Model For Rubber-like Materials Research Articles

A viscoelastic constitutive modeling of rubber-like materials with the Payne effect

• A damage visco-hyperelastic constitutive model is proposed. • The non-monotonous variation of loss modulus of rubbery materials is explained. • The thermodynamic dissipation processes behind Payne effect are studied. • An effective and simple method is developed to identify material parameters. This paper aims to present a viscoelastic constitutive model of rubber-like materials, which can capture the Payne effect under dynamic cyclic loadings. The Payne effect is induced by a damage process of bond rupture inside the rubber-like materials, which leads to the storage and loss moduli changing with the dynamic strain amplitude. A viscoelastic constitutive relation is established based on the nonequilibrium thermodynamics for the rubber-like materials by constructing the Helmholtz free energy as the superposition of a hyperelastic model and a convolution viscous model. The neo-Hookean hyperelastic model and the convolution viscous model in terms of the Prony series are then employed in a modification that the material parameters concerned are treated as internal variables and can be identified through a simple but effective approach. At last, the Payne effect is effectively predicted in a good agreement with experimental results.

Methodical fitting for mathematical models of rubber-like materials

A great variety of models can describe the nonlinear response of rubber to uniaxial tension. Yet an in-depth understanding of the successive stages of large extension is still lacking. We show that the response can be broken down in three steps, which we delineate by relying on a simple formatting of the data, the so-called Mooney plot transform. First, the small-to-moderate regime, where the polymeric chains unfold easily and the Mooney plot is almost linear. Second, the strain-hardening regime, where blobs of bundled chains unfold to stiffen the response in correspondence to the ‘upturn’ of the Mooney plot. Third, the limiting-chain regime, with a sharp stiffening occurring as the chains extend towards their limit. We provide strain-energy functions with terms accounting for each stage that (i) give an accurate local and then global fitting of the data; (ii) are consistent with weak nonlinear elasticity theory and (iii) can be interpreted in the framework of statistical mechanics. We apply our method to Treloar's classical experimental data and also to some more recent data. Our method not only provides models that describe the experimental data with a very low quantitative relative error, but also shows that the theory of nonlinear elasticity is much more robust that seemed at first sight.

HIA: A Hybrid Integral Approach to model incompressible isotropic hyperelastic materials—Part 1: Theory

This paper provides a new constitutive model for rubber-like materials. The model adds to the 8-chain density introduced by Arruda and Boyce, two phenomenological components: an original part made of an integral density and an interleaving constraint part represented by a logarithmic function as proposed by Gent and Thomas. The model contains six rheological parameters connected to the polymer chemistry and to the macroscopic behavior. Four sets of experimental data from the literature are used to identify the rheological parameters and to assess the proposed model. The model is able to reproduce with a good accuracy experimental data performed under different loading conditions such as uniaxial and equibiaxial tension, uniaxial compression, pure and simple shear as well as the Mooney plot.

Complex step derivative approximation of consistent tangent operators for viscoelasticity based on fractional calculus

In this contribution, the convergence behaviour of simulations with nonlinear viscoelastic models for rubber-like materials using fractional derivatives is investigated. Based on the complex step d...

New compressible hyper-elastic models for rubberlike materials

Multi-axial hyper-elastic models for large strain rubberlike elasticity are established in a broad sense free of the commonly assumed constraint of incompressibility. Results are derived directly from uniaxial stress–strain relations by means of certain explicit procedures. Novelties in four respects are incorporated in the new models: (i) constitutive parameters of direct physical meanings may be introduced to represent features of rubberlike elasticity; (ii) the usual incompressibility constraint is no longer assumed and relevant issues may be rendered irrelevant; (iii) the incompressibility case may be derived as a natural limit; and (iv) accurate agreement with benchmark data for several deformation modes may be achieved, including uniaxial extension, equi-biaxial extension as well as plane-strain extension and others.

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.

Nonlinear Viscosity Law in Finite-Element Analysis of High Damping Rubber Bearings and Expansion Joints

AbstractA simple computational strategy for finite-element implementation of a finite-strain viscohyperelasticity model for rubber-like materials was developed. The constitutive model has had a strong physical significance because of the explicit consideration of the nonlinear dependence of viscosity through internal variables (e.g., past maximum overstress and current deformation). To simulate the stress-strain response for particular one-dimensional boundary value problems, a scheme for solving the first-order differential equation representing the viscosity-induced strain-rate effect of rubber was proposed. The scheme was successful in reproducing experimental results obtained from high-damping rubber specimens. In addition, the wider applicability of the proposed strategy in simulation was tested by verifying the numerical results with independent experiments on full-scale high-damping rubber bearings with different geometries and loading rates. The effect of shape factor on bearing responses was exam...

Approximation of the inverse Langevin function revisited

The main purpose of this paper is to provide an easy-to-use approximation formula for the inverse Langevin function. The mathematical complexity of this function makes it unfeasible for an analytical manipulation and inconvenient for computer simulation. This situation has motivated a series of papers directed on its approximation. The best known solution is given by Cohen. It is used in a lot of statistically based models of rubber-like materials. The formula is derived from rounded Padé approximation [3/2]. The main idea of the presented approach in this paper relies on improvement of the precision of approximation formula for the inverse Langevin function by using multipoint Padé approximation method. We focused our study strongly on obtaining a simple and accurate approximation. It is assumed that the proposed approximation formula may be considered a useful tool for verification of the results obtained in other ways. Our results are supported by investigating several numerical examples. The paper also presents a few applications of computer software named Mathematica which can be used to calculate symbolically one point Padé approximants and numerically multipoint Padé approximants. Using this software, we showed also how to compute higher order derivatives of the inverse function in a simple and elegant way. This issue was discussed by Itskov et al.

The Gent model for rubber-like materials: An appraisal for an ingenious and simple idea

We review the main aspects of the celebrated Gent constitutive model for rubberlike materials. Emphasis is placed on the case of damageable materials describing possible damage and deformation localization.

Hyperelasticity with rate-independent microsphere hysteresis model for rubberlike materials

The mechanical behavior of elastomers strongly differs from one to another. Among these differences, hysteresis upon cyclic load can take place, and can be either rate-dependent or rate-independent. In the present paper, a microsphere model taking into account rate-independent hysteresis is proposed and applied to model filled silicone rubbers behavior. The hysteresis model is based on a combination of monodimensional constitutive equations distributed in space. The behavior of each direction is described by a collection of parallel spring slider elements. The sliders are Coulomb dampers with non-zero break-free force in tension. This model is tested on a filled silicone rubber by the way of uniaxial tensile and pure shear tests. The mechanical response of the material is well predicted for such tests. Finally, the constitutive equations are implemented in the finite element software ABAQUS. Calculation results highlight good performances of the proposed model.

A visco-hyperelastic constitutive model for rubber-like materials: A rate-dependent relaxation time scheme

A three-dimensional visco-hyperelastic constitutive model is developed to describe the rate-dependent behavior of rubber-like materials at large deformations. The model encompasses a hyperelastic part which uses the “Exp–Ln” strain energy function to characterize the equilibrium response and a viscous part capturing the rate sensitivity using a hereditary integral form which links the overstress to the history of stored strain energy. A physically consistent rate-dependent relaxation time scheme is introduced which reduces the number of required material parameters and also facilitates the calibration process. The proposed model is verified using various uniaxial experimental data in different rate ranges. Furthermore, the model is incorporated via VUMAT in ABAQUS/Explicit to examine its performance in three-dimensional deformations. To this end, finite element analysis of an elastomeric bushing is performed and the results are compared to those of experiment. It is then concluded that, the proposed constitutive relations are quite efficient in predicting the behavior of rubber-like materials in different states of deformation and also in wide ranges of strain rate.

More hyperelastic models for rubber-like materials: consistent tangent operators and comparative study

AbstractRubber-like materials can deform largely and nonlinearly upon loading, and they return to the initial configuration when the load is removed. Such rubber elasticity is achieved due to very flexible long-chain molecules and a three-dimensional network structure that is formed via cross-linking or entanglements between molecules. Over the years, to model the mechanical behavior of such randomly oriented microstructures, several phenomenological and micromechanically motivated network models for nearly incompressible hyperelastic polymeric materials have been proposed in the literature. To implement these models for polymeric material (undoubtedly with widespread engineering applications) in the finite element framework for solving a boundary value problem, one would require two important ingredients, i.e., the stress tensor and the consistent fourth-order tangent operator, where the latter is the result of linearization of the former. In our previous work, 14 such material models are reviewed by deriving the accurate stress tensors and tangent operators from a group of phenomenological and micromechanical models at large deformations. The current contribution will supplement some further important models that were not included in the previous work. For comparison of all selected models in reproducing the well-known Treloar data, the analytical expressions for the three homogeneous defomation modes, i.e., uniaxial tension, equibiaxial tension, and pure shear, have been derived and the performances of the models are analyzed.

A hyperelastic constitutive model for rubber-like materials

Hyperelastic behavior of isotropic incompressible rubbers is studied to develop a strain energy function which satisfies all the necessary characteristic properties of an efficient hyperelastic model. The proposed strain energy function includes only three material parameters which are somehow related to the physical quantities of the material molecular network. Moreover, the model benefits from mathematical simplicity, well suitting in all ranges of stretch and possessing the property of deformation-mode-independency. This reduces the required number of experimental tests for parameter calibration of the model. Results of the proposed model are compared with results of some available models as well as experimental data. Moreover, complete analysis of the Mooney plot over a wide range of stretch in extension–compression is carried out. It is found that the proposed model gives reasonable predictions in comparison with those of experiments.

Unified constitutive modeling of rubber-like materials under diverse loading conditions

This paper presents a new constitutive model that unifies the behavioral characterizations of rubber-like materials in a broad range of loading regimes. The proposed model combines a selection of existing components that are known to reflect, with suitable accuracy, two fundamental aspects of rubber behavior in finite strain: (i) rate-independent softening under deformation, also known as the Mullins effect, and (ii) hyper-viscoelasticity, including at high strain rates. The evolution model is further generalized to account for multiple rates of internal dissipation (or material time-scales). Suitable means of identifying the system’s parameters from simple uniaxial extension tests are explored. Several aspects of the model’s behavior are shown in virtual experiments of uniaxial extension, at different stretch rates. A possible directional approach extending the model to handle softening induced anisotropy is briefly discussed.

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.

The trousers test for tearing of soft biomaterials

The mechanical modeling of rubber-like materials within the framework of nonlinear elasticity theory is well established. The application of such modeling to soft biomaterials is currently the subject of intense investigation. For soft biomaterials it is well known that exponential strain energy density models are particularly useful as they reflect the typical J-shaped stress-stretch stiffening response that is observed experimentally. The most celebrated of these models for isotropic hyperelastic materials are those of Fung and Demiray which depends only on the first strain invariant and its generalization by Vito that depends on both strain invariants. In the limit as the strain-stiffening parameter tends to zero, one recovers the neo-Hookean and Mooney–Rivlin models that are linear functions of the invariants. Here we apply these models to the analysis of the fracture or tearing of soft biomaterials. Attention is focused on a particular fracture test namely the trousers test where two legs of a cut specimen are pulled horizontally apart out of the plane of the test piece. It is shown that, in general, the location of the cut in the specimen plays a key role in the fracture analysis, and that the effect of the cut position depends crucially on the constitutive model employed. This dependence is characterized explicitly for the strain-stiffening exponential constitutive models considered. In contrast to the situation for rubber, our findings show that the critical driving force and fracture toughness in tearing of some soft biomaterials in the trousers test are virtually independent of the cut position.

Nonhyperelastic Nature of an Elastomeric High Strain Material

Rubber materials have mostly been modeled by utilizing hyperelasticity, which have led to greater understanding and acceptable predictability of their stress-strain response. However, inherent inelastic behavior excluded by approximation has never been characterized by time-dependent parameters such as time, strain-rate, and strain history. We hypothesized that time, stretch rate, and stretch history were prominent factors that induce the inelasticity, and we characterized the inelasticity in terms of those factors using a randomized stretch-controlled testing protocol. We applied the custom randomized testing protocol with the fundamental statistical theory to characterize inelastic behavior imbedded in the high strain material. We hypothesized that time spent testing (T), rate-related stretch history (Ht2), and long-term stretch history (Ht1) give rise to the inelastic deviation from hyperelasticity. We examined the significance, effectiveness, and differences of T, Ht2, and Ht1 by looking at the derived multivariable linear regression models. Distribution of prediction deviation was also examined to see if we missed any other significant variable. Predictability of the multivariable linear regression models was validated by utilizing the unused data from the randomized testing protocol and data from the conventional cyclic testing protocol. We found that the inelasticity of the rubber-like material is highly related to T, Ht2, and Ht1, but not equally influential to all stretches. At smaller deformations, greater inelastic deviation occurs. Inelasticity exponentially decreased over stretch and was nonlinearly related to time. This study successfully determined the elastic/inelastic responses and factors that induce the inelastic response of the rubber-like material. This investigation suggests a way to better describe the elastic/inelastic properties and phenomenological models of rubber-like materials.

An 8-chain Model for Rubber-like Materials Accounting for Non-affine Chain Deformations and Topological Constraints

Several industrial applications involve rubber and rubber-like materials, and it is important to be able to predict the constitutive response of these materials. In the present paper, a new constitutive model for rubber-like solids is proposed. The model is based on the 8-chain concept introduced by Arruda and Boyce (J. Mech. Phys. Solids 41, 389–412, 1993) to which two new components are added. Real polymer networks do not deform affinely, and in the proposed model this is accounted for by the inclusion of an elastic spring, acting in series with the representative polymer chain. Furthermore, real polymer chains are not completely free to move, which is modelled by imposing a topological constraint on the transverse motions of the representative polymer chain. The model contains five model parameters and these need to be determined on the basis of experimental data. Three experimental studies from the literature were used to assess the proposed model. The model was able to reproduce experimental data performed under conditions of uniaxial tension, generalised plane deformation, and biaxial tension with an excellent accuracy. The strong predictive abilities together with the numerically efficient structure of the model make it suitable for implementation in a finite element context.

Constitutive modeling of rubberlike materials based on consistent strain energy density functions

AbstractRubberlike materials are characterized by high deformability and reversibility of deformation. From the continuum viewpoint, a strain energy density function is postulated for modeling the behavior of these materials. In this paper, a general form for the strain energy density of these materials is proposed from a phenomenological point of view. Based on the Valanis‐Landel hypothesis, the strain energy density of incompressible materials is expressed as the sum of independent functions of the principal stretches meeting the essential requirements on the form of the strain energy density. It is cleared that the appropriate mathematical expressions for constitutive modeling of these materials are polynomial, logarithmic, and particularly exponential functions. In addition, the material parameters are calculated using a novel procedure that is based on the correlation between the values of the strain energy density (rather than the stresses) cast from the test data and the theory. In order to evaluate the performance of the proposed strain energy density functions, some test data of rubberlike materials with pure homogeneous deformations are used. It is shown that there is a good agreement between the test data and predictions of the models for incompressible isotropic materials. POLYM. ENG. SCI., 2010. © 2010 Society of Plastics Engineers

A Review of Constitutive Models for Rubber-Like Materials

Problem statement: This study reviewed the needs of different constitutive models for rubber like material undergone large elastic deformation. The constitutive models are widely used in Finite Element Analysis (FEA) packages for rubber components. Most of the starting point for modeling of various kinds of elastomer is a strain energy function. In order to define the hyperelastic material behavior, stress-strain response is required to determine material parameters in the strain energy potential and also proper selection of rubber elastic material model is the first attention. Conclusion: This review provided a sound basis decision to engineers and manufactures to choose the right model from several constitutive models based on strain energy potential for incompressible and isotropic materials.