The Payne effect in finite viscoelasticity: constitutive modelling based on fractional derivatives and intrinsic time scales

As we know from experimental testing, the stiffness behaviour of carbon black-filled elastomers under dynamic deformations is weakly dependent on the frequency of deformation but strongly dependent on the amplitude. Increasing strain amplitudes lead to a decrease in the dynamic stiffness, which is known as the Payne effect. In this essay, we develop a constitutive approach of finite viscoelasticity to represent the Payne effect in the context of continuum mechanics. The starting point for the constitutive model resulting from this development is the theory of finite linear viscoelasticity for incompressible materials, where the free energy is assumed to be a linear functional of the relative Piola strain tensor. Motivated by the weak frequency dependence of the dynamic stiffness of reinforced rubber, the memory kernel of the free energy functional is of the Mittag Leffler type. We demonstrate that the model is compatible with the Second Law of Thermodynamics and equal to a fractional differential equation between the overstress of the Second Piola Kirchhoff type and the Piola strain tensor. In order to represent the dependence of the dynamic stiffness on the amplitude of strain, we replace the physical time by an intrinsic time variable. The temporal evolution of the intrinsic time is driven by an internal variable, which is a measure for the current state of the material's microstructure. The material constants of the model are estimated using a stochastic identification algorithm of the Monte Carlo type. We demonstrate that the constitutive approach pursued here represents the combined frequency and amplitude dependence of filler-reinforced rubber. In comparison with the micromechanical Kraus model developed for sinusoidal strains, the theory set out in this essay allows the representation of the stress response under arbitrary loading histories.     

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。     

Shape memory behaviour: modelling within continuum thermomechanics

A phenomenological material model to represent the multiaxial material behaviour of shape memory alloys is proposed. The material model is able to represent the main effects of shape memory alloys: the one-way shape memory effect, the two-way shape memory effect due to external loads, the pseudoelastic and pseudoplastic behaviour as well as the transition range between pseudoelasticity and pseudoplasticity.The material model is based on a free energy function and evolution equations for internal variables. By means of the free energy function, the energy storage during thermomechanical processes is described. Evolution equations for internal variables, e.g. the inelastic strain tensor or the fraction of martensite are formulated to represent the dissipative material behaviour. In order to distinguish between different deformation mechanisms, case distinctions are introduced into the evolution equations. Thermomechanical consistency is ensured in the sense that the constitutive model satisfies the Clausius–Duhem inequality.Finally, some numerical solutions of the constitutive equations for isothermal and non-isothermal strain and stress processes demonstrate that the various phenomena of the material behaviour are well represented. This applies for uniaxial processes and for non-proportional loadings as well.     

Compressible rubber materials: experiments and simulations

Porous rubber materials are often used in automotive industries. In this paper, a carbon black-filled one is investigated, which is used, for example, as sealing. Such materials are distinguished by viscoelastic behaviour and by a structural compressibility induced by the porous structure. To identify the material behaviour, uniaxial tension tests and hydrostatic compression tests are performed. Therein the main focus of attention lies on the basic elasticity and on the viscoelasticity in the whole loading range. An important observation of these tests is the viscoelastic behaviour under hydrostatic compression, which has to be included in the material model. Because of the two-phase character of cellular rubber, the theory of porous media is taken into account. To model the structural compressibility, a volumetric–isochore split of the deformation gradient is used. Therein the volumetric part includes the aspect of the point of compaction. Finally, the concept of finite viscoelasticity is applied introducing an intermediate configuration. Because of the viscoelastic behaviour under hydrostatic compression, the volumetric–isochore split is taken into account for the nonequilibrium parts, too. Nonlinear relaxation functions are used to model the process-dependent relaxation times and the highly nonlinear behaviour with respect to the deformation and feedrate. The material parameters of the model are estimated using a stochastic identification algorithm.     

Mechanical behavior of amorphous polymers in shear

Based on the non-equilibrium thermodynamic theory, a new thermo-viscoelastic constitutive model for an incompressible material is proposed. This model can be considered as a kind of generalization of the non-Gaussian network theory in rubber elasticity to include the viscous and the thermal effects. A set of second rank tensorial internal variables was introduced, and in order to adequately describe the evolution of these internal variables, a new expression of the Helmholtz free energy was suggested. The mechanical behavior of the thermo-viscoelastic material under simple shear deformation was studied, and the “ viscous dissipation induced“ anisotropy due to the change of orientation distribution of molecular chains was examined. Influences of strain rate and thermal softening produced by the viscous dissipation on the shear stress were also discussed. Finally, the model predictions were compared with the experimental results performed by G‘ Sell et al. , thus the validity of the proposed model is verified.     

A thermodynamic formulation of finite-deformation elastoplasticity with hardening based on the concept of material isomorphism

This paper presents a thermodynamic formulation of a model for finite deformation of materials exhibiting elastoplastic material behaviour with non-linear isotropic and kinematic hardening. Central to this formulation is the notion that the form of the elastic constitutive relation be unaffected by the plastic deformation or transformation in the material, as commonly assumed in particular in the context of crystal plasticity. When generalized to the phenomenological context, this implies that the internal variable representing plastic deformation is an elastic material isomorphism. Among other things, this requirement on the plastic deformation leads directly to the standard elastoplastic multiplicative decomposition of the deformation gradient. In addition, a dependence of the plastic part of the free energy on the plastic deformation itself yields a thermodynamic form for the centre of the elastic range of the material, i.e. the back stress. Finally, we show how this approach can be applied to formulate thermodynamic forms for linear, and non-linear Armstrong-Frederick, kinematic hardening models.     

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.     

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.     

Nonlinear Fractional Order Viscoelasticity at Large Strains

In this paper, we formulate a fractional order viscoelastic model for large deformations and develop an algorithm for the integration of the constitutive response. The model is based on the multiplicative split of the deformation gradient into elastic and viscous parts. Further, the stress response is considered to be composed of a nonequilibrium part and an equilibrium part. The viscous part of the deformation gradient (here regarded as an internal variable) is governed by a nonlinear rate equation of fractional order. To solve the rate equation the finite element method in time is used in combination with Newton iterations. The method can handle nonuniform time meshes and uses sparse quadrature for the calculations of the fractional order integral. Moreover, the proposed model is compared to another large deformation viscoelastic model with a linear rate equation of fractional order. This is done by computing constitutive responses as well as structural dynamic responses of fictitious rubber materials.     

Nonlinear Fractional Order Viscoelasticity at Large Strains

In this paper, we formulate a fractional order viscoelastic model for large deformations and develop an algorithm for the integration of the constitutive response. The model is based on the multiplicative split of the deformation gradient into elastic and viscous parts. Further, the stress response is considered to be composed of a nonequilibrium part and an equilibrium part. The viscous part of the deformation gradient (here regarded as an internal variable) is governed by a nonlinear rate equation of fractional order. To solve the rate equation the finite element method in time is used in combination with Newton iterations. The method can handle nonuniform time meshes and uses sparse quadrature for the calculations of the fractional order integral. Moreover, the proposed model is compared to another large deformation viscoelastic model with a linear rate equation of fractional order. This is done by computing constitutive responses as well as structural dynamic responses of fictitious rubber materials.     

Dissipative heating of elastomers: a new modelling approach based on finite and coupled thermomechanics

Especially in the automotive industries, elastomers take an important role. They are used in different types of bearings, where they inhibit vibration propagation and thereby significantly enhance driving performance and comfort. That is why several models have already been developed to simulate the material’s mechanical response to stresses and strains. In many cases, these models are developed under isothermal conditions. Others include the temperature-dependent mechanical behaviour to represent lower stiffness’s for higher temperatures. In this contribution it is shown by some exemplary experiments that viscoelastic material heats up under dynamic deformations. Hence, the material’s properties change due to the influence of the temperature without changing the surrounding conditions. With some of these experiments, it is shown that a fully coupled material model is necessary to predict the behaviour of bearings under dynamic loads. The focus of this contribution lies on the modelling of the thermoviscoelastic behaviour of elastomers. In a first step, a twofold multiplicative split of the deformation gradient is performed to be able to describe both mechanical and thermal deformations. This concept introduces different configurations. The stress tensors existing on these configurations are used to formulate the stress power in the first law of thermodynamics which allows to simulate the material’s self-heating. To formulate the temperature dependency of the mechanical behaviour, the non-equilibrium part of the Helmholtz free energy function is formulated as a function of the temperature and the deformation history. With the introduced model, some FE calculations are carried out to show the model’s capability to represent the thermoviscoelastic behaviour including the coupling in both directions.     

Representation of the glass-transition in mechanical and thermal properties of glass-forming materials: A three-dimensional theory based on thermodynamics with internal state variables

In the vicinity of the glass transition, glass-forming materials exhibit pronounced frequency-dependent changes in the mechanical material properties, the thermal expansion behaviour and the specific heat. The frequency dependence becomes apparent under harmonic stress, strain or temperature excitations. The Prigogine-Defay ratio is a characteristic number which connects the changes in magnitude of these quantities at the glass transition. In order to represent the thermoviscoelastic properties of glass-forming materials in continuum mechanics, a three-dimensional approach which is based on the Gibbs free energy as thermodynamic potential is developed in this article. The Gibbs free energy depends on the stress tensor, the temperature and a set of internal variables which is introduced to take history-dependent phenomena into account. In the vicinity of an equilibrium reference state, the specific Gibbs free energy is approximated up to second order terms. Evaluating the Clausius-Duhem inequality, the constitutive relations for the strain tensor, the entropy and the internal variables are derived. In comparison with other approaches, the entropy, the strain tensor and the internal variables are functionals not only of the stress tensor but also of the temperature. Applying harmonic temperature- or stress-controlled excitations, complex frequency-dependent relations for the specific heat under constant stress, for the thermal expansion coefficients as well as for the dynamic mechanical compliance are obtained. The frequency-dependence of these quantities depicts the experimentally observed behaviour of glass-forming materials as published in literature. Under the assumption of isotropic material behaviour, it is shown that the developed theory is compatible with the Prigogine-Defay inequality for arbitrary values of the material parameters.     

基于修正偶应力和高阶剪切变形理论的变截面微梁的自由振动

基于修正偶应力和高阶剪切理论建立了仅含有一个尺度参数的Reddy变截面微梁的自由振动模型,研究了变截面微梁自由振动问题的尺度效应和横向剪切变形对自振频率计算的影响。基于哈密顿原理推导了动力学方程与边界条件,并采用微分求积法求解了各种边界条件下的自振频率。算例结果表明,基于偶应力理论预测的变截面微梁的自振频率均大于经典梁理论的预测结果,即捕捉到了尺度效应。另外,梁的几何尺寸与尺度参数越接近,尺度效应就越明显,而梁的长细比越小,横向剪切变形对自振频率的影响就越明显。     

A theory of finite deformation magneto-viscoelasticity

This paper deals with the mathematical modelling of large strain magneto-viscoelastic deformations. Energy dissipation is assumed to occur both due to the mechanical viscoelastic effects as well as the resistance offered by the material to magnetisation. Existence of internal damping mechanisms in the body is considered by decomposing the deformation gradient and the magnetic induction into ‘elastic’ and ‘viscous’ parts. Constitutive laws for material behaviour and evolution equations for the non-equilibrium fields are derived that agree with the laws of thermodynamics. To illustrate the theory the problems of stress relaxation, magnetic field relaxation, time dependent magnetic induction and strain are formulated and solved for a specific form of the constitutive law. The results, that show the effect of several modelling parameters on the deformation and magnetisation process, are illustrated graphically.     

A micromechanically motivated material model for the thermo-viscoelastic material behaviour of rubber-like polymers

The material behaviour of rubber at the micro level is usually described by means of statistical mechanics. In particular, the Neo-Hooke model has been derived in this fashion. The micromechanical modelling can be extended to include also the breaking and reforming of chains. One possible approach at this level is the so-called transient network theory. Using certain assumptions for the chain distributions, one arrives at a continuum mechanical model of finite viscoelasticity which is based on the multiplicative decomposition of the deformation gradient. This means that the inelastic part of the deformation is regarded as an elastic isomorphism. Further, the considerations at the micro level give information about the temperature dependence of the mechanical material parameters. For instance, it can be shown easily that the shear modulus depends approximately linearly on the temperature. This fact has important consequences for thermo-mechanical coupling which have not yet been discussed in detail in the literature.     

Visco-hyperelastic model with internal state variable coupled with discontinuous damage concept under total Lagrangian formulation

The silica-filled rubber material presented in this paper exhibits nonlinear elasticity, nonlinear rate dependence and stress-softening effect under cyclic loading. In order to model the material behavior in a finite element code, the internal state variable concept is considered for finite deformation viscoelasticity. Moreover, the so-called Mullins' effect is taken into account by using a discontinuous damage concept. A total Lagrangian formulation with incompressibility constraint is adopted in the finite element code. The constitutive equations with their optimized set of parameters are validated by comparing the simulated results with experimental data. This result is very useful for the fatigue lifetime analysis of the investigated silica-filled rubber material.     

Modelling of frequency- and amplitude-dependent material properties of filler-reinforced rubber

Filler-reinforced elastomers are extremely complicated materials with pronounced deformation and temperature history-dependent material properties. In the current paper, the dynamic material behaviour is investigated and modelled. To this end, a carbon black-filled rubber compound is loaded with harmonic deformations under different frequencies and amplitudes and the stationary stress response is evaluated in terms of the storage and the loss modulus or, equivalently, in terms of the dynamic modulus and the loss angle. In this essay, detailed experimental investigations of the dynamic material properties of carbon black-filled elastomers are provided and a new three-dimensional constitutive approach of finite nonlinear viscoelasticity to represent the observed material behaviour is developed.     

Stress relaxation,creep and set recovery of elastomers

The stress relaxation, creep and recovery behaviour of a cross-linked unfilled natural rubber has been investigated at moderate stresses in tension. The aim being to extend the idea, initially developed by Alan Gent in his seminal 1962 paper on the relaxation behaviour of rubber, in order to understand and examine the time dependent mechanisms that are present in elastomers under strain. A method based upon the Boltzmann superposition principle was used to compare the creep compliance with a measurement of its recovery after release from a range of constant loads held for different times. The creep behaviour was seen to exhibit the usual linear dependence on the logarithm of time. The recovery data was also seen to reduce onto a single recovery curve for any given applied tensile stress for a range of loading times using the Boltzmann superposition principle. The differences between the relative rates of the creep and the recovery behaviour can in part be attributed to the non-linearity in the stress–strain behaviour exhibited in tension of the elastomer.