Behavior Of Rubber-like Materials Research Articles

A neural network peridynamic method for modeling rubber-like 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.

A Visco-Hyperelastic Constitutive Model to Characterize the Stress-Softening Behavior of Ethylene Propylene Diene Monomer Rubber.

Uniaxial and biaxial cyclic tensile tests and stress relaxation tests were performed on the ethylene propylene diene monomer (EPDM) material to investigate its stress-softening effect. The experimental results reveal that the EPDM material presents a significant Mullins effect during the cyclic stretching processes. Furthermore, it is found that the deformation of the EPDM material does not return to zero simultaneously with the stress, due to the viscoelasticity of the EPDM material. Therefore, this study combines pseudo-elasticity theory and viscoelastic theory to propose a visco-hyperelastic constitutive model. The proposed model is used to fit and analyze the uniaxial and biaxial cyclic test results of EPDM and a comparison is conducted with the corresponding hyper-elastic constitutive model. The results show that the proposed model is in good agreement with the experimental data and superior to the hyper-elastic constitutive model, especially when it comes to the stress-softening unloading process. This work is conducive to accurately characterizing the stress-softening behavior of rubber-like materials at large deformation and can provide some theoretical guidance for their widespread application in industry.

A model for hyperelastic rubber-like materials based on micro-mechanical elements

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.

A micro-mechanically-based constitutive model for hyperelastic rubber-like materials considering the topological constraints

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.

STRAIN RATE DEPENDENCY OF NEOHOOKEAN MATERIAL USING KELVIN-VOIGT VISCOELASTIC MODEL

This research paper examines the strain rate dependency of a Neo-Hookean material using the Kelvin-Voigt viscoelastic model. The Neo-Hookean model is widely used to describe the mechanical behavior of rubber-like materials, but it fails to consider the timedependent response observed in many materials. To overcome this limitation, the Kelvin-Voigt viscoelastic model, which incorporates a viscosity term into the NeoHookean model, is employed to capture the strain rate dependency. The paper commences by introducing the Neo-Hookean material model and its assumptions. It then elucidates the rationale behind incorporating the Kelvin-Voigt model to account for the time-dependent behavior. To validate the strain rate dependency of the KelvinVoigt model, tensile test is performed using experimental data. The experimental data comprises stress-strain curves, from which the material parameters of both the Neo-Hookean and Kelvin-Voigt models are determined using curve-fitting techniques. The study presents and analyzes the results, with a specific focus on the material's strain rate sensitivity. The stress-strain curves obtained from the Kelvin-Voigt model are compared with the experimental data, and the model's accuracy is evaluated. Additionally, the material parameters obtained from the curve-fitting process are examined to explore the influence of strain rate on the mechanical response of the material.

Development and Validation of a Nonlinear Model to Describe the Tension–Compression Behavior of Rubber-Like Base Isolators

The fractional-order derivative Zener (FDZ) model can reasonably predict the frequency dependence behavior of rubber-like materials. However, its capability to capture nonlinear behaviors is limited. In our previous work, we built a horizontal shear model of a multidimensional rubber-like base isolation bearing, modifying the FDZ model to compensate for its inability to reproduce the nonlinear behaviors of amplitude dependence and slow stabilization. In this paper, the tension-compression behavior of the same bearing in the vertical direction is studied, where strain-stiffening nonlinear behavior is exhibited. A different strategy is used to establishing the modified tension-compression FDZ (CFDZ) model, which is shown to have good agreement with experimental results. State-space representation of the CFDZ model is presented and included in modelling a train-bridge-bearing system for dynamic analysis. Simulation results demonstrate that the proposed bearing achieves a high base isolation performance in the vertical direction.

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.

A Modified Fractional-Order Derivative Zener Model for Rubber-Like Devices for Structural Control

AbstractFractional-order derivative (FD) models are widely used to describe the dynamic behavior of rubber-like materials. However, experiments have shown that FD models cannot reproduce the amplit...

Comparison between 3D-reconstruction optical methods applied to bulge-tests through a feed-forward neural network

The mechanical behaviour of rubber-like materials can be investigated through numerous techniques that differ from each other in costs, execution times and parameters described. Bulge test method proved helpful for hyperelastic membranes under plane and equibiaxial stress state. In the present study, bulge tests in force control were carried out on SBR 20% CB-filled specimens. 3D reconstructions of the dome were achieved through two different stereoscopic techniques, the epipolar geometry and the Digital Image Correlation. Through a Feed-Forward Neural Network (FFNN), these reconstructions were compared with the measurements by a laser triangulation sensor taken as reference. 3D-DIC reconstruction was found to be more accurate. Indeed, bias errors of the 3D-DIC and epipolar techniques with respect to the relative reference values, under creep condition, were 0.53 mm and 0.87 mm, respectively.<br /><br />

A model for the dynamic friction behaviour of rubber-like materials

Rubber friction has received much attention because of its practical importance for modelling, for example, tyre-road friction. Many experiments were carried out and a number of theories were proposed to investigate and explain its behaviour and dependency on external conditions such as temperature and velocity. However, only a few published articles describe the observed dynamic friction behaviour using compact analytical models. Models for dry friction, such as the Dahl and LuGre models and their several variations, have been proposed to describe rubber friction. Unfortunately, they cannot reproduce many important and specific phenomena observed in experiments focused on rubber friction. In this paper, we propose a simple analytical friction model suitable to simulate the dynamic friction behaviour of rubber-like materials on rigid surface. Phenomena such as pre-sliding, sticking/sliding transitions, relaxation of the friction force at standstill and stick-slip effects can be accurately modelled. We validated the model with several experiments, obtaining a good agreement between simulation and measurements. In addition, we propose a two-dimensions (2D) extension of the model, to broaden the field of cases, in which it can be used. The capability of the model to simulate 2D anisotropic friction behaviour was validated with measurements of a vehicle tyre.

On a new class of non-Gaussian molecular-based constitutive models with limiting chain extensibility for incompressible rubber-like materials

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 , 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 (the chain stretch) of the inverse Langevin function. A similar approach for devising a general term as an adjunct to the part of the model will also be presented, for applications where the addition of an 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 and has no direct connection to Padé approximations of any order in , 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.

A note on a class of generalized neo-Hookean models for isotropic incompressible hyperelastic materials

In a recent paper in this journal by Anssari-Benam and Bucchi (2021), the authors have proposed a new two-parameter constitutive model for isotropic incompressible hyperelastic generalized neo-Hookean materials. The model reflects the limiting chain extensibility characteristic of non-Gaussian molecular models for rubber. A major contribution of Anssari-Benam and Bucchi (2021) is in showing that the model proposed there is superior to the well-known two-parameter Gent model when fitting with a large variety of experimental data for rubber for the homogeneous deformations of uniaxial, equi-biaxial and pure shear. Moreover, for all of these deformations, a fitting with data is achieved with a single set of values for the parameters with a narrow range of variation. In the present note, we establish a simple direct relation between the new model and the classical Gent model. The large body of research results on the mechanical behavior of rubber-like materials based on the latter model are thus readily applicable to the new model. Some other directions for widening the applicability of the new model are also suggested.

An extended eight-chain model for hyperelastic and finite viscoelastic response of rubberlike materials: Theory, experiments and numerical aspects

Rubberlike materials exhibit strong rate-dependent mechanical response which manifests itself in creep and relaxation tests as well as in the hysteresis curves under cyclic loading. Unlike linear viscoelasticity, creep and relaxation response of rubber is nonlinear and amplitude-dependent. Within this context, the contribution of this work is three fold. (i) On the experimental side, the characterization of equilibrium and non-equilibrium responses are carried out by means of uniaxial and equibiaxial extension tests. Also performed are the creep and relaxation experiments at various stress and strain levels. (ii) On the theoretical side, we extend the well-known eight-chain model via incorporating a simple yet instrumental tube-constraint term composed of the second invariant into the non-affine network contribution reflecting the ground state equilibrium response. For the non-equilibrium response, we propose a new evolution equation for the creep/relaxation behavior of rubberlike materials based on a relaxation kinetics of a single polymer chain. The geometric non-linearity is incorporated via the finite deformation kinematics based on the multiplicative split of the deformation gradient into elastic and viscous parts, whereas the volumetric and isochoric split of the deformation gradient is entirely discarded. The rheology uses a nonlinear spring responsible for equilibrium elastic response in parallel to n number of Maxwell elements, leading to a generalized Maxwell-Wiechert viscoelastic solid. (iii) The algorithmic implementation of the model features the spectral decomposition of the respective terms and is demonstrated within the context of the finite element method. The developed model is validated by fitting both the elastic and viscoelastic model responses with respect to the experimental data in the sense of uniaxial, (equi)biaxial extensions, and pure shear tests. Relaxation and creep behavior of the model are thoroughly assessed. Also presented in the manuscript is the capability and the performance of the model in the face of a relevant non-homogeneous boundary value problem. The quality of the findings earns the model vast utilization areas from an engineering perspective.

A new approach to describe the elastic behavior of filled rubber-like materials under complex uniaxial loading

The construction of a model capable of describing viscoelastic behavior of rubber-like materials is a challenging problem. Although this task can be simplified, if we restrict our consideration only to elastic behavior of the material, excluding all temporary (rate-dependent) effects. In practice, it can be achieved by testing polymer samples at very low loading rates. However, our research is devoted to the performance of complex experiments on uniaxial cyclic loading with different degrees of deformation of the samples, in which the arising stresses are considered as the sum of equilibrium (elastic) and dissipative (inelastic) terms. A series of experiments have been carried out on the samples of materials, which have identical matrices (styrene-butadiene rubber) but different degrees of filling with carbon black. The result of our research is the construction of a constitutive model, which can describe the elastic behavior of rubber-like materials taking account of the Mullins softening effect.

Characterization of PVA hydrogels’ hyperelastic properties by uniaxial tension and cavity expansion tests

The mechanical behavior of rubber-like materials is dominantly non-linear elastic. Their mechanical response is measured by numerous conventional techniques including the universal loading machines (tension and compression) and indentation. Recently, an unconventional technique based on cavitation rheology has been implemented to measure the mechanics of these materials. The loading mechanism of this technique is different from the axial force–displacement mechanism observed in the other techniques thus the aim of this study is to investigate the difference between the cavity expansion technique and one of the conventional techniques, uniaxial stretching, to understand the difference in material responses under each loading mechanism. PVA hydrogel was used as a representative to hyperelastic materials. The gels were loaded by both techniques, and hyperelastic models (Yeoh, Ogden, and Arruda–Boyce) were used to model their behavior. Finite Element (FE) simulations were performed to reproduce the experimental data. It was observed that the tension stresses generated during the cavity expansion test were similar to those generated in the uniaxial tension to a strain level of 45%, afterward, the cavity tension stresses increased exponentially exceeding those generated in the uniaxial tension. When the von Mises stresses, from both tests, were compared against the major tension stress data, it was observed that the cavity test imposed tension stresses that were twice of those generated during the uniaxial tensile test. In addition, the hoop stresses were significantly larger than the applied pressure. These observations indicate to the equi-biaxial nature of the cavity expansion test.

Characterization of hyperelastic deformation behavior of rubber-like materials

This paper proposes a new hyperelastic model for rubber-like materials applicable over the entire modes of the large deformations. The whole material parameters have been identified using the mathematical approach combined with the genetic algorithm for the best method for the structure of hyperelastic models applicable to incompressible rubber-like materials. The step-by-step method is used on the restricted form by gradually adding a term of the highest degree of the hyperelastic model when the deformation increases. This combination greatly minimizes the difference between experimental data and analytical solutions. It is about a method that is applied to nonlinear behaviors leading to an optimal solution. The power of the model is demonstrated via comparisons with different phenomenological hyperelastic models using the experimental data from the Treloar and Nunes works. According to the litterature, many authors have proposed a model based on both strain invariants but unable to reproduce the equibiaxial deformation mode. This work offers the constitutive hyperelastic model based on both strain invariants $$I_1$$ and $$I_2$$ with a minimal number of rheological parameters.

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.

A CONSTITUTIVE MODEL FOR BOTH LOW AND HIGH STRAIN NONLINEARITIES IN HIGHLY FILLED ELASTOMERS AND IMPLEMENTATION WITH USER-DEFINED MATERIAL SUBROUTINES IN ABAQUS

ABSTRACT Strain energy functions (SEFs) are used to model the hyperelastic behavior of rubberlike materials. In tension, the stress–strain response of these materials often exhibits three characteristics: (i) a decreasing modulus at low strains (<20%), (ii) a constant modulus at intermediate strains, and (iii) an increasing modulus at high strains (>200%). Fitting an SEF that works in each regime is challenging when multiple or nonhomogeneous deformation modes are considered. The difficulty increases with highly filled elastomers because the small strain nonlinearity increases and finite-extensibility occurs at lower strains. One can compromise by fitting an SEF to a limited range of strain, but this is not always appropriate. For example, rubber seals in oilfield packers can exhibit low global strains but high localized strains. The Davies–De–Thomas (DDT) SEF is a good candidate for modeling such materials. Additional improvements will be shown by combining concepts from the DDT and Yeoh SEFs to construct a more versatile SEF. The SEF is implemented with user-defined material subroutines in Abaqus/Standard (UHYPER) and Abaqus/Explicit (VUMAT) for a three-dimensional general strain problem, and an approach to overcome a mathematically indeterminate stress condition in the unstrained state is derived. The complete UHYPER and VUMAT subroutines are also presented.

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 UNIFIED INELASTICITY AND DAMAGE MECHANICS FOR RUBBER-LIKE MATERIALS

The nonlinearities observed in the behavior of rubber and polymeric materials are influenced not only by the far-field stress conditions but also by the morphology, microstructures, and changes at the meso-level such as kinks, crosslinking, and micro tearing. Considering constants temperature applications, the occurrence of microtearing has a significant contribution on the performance of polymers, soft tissues and other rubber-like materials. In this paper, a unified damage mechanics and nonlinear elasticity approach is presented to model nonlinear behavior of rubber-like materials under constant temperatures. The theory is cast within the general framework of the internal variable theory of thermodynamics with large deformations where the Clausius-Duhem inequality is provoked to develop general damage potential. The strain energy density function is formulated in terms of an effective Lagrangian strain tensor that evolves with cumulative damage as cracking and micro-tearing take places. Piola-Kirchhoff (PK) stress tensor is presented and a new form of the damage response tensor is proposed. The model prediction is illustrated against experimental results with good agreement.