Model For Rubber-like Materials Research Articles

New two-parameter constitutive models for rubber-like materials: Revisiting the relationship between single chain stretch and continuum deformation

The connection between macroscopic deformation and microscopic chain stretch is a key element in constitutive models for rubber-like materials that are based on the statistical mechanics of polymer chains. A new micro-macro chain stretch relation is proposed, using the Irving-Kirkwood-Noll procedure to construct a Cauchy stress tensor from forces along polymer chains. This construction assumes that the deformed polymer network remains approximately isotropic for low to moderate macroscopic stretches, a starting point recently adopted in the literature to propose a non-affine micro-macro chain stretch relation (Amores, V.J., Nguyen, K., Montáns, F.J., 2021. On the network orientational affinity assumption in polymers and the micro–macro connection through the chain stretch. Journal of the Mechanics and Physics of Solids 148, 104279). Requiring the constructed Cauchy stress to be consistent with the stress tensor derived from the strain energy density results in a new chain stretch relation involving the exponential function. A hybrid chain stretch relation combining the new chain stretch with the well-known affine relation is then proposed to account for the whole range of stretches in experimental datasets. Comparison of the model predictions to experimental data in the literature shows that the two new micro-macro chain stretch relations in this work result in two-parameter constitutive models that outperform those based on existing chain stretches with no increase in the number of fitting parameters used.

A second gradient theory of thermoviscoelasticity

This paper is concerned with a rate-type theory of thermoviscoelasticity in which the second gradient of the displacement and the second temperature gradient are added to the classical set of independent constitutive variables. Viscoelasticity and related phenomena are of great importance in the study of biological materials. An adequate modeling of rubber-like materials and of biological soft tissues requires the use of the theory of viscoelasticity. Introduction of the concept of thermal displacement and the theory of multipolar continua allows us to show that Green-Naghdi thermomechanics can be used to derive a second gradient theory. The basic equations of the theory are established and the boundary conditions associated to nonsimple materials are investigated. The stress tensor and hyperstress tensor are shown to depend on the first and second temperature gradients. For rigid heat conductors we find that the temperature satisfies a fourth order equation. The boundary-initial-value problems are formulated. A uniqueness result in the dynamic theory of thermoviscoelastic materials is presented. We establish an existence result and prove the analyticity of the solutions. As a consequence, the exponential decay of the solutions and their impossibility of localization are obtained.

A second gradient theory of thermoviscoelasticity

This paper is concerned with a rate-type theory of thermoviscoelasticity in which the second gradient of the displacement and the second temperature gradient are added to the classical set of independent constitutive variables. Viscoelasticity and related phenomena are of great importance in the study of biological materials. An adequate modeling of rubber-like materials and of biological soft tissues requires the use of the theory of viscoelasticity. Introduction of the concept of thermal displacement and the theory of multipolar continua allows us to show that Green-Naghdi thermomechanics can be used to derive a second gradient theory. The basic equations of the theory are established and the boundary conditions associated to nonsimple materials are investigated. The stress tensor and hyperstress tensor are shown to depend on the first and second temperature gradients. For rigid heat conductors we find that the temperature satisfies a fourth order equation. The boundary-initial-value problems are formulated. A uniqueness result in the dynamic theory of thermoviscoelastic materials is presented. We establish an existence result and prove the analyticity of the solutions. As a consequence, the exponential decay of the solutions and their impossibility of localization are obtained.

A general hyperelastic model for rubber-like materials incorporating strain-rate and temperature

A comprehensive hyperelastic model that precisely forecasts the mechanical characteristics of materials with rubber-like qualities is presented. This model relies on the well-established five-parameter Mooney-Rivlin model, which is then extended to incorporate strain rate dependent and temperature dependent term. To validate its accuracy, experimental data from ethylene propylene diene monomer (EPDM) rubber materials was utilized to compare with the model prediction. Hyperelastic stress-strain curves were collected from a variety of materials that were subjected to varying temperatures and strain rates to improve the model’s applicability. Its prediction results are compared against the collected experimental data, resulting in consistent and reliable outcomes. The simplified form of the model not only establishes an effective framework for characterizing and predicting the mechanical response of materials that resemble rubber under different working conditions but makes the coding and implementation of finite element analysis easier.

Using the Mooney Space to Characterize the Non-Affine Behavior of Elastomers.

The formulation of the entropic statistical theory and the related neo-Hookean model has been a major advance in the modeling of rubber-like materials, but the failure to explain some experimental observations such as the slope in Mooney plots resulted in hundreds of micromechanical and phenomenological models. The origin of the difficulties, the reason for the apparent need for the second invariant, and the reason for the relative success of models based on the Valanis-Landel decomposition have been recently explained. From that insight, a new micro-macro chain stretch connection using the stretch tensor (instead of the right Cauchy-Green deformation tensor) has been proposed and supported both theoretically and from experimental data. A simple three-parameter model using this connection has been suggested. The purpose of this work is to provide further insight into the model, to provide an analytical expression for the Gaussian contribution, and to provide a simple procedure to obtain the parameters from a tensile test using the Mooney space or the Mooney-Rivlin constants. From different papers, a wide variety of experimental tests on different materials and loading conditions have been selected to demonstrate that the simple model calibrated only from a tensile test provides accurate predictions for a wide variety of elastomers under different deformation levels and multiaxial patterns.

On the origin of Sanchez-Lacombe equation of state theory in hydrostatic strain energy model for rubber-like materials

To capture the volume deformation of rubber-like materials, the strain energy density (SED) function of a compressible hyperelastic model is formulated as the hydrostatic/liquid-like term from interchain interaction changes plus the compressible elastic term owing to elastic deformation of crosslinking network. Notably, the former term dominates volume responses. Although the physics of the hydrostatic term is relatively clear, to our knowledge, few physically-based models are available. However, in our previous work (Liu and Lu, 2023), we constructed a physically-based hydrostatic SED function inspired by the Flory-Orwoll-Vrij equation of state (EOS) theory for pure polymer fluids, and proposed the general strategy for developing hydrostatic functions according to the EOS theories. Sanchez-Lacombe EOS theory (note: a type of lattice-fluid EOS theory) inspired this work. We assume: (i) a rubber-like material consists of polymer segments occupying lattice sites and unoccupied/vacant lattice sites, and the compressibility of the material corresponds to changes in vacant sites fraction; (ii) the hydrostatic strain energy is associated with the interchain interaction energy change and the mixing entropy change between network and vacant sites. According to the Helmholtz free energy in Sanchez-Lacombe EOS theory and limiting our attention to the isothermal condition, another physically-based hydrostatic SED function is constructed; a specific compressible hyperelastic model is provided by further combining compressible 8-chain elastic SED function. The basic framework of this compressible model is inherently consistent with the Flory-Rehner framework for swollen elastomers. Our model provides good predictions for volume deformation data of ten different rubber-like materials in hydrostatic compression (HC), uniaxial tension (UT), and constrained uniaxial compression (CUC). The proposed model reveals that HC, CUC, or uniaxial compression correspond to the entropy-decreasing process and they all are controlled by entropy changes; while for UT, equibiaxial tension, or pure shear, the entropy first increases and then decreases, and the interchain interaction energy and entropy changes control the responses for initial small stretches and remaining large stretches, respectively. This study aims to provide a new physical insight and a valid physically-based hydrostatic SED function for the compressibility of rubber-like materials.

A physically-based hydrostatic strain energy model for rubber-like materials inspired by Flory-Orwoll-Vrij equation of state theory

In a compressible hyperelastic model for rubber-like materials, the strain energy function is generally composed of the hydrostatic part corresponding to changes in interchain interactions and the compressible elastic part attributable to the network structure. It is well known that the hydrostatic part dominates the compressibility of the material. In this study, inspired by the Flory-Orwoll-Vrij (FOV) equation of state (EOS) theory for pure polymer liquids (note: a cell-like EOS theory), we assume that (i) the compressibility of rubber-like materials corresponds to changes in free volume of polymer chain segments; (ii) the hydrostatic strain energy of a rubber-like solid is attributable to changes in interchain interaction energy and chain segments’ external motion degrees of freedom (the latter solely depends on interchain forces). With a focus on the reversible isothermal deformation process, we construct a physically-based hydrostatic strain energy function based on the Helmholtz free energy formulation in FOV EOS theory. With a view towards applications, we provide a specific compressible hyperelastic model by incorporating the new hydrostatic strain energy function and compressible eight-chain model, where the latter is utilized as the elastic part of the strain energy function. Our model is capable of predicting various volume data of rubber-like materials from the literature, such as the nonlinear pressure-volume response at finite volume changes in hydrostatic compression (HC), the volume change-stretch and stress-stretch data in uniaxial tension (UT), and the stress-volume data in constrained uniaxial compression (CUC). Given the severely limited volume change data for finite stretches in UT and other modes of deformation, we simulate the volume change-volume modulus-stretch responses in UT, equibiaxial tension (ET), pure shear (PS), and uniaxial compression (UC) over their respective theoretical range of stretch and successfully predict some available (qualitative) experimental observations. Together with the simulations of the volume change-stretch responses in UT, ET, PS, and UC based on the Ogden’s and Bischoff et al.’s compressible models, we summarize the characteristics of these responses and analyze the possible deformation mechanisms. These simulations can provide some guidance for future corresponding experiments. Finally, we analyze the deformation mechanisms of HC, CUC, UT, ET, PS, and UC by simulating the responses for total strain energy and its components. This study provides a new physical picture and corresponding theoretical model for the hydrostatic strain energy function of rubber-like materials and finally proposes the general research strategy for constructing new hydrostatic strain energy functions based on the EOS theories for pure polymer liquids.

A multiscale phase field fracture approach based on the non-affine microsphere model for rubber-like materials

Rubber-like materials have a broad scope of applications due to their unique properties like high stretchability and increased toughness. Hence, computational models for simulating their fracture behavior are paramount for designing them against failures. In this study, the phase field fracture approach is integrated with a multiscale polymer model for predicting the fracture behavior in elastomers. At the microscale, damaged polymer chains are modeled to be made up of a number of elastic chain segments pinned together. Using the phase field approach, the damage in the chains is represented using a continuous variable. Both the bond stretch internal energy and the entropic free energy of the chain are assumed to drive the damage, and the advantages of this assumption are expounded. A framework for utilizing the non-affine microsphere model for damaged systems is proposed by considering the minimization of a hypothetical undamaged free energy, ultimately connecting the chain stretch to the macroscale deformation gradient. At the macroscale, a thermodynamically consistent formulation is derived in which the total dissipation is assumed to be mainly due to the rupture of molecular bonds. Using a monolithic scheme, the proposed model is numerically implemented and the resulting three-dimensional simulation predictions are compared with existing experimental data. The capability of the model to qualitatively predict the propagation of complex crack paths and quantitatively estimate the overall fracture behavior is verified. Additionally, the effect of the length scale parameter on the predicted fracture behavior is studied for an inhomogeneous system.

A finite strain visco-hyperelastic damage model for rubber-like materials: theory and numerical implementation

A finite strain visco-hyperelastic damage model for rubber-like materials: theory and numerical implementation

A model for rubber-like materials with three parameters obtained from a tensile test

Based on our recent findings on the chains orientational distribution in deformed rubber-like materials, we present a novel, accurate and simple non-affine model for isotropic, incompressible polymers. The model has only three material parameters that are easily calibrated from the characteristic behaviour of a tensile test: the initial slope, the slope of the mid-range, and the locking stretch. Once calibrated, the proposal reproduces accurately the behaviour of the material under the general deformation states for moderately large stretches measured by Kawabata (rubber) and Kawamura (silicone), and at least for three deformation states approaching the locking behaviour (Treloar’s tests on rubber).

Constitutive modeling of rubber-like materials: an alternative material model

ABSTRACT This article proposes an alternative strain energy density function to describe the hyperelastic behavior of rubber-like materials under various deformations. This proposed strain energy function fulfilled all the necessary characteristics of a hyperelastic material model. This function consists of four material parameters and represents an invariant-based material model. Also, the material parameters are kept constant for the different modes of loading conditions. The model solution based on the proposed energy function is mathematically simple and well suited for all stretching ranges. The model solution is further tested with the existing experimental data and compared with the results of available model solutions. The findings of the model solution agree well with both experimental and theoretical results available in the literature.

Strain mode-dependent weighting functions in hyperelasticity accounting for verification, validation, and stability of material parameters

Optimized material parameters obtained from parameter identification for verification wrt a certain loading scenario are amenable to two deficiencies: Firstly, they may lack a general validity for different loading scenarios. Secondly, they may be prone to instability, such that a small perturbation of experimental data may ensue a large perturbation for the material parameters. This paper presents a framework for extension of hyperelastic models for rubber-like materials accounting for both deficiencies. To this end, an additive decomposition of the strain energy function is assumed into a sum of weighted strain mode related quantities. We propose a practical guide for model development accounting for the criteria of verification, validation and stability by means of the strain mode-dependent weighting functions and techniques of model reduction. The approach is successfully applied for 13 hyperelastic models with regard to the classical experimental data on vulcanized rubber published by Treloar (Trans Faraday Soc 40:59–70, 1944), showing both excellent fitting capabilties and stable material parameters.

On the Performance of Isotropic Hyperelastic Constitutive Models for Rubber-Like Materials: A State of the Art Review

Abstract Besides the well-known landmark models for the hyperelastic response of rubberlike materials, many new hyperelastic constitutive models have emerged over the last decade. Despite many reviews on constitutive modeling or elastomers, it is still a challenging endeavor for engineers to decide for a constitutive model for the specific rubber compound and application. In this work, we have reviewed 44 hyperelastic constitutive models for elastomers and assessed their strength and weaknesses under uniaxial, pure shear, and (equi)biaxial deformations. To this end, we first present a novel parameter identification methodology based on various multi-objective optimization strategies for the selection of the best constitutive models from a given set of uniaxial tension, pure shear, and (equi)biaxial tension experiments. We utilize a hybrid multi-objective optimization procedure using a genetic algorithm to generate multiple initial points for gradient-based search algorithm, Fmincon utility in matlab. The novelty of the approach is (i) simultaneous fitting with variable weight factors for uniaxial, equibiaxial, and pure shear data, and (ii) the sorting of the models based on objective normalized quality of fit metric. For the models incapable of simultaneously fitting the three distinct deformation data, the validity range is assessed through a threshold value for the quality of fit measure. Accordingly, 44 hyperelastic models are sorted with respect to their simultaneous fitting performance to the experimental dataset of Treloar and Kawabata. Based on the number of material parameters, and their fitting performance to experimental data, a detailed discussion is carried out.

A generalized tube model of rubber elasticity.

In the present paper, a new type of micro-mechanically motivated chain network model for rubber-like materials is proposed. The model captures topological constraints of polymer network chains, in particular, entanglements. The model demonstrates how the local molecular packing constraints modify under deformation and shows the impact of these changes on the macroscopic elasticity of the material. To this end, we combine concepts of a confining tube and a slip-link (reptation) model. In these models, entanglements of polymer chains play an important role. The nature of entanglements is discussed, and relationships governing entanglements are formulated in terms of molecular physics. In the context of nonlinear elasticity, we apply a non-affine concept which captures the liquid-like behavior of polymer networks at smaller scales in a more realistic way. Model predictions show good agreement with experimental results from uniaxial and biaxial tension tests.

An elastic model for rubber-like materials based on a force-equivalent network

In this paper, a new network and then a corresponding constitutive model of rubber-like materials are proposed. The materials are assumed to be composed of equal-length helical subnetworks which can slide along their occupied space. According to equivalence of external force, the network of this kind of materials is simplified to a sample network which only has subnetworks in three principal directions. By virtue of chain force calculated from Boltzmann's equation, the expression of principal stress is derived according to the network structure. The model is fitted to experimental data and compared with other models, which show the model can describe rubber-like materials well. The proposed network structure and corresponding constitutive model can also give reasonable explanations of the Ogden model and the Arruda-Boyce model.

A complementary energy approach accommodates scale differences in soft tissues

The mechanics of biological entities, from single molecules to the whole organ, has been extensively analyzed during the last decades. At the smaller scales, statistical mechanics has fostered successful physical models of proteins and molecules, which have been later incorporated within constitutive models of rubber-like materials and biological tissues. At the macroscopic scale, the additive decomposition of energy functions i.e., a parallel arrangement of the tissue constituent, has been recurrently used to account for the internal heterogeneity of soft biological materials. However, it has not yet been possible to unite the mechanics at the tissue level with the actual response of the tissue components. Here, we exemplify our approach using cardiovascular tissue where the mechanical response at the tissue scale is in the range of kPa whereas the elastic modulus of collagen, the main component of the vascular tissue, is in the range of MPa GPa. In this work we develop a novel theoretical framework based on a complementary strain energy function that builds-up on a full network model. The complementary strain energy function introduces naturally an additive decomposition of the deformation gradient for the tissue constituents, i.e an arrangement in series of the constituents. We demonstrate that the macroscopic response of the tissue can be reproduced by just introducing the underlying mechanical and structural features of the micro-constituents, improving in a fundamental manner previous attempts in the mechanical characterization of soft biological tissues. The proposed theoretical framework unveils a new direction in the mechanical modeling of soft tissues and biological networks.

A hyperelastic constitutive model for rubber-like materials

In this contribution, a new form of the strain energy function is proposed to describe the hyperelastic behavior of rubber-like materials under various deformation. The proposed function represents an invariant-based model and contains two material parameters. The model was tested with the experimental data of vulcanized rubbers, collagen and fibrin. The material parameters are kept constant for a material subjected to different types of loading. Good agreement between model and experimental data was obtained for all materials.

Modeling and Verification of a New Hyperelastic Model for Rubber-Like Materials

The essential evaluation criterion for the hyperelastic model is its ability to describe the mechanical behavior of rubber-like materials under different deformation modes over a large deformation range accurately. Based on the Seth strain tensor invariant, a new hyperelastic model for isotropic and incompressible rubber-like materials is proposed. In order to investigate the prediction ability of the new model, the parameters of the new model, the Yeoh model, and the Carroll model are identified by test data of 8% vulcanized rubber and two different types of carbon black filled rubber, respectively. To this end, the data of uniaxial tension and equibiaxial tension are used simultaneously. Then, the same set of model parameters is used for prediction of pure shear (plane tension) deformation. The results show that the new model not only can predict the test data of pure shear (or plane tension) accurately, but also can be reliable to describe the response of various rubber materials over a large deformation range. Finally, the finite element simulation and experiment on static stiffness of rubber bushing are carried out based on the new model. By comparison of the experimental data with the simulation data, the new model can accurately reflect the mechanical behavior of rubber bushing. The new model can be used for performance analysis of rubber products and has better application value.

A fully equilibrated microsphere model with damage for rubberlike materials

A non-affine microsphere model for rubberlike materials is proposed, based on a local minimization of the network free energy under a maximal advance path constraint. It accounts for any chain weight distribution and for damage such as Mullins softening observed in filled rubber materials. The non-affine equal-force model is compared to the common affine model and a hybrid equal-force model from the literature, when considering the isotropic hyperelastic behavior without damage of rubber materials presenting chains of various lengths. The non-affine model shows an improved deformability compared to the affine model limited by the maximal extension of the shorter chains and a significantly softer behavior. Possible damage is introduced by increasing the chain lengths according to the submitted maximal chain traction force. Each chain is impacted independently resulting in a directional softening that introduces the evolution of the stress-free configuration that needs to be assessed over the loadings. The model was successfully tested on the cyclic uniaxial tension stretch-stress responses of carbon-black filled styrene butadiene rubbers that were well fitted with three parameters only.

A Network Decomposition Model for Rubber-Like Materials Considering Topological Constraints

A molecular network constitutive model is proposed in this study. Based on the concept of molecular network decomposition, the molecular network was decomposed into a James–Guth 3-chain network model and an Arruda–Boyce 8-chain network model. Considering that the single molecular chain of rubber is constrained by the surrounding molecular networks, the 3-chain model and the 8-chain model were consequently modified using the tube theory. The proposed model contains four material parameters that were determined by fitting the uniaxial tensile test data from rubber-like materials, and the values of these parameters were utilized to determine the pure shear and the biaxial extension. The proposed model agrees well with the experimental data and can predict the test results of experiments performed under the conditions of pure shear and equi-biaxial deformation with acceptable accuracy.