A comparison of limited-stretch models of rubber elasticity

In this paper we describe various limited-stretch models of non-linear rubber elasticity, each dependent on only the first invariant of the left Cauchy–Green strain tensor and having only two independent material constants. The models are described as limited-stretch, or restricted elastic, because the strain energy and stress response become infinite at a finite value of the first invariant. These models describe well the limited stretch of the polymer chains of which rubber is composed. We discuss Gent׳s model which is the simplest limited-stretch model and agrees well with experiment. Various statistical models are then described: the one-chain, three-chain, four-chain and Arruda–Boyce eight-chain models, all of which involve the inverse Langevin function. A numerical comparison between the three-chain and eight-chain models is provided. Next, we compare various models which involve approximations to the inverse Langevin function with the exact inverse Langevin function of the eight-chain model. A new approximate model is proposed that is as simple as Cohen׳s original model but significantly more accurate. We show that effectively the eight-chain model may be regarded as a linear combination of the neo-Hookean and Gent models. Treloar׳s model is shown to have about half the percentage error of our new model but it is much more complicated. For completeness a modified Treloar model is introduced but this is only slightly more accurate than Treloar׳s original model. For the deformations of uniaxial tension, biaxial tension, pure shear and simple shear we compare the accuracy of these models, and that of Puso, with the eight-chain model by means of graphs and a table. Our approximations compare extremely well with models frequently used and described in the literature, having the smallest mean percentage error over most of the range of the argument.     

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).     

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.     

Problems in determining the elastic strain energy function for rubber

Lightly crosslinked natural rubber can be stretched by 600% or more, and recovers almost completely. It is often regarded as a model highly elastic material and characterized by a strain energy function to describe its stress-strain behavior under various types of deformation. A number of such functions have been proposed; some of them appear in current finite element programs. They are usually validated by comparison with measured stress-strain relations by Treloar [7] [L.R.G. Treloar, Stress-strain data for vulcanized rubber under various types of deformation, Trans. Faraday Soc. 40 (1944) 59-70] and Jones and Treloar [15] [D.F. Jones, L.R.G. Treloar, The properties of rubber in pure homogeneous strain, J. Phys. D Appl. Phys. 8 (1975) 1285-1304]. But Treloar pointed out that the relations at high strains became markedly irreversible, and he did not assign a strain energy function for strains greater than about 300%. Rivlin's universal relation between torsional stiffness and tensile stress [14] [R.S. Rivlin, Large elastic deformations of isotropic materials. Part V1: further results in the theory of torsion, shear and flexure, Philos. Trans. R. Soc. A 243 (1949) 251-288] is applied here to show that a typical elastic solid cannot be described by any strain energy function at strains greater than about 300%. Elastic strain energy functions for higher strains, or for other rubbery materials, are thus of doubtful value unless evidence for reversibility of stress-strain relations is adduced or the applicability of a strain energy function is demonstrated.     

A Strain Energy Function for Vulcanized Rubbers

A three-parameter strain energy function is developed to model the nonlinearly elastic response of rubber-like materials. The development of the model is phenomenological, based on data from the classic experiments of Treloar, Rivlin and Saunders, and Jones and Treloar on sheets of vulcanized rubber. A simple two-parameter version, similar to the Mooney-Rivlin and Gent-Thomas strain energies, provides an accurate fit with all of the data from Rivlin and Saunders and Jones and Treloar, as well as with Treloar’s data for deformations for which the principal deformation invariant I ₁ has values in the range 3≤I ₁≤20.     

炭黑填充橡胶材料改进Mooney模型

本文讨论了炭黑填充橡胶材料的唯象本构模型。考虑到Mooney模型无法表征橡胶类材料大变形阶段的力学特性,首先利用实验数据,对Mooney模型进行了分析,讨论了炭黑含量与Mooney模型准确表征橡胶材料应变区间大小的关系,Mooney模型对纯剪切和等比双向拉伸等复杂变形的预测能力,同时也分析了材料参数对Mooney模型的影响。最后在Mooney模型的基础上添加了一个修正项,且改进后的Mooney模型满足Treloar和Ogden六项假设。通过与实验数据对比分析,改进Mooney模型可以较好地描述橡胶材料大变形阶段的应力应变关系,同时提高了预测橡胶材料复杂变形的能力。     

Hyperelastic models for rubber-like materials: consistent tangent operators and suitability for Treloar’s data

Rubber-like materials consist of chain-like macromolecules that are more or less closely connected to each other via entanglements or cross-links. As an idealisation, this particular structure can be described as a completely random three-dimensional network. To capture the elastic and nearly incompressible mechanical behaviour of this material class, numerous phenomenological and micro-mechanically motivated models have been proposed in the literature. This contribution reviews fourteen selected representatives of these models, derives analytical stress–stretch relations for certain homogeneous deformation modes and summarises the details required for stress tensors and consistent tangent operators. The latter, although prevalently missing in the literature, are indispensable ingredients in utilising any kind of constitutive model for the numerical solution of boundary value problems by iterative approaches like the Newton–Raphson scheme. Furthermore, performance and validity of the models with regard to the classical experimental data on vulcanised rubber published by Treloar (Trans Faraday Soc 40:59–70, 1944) are evaluated. These data are here considered as a prototype or worst-case scenario of highly nonlinear elastic behaviour, although inelastic characteristics are clearly observable but have been tacitly ignored by many other authors.     

Probabilistic and stochastic aspects of rubber hyperelasticity

The main aim of this work is to compare various models of rubber elasticity, i.e. neo-Hookean, Mooney-Rivlin, Yeoh, Gent, Arruda-Boyce as well as the Extended Tube model in terms of their application to the probabilistic analysis. Some discussions concerning failure analysis of the rubbers according to these models is provided also. Constitutive relations following these theories are tested for the case of uniaxial tension of the incompressible material, where deformation of a rubber specimen is treated as Gaussian random variable having a priori given expectation and standard deviations varying in some interval with bounds driven by various experimentation techniques. Probabilistic analysis is provided here in two alternative ways—via traditional Monte-Carlo technique as well as using higher order stochastic perturbation method implemented both in the symbolic computer algebra software. An application of non-Gaussian distributions relevant to the considered deformation, like lognormal one for instance has been also considered. This analysis includes computational determination of the first four basic probabilistic characteristics, i.e. expectation, coefficient of variation, skewness and kurtosis, and is provided to verify the resulting probabilistic distribution of the induced stress and its entropy. Some conclusions are drawn for the generalization of this method to other stress softening materials.     

基于单轴数据决定橡胶类材料多轴刚性化有界超弹性势

指出表征橡胶类材料应变刚化效应的现有超弹性模型涉及应变能无穷发散困难,提出新方法解决该困难。基于对数应变不变量的多轴扩张和多轴匹配步骤,建议直接构造橡胶类材料大变形弹性势的显式直接方法。该方法从单轴应力-应变关系直接得到多轴弹性势,所得结果避免了现有各方法决定待定参数组的复杂数值计算,能够准确描述应变刚性化效应,且给出有界弹性应变能,从而避免了前述发散困难。数值结果表明,从单轴数据所得到的弹性势可同时很好的拟合平面应变拉伸(剪切)数据以及等双轴拉伸数据。     

Composite and filament models for the mechanical behaviour of elastomeric materials

In this work we present a composite model, which combines the approach of Poisson's function with the filament theory and requires three material parameters. We also suggest the form for a strain-energy function that approximates the constitutive equations of the composite model. Furthermore, a simple asymptotic analysis allows us to reduce the number of material constants to only two, thus, forming a new filament model. The predictive capability of the two models to reproduce the mechanical behaviour of elastomeric materials in deformation experiments is evaluated against the extensive data of Kawabata et al. (Macromolecules 14 (1981) 154). The models give excellent agreement in not only uniaxial and equibiaxial but also non-equibiaxial extension. Although being rather more simplistic in comparison with some successful network models involving non-Gaussian chain statistics, the two models conform much more closely to the classical experimental data of Treloar (Trans. Faraday Soc. 40 (1944) 59).     

Three-dimensional modeling of tires

Modeling the stress-strain state of pneumatic tires in the conditions of steady-state and transient rolling is of interest for mechanics of composites and computational mechanics and important from the applied point of view. Mechanical models of various levels of complexity can be used for numerical modeling. In quite a few papers, the corresponding models are derived from the theory of orthotropic shells [1]. However, more thorough and accurate studies of the stress-strain state can be carried out on the basis of three-dimensional models based on the elasticity or viscoelasticity equations. As far as Russian authors are concerned, this approach has first been suggested and implemented in [2]. Another, combined approach uses both the shell theory and the three-dimensional equations of elasticity theory [3, 4]. This approach is reasonable, because the tire structure includes both volumes filled with rubber and thin layers of the rubber cord. The rubber cord layers can be considered as a composite whose structural components possess very different properties. Also, it is quite admissible to consider the rubber cord as a structure periodic in the horizontal projection. Note that the mathematical theory of periodic composites has been developed in [5]. Owing to strong anisotropy and inhomogeneity of the material, large shape distortions of the tire, and, in some cases, its large deformations, viscoelastic properties of rubber play an important role, so that the mechanic model of the tire turns out to be quite complex. The large property differences between various structural components make the matrix of the resulting system of linear equations ill-conditioned, which complicates its numerical solution [6].In this paper, theoretical aspects of a three-dimensional tire model and its numerical implementation are considered.     

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.     

The remarkable Gent constitutive model for hyperelastic materials

In 1996, Alan Gent published a short paper that proposed the use of a very simple two parameter phenomenological constitutive model for hyperelastic isotropic incompressible materials. The model is empirical but has the advantages of mathematical simplicity, reflects the severe strain-stiffening at large strains observed experimentally, reduces to the classic neo-Hookean model for small strains and involves just two material parameters namely the shear modulus for infinitesimal deformations and a parameter that measures a maximum allowable value of strain. The model reflects the limiting chain extensibility characteristic of non-Gaussian molecular models for rubber. Here we review some of the numerous developments, extensions and widespread applications that have resulted from that groundbreaking paper not only in rubber elasticity but also in the area of biomechanics of soft biomaterials. The Gent model is remarkably robust: its mathematical simplicity combined with physical basis has ensured that it has reached status as a fundamental canonical phenomenological constitutive model for hyperelastic materials.     

On a molecular statistical basis for Ogden's model of rubber elasticity

In this paper, a link is established between the statistical theory of long chain molecules and Ogden's phenomenological model of rubber elasticity. It has been shown by several authors in the past that many invariant-based phenomenological models for rubber-like materials are related to the classical statistical theories. The essential means to reach this reconciliation were methods to account for a non-affine deformation of polymer chains in the network, appropriate techniques to calculate their averaged response, and an approximation of the inverse Langevin function appearing in the non-Gaussian statistical theory. It is shown in this paper that the very same approach, if appropriately implemented, allows to express the strain-energy function of Ogden's material in terms of physical constants characterising the polymer chain and network, together with few additional parameters that account for the non-affine deformation of the polymer chains. Particularly, it is shown that Ogden's model can be represented as a non-affine non-Gaussian 3-chain model with topological constraints.     

Pressure Dependent Properties of a Compressible Polymer

This paper deals with the investigations of a porous carbon black-filled rubber, tested with regard to its pressure and tension behaviour. In the tension range only uniaxial tests are performed while in the pressure range uniaxial as well as hydrostatic tests are performed. The uniaxial experiments are carried out in a custom-made uniaxial device and the hydrostatic tests in a pressure chamber which is specially developed for this application. The construction and use of the pressure chamber is clearly described in this paper. All experiments are related to the basic elasticity of the material. The viscoelastic behaviour is completely disregarded at this point. Not only the experiments are discussed, also the modelling of the material is looked at. The tested cellular rubber is composed of an incompressible solid phase and a compressible gas phase. For that reason a so-called structural compressibility is observed. The compressible behaviour of cellular rubber is an important property. So the main focus of the paper is on the pressure tests and the simulation of these. The existing material models for rubber like materials only deal with incompressible rubber structures. To represent the compressible behaviour, the Theory of Porous Media is used. The constitutive model is based on a polynomial approach for an incompressible material. This is complemented by a volumetric expansion term with a point of compaction to model the structural compressibility.     

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.     

A new constitutive theory for fiber-reinforced incompressible nonlinearly elastic solids

We consider an incompressible nonlinearly elastic material in which a matrix is reinforced by strong fibers, for example fibers of nylon or carbon aligned in one family of curves in a rubber matrix. Rather than adopting the constraint of fiber inextensibility as has been previously assumed in the literature, here we develop a theory of fiber-reinforced materials based on the less restrictive idea of limiting fiber extensibility. The motivation for such an approach is provided by recent research on limiting chain extensibility models for rubber. Thus the basic idea of the present paper is simple: we adapt the limiting chain extensibility concept to limiting fiber extensibility so that the usual inextensibility constraint traditionally used is replaced by a unilateral constraint. We use a strain-energy density composed with two terms, the first being associated with the isotropic matrix or base material and the second reflecting the transversely isotropic character of the material due to the uniaxial reinforcement introduced by the fibers. We consider a base neo-Hookean model plus a special term that takes into account the limiting extensibility in the fiber direction. Thus our model introduces an additional parameter, namely that associated with limiting extensibility in the fiber direction, over previously investigated models. The aim of this paper is to investigate the mathematical and mechanical feasibility of this new model and to examine the role played by the extensibility parameter. We examine the response of the proposed models in some basic homogeneous deformations and compare this response to those of standard models for fiber reinforced rubber materials. The role of the strain-stiffening of the fibers in the new models is examined. The enhanced stability of the new models is then illustrated by investigation of cavitation instabilities. One of the motivations for the work is to apply the model to the biomechanics of soft tissues and the potential merits of the proposed models for this purpose are briefly discussed.