it is not included in the material model. The constitutive model for the basic elasticity is based on a polynomial approach for an incompressible material which is expanded by a volumetric term to include the structural compressibility. Finally, the concept of finite viscoelasticity is applied introducing an intermediate configuration. Nonlinear relaxation functions are used to model the process dependent relaxation times, to simulate the thixotropy and the highly nonlinear behaviour concerning the deformation and feedrate. The material parameters of the model are estimated using a stochastic identification algorithm.     

多孔硅橡胶有限变形的粘弹性行为

针对孔隙度较大（孔隙度大于50％）的硅橡胶材料在有限变形时的粘弹性行为，从建立描述材料粘弹性特征的松驰函数和变形特征的应变能函数出发，提出了适合多孔隙、可压硅橡胶材料的非线性粘弹性力学行为的本构关系，松驰函数和应变能函数可解耦为等容和体积变形两部分，并引入了拟时间的概念来反映变形对材料特征时间的影响，利用硅橡胶材料的单轴压缩松驰实验与材料模型进行了对比，讨论了多孔硅橡胶的等容变形和体积变形对应力松驰的影响。     

Pressure Dependent Properties of a Compressible Polymer

This paper deals with the investigations of a porous carbon black-filled rubber, tested with regard to its pressure and tension behaviour. In the tension range only uniaxial tests are performed while in the pressure range uniaxial as well as hydrostatic tests are performed. The uniaxial experiments are carried out in a custom-made uniaxial device and the hydrostatic tests in a pressure chamber which is specially developed for this application. The construction and use of the pressure chamber is clearly described in this paper. All experiments are related to the basic elasticity of the material. The viscoelastic behaviour is completely disregarded at this point. Not only the experiments are discussed, also the modelling of the material is looked at. The tested cellular rubber is composed of an incompressible solid phase and a compressible gas phase. For that reason a so-called structural compressibility is observed. The compressible behaviour of cellular rubber is an important property. So the main focus of the paper is on the pressure tests and the simulation of these. The existing material models for rubber like materials only deal with incompressible rubber structures. To represent the compressible behaviour, the Theory of Porous Media is used. The constitutive model is based on a polynomial approach for an incompressible material. This is complemented by a volumetric expansion term with a point of compaction to model the structural compressibility.     

On the representation of chemical ageing of rubber in continuum mechanics

In order to represent the chemical ageing behaviour of rubber under finite deformations a three-dimensional theory is proposed. The fundamentals of this approach are different decompositions of the deformation gradient in combination with an additive split of the Helmholtz free energy into three parts. Its first part belongs to the volumetric material behaviour. The second part is a temperature-dependent hyperelasticity model which depends on an additional internal variable to consider the long-term degradation of the primary rubber network. The third contribution is a functional of the deformation history and a further internal variable; it describes the creation of a new network which remains free of stress when the deformation is constant in time. The constitutive relations for the stress tensor and the internal variables are deduced using the Clausius–Duhem inequality. In order to sketch the main properties of the model, expressions in closed form are derived with respect to continuous and intermittent relaxation tests as well as for the compression set test. Under the assumption of near incompressible material behaviour, the theory can also represent ageing-induced changes in volume and their effect on the stress relaxation. The simulations are in accordance with experimental data from literature.     

多孔硅橡胶有限变形的弹性行为

针对孔隙度较大 (孔隙度大于 5 0 % )的硅橡胶材料在压缩情况下的大变形 ,提出了可描述此类可压橡胶材料力学行为的应变能密度函数 ,推导了硅橡胶材料的本构方程。利用硅橡胶材料的单轴压缩实验进行了材料参数拟合 ,讨论了多孔硅橡胶的孔隙度和体积变形对压缩性能的影响     

The mechanical behaviour of rubber under hydrostatic compression and the effect on the results of finite element analyses

Using finite element tools for the dimensioning of rubber components is state of the art. When conducting finite element simulations, a split of the strain energy function, which results in stresses, into a deviatoric and a volumetric part is made. The mechanical behaviour of reinforced natural rubber under hydrostatic pressure is determined to prove the validity of this assumption. Furthermore, a nearly incompressible material behaviour is assumed in simulations of rubber, which may cause an insufficient outcome quality especially for rubber components that are exposed to hydrostatic pressure like highly confined bushings. In this paper, a method is presented to determine the compressibility, or its reciprocal the bulk modulus of rubber. The effect of the bulk modulus of a natural rubber on the simulation results of a bearing is pointed out. The obtained results are compared to test data to show the significance of the value of the bulk modulus for achieving a satisfactory outcome quality. Therefore, reliable information about the in-use behaviour of rubber components is obtained to reduce the costs and the effort in the dimensioning process.     

Modelling the slight compressibility of anisotropic soft tissue

In order to avoid the numerical difficulties in locally enforcing the incompressibility constraint using the displacement formulation of the Finite Element Method, slight compressibility is typically assumed when simulating transversely isotropic, soft tissue. The current standard method of accounting for slight compressibility of hyperelastic soft tissue assumes an additive decomposition of the strain-energy function into a volumetric and a deviatoric part. This has been shown, however, to be inconsistent with the linear theory for anisotropic materials. It is further shown here that, under hydrostatic tension or compression, a transversely isotropic cube modelled using this additive split is simply deformed into another cube regardless of the size of the deformation, in contravention of the physics of the problem. A remedy for these defects is proposed here: the trace of the Cauchy stress is assumed linear in both volume change and fibre stretch. The general form of the strain-energy function consistent with this model is obtained and is shown to be a generalisation of the current standard model. A specific example is used to clearly demonstrate the differences in behaviour between the two models in hydrostatic tension and compression.     

橡胶粘结颗粒材料粘弹性性能的试验研究与离散元模拟

制备了颗粒规则四方排列和六方排列的橡胶粘接颗粒材料试样,实验测试了所制备试样在单向拉伸载荷下的应力松弛曲线和不同应变率时的应力应变曲线。基于所测试的应力松弛曲线,采用曲线拟合方法得到了所测试材料的宏观Burger’s粘弹性本构模型参数。采用离散元模型中单元间连结模型代表颗粒间橡胶粘接剂的作用,并基于试样的宏观Burger’s模型参数与离散元模型中细观Burger’s连结模型参数间的关系,建立了橡胶粘接颗粒材料的无厚度胶结离散元分析模型。最后采用所建立的离散元模型计算了所测试试样的松弛和拉伸力学性能。离散元预测结果与实验结果的对比表明,采用无厚度胶结离散元模型能较好的计算颗粒规则排列的橡胶粘接颗粒材料松弛和拉伸力学性能,但基于应力松弛实验拟合而来参数不能准确反应橡胶粘接剂在高应变率条件下其力学性能的应变率相关性。     

Theoretical and experimental analysis of the contact between a solid-rubber tire and a chassis dynamometer

We consider the problem of modeling the test where a solid-rubber tire runs on a chassis dynamometer for determining the tire rolling resistance characteristics.We state the problem of free steady-state rolling of the tire along the test drum with the energy scattering in the rubber in the course of cyclic deformation taken into account. The viscoelastic behavior of the rubber is described by the Bergströ m–Boyce model whose numerical parameters are experimentally determined from the results of compression tests with specimens. The finite element method is used to obtain the solution of the three-dimensional viscoelasticity problem. To estimate the adequacy of the constructed model, we compare the numerical results with the results obtained in the solid-rubber tire tests on the Hasbach stand from the values of the rolling resistance forces for various loads on the tire.     

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.     

Finite element simulation of deformation behaviour of cellular rubber components

This paper presents a new prospect of investigating the mechanical behaviour of cellular rubber using a porous hyperelastic material model within the framework of homogenisation method to consider pore volume fraction. There are number of hyperelastic material models to describe the behaviour of homogeneous elastomer, but very few to characterise the complex properties of cellular rubber. The analysis of dependence of material behaviour on pore density using the new material model is supported with experiments to characterise the actual material behaviour. The finite element simulations are then followed by compression load tests to validate the material model.     

聚硅氧烷硅胶的黏超弹性力学行为研究

聚硅氧烷硅胶是一类以Si——O键为主链、硅原子上直接连接有机基团的无色透明高分子聚合物, 因其具有优异的超弹性性能而广泛应用于精密减震结构、柔性电子器件等领域. 在聚硅氧烷硅胶减震结构和柔性电子器件的设计中, 材料在大变形和动态加载下的黏超弹性力学行为的精确描述至关重要. 本文针对该问题进行了系统的研究:首先, 将该硅胶的超弹性和黏弹性行为进行解耦, 确定其黏超弹性本构方程的基本框架;其次, 基于单轴拉压、平面拉伸试验确定其准静态超弹性模型的各项参数;再次, 利用霍普金森压杆冲击试验确定其黏弹性模型的各项参数;在此基础上, 将超弹性和黏弹性模型合并为适用于大应变和大应变率的黏超弹性动态本构模型;最后, 利用落锤冲击试验对该硅胶薄片的冲击变形行为进行了研究, 并利用上述建立的动态本构模型对落锤冲击过程进行了有限元模拟. 结果表明:本文建立的黏超弹性本构模型可有效预测该硅胶在冲击载荷下的力学行为, 从而为聚硅氧烷硅胶减震结构和柔性电子器件的优化设计提供了理论和应用基础.      

填充橡胶温度依赖的非弹性力学行为实验研究

填充橡胶具有复杂的非弹性力学行为，主要包括应变率依赖的粘弹性效应和变形历史依赖的Mullins效应。当前大多数对填充橡胶的实验研究集中于室温条件，针对以上问题，本文通过单轴压缩实验系统地研究了温度对氟橡胶粘弹性和Mullins效应这两种非弹性行为的影响。首先采用多次循环加载获得了完全消除了Mullins效应的预处理试样。通过对原试样和预处理试样的单轴加卸载实验应力响应进行对比，发现Mullins效应不受变形温度和应变率的影响。通过对消除Mullins效应橡胶材料应力松弛实验结果分析，发现粘弹性行为不仅与变形的温度、应变率相关，还受加载应变的影响，表现为较大的加载应变会抑制氟橡胶的应力松弛。     

Characterization of Polyurethane Rubber at High Deformation Rates

Polyurethane rubber materials have widespread usage in large-deformation energy absorption and dissipation applications. Accurate design modeling with these materials requires an appropriate constitutive material model that accounts for both static (low strain rate) and dynamic (high strain rate) responses. A common modeling approach is the use of hyper-viscoelastic formulations, which couple quasi-static hyperelastic with dynamic viscoelastic responses and describe the material response over a range of deformation rates. In this work the effectiveness of two models, the Modified Quasi-Linear Viscoelastic and Non-Linear Hyper-Viscoelastic, are investigated to describe the high-rate behaviour of two different grades of polyurethane rubber. From quasi-static, uniaxial compression tests, a Rivlin hyperelastic formulation was found to describe the low-rate response well. High-rate, uniaxial compressions test were performed using a Polymeric Split Hopkinson Pressure Bar (PSHPB), supported by high-speed photography. In general, it was found that the Modified Quasi-Linear Viscoelastic model did not fit the experimental data well due to its limited non-linear terms, while the Non-Linear Hyper-Viscoelastic provided very good agreement.     

Thermomechanical characterisation of cellular rubber

This contribution discusses an experimental possibility to characterise a cellular rubber in terms of the influence of multiaxiality, rate dependency under environmental temperature and its behaviour under hydrostatic pressure. In this context, a mixed open and closed cell rubber based on an ethylene propylene diene monomer is investigated exemplarily. The present article intends to give a general idea of the characterisation method and the considerable effects of this special type of material. The main focus lies on the experimental procedure and the used testing devices in combination with the analysis methods such as true three-dimensional digital image correlation. The structural compressibility is taken into account by an approach for a material model using the Theory of Porous Media with additional temperature dependence.     

A viscoelastic constitutive model of rubber under small oscillatory load superimposed on large static deformation

Summary  A viscoelastic constitutive equation of rubber that is under small oscillatory load superimposed on large static deformation is proposed. The model is derived through linearization of Simo's nonlinear viscoelastic constitutive model and reference configuration transformation. Most importantly, in this model, static deformation correction factor is introduced to consider the influence of pre-strain on the relaxation function. Natural statically pre-deformed state is served as reference configuration. The proposed constitutive equation is extended to a generalized viscoelastic constitutive equation that includes widely used Morman's model as a special case using objective stress increment. The proposed constitutive model is tested for dynamic behavior of rubber specimens with different carbon black content. It is concluded from the test that the assumption that the effects of static deformation can be separated from time effects, which is the basis of Morman's model, is only applicable to unfilled rubber. The viscoelastic constitutive equation for filled rubber must include, therefore, the influence of the static deformation on the time effects. The suggested constitutive equation with static deformation correction factor shows good agreement with test values. Received 4 January 2001; accepted for publication 13 June 2001     

Application of fractional exponential hereditary kernels in the nonlinear theory of viscoelasticity

It is proved that fractional exponential hereditary kernels of nonlinear viscoelasticity can be used to evaluate creep strains and stress relaxation. A nonlinear theory of viscoelasticity with time-independent nonlinearity described as a nonlinear curve of instantaneous elastoplastic deformation is used. The calculated results are validated against experimental data on the viscoelastic deformation of laminated and unidirectional fibrous composites and their components under the conditions of constant stresses, complete unloading, incremental loading, pure torsion, and constant strains     

Chemorheological response of elastomers at elevated temperatures: Experiments and simulations

An experimental study and a method for simulating the constitutive response of elastomers at temperatures in the chemorheological range (90-150 °C for natural rubber) are presented. A comprehensive set of uniaxial experiments for a variety of prescribed temperature histories is performed on natural rubber specimens that exhibit finite elasticity, entropic stiffening with temperature, viscoelasticity, scission, and oxygen diffusion/reaction effects. The simulation approach is based on a multi-network framework for finite elasticity, isothermal incompressibility, thermal expansion, and temperature-induced degradation. The model extends previous work to account for kinetics of scission for arbitrary time-varying temperature histories and incorporates the effects of viscoelastic relaxation and diffusion-limited oxidative scission. The model is calibrated to experiments performed on a commercially-available filled natural rubber material, and numerical simulations are compared favorably to experiments for a variety of temperature histories.     

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.