https://sam.ensam.eu:443 The DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material. Mon, 13 Jul 2026 06:32:39 GMT 2026-07-13T06:32:39Z http://hdl.handle.net/10985/8156 Constitutive modeling of the anisotropic behavior of Mullins softened filled rubbers MERCKEL, Yannick; DIANI, Julie; BRIEU, Mathias; CAILLARD, Julien Original constitutive modeling is proposed for filled rubber materials in order to capture the anisotropic softened behavior induced by general non-proportional pre-loading histo-ries. The hyperelastic framework is grounded on a thorough analysis of cyclic experimental data. The strain energy density is based on a directional approach. The model leans on the strain amplification factor concept applied over material directions according to the Mul-lins softening evolution. In order to provide a model versatile that applies for a wide range of materials, the proposed framework does not require to postulate the mathematical forms of the elementary directional strain energy density and of the Mullins softening evo-lution rule. A computational procedure is defined to build both functions incrementally from experimental data obtained during cyclic uniaxial tensile tests. Successful compari-sons between the model and the experiments demonstrate the model abilities. Moreover, the model is shown to accurately predict the non-proportional uniaxial stress-stretch responses for uniaxially and biaxially pre-stretched samples. Finally, the model is effi-ciently tested on several materials and proves to provide a quantitative estimate of the anisotropy induced by the Mullins softening for a wide range of filled rubbers. Sun, 01 Jan 2012 00:00:00 GMT http://hdl.handle.net/10985/8156 2012-01-01T00:00:00Z MERCKEL, Yannick DIANI, Julie BRIEU, Mathias CAILLARD, Julien Original constitutive modeling is proposed for filled rubber materials in order to capture the anisotropic softened behavior induced by general non-proportional pre-loading histo-ries. The hyperelastic framework is grounded on a thorough analysis of cyclic experimental data. The strain energy density is based on a directional approach. The model leans on the strain amplification factor concept applied over material directions according to the Mul-lins softening evolution. In order to provide a model versatile that applies for a wide range of materials, the proposed framework does not require to postulate the mathematical forms of the elementary directional strain energy density and of the Mullins softening evo-lution rule. A computational procedure is defined to build both functions incrementally from experimental data obtained during cyclic uniaxial tensile tests. Successful compari-sons between the model and the experiments demonstrate the model abilities. Moreover, the model is shown to accurately predict the non-proportional uniaxial stress-stretch responses for uniaxially and biaxially pre-stretched samples. Finally, the model is effi-ciently tested on several materials and proves to provide a quantitative estimate of the anisotropy induced by the Mullins softening for a wide range of filled rubbers. http://hdl.handle.net/10985/17617 A Mullins softening criterion for general loading conditions MERCKEL, Yannick; BRIEU, Mathias; DIANI, Julie; CAILLARD, Julien Samples of carbon-black filled styrene butadiene rubbers (SBRs) were submitted to successive nonproportional loadings in order to define a general criterion for the Mullins softening. For this purpose, each sample was initially submitted to uniaxial or biaxial preloadings followed by a cyclic uniaxial tension test. An original experimental analysis aimed at defining the activation threshold for the Mullins softening during cyclic uniaxial loadings. The experimental data provide substantial evidences establishing the surface of the maximum directional stretch undergone by the material as a relevant Mullins softening criterion. The latter was used to successfully predict the Mullins softening surfaces for additional loading cases. Sun, 01 Jan 2012 00:00:00 GMT http://hdl.handle.net/10985/17617 2012-01-01T00:00:00Z MERCKEL, Yannick BRIEU, Mathias DIANI, Julie CAILLARD, Julien Samples of carbon-black filled styrene butadiene rubbers (SBRs) were submitted to successive nonproportional loadings in order to define a general criterion for the Mullins softening. For this purpose, each sample was initially submitted to uniaxial or biaxial preloadings followed by a cyclic uniaxial tension test. An original experimental analysis aimed at defining the activation threshold for the Mullins softening during cyclic uniaxial loadings. The experimental data provide substantial evidences establishing the surface of the maximum directional stretch undergone by the material as a relevant Mullins softening criterion. The latter was used to successfully predict the Mullins softening surfaces for additional loading cases. http://hdl.handle.net/10985/8196 Effects of the Amount of Fillers and of the Crosslink Density on the Mechanical Behavior of Carbon-Black Filled Styrene Butadiene Rubbers MERCKEL, Yannick; DIANI, Julie; BRIEU, Mathias; CAILLARD, Julien Several carbon-black filled styrene-butadiene rubbers are subjected to monotonic uniaxial tension tests in order to investi-gate the effects of the amount of fillers and of the crosslink density on their mechanical properties. The Young modulus, the volume changes associated with material damage and the stretch to failure are extracted and discussed. Results compare well to the literature results when exist and quantitative analysis are proposed when possible. Results show that filled rubbers are not incompressible when submitted to uniaxial tension tests and their volume changes are strongly dependent of the amount of fillers but are unaffected by the crosslink density. The latter shows strong impact on the filled rubbers stretch to failure but more interestingly this impact is com-parable to what is encountered in unfilled rubbers. The stretch to failure is improved by the addition of fillers with an optimum for material filled around 30 phr. Tue, 01 Jan 2013 00:00:00 GMT http://hdl.handle.net/10985/8196 2013-01-01T00:00:00Z MERCKEL, Yannick DIANI, Julie BRIEU, Mathias CAILLARD, Julien Several carbon-black filled styrene-butadiene rubbers are subjected to monotonic uniaxial tension tests in order to investi-gate the effects of the amount of fillers and of the crosslink density on their mechanical properties. The Young modulus, the volume changes associated with material damage and the stretch to failure are extracted and discussed. Results compare well to the literature results when exist and quantitative analysis are proposed when possible. Results show that filled rubbers are not incompressible when submitted to uniaxial tension tests and their volume changes are strongly dependent of the amount of fillers but are unaffected by the crosslink density. The latter shows strong impact on the filled rubbers stretch to failure but more interestingly this impact is com-parable to what is encountered in unfilled rubbers. The stretch to failure is improved by the addition of fillers with an optimum for material filled around 30 phr. http://hdl.handle.net/10985/6521 Volume changes in a filled elastomer studied via digital image correlation DE CREVOISIER, Jordan; BESNARD, Gilles; MERCKEL, Yannick; ZHANG, Huan; CAILLARD, Julien; VION-LOISEL, Fabien; BERGHEZAN, Daniel; CRETON, Costantino; DIANI, Julie; BRIEU, Mathias; HILD, François; ROUX, Stéphane Volumetric strains in a filled SBR specimen subjected to cyclic uniaxial tension with increasing extensions are studied. Digital image correlation is used to follow the kinematics of two orthogonal free faces. A volume expansion is observed past a critical elongation, which can be interpreted as the onset of cavitation. Under unloading, the volume returns to its original value and remains constant upon reloading. Increasing the elongation to higher values than the previous cycle leads again to a volumetric expansion. Sun, 01 Jan 2012 00:00:00 GMT http://hdl.handle.net/10985/6521 2012-01-01T00:00:00Z DE CREVOISIER, Jordan BESNARD, Gilles MERCKEL, Yannick ZHANG, Huan CAILLARD, Julien VION-LOISEL, Fabien BERGHEZAN, Daniel CRETON, Costantino DIANI, Julie BRIEU, Mathias HILD, François ROUX, Stéphane Volumetric strains in a filled SBR specimen subjected to cyclic uniaxial tension with increasing extensions are studied. Digital image correlation is used to follow the kinematics of two orthogonal free faces. A volume expansion is observed past a critical elongation, which can be interpreted as the onset of cavitation. Under unloading, the volume returns to its original value and remains constant upon reloading. Increasing the elongation to higher values than the previous cycle leads again to a volumetric expansion. http://hdl.handle.net/10985/18692 Constitutive modeling of the anisotropic behavior of Mullins softened filled rubbers MERCKEL, Yannick; DIANI, Julie; BRIEU, Mathias; CAILLARD, Julien Original constitutive modeling is proposed for filled rubber materials in order to capture the anisotropic softened behavior induced by general non-proportional pre-loading histories. The hyperelastic framework is grounded on a thorough analysis of cyclic experimental data. The strain energy density is based on a directional approach. The model leans on the strain amplification factor concept applied over material directions according to the Mullins softening evolution. In order to provide a model versatile that applies for a wide range of materials, the proposed framework does not require to postulate the mathematical forms of the elementary directional strain energy density and of the Mullins softening evolution rule. A computational procedure is defined to build both functions incrementally from experimental data obtained during cyclic uniaxial tensile tests. Successful comparisons between the model and the experiments demonstrate the model abilities. Moreover, the model is shown to accurately predict the non-proportional uniaxial stress-stretch responses for uniaxially and biaxially pre-stretched samples. Finally, the model is efficiently tested on several materials and proves to provide a quantitative estimate of the anisotropy induced by the Mullins softening for a wide range of filled rubbers. Tue, 01 Jan 2013 00:00:00 GMT http://hdl.handle.net/10985/18692 2013-01-01T00:00:00Z MERCKEL, Yannick DIANI, Julie BRIEU, Mathias CAILLARD, Julien Original constitutive modeling is proposed for filled rubber materials in order to capture the anisotropic softened behavior induced by general non-proportional pre-loading histories. The hyperelastic framework is grounded on a thorough analysis of cyclic experimental data. The strain energy density is based on a directional approach. The model leans on the strain amplification factor concept applied over material directions according to the Mullins softening evolution. In order to provide a model versatile that applies for a wide range of materials, the proposed framework does not require to postulate the mathematical forms of the elementary directional strain energy density and of the Mullins softening evolution rule. A computational procedure is defined to build both functions incrementally from experimental data obtained during cyclic uniaxial tensile tests. Successful comparisons between the model and the experiments demonstrate the model abilities. Moreover, the model is shown to accurately predict the non-proportional uniaxial stress-stretch responses for uniaxially and biaxially pre-stretched samples. Finally, the model is efficiently tested on several materials and proves to provide a quantitative estimate of the anisotropy induced by the Mullins softening for a wide range of filled rubbers. http://hdl.handle.net/10985/6804 Charcaterization of the Mullins effect of carbon-black filled rubbers MERCKEL, Yannick; DIANI, Julie; BRIEU, Mathias; GILORMINI, Pierre; CAILLARD, Julien Several carbon-black filled styrene-butadiene rubbers showed different sensibilities to the Mullins softening when submitted to cyclic uniaxial tension. In order to quantify this softening, a damage parameter was introduced. It is defined by using a classic damage approach and can be estimated by using either the strain amplification factor method or the tangent modulus at zero stress. The proposed parameter is used to study the effects of crosslink density and filler amount on the Mullins softening. The latter is shown to remain unaffected by a change of crosslink density and to increase with an increase of filler amount. The damage parameter exhibits mere linear dependences on the maximum Hencky strain applied and on the filler volume fraction. A simple linear expression is given finally to predict the Mullins softening of filled rubbers. The parameter also provides an objective analysis for the Mullins softening that supports comments on a better understanding of this effect. Publisher version : http://rubberchemtechnol.org/doi/abs/10.5254/1.3592294?journalCode=rcat Sat, 01 Jan 2011 00:00:00 GMT http://hdl.handle.net/10985/6804 2011-01-01T00:00:00Z MERCKEL, Yannick DIANI, Julie BRIEU, Mathias GILORMINI, Pierre CAILLARD, Julien Several carbon-black filled styrene-butadiene rubbers showed different sensibilities to the Mullins softening when submitted to cyclic uniaxial tension. In order to quantify this softening, a damage parameter was introduced. It is defined by using a classic damage approach and can be estimated by using either the strain amplification factor method or the tangent modulus at zero stress. The proposed parameter is used to study the effects of crosslink density and filler amount on the Mullins softening. The latter is shown to remain unaffected by a change of crosslink density and to increase with an increase of filler amount. The damage parameter exhibits mere linear dependences on the maximum Hencky strain applied and on the filler volume fraction. A simple linear expression is given finally to predict the Mullins softening of filled rubbers. The parameter also provides an objective analysis for the Mullins softening that supports comments on a better understanding of this effect. http://hdl.handle.net/10985/6805 Effect of the microstructure parameters on the Mullins softening in carbon-black filled SBRs MERCKEL, Yannick; DIANI, Julie; BRIEU, Mathias; GILORMINI, Pierre; CAILLARD, Julien A quantitative estimate of the Mullins softening is proposed and tested on various carbon-black filled styrene-butadiene rubbers. In order to model the behaviour of elastomeric materials, some constitutive equations reported in the literature are based on the account of a strain amplification factor, which evolves with the maximum strain history. The amplification factor is grounded on the representation of filled rubbers as heterogeneous materials made of hard rigid domains and soft deformable domains. In the present work, this factor is splitted into two parts with opposite effects that account for the Mullins softening and for the filler reinforcement, respectively. Evolutions of both parts are obtained through a direct analysis of cyclic uniaxial tensile tests performed on a series of materials. The Mullins softening part is shown to linearly depend on the filler volume fraction and on the maximum strain applied, when defined as the first invariant of the Hencky tensor. Its changes with the gum crosslink density parameter are insignificant. The reinforcement part of the amplification factor shows quadratic dependence on the filler volume fraction. [The definitive version is available at www3.interscience.wiley.com] Sun, 01 Jan 2012 00:00:00 GMT http://hdl.handle.net/10985/6805 2012-01-01T00:00:00Z MERCKEL, Yannick DIANI, Julie BRIEU, Mathias GILORMINI, Pierre CAILLARD, Julien A quantitative estimate of the Mullins softening is proposed and tested on various carbon-black filled styrene-butadiene rubbers. In order to model the behaviour of elastomeric materials, some constitutive equations reported in the literature are based on the account of a strain amplification factor, which evolves with the maximum strain history. The amplification factor is grounded on the representation of filled rubbers as heterogeneous materials made of hard rigid domains and soft deformable domains. In the present work, this factor is splitted into two parts with opposite effects that account for the Mullins softening and for the filler reinforcement, respectively. Evolutions of both parts are obtained through a direct analysis of cyclic uniaxial tensile tests performed on a series of materials. The Mullins softening part is shown to linearly depend on the filler volume fraction and on the maximum strain applied, when defined as the first invariant of the Hencky tensor. Its changes with the gum crosslink density parameter are insignificant. The reinforcement part of the amplification factor shows quadratic dependence on the filler volume fraction.