nanoclay rubber composites: An experimental and theoretical investigation

Composites Part B 176 (2019) 107312

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Mixed-mode fracture in EPDM/SBR/nanoclay rubber composites: An experimental and theoretical investigation M.R. Ayatollahi a, *, M. Heydari-Meybodi a, F. Berto b, M. Yazid Yahya c a

Fatigue and Fracture Research Laboratory, Center of Excellence in Experimental Solid Mechanics and Dynamics, School of Mechanical Engineering, Iran University of Science and Technology, Narmak, 16846, Tehran, Iran b NTNU Department of Engineering Design and Materials, Richard Birkelands vei 2b, 7491, Trondheim, Norway c Center for Advanced Composite Materials, Universiti Teknologi Malaysia, 81310, Johor Bahru, Malaysia

A R T I C L E I N F O

A B S T R A C T

Keywords: Particle-reinforcement Fracture Finite element analysis (FEA) Mechanical testing Rubber/clay nanocomposites

Fracture analysis of rubber nanocomposites weakened by a crack and loaded in mixed-mode (I/II) condition is investigated in the current research for the first time. In fact, no study was devoted in the past to investigate the fracture of rubbers reinforced with nanoparticles and subjected to mixed-mode loading, neither experimentally nor theoretically. To fill this gap, in the experimental phase of study, ethylene–propylene-diene monomer (EPDM)/styrene-butadiene rubber (SBR) blends reinforced with CLOISITE 15 nanoclay are prepared and some uniaxial tensile experiments and fracture tests are conducted. In the theoretical field of contribution, due to the undeniable importance of presentation of a fracture criterion, an energy-based criterion, namely averaged strain energy density (ASED), is adopted to predict the rupture of tested rubber/nanoclay composites. To apply the criterion, some non-linear finite element analyses considering the Ogden hyperelastic material model are also performed. The results highlight the success of this criterion to assess the fracture of nano-reinforced rubbers containing a mixed-mode crack.

1. Introduction Nano-fillers can indirectly modify the physical properties of matrix in nano-composites. Indeed, nano-fillers can effectively adsorb activa­ tors and crosslinking agents and as a result, increase the crosslinking density of the matrix in the vicinity of nano-fillers [1]. One of the widely used nanoparticles is nanoclay which owning to its high aspect ratio, has been utilized as an effective reinforcing filler in elastomers and rubbers. Rubber-like materials, due to their very special properties, have been utilized in many applications such as seals, shoes, pipes and tires. Thanks to these industrial interests, as well as impressive development of nano-fillers and nano-scale characterization techniques, the investi­ gation of the mechanical properties of nanoparticle-filled rubbers has been extensively studied during the recent years, see among others, Refs. [2–13]. Among the various aspects of mechanical properties, investigation of the fracture characteristic of a material is of paramount importance. In fact, the existence of any flaw like a crack in a material can significantly affect its performance. On the other hand, the crack can be loaded either in a pure-mode loading (e.g. mode-I, mode-II) or in a combined-mode

condition (e.g. mixed-mode I/II). Some energy-based fracture criteria have been developed in the past for mode-I fracture assessment of rubber-like materials. The first one which is an extension of Griffith’s approach is called tearing energy as suggested by Rivlin and Thomas [14]. Tearing energy approach is related to an incremental increase in the fracture surface area [15]. According to this criterion, the critical energy required for crack prop­ agation (Tcr) is a material property and independent of the geometry of sample. This criterion was used in the rupture analysis of rubbers in some studies like [16,17]. The energy limiters approach is another useful energy-based criterion which was utilized to study the structural instability in the presence of a mode-I crack [18,19]. The key idea un­ derlying this method is to limit the capability of a material model to accumulate energy without any failure by introducing a limiter for the strain energy density (SED) of material. SED criterion, introduced by Sih [20], is the next energetic approach which was extended for being used in rubber-like materials. Hocine et al. [21] have shown that the mode-I crack propagation can be assessed by using the SED criterion. Mzabi et al. [22] have utilized another energy-based criterion for hyperelastic materials which is called the local energy release rate criterion. The results confirmed that the crack growth in rubber is controlled by the

* Corresponding author. E-mail address: [email protected] (M.R. Ayatollahi). https://doi.org/10.1016/j.compositesb.2019.107312 Received 19 March 2019; Received in revised form 16 July 2019; Accepted 11 August 2019 Available online 14 August 2019 1359-8368/© 2019 Elsevier Ltd. All rights reserved.

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Nomenclature and abbreviations E KIc Rc UOgden

W Wc Wc; F Wc; UT

ν σt λ1 ; λ2 ; λ3

λc; UT μi , αi θ ASED EPDM EMP FEM NRR phr SBR SED

Elastic modulus Fracture toughness Critical radius of control volume The Ogden hyperelastic material model

Amount of strain energy stored in a volume Critical value of SED Critical SED in a fracture test Critical SED in a uniaxial tensile testing Poisson’s ratio Ultimate tensile stress Principal invariants of the right Cauchy–Green

local strain energy stored in a highly strained zone near the apex of crack and is not dependent on the total strain energy of tested sample. However, in practice, cracks very often experience tension and shear loadings simultaneously which lead to a mixed-mode I/II condition. Moreover, the state of stress is generally more complex in mixed-mode loading condition compared to a pure mode-I case. Thus, development of a suitable criterion which can characterize the mixed-mode crack initiation is very important. The importance of the work is multiplied when the aim is to develop the criterion for materials having non-linear behaviors. This complex situation exists for rubbers due to their both geometry and material non-linearities. In the case of energy-based fracture criteria developed for rupture assessment of elastomers subjected to the mixed-mode loading condi­ tion, one can refer to Ref. [23]. They examined SED criterion for rupture prediction of elastomers under mixed-mode (I/II) loading. Their results suggested that the SED concept could not be used as a fracture criterion for estimating mixed-mode fracture loads in rubbers. In addition, Pidaparti et al. [24] applied the tearing energy criterion to predict the fracture loads of cracked elastomers subjected to mixed-mode (I/II) loading. In a series of studies performed on mixed-mode puncture/cut­ ting of elastomers by pointed blades, Triki et al. [15,25–27] have developed an energy-based criterion constructed on the basis of Rivlin and Thomas’s [14] and Lake and Yeoh’s [16] theories. Their newly presented criterion, which is usually called as true fracture energy, sug­ gests that the energy requires to puncture or cut of elastomeric samples are controlled by the fracture and the friction mechanisms: GTo­ tal ¼ GFracture þ GFriction. Recently, they adopted finite element modeling (FEM) to investigate the puncture-cutting response of soft material by a pointed blade [28]. Their results show that the true energy is nearly constant and can be regarded as an intrinsic property of elastomer. In the case of fracture analysis of nano-reinforced rubber (NRR),

deformation tensor Critical stretch in a uniaxial tensile test Material parameters of the Ogden model Crack angle Averaged strain energy density Ethylene–propylene-diene monomer (rubber) Equivalent material properties Finite element modeling Nano-reinforced rubber Parts per hundred parts of rubber Styrene butadiene rubber Stain energy density

some limited studies have been accomplished so far. Dong et al. [29] analyzed the response of mode-I cracked natural rubber composites reinforced with carbon nanotube bundles (CNTBs) on the basis of frac­ ture mechanical methods. They used J-integral tests to characterize the fracture resistance of tested nanocomposite rubbers. In another experi­ mental work [30], Dong et al. studied the fracture resistance of styrene-butadiene rubber (SBR) composite reinforced with carbon nanotubes (CNTs) subjected to pure mode-I loading. They adopted the digital image correlation (DIC) method in conjunction with the J-testing approach to explore the fracture resistance mechanisms. The fracture resistance of different rubbers containing various nanofillers such as nanoclay, multiwall carbon nanotube (MWCNT), and silica was deter­ mined by Agnelli et al. [31]. They performed the experiments by the J-integral method and making use of the single edge notched tensile (SEN-T) specimen loaded in mode-I condition. A precise review of previous studies in the field of fracture analysis of nano-toughened rubbers reveals that all of them analyzed the cracked rubbers subjected to pure mode-I loading. Moreover, the previous studies deal only with the fracture resistance of reinforced rubber with nano-particles (usually with J-integral method) and in other words, no attempts were made by the researchers to develop a fracture criterion for rupture prediction of cracked rubber nanocomposites. Therefore and to fill this gap, the current research is devoted to investigate the rupture assessment of NRRs containing an edge crack and subjected to mixedmode (I/II) loading condition. The goal of this research is twofold, as follows: � First and in the experimental phase of study, some fracture tests are conducted on nanoclay-reinforced rubber subjected to mixed-mode (I/II) loading condition. This set of experiments can enlarge the scarce data in this field and may be useful for future studies.

Fig. 1. A schematic representation of ASED criterion for a mixed-mode cracked component. 2

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� In the second phase, an attempt is made to assess the rupture of cracked rubber nanocomposites by using a reliable criterion. In this study, an energy-based criterion, namely the averaged strain energy density (ASED) criterion, is adopted. This criterion has been exten­ sively used in the previous studies because of its simplicity and high accuracy. Moreover, due to the dependence of the mechanical properties of rubbers to the SED function and in addition, laying the foundation of the ASED criterion on the basis of SED, it is expected that this criterion can be successfully utilized for rupture prediction of rubber nanocomposites. The applicability of the ASED criterion in mixed-mode cracked NRRs is verified in the last part of this study.

These functions can be written generally in the following form in terms of principal invariants of the right Cauchy–Green deformation tensor [38]: One of the widely used hyperelastic material models is the Ogden formulation [39]. The form of the Ogden strain energy potential under the assumption of incompressibility can be described as follows [40]: UOgden ¼

2.1. Use of ASED criterion in hyperelastic materials The ASED criterion, classified as an energy-based failure model, deals with the amount of strain energy stored in a limited volume (W). Indeed, it estimates the crack initiation in a sample when the strain energy stored in a control volume surrounding the crack tip reaches its critical value Wc . As it is shown in Fig. 1, the shape of control volume for a mixed-mode cracked brittle specimen is a circle centered at the crack tip. The radius of control volume (Rc ) and the critical value of SED (Wc ) constitute the main parameters of the criterion. In this regard, previous studies have provided some relations for these two values as follows [32–34]:

Rc ¼

σ 2t

> > > > :

ð5

� KIc 3νÞ 2σ t

� �2 KIc 8νÞ 2σ t �2

ðλα1 i þ λα2 i þ λ1 αi λ2 αi

(4)

3Þ;

(5)

Wc ¼ Wc; F ¼ Wc; UT

where Wc; F and Wc; UT are the critical values of SED in a fracture test and in a uniaxial tensile test, respectively. On the other hand, the critical value of SED in a uniaxial tensile test is related to its critical strain/stretch. If the critical stretch in a uniaxial tensile test of a rubber is denoted by λc; UT and in addition, the Ogden hyperelastic material model (Eq. (4)) is adopted, one can write the critical SED in the following form:

(1)

ð1 þ νÞð5

i

α2i

where μi and αi are two material parameters and should be obtained based on the behavior of the tested rubber. Moreover, λ1 and λ2 are the first and second principal invariants of the right Cauchy–Green defor­ mation tensor. After the first property of rubber mentioned above, the second feature of rubber-like materials which can help us to define the value of Wc in hyperelastic materials is the almost uniaxial state of stress field in the vicinity of the crack tip. This remarkable characteristic of cracked rubbers have been reported both in mode-I and mixed-mode loading conditions [41–44]. This point can facilitate the determination of the value of Wc in a rubber weakened by a crack, as explained below. Because of the uniaxiality near the crack apex, it is expected that the same condition occurs between the uniaxial tensile test of a rubber and its rupture test due to the existence of a crack. In other words, it can be suggested that the values of critical SED, Wc , in these two cases are the same. This finding can be written as follows:

The basis of ASED criterion and its use in the hyperelastic materials toughened with nano-particles are presented in the following subsections.

2E 8 > > > > <

n X 2μ i¼1

2. Analytical framework

Wc ¼

(3)

U ¼ Uðλ1 ; λ2 ; λ3 Þ

Plane Strain (2)

n X � 2μi αi αi λc; UT þ 2λc; 0:5 Wc; UT ¼ UOgden �@λc; UT ¼ UT 2

Plane Stress

i¼1

where σt , E, ν and KIc in the above equations respectively refer to ulti­ mate tensile stress, elastic modulus, Poisson’s ratio and fracture tough­ ness of target material. It should be noted that the above relations are generally restricted to materials having linear elastic and small strain behavior. Thus, they cannot be directly utilized in the cases where any kind of non-linearity exists; just like that occurs in rubber-like materials. Indeed, due to the geometry and material non-linear nature of rubbers, some new methods should be adopted for obtaining these two main parameters (i.e., Rc and Wc ). To this end, the present authors have extended the use of the ASED criterion to hyperelastic materials by reformulation of the parameters of the criterion, i.e., Wc and Rc [35–37]. For determination of the value of Wc in rubber, two key features of these types of materials have been considered. The first one is the strain energy potential function in hyperelastic materials which plays an important role in mechanical characterization of these materials. In other words, the SED function of a hyperelastic material, U, describes the strain energy stored in the ma­ terial per unit of its initial volume as a function of the strain/stretch. It is worth noting that based on the points presented in the beginning of this sub-section, there is a similarity between the basis of the ASED criterion and SED function in hyperelastic materials. As a result, this link may be a good sign of success for the ASED criterion in rupture prediction of rubber-like materials. Because of the remarkable role of SED function in hyperelastic ma­ terials, various potential functions have been developed during the past.

αi

� 3 ;

(6)

Therefore, the value of Wc; F can be achieved. For determination of the value of Rc , it should be firstly noted that as stated in Ref. [45], it is difficult to present a closed form solution for Rc in rubbers. However, the present authors have presented a coupled experimental-numerical method to achieve the critical radius Rc . In this regard, an arbitrary value for the radius of control volume should be assumed and the value of SED averaged in this volume should be ob­ tained. Then, the procedure should be repeated with another control radius (i.e., the radius should be increased/decreased) till the average value of SED in the volume (W) reaches Wc . This corresponding radius can be regarded as the critical radius Rc for the tested rubber. 2.2. Extension of ASED criterion to nano-reinforced rubber Modeling of nano-reinforced materials is generally an ongoing field of challenge among the researchers. Although the nano-mechanical simulation can result in a better understanding of the nano-composite behavior, this needs a high resolution and has a significant computa­ tional cost. As it is claimed, the precise modeling is currently and in the near future not possible [46]. To overcome this issue, some alternative methods for modeling the nano-toughened materials have been suggested by the researchers, like hierarchical and concurrent multi-scale methods [46], and equivalent material properties (EMP) approach [47,48]. The former methods usu­ ally incorporates a representative volume element (RVE) for 3

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Table 1 Formulation of prepared rubbers containing 0 phr and 5 phr nanoclay.

Fig. 2. A typical mesh pattern used in the FE simulations.

Material

Content (phr)

EPDM SBR Calcium Carbonate Zinc Oxide Stearic Acid Sulfur Accelerator Accelerator Nanoclay (CLOISITE 15)

90 10 25 5 2 3 1.5 2 0/5

� Organo-modified layered nanosilicates CLOISITE 15, a bis(hydro­ genated tallow alkyl)dimethyl, salt with bentonite, purchased from BYK Additives Inc. in Germany.

computations, while the later approach can effectively model the overall behavior of nanocomposite with less complexity and lower computa­ tional cost. The EMP approach is especially applicable in the cases where the damage mechanisms is not a concern in the modeling of nano­ composites. According to the EMP approach, the stress-strain of a nanocomposite can be obtained experimentally and then, this data can be used to model the nanocomposite behavior in a finite element code. As a result, since the damage analysis in the target rubber nano­ composites is not a concern in the current research, one can hope that modeling of nanocomposite by using the EMP method can be applicable and give acceptable results. It might be helpful here to describe some important steps towards finite element modeling of rubber nanocomposite. The first point is that as recommended by Ref. [40], the crack tip in rubber-like materials should be modeled as a small radius groove (see Fig. 8). This point can lead to a better convergence of FEM, provided that the crack tip radius is very small. In the current study, the crack tip radius was considered to be equal to 0.005 mm. The second point which should be emphasized regarding FEM, is related to both material and geometry non-linearity of rubber. To do so, the Nlgeom option was selected in ABAQUS® finite element code to consider the geometry non-linearity. In this condition, the elements are formulated in the current configuration using their current nodal positions. Therefore, the updated Lagrangian formulation is used when Nlgeom is specified [40]. Moreover, the Ogden material model which consider the material non-linearity of rubber, as described in Section 4.2, was chosen for the target NRRs. The last point is related to the mesh pattern near the crack apex. Due to high deformation of rubber during the analysis, one needs to consider more refined meshes around the crack tip. This point was also applied in the simulation. However, as described more precisely in Section 4.3, the results of utilized ASED criterion is nearly independent of mesh sizes considered around the crack tip. It is also useful to mention here that two-dimensional models with eight-node plane-stress quadrilateral elements were used for the entire FE simulations. Moreover, the reduced integration technique was cho­ sen to remove the likely over-stiffening effects due to volumetric locking which may occur during the analysis of almost incompressible materials like rubbers [40]. These elements are usually named in ABAQUS as CPS8R. A typical mesh pattern used in the FE simulation is depicted in Fig. 2.

For preparation of rubber nanocomposites, it is primarily important to efficiently disperse the nanoclay particles in the rubber matrix. This goal can be achieved by applying high shearing force to the nanoclay/ rubber blend in order to overcome the nanoclay bonds. One of the most common instruments used in rubber industries for dispersion of nanoparticles in rubber blends is the rubber internal mixer. This apparatus, which consists of an enclosed mixing chamber with two rotors, is uti­ lized for dispersing raw rubber with its ingredients. Because of providing high shearing force during the process of mixing, this devise can also be used for dispersion of nanoclay particles into the rubber matrix [8]. In the current study, the matrix including EPDM/SBR with the weight ratio of 90/10, was mixed with 5 phr nanoclay particles in Brabender Plas­ ticator internal mixer, made in Germany. The temperature and rotor speed during the process was set to be equal to 150 � C and 80 rev/min, respectively. After the end of mixing in the internal mixer, the other curing ingredients were added to the obtained nanoclay/EPDM/SBR gum and next, were mixed on a two roll mill. The detailed formulation of the NRR ingredients are reported in Table 1. It is important to note that in addition to the NRR samples toughened with 5 phr CLOISITE 15, some samples with no nanoclay (i.e., 0 phr CLOISITE 15) were prepared to serve as a comparative basis. For curing of a rubber compound, it is essential to determine the required curing time during the hot pressing. This time, which is usually named as optimum cure time (t90), can be found by using an oscillating disc rheometer analysis. It may be useful to note that a rheometer de­ scribes the curing and processing characteristics of vulcanizable rubber compounds. With the aim of determining the optimum cure time t90 for both series of samples (i.e., the samples with 0 phr and 5 phr nanoclay), Monsanto 100S UK rheometer was utilized and after analysis, the time was obtained equal to 1420 s and 870 s, respectively for 0 phr and 5 phr samples. Finally, each sample was laid in a 1.5-mm-thickness mold and cured in a hydraulic hot press with the corresponding obtained time. After preparation of rubber sheets reinforced with 0 phr/5 phr CLOISITE 15 nanoclay, the samples required for mechanical testing were provided as follows: � Tensile test sample: The response of this sample under the uniaxial tensile loading is needed in order to obtain the mechanical properties of tested rubber, especially the hyperelastic material model which correlates well with the rubber. The uniaxial tensile sample, often named as dog-bone shaped sample, was prepared according to ASTM D 412 standard test method. Fig. 3 presents the geometry and di­ mensions of the specimen. It is important to note that two points painted in Fig. 3 can be utilized for accurate analysis of strain in the sample during the tensile test by using a camera. The locations of these two points painted on the dog-bone shaped sample are arbi­ trarily chosen, since the strain of sample is independent of the lo­ cations. However, the points should be located within the gauge

3. Experimental procedures The materials used as the main ingredients of the NRR components are as follows: � Ethylene–propylene-diene monomer rubber (EPDM, KEP 270, South Korea) with Mooney viscosity of ML (1 þ 4, 125 � C) ¼ 55 � Styrene butadiene rubber (SBR1502, Bandar Imam Petrochemical Co. Iran) with 23.5% styrene content and Mooney viscosity of ML (1 þ 4, 100 � C) ¼ 59 4

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Fig. 5. Stress-stain curves of tested rubbers containing 0 phr and 5 phr CLOI­ SITE 15, subjected to uniaxial tensile loading. Fig. 3. Geometry and dimensions of tensile test sample (dimensions are in mm).

Fig. 4. A schematic of mixed-mode cracked sample used in this study (di­ mensions are in mm).

length region. More descriptions regarding this note are presented in Section 4.1. � Fracture test samples: These samples have an angled edge-crack and subjected to the tensile loading which will lead to a mixed-mode (I/ II) loading condition. The crack length in all the samples were kept constant to be equal to a ¼ 8 mm, while the initial cracks were introduced into the rubbers in three different angles, i.e., 15, 30, 45 deg. Moreover, as described later, a mode-I cracked sample with the crack angle of θ ¼ 0� was also prepared in both types of rubbers (i.e., 0 and 5 phr nanoclay) in order to obtain the value of Rc needed for adopting the ASED criterion. A schematic of geometry and di­ mensions of the fracture sample is shown in Fig. 4. It should be further noted that in all of the mechanical testing, the crosshead speed of 2 mm/min and the condition of displacement-control were considered.

Fig. 6. Some frames of a typical cracked sample captured during the defor­ mation process.

verification of adopting the ASED criterion in the mixed-mode cracked samples made of rubber nanocomposites is also accomplished in the end of this section.

4. Results and discussions Various results concluded from the experiments and finite element analyses in conjunction with related discussions are collected here. The 5

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Table 2 Critical displacement δcr of tested mixed-mode samples containing 0 phr and 5 phr nanoclay. Crack angle (deg.) θ ¼ 15 θ ¼ 30 θ ¼ 45

Containing 0 phr nanoclay

Containing 5 phr nanoclay

1st sample

2nd sample

Average (mm)

1st sample

2nd sample

3rd sample

Average (mm)

8.0 9.17 11.67

9.17 10.33 9.67

8.59 9.75 10.67

9.33 10.6 10.93

8.1 9.6 10.63

10.34 9.77 13.0

9.3 10.0 11.5

4.1. Mechanical testing The results of mechanical testing including uniaxial tensile test and mixed-mode (I/II) fracture test for both samples, with or without nanoclay fillers, are presented and discussed in this subsection. Before presentation of the data, it is important to note that due to the high deformation nature of rubber subjected to the tensile loading, utilization of a camera during the tests for specified purposes is usually inevitable. In the uniaxial tensile loading, the camera can help one to obtain the accurate values of strain in each step time. In other words, the motion of two transparent points presented in Fig. 3 can be recorded by the camera and afterwards, relative motion of these points can be related to the strain in the test sample. On the other hand, the initiation of crack propagation in the fracture tests can be determined by using the camera (see Fig. 6 for more information). In the current study, a Canon CMOS camera was used during all of the mechanical testing. The variation of engineering stress values versus the engineering strain ones for the dog-bone shaped tensile samples containing 0 phr and 5 phr nanoclay are shown in Fig. 5. As it is clear from this figure, the rupture strain of 0 phr EPDM/SBR sample is equal to 280%, while the corresponding value for the 5 phr NRR is increased to 420%. Moreover, the rupture engineering stress for 0 phr and 5 phr nanoclay reinforced EPDM/SBR samples are 2.13 MPa and 4.9 MPa, respectively. These re­ sults highlight that by addition of 5 phr CLOISITE 15 to the neat EPDM/ SBR rubber, an increase of 50% occurs in the rupture strain. In addition, this improvement for the final engineering stress during the uniaxial tensile test is about 130% by addition of nanoclay to the tested rubber. In the case of mixed-mode (I/II) fracture tests, each test was continued until the final rupture of the sample and as was described earlier, the rupture initiation in each sample was detected by using the camera. In more details, we tracked the images taken by the camera and detected the image in which the crack initiation occurred. Afterwards, by knowing the sequential number of the above-mentioned image taken by the camera, the cross-head speed of uniaxial testing machine as well as the camera time interval used for capturing each two consecutive images, we could determine the critical displacement of tested sample as follows:

Fig. 7. Comparison between the stress–strain curves based on the prediction of Ogden hyperelastic model and corresponding experiments for EPDM/SBR-0 phr nanoclay sample.

blunting) (2) a small increase in the crack length and (3) the abrupt propagation of the crack through the sample till its complete rupture. Following the above-mentioned procedure, the critical displacement δcr (i.e., the displacement at which the crack initiation occurs) for all the cracked samples, with or without the presence of CLOISITE 15, were achieved and the summary is reported in Table 2. As can be seen from Table 2, by addition of 5 phr nanoclay to the tested EPDM/SBR, the critical displacements of cracked samples in all the crack angles increase. This finding also confirm the positive effect of addition of nanoclay to the rubber. The improvement can be attributed to the higher crosslinking density of the matrix in the presence of nanofillers which can increase the filler–rubber interaction and thus, make the NRR stronger against the crack initiation [49]. As a conclusion of this subsection, the addition of nanoclay to our target rubber has a significant effect on improvement of its mechanical properties.

Critical displacement¼ (ImagNum-1)*(MachSpeed)*(CamTimeInter)

4.2. Selection of a suitable hyperelastic material model

where:

As it was stated before, determination of an appropriate hyperelastic model for modeling the rubber and subsequently, utilization of the ASED criterion, is required. To this end, for target rubbers with or without the presence of nanoclay, the Ogden hyperelastic material model which has the best agreement with the obtained stress-strain curve in both samples was selected. Next, by using a curve fitting analysis, the material con­ stants of the Ogden model (Eq. (4)) were determined. The constants for EPDM/SBR-0 phr nanoclay sample considering two-term Ogden model (i ¼ 2 in Eq. (4)) are as follows: � μ1 ¼ 5:746e 4 ; α1 ¼ 7:236 (7) μ2 ¼ 1:380 ; α2 ¼ 2:392

ImagNum: the number of image in which the crack initiation occurred. MachSpeed: the cross-head speed of uniaxial testing machine. CamTimeInter: camera time interval for capturing each two consecutive images. Fig. 6 shows four consecutive frames of a typical cracked sample captured during the deformation process. It might be useful to notice that as can be seen in Fig. 6, some white speckle points were painted near the crack tip in order to facilitate the detection of crack propaga­ tion. The zoomed image beneath the 3rd frame in Fig. 6 refers to this point. According to Fig. 6, the major mechanisms of fracture for different stages of failure in the cracked rubber can be suggested as: (1) the in­ crease in the crack opening angle (which is usually named as crack

Moreover, by considering the three-term Ogden material model (i ¼ 3 in Eq. (4)), the following constants are achieved for the nano­ composite rubber containing 5 phr CLOISITE 15:

6

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Fig. 8. Comparison of stress-strain data between the experiments and predic­ tion of the fitted hyperelastic model for EPDM/SBR sample containing 5 phr nanoclay.

Fig. 11. Iso-strain energy density contours (in MJ/m3) plotted in the vicinity of crack tip of a typical NRR.

8 < μ1 ¼ 1:184 ; α1 ¼ 1:375 μ ¼ 0:323 ; α2 ¼ 2:976 : 2 μ3 ¼ 2:437 ; α3 ¼ 1:808

(8)

The comparison of stress-strain curve obtained from the fitted Ogden model with the corresponding experiments for 0 phr and 5 phr samples are drawn in Fig. 7 and Fig. 8, respectively. As it is obvious from Figs. 7 and 8, very good agreement in both cases exists between the experimental uniaxial stress-strain data with those predicted by the Ogden hyperelastic material models. It may be useful to emphasize here that the Ogden material model was selected for finite element simulations of tested rubbers, since this model had the best agreement with the experimental data. To shed more light on this issue, the stress-strain curves obtained from two other hyperelastic material models, namely the Mooney-Rivlin and NeoHooke models, for both samples (with or without the nanoclay) in conjunction with that of the Ogden model have been compared with the corresponding experimental curve and the results are shown in Figs. 9 and 10. As can be seen from Figs. 9 and 10, the selected Ogden model has the best agreement with the corresponding experimental curve in compar­ ison with the two other models. Therefore, the Ogden model was utilized in finite element simulations performed in the current study.

Fig. 9. The stress–strain curves based on the prediction of Ogden, MooneyRivlin and Neo-Hookean hyperelastic models in comparison with the experi­ mental curve obtained for EPDM/SBR-0 phr nanoclay sample.

4.3. Results achieved from finite element modeling Two main findings, which can be achieved from FEM, are discussed in this subsection in more details. First of all, the shape of control volume around the crack tip of a nanocomposite sample loaded under mixedmode (I/II) condition is obtained via FEM. As it is stated in Ref. [33], the shape of control volume required in the ASED criterion, can be found if the iso-strain energy density contours are plotted in the vicinity of stress concentration region. This point is not dependent on the type of stress concentrator as well as the fracture mode. Considering this point and as a typical example, the SED contours are plotted for 5 phr component with the crack angle of 45 deg in Fig. 11. As can be seen from Fig. 11, the shape of control volume for the NRR in the undeformed configuration is nearly a circle centered at the crack tip. The second issue which should be tracked by FEM is the indepen­ dency of analysis to the element number (or equivalently the element size). To shed more light on this issue in the current analysis, the effect from the number of elements on the value of SED averaged in a typical volume (W) is investigated. Without loss of generality of the problem, a semicircle in the undeformed configuration with the radius of 1 mm

Fig. 10. The stress–strain curves based on the prediction of Ogden, MooneyRivlin and Neo-Hookean hyperelastic models in comparison with the experi­ mental curve obtained for EPDM/SBR-5 phr nanoclay sample.

7

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Table 3 Influence of mesh pattern on value of strain energy density averaged in a region. Mesh Pattern

Number of elements within the selected region

W (MJ/ m3)

Variation (%)

7200

1.135

–

Table 4 Key parameters of two types of tested rubbers required for adopting the ASED criterion. Sample

λc; UT

Wc (MJ/m3)

Rc (mm)

EPDM/SBR – 0 phr nanoclay EPDM/SBR – 5 phr nanoclay

3.8 5.2

3.679 9.356

0.209 0.086

Table 5 A comparison between the critical displacements obtained experimentally and those predicted using the ASED criterion for both types of rubbers. 3600

1.136

0.09

1800

1.136

0.09

900

1.140

0.44

450

1.140

0.44

216

1.151

1.41

108

1.151

1.41

Crack angle (deg.)

θ ¼ 15

Content of CLOISITE 15 Experiment ASED estimation Discrepancy (%)

0 phr 8.59 9.08 5.7

θ ¼ 30 5 phr 9.3 9.13 1.8

0 phr 9.75 9.98 2.3

θ ¼ 45 5 phr 10.0 10.04 0.4

0 phr 10.67 12 12.4

5 phr 11.5 12.1 5.2

coarsest mesh) has led to varying the value of W by only 1.41%. As a result, it can be inferred that the value of SED averaged over a control volume is nearly independent of the element size and in fact, this point can be regarded as one of the great advantages of the ASED criterion which reduces its computational cost of modeling. 4.4. Verification of ASED criterion for mixed-mode cracked rubber nanocomposite Prior to adopting the ASED criterion, it is essential to determine its main parameters, including the value of critical SED (Wc ) and the critical radius of control area (Rc ), for both 0 phr and 5 phr nanoclay reinforced rubbers. First, the value of Wc can be achieved by knowing the rupture stretch of selected rubber under the uniaxial tensile loading (λc; UT ) and its substitution into Eq. (6). After determination of Wc , the value of Rc for each sample can be determined by adopting the method described in Section 2.1 for cracked-samples loaded in pure mode-I condition. Doing these procedures, the corresponding values for each target rubber were obtained and the results are collected in Table 4. An interesting point can be concluded from Table 4 regarding the comparison of the values of Wc for rubbers with or without the presence of nanoclay. As can be seen, the value of Wc for reinforced-nanoclay rubber is about 2.5 times larger than the corresponding value of the pure EPDM/SBR. Since the value of Wc can be regarded as an indicator of material resistance against the crack initiation, it may be true to consider this value as a representative of fracture toughness in rubberlike materials. The obtained critical rupture displacements of tested rubbers presented in Table 2 can be evidence for this claim. Indeed, based on the data presented in Table 2, cracked-NRRs have higher rupture displacements rather than the 0 phr nanoclay-rubbers and so, NRRs have higher fracture toughness. After determination of the underlying values of the ASED criterion, this criterion can now be utilized for prediction of the rupture loads of mixed-mode fracture samples in both 0 phr and 5 phr nanoclay rubbers. For each cracked sample, an FEM was performed and the displacement applied to the model was changed till the SED value averaged over the predefined circular control volume reached the value of Wc . In this condition, the exerted displacement was considered as the theoretical prediction of rupture load of target sample based on the ASED criterion. Table 5 presents a comparison between the results of critical displace­ ments obtained experimentally and those predicted by using the ASED criterion for both types of rubbers. As it is clear from Table 5, the ASED criterion could precisely predict the rupture displacements of NRRs subjected to mixed-mode (I/II) loading; so that the mean discrepancy between the experiments and theoretical values for 5 phr nanoclay reinforced rubber is less than 2.5%.

located in the vicinity of the crack tip was considered as the typical region. Moreover, the Ogden hyperelastic model with the material constants specified in Eq. (8) and crack angle of 45� were selected in the case of investigation. Next, the number of elements within this region was changed and the value of W in each case was computed. The results are summarized in Table 3. As can be seen from Table 3, the change in the element numbers from 7200 elements (as the finest mesh) to about 100 elements (as the

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Fig. 12. The ratio of Max. to Min. principal stresses along the circumferential path.

This result can firstly confirm the efficacy of the ASED criterion in rupture assessment of cracked rubber nanocomposites in conjunction with the pure rubbers. Moreover, the good agreement between the theoretical and experimental data can support acceptable performance of the EMP method adopted to model the rubber nanocomposite. 4.5. Further discussions As stated earlier, the main characteristic of the nano-reinforced rubber which allowed the extension of the ASED criterion to them is the nearly-uniaxial state of stress field next to the crack tip. This key feature is examined here via the finite element results. To investigate the state of stress field next to the crack apex, the nano-reinforced sample with the initial crack angle of 30 deg. is selected as a sample and the ratio of maximum in-plane principal stress value (σ Max Princ) to its minimum one (σ Min Princ) was determined along a circumferential path. This path is located at r ¼ 0.1 mm as a typical path close to the crack apex. It may be important to note that since the almostuniaxial state of stress fields should be established in the control volume, the path is considered much close to the obtained critical radius (i.e., Rc ¼ 0.086 mm) for the tested nano-reinforced rubber. By applying the averaged critical displacement obtained from the experiments (see Table 2) in the FE model, the stress ratio was computed along the path and the result is depicted in Fig. 12. As it is clear from Fig. 12, the stress ratio varies between 13 and 43. From this result, it can be confirmed that the nearly-uniaxial state of stress in the proximity of the crack apex is also established in a nanoreinforced rubber. After examination of the state of stress field ahead of the crack tip in the rubber, it is interesting to investigate the effect of crack angle as well as nanoclay on the mode of fracture. It should first be noted that the mode of fracture in our study (i.e., the tensile loading of an edge inclined crack) is mixed-mode I/II based on the concepts of fracture mechanics. To determine which mode is predominant in mixed mode loading, it is common for linear elastic materials to compute the ratio of KII to KI (i.e., the mode II to mode I stress intensity factors (SIF), respectively). The parameters KI and KII appear in the Williams series solution for describing the stress fields ahead of the crack tip in an isotropic and linear elastic material under small strain condition. Because of the important role

Fig. 13. A schematic representations of rþ 0 and r 0 as well as the selection of X-Y coordinate system for determining the relative mode II over mode I deformation.

of SIFs in linear elastic fracture mechanics (LEFM), the commercial finite element codes (like ABAQUS software) often have built-in routines to automatically compute these values in crack problems. However, since rubbers obey the hyperelastic strain energy density functions and have both material and geometry non-linearities, the use of Williams’ series solution is not applicable for rubbers. Thus, we cannot use the values which are given by the ABAQUS code for KI and KII in the present study. Therefore, we use an alternative and more general method which is considered as an indicator of mode II to mode I ratio. In this method, the contribution from each mode of deformation is related to the relative displacements of crack edges. Considering that in pure mode I there is no crack-edge relative sliding, and in pure mode II there is no relative crack-edge opening, the ratio of mode II to mode I can be suggested to be calculated from the displacements ux and uy along the upper and lower crack faces as follows [55]:

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Table 6 The mode II/mode I ratio for both types of tested samples and for three different crack angles.

Table 8 The value of SED in the control volume when critical displacement was applied to the finite element code for EPDM/SBR – 5 phr nanoclay samples.

Crack Angle (deg.)

0 phr nanoclay

5 phr nanoclay

Crack Angle (deg.)

SED in the control volume (MJ/m3)

θ ¼ 15 θ ¼ 30 θ ¼ 45

0.20 0.41 0.65

0.19 0.39 0.62

θ ¼ 15 θ ¼ 30 θ ¼ 45

9.612 9.304 8.713

Table 7 The value of SED in the control volume when critical displacement was applied to the finite element code for EPDM/SBR – 0 phr nanoclay samples. Crack Angle (deg.)

SED in the control volume (MJ/m3)

θ ¼ 15 θ ¼ 30 θ ¼ 45

3.439 3.576 3.173

� Mode II ux rþ 0 ¼ þ uy ðr0 Þ Mode I

� ux r 0 uy ðr0 Þ

Table 9 Comparison of SED value in the control volume when critical displacement was applied to the finite element code with the critical SED, Wc obtained from Eq. (6). EPDM/SBR – 0 phr nanoclay sample

(9)

Discrepancya (%)

Crack Angle (deg.)

Discrepancya (%)

θ ¼ 15 θ ¼ 30 θ ¼ 45

6.5 2.8 13.7 Average: 7.6

θ ¼ 15 θ ¼ 30 θ ¼ 45

2.7 0.5 6.8 Average: 3.3

a

where r0 is a small distance from the crack tip and the signs þ and – correspond to the upper and lower faces of the crack, respectively. Fig. 13 illustrates a schematic representation of rþ 0 and r0 as well as the selection of X-Y coordinate system. It is important to note that since in a non-linear problem (such as the case in our study) the principal of superposition is not applicable, the above relation in rubbers depends on the value of applied load in FE model. However, for a brittle material with linear elastic behavior, the relation is independent of the value of the load applied to the finite element models. To overcome this problem in our modeling and to provide almost similar conditions in all cases studied, we consider a constant arbitrary far-field displacement of 8 mm applied to all the samples and then the mode II to mode I ratio is computed according to this constant far-field displacement. Finite element modeling was performed for three different crack angles to investigate the influence of crack angle on the relative mode of fracture. Moreover, both types of samples (0 phr and 5 phr nanoclay) were modeled in order to account for the effect of nanoclay on the fracture mode. According to the above formulation, the mode II/mode I ratio for each sample was computed at a distance r0~ 0.8 � 10 3 mm from the crack tip. The results are summarized in Table 6. From Table 6, two results can be deduced:

EPDM/SBR – 0 phr nanoclay sample

Crack Angle (deg.)

The discrepancy was computed as.

jSED Wcj Wc

Fig. 14. The values of SED as a function of the crack angle for EPDM/SBR-0 phr nanoclay sample.

volume for each type of rubbers is in accordance with the data presented in Table 4. The results of above-mentioned computations are summa­ rized in Tables 7 and 8. Moreover, it is useful now to compare the values of SED in Tables 7 and 8 with the corresponding critical SED (Wc obtained from Eq. (6)) which was considered for each type of samples (i.e., Wc ¼ 3.679 MJ/m3 for 0 phr sample and Wc ¼ 9.356 MJ/m3 for 5 phr nano-reinforced rubber). The comparison is conducted and the data are reported in Table 9: As can be seen from Table 9, the average discrepancy between the SED averaged in the control volume (when critical displacement was applied to the finite element code) with the critical value of SED (Wc) is 7.6% and 3.3% for 0 phr and 5 phr nanoclay reinforced rubbers, respectively. This result can confirm that the SED is almost independent of the crack angle introduced into the samples. Indeed, the above-achieved finding (i.e., the nearly constant value of SED for various crack angles) was predictable, since the independency of SED to the crack angle is one of the main assumptions of the ASED criterion. This independency was previously and extensively assessed in the fracture analysis of brittle materials with small strains (see, among others, Refs. [45,50–54]). The value of SED in these references was nearly independent of crack (or notch) geometry and in all cases, the

✓ Regarding the effect of crack angle on fracture mode: As the crack angle increases, the mode II/mode I ratio increases in both types of samples. Since the higher ratio shows the dominance of mode II, it can be concluded that by increasing the crack angle, mode II has more contribution than mode I in the crack deformation. ✓ Regarding the effect of nanoclay on fracture mode: As can be seen from Table 6, in a constant crack angle, the mode II/mode I ratio is nearly the same for both types of rubbers, although, the ratio for 0 phr nanoclay is slightly higher than that of 5 phr nanoclay. Therefore, one can suggest that the addition of nanoclay to the target rubber has nearly no effect on the fracture mode and in fact, the fracture mode is mainly governed by the crack angle and more generally, by the geometry of sample, not by the material property of components. As the final part of this section, it may be useful to investigate if the critical SED can be considered as an intrinsic property of material? To do so, we computed the value of SED averaged in the control volume for both 0 phr and 5 phr reinforced nanoclay EPDM/SBR rubbers when the average far-filed critical displacements (given in Table 2) were applied to the finite element code. Moreover, the radius of control 10

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nanocomposites was successfully modeled in the finite element code with a low computational cost. Then, by adopting the finite element analyses, the predictions of the ASED criterion were evaluated for the tested cracked EPDM/SBR/CLOISITE 15 samples loaded in mixed-mode (I/II) condition. The results highlight the success of the ASED criterion in predicting the rupture of rubber nanocomposites weekend by crack and subjected to mixed-mode loading. Funding sources This research was funded by Universiti Teknologi Malaysia under grant number of HIR - QJ130000.2424.04G40. Data availability statement It is declared that the data provided in the present manuscript is not confidential and is available to whom it may concern. Fig. 15. The values of SED as a function of the crack angle for EPDM/SBR-5 phr nanoclay sample.

References [1] Wolff S. Chemical aspects of rubber reinforcement by fillers. Rubber Chem Tech 1996;69:325–45. [2] Rooj S, Das A, Morozov IA, St€ ockelhuber KW, Stocek R, Heinrich G. Influence of ‘‘expanded clay’’ on the microstructure and fatigue crack growth behavior of carbon black filled NR composites. Compos Sci Technol 2013;76:61–8. [3] Galimberti M, Coombs M, Riccio P, Ricco T, Passera S, Pandini S, et al. The role of CNTs in promoting hybrid filler networking and synergism with carbon black in the mechanical behavior of filled polyisoprene. Macromol Mater Eng 2013;298(2): 241–51. [4] Konstantinovich GO, Vasil’evich SV, L’vovich SA, Konstantinovich SA, St€ ockelhuber KW. Visco-elastic-plastic properties of natural rubber filled with carbon black and layered clay nanoparticles. Experiment and simulation. Polym Test 2017;63:133–40. [5] Valentini L, Bon SB, Lopez-Manchado MA, Verdejo R, Pappalardo L, Bolognini A, et al. Synergistic effect of graphene nanoplatelets and carbon black in multifunctional EPDM nanocomposites. Compos Sci Technol 2016;128:123–30. [6] Tang Z, Zhang C, Wei Q, Weng P, Guo B. Remarkably improving performance of carbon black-filled rubber composites by incorporating MoS2 nanoplatelets. Compos Sci Technol 2016;132:93–100. [7] Peddini SK, Bosnyak CP, Henderson NM, Ellison CJ, Paul DR. Nanocomposites from styrene-butadiene rubber (SBR) and multiwall carbon nanotubes (MWCNT) part 2: mechanical properties. Polymer 2015;56:443–51. [8] Nematollahi M, Jalali-Arani A, Golzar K. Organoclay maleated natural rubber nanocomposite. Prediction of abrasion and mechanical properties by artificial neural network and adaptive neuro-fuzzy inference. Appl Clay Sci 2014;97–98: 187–99. [9] Fang Q, Song B, Tee T-T, Sin LT, Hui D, Bee S-T. Investigation of dynamic characteristics of nano-size calcium carbonate added in natural rubber vulcanizate. Compos B Eng 2014;60:561–7. [10] Chen Y, Yin Q, Zhang X, Zhang W, Jia H, Ji Q, et al. Rational design of multifunctional properties for styrene-butadiene rubber reinforced by modified Kevlar nanofibers. Compos B Eng 2019;166:196–203. [11] Liu P, Zhang X, Jia H, Yin Q, Wang J, Yin B, et al. High mechanical properties, thermal conductivity and solvent resistance in graphene oxide/styrene-butadiene rubber nanocomposites by engineering carboxylated acrylonitrile-butadiene rubber. Compos B Eng 2017;130:257–66. [12] Zhang X, Xue X, Yin Q, Jia H, Wang J, Ji Q, et al. Enhanced compatibility and mechanical properties of carboxylated acrylonitrile butadiene rubber/styrene butadiene rubber by using graphene oxide as reinforcing filler. Compos B Eng 2017;111:243–50. [13] Yan F, Zhang X, Liu F, Li X, Zhang Z. Adjusting the properties of silicone rubber filled with nanosilica by changing the surface organic groups of nanosilica. Compos B Eng 2017;75:47–52. [14] Rivlin RS, Thomas AG. Rupture of rubber. I. characteristic energy for tearing. J Polym Sci 1953;10(3):291–318. [15] Triki E, Gauvin C. Analytical and experimental investigation of puncture-cut resistance of soft membranes. Mech Soft Mater 2019. https://doi.org/10.1007/s4 2558-019-0007-z:1-6. [16] Lake GJ, Yeoh OH. Effect of crack tip sharpness on the strength of vulcanized rubbers. J Polym Sci B Polym Phys 1987;25:1157–90. [17] Lake GJ. Aspects of fatigue and fracture of rubber. Progr Rubber Chem 1983;45: 89–143. [18] Trapper P, Volokh KY. Cracks in rubber. Int J Solids Struct 2008;45:6034–44. [19] Volokh K. Review of the energy limiters approach to modeling failure of rubber. Rubber Chem Tech 2013;86(3):470–8. [20] Sih GC. Strain energy density factor applied to mixed mode crack problems. Int J Fract 1974;10(3):305–21. [21] Hocine NA, Abdelaziz MN, Imad A. Fracture problems of rubbers: J-integral estimation based upon η factors and an investigation on the strain energy density distribution as a local criterion. Int J Fract 2002;117(1):1–23.

obtained values were placed in a reliable band with a scatter of nearly � 20% relative to the value of critical SED (Wc). To do a similar analysis for our tested rubbers, we plotted the data obtained in Tables 7 and 8 (i.e., the variation of SED for samples versus the crack angle) in Figs. 14 and 15 for both types of rubbers. It can be seen from Figs. 14 and 15 that the SED of all samples (with the exception of one sample) are inside a reliable scatter of �10%, ranging from 90% to 110%. Therefore, SED is substantially independent of the crack angle in rubber-like materials, just like that was seen before in several brittle materials. Here, it may be interesting to justify this �10% scatter band. As it was prescribed earlier, if the state of stress field near the crack tip in a rubber remains precisely uniaxial, one can find the value of critical SED (Wc) in rubber-like materials in accordance with the value obtained from the uniaxial tensile test. On the other hands, as it was proved in the first part of this subsection, the state of stress field next to crack tip in 5 phr nanoclay reinforced rubbers was nearly uniaxial. Thus, a small scatter of �10% for the value of SED compared to the value of Wc (as shown above in Figs. 14 and 15) is reasonable. 5. Conclusions Rupture assessment of cracked rubber nanocomposites loaded in mixed-mode condition was investigated for the first time. Since there was no experiments in the previous published experiments regarding the rupture of cracked rubbers containing nanoparticles and subjected to mixed-mode (I/II) condition, a set of experiments was designed in this research. The EPDM/SBR blends reinforced with CLOISITE 15, as nanoclay particles, were prepared. Afterwards, three different crack angles were introduced into the rubber nanocomposites and tested under tensile loading which led to mixed-mode condition. In addition to the fracture experiments, some uniaxial tensile tests were also accom­ plished for determination of the mechanical properties of target rubbers. The experimental results show that by addition of 5 phr nanoclay to the tested EPDM/SBR, increases of about 50% and 130% were obtained respectively in the rupture strain and rupture stress of dog-bone shaped samples during the uniaxial tensile test. Due to the importance of developing a suitable fracture criterion and considering that no study has been devoted in the past to present a criterion for fracture assessment of mixed-mode cracked rubber nano­ composites, an attempt was made to fill this gap. To do so, the averaged strain energy density (ASED) criterion, which has attracted much in­ terest during the recent years, was utilized here. The ASED criterion was extended for being used in rubber nanocomposites. By using the equivalent material properties (EMP) method, the behavior of rubber 11

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