The position vector X labels the particle P at a given time t in the reference or undeformed configuration. When the region moves to a new configuration, known as current or deformed, the particle P corresponds to the new position x. The state of deformation at each material point is described by a deformation gradient F , Equation 1.
Other important tensors that are useful to describe the deformation of the material are the left Cauchy—Green deformation tensor B , Equation 2 and the right Cauchy—Green deformation tensor C , Equation 3 , both obtained by multiplying F by its transpose:. Equations 4 — 6 define the three invariants of B :. For the uniaxial test of an incompressible material, the principal stretches are given by Equation 7 and the deformation gradient is given by Equation 8 :.
In addition, a Sensitivity Analysis was conducted on some models.
Generalization of one-dimensional material models for the finite element method. Mars, M. Meunier, L. Bojtar, Mousseau, A. Corre, M. The formula Rivlin polynomial and Ogden theoretically provide, the of this model cannot be used, when the deformation of same results.
A variance-based analysis is one of the most accurate techniques for these nonlinear theories [ 37 ]. It allows the full exploration of the input space global method and the estimation of the covariance matrix provides an exploratory study of the inter-relationship between variables how the variables change in relation to one another. Different models exist for different loading conditions to account for the strong nonlinearity of the filled vulcanized rubber. For the case of a cyclic loading condition with small strain amplitude, the filled rubber shows a dependence of the viscoelastic storage modulus on the magnitude of the applied strain.
This effect is known as the Fletcher—Gent effect or the Payne effect and this has been extensively studied previously [ 4 , 26 , 38 , 39 ]. One of the modelling techniques is based on the use of the fractional derivatives [ 40 , 41 ]. In particular Lion, [ 40 ], proposed a model with six parameters, which combines the fractional Maxwell model with the concept of an intrinsic time z t. A finite strain visco-plastic-elastic model for rubber with reinforcing filler was proposed by Osterlof [ 26 , 27 ].
It assumes small strains and can capture the amplitude dependency and frequency dependency with a multiplicative split of the deformation gradient into elastic, plastic and viscoelastic components. Another important phenomenon is the Mullins effect, which is observed at larger strains. It includes the cyclic stress softening Figure 15 and the ability of the material to recover and return towards the virgin stress—strain path when the load rises over the previous maximum applied stress value Figure Many experimental data show that the Mullins effect is strongly anisotropic and produces a significant permanent set [ 31 , 42 , 43 ].
The next section describes in detail some models that can be used to characterise some of these nonlinear effects Figure 4 , Figure 5 , Figure 6 , Figure 7 , Figure 8 , Figure 9 and Figure In particular, the focus is on the ability to predict the Mullins Effect, the different unloading and reloading curves and the pre-strain effect. The phenomenological models can roughly be classified into three categories: the damage models, the rheological models with serial and parallel combination of elastic and viscous elements and the constitutive equations based on a rubber elasticity model. Viscoelastic damage models use a softening parameter, called the damage variable.
The hyperelastic strain energy density W 0 is penalised by a damage function D , depending on the maximum deformation, Equation 11 :. Simo [ 44 ] was the first to adopt such a continuum damage mechanics theory.
Over the subsequent years, various models have been defined [ 32 , 45 , 46 ]. In these works, the damage is not related to physical considerations, but it is purely phenomenological. Such model, however, neglects residual strains. W d e v m a x is the maximum deviatoric strain energy, W d e v is the deviatoric strain energy and r , m and q are the material parameters. In their work, Guo and Sluys [ 49 ] discussed limitations and advantages of the continuum mechanics model and the pseudo-elastic model.
Experimental data of Natural Rubber with 60 phr of carbon black are used to fit the models and compare the results following the time history of the damage parameter. This approach is useful as it helps examine the limitations of such models in predicting phenomena such as the Mullins effect and the cyclic stress relaxation. The theoretical stress—strain response returns to the virgin stress—strain path without any transition Figure The unloading and reloading for a new cycle paths are on the same curve, which disagrees with the experiments when, for example, after a large strain the material is reloaded at a smaller amplitude.
In addition, these laws can not reproduce the hysteresis during these cycles. Furthermore, the models fail to capture the stress—strain curve for cyclic uniaxial stress with uniform maximum strain amplitude, Figure The theoretical stress for the maximum strain is the same for the first and following loadings, which ignores the cyclic stress-softening. The hysteresis for the cycles after the first is also ignored. The limitations of these models are clear when the theoretical softening function [ 45 ] is compared with the real damage estimated as ratio between the experimental stress and the Yeoh model fitted for the first loading curve Figure 21 and Figure The real damage can be considered also to be the ratio of the experimental stress in a subsequent cycle to the experimental stress at the same strain during the first loading curve.
It is possible to see that the real experimentally observed damage function shows the following features:.
It is not constant after significant peak strain when the material is reloaded to a smaller strain. It never returns to the value 1 there is always a softening when the material is reloaded even at a fixed maximum strain amplitude. Comparison of the evolution of theoretical softening function [ 45 ] with a real damage parameter estimate as the ratio between the experimental stress and the hyperelastc stress for NR60 with cyclic uniaxial stress with different maximum strain amplitude in step up and step down phases MAXAMP-UPDOWN Figure 3.
Comparison of the evolution of theoretical softening function [ 45 ] with real damage parameter estimate as a ratio of the experimental stress to the hyperelastic stress for NR60 with cyclic uniaxial test with uniform maximum strain amplitude MAXAMP-CONST Figure 1. In Figure 21 and Figure 22 , a conflict between the theory and experimental data is evident, which has already been highlighted by Diani [ 50 ].
Constitutive Models for Rubber X: Proceedings of the European Conference on Constitutive Models for Rubbers X (Munich, Germany, Summary. The unique properties of rubber make it ideal for use in a wide variety of engineering applications such as tyres, engine mounts, shock absorbers.
The damage function must be increasing monotonically during the loading to satisfy the Clausius—Duhem inequality. However, the experimental results indicate that the softening variable shows a clear non-monotonic trend. Some of the viscoelastic models split the stress into an equilibrium part that is rate-independent and an additional viscous part. This approach assumes a multiplicative decomposition of the deformation gradient into an elastic and inelastic time dependent part [ 27 , 38 , 51 , 52 , 53 , 54 ]:. One of the most widely adopted models is the micro-mechanism inspired model proposed by Bergstrom and Boyce [ 51 , 55 , 56 , 57 ], which is already widely implemented in various Finite Element Software, such as ABAQUS.
It is a hyper-viscoelastic model with a superposition of viscoelastic and hyperelastic networks in parallel, Figure 23 that is able to predict hysteresis and strain rate dependence for slowly applied loads. It has a remarkable correspondence to the equilibrium state after many cycles Figure Rheological model: hyperelastic model in parallel with viscoelastic network with Bergstrom Boyce flow [ 51 ]. Comparison of Bergstrom Boyce model [ 51 ] with experimental data for NR50 in cyclic uniaxial test.
In the last decade, a finite-strain constitutive framework known as a Parallel Rheological Framework [ 23 , 58 ] has been developed to model nonlinear viscoelasticity, Mullins Effect and a permanent set in elastomers. It consists of multiple networks connected in parallel Figure An arbitrary number of viscoelastic networks are used in parallel with an equilibrium network that can be purely elastic or elastoplastic.
Figure 26 shows the efficiency of this framework with three viscoelastic networks in parallel with a hyperelastic network, Figure The model fits well with the experimental response of Natural Rubber filled with 60 phr carbon black under cyclic loading at uniform strain amplitude. However, the model introduces a non-realistic shape in the first loading path.
In addition, this approach requires a significant number of parameters that are difficult to derive and which change erratically with small changes in the input test data.
The MORPH model is defined by the characterization of 8 p i material parameters assuming incompressible behaviour and by introducing a history function B T H. B is the left Cauchy strain and B T is a special invariant given by the maximum difference of two eigenvalues. The model fits the hysteresis and the influence of the loading history for rubber very well for materials with a small amount of filler, as is shown in Figure A good fit is also shown for the cyclic uniaxial cyclic test at constant strain amplitude.
The model can predict the stress softening between the first and the second cycle Figure However, the model is less able to work well when the amount of filler is increased. Figure 31 shows an unrealistic effect, especially in the area of small strain. In , Freund [ 60 ] proposed a new algorithm for the 3D generalization of one-dimensional model and tested the algorithm on the MORPH model.
The idea is based on the concept of representative directions.
It is a continuum mechanics generalization without any reference to the physical structure and applies to any model, even to inelastic materials without a known free energy:. A different approach was proposed by Liu [ 54 ]. This model is based on uniaxial test data and assumes that the materials is isotropic and incompressible and deforms in an isothermal process.
dilrefara.tk The total stress is the sum of a primary hyperelastic stress and an additional viscoelastic stress term. The main response is taken as the average stress—strain response in the hysteresis produced during cycling loading, which changes with strain amplitude and rate.
The additional viscoelastic stress is the difference between the loading stress and the primary stress. The model by Liu can predict the steady state response after multiple loading cycles. In , a different approach was proposed by Rickaby [ 61 ].
He developed a model able to reproduce the inelastic behaviours cyclic stress softening, Mullins effect, residual strain for a uniaxial tension of an isotropic rubber that utilised a large number of parameters. In this model inelastic features can be either excluded or added without compromising the integrity of the model. Several researchers have sought a physical understanding of the nonlinear and inelastic behaviour of reinforced elastomers. An alternative to the damage or rheological models is given by the model derived from micromechanical observations.
The rubber-like materials are treated as a network of long chains connected by links [ 62 ]. The types of behaviour are considered to be a consequence of the molecular motion of these polymer chains and the breakage and reformation of the links. During vulcanisation, the filler particles flocculate together to form clusters.
The constitutive hypothesis of DFM is derived from the thermodynamic equilibrium state at an infinitely slow rate of deformation. The stress response of filled rubber can be derived by two mesoscopic phenomena: hydrodynamic reinforcement of the rubber matrix with strong filler-filler bonds and cyclic break down and re-aggregation of softer filler clusters and hence weaker filler-filler bonds. Any type of cluster that is stretched stores energy that is dissipated when the cluster breaks. Lorenz [ 65 , 66 ] suggested a broader generalization of the framework.