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ABSTRACT A physical phenomenon called “crack removal” is investigated for highly deformable rubbery materials containing a pre‐existing crack under uniaxial pure mode‐I loading. In this case, the stress concentration significantly diminishes and in a specific stage of loading, the specimen behaves as a sample which has no apparent effect from the initial sharp crack and thus nearly undergoes uniaxial state of stress until its final rupture. To explore this phenomenon that has not been previously explored, theoretical and experimental investigations were carried out in the current study. In the experimental part, due to the lack of pertinent experimental data for crack removal phenomenon, a series of mode‐I fracture tests were carried out on nitrile butadiene rubber (NBR) containing 15 phr carbon black. In the theoretical part, an energy‐based criterion was proposed for detecting the crack removal load and subsequently, for predicting the final rupture load in such materials. For implementation of this criterion, non‐linear finite element analyses were performed. The results confirm very good ability of the proposed criterion for predicting the crack removal and final rupture phases in rubber‐like materials having high deformability. [ABSTRACT FROM AUTHOR]

The study of polymers has continued to gain substantial attention due to their expanding range of applications, spanning essential engineering fields to emerging domains like stretchable electronics, soft robotics, and implantable sensors. These materials exhibit remarkable properties, primarily stemming from their intricate polymer chain network, which, in turn, increases the complexity of precisely modeling their behavior. Especially for modeling elastomers and their fracture behavior, accurately accounting for the deformations of the polymer chains is vital for predicting the rupture in highly stretched chains. Despite the importance, many robust multiscale continuum frameworks for modeling elastomer fracture tend to simplify network deformations by assuming uniform behavior among chains in all directions. Recognizing this limitation, our study proposes a multiscale fracture model that accounts for the anisotropic nature of elastomer network responses. At the microscale, damage in the chains is assumed to be driven by both the chain's entropy and the internal energy due to molecular bond distortions. In order to bridge the stretching in the chains to the macroscale deformation, we employ the maximal advance path constraint network model, inherently accommodating anisotropic network responses. As a result, chains oriented differently can be predicted to exhibit varying stretch and, consequently, different damage levels. To drive macroscale fracture based on damages in these chains, we utilize the micromorphic regularization theory, which involves the introduction of dual local‐global damage variables at the macroscale. The macroscale local damage variable is obtained through the homogenization of the chain damage values, resulting in the prediction of an isotropic material response. The macroscale global damage variable is subjected to nonlocal effects and boundary conditions in a thermodynamically consistent phase field continuum formulation. Moreover, the total dissipation in the system is considered to be mainly due to the breaking of the molecular bonds at the microscale. To validate our model, we employ the double‐edge notched tensile test as a benchmark, comparing simulation predictions with existing experimental data. Additionally, to enhance our understanding of the fracturing process, we conduct uniaxial tensile experiments on a square film made up of polydimethylsiloxane (PDMS) rubber embedded with a hole and notches and then compare our simulation predictions with the experimental observations. Furthermore, we visualize the evolution of stretch and damage values in chains oriented along different directions to assess the predictive capacity of the model. The results are also compared with another existing model to evaluate the utility of our model in accurately simulating the fracture behavior of rubber‐like materials. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this research, the hyperelastic strain energy density function based on the exponential–logarithmic invariant is extended to the visco-hyperelastic constitutive model to describe the mechanical characteristics of the rate dependence and large deformations of rubber-like materials. On the basis of the general regression neural network (GRNN) technique, a parameter identification approach for the visco-hyperelastic model is designed. In addition, the proposed research scheme is verified using various uniaxial experimental data of rubber-like materials. The comparison results reveal that the predicted stress responses agree well with the experimental data under different loading conditions. This paper concludes that the present model can describe the mechanical behavior of rubber-like materials and that the GRNN-based approach is practicable for parameter identification of complex visco-hyperelastic constitutive models. [ABSTRACT FROM AUTHOR]

Nonlinear finite element analysis is widely used for structural optimization of the design and the reliability analysis of complex elastomeric components. However, high-precision numerical results cannot be achieved without reliable strain energy functions (SEFs) of the rubber or rubber nanocomposite material. Although hyperelastic constitutive models have been studied for nearly 80 years, selecting one that accurately describes rubber's mechanical response is still a challenge. This work reviews 85 isotropic SEFs based on both the phenomenological theory and the micromechanical network theory proposed from the 1940s to 2019. A fitting algorithm which can realize the automatic fitting optimization and determination of the parameters of all SEFs reviewed is developed. The ability of each SEF to reproduce the experimental data of both the unfilled and highly filled rubber nanocomposite is quantitatively assessed based on a new proposed evaluation index. The top 30 SEFs for the unfilled rubber and the top 14 SEFs for the highly filled rubber nanocomposite are presented in the ranking lists. Finally, some suggestions on how to select an appropriate hyperelastic constitutive model are given, and the perspective on the future progress of constitutive models is summarized.

This work is a comprehensive literature overview in the area of probabilistic methods related to composite materials with components exhibiting hyper-elastic constitutive behavior. A practical area of potential applications is seen to be rubber, rubber-like, or even rubber-based heterogeneous media, which have a huge importance in civil, mechanical, environmental, and aerospace engineering. The overview proposed and related discussion starts with some general introductory remarks and a general overview of the theories and methods of hyper-elastic material with a special emphasis on the recent progress. Further, a detailed review of the current trends in probabilistic methods is provided, which is followed by a literature perspective on the theoretical, experimental, and numerical treatments of interphase composites. The most important part of this work is a discussion of the up-to-date methods and works that used the homogenization method and effective medium analysis. There is a specific focus on random composites with and without any interface defects, but the approaches recalled here may also serve as well in sensitivity analysis and optimization studies. This discussion may be especially helpful in all engineering analyses and models related to the reliability of elastomers, whose applicability range, which includes energy absorbers, automotive details, sportswear, and the elements of water supply networks, is still increasing, as well as areas where a stochastic response is the basis of some limit functions that are fundamental for such composites in structural health monitoring. [ABSTRACT FROM AUTHOR]

A statistically based strain energy is proposed for rubber-like materials at large stretches. It is based on the micro-mechanically vectorial modeling of a single polymer chain, and its entropy is used in order to account for the entropic elasticity of rubbery macromolecules. We propose a framework for derivation of a microscopic free energy function based on a multidimensional form of a generic normal (Gauss) probability distribution function (pdf). Homogenization of the microscopic free energy by means of statistical tools renders a macroscopic free energy. The random variables of the general formulation are specified as bond angle differences, representing bending and torsion, respectively, for each bead of the single chain. A further step is a formulation of both quantities in terms of the applied stretch, which eventually renders the macroscopic strain energy as a hyperelastic energy function. Additionally, we propose a methodology to satisfy a normalization condition for the related integral of the pdf over the constraint statistic domain. A numerical example illustrates the capability of the proposed energy function to simulate the S-shape behavior of the well-known experimental data for vulcanized rubbers by Treloar (Trans Faraday Soc 40:59–70, 1944). [ABSTRACT FROM AUTHOR]

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 et al., 2021). 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. • New chain stretch relations are proposed for modelling rubber-like materials. • The Irving–Kirkwood–Noll procedure gives these relations a physical basis. • Full-network models using these relations capture experimental data accurately. • Improvements in accuracy are achieved when using only two model parameters. • The model parameters have molecular interpretations. [ABSTRACT FROM AUTHOR]

A three parameter novel hyperelastic strain energy function is introduced in this paper for soft and rubber-like materials. The function integrates a non-separable exponential component with a single term Ogden-type polynomial-like function, resulting in an exponential-polynomial based strain energy function. This helps in capturing both small and large deformation (stretch) behaviours of hyperelastic materials. The structure of the model is simple and validated against several experimental datasets including rubbers, hydrogel, and soft tissues. The model is reported to capture key material behaviors, including strain stiffening and various deformation paths. Through comparative studies with well-known models like the Ogden (six parameters) and Yeoh (three parameters), the model's effectiveness is established. Furthermore, the model successfully addresses pressure-inflation instability in thin spherical balloons. It's applicability extends to biological materials, as evidenced by its effectiveness in characterizing porcine brain tissue and a monkey's bladder. [Display omitted] • A three parameter strain energy function for soft and rubber-like materials has been developed. • Combined an exponential component with an Ogden-type polynomial term. • Experimental data of rubbers, hydrogel, and soft tissue are used to validate the model. • The model effectively addresses pressure-inflation instability in thin spherical balloons. • Advantages of the model are illustrated through comparisons with Ogden and Yeoh models. [ABSTRACT FROM AUTHOR]

The application of a newly proposed generalised neo-Hookean strain energy function to the inflation of incompressible rubber-like spherical and cylindrical shells is demonstrated in this paper. The pressure (P ) – inflation (λ or v ) relationships are derived and presented for four shells: thin- and thick-walled spherical balloons, and thin- and thick-walled cylindrical tubes. Characteristics of the inflation curves predicted by the model for the four considered shells are analysed and the critical values of the model parameters for exhibiting the limit-point instability are established. The application of the model to extant experimental datasets procured from studies across 19th to 21st century will be demonstrated, showing favourable agreement between the model and the experimental data. The capability of the model to capture the two characteristic instability phenomena in the inflation of rubber-like materials, namely the limit-point and inflation-jump instabilities, will be made evident from both the theoretical analysis and curve-fitting approaches presented in this study. A comparison with the predictions of the Gent model for the considered data is also demonstrated and is shown that our presented model provides improved fits. Given the simplicity of the model, its ability to fit a wide range of experimental data and capture both limit-point and inflation-jump instabilities, we propose the application of our model to the inflation of rubber-like materials. [ABSTRACT FROM AUTHOR]

We obtain linear elasticity as Γ -limit of finite elasticity under incompressibility assumption and Dirichlet boundary conditions. The result is shown for a large class of energy densities for rubber-like materials. [ABSTRACT FROM AUTHOR]

Recent research studies on industrial cyber-physical systems (ICPSs) have witnessed vast patronage with emphasis on data utility for improved design, maintenance, and high-level decision making. The design of proton-exchange membrane fuel cells (PEMFC) is geared towards improving performance and extending life cycles. More often, material selection of PEMFC components contributes a major determining factor for efficiency and durability with the seal/gasket quality being one of the most critical components. Finite element analysis (FEA) offers a simulated alternative to real-life stress analysis of components and has been employed on different rubber-like gasket materials for hydrogen fuel cells for determining an optimal strain energy density function using different hyperelastic models following uniaxial tensile testing. The results show that the Mooney–Rivlin, Ogden, and Yeoh models were the most fitting model with the best stress–strain fit following a weighted error evaluation criteria which returned 18.54%, 19.31%, and 21.96% for 25% displacement, and 22.1%, 21.7%, and 21.17% for 40% displacements, respectively. Further empirical analysis using the multi-metric regression technique for compatibility testing (curve similarity) between the hyperelastic model outputs and the tensile data reveal that the Yeoh model is the most consistent as seen in the marginal error difference amidst increasing displacement while the Arruda–Boyce model is most inconsistent as shown in the high error margin as the displacement increases from 25% to 40%. Lastly, a comparative assessment between different rubber-like materials (RLM) was presented and is expected to contribute to improved decision-making and material selection. [ABSTRACT FROM AUTHOR]

In constitutive modelling of rubber-like materials, the strain-hardening effect at large deformations has traditionally been captured successfully by non-Gaussian statistical molecular-based models involving the inverse Langevin function, as well as the phenomenological limiting chain extensibility models. A new model proposed by Anssari-Benam and Bucchi (Int. J. Non Linear Mech. 2021; 128; 103626. DOI: 10.1016/j.ijnonlinmec.2020.103626), however, has both a direct molecular structural basis and the functional simplicity of the limiting chain extensibility models. Therefore, this model enjoys the benefits of both approaches: mathematical versatility, structural objectivity of the model parameters, and preserving the physical features of the network deformation such as the singularity point. In this paper we present a systematic approach to constructing the general class of this type of model. It will be shown that the response function of this class of models is defined as the [1/1] rational function of I 1 , the first principal invariant of the Cauchy–Green deformation tensor. It will be further demonstrated that the model by Anssari-Benam and Bucchi is a special case within this class as a rounded [3/2] Padé approximant in λ c (the chain stretch) of the inverse Langevin function. A similar approach for devising a general I 2 term as an adjunct to the I 1 part of the model will also be presented, for applications where the addition of an I 2 term to the strain energy function improves the fits or is otherwise required. It is concluded that compared with the Gent model, which is a [0/1] rational approximation in I 1 and has no direct connection to Padé approximations of any order in λ c , the presented new class of the molecular-based limiting chain extensibility models in general, and the proposed model by Anssari-Benam and Bucchi in specific, are more accurate representations for modelling the strain-hardening behaviour of rubber-like materials in large deformations. [ABSTRACT FROM AUTHOR]

The Mullins effect represents a softening phenomenon observed in rubber-like materials and soft biological tissues. It is usually accompanied by many other inelastic effects like for example residual strain and induced anisotropy. In spite of the long term research and many material models proposed in literature, accurate modeling and prediction of this complex phenomenon still remain a challenging task. In this work, we present a novel approach using deep symbolic regression (DSR) to generate material models describing the Mullins effect in the context of nearly incompressible hyperelastic materials. The two step framework first identifies a strain energy function describing the primary loading. Subsequently, a damage function characterizing the softening behavior under cyclic loading is identified. The efficiency of the proposed approach is demonstrated through benchmark tests using the generalized the Mooney–Rivlin and the Ogden–Roxburgh model. The generalizability and robustness of the presented framework are thoroughly studied. In addition, the proposed methodology is extensively validated on a temperature-dependent data set, which demonstrates its versatile and reliable performance. • Deep symbolic regression is applied to automatically generate accurate analytical models capturing the Mullins effect in elastomers. • Highly specific damage models accurately representing complex characteristics of the Mullins effect, including temperature-dependent effects are generated. • Validation of the framework with multiple data sets, including temperature-dependent experimental results. • Robustness and generalizability of the proposed framework under sparse data conditions are demonstrated. [ABSTRACT FROM AUTHOR]

• A neural network peridynamics is proposed for rubber-like materials. • A neural network is used to establish the mapping relation to force density vectors. • The developed method is free of the zero-energy modes. • The developed method significantly improves the computational efficiency. • The proposed method can be applied to various hyperelastic materials. Peridynamic (PD) is a powerful tool for simulating the large deformation and failure process of many types of materials. However, its use in modeling rubber-like materials is limited due to the complex constitutive nature of the material, and low efficiency and numerical oscillation caused by PD. To address this issue, a neural network (NN) non-ordinary state-based peridynamics (NOSB PD) method is developed to model the large deformation and failure behavior of rubber-like materials. This method is free of the zero-energy modes, and can significantly improve the computational efficiency. Unlike the traditional NOSB PD method that formulates the force density vector based on the deformation gradient, this method uses a deep NN to map the bond related quantities to the force density vector. The accuracy and efficiency of the proposed method are demonstrated through a series of numerical examples. Additionally, this method can be applied to various hyperelastic materials for which analytical constitutive models exist. [Display omitted] Modeling rubber-like materials through a neural network facilitated peridynamic method. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a micro-mechanically based constitutive model is presented to describe stress softening and self-healing in alginate-polyacrylamide (PAAm) double network (DN) hydrogels. The stress softening phenomenon in alginate-PAAm DN hydrogels under cyclical deformation is assumed to be the result of the rupture of chain linkages. Therefore, the network evolution method [Dargazany and Itskov, International Journal of Solids and Structures , 2009, 46 , 2967] is used to characterize stress softening. The polymer matrix is initially decomposed into reversible and irreversible polymer networks. To model stress softening, the entropic energy of a polymer chain and the chain distribution are taken into account for each network. Unlike conventional DN hydrogels, after deformation alginate-PAAm hydrogels show self-healing. The rate of self-healing is associated with both intermolecular forces and the duration of storage of the samples in a thermal chamber. Broken chain linkages are assumed to rebond due to intermolecular forces and heating. Chemical reaction kinetics and heat transfer equations are utilized to calculate the quantity of the reversible cross-linking rebonding. This model contains few material parameters and demonstrates good agreement with experimental data in stress softening and self-healing. [ABSTRACT FROM AUTHOR]

Highlights • A nonlinear visco-hyperelastic model is numerically studied. • Genetic algorithms are employed to fully characterize the material. • Single and multi objectives versions are considered. • A benchmarking analysis is performed using a known solution. • Uniaxial, biaxial and shear tests are designed to compare the proposed candidates. Abstract Rubber-like materials are widely deployed in industry because of their outstanding properties of elasticity and energy dissipation capacity. In order to analyze and improve the behavior of components made of rubber, suitable models for the material behavior must be formulated before their usage in complex evaluations. The aim of such models is to predict the mechanical response of these materials under external loads, using visco-hyperelastic descriptions. In this paper we are interested in reproducing the nonlinear elastic behavior of rubbers as well as the strain-dependent part (viscoelasticity). Typical viscoelastic models are based on linear assumptions, being the viscous part time-dependent but no strain-dependent. Therefore, nonlinear elasticity and linear viscoelasticity are mixed up in the available models. In our work, we propose a model with both parts described as nonlinear. We use a classical hyperelastic model (Mooney–Rivlin) combined with a nonlinear viscous part including both dependencies in strain and time. We also develop a genetic algorithm for searching (optimizing) the model parameters that describe the behavior of this type of materials. We describe simple and multiple objective genetic algorithms for the optimization based on uniaxial and biaxial stresses. We present benchmarking tests for the algorithms, which reproduce different tests with high accuracy and we also discuss the reliability of the model for different stress sates, strain ranges and strain rates. Finally, a real specimen is tested and studied with the algorithm to define its properties according to the material model developed. The study highlights the capabilities of the model to describe complex stress states based on limited information, and it also shows the possible limitations. The procedure we develop can be used as a design tool that allows implementing any simply analyzed material into a realistic material model for further mechanical evaluations, as it is finite elements or similar. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract: A novel type of polymeric material with tunable mechanical properties is fabricated from polymers and small molecules that can form hydrogen‐bonded intermolecular complexes (IMCs). In this work, poly(vinyl alcohol) (PVA)–glycerol hydrogels are first prepared, and then they are dried to form IMCs. The tensile strengths and moduli of IMCs decrease dramatically with increasing glycerol content, while the elongations increase gradually. The mechanical properties are comparable with or even superior to those of common engineering plastics and rubbers. The IMCs with high glycerol content also show excellent flexibility and cold‐resistance at subzero temperatures. Cyclic tensile and stress relaxation tests prove that there is an effective energy dissipation mechanism in IMCs and dynamic mechanical analysis confirms their physical crosslinking nature. FTIR and NMR characterizations prove the existence of hydrogen bonding between glycerol and PVA chains, which suppresses the crystallization of PVA from X‐ray diffraction measurement. These PVA–glycerol IMCs may find potential applications in barrier films, biomedical packaging, etc., in the future. [ABSTRACT FROM AUTHOR]

Dynamic cavitation is known to be a typical failure mechanism in rubber-like solids. While the mechanical behaviour of these materials is generally rate-dependent, the number of theoretical and numerical works addressing the problem of cavitation using nonlinear viscoelastic constitutive models is scarce. It has been only in recent years when some authors have suggested that cavitation in rubber-like materials is a dynamic fracture process strongly affected by the rate-dependent behaviour of the material because of the large strains and strain rates that develop near the cavity. In the present work we further investigate previous idea and perform finite element simulations to model the dynamic expansion of a spherical cavity embedded into a rubber-like ball and subjected to internal pressure. To describe the mechanical behaviour of the rubber-like material we have used an experimentally calibrated constitutive model which includes rate-dependent effects and material failure. The numerical results demonstrate that inertia and viscosity play a fundamental role in the cavitation process since they stabilize the material behaviour and thus delay failure. [ABSTRACT FROM AUTHOR]

In this work, we propose a microstructurally motivated hyperelastic model to describe the behavior of rubber-like materials. At the scale of the Representative Volume Element (RVE), we assume that, for each macromolecular chain, the segments of the chains are deformable and that there is a bending energy between two consecutive segments. We propose to model each macromolecular chain using micro-mechanical elements: elastic bars to represent the segments between cross-linking points and elastic spire to illustrate the flexibility of rotations around the cross-linking points. We thus suggest to model the behavior of the macrochain, using a quadratic spring like potential for a linear behavior of the chain segments, associated with a nonlinear elastic sigmoidal behavior at the connection points between Kuhn segments. Numerical simulation, on different RVEs, show that the proposed modeling represent the response of hyperelastic rubber-like materials in uniaxial extension, simple shear, pure shear and biaxial extension. In order to validate the proposed model, the results obtained in the case of four RVEs will be compared with the experimental data of Treloar (1944). These comparisons show that the proposed model is able to reproduce the experimental behavior of rubber-like materials. • The paper proposes a new microstructurally motivated hyperelastic model to describe the behavior of rubber-like materials. • Numerical simulation, on different RVEs, show that the proposed modeling represents the response of hyperelastic rubber-like materials in uniaxial tension, simple shear, pure shear and biaxial tension. • Only three parameters should be identified in the proposed model. • The proposed model is validated by means of a comparison with experimental data of Treloar (1944). [ABSTRACT FROM AUTHOR]

Hyperelastic rubber-like materials are described by their nonlinear elastic behavior subjected to large deformations. The one-dimensional strain energy function for polymeric material was introduced in Mahnken and Mirzapour (2022). This follow-up contribution presents a three-dimensional statistical micro-mechanical-based constitutive model incorporating the topological constraint's effects. The micro strain energy corresponding to the stored energy in a single polymer chain is mainly entropic and calculated from the entropic part of the Helmholtz free energy based on the Boltzmann equation. Then, the macro strain energy corresponding to the polymer networks is obtained by homogenizing the micro strain energy over the unit volume. At the same time, it can be decomposed into cross-linked and entangled strain energies. The current model is developed on the different angles of the polymer chain. The random nature of the polymer chain's angles is defined by proper probability distribution functions (PDFs) representing different angles' distribution. I incorporate the entangled chains based on the statistical degree of freedom introduced as polymer angles. Numerical examples to calibrate and verify the proposed model illustrate the capability of the current proposed model to reproduce the highly nonlinear behavior of rubber-like materials under different loading conditions. • An extension of one-dimensional strain energy to three-dimensional. • Incorporating the effects of the entangled chains. • Physically-based evolution of different angles of the polymer chain. • Considering the tension and compression phases of the polymer chain. [ABSTRACT FROM AUTHOR]

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). • A simple model for rubber-like materials with 3 parameters obtained from a tensile test. • Physically interpretable material parameters linked to the micro-structure. • Analytical asymptotic behaviour of the model for very large and for stretches close to 1. • The model is a full network model with non-Gaussian statistics. • Accurate predictions are obtained for any loading condition in rubber and silicone. [ABSTRACT FROM AUTHOR]

Kauçuk türü malzemelerin mekanik özellikleri malzemeye uygulanan şekil değiştirme hızıyla değiştirmektedir. Uygulamadaki deformasyon proseslerinde kauçuk türü malzemeler konvansiyonel test cihazlarında test edilen şekil değiştirme hızlarından daha yüksek hızlarda deforme edilmektedir. Bu sebeple konvansiyonel test cihazlarından elde edilen mekanik özellikler yüksek şekil değiştirme hızları için kullanılamaz. Dolayısıyla malzemenin deformasyonundaki şekil değiştirme hızı bilinmeli bu hızda malzemenin mekanik özellikleri belirlenmelidir. Özellikle bu durum malzemenin davranışını matematiksel olarak modelleyebilmek için önem arz etmektedir. Hazırlanan çalışmada, öncelikle, kauçuk türü malzemelerin çekme ve basma deformasyonu altındaki mekanik özelliklerini belirlemek için uygulanan deneysel teknik ve cihazlar sonrasında ise literatürde kauçuk türü malzemeler için yapılan deneysel çalışmalar incelenmiştir. Deneysel çalışmalar yarı-statik, düşük hızdaki dinamik, yüksek hızdaki dinamik ve yüksek hızdaki darbeli deneysel çalışmalar olarak ayrılmıştır.

The paper contains the results of investigation of elastic and viscous properties of rubber-like materials with orientational order. The elastic properties are described with the help of the introduced microscopic Cauchy-Green tensor. The obtained results describe the nonlinear elastic properties well and predict the existence of the soft modes. In the theory of viscous properties the approved model for the time correlation function of the microscopic stress tensor is used.

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. [ABSTRACT FROM AUTHOR]

We test the predictability of our (DPS) microstructure inspired constitutive approach for damage and hysteresis in rubberlike materials by comparing it with the widely adopted Ogden and Roxburgh (OR) and Beatty and Krishnaswamy (BK) models. The experimental validation is analyzed by uniaxial cyclic tests with increasing maximum strains of high damping EPDM rubber specimens adopted in seismic isolation. The numerical identification of the material parameters is obtained by a careful population-based stochastic optimization approach. After testing the stability of our numerical approach, based on a large number of numerical experiments, we deduce that our constitutive model and the OR model are the most accurate. However, differently from our model, we find that the OR model exhibits a strong numerical instability, with a very high variance of the optimal material parameters for the same experimental data sets. We believe that the obtained results are important both for rubber constitutive models and for material parameters optimization approaches. [ABSTRACT FROM AUTHOR]

This paper discusses the different approaches that can be used to determine the strain energy density of a given rubber-like material based on tension-torsion experimental results. More precisely, the aim is to answer the question: how to handle the measured macroscopic quantities, i.e. load and torque, to determine the constitutive equation with the less possible assumptions? The method initially proposed by Penn and Kearsley [Trans. Soc. Rheol. 20 (1976) 227-238] is adopted: the strain energy derivatives with respect to kinematical quantities have to be calculated in terms of the measured load and torque. Here, we propose to consider different sets of kinematical quantities to overcome the incoherence encountered with the classical Cauchy-Green strain invariants I1 and I2. Two new sets are considered: the principal stretch ratios and two specific invariants of the logarithmic (true) Hencky strain tensor. The corresponding derivations coupled with new experimental results permit (i) to calculate the Cauchy stress tensor on the outer surface of the cylindrical samples, and (ii) to demonstrate that a well-conditioned set of kinematical quantities must be adopted to determine the strain energy density. It is proved here that the principal stretch ratios are good candidates to express and determine the strain energy density with tension-torsion experiments. [ABSTRACT FROM AUTHOR]

Highlights: [•] Filled styrene butadiene and natural rubbers are tested under cyclic uniaxial loading. [•] Temperature measurements are performed using infrared thermography. [•] Calorimetric response of both materials is characterized during stress softening. [•] Mechanical dissipation due to Mullins effect is determined. [•] Results provide the first database for thermomechanical modeling of Mullins effect. [ABSTRACT FROM AUTHOR]

We present a model for incompressible finite strain orthotropic hyperelasticity using logarithmic strains. The model does not have a prescribed shape. Instead, the energy function shape and the material data of the model are obtained solving the equilibrium equations of the different experiments. As a result the model almost exactly replicates the given experimental data for all six tests needed to completely define our nonlinear orthotropic material. We derive the constitutive tensor and demonstrate the efficiency of the finite element implementation for complex loading situations. [ABSTRACT FROM AUTHOR]

Highlights: [•] We develop a practical approach for building adequate hyperelastic models. [•] Foundations of original attractive models building-strategy have been investigated. [•] Stable parameter’s values and the validity of a model are established from data. [•] We construct a new constitutive model generalizing that of Hart-Smith. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, high-order tetrahedral finite elements are employed to analyze structures and solids composed of functionally graded rubber-like materials under finite displacements, finite strains, statically applied forces and isothermal conditions. In order to do so, the following concepts are used: geometrically nonlinear analysis, Green–Lagrange strain tensor, second Piola–Kirchhoff stress tensor, hyperelastic constitutive relations, isoparametric solid tetrahedral finite elements of any order of approximation, and functionally graded materials. The equilibrium of the body is achieved via the Principle of the Stationary Total Potential Energy. The elements are fully integrated via Gaussian quadratures, and the resultant processing time is reduced by means of parallel techniques. To solve the nonlinear system of equations, the Newton–Raphson iterative procedure is employed. The proposed formulation is validated by benchmark problems such as: the Cook’s membrane and the thick cylinder. Other interesting simulation, the Cook’s block is proposed in order to evaluate high strain gradient situations. The results show that, in the context of the present study, locking-free behavior is obtained with simple mesh refinement. [Copyright &y& Elsevier]

Abstract: In this paper we extend the Sussman–Bathe spline-based hyperelastic isotropic model to predict the behavior of transversely isotropic isochoric materials. The model is based on an uncoupled decomposition of the stored energy function and a generalization of the inversion formula used by Sussman and Bathe. The present extension introduces some approximations that, in all studied cases, do not yield relevant deviations from the experimental data. The isotropic model results in a particular case of the present formulation. Several possibilities of user-prescribed experimental data are addressed. The model is used to predict experiments of calendered rubber and of biological tissues. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

Since the pioneer works of Gough and Joule, the thermal characterisation of elastomers under mechanical loading has been investigated by numerous research teams. This is not surprising as the thermal signature of rubber is very useful data to investigate the dissipation mechanisms as well as the thermodynamical variables and couplings. In former recent studies dealing with fatigue investigations, an experimental protocol was developed. This protocol imposes cyclic loading to hourglass shaped samples, takes into account the large displacements and permits dissociation between the intrinsic dissipation, responsible for the mean temperature variation (called heat build-up in the literature) and thermomechanical couplings responsible for the temperature variation around this mean value during one cycle. Up to now, the mean temperature has been investigated in order to feed an energetic fatigue criterion. The aim of the present study is to investigate the thermomechanical couplings and the ability of thermal measurements to exhibit some specific thermomechanical properties observed for rubberlike materials. The materials studied are natural rubber and styrene butabiène rubber compounds filled with several amounts of carbons blacks. The experimental data clearly exhibit interesting features such as the thermoelastic inversion point and difference in the temperature signal between mechanical loading and unloading. This rich database is analysed and correlated to other results from the literature. The main results obtained are dealing with the ability of accurate measurements to characterise the thermodynamic couplings and to detect the stress induced crystallisation. [ABSTRACT FROM AUTHOR]