Abstract
The stress softening phenomenon of Mullins effect is studied by investigating mechanical properties of Fe (abbreviation of Iron) powder-modified rubber composites via experiments and numerical simulation methods. The distinct reinforcements of composites are found experimentally on the elastic modulus and the softened stress of Mullins effect. Then, a representative volume method: unit cell model method is selected to simulate the particle-reinforced composites, and investigate the intrinsic mechanism in the viewpoint of mechanics. It is found that due to the existence of rigid inclusion, the surrounding strain distribution field is changed and the continuous increasing stress characteristic of the matrix acts an important role in the mechanical reinforcement. The proposed mechanism based on it, can account for the increasing softened stress of Mullins effect for the composites. This finding can be also extended to other rigid particle-reinforced rubber composite, which is confirmed by numerically simulating.
Keywords
Introduction
Rubbers are widely used polymers in engineering applications as dampers and tires for their merits like large elasticity and high failure strains [1, 2]. With the modification of particle, the hydrodynamics/volume fraction effects, reinforcing fillers such as carbon blacks or rigid silica impart great improvement in mechanical properties of rubber matrix composites. They present an evident stress softening phenomenon when subjected to cycling stretching loads, which is known as the Mullins effect [3]. It was first attributed to the transformation of hard domains within the rubber into soft domains while stretching, and later linked to the rupture of bonds between particles [3, 4]. Researchers also had developed different damage mechanisms, like slippage of molecules and chain disentanglement, filler fragmentation and loss of bonding between filler and matrix to explain the Mullins effect [5-8]. Some methodologies were also developed based on the finite element method to investigate the influence of physical interactions [9, 10]. The mechanism was further investigated by focusing on filler network interactions [11] and molecular chain disentanglement view [12]. Whereas, limited literature is currently available to find a definite reinforcement mechanism for the softened stress of Mullins effects. It may be attributed to the complex microstructure of molecules and different chains segments, and even their coupling chemical and physical interactions and evolution during deformation [13-15].
The intrinsic mechanisms should be well known in advance for efficiently designing the rubber matrix composites. The representative volume elements methods, like the unit cell model, offers a good method, which has been getting extensive popularity in analysing the mechanism for composites [16], especially the particle-reinforced composites. Based on the unit cell modelling, Li et al. [17] numerically studied the influence of particle volume characteristics on the metal matrix composites, and found that rate-dependent flow stress was influenced by particle aspect ratio and particle shape. Since then, various unit cell methods were extensively employed to investigate thermo/mechanical behaviours of particle-modified composites, such as ceramic-particle-modified polymer [18], the elastic–plastic behaviour of a porous hierarchical scaffold [19] and the thermomechanical behaviour of nanocomposites [20]. It was even used to investigate the particle size dependence of the elastic–plastic stress–strain response of polymer/clay nanocomposites [21], and reinforcement dependence on the constitutive characteristics of matrix [22]. Therefore, the unit cell method is an efficient way to study the potential reinforcement effect and the intrinsic mechanisms.
In this work, the stress softening phenomenon from the Mullins effect is studied, with aims to understand the mechanical characteristics and the mechanisms, and the unit cell method is selected. The rigid Fe powder is selected to reinforce the rubber matrix. Their mechanical characteristics are analysed in aspects of the elastic modulus and softened stress from the Mullins effect. The mechanism is proposed, with the aid of unit cell modelling, for the rigid particle-reinforced composites, and also is used to further explain the dependence of softened stress on the particle fraction and strain.
Materials and experiments
Material preparation
The investigated rubber is a vulcanised Ethylene Propylene Diene Monomer (short for EPDM) Rubber bought from Shanghai Fuyou international trading Co., Ltd. Fe powder is obtained from China Institute of metallurgy, with the nominal diameter 500 nm. An open mill machine is used to mix Fe powder into the rubber. Then, the mixture is cured in a mold for 10 min under the coupling ambient conditions of 15 MPa and 150°C, then the rubber and Fe-reinforced rubber are obtained in plate form with ∼2 mm in the thickness. Different composites are made with three volume fractions: 0, 2.5 and 7.0%, respectively. The specimens are cut from the plate using a cutting press with a dumbbell shape as shown in Figure 1(a). The gauge length section is designed to be 5 mm in width and 20 mm in length as shown in Figure 1(b).
Dumbbell punch and specimen.
Mechanical experiments
The uniaxial tension tests are conducted by Micro Instron 5848 machine with the load cell of the capacity 100 N, and the attached video extensometer to in-situ measure strain of specimen. The cyclic tests are designed with the increasing maximum strain to investigate the softening from the Mullins effect. The nominal strain rate 0.1 s−1 is achieved by controlling the loading velocity of the crosshead. The repeating tests are conducted to ensure the experimental reliability and the dispersion of mechanical properties of specimen.
Unit cell analysis formulation
Unit cell modelling analysis is conducted using Abaqus explicit 6.14-2 software. Two-dimensional axisymmetric model is selected to model the particle-reinforced composites, with the schematic model as shown in Figure 2. The periodic boundary is introduced on the line BC, which is used to ensure the cell to remain parallel during shrinking induced by tension. The symmetric boundary is assigned on line OB. With the aids of the boundaries, the unit cell can be extended in three dimensions directions to establish a bulk composites. It is a normalised analytical model, and the actual size such as the radius and height are 5 mm in actual simulating, but presents little significance on the simulation results. The uniformly distributed velocity is applied on the top line AC, with the tension velocity 0.01 m s−1. The automatic incrementation type is selected in the Abaqus edit step, and the time period is assigned to be 2. The inclusion zone size is calculated based on the investigated volume fraction of Fe particle, and the inclusion is partitioned out from the unit cell. The approximate global size is 0.0005 in the sizing control, and the cell is meshed into at least 2700 elements, with the type of CAX4R (a 4-node bilinear axisymmetric quadrilateral, reduced integration, hourglass control). The Ogden model is selected for the rubber matrix, and the uniaxial test data are from experimental results of pure rubber as shown in Figure 5(a). Its Poisson's ratio is assumed to be 0.49, and the density is 900 kg m−3. The Fe particle as inclusion is rather more rigid than the rubber matrix and assumed to linear elastic constitutive relation with the elastic modulus 210 GPa, Poisson ratio 0.3, and density 7830 kg m−3 [22]. Additionally, the analyses of mesh sensitivity are conducted to ensure that the stress–strain relations have little distinction after comparing the further mesh refinements. The true stress–strain curves of the composites are then calculated based on the reaction forces and displacement from the simulation results.
Schematics of two-dimensional unit cell model.
Results and discussion
After performing the tension experimenting on rubber and its composites, the stress–strain curves can be calculated based on the load and strain from the video extensometer, or the displacement of the machine. With the constant volume assumption for specimen during tension process, the true strain
and stress
are obtained as a function of engineering strain
and stress
according to the following equation:
The obtained true stress–strain curves are compared as shown in Figure 3, with the strain from the video extensometer and machine crosshead displacement. It is clear to find that the difference is distinct. While loading, the rigid components of the machine will produce certain displacement which will be misconstrued to a part of the crosshead displacement, and the specimen strain measurement will be then interfered with it. Then the strain from the video extensometer is used due to its in-situ strain measurement with a high accuracy. The repeating experimental results for the Mullins effect are shown in Figure 4. It indicates that the curves are matching well, which help to ensure the experimental reliability and the little dispersion in the mechanical properties of the composites specimen.
Comparisons of true stress–strain curves of a polymer material with strain from video extensometer and crosshead displacement of Instron 5848 machine. Repeating experimental results: true stress–strain curves of composites.
Reinforced mechanical properties
The true stress–strain curves of the rubber and composites are shown in Figure 5(a–c), respectively. The curves from cycling tests are also obtained for the reinforced Mullins effect, and shown in Figure 6(a–c), respectively. The softened stress is defined by the differential stress between the tension stress and recovering stress at certain strain, which is labelled
True stress–strain curves of rubber and composites under strain rate of 0.1 s−1. Mullins effect curves of rubber and composites under strain rate of 0.1 s−1.
σεi as depicted in Figure 6(c). The stress–strain curves of rubber and its composites, as shown in Figure 5, demonstrate the same trend: the initial linear elasticity and non-linearly concave increment. The particle inclusion presents a distinct reinforcement in the elastic modulus from 0.84 ± 0.04, 1.69 ± 0.17 to 2.05 ± 0.24 MPa corresponding to pure rubber, 2.5 and 7% fraction, respectively. The reinforcement is also found on the softened stress as indicated from Figure 6, from 0.62 ± 0.02, 0.85 ± 0.03a and 2.70 ± 0.03 MPa corresponding to pure rubber, 2.5 and 7% fraction at true strain 120%, respectively. The similar trend is also found of the softened stress dependence on the fraction in reference [7, 12].
Mechanism investigation
A mechanic viewpoint is selected to analyse the potential mechanism. After simulating composites with different fraction using the unit cell model, the loads and displacements are output, then the true stress–strain curves are obtained. The elastic modulus is obtained to be 2.16 MPa for 7% volume fraction. The error is 5.4% after comparing with the experimental result of 2.05 MPa. Thus, it demonstrates the modelling reliability.
Mechanism for stress–strain curves
The deformation contours are outputted are shown in Figure 7(b) in the pattern of LE22 true strain of the unit cell, where the model is for 2.5% fraction composite when the overall strain of the cell is 15%. The overall strain is the unit cell overall strain calculated from the extension displacement and Equation (1). The strain distributions are not uniform as shown in Figure 7(b). A strain localisation zone is observed outside of the inclusion north pole. It is just for the severely mismatched mechanical properties of Fe particle and the rubber matrix (210 GPa/0.88 MPa in elastic modulus aspect), and then the deformation of the rigid inclusion is tiny even though the overall strain of composite is considerable. The strain along Y axial in the centre axle line is output and normalised by the overall strain of the cell, which is listed in Figure 7(a). The distance is also normalised by the radius of inclusion, and plotted in X axial as shown in Figure 7(a). The normalised maximum strain in the matrix can be up to 1.82 for 2.5% fraction composite. The constitutive behaviour of the matrix rubber can account for it. It is predicted from the concave stress–strain curve of rubber in Figure 5(a), that the strain localisation zones have come into higher strain and stand a higher stress level than the rest matrix zone, then the localisation zone presents a higher deformation resistance and contribute a lot to the reinforcement in the elastic modulus and the overall stress–strain curve of the composites. So, it is concluded that it is the higher localisation strain that acts a distinct reinforcement role for the rigid particle-reinforced composites. The localised strain is more severe with the volume fraction increasing, due to less matrix participation into the deformation. It is confirmed by Figure 8(b) for 7.0% fraction composite. Its strain concentration is more severe with the normalised maximum strain up to 2.18 as shown in by Figure 8(a). Thus, more matrix has come into the higher strain deformation, and contribute more to the overall mechanical properties of the cell. It is predicted that with overall deformation increasing, the reinforcement effect is more and more distinct due to the concave characteristic of matrix rubber, which is confirmed experimentally by the higher stress increasing characteristics of curves at subsequent deformations in Figure 5(b,c).
True strain from the elements on the left hand edge of the unit cell model (a) and strain distribution contour for 2.5% fraction composite under the overall strain of unit cell 15% (b). True strain from the elements on the left hand edge of the unit cell model (a) and strain distribution contour for 7.0% fraction composite under the overall strain of unit cell 15% (b).
Mechanism for the softened stress
It is indicated from Figure 6 that the softened stress presents a high dependence on the strain and particle fraction. The aforementioned mechanism can contribute to it too. The stress increment of composites is found to be more and more distinct with strain increasing as shown in Figure 5(b,c), due to the concave characteristic of matrix rubber. Thus, for composites (e.g. 2.5% fraction composite), the localised strain zone has come into larger strain deformation, even up to 27% around (estimated based on the normalised strain 1.82 and over strain of composite 0.15 as shown in Figure 7). Then the Mullins effect due to the damage at maximum strain 27% induces the more severe stress softening than that of the pure rubber at strain 15%. With the strain increasing, the softened stress would be much more distinct due to the concave mechanical characteristics. For the composite with the fraction 7.0%, the localisation strains (even up to maximum 33%), induces more severe damage on rubber and results in an increased softened stress. It is just the concave mechanical behaviour of the rubber that determines the dependence of the softened stress on the strain and particle fraction, which is functioned by the strain localisation induced by the rigid inclusion.
For the multi-characteristic matrix epoxy, there will be no further reinforcement if the specific elastic modulus of rigid inclusion to matrix is over 3 [22]. Whereas, for the rubber with an increasing stress characteristics, further works should be conducted to investigate the particle influence on the mechanical properties. In the aforementioned investigations of experiments and simulations, the inclusion is an extreme case of Fe particle as a reinforcement filler for the matrix rubber. The inclusion as a reinforced filler in the simulations is assumed to own the following mechanical properties: twice, threefold and five times of that of matrix rubber, and labelled as Specific ratio 2, 3 and 5. Whereas, their Poisson's ratio and density are remained. Hereby, the composite with 7.0% volume fraction is simulated, and the elastic moduli of composites are obtained to be: 2.07, 2.12, 2.13 times of the rubber matrix. It is found that the elastic moduli remain a few increasing when the inclusion elastic modulus is 2 times than that of matrix. The strain contours are selected and shown in Figure 9, with strain from 0.06, 0.18 to 0.30, and specific ratio from 2, 3 to 5, respectively. For each strain, there are little difference in the strain distribution fields for all the specific ratios. For the carbon black, or other inclusions used for reinforcing rubber, they present rather rigid than rubber, and their reinforcement characteristics and mechanism could be partially attributed to the above-analysed.
True strain distribution contours for different strain and specific ratio of particle/matrix in aspect of stress–strain curve, where the strain (0.06, 0.18 and 0.30) in the vertical axial is the overall strain of the simulated unit cell model.
In the simulation, the boundary is assumed to be perfect between the matrix rubber and the Fe filler particle. There is no debonding happening during deformation, whereas, it is actually not for a larger extension strain. After debonding, the strain filed around the inclusion will change, but the strain characteristics and trends are similar as those of the perfect interface which is predicted based on the comparison results of the experiments and simulations including different interfacial debondings [23]. Therefore, the mechanism will work and contribute a certain degree.
Conclusions
The mechanical properties of the rigid particle-reinforced rubber composites are investigated by using experiments: Instron mechanical machine with a digit extensometer function, and finite element simulations: unit cell modelling. The analyses on the distinct reinforcements are performed and it is found that the elastic modulus and the softened stress of Mullins effect are attributed to the same mechanism in the mechanical viewpoint: the introduction of rigid inclusion changes its surrounding strain distribution field, and strain concentration zone is generated. Meanwhile, the increasing stress characteristic of the matrix rubber acts a major role in the composites mechanical properties, like the elastic modulus and the softened stress from the Mullins effect.
