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.     

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.     

The mullins effect in uniaxial extension and its influence on the transverse vibration of a rubber string

This paper presents a study of the stress softening effect encountered in uniaxial extension and explores its effect on the small amplitude transverse vibration of a stretched rubber cord. An idealization of the uniaxial stressstrain behavior of a stress softening material is presented, the importance of the deformation history is emphasized, and parameters are introduced to track the deformation history. An extended investigation of a model proposed by Mullins and Tobin to quantify stress softening by introduction of a strain amplification factor is then presented. A major result derived from this model is shown to be consistent with results reported by others. The uniaxial stress softening theory is used to describe the transverse vibration behavior of a rubber string subjected to repeated stretching. This appears to be the first application of the softening model of Mullins and Tobin to a dynamical problem. Analytical results are compared with uniaxial stress-strain and transverse vibration experiments performed with buna-n, neoprene, and silicone rubber cords. Both types of experiments provide a simple and novel method to evaluate the predictive success of our uniaxial theory without the need for a specific constitutive model. The pseudoelastic response found in biological tissues is discussed in light of results obtained in the transverse vibration experiments.     

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.     

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.     

混凝土单轴受拉的非局部本构模型

混凝土受拉本构行为存在很强的局部软化现象,使得单轴受拉试验无法给出应力-应变关系,而只能给出应力-位移关系。本文根据内变量理论和等效应变假设建立了基于真实应变的混凝土单轴受力本构方程,并根据Weibull分布可以描述混凝土等脆性材料断裂过程的试验现象,建立了关于弹性应变的损伤演化规律。然后,通过假设平均应变与真实弹性应变的函数关系,在应力-平均应变的本构关系中采用平均弹性应变以描述其非局部行为,而在材料的损伤演化规律中采用真实弹性应变以描述其局部行为,由此建立了单轴受拉荷载条件下的非局部本构模型。最后,对一个单调受拉试验和一个反复受拉试验的仿真结果表明所提出的非局部本构模型可以准确地模拟试验结果。     

Modeling of anisotropic softening phenomena: Application to soft biological tissues

A thermodynamically consistent dissipative model is proposed to describe softening phenomena in anisotropic materials. The model is based on a generalized polyconvex anisotropic strain energy function represented by a series. Anisotropic softening is considered by evolution of internal variables governing the anisotropic properties of the material. Accordingly, evolution equations are formulated and anisotropic conditions for the onset of softening are defined. In numerical examples, the model is applied to simulate the preconditioning behavior of soft biological tissues subjected to cyclic loading experiments. The results suggest that the general characteristics of preconditioning with different upper load limits are well captured including hysteresis and residual deformations. A model for the Mullins effect is obtained as a special case and shows very good agreement with experimental data on mouse skin.     

半晶态聚合物拉伸变形的微观机理

应用大规模分子动力学方法,采用粗粒化聚乙烯醇模型,模拟了晶区与非晶区随机交杂的半晶态聚合物模型系统,研究了半晶态聚合物在单轴拉伸变形过程中的应力-应变行为和微观结构演变.应力-应变曲线表现出4个典型变形阶段:弹性变形、屈服、应变软化和应变强化.在拉伸变形过程中,主要存在晶区折叠链之间的滑移、晶区破坏、非晶区的解缠结,以及分子链沿拉伸方向重新取向等4种主要的微结构演变形式.在屈服点附近,晶区分子链之间排列紧密程度减小而发生滑移,之后晶区变化需要的应力变小,从而形成应变软化现象.随着应变的增大,经各分子链段协同作用使非晶区分子链的解缠结和重新取向行为扩展到相对宏观尺度,导致拉伸应力增大而形成应变强化现象.      

Discrete dislocation dynamics modelling of mechanical deformation of nickel-based single crystal superalloys

Discrete dislocation dynamics (DDD) has been used to model the deformation of nickel-based single crystal superalloys with a high volume fraction of precipitates at high temperature. A representative volume cell (RVC), comprising of both the precipitate and the matrix phase, was employed in the simulation where a periodic boundary condition was applied. The dislocation Frank-Read sources were randomly assigned with an initial density on the 12 octahedral slip systems in the matrix channel. Precipitate shearing by superdislocations was modelled using a back force model, and the coherency stress was considered by applying an initial internal stress field. Strain-controlled loading was applied to the RVC in the [0 0 1] direction. In addition to dislocation structure and density evolution, global stress-strain responses were also modelled considering the influence of precipitate shearing, precipitate morphology, internal microstructure scale (channel width and precipitate size) and coherency stress. A three-stage stress-strain response observed in the experiments was modelled when precipitate shearing by superdislocations was considered. The polarised dislocation structure deposited on the precipitate/matrix interface was found to be the dominant strain hardening mechanism. Internal microstructure size, precipitate shape and arrangement can significantly affect the deformation of the single crystal superalloy by changing the constraint effect and dislocation mobility. The coherency stress field has a negligible influence on the stress-strain response, at least for cuboidal precipitates considered in the simulation. Preliminary work was also carried out to simulate the cyclic deformation in a single crystal Ni-based superalloy using the DDD model, although no cyclic hardening or softening was captured due to the lack of precipitate shearing and dislocation cross slip for the applied strain considered.     

A physically-based constitutive model for anisotropic damage in rubber-toughened glassy polymers during finite deformation

The present work focuses on the development of a physically-based model for large deformation stress-strain response and anisotropic damage in rubber-toughened glassy polymers. The main features leading to a microstructural evolution (regarding cavitation, void aspect ratio, matrix plastic anisotropy and rubbery phase deformation) in rubber-toughened glassy polymers are introduced in the proposed constitutive model. The constitutive response of the glassy polymer matrix is modelled using the hyperelastic-viscoplastic model of [Boyce et al., 1988] and [Boyce et al., 2000]. The deformation mechanisms of the matrix material are accounted for by two resistances: an elastic-viscoplastic isotropic intermolecular resistance acting in parallel with a visco-hyperelastic anisotropic network resistance, each resistance being modified to account for damage effects by void growth with a variation of the void aspect ratio. The effective contribution of the hyperelastic particles to the overall composite behaviour is taken into account by treating the overall system in a composite scheme framework. The capabilities of the proposed constitutive model are checked by comparing experimental data with numerical simulations. The deformation behaviour of rubber-toughened poly(methyl methacrylate) was investigated experimentally in tension at a temperature of 80 °C and for different constant true strain rates monitored by a video-controlled technique. The reinforcing phase is of the soft core-hard shell type and its diameter is of the order of one hundred nanometers. The particle volume fraction was adjusted from 15% to 45% by increments of 5%. The stress-strain response and the inelastic volumetric strain are found to depend markedly on particle volume fraction. For a wide range of rubber volume fractions, the model simulations are in good agreement with the experimental results. Finally, a parametric analysis demonstrates the importance of accounting for void shape, matrix plastic anisotropy and rubber content.     

The role of texture in tension–compression asymmetry in polycrystalline NiTi

The purpose of the present study is to thoroughly understand the stress–strain behavior of polycrystalline NiTi deformed under tension versus compression. To do this, a micro-mechanical model is used which incorporates single crystal constitutive relationships and experimentally measured polycrystalline texture into the self-consistent formulation. For the first time it is quantitatively demonstrated that texture measurements coupled with a micro-mechanical model can accurately predict tension/compression asymmetry in NiTi shape memory alloys. The predicted critical transformation stress levels and transformation stress–strain slopes under both tensile and compressive loading are consistent with experimental results. For textured polycrystalline NiTi deformed under tension it is demonstrated that the martensite evolution is very abrupt, consistent with the Luders type deformation experimentally observed. The abrupt transformation under tension is attributed to the fact that the majority of the grains are oriented along the [111] crystallographic direction, which is soft under tensile loading. Since single crystals of the [111] orientation are hard under compression it is also demonstrated that under compression the martensite in textured polycrystalline NiTi evolves relatively slower.     

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.     

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.     

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.     

炭黑填充橡胶材料的修正三链模型和Mullins模型

本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。     

Rate dependent finite strain constitutive model of polyurea

Continuous loading and unloading experiments are performed at different strain rates to characterize the large deformation behavior of polyurea under compressive loading. In addition, uniaxial compression tests are carried out with multistep strain history profiles. The analysis of the experimental data shows that the concept of equilibrium path may not be applied to polyurea. This finding implies that viscoelastic constitutive models of the Zener type are no suitable for the modeling of the rate dependent behavior of polyurea. A new constitutive model is developed based on a rheological model composed of two Maxwell elements. The soft rubbery response is represented by a Gent spring while nonlinear viscous evolution equations are proposed to describe the time-dependent material response. The eight material model parameters are identified for polyurea and used to predict the experimentally-measured stress-strain curves for various loading and unloading histories. The model provides a good prediction of the response under monotonic loading over wide range of strain rates, while it overestimates the stiffness during unloading. Furthermore, the model predictions of the material relaxation and viscous dissipation during a loading-unloading cycle agree well with the experiments.     

A viscoplastic constitutive model incorporating dynamic strain aging effect during cyclic deformation conditions

In this paper, a viscoplastic constitutive model previously proposed by the authors was extended to apply to the cyclic deformation analysis of the modified 9Cr-1Mo steel. A series of cyclic deformation tests were conducted on modified 9Cr-1Mo steel at various temperatures, including those under anisothermal conditions. Furthermore, cyclic evolution of state variables used in the authors' constitutive model was experimentally measured. Based on the test results, cyclic softening behavior depending on the temperature and its history was introduced into the constitutive model. The extended model was applied to simulations of inelastic deformation behavior under monotonic tension, stress relaxation, creep, isothermal cyclic deformations including stress relaxation and anisothermal cyclic deformations. It was found that the present constitutive model has a capability of describing the inelastic deformation behavior of modified 9Cr-1Mo steel adequately at various loading conditions.     

Micromechanical modeling of the elasto-viscoplastic behavior of semi-crystalline polymers

A micromechanically based constitutive model for the elasto-viscoplastic deformation and texture evolution of semi-crystalline polymers is developed. The model idealizes the microstructure to consist of an aggregate of two-phase layered composite inclusions. A new framework for the composite inclusion model is formulated to facilitate the use of finite deformation elasto-viscoplastic constitutive models for each constituent phase. The crystalline lamellae are modeled as anisotropic elastic with plastic flow occurring via crystallographic slip. The amorphous phase is modeled as isotropic elastic with plastic flow being a rate-dependent process with strain hardening resulting from molecular orientation. The volume-averaged deformation and stress within the inclusions are related to the macroscopic fields by a hybrid interaction model. The uniaxial compression of initially isotropic high density polyethylene (HDPE) is taken as a case study. The ability of the model to capture the elasto-plastic stress-strain behavior of HDPE during monotonic and cyclic loading, the evolution of anisotropy, and the effect of crystallinity on initial modulus, yield stress, post-yield behavior and unloading-reloading cycles are presented.     

Computational modelling of the stress-softening phenomenon of rubber-like materials under cyclic loading

When a rubber specimen is subjected to cyclic loading, not only non-linear behaviour but also damage-induced stress-softening phenomena (the Mullins effect) have been observed. Applications of a continuum damage mechanics model and Ogden and Roxburgh's pseudo-elastic model to describe the Mullins effect in elastomers have been considered. Both models together with Gao's elastic law were implemented to describe the mechanical behaviour of rubber-like materials including the stress-softening phenomenon. Two sets of experimental data (a simple tension test and a simple tension and pure shear test) are used to validate the constitutive models. Model parameters are estimated via an inverse technique. Computational results show that both constitutive models together with Gao's elastic law can describe the typical Mullins effect. From engineering point of view, the pseudo-elastic model has the advantages that (i) the model is simple and practical, since it considers that the stress-softening function is only activated on unloading or reloading paths, (ii) the model with a slight modification of the damage variable is very stable in finite element calculations, and (iii) the numerical results agree very well with experimental data in both simple tension and pure shear deformation. Two applications illustrate the capability of combining the pseudo-elastic model with Gao's elastic law in describing the Mullins effect. It is emphasized that both models are applicable to multiaxial states of stress and strain because both models are energy-based and not strain-based.