Experimentally determined properties of softening functions in pseudo-elastic models of the Mullins effect

This paper aims to identify the true source of limitations of pseudo-elastic models for describing the stress-softening phenomenon in elastomers which were recently proposed in the literature [Ogden, R.W., Roxburgh, D.G., 1999. A pseudo-elastic model for the Mullins effect in filled rubber. Proceedings of the Royal Society of London A 455 (1988), 2861–2877; Elías-Zúñiga, A., Beatty, M.F., 2002. A new phenomenological model for stress-softening in elastomers. Zeitschrift für angewandte Mathematik und Physik (ZAMP) 53 (5), 794–814]. These models as well as their modified versions [Mars, W.V., 2004. Evaluation of pseudo-elastic model for the Mullins effect. Tire Science and Technology, TSTCA 32 (3), 120–145; Elías-Zúñiga, A., 2005. A phenomenological energy-based model to characterize stress-softening effect in elastomers. Polymer 46 (10), 3496–3506] fail to give fully satisfactory coincidence of experimental data and theoretical predictions. In this paper a suitable analysis of experimental data published in the open literature is presented. This analysis shows several interesting features regarding the nature of the stress-softening phenomenon (widely known as the Mullins effect). In particular, it is shown that the magnitude of stress softening varies with strain in a non-monotonous manner and this non-monotonous character of the stress-softening phenomenon strongly depends on magnitude of the pre-strain. This experimental fact is in contradiction with the basic assumption used in pseudo-elastic models that the stress softening is described by a monotonously increasing function of strain. The common theoretical basis of pseudo-elastic models of stress softening and the source of this conflict are clarified.     

A theory of network alteration for the Mullins effect

This paper reports on the development of a new network alteration theory to describe the Mullins effect. The stress-softening phenomenon that occurs in rubber-like materials during cyclic loading is analysed from a physical point of view. The Mullins effect is considered to be a consequence of the breakage of links inside the material. Both filler-matrix and chain interaction links are involved in the phenomenon. This new alteration theory is implemented by modifying the eight-chains constitutive equation of Arruda and Boyce (J. Mech. Phys. Solids 41 (2) (1993) 389). In the present method the parameters of the eight-chains model, denoted C_R and N in the bibliography, become functions of the maximum chain stretch ratio. The accuracy of the resulting constitutive equation is demonstrated on cyclic uniaxial experiments for both natural rubbers and synthetic elastomers.     

Modeling the low-cycle fatigue behavior of visco-hyperelastic elastomeric materials using a new network alteration theory: Application to styrene-butadiene rubber

Although several theories were more or less recently proposed to describe the Mullins effect, i.e. the stress-softening after the first load, the nonlinear equilibrium and non-equilibrium material response as well as the continuous stress-softening during fatigue loading need to be included in the analysis to propose a reliable design of rubber structures. This contribution presents for the first time a network alteration theory, based on physical interpretations of the stress-softening phenomenon, to capture the time-dependent mechanical response of elastomeric materials under fatigue loading, and this until failure. A successful physically based visco-hyperelastic model is revisited by introducing an evolution law for the physical material parameters affected by the network alteration. The general form of the model can be basically represented by two parallel networks: a nonlinear equilibrium response and a time-dependent deviation from equilibrium, in which the network parameters become functions of the damage rate (defined as the ratio of the applied cycle over the applied cycle to failure). The mechanical behavior of styrene-butadiene rubber was experimentally investigated, and the main features of the constitutive response under fatigue loading are highlighted. The experimental results demonstrate that the evolution of the normalized maximum stress only depends on the damage rate endured by the material during the fatigue loading history. The average chain length and the average chain density are then taken as functions of the damage rate in the proposed network alteration theory. The new model is found to adequately capture the important features of the observed stress-strain curves under loading-unloading for a large spectrum of strain and damage levels. The model capabilities to predict variable amplitude tests are critically discussed by comparisons with experiments.     

A damage directional constitutive model for Mullins effect with permanent set and induced anisotropy

A stress softening, commonly known as the Mullins effect, is observed usually in filled rubberlike materials after the first load. Experimental observations have shown that the Mullins effect induces a permanent set and some anisotropy. In order to account for these two features, a strain energy density is proposed, which is based on material directions and coupled with a theory for network alteration. The constitutive law is based on a macromolecular approach, and the network alteration theory is motivated physically. The model has been compared to experimental data successfully.     

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.     

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。     

On the relevance of Continuum Damage Mechanics as applied to the Mullins effect in elastomers

The present paper reports and rationalizes the use of Continuum Damage Mechanics (CDM) to describe the Mullins effect in elastomers. Thermodynamics of rubber-like hyperelastic materials with isotropic damage is considered. Since it is demonstrated that stress-softening is not strictly speaking a damage phenomenon, the limitations of the CDM approach are highlighted. Moreover, connections with two-network-based constitutive models proposed by other authors are exhibited through the choice of both the damage criterion and the measure of deformation. Experimental data are used to establish the evolution equation of the stress-softening variable, and the choice of the maximum deformation endured previously by the material as the damage criterion is revealed as questionable. Nevertheless, the present model agrees qualitatively well with experiments except to reproduce the strain-hardening phenomenon that takes place as reloading paths rejoin the primary loading path. Finally, the numerical implementation highlights the influence of loading paths on material response and thereby demonstrates the importance of considering the Mullins effect in industrial design.     

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.     

An invariant based damage model of stress-softening

A phenomenological model to predict the Mullins stress-softening effect in an isotropic, incompressible, hyperelastic rubber-like material is proposed which describes deformation induced microstructural damage and the same is characterised by a simple exponential softening function. The proposed isotropic damage function depends on the maximum previous value of the first invariant of the left Cauchy–Green deformation tensor. The proposed model of softening is illustrated with the theory of Gent material model and finally it is validated with experimental data provided in the literature. The model shows a simple functional form and brings out the interrelation between other models of this type.     

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.     

A finite strain kinematic hardening constitutive model based on Hencky strain: General framework, solution algorithm and application to shape memory alloys

The logarithmic or Hencky strain measure is a favored measure of strain due to its remarkable properties in large deformation problems. Compared with other strain measures, e.g., the commonly used Green-Lagrange measure, logarithmic strain is a more physical measure of strain. In this paper, we present a Hencky-based phenomenological finite strain kinematic hardening, non-associated constitutive model, developed within the framework of irreversible thermodynamics with internal variables. The derivation is based on the multiplicative decomposition of the deformation gradient into elastic and inelastic parts, and on the use of the isotropic property of the Helmholtz strain energy function. We also use the fact that the corotational rate of the Eulerian Hencky strain associated with the so-called logarithmic spin is equal to the strain rate tensor (symmetric part of the velocity gradient tensor). Satisfying the second law of thermodynamics in the Clausius-Duhem inequality form, we derive a thermodynamically-consistent constitutive model in a Lagrangian form. In comparison with the available finite strain models in which the unsymmetric Mandel stress appears in the equations, the proposed constitutive model includes only symmetric variables. Introducing a logarithmic mapping, we also present an appropriate form of the proposed constitutive equations in the time-discrete frame. We then apply the developed constitutive model to shape memory alloys and propose a well-defined, non-singular definition for model variables. In addition, we present a nucleation-completion condition in constructing the solution algorithm. We finally solve several boundary value problems to demonstrate the proposed model features as well as the numerical counterpart capabilities.     

A network evolution model for the anisotropic Mullins effect in carbon black filled rubbers

To the best of our knowledge, there are no constitutive models that properly describe experimental data on anisotropy of the Mullins effect. In this paper, such a micro-mechanical model is proposed for carbon black filled rubbers. The model describes the deformation induced anisotropy and permanent set as well. Damage of the polymer-filler network is considered as a consequence of chain sliding on or debonding from aggregates. In contrast to previous works on anisotropy of the Mullins effect we do not introduce any phenomenological damage function. Damage in different directions is governed by a network evolution concept which describes the changes in the inter-aggregate distribution of polymer chains. The model includes a few number of physically motivated material constants and demonstrates good agreement with own experimental data on subsequent uniaxial tensions in two orthogonal directions.     

Direction dependent orthotropic model for Mullins materials

The motivating key for this work was the absence of a phenomenological model that can reasonably predict a variety of non-proportional experimental data on the anisotropic Mullins effect for different types of rubber-like materials. Hence, in this paper, we propose a purely phenomenological direction dependent orthotropic model that can describe the anisotropic Mullins behaviour with permanent set and, has orthotropic invariants that have a clear physical interpretation. The formulation is based on an orthotropic principal axis theory recently developed for nonlinear elastic problems. A damage function and a direction dependent damage parameter are introduced in the formulation to facilitate the analysis of anisotropic stress softening in rubber-like materials. A direction dependent free energy function, written explicitly in terms of principal stretches, is postulated. The proposed theory is able to predict and compares well with experimental data available in the literature for different types of rubberlike materials.     

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.     

A three-dimensional nonlinear model for dissipative response of soft tissue

A set of three-dimensional constitutive equations is proposed for modeling the nonlinear dissipative response of soft tissue. These constitutive equations are phenomenological in nature and they model a number of physical features that have been observed in soft tissue. The equations model the tissue as a composite of a purely elastic component and a dissipative component, both of which experience the same total dilatation and distortion. The stress response of the purely elastic component depends on dilatation, distortion and the stretch of material fibers, whereas the stress response of the dissipative component depends on distortional deformation only. The equations are hyperelastic in the sense that the stress is obtained by derivatives of a strain energy function, and they are properly invariant under superposed rigid body motions. In contrast with standard viscoelastic models of tissues, the proposed constitutive model includes the total deformation rate in evolution equations that can reproduce the observed physical feature that the hysteresis loops of most biological soft tissues are nearly independent of strain rate (Biomechanics, Mechanical Properties of Living Tissues, second ed. (1993)). Material constants are determined which produce good agreement with uniaxial stress experiments on superficial musculoaponeurotic system and facial skin.     

Anisotropic Mullins stress softening of a deformed silicone holey plate

Rubber like materials parts are designed using finite element code in which more and more precise and robust constitutive equations are implemented. In general, constitutive equations developed in literature to represent the anisotropy induced by the Mullins effect present analytical forms that are not adapted to finite element implementation. The present paper deals with the development of a constitutive equation that represents the anisotropy of the Mullins effect using only strain invariants. The efficiency of the modeling is first compared to classical homogeneous experimental tests on a filled silicone rubber. Second, the model is tested on a complex structure. In this aim, a silicone holey plate is molded and tested in tension, its local strain fields are evaluated by means of digital image correlation. The experimental results are compared to the simulations from the constitutive equation implemented in a finite element code. Global measurements (i.e. force and displacement) and local strain fields are successfully compared to experimental measurements to validate the model.     

Effect of Stress-softening on the Dynamics of a Load Supported by a Rubber String

The Mullins effect in the oscillatory motion of a load under gravity and attached to a stress-softening, neo-Hookean rubber string is investigated. Equations for the small amplitude vertical oscillations of the load superimposed on the finite static stretch of both the virgin and stress-softened cords, the latter subjected to varying degrees of preconditioning, are derived. The vibrational frequency of the small motion exhibits behavior similar to that observed in experiments by others on postmortem, human aortic tissue for which no stress-softening is reported. Standard numerical methods are applied to study the finite amplitude motion of the load in the stress-softened case. The resultant motions and their various physical aspects under free-fall and general initial conditions are described in several examples. Oscillations that engage all three phases of motion consisting of the suspension, the free-flight, and the retraction of the load in its general vertical motion are illustrated. Effects due to the degree of stress-softening are discussed; and the motion response for two values of the model softening parameter is compared in several examples. All results are illustrated graphically and numerous tabulated numerical results are provided.      

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.     

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.     

金属材料的率-温耦合响应与动态本构关系综述

金属材料在冲击、爆炸等高应变率加载下的塑性流动行为具有不同于静载下的率-温耦合性和微观机制。航空航天、航海、能源开采、核工业、公共安全、灾害防治等方面的金属结构设计与性能评估需要进行大量的动载实验和数值模拟,建立准确的材料动态本构模型是结构数值模拟可靠性的基础和关键。本文中,总结了金属材料的率-温耦合变形行为及内在机理,回顾了金属动态本构关系研究的起源与发展脉络,分别针对唯象模型、具有物理基础的模型和人工神经网络模型进行了详细介绍和横向比较。唯象模型和人工神经网络模型分别因易应用和高预测精度而受到青睐,基于物理概念的宏观连续介质模型可以描述体现内部演化的真实物理量,从而涵盖更大的应变范围,更好地反映应变率、温度和应变的影响机制。