A constitutive equation for the Mullins effect in stress controlled uniaxial extension experiments

A one-dimensional constitutive equation for the Mullins effect in rubberlike materials, which is motivated by the two phase microstructural material model proposed by Mullins and Tobin [I], is developed in [2]. The constitutive equation is used in [2] to predict the effect of stress softening on the small transverse vibration of a rubber string loaded in uniaxial extension. The two functions which comprise the constitutive equation were assumed to be monotone, but no further analysis of the actual nature of these functions was necessary.In this paper, we examine more closely how the results of a stress controlled uniaxial extension experiment can be used to gain insight into the specific nature of the microstructural strain and the strain amplification functions which comprise the constitutive equation. We examine experimental representations of the two functions which are independent of any special microstructural interpretations. Stress controlled uniaxial extension experiments with buna-n, neoprene, and silicone rubber cords are examined. We demonstrate how the experimental data can be applied to yield representations of the functions of interest to within a multiplicative constant; but no attempt is made to find specific analytical representations of these functions. For buna-n and neoprene samples, we observe behavior consistent with our monotone assumptions, while anomalous behavior is observed with silicone rubber.     

Constitutive model for stretch-induced softening of the stress-stretch behavior of elastomeric materials

Elastomeric materials experience stretch-induced softening as evidenced by a pre-stretched material exhibiting a significantly more compliant response than that of the virgin material. In this paper, we propose a fully three-dimensional constitutive model for the observed softening of the stress-strain behavior. The model adopts the Mullins and Tobin concept of an evolution in the underlying hard and soft domain microstructure whereby the effective volume fraction of the soft domain increases with stretch. The concept of amplified strain is then utilized in a mapping of the macroscopic deformation to the deformation experienced by the soft domain. The strain energy density function of the material is then determined from the strain energy of the soft domain and thus evolves as the volume fraction of soft domain evolves with deformation. Comparisons of model results for cyclic simple extension with the experimental data of Mullins and Tobin show the efficacy of the model and suggest that an evolution in the underlying soft/hard domain microstructure of the elastomer captures the fundamental features of stretch-induced softening. Model simulations of the cyclic stress-strain behavior and corresponding evolution in structure with strain for uniaxial tension, biaxial tension and plane strain tension are also presented and demonstrate three-dimensional features of the constitutive model.     

The Mullins Effect in a Pure Shear

The Mullins effect in a rubberlike material subjected to a pure shear deformation is studied in the context of a recent theory of stress-softening for incompressible materials proposed by Beatty and Krishnaswamy. Some general technical results characterizing the mechanical response are presented. These show that the theory delivers results consistent with the overall behavior expected of a Mullins material, but usually exhibited in uniaxial extension or equibiaxial stretch experiments. The extent of stress-softening in a pure shear is shown to be much less than that due to an equibiaxial deformation, and only slightly greater than the degree of stress-softening induced by an uniaxial deformation, all to the same stretch. The Mullins effect in an equivalent simple shear deformation, even one having a rather large angle of shear, is small. The simple shear is the least damaging deformation among all of those mentioned here. Some graphical results, based on a special class of stress-softening materials applied to two parent material models – the familiar Mooney–Rivlin and a certain biotype material model, illustrate the general conclusions obtained for arbitrary Mullins materials. The inflation of a biomaterial membrane preconditioned in a pure shear deformation demonstrates the familiar stress-softening phenomenon observed in the inflation of a balloon.     

炭黑填充橡胶材料的修正三链模型和Mullins模型

论文探讨了炭黑颗粒填充橡胶材料的本构模型.考虑到橡胶单个分子链与周围分子网络的约束作用和炭黑颗粒对橡胶的补强作用,提出了一种修正三链模型,用Edwards管模型描述分子链之间的相互作用和约束,采用应变放大因子来考虑炭黑含量的影响.并在修正三链模型的基础上,利用橡胶分子网络重构理论,提出了一种适合表征橡胶Mullins现象的本构模型.通过与实验数据比较分析,修正三链模型可较准确地表征未填充橡胶材料不同变形模式的力学性能和炭黑颗粒填充橡胶材料的单向拉伸力学行为,Mullins模型也可较好地描述橡胶材料的Mullins现象.     

Observation and modeling of the anisotropic visco-hyperelastic behavior of a rubberlike material

The visco-hyperelastic behavior of a filled rubberlike material has been studied experimentally by large deformation cyclic uniaxial loadings, and an anisotropy induced by the Mullins effect has been demonstrated. By applying a generalized Maxwell model to a set of material directions, damage could be included in order to reproduce the stress softening due to the Mullins effect. This induces also an anisotropic mechanical response, and the model compares favorably with the experimental measures.     

Constitutive modeling of strain-induced softening in swollen elastomers

Under cyclic loading, elastomeric material exhibits strong inelastic responses such as stress-softening due to Mullins effect, hysteresis and permanent set. The corresponding inelastic responses are observed in both dry and swollen rubbers. Moreover, it is observed that inelastic responses depend strongly on the swelling level. For engineering applications involving the interaction and contact between rubber components and solvent, the understanding and consideration of swelling are essential pre-requisites for durability analysis. In this paper, a simple phenomenological model describing Mullins effect in swollen rubbers under cyclic loading is proposed. More precisely, the proposed model adopts the concept of evolution of soft domain microstructure with deformation originally proposed by Mullins and Tobin. The swollen rubbers are obtained by immersing dry ones in solvent until desired degrees of swelling are achieved. Subsequently, their mechanical responses, in particular Mullins effect, under cyclic loading are investigated. These experimental data are used to assess the efficiency of the proposed model. Results show that the model agrees qualitatively well with experiments. Furthermore, the model captures well the fundamental features of strain-induced softening.     

A constitutive model for carbon black filled rubber: Experimental investigations and mathematical representation

Abspract The stress-strain behavior of carbon black filled rubber is recognized to be nonlinearly elastic in its main part (see e.g. Gent [1]). In addition, inelastic effects occur under monotonic and cyclic processes. The inelastic behavior includes nonlinear rate dependence as well as equilibrium hysteresis. Moreover, the first periods of a stress-strain curve differ significantly from the shape of subsequent cycles; a characteristic feature, which is called the Mullins effect, because it has been pointed out by Mullins [2]. All inelastic phenomena are strongly influenced by the volume fraction of the filler particles (see e.g. Payne [3], So and Chen [4], Meinecke and Taftaf [5]).The aim of the present paper is to design a constitutive model, representing this kind of material behavior as a phenomenological theory of continuum mechanics. In order to motivate the basic structure of the constitutive theory, a series of uniaxial experiments between 100% in tension and 30% in compression are presented and analyzed. First of all, monotonic strain controlled experiments show the nonlinear rate dependence of the stress response. Then, a series of inserted relaxation periods at constant strain yields the monotonic equilibrium stress-strain curve, which is strongly nonlinear and unsymmetric with respect to the origin. Finally, cyclic experiments under strain control display pronounced hysteresis behavior. The hysteresis effects are mainly rate dependent, but there exists also a weak equilibrium hysteresis (compare to similar observations of Orschall and Peeken [6]). The Mullins effect corresponds to a softening phenomenon during the first few cycles. By means of an appropriate preprocess, this effect was excluded during the above experiments. Apart from the Mullins effect, neither hardening nor significant softening phenomena were observed in the context of cyclic loadings.These results motivate the structure of a constitutive model of finite strain viscoplasticity: The total stress is decomposed into an equilibrium stress and an overstress, where the overstress is a rate dependent functional of the strain history. The overstress represents the rate dependence of the material behavior and tends asymptotically to zero during relaxation processes. The nonlinearity of the rate dependence is incorporated by means of a stress dependent relaxation time. The equilibrium stress is assumed to be a rate independent functional of the strain history. For this quantity, we make use of an arclength representation, which was originally introduced by Valanis [7]. In case of vanishing equilibrium hysteresis and vanishing rate dependence our constitutive model reduces to finite strain hyperelasticity, which is the first approximation of the constitutive properties. In more general cases the main shape of a stress-strain curve is determined by hyperelasticity, superimposed by rate dependent and equilibrium hysteresis. The representation of the Mullins effect is incorporated by a continuum damage model.Some numerical simulations at the end of the paper demonstrate that the presented theory is able to represent the observed phenomena qualitatively and quantitatively with sufficient approximation.     

N330炭黑增强天然橡胶材料力学性能的实验研究

为了研究炭黑对橡胶材料力学性能的影响,对9种不同体积含量的炭黑填充橡胶材料进行了准静态力学实验研究。利用循环拉伸加卸载实验分析了炭黑对橡胶Mullins效应及能量损耗的影响,通过单轴拉伸实验研究了炭黑对橡胶材料刚度和起始模量的影响,采用多步松弛拉伸加卸载实验研究了炭黑对橡胶材料应力行为的应变历史相关性的影响。实验结果表明:炭黑填充量越高,橡胶材料的刚度越大,初始模量越大,Mullins效应也越明显;随着炭黑填充量的增加,橡胶在加卸载循环中所产生的迟滞损耗、Mullins效应相对能量损耗以及残余应变都呈现出非线性增长趋势;随着炭黑填充量的升高,橡胶在加卸载过程中的应力松弛现象越明显,其平衡态迟滞损耗以及与时间相关部分的迟滞损耗也越大。     

Brain tissue deforms similarly to filled elastomers and follows consolidation theory

Slow, large deformations of human brain tissue—accompanying cranial vault deformation induced by positional plagiocephaly, occurring during hydrocephalus, and in the convolutional development—has surprisingly received scarce mechanical investigation. Since the effects of these deformations may be important, we performed a systematic series of in vitro experiments on human brain tissue, revealing the following features. (i) Under uniaxial (quasi-static), cyclic loading, brain tissue exhibits a peculiar nonlinear mechanical behaviour, exhibiting hysteresis, Mullins effect and residual strain, qualitatively similar to that observed in filled elastomers. As a consequence, the loading and unloading uniaxial curves have been found to follow the Ogden nonlinear elastic theory of rubber (and its variants to include Mullins effect and permanent strain). (ii) Loaded up to failure, the “shape” of the stress/strain curve qualitatively changes, evidencing softening related to local failure. (iii) Uniaxial (quasi-static) strain experiments under controlled drainage conditions provide the first direct evidence that the tissue obeys consolidation theory involving fluid migration, with properties similar to fine soils, but having much smaller volumetric compressibility. (iv) Our experimental findings also support the existence of a viscous component of the solid phase deformation.Brain tissue should, therefore, be modelled as a porous, fluid-saturated, nonlinear solid with very small volumetric (drained) compressibility.     

A constitutive model for the Mullins effect with changes in material symmetry

When an unfilled or particle reinforced rubber is subjected to cyclic loading–unloading with a fixed amplitude from its natural reference configuration, the stress required on reloading is less than on the initial loading for a deformation up to the maximum value of the stretches achieved. The stress differences in successive loading cycles are largest during the first and second cycles and become negligible after about 4–6 cycles. This phenomenon is known as the Mullins effect. In this paper new experimental data are reported showing the change in material symmetry for an initially undamaged and isotropic material subjected to uniaxial and biaxial extension tests. The effect of preconditioning in one direction on the mechanical response when loaded in a perpendicular direction is discussed. A simple phenomenological model is derived to account for stress softening and changes in material symmetry. The formulation is based on the theory of pseudo-elasticity, the basis of which is the inclusion of scalar variables in the energy function. When active, these variables modify the form of the energy function during the deformation process and therefore change the material response. The general formulation is specialized to pure homogeneous deformation in order to fit the new data. The numerical results are in very good agreement with the experimental data.     

Dislocation density based constitutive model for ultrasonic assisted deformation

The mechanical behaviour of metallic materials subjected to plastic deformation is altered with the superposition of ultrasonic vibrations. A significant effect is the reduction of flow stress or acoustic softening. This phenomenon is utilized in metal forming and other deformation based manufacturing processes. Experimental investigations on ultrasonic assisted tensile tests focus on the effect of ultrasonic vibrations along the longitudinal axis of the specimen, whereas the manufacturing processes employs in transverse directions. In the present work, transverse ultrasonic vibrations are imposed during a uniaxial tensile test using an aluminium alloy. The trend of acoustic softening due to transverse direction vibrations is similar to that along longitudinal direction. A dislocation density based constitutive model is extended to model the softening due to ultrasonic effect. The predicted results agree well with the experimental observations.     

Stress-softening Effects in the Transverse Vibration of a Non-Gaussian Rubber String

The Mullins effect in the small amplitude transverse vibration of a rubber cord is investigated. The fundamental frequency is determined for a specific class of stress-softening materials. Analytical relations for the cord vibration frequency are illustrated graphically for three phenomenological models. These results demonstrate the role of the material parameters and exhibit response characteristic of those reported in experiments by others and subsequently described here in new experiments. Frequency versus stretch results for two kinds of non-Gaussian molecular network models for rubber elasticity are compared with experimental data for four varieties of rubber cords, for each of which only three experimentally determined material constants are needed. It is shown that the theoretical predictions stand in excellent agreement with test data.     

Cyclic stress-softening model for the Mullins effect in compression

This paper models the cyclic stress softening of an elastomer in compression. After the initial compression the material is described as being transversely isotropic. We derive non-linear transversely isotropic constitutive equations for the elastic response, stress relaxation, residual strain, and creep of residual strain in order to model accurately the inelastic features associated with cyclic stress softening. These equations are combined with a transversely isotropic version of the Arruda–Boyce eight-chain model to develop a constitutive relation that is capable of accurately representing the Mullins effect during cyclic stress softening for a transversely isotropic, hyperelastic material, in particular a carbon-filled rubber vulcanizate. To establish the validity of the model we compare it with two test samples, one for filled vulcanized styrene–butadiene rubber and the other for filled vulcanized natural rubber. The model is found to fit this experimental data extremely well.     

Mullins effect and cyclic stress softening of filled elastomers by internal sliding and friction thermodynamics model

Elastomers are characterized by their ability to undergo large elastic deformation. Nevertheless, their behavior exhibits stress softening, hysteresis and cyclic softening. The first phenomenon, known as Mullins effect, is commonly assumed to be either the result of an evolution in the hard and soft domain microstructure whereby the effective volume fraction of the soft domain increases with stretch or the result of irreversible damage in the material or combination of both. Hysteresis and cyclic stress softening are often considered as the result of the effect of stress relaxation. Based on the physical structure of filled elastomers, the present study shows that the Mullins effect, hysteresis and cyclic softening can be modeled by dissipative friction phenomena due to internal sliding of the macromolecular chains and to sliding of the connecting chains on the reinforcing filler particles. This implies that the three effects are in fact related to one single deformation process. The proposed analysis allows to identify the state variables and to build a thermodynamic potential which accounts for the nonlinearity of the material behavior and for a time independent hysteresis. The constitutive model is 3D. Written in a rate form it applies to complex loadings: monotonic, cyclic, random fatigue, etc. Filled elastomers hysteresis loops and cyclic softening are represented with no need to introduce neither damage nor viscosity. The model was implemented in a Finite Element software to simulate a metal/elastomer lap joint. Good agreement with experiment was achieved.     

Experimental investigation of the transverse mechanical properties of a single Kevlar® KM2 fiber

A new experimental setup is developed to investigate the transverse mechanical properties of Kevlar^® KM2 fibers, which has been widely used in ballistic impact applications. Experimental results for large deformation reveal that the Kevlar^® KM2 fibers possess nonlinear, pseudo-elastic transverse mechanical properties. A phenomenon similar to the Mullins effect (stress softening) in rubbers exists for the Kevlar^® KM2 fibers. Large transverse deformation does not significantly reduce the longitudinal tensile load-bearing capacity of the fibers. In addition, longitudinal tensile loads stiffen the fibers' transverse nominal stress–strain behaviors at large transverse deformation. Loading rates have insignificant effects on their transverse mechanical properties even in the finite deformation range. An analytical relationship between transverse compressive force and displacement is derived at infinitesimal strain level. This relation is used to estimate the transverse elastic modulus of the Kevlar^® KM2 fibers, which is 1.34 ± 0.35 GPa.     

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,本论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进行分析,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.     

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.     

Modelling of stress softening in elastomeric materials: foundations of simple theories

This work is concerned with formulation of constitutive relations for materials exhibiting the stress softening phenomenon (known as the Mullins effect) typical observed in elastomeric and other amorphous materials during loading–reloading cycles. It is assumed that microstructural changes in such materials during the deformation process can be represented by a single scalar-valued softening variable whose evolution is accompanied by microforces satisfying their own law of balance, besides the classical laws of mechanics underlying macroscopic deformation of a material. The constitutive equations are then derived in consistency with thermodynamics of irreversible processes with the restriction to purely mechanical theory. The general form of the derived constitutive equations is subsequently simplified through introduction of additional assumptions leading to various models of the stress softening phenomenon. As an illustration of the general theory, it is shown that the so-called pseudo-elastic model proposed in the literature may be derived without an ad hoc postulate of the variational principle.