A constitutive model dealing with damage due to cavity growth and the Mullins effect in rubber-like materials under triaxial loading

In this work, we attempted to describe the evolution of damage in rubber-like materials due to the Mullins effect and the cavity growth process. To this end we introduced two distinct internal variables into the constitutive laws; the first one essentially describes the Mullins damage and the second describes the cavity growth. The Mullins effect was considered as a continuous type of damage that can be modelled within the continuum damage theory. The cavity growth, being discontinuous at the microscopic scale, was also modelled by a continuous variable after a homogenization procedure. These analyses allow the establishment of a compressible constitutive law describing the strain-softening phenomena for rubber-like materials. In order to identify the material parameters and to verify the efficiency of the model, we carried out experimental studies involving uniaxial, biaxial, and hydrostatic tensions under monotonic and cyclic loading. Comparison between the model-predicted results and the experimental data shows that the present model can efficiently describe both the Mullins damage and the porosity evolution of rubber-like materials under triaxial monotonic or cyclic loading with a satisfactory accuracy. The proposed concept is simple and easy to apply to engineering calculations.     

A micro-macro approach to rubber-like materials. Part III: The micro-sphere model of anisotropic Mullins-type damage

A micromechanically based non-affine network model for finite rubber elasticity and viscoelasticity was discussed in Parts I and II [Miehe, C., Göktepe, S., Lulei, F., 2004. A micro-macro approach to rubber-like materials. Part I: The non-affine micro-sphere model of rubber elasticity. J. Mech. Phys. Solids 52, 2617-2660; Miehe, C., Göktepe, S., 2005. A micro-macro approach to rubber-like materials. Part II: Viscoelasticity model for polymer networks. J. Mech. Phys. Solids, published on-line, doi:10.1016/j.jmps.2005.04.006.] of this work. In this follow-up contribution, we further extend the micro-sphere network model such that it incorporates a deformation-induced softening commonly referred to as the Mullins effect. To this end, a continuum formulation is constructed by a superimposed modeling of a crosslink-to-crosslink (CC) and a particle-to-particle (PP) network. The former is described by the non-affine elastic network model proposed in Part I. The Mullins-type damage phenomenon is embedded into the PP network and micromechanically motivated by a breakdown of bonds between chains and filler particles. Key idea of the constitutive approach is a two-step procedure that includes (i) the set up of micromechanically based constitutive models for a single chain orientation and (ii) the definition of the macroscopic stress response by a directly evaluated homogenization of state variables defined on a micro-sphere of space orientations. In contrast to previous works on the Mullins effect, our formulation inherently describes a deformation-induced anisotropy of the damage as observed in experiments. We show that the experimentally observed permanent set in stress-strain diagrams is achieved by our model in a natural way as an anisotropy effect. The performance of the model is demonstrated by means of several numerical experiments including the solution of boundary-value problems.     

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.     

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.     

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.     

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.     

显式模拟类橡胶材料应力软化引起的不可恢复变形及其各向异性特征

类橡胶材料在经过初次加载后会产生应力软化现象, 也就是Mullins效应. 实验证明应力软化现象会导致材料产生不可恢复变形, 同时引入各向异性特征. 本文基于对数应变构造一个多轴可压缩应变能函数, 先引入耗散来表征应力软化现象, 再引入依赖耗散大小的不可恢复变形量以及各向异性特征量, 使得新模型既可以表征Mullins效应, 又能模拟应力软化作用下产生的不可恢复变形和各向异性特征. 本文在各向同性形函数的基础上, 通过球坐标系的思想, 进一步发展并提出了一个任意方向适用的各向异性形函数. 新模型在材料尚未发生软化(耗散为0)的情况下, 表现出各向同性; 一旦发生应力软化(耗散大于0), 则变为各向异性. 随着加载?卸载循环的累积, 耗散逐渐变大, 不可恢复变形也随之变大直到达到一个稳定的值, 各向异性特性也逐渐变得明显. 新方法得到的结果可以精确匹配经典的实验数据, 并预测不同方向的应力软化现象以及由此产生的不可恢复变形和各向异性特征.      

Constitutive and damage model for 63Sn37Pb solder under uniaxial and torsional cyclic loading

A constitutive model with Ohno–Wang kinematic hardening rule is developed and employed to simulate the isothermal cyclic behavior of Sn–Pb solder under uniaxial and torsional loading. An implicit constitutive integration scheme is presented for inelastic flow of solder. Then a modified low cycle fatigue life prediction model is put forward in which the sum of maximum shear strain range and normal strain range based on the critical plane concept is adopted to replace the uniaxial strain range used by Stolkarts et al. [Stolkarts, V., Keer, L.M., Fine, M.E., 1999. Damage evolution governed by microcrack nucleation with application to the fatigue of 63Sn–37Pb solder. J. Mech. Phys. Solids 47, 2451–2468]. Comparison of the experimental results and simulation verifies that the stress strain hysteresis loops and peak stress decline curve of solder can be reasonably modeled over a wide range of loading conditions with implement of damage coupled constitutive model, and the lifetime estimations of 63Sn37Pb solder based on the assumption of microcrack nucleation governed damage is effective to provide a conservative prediction.     

A symmetric over-nonlocal microplane model M4 for fracture in concrete

This paper analyzes the effectiveness of a nonlocal integral-type formulation of a constitutive law such as microplane model M4 in which the yield limits soften as a function of the total strain for prediction of fracture propagation. For a correct regularization of the mathematical problems caused by the softening behavior, an “over-nonlocal” generalization of the type proposed by Vermeer and Brinkgreve [Vermeer, P.A., Brinkgreve, R.B.J., 1994. A new effective non-local strain measure for softening plasticity. In: Chambon, R., Desrues, J., Vardoulakis, I. (Eds.), Localization and Bifurcation Theory for Soil and Rocks, Balkema, Rotterdam, pp. 89–100.] is adopted. Moreover, the symmetric weight function, proposed by Borino et al. [Borino, G. Failla, B., Parrinello, F., 2003. A symmetric nonlocal damage theory. International Journal of Solids and Structure 40, 3621–3645.] for damage mechanics, is introduced for the calculation of the nonlocal averaging of the total strain upon which the yield limits depend. The capability of the proposed model for reproducing the stress and strain fields in the vicinity of a notch is also investigated. Finally, the symmetric over-nonlocal generalization of microplane model M4 has been applied for the simulation of a mixed-mode fracture test such as the four-point-shear test and the test of axial tension at constant shear force [Nooru-Mohamend, M.B., 1992. Mixed-mode fracture of concrete: an experimental approach. Doctoral Thesis Delft University of Thechnology, Delft, The Netherlands.]     

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.     

Development of new constitutive equations for the Mullins effect in rubber using the network alteration theory

During cyclic loading, both natural and synthetic elastomers exhibit a stress-softening phenomenon known as the Mullins effect. In the last few years, numerous constitutive equations have been proposed. The major difficulty lies in the development of models which are both physically motivated and sufficiently mathematically well defined to be used in finite element applications. An attempt to reconcile both physical and phenomenological approaches is proposed in this paper. The network alteration theory of Marckmann et al. [Marckmann, G., Verron, E., Gornet, L., Chagnon, G., Charrier, P., Fort, P., 2002. A theory of network alteration for the Mullins effect. J. Mech. Phys. Solids 50, 2011–2028] is considered and modified. The equivalence between three different strain energy functions is then used to develop two new constitutive equations. They are founded on phenomenological strain energy densities which ensure simple numerical use, but the evolution of their material parameters during stress-softening is based on physical considerations. Basic examples illustrate the efficiency of this approach.     

Stress–softening effect of SBR/nanocomposites by a phenomenological Gent–Zener viscoelastic model

An experimental study of a tensile loading–unloading procedure, as well as multi-cyclic response in a strain-controlled program of a Styrene-Butadiene (SBR) elastomer reinforced with four different weight fractions of carbon nanotubes (CNTs) has been performed. The Mullins effect features, namely hysteresis, damage and residual strain, exhibited by the SBR/nanocomposites were analyzed by a modified Gent–Zener rheological model, and a damage function. Especially for the multi-cyclic stress–strain curves, phenomenological equation of the model parameters evolution with strain were also introduced. The same loading procedure was applied in pre-stressed materials, revealing a different stress–strain response due to strain prehistory. The model has been proven to accurately capture the loading–unloading behavior, the residual strain, hysteresis loops as well as the multi-cyclic behavior of the SBR/CNT nanocomposites.     

Gent models for the inflation of spherical balloons

We revisit an iconic deformation of non-linear elasticity: the inflation of a rubber spherical thin shell. We use the 3-parameter Mooney and Gent-Gent (GG) phenomenological models to explain the stretch–strain curve of a typical inflation, as these two models cover a wide spectrum of known models for rubber, including the Varga, Mooney–Rivlin, one-term Ogden, Gent-Thomas and Gent models. We find that the basic physics of inflation exclude the Varga, one-term Ogden and Gent-Thomas models. We find the link between the exact solution of non-linear elasticity and the membrane and Young–Laplace theories often used a priori in the literature. We compare the performance of both models on fitting the data for experiments on rubber balloons and animal bladder. We conclude that the GG model is the most accurate and versatile model on offer for the modelling of rubber balloon inflation.     

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.     

Heuristic search for a predictive strain-energy function in nonlinear elasticity

In this work, a new, quasi-structural model – bootstrapped eight-chain model – is proposed as a modification to the strain energy of eight-chain model [Arruda, E.M., Boyce, M.C., 1993. A three-dimensional constitutive model for the large stretch behaviour of rubber elastic materials. J. Mech. Phys. Solids 41, 389—412] that invokes the Langevin chain statistics. This development has been led to by our heuristic search into how the strain energy of eight-chain model may be adapted in order to account better for the mechanical behaviour of elastomeric materials in both linear and nonlinear elastic regimes [Treloar, L.R.G., 1944. Stress–strain data for vulcanised rubber under various types of deformation. Trans. Faraday Soc. 40, 59–70]. The eight-chain model appears to produce very similar results in predicting biaxial stress to those of a first stretch-invariant model that gives a good fit in uniaxial extension and, thus, it is shown that the former can not be significantly enhanced within the limitation of the latter. Evaluation of predictive capability for an additive invariant-separated form of strain energy shows that an explicit inclusion of a second stretch-invariant function would not work and that any thus added term ought to be dependent on both the first and second stretch-invariants of deformation tensor, and hints that an improvement is possibly needed at low strain. The composite and filament models [Miroshnychenko, D., Green, W.A., Turner, D.M., 2005. Composite and filament models for the mechanical behaviour of elastomeric materials. J. Mech. Phys. Solids 53 (4), 748–770] have their strain-energy functions in that suggested form and cope very well with predicting the experimental data of Treloar (1944). We use the form of strain energy for the filament model, that proved to be successful, to bootstrap the strain energy of eight-chain model in order to improve the performance of the latter at low strain. Thus, we derive a new model – bootstrapped eight-chain model – that requires only two material parameters – a rubber modulus and a limiting chain extensibility. The proposed model is quasi-structural due to bootstrapping and it retains the best traits and corrects the faults of the eight-chain model, conforming more closely to the classical experimental data of Treloar (1944).     

Modeling of ductile fracture: Application of the mechanism-based concepts

This paper summarizes our recent studies on modeling ductile fracture in structural materials using the mechanism-based concepts. We describe two numerical approaches to model the material failure process by void growth and coalescence. In the first approach, voids are considered explicitly and modeled using refined finite elements. In order to predict crack initiation and propagation, a void coalescence criterion is established by conducting a series of systematic finite element analyses of the void-containing, representative material volume (RMV) subjected to different macroscopic stress states and expressed as a function of the stress triaxiality ratio and the Lode angle. The discrete void approach provides a straightforward way for studying the effects of microstructure on fracture toughness. In the second approach, the void-containing material is considered as a homogenized continuum governed by porous plasticity models. This makes it possible to simulate large amount of crack extension because only one element is needed for a representative material volume. As an example, a numerical approach is proposed to predict ductile crack growth in thin panels of a 2024-T3 aluminum alloy, where a modified Gologanu–Leblond–Devaux model [Gologanu, M., Leblond, J.B., Devaux, J., 1993. Approximate models for ductile metals containing nonspherical voids – Case of axisymmetric prolate ellipsoidal cavities. J. Mech. Phys. Solids 41, 1723–1754; Gologanu, M., Leblond, J.B., Devaux, J., 1994. Approximate models for ductile metals containing nonspherical voids – Case of axisymmetric oblate ellipsoidal cavities. J. Eng. Mater. Tech. 116, 290–297; Gologanu, M., Leblond, J.B., Perrin, G., Devaux, J., 1995. Recent extensions of Gurson’s model for porous ductile metals. In: Suquet, P. (Ed.) Continuum Micromechanics. Springer-Verlag, pp. 61–130] is used to describe the evolution of void shape and void volume fraction and the associated material softening, and the material failure criterion is calibrated using experimental data. The calibrated computational model successfully predicts crack extension in various fracture specimens, including the compact tension specimen, middle crack tension specimens, multi-site damage specimens and the pressurized cylindrical shell specimen.     

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.     

RVE-based studies on the coupled effects of void size and void shape on yield behavior and void growth at micron scales

The present paper extends the Gurson and GLD models [Gurson, A.L., 1977. Continuum theory of ductile rupture by void nucleation and growth, Part I—yield criteria and flow rules for porous ductile media. J. Mech. Phys. Solids 99, 2–15; Gologanu, M., Leblond, J.B., Devaux, J., 1993. Approximate models for ductile metals containing non-spherical voids—case of axisymmetric prolate ellipsoidal cavities. J. Mech. Phys. Solids 41, 1723–1754; Gologanu, M., Leblond, J.B., Devaux, J., 1994. Approximate models for ductile metals containing non-spherical voids—case of axisymmetric oblate ellipsoidal cavities. J. Eng. Mater. Technol. 116, 290–297] to involve the coupled effects of void size and void shape on the macroscopic yield behavior of non-linear porous materials and on the void growth. A spheroidal representative volume element (RVE) under a remote axisymmetric homogenous strain boundary condition is carefully analyzed. A wide range of void aspect ratios covering the oblate spheroidal, spherical and prolate spheroidal void are taken into account to reflect the shape effect. The size effect is captured by the Fleck–Hutchinson phenomenological strain gradient plasticity theory [Fleck, N.A., Hutchinson, J.W., 1997. Strain gradient plasticity. In: Hutchinson, J.W., Wu, T.Y. (Eds.), Advance in Applied Mechanics, vol. 33, Academic Press, New York, pp. 295–361]. A new size-dependent damage model like the Gurson and GLD models is developed based on the traditional minimum plasticity potential principle. Consequently, the coupled effects of void size and void shape on yield behavior of porous materials and void growth are discussed in detail. The results indicate that the void shape effect on the yield behavior of porous materials and on the void growth can be modified dramatically by the void size effect and vice versa. The applied stress triaxiality plays an important role in these coupled effects. Moreover, there exists a cut-off void radius r_c, which depends only on the intrinsic length l₁ associated with the stretch strain gradient. Voids of effective radius smaller than the critical radius r_c are less susceptible to grow. These findings are helpful to our further understanding to some impenetrable micrographs of the ductile fracture surfaces.     

Electro-elastomers: Large deformation analysis of silicone membranes

Electro-elastomers are large strain smart materials capable of both sensing and actuation. Typical electro-elastomer setups consist of either a silicone or acrylic membrane sandwiched between two compliant grease electrodes. Silicone electro-elastomers have maximum elastic strains between 200% and 350%. Acrylic electro-elastomers are more widely employed due to larger actuation strains but are softer than silicone and have a lower force output [Goulbourne, N.C., Frecker, M., Mockensturm, E.M., Snyder, A.J., 2003. Modeling of a dielectric elastomer diaphragm for a prosthetic blood pump. In: Proceedings of SPIE, Smart Structures and Materials: EAPAD, San Diego; Goulbourne, N.C., Mockensturm, E.M., Frecker, M., 2005b. Quasi-static and dynamic inflation of a dielectric elastomer membrane. In: Proceedings of SPIE, Smart Structures and Materials: EAPAD, San Diego]. A numerical formulation for the large deformation response of electro-elastomer membranes subject to electromechanical loading is derived in this paper. The approach is based on modifying the elastic membrane theory of Green, Adkins, and Rivlin [Adkins, J.E., Rivlin, R.S., 1952. Large elastic deformations of isotropic materials IX. The deformation of thin shells. Philosophical Transactions of the Royal Society of London. Series A Mathematical and Physical Sciences 244, 505–531; Green, A.E., Adkins, J.E., 1970. Large Elastic Deformations. Oxford University Press, London]. The electro-elastic stress state is defined as the combination of the electrical Maxwell stress and the mechanical stress for hyperelastic materials [Goulbourne, N.C., Mockensturm, E.M., Frecker, M., 2005a. A nonlinear model for dielectric elastomer membranes. ASME Journal of Applied Mechanics 72, (6) 899–906]. This paper augments our previous work by presenting a mathematical solution procedure for simulating the field responsive behavior of silicone electro-elastomers configured for both in-plane and out-of-plane deformation. Thin axisymmetric membranes subject to electromechanical loads are the focus of this investigation. The numerical analysis shows that there is a delicate balance between the electrical and the mechanical portions of the stress, which must be maintained for the overall stress to remain tensile and by extension the electro-elastomer to remain stable. It is shown that at very high voltages the stress can become negative ultimately leading to transducer failure. For sensing applications, the varying capacitive behavior of electro-elastomers is used to extract information about the membrane’s deformed state.     

显式模拟类橡胶材料Mullins效应滞回圈

通过显式、直接的方法提出一个多轴可压缩应变能函数,用来模拟类橡胶材料在加载——卸载作用下,由于Mullins效应而产生的应力——应变滞回圈. 本文的创新点在于将表征能量耗散的变量引入到应变能函数.新的弹性势具有以下两个特点:第一,在加载情况下,新引入的变量不会对弹性势产生任何影响,因此,只要给出合适的形函数显式表达,3个基准实验,包括单轴拉伸和压缩,等双轴拉伸和压缩,以及平面应变,都可精确模拟;第二,新引入的变量在卸载情况下将被激活.在不同的卸载应力下,变量将发生改变,从而影响弹性势,使其最终产生不同的应力——应变关系卸载曲线,与对应的加载曲线共同构成应力——应变滞回圈.通过对Mullins效应实验数据进行分析和研究,得出了卸载形函数在不同卸载应力下变化的规律,并预测不同卸载应力下的应力——应变关系.最后,我们将得到精确匹配实验数据的数值模拟结果,从而证明本文方法不仅可以精确匹配至少3个基准实验,还可以模拟和预测类橡胶材料在加载——卸载作用下由于Mullins效应而产生的滞回圈.