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Nigel Scott - One of the best experts on this subject based on the ideXlab platform.
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Orthotropic cyclic Stress-Softening model for pure shear during repeated loading and unloading
IMA Journal of Applied Mathematics, 2014Co-Authors: Stephen R. Rickaby, Nigel ScottAbstract:We derive an orthotropic model to describe the cyclic Stress Softening of a carbon-filled rubber vulcanizate through multiple Stress-strain cycles with increasing values of the maximum strain. We specialize the deformation to pure shear loading. As a result of strain-induced anisotropy following on from initial primary loading, the material may subsequently be described as orthotropic because in pure shear there are three different principal stretches so that the strain-induced anisotropy of the Stress response is different in each of these three directions. We derive non-linear orthotropic models for the elastic response, Stress relaxation and residual strain to model accurately the inelastic features associated with cyclic Stress Softening. We then develop an orthotropic version of the Arruda-Boyce eight-chain model of elasticity and then combine it with the ideas previously developed in this paper to produce an orthotropic constitutive relation for the cyclic Stress Softening of a carbon-filled rubber vulcanizate. The model developed here includes the widely occurring effects of hysteresis, Stress-relaxation and residual strain. The model is found to compare well with experimental data.
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A cyclic Stress Softening model for the Mullins effect
International Journal of Solids and Structures, 2013Co-Authors: Stephen R. Rickaby, Nigel ScottAbstract:Abstract In this paper the inelastic features of Stress relaxation, hysteresis and residual strain are combined with the Arruda–Boyce eight-chain model of elasticity, in order to develop a model that is capable of describing the Mullins effect for cyclic Stress-Softening of an incompressible hyperelastic material, in particular a carbon-filled rubber vulcanizate. We have been unable to identify in the literature any other model that takes into consideration all the above inelastic features of the cyclic Stress-Softening of carbon-filled rubber. Our model compares favourably with experimental data and gives a good description of Stress-Softening, hysteresis, Stress relaxation, residual strain and creep of residual strain.
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Cyclic Stress-Softening model for the Mullins effect in compression
International Journal of Non-Linear Mechanics, 2013Co-Authors: Stephen R. Rickaby, Nigel ScottAbstract:Abstract This paper models the cyclic Stress Softening of an elastomer in compression. After the initial compression the material is described as being transversely isotropic. We derive non-linear transversely isotropic constitutive equations for the elastic response, Stress relaxation, residual strain, and creep of residual strain in order to model accurately the inelastic features associated with cyclic Stress Softening. These equations are combined with a transversely isotropic version of the Arruda–Boyce eight-chain model to develop a constitutive relation that is capable of accurately representing the Mullins effect during cyclic Stress Softening for a transversely isotropic, hyperelastic material, in particular a carbon-filled rubber vulcanizate. To establish the validity of the model we compare it with two test samples, one for filled vulcanized styrene–butadiene rubber and the other for filled vulcanized natural rubber. The model is found to fit this experimental data extremely well.
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Transversely isotropic cyclic Stress-Softening model for the Mullins effect
Proceedings of the Royal Society A: Mathematical Physical and Engineering Sciences, 2012Co-Authors: Stephen R. Rickaby, Nigel ScottAbstract:This paper models Stress Softening during cyclic loading and unloading of an elastomer. The paper begins by remodelling the primary loading curve to include a Softening function and goes on to derive nonlinear transversely isotropic constitutive equations for the elastic response, Stress relaxation, residual strain and creep of residual strain. These ideas are combined with a transversely isotropic version of the Arruda–Boyce eight-chain model to develop a constitutive relation that is capable of accurately representing the Mullins effect during cyclic Stress Softening for a transversely isotropic, hyperelastic material, in particular, a carbon-filled rubber vulcanizate.
Grégory Chagnon - One of the best experts on this subject based on the ideXlab platform.
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Experimental investigation and theoretical modelling of induced anisotropy during Stress-Softening of rubber
International Journal of Solids and Structures, 2016Co-Authors: Gilles Marckmann, Grégory Chagnon, M. Le Saux, Pierre CharrierAbstract:The Mullins effect refers to a Stress-Softening phenomenon of rubber-like materials during cyclic loading. Anisotropy of the material behaviour is generally observed after stretching. In this paper, a large set of original suitable experiments are reported to characterise this effect under several deformation conditions. Then, a phenomenological model is derived to capture the anisotropic distribution. For that, the affine micro-sphere model (Miehe et al., 2004) is amended with a directional network alteration in order to describe anisotropy. The alteration process, involving the breakage and the slippage of the links embedded in the macromolecular network, is modelled by the evolution of the average number of monomer segments per chain during stretching. The average chain length and the chain density are incrementally described by functions to allow both Softening and stiffening , depending to the maximum and the minimum stretch rates and levels endured in each direction. The good capacity of the model to reproduce * Corresponding author: gilles.marckmann@ec-nantes.fr experimental observations validates the above assumptions.
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Experimental investigation and theoretical modelling of induced anisotropy during Stress-Softening of rubber
International Journal of Solids and Structures, 2016Co-Authors: Gilles Marckmann, Grégory Chagnon, M. Le Saux, Pierre CharrierAbstract:The Mullins effect refers to a Stress-Softening phenomenon of rubber-like materials during cyclic loading. Anisotropy of the material behaviour is generally observed after stretching. In this paper, a large set of original suitable experiments are reported to characterise this effect under several deformation conditions. Then, a phenomenological model is derived to capture the anisotropic distribution. For that, the affine micro-sphere model (Miehe et al., 2004) is amended with a directional network alteration in order to describe anisotropy. The alteration process, involving the breakage and the slippage of the links embedded in the macromolecular network, is modeled by the evolution of the average number of monomer segments per chain during stretching. The average chain length and the chain density are incrementally described by functions to allow both Softening and stiffening, depending to the maximum and the minimum stretch rates and levels endured in each direction. The good capacity of the model to reproduce experimental observations validates the above assumptions.
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Theory and identification of a constitutive model of induced anisotropy by the Mullins effect
Journal of the Mechanics and Physics of Solids, 2014Co-Authors: Guilherme Machado, Grégory Chagnon, Denis FavierAbstract:Rubber-like materials present a Stress Softening phenomenon after a first loading known as the Mullins effect. Some recent experimental data on filled silicone rubber is presented in literature, using uniaxial and biaxial tests to precondition samples thus induce some primary Stress Softening. A generic modeling based on the polymer network decomposition into an isotropic hyperelastic one, and a Stress-Softening evolution one, is proposed taking into account the contribution of many spatial directions. A new Stress Softening criterion tensor is built by means of a tensor that measures the repartition of energy in space. A general form of the Stress Softening function associated to a spatial direction is written by the way of two variables: one, the maximal eigenvalue of the energy tensor; the other, the energy in the considered direction. Finally, a particular form of constitutive equation is proposed. The model is fitted and compared to experimental data. The capacities of such modeling are finally discussed.
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Extension of classical viscoelastic models in large deformation to anisotropy and Stress Softening
International Journal of Non-Linear Mechanics, 2014Co-Authors: Marie Rebouah, Grégory ChagnonAbstract:In this paper, an extension of two classical viscoelastic models adapted in large deformation for incompressible rubber like materials or soft tissues is proposed. These models are built by using a three dimensional homogeneization by means of a sphere unit approach. Thus several comparisons between classical formulations and homogeneization on a sphere unit formulation is proposed. An adaptation of those models to describe anisotropy is proposed. Finally an extension of those models to take into account Stress Softening is described.
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Analysis of the isotropic models of the Mullins effect based on filled silicone rubber experimental results
Mechanics of Materials, 2010Co-Authors: Guilherme Machado, Grégory Chagnon, Denis FavierAbstract:The Mullins effect of rubber like material is classically defined as the Stress Softening during initial loading cycles. This effect is not accounted when the mechanical properties of material are modeled by a simple hyperelastic strain-energy function. In order to capture the Stress Softening it is necessary to define a set of supplementary variables as well a dissipation function, which evolves with the deformation history. In this paper we first describe experimental results that illustrate Stress Softening in particle-reinforced silicone rubber for uniaxial, planar and equibiaxial traction. The results allow to analyze the Stress Softening for the three different load cases. First, with respect to the choice of a Stress-Softening measure, the energy loss was evaluated by comparing the stored elastic energy for the first and the second loadings. The results point out that the virgin energy and the first invariant parameters are the best choice. Nevertheless, the maximum principal elongation, classically used in Mullins effect modeling, is not able to describe the different load cases. Furthermore, the ability of different class of models to describe filled silicone rubber was studied. The results show that models with a non-proportional and non-homothetical second load paths seem to be more efficient.
Millard F. Beatty - One of the best experts on this subject based on the ideXlab platform.
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Small amplitude, free longitudinal vibrations of a load on a finitely deformed Stress-Softening spring with limiting extensibility
Zeitschrift für angewandte Mathematik und Physik, 2009Co-Authors: Millard F. Beatty, Ranjan Bhattacharyya, Somnath SarangiAbstract:A constitutive theory for a general class of incompressible, isotropic Stress-Softening, limited elastic rubberlike materials is introduced. The model is applied to study the small amplitude, free longitudinal vibrational frequency of a load about a suspended static equilibrium stretch of a finitely deformed, Stress-Softening spring with limiting extensibility. A number of physical results, including bounds on the frequency, are reported. It is proved, for example, that the normalized vibrational frequency for the ideally elastic neo-Hookean oscillator is a lower bound for the normalized frequency of every incompressible, isotropic Stress-Softening, limited elastic oscillator within the general class. All results are illustrated for the special limited elastic Gent and the purely elastic Demiray biomaterial models, both with Stress-Softening characterized by a Zuniga–Beatty front factor damage function. The results for both Stress-Softening models are compared with experimental data for several gum rubbers and thoracic aortic tissue provided by others; and, overall, it is found that the Stress-Softening, limited elastic Gent model best characterizes the data.
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Stress-Softening Effects in the Transverse Vibration of a Non-Gaussian Rubber String
Meccanica, 2003Co-Authors: Alex Elías-zúñiga, Millard F. BeattyAbstract:The Mullins effect in the small amplitude transverse vibration of a rubber cord is investigated. The fundamental frequency is determined for a specific class of Stress-Softening materials. Analytical relations for the cord vibration frequency are illustrated graphically for three phenomenological models. These results demonstrate the role of the material parameters and exhibit response characteristic of those reported in experiments by others and subsequently described here in new experiments. Frequency versus stretch results for two kinds of non-Gaussian molecular network models for rubber elasticity are compared with experimental data for four varieties of rubber cords, for each of which only three experimentally determined material constants are needed. It is shown that the theoretical predictions stand in excellent agreement with test data.
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Stress-Softening Effects in the Vibration of a Non-Gaussian Rubber Membrane
Mathematics and Mechanics of Solids, 2003Co-Authors: Millard F. Beatty, Alex Elías-zúñigaAbstract:The Mullins effect in the small amplitude transverse vibration of a stretched rubber membrane is investigated. The fundamental frequency, which decreases with increasing Softening, is determined for a specific class of Stress-Softening materials. Analytical relations for the membrane vibration frequency are illustrated graphically for three phenomenological models and two kinds of non-Gaussian molecular network models for rubber elasticity. The results demonstrate the role of the material parameters and, although no experimental data for the vibration of a rubber membrane currently are known, the theoretical predictions are characteristic of the frequency-stretch response reported in vibration experiments on rubber cords.
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A new phenomenological model for Stress-Softening in elastomers
Zeitschrift für angewandte Mathematik und Physik, 2002Co-Authors: A. E. Zúñiga, Millard F. BeattyAbstract:A new phenomenological model for Stress-Softening of isotropic, incompressible hyperelastic rubberlike materials is presented. For any specified virgin material constitutive equation, the Stress-softened material response due to microstructural damage is characterized by an exponential Softening function that depends on the current magnitude of strain and its maximum previous value in a deformation of the virgin material. The theory is illustrated for a neo-Hookean material; and it is shown that results derived for two non-Gaussian molecular network material models compare most favorably with uniaxial extension data provided by others.
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Damage Induced Stress‐Softening in the Torsion, Extension and Inflation of a Cylindrical Tube
The Quarterly Journal of Mechanics and Applied Mathematics, 2001Co-Authors: Shankar Krishnaswamy, Millard F. BeattyAbstract:The non-homogeneous deformation of combined torsion, extension and inflation of a Stress-Softening cylindrical tube is discussed within the framework of a theory of isotropic Stress-Softening in incompressible isotropic materials. The theory is based on the idea that the Stress-Softening material is an inelastic material that remembers only the maximum previous deformation to which it has been subjected. This selective memory dependence is incorporated within general material response functions that are monotone decreasing functions of a Stress-Softening variable; the latter is a monotone increasing function of the maximum previous strain experienced by the material. Results demonstrating the effects of Stress-Softening are obtained for general Stress-Softening materials in combined torsion, extension and inflation. A special analytical model is used to illustrate some general results as well as to provide graphical examples. By deforming a cylinder successively, first in simple torsion and then in uniaxial extension, the concept of deformation-induced inhomogeneity is presented. Finally, it is shown that the overall effects of Stress-Softening in pure torsion are much smaller than the corresponding effects in uniaxial extension.
Alex Elías-zúñiga - One of the best experts on this subject based on the ideXlab platform.
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On the Rule of Mixtures for Predicting Stress-Softening and Residual Strain Effects in Biological Tissues and Biocompatible Materials
Materials (Basel Switzerland), 2014Co-Authors: Alex Elías-zúñiga, Oscar Martínez-romero, Karen Baylón, I. Ferrer, Lídia Serenó, Maria Luisa Garcia-romeu, Isabel Bagudanch, J. Grabalosa, Tania Pérez-recio, Wendy Ortega-laraAbstract:In this work, we use the rule of mixtures to develop an equivalent material model in which the total strain energy density is split into the isotropic part related to the matrix component and the anisotropic energy contribution related to the fiber effects. For the isotropic energy part, we select the amended non-Gaussian strain energy density model, while the energy fiber effects are added by considering the equivalent anisotropic volumetric fraction contribution, as well as the isotropized representation form of the eight-chain energy model that accounts for the material anisotropic effects. Furthermore, our proposed material model uses a phenomenological non-monotonous Softening function that predicts Stress Softening effects and has an energy term, derived from the pseudo-elasticity theory, that accounts for residual strain deformations. The model’s theoretical predictions are compared with experimental data collected from human vaginal tissues, mice skin, poly(glycolide-co-caprolactone) (PGC25 3-0) and polypropylene suture materials and tracheal and brain human tissues. In all cases examined here, our equivalent material model closely follows Stress-Softening and residual strain effects exhibited by experimental data.
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Stress-Softening and Residual Strain Effects in Suture Materials
Advances in Materials Science and Engineering, 2013Co-Authors: Alex Elías-zúñiga, Beatriz Montoya, Wendy Ortega-lara, Eduardo Flores-villalba, Ciro A. Rodríguez, Héctor R. Siller, José Antonio Díaz-elizondo, Oscar Martínez-romeroAbstract:This work focuses on the experimental characterization of suture material samples of MonoPlus, Monosyn, polyglycolic acid, polydioxanone 2–0, polydioxanone 4–0, poly(glycolide-co-epsilon-caprolactone), nylon, and polypropylene when subjected to cyclic loading and unloading conditions. It is found that all tested suture materials exhibit Stress-Softening and residual strain effects related to the microstructural material damage upon deformation from the natural, undistorted state of the virgin suture material. To predict experimental observations, a new constitutive material model that takes into account Stress-Softening and residual strain effects is developed. The basis of this model is the inclusion of a phenomenological nonmonotonous Softening function that depends on the strain intensity between loading and unloading cycles. The theory is illustrated by modifying the non-Gaussian average-stretch, full-network model to capture Stress-Softening and residual strains by using pseudoelasticity concepts. It is shown that results obtained from theoretical simulations compare well with suture material experimental data.
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A non-monotonous damage function to characterize Stress-Softening effects with permanent set during inflation and deflation of rubber balloons
International Journal of Engineering Science, 2010Co-Authors: Alex Elías-zúñiga, Ciro A. RodríguezAbstract:A non-monotonous Stress-Softening phenomenological model is applied to study the Mullins effect with residual strains to characterize the inflation and deflation of rubber balloons. It is shown that analytical predictions based on our proposed non-monotonous Softening function and the modified Stress-Softening non-Gaussian average-stretch full-network constitutive equation that accounts for residual strains are consistent with experimental data. Also, we use the constitutive equation for equibiaxial extension to predict Stress-Softening behavior in a kinematically equivalent simple compression deformation state.
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A phenomenological energy-based model to characterize Stress-Softening effect in elastomers
Polymer, 2005Co-Authors: Alex Elías-zúñigaAbstract:Abstract A phenomenological energy-based model for Stress-Softening of isotropic, incompressible hyperelastic rubberlike materials is derived here. In this model, the microstructural damage is characterized by an exponential Softening function that depends on the current magnitude of the strain–energy function and its maximum previous value in a deformation of the virgin material. Theoretical models are presented for uniaxial, equibiaxial and pure shear deformations by using Gaussian and non-Gaussian material molecular network models. The accuracy of the resulting constitutive equations is demonstrated on uniaxial, equibiaxial and pure shear experimental data provided in the literature. Comparisons between the energy-based model and the strain intensity based phenomenological model described in [Elias-Zuniga A, Beatty MF. ZAMP 2002;53:794–814. [1] ] show that the model developed here is slightly superior in following experimental data.
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Stress-Softening Effects in the Transverse Vibration of a Non-Gaussian Rubber String
Meccanica, 2003Co-Authors: Alex Elías-zúñiga, Millard F. BeattyAbstract:The Mullins effect in the small amplitude transverse vibration of a rubber cord is investigated. The fundamental frequency is determined for a specific class of Stress-Softening materials. Analytical relations for the cord vibration frequency are illustrated graphically for three phenomenological models. These results demonstrate the role of the material parameters and exhibit response characteristic of those reported in experiments by others and subsequently described here in new experiments. Frequency versus stretch results for two kinds of non-Gaussian molecular network models for rubber elasticity are compared with experimental data for four varieties of rubber cords, for each of which only three experimentally determined material constants are needed. It is shown that the theoretical predictions stand in excellent agreement with test data.
Stephen R. Rickaby - One of the best experts on this subject based on the ideXlab platform.
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transversely isotropic cyclic Stress Softening model for the mullins effect
arXiv: Soft Condensed Matter, 2020Co-Authors: Stephen R. Rickaby, N H ScottAbstract:This paper models Stress Softening during cyclic loading and unloading of an elastomer. The paper begins by remodelling the primary loading curve to include a Softening function and goes on to derive non-linear transversely isotropic constitutive equations for the elastic response, Stress relaxation, residual strain and creep of residual strain. These ideas are combined with a transversely isotropic version of the Arruda-Boyce eight-chain model to develop a constitutive relation that is capable of accurately representing the Mullins effect during cyclic Stress-Softening for a transversely isotropic, hyperelastic material, in particular a carbon-filled rubber vulcanizate. Keywords: Mullins effect, Stress-Softening, hysteresis, Stress relaxation, residual strain, creep of residual strain, transverse isotropy. MSC codes: 74B20, 74D10, 74L15
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Orthotropic cyclic Stress-Softening model for pure shear during repeated loading and unloading
IMA Journal of Applied Mathematics, 2014Co-Authors: Stephen R. Rickaby, Nigel ScottAbstract:We derive an orthotropic model to describe the cyclic Stress Softening of a carbon-filled rubber vulcanizate through multiple Stress-strain cycles with increasing values of the maximum strain. We specialize the deformation to pure shear loading. As a result of strain-induced anisotropy following on from initial primary loading, the material may subsequently be described as orthotropic because in pure shear there are three different principal stretches so that the strain-induced anisotropy of the Stress response is different in each of these three directions. We derive non-linear orthotropic models for the elastic response, Stress relaxation and residual strain to model accurately the inelastic features associated with cyclic Stress Softening. We then develop an orthotropic version of the Arruda-Boyce eight-chain model of elasticity and then combine it with the ideas previously developed in this paper to produce an orthotropic constitutive relation for the cyclic Stress Softening of a carbon-filled rubber vulcanizate. The model developed here includes the widely occurring effects of hysteresis, Stress-relaxation and residual strain. The model is found to compare well with experimental data.
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A cyclic Stress Softening model for the Mullins effect
International Journal of Solids and Structures, 2013Co-Authors: Stephen R. Rickaby, Nigel ScottAbstract:Abstract In this paper the inelastic features of Stress relaxation, hysteresis and residual strain are combined with the Arruda–Boyce eight-chain model of elasticity, in order to develop a model that is capable of describing the Mullins effect for cyclic Stress-Softening of an incompressible hyperelastic material, in particular a carbon-filled rubber vulcanizate. We have been unable to identify in the literature any other model that takes into consideration all the above inelastic features of the cyclic Stress-Softening of carbon-filled rubber. Our model compares favourably with experimental data and gives a good description of Stress-Softening, hysteresis, Stress relaxation, residual strain and creep of residual strain.
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Cyclic Stress-Softening model for the Mullins effect in compression
International Journal of Non-Linear Mechanics, 2013Co-Authors: Stephen R. Rickaby, Nigel ScottAbstract:Abstract This paper models the cyclic Stress Softening of an elastomer in compression. After the initial compression the material is described as being transversely isotropic. We derive non-linear transversely isotropic constitutive equations for the elastic response, Stress relaxation, residual strain, and creep of residual strain in order to model accurately the inelastic features associated with cyclic Stress Softening. These equations are combined with a transversely isotropic version of the Arruda–Boyce eight-chain model to develop a constitutive relation that is capable of accurately representing the Mullins effect during cyclic Stress Softening for a transversely isotropic, hyperelastic material, in particular a carbon-filled rubber vulcanizate. To establish the validity of the model we compare it with two test samples, one for filled vulcanized styrene–butadiene rubber and the other for filled vulcanized natural rubber. The model is found to fit this experimental data extremely well.
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Transversely isotropic cyclic Stress-Softening model for the Mullins effect
Proceedings of the Royal Society A: Mathematical Physical and Engineering Sciences, 2012Co-Authors: Stephen R. Rickaby, Nigel ScottAbstract:This paper models Stress Softening during cyclic loading and unloading of an elastomer. The paper begins by remodelling the primary loading curve to include a Softening function and goes on to derive nonlinear transversely isotropic constitutive equations for the elastic response, Stress relaxation, residual strain and creep of residual strain. These ideas are combined with a transversely isotropic version of the Arruda–Boyce eight-chain model to develop a constitutive relation that is capable of accurately representing the Mullins effect during cyclic Stress Softening for a transversely isotropic, hyperelastic material, in particular, a carbon-filled rubber vulcanizate.
