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Experimental characterization and finite element modeling the deformation behavior of rubbers with geometry defects
Polymer Testing 80 (2019) 106111
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Test Method
Experimental characterization and finite element modeling the deformation behavior of rubbers with geometry defects Huxiao Yang a, Rui Xiao a, b, *, Zhou Yang c, Dong Lei a a
Department of Engineering Mechanics, College of Mechanics and Materials, Hohai University, Nanjing, Jiangsu, 210098, China Key Laboratory of Soft Machines and Smart Devices of Zhejiang Province, Department of Engineering Mechanics, Zhejiang University, Hangzhou, Zhejiang, 310027, China c Department of Mechanical Engineering, University of Washington, Seattle, WA, 98195, USA b
A R T I C L E I N F O
A B S T R A C T
Keywords: Failure Rubber DIC Finite element modeling
Rubbers have broad applications in various areas. The engineering application requires accurately capturing the response of these materials under large deformation. In this work, we investigate the deformation behavior of two engineering rubbers (EPDM and Silicone rubbers) until failure occurs. The digital image correlation tech nique is used to capture the strain field of rubber specimens with different types of geometry defects. The results show that EPDM rubbers are more ductile than Silicone rubbers. For both rubbers a homogenous deformation behavior can be observed for specimens without defects, while strain localization behaviors occur for specimens with geometry defects. Finite element analysis (FEA) is also performed. The Arruda-Boyce eight chain model is used to represent the constitutive relationship of both rubbers with the parameters determined from the stressstrain response of specimens without geometry defects. The strain field obtained from FEA shows good agree ment with the experimental data even in the large strain region close to failure. However, the simulated forcedisplacement is consistent with the experimental results only for specimens without geometry defects. For specimens with geometry defects, discrepancies exist in all the cases, which indiates that more mechanisms need to be incorporated into the constitutive relationship.
1. Introduction Rubbers, which can sustain large deformations, have been widely applied in various areas, including tires, sealing materials and shock absorbers. However, natural rubbers are relatively soft, which are not suitable for applications requiring a high strength material. To overcome this limitation, the most common method is adding nanoparticles into rubber matrix to form rubber composites [1]. For example, carbon black filled rubbers are widely used in tires. For practical applications, rubbers are also required to have a high tolerance to geometry defects. Thus, various works have been done to investigate deformation behaviors of engineering rubbers in presence of different geometry imperfections [2–5]. The geometry defects can induce an inhomogeneous deformation state, which is a challenge for mechanical characterization. The recent development of digital image correlation (DIC) technique has provided several advantages to obtain the strain field of materials compared with traditional methods [6–11]. The traditional stiff strain gage may
influence strain distributions when attached to a soft rubbery material. In contrast, the DIC technique is a non-contact method, which eliminates this drawback. Also the DIC technique can measure strain distributions over a wide strain region, while the traditional methods typically can only measure the response of materials in a small strain region. More over, the measurement length-scale of the DIC technique ranges from a size from nanometer to megameter [12,13]. For all these advantages, the DIC technique has been widely used to characterize the deformation behaviors of materials. Due to abundant works in the literature, in the following we will only review the appli cation of the DIC technique to rubbery materials [2–4]. Mzabi et al. [14] applied DIC to measure the strain field around a crack tip of filled rubbers in the cyclic loading tests, while Major and Lang [15] and Martinez et al. [16] combined the DIC technique and infrared ther mography to characterize the fracture behaviors of rubbers in mono tonic and cyclic loading conditions respectively. Brighenti et al. [5] analyzed the defect tolerance of silicone rubbers with a central crack while the DIC technique was used to measure the strain distribution of
* Corresponding author. Department of Engineering Mechanics, College of Mechanics and Materials, Hohai University, Nanjing, Jiangsu, 210098, China. E-mail address: [email protected] (R. Xiao). https://doi.org/10.1016/j.polymertesting.2019.106111 Received 24 June 2019; Received in revised form 21 August 2019; Accepted 19 September 2019 Available online 25 September 2019 0142-9418/© 2019 Elsevier Ltd. All rights reserved.
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engineering rubbers, EPDM and Silicone rubbers. The DIC technique is used to measure the strain distribution of specimens with different types of geometry defects. The FEA is also performed to analyze the effects of geometry imperfections on the mechanical response. The paper is ar ranged as follows. The experimental methods are firstly described. Section 3 presents some representative experimental results. The constitutive relationship of the model and the methods of parameter determinations are then shown in Section 4. The followed section compares the experimental and simulated response of two rubbers. The conclusion part summarizes the main findings in this paper. 2. Experimental methods Fig. 1. The geometry of the specimens used in uniaxial tension tests. The unit of the dimension is mm. The marked area in the first subfigure is the measured area using the DIC technique.
The EPDM and Silicone rubbers used in this work were directly ob tained from the McMaster-Carr Supply Company and used as the inreceived condition. The filler for both rubbers is carbon black. Howev er, the content of the filler is not provided by the supplier. The film with a thickness of 1.59 mm (1/16 inch) was then manufactured to the experimental shape using laser cutting. Since Silicone rubbers can withstand a high temperature (250 � C), the laser cutting seems to have a limited influence on the property of the specimens. However, for EPDM rubbers the power of laser needs to be carefully controlled to minimize the influence of laser cutting. As shown in Fig. 1, three different ge ometry defects were also introduced to the dog-bone shaped specimens: a circular defect in the middle, a symmetric distributed half-circular defect on each edge, an antisymmetric distributed half-circular defect on each edge. Before tests, the surfaces of the EPDM specimens were sprayed with white speckles while the black speckles were used for Silicone speci mens. All experiments were performed on an Instron 3367 Series Testing Machine at room temperature with a 1 kN load cell using a uniaxial tension mode. The specimens were deformed at a rate of 15 mm/min until failure. A preliminary study showed that the performance of both rubbers had negligible dependence on the loading rate, which indicated the viscoelastic effects could be neglected for EPDM and Silicone rubbers at room temperature. During the tests, a CCD camera was used to record the surface of the specimens. The strain distribution was calculated using the commercial software VIC 2D.
specimens with different crack sizes and loading conditions. Schwan et al. [17] used the DIC technique to characterize the strain localization behaviors of aerogels in uniaxial compression tests, which were mainly caused by the local fracture of gels. Meunier et al. [18] combined ex periments and numerical simulation to investigate the deformation be haviors of unfilled silicone rubbers in tensile, compression, shear and bulge tests, which provided an insight on the choice of the constitutive models for rubbers. To obtain the strain field in large deformation re gion, the incremental DIC method is often used to guarantee a good correlation of images. Goh et al. [19] developed a single-step image correlation method and demonstrated this method could also provide an accurate calculation of strain field. In addition to experiments, numerical models are also widely employed to characterize the mechanical response of rubbers and soft materials [20,21]. For rubbers, the hyperelasticity should be first considered. It is now well accepted that the hyperelastic behaviors of rubbers origin from the entropy change during the deformation. Based on statistical thermodynamics, various models have been developed to describe the stress response of rubbers in the finite deformation region, such as three chain model, four chain model, eight chain models and full chain models [22–24]. Alternatively, phenomenological models have also been proposed based on the requirement of continuum mechanics [25,26]. Some rubbers exhibit a rate-dependent viscoelastic response and a deformation history dependent softening behavior. In these cases, the viscoelastic and damage mechanisms need to be incorporated into the constitutive relationships [27]. To capture the response of rubbers with complex geometry and loading conditions, the constitutive model needs to be implemented into a finite element suite. The finite element analysis (FEA) is a standard method and has already been widely used to characterize rubbery materials [27–31]. In this paper, we investigate the deformation behaviors of two
3. Experimental results In this section, we briefly summarize the main experimental obser vations. Fig. 2 plots the force-displacement relationships for EPDM and Silicone rubbers. Both types of rubbers exhibits good ductility, though EPDM rubbers are more ductile in all the conditions. The failure displacement of EPDM rubbers specimens without geometry defects is around 110 mm, which is larger than the failure displacement of Silicone rubbers with a value of less than 75 mm. Geometry defects reduce the
Fig. 2. The force as a function of the displacement for: (a) EPDM rubbers and (b) Silicone rubbers. 2
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Fig. 3. Representative strain contours of Eyy for EPDM rubber specimens: (a) without geometry defects and (b) with a symmetric distributed half-circular defect.
Fig. 4. The Eyy distribution along the middle line of the specimens for (a) EPDM rubbers without geometry defects and (b) EPDM rubbers with a symmetric distributed half-circular defect.
failure displacement for both rubbers. For EPDM rubbers, the specimens with a circular defect in the middle exhibits the lowest failure displacement with a value of around 75 mm. However, compared with plastics, geometry imperfections have a less pronounced effect on the failure displacement in rubbers [32]. Shi et al. [32] found that the failure displacement of specimens with a geometry defect for thermoplastics was reduced by 4–10 times compared with specimens without geometry imperfections. For EPDM rubbers, the force shows a nearly linear in crease with the displacement. In contrast, the force-displacement of Silicone rubbers exhibites a linear region till a displacement of around 5 mm. The slope then decreases to a lower value. The geometry defects do not change the shape of the force-displacement curves, though the force is smaller for specimens with geometry defects. For all the condi tions, the tests were repeated at least twice. For EPDM rubbers, good repeatability has been observed. However, for Silicone rubbers, small variances can be observed for the repeated tests, as shown in Fig. S1 in the supplementary material. The lowest failure displacement may occur for the specimens with different geometry defects. Fig. 2 and Fig. S1 also shows that different locations of geometry defects seem to have an insignificant influence on the force response, though small variances can be observed for Silicone rubbers. The similar force response is probably because the same size of geometry defects are adopted in all the con ditions. To further validate the influence of the geometry size on the force response, the size of the circular defects was changed to a radius of
3 mm. As shown in Fig. S2 in the supplementary material, the specimens with the same size of geometry defects again exhibit a nearly identical force response. The DIC technique was employed to obtain the strain distribution. For the uniaxial deformation case, we show the engineering strain in the tension direction (Fig. 1), defined as Eyy ¼ ∂u=∂y, where u is the displacement in the tension direction. Fig. 3 plots two representative strain contours for EPDM rubbers. For specimens without geometry defects, a nearly homogenous strain field is observed as shown in Fig. 3a, though inhomogeneous strain appears at the two ends of the measured area. This is because the shoulders influence the strain distribution and can induce an inhomogeneous strain field for the middle part close to the shoulders. The strain contour before failure, which is the last subfigure in Fig. 3-a, shows that the strain distribution falls into a narrow region. This is different from glassy polymers, which exhibit a strain localization behavior for specimens without geometry defects [33,34]. To quanti tatively investigate the deformation behaviors, we also plot the strain along the middle line of the specimens in Fig. 4-a, which clearly shows a nearly constant strain value along the middle line for specimens without defects. Geometry defects can significantly affect the strain distribution. Fig. 3-b shows the strain contours of EPDM rubbers with a symmetric distributed half-circular defect on each edge. The strain around the defect is larger than the other parts and the specimens also fail at the 3
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Fig. 5. Comparison between measured and predicted stress-strain relationship of two rubbers: (a) EPDM rubbers and (b) Silicone rubbers.
with strain. A clear strain stiffening response can be observed under large deformation. As a consequence, the strain difference of rubber specimens with defects decreases with deformation. All the other experimental results of strain contours and strain evolution are showed in Figs. S3–S12 in the supplementary material.
defect region eventually. However, the strain in other regions is not negligible. For example, before failure the minimum engineering strain is 1.54 occurring in the undisturbed area and the maximum engineering strain is 2.29 around the geometry defects. This can also been observed from the strain profile as shown in Fig. 4-b. The strain reaches a maximum value in the middle point and gradually decreases to a steadystate value. This phenomenon significantly differs from the glassy polymers [32]. For glassy polymers, the deformation mainly occurs around the defect region. The strain far from the defect region is much smaller and remains in the elastic region. Thus, the strain field exhibits a very sharp peak. The different response to geometry defects arises from the distinct stress-strain relationship of rubbers and glassy polymers. For glassy polymers, the stress-strain relationship has a stiff elastic region followed by a strain softening region to reach a steady-state flow stress. The region around the defect first reaches the strain softening region and results in a strong localization region. As a consequence, the other parts are still in the small strain elastic region. Rubbers do not exhibit the stiff elastic and strain softening behaviors. The stress continuously increases
4. Numerical methods We also applied finite element simulation to investigate the effects of geometry defects on the deformation behaviors. To obtain the stressstrain relationship, we tracked the displacement of two points sepa rated by 5 mm in the middle part of the specimens without geometry defects. The correspondingly true strain was then calculated. Thus, this true strain is actually an averaging strain. From Fig. 4, it can be seen that the true strain obtained using this method can well represent the actual deformation state due to a nearly homogenous deformation behavior in the middle region. The true stress can be calculated by assuming volu metric incompressible. The response of both rubbers shows negligible
Fig. 6. Comparison between the measured and model predicted force-displacement relationship for EPDM rubber with different geometry defects: (a) no defect, (b) a circular defect in the middle, (c) a symmetric distributed half-circular defect, and (d) an antisymmetric distributed half-circular defect. 4
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Fig. 7. Comparison between the measured and model predicted force-displacement relationship for Silicone rubbers with different geometry defects: (a) no defect, (b) a circular defect in the middle, (c) a symmetric distributed half-circular defect, and (d) an antisymmetric distributed half-circular defect.
dependence on the loading rate. So the Arruda-Boyce eight chain hyperelastic model was used to fit the stress response of both rubbers. For finite deformation, we first define the deformation gradient as F ¼ ∂x=∂X, which maps the point X in the reference configuration to the point x in the current configuration. The left Cauchy deformation tensor b ¼ FFT and the right Cauchy-Green deformation tensor C ¼ FT F are also defined. The corresponding deviatoric part of Cauchy-Green deformation tensors can be calculated as C ¼ J 2=3 C and b ¼ J 2=3 b, where J ¼ detðFÞ. The Helmholtz free energy density of a quasiincompressible Arruda-Boyce model has the following form: � � � �� � x �� � λeff 1 y κ Ψ ¼ μN λ2L þ J 2 2lnJ 1 ; x þ ln μN λ2L y þ ln sinhx λL sinhy 2 λL (1)
are μN ¼ 1:194 MPa, λL ¼ 5:3 and κ ¼ 119:4 MPa. The values of the bulk modulus κ were chosen as two orders larger of the shear modulus to ensure the volumetric quasi-incompressible condition. As shown in Fig. 5, the Arruda-Boyce model can accurately describe the experi mentally measured stress for EPDM rubbers. For Silicone rubbers, the Arruda-Boyce model fails to capture the stress response in the small strain region, where Silicone rubbers exhibits a slightly stiffer response within a true strain of 20%. A more robust approach to obtain the hyperelastic constitutive relationships of rubbers can be found in Li and Liu [35] and Han et al. [36]. 4.1. Comparison between experiments and simulation The commercial finite element package ABAQUS was employed to simulate the response of two rubbers with or without geometry defects. The mesh was discretized using the C3D8H element. A convergence study was also applied until the simulation results showed negligible change with increasing the mesh density. We first compare the experi mentally measured and simulated global force-displacement relation ships under various geometry conditions. As shown in Fig. 6, for EPDM rubbers the simulated force-displacement shows a good agreement with the measured response for specimens without geometry defects. For the other cases, the simulation results overestimates the measured force response. For Silicone rubbers, discrepancies can be observed between experiments and simulations. For specimens without geometry defects, the simulation underestimates the experimentally measured force response. While for the specimens with geometry defects, compared with the measured response, the predicted force is initially smaller in the small region and larger in the large strain region. This discrepancy may be caused by the Arruda-Boyce model not fully capturing the constitu tive relationship of Silicone rubbers. For each rubber, the simulated force-displacement relationships with different geometry conditions are shown in Fig. S13 of the supplementary material. The general trend is consistent with the experimental data in Fig. 2. We then compare the strain contours from DIC and FEA. Fig. 8 plots
where μN is the characteristic chain stiffness of polymer chain, λL is the qffiffiffiffiffiffi 1 stretch limit, κ is the bulk modulus, λeff ¼ 3I1 , I 1 ¼ C : I, x ¼ � � � � L 1 λλeffL and y ¼ L 1 λ1L . The L ðuÞ ¼ cothu u 1 is the Langevin function. The Cauchy stress can which gives � �� 1μ λ λ σ ¼ N L L 1 eff b J 3 λeff λL
then be calculated through σ ¼ � 1 1 I 1 1 þ κðJ 3 J
1Þ1:
2 ∂Ψ T J F ∂CF ,
(2)
The stress response for uniaxial deformation tests can be obtained by assuming the deformation gradient only has three principal components F ¼ λ1 e1 � e1 þ λ2 e2 � e2 þ λ3 e3 � e3 . Given the stretch ratio λ1 in the tension direction, λ2 and λ3 can then be calculated based on σ2 ¼ 0 and σ 3 ¼ 0, which gives the corresponding stress σ1 . To obtain the model parameters, an iterative method was used to achieve the minimum value P of Ni ðσexp σsimu Þ2 . In total, 50 pairs of experimentally measured stressstrain data were used. The obtained μN , λL and κ for EPDM rubbers are 1.047 MPa, 3.47 and 104.7 MPa, while these values for Silicone rubbers 5
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Fig. 8. Comparison of the strain contour from DIC and finite element simulation of specimens without geometry defects: (a) EPDM rubbers and (b) Silicone rubbers.
Fig. 9. Comparison between measured and simulated Eyy distribution along the middle line of the specimen for (a) EPDM rubber without geometry defects and (b) Silicone rubber without geometry defects.
the strain contours of specimens without geometry defects. The results from both DIC and FEA show a homogenous strain field for both rubbers. To quantitatively evaluate the performance of finite element model, we also show the strain distribution along the middle line of the specimen in Fig. 9. The predicted engineering strain exhibits a good agreement with the measured value. The largest discrepancy occurrs for Silicone speci mens, showing that the simulated strain is smaller than the measured strain in the middle region. This is because the Arruda-Boyce model predicts a softer response in the small region, which leads to an over estimate of the strain on the shoulders of specimens and an underesti mate of the strain in the middle region. We also investigate the effects of geometry defects on the deforma tion behaviors. Fig. 10 directly compares the strain field of specimens at different deformation stages. In general, the simulation can well repro duce the main features observed in experiments. Specifically, simulation captures the shape of circular defects changes to an elliptical shape with deformation. The finite element simulation can also describe the for mation of different necked shapes of specimens with a half circular defect distributed symmetrically or antisymmetrically on each edge. The strain distribution around and far from the defect is quantitatively captured. The strain region from DIC and FEA is in general consistent. However, several inconsistencies also exist between experiments and simulation. For example, at the same displacement the elliptical shape predicted by finite element method is smaller than the measured shape. The strain field from DIC results does not show the small localized region
at the top and bottom of the elliptical shape. The strain localization zones in Fig. 10-c and Fig. 10-d is also smaller in the simulation than the zones measured in the experiment. The force-displacement relations in Figs. 6 and 7 show that when failure occurs the simulated force is larger than the measured force for specimens with an antisymmetric distrib uted half-circular defect. However, Fig. 10-c and d show that the maximum Eyy from the simulated contours is slightly smaller than the experimental data. This is probably because measuring the maximum strain for specimens with antisymmetric distributed defects is more challenge, which is more sensitive to the geometry inhomogeneous. To quantitatively evaluate the performance of numerical methods, we also compare the strain distribution of specimens with geometry defects from DIC and FEA. As shown in Fig. 11, the simulation results exhibit a good agreement with experimental results, which captures the maximum strain as well as the strain profile with locations. For speci mens with geometry defects, the crosssection is reduced by 40%. Due to strain stiffening, the average strain in the undisturbed crosssection should be more than 60% of the strain in the reduced crosssection, which is consistent with the observation in Fig. 11. However, the prin cipal strain around the defect is not always in the tension direction. Thus, the minimum strain Eyy from the contours is often smaller than 60% of the maximum Eyy from the contours as shown in Fig. 10. Fig. 11 also demonstrates some discrepancies between experiments and simu lation. For example, the predicted maximum strain at a displacement of 15 mm is smaller than the measured value for Silicone rubbers. This is 6
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Fig. 10. Comparison of the strain contours from DIC and finite element simulation: (a) EPDM rubbers with a circular defect, (b) Silicone rubbers with a circular defect, (c) EPDM rubbers with an antisymmetric distributed half-circular defect, (d) Silicone rubbers with an antisymmetric distributed half-circular defect, (e) EPDM rubbers with a symmetric distributed half-circular defect, and (f) Silicone rubbers with a symmetric distributed half-circular defect.
caused by the inaccuracy of the hyperelastic model to describe the stress response of Silicone rubbers in the small strain region. Among the three different defect positions, both experimental and simulation results show that the maximum Eyy for specimens with a circular defect in the middle is the highest at the same displacement, while the maximum Eyy for specimens with an antisymmetric distributed half-circular defect on each edge is the lowest at the same dispalcement. Both rubbers exhibit an abrupt failure when the local strain reaches a critical value. For EPDM rubbers, the failure strain is close for specimens with different geometry defects. However, for Silicone rubbers, the specimen with a symmetric distributed half-circular defect exhibits the lowest failure strain since the corresponding failure displacement is also the lowest in this set of experiments. Due to the variance in specimens, the lowest failure strain can also occur in specimens with other geometry defects (Fig. S1 in the supplementary material). The macroscopic response of materials is determined by the geom etry, loading conditions and constitutive relationship of the material. In this work, we investigate the mechanical response of two filled rubbers with different geometry defects. Both experiments and simulation were
performed to fully understand the mechanical response of these mate rials. The DIC method provides an accurate measurement of the strain distribution under large deformation, while the finite element analysis can provide a mechanism to explain the experimental observations. The presence of the geometry defects can induce strain localizations. The monotonic stress response of two rubbers has a restriction on the degree of localization. Thus, in general no severe localization occurs, which leads to a ductile response even for specimens with geometry defects. However, we only investigate the circular defects. In fact, the shape of defects has a significant influence on the deformation behaviors. The crack-type defects are most commonly used in the literature to charac terize the deformation behaviors of rubbers [3,4,31]. A strong strain localization zone can be observed around the crack tip. When beyond a certain threshold, the crack propagates, which further results in the failure of rubbers. For a circular defect, we do not observe a clearly damage formation and propagation around the defect. Instead, rubbers exhibit an abrupt failure when the local strain reaches a critical value. The polymer types also determine the deformation behaviors. In our previous works, we mainly focused on characterization the mechanical 7
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Fig. 11. Comparison between measured and simulated Eyy distribution along the middle line of the specimen for: (a) EPDM rubbers with an antisymmetric distributed half-circular defect, (b) Silicone rubbers with an antisymmetric distributed half-circular defect, (c) EPDM rubbers with an symmetric distributed halfcircular defect, (d) Silicone rubbers with a symmetric distributed half-circular defect.
localization behaviors and the final failure strain. We also applied FEA to investigate the experimental observations. The Arruda-Boyce eight chain model was adopted for both rubbers. The measured and simulated strain distributions show good agreement especially in the large defor mation region. However, discrepancies still exist between the predicted and measured force-displacement relationship, which indicates that other mechanisms need to be incorporated into the hyperelastic model. We should also emphasize that the characterization methods in this work are limited. We only investigated the influence of circular defects on the deformation behaviors. Also only uniaxial tension tests were performed. More tests need to be carried out to fully understand the effects of geometry imperfections on the deformation behaviors of en gineering rubbers.
behaviors of glassy polymer, such as amorphous thermoplastic poly (ethylene terephthalate)-glycol (PETG). The deformation behavior of PETG is inherently different from the two rubbers investigated in this work. Strain localization always occurs even for specimens without geometry defects due to strain softening [34]. The geometry defects can cause a strong localization zone around the defect. The strain remains at a low value for positions away from the defects. Thus, the failure displacement of specimens with defects is much smaller [32]. 5. Conclusions The deformation behaviors of rubbers play an essential role in their applications. The main challenge of characterization the behaviors of rubbers is that large deformation has to be considered. The recent development of the DIC technique has provided a powerful tool to measure the strain field of rubbers in the large deformation condition. In this work, we employed the DIC technique to obtain the uniaxial deformation behavior of two engineering rubbers, EPDM and Silicone rubbers. The results show that homogenous deformation occurs for specimens without geometry defects. In contrast, the geometry imper fections induce a localization phenomenon and a smaller failure displacement. All the specimens exhibit an abrupt failure when the local strain reaches a critical value. EPDM rubbers exhibit similar critical failure strains. However, variances can be observed in Silicone rubbers. In the repeated tests, the lowest critical failure strain may occur for specimens with different geometry defects. Compared with glassy polymers, the geometry defects have a less pronounced effect on the
Data availability The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study. Acknowledgements The authors acknowledge the Fundamental Research Funds for Central Universities, Hohai University (Grant No. 2019B85814), the project (No: 19HNKLE01) supported the Hunan Province Key Labora tory of Engineering Rheology, Central South University of Forestry and Technology.
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Appendix A. Supplementary data
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Polymer Testing journal homepage: http://www.elsevier.com/locate/polytest
Test Method
Experimental characterization and finite element modeling the deformation behavior of rubbers with geometry defects Huxiao Yang a, Rui Xiao a, b, *, Zhou Yang c, Dong Lei a a
Department of Engineering Mechanics, College of Mechanics and Materials, Hohai University, Nanjing, Jiangsu, 210098, China Key Laboratory of Soft Machines and Smart Devices of Zhejiang Province, Department of Engineering Mechanics, Zhejiang University, Hangzhou, Zhejiang, 310027, China c Department of Mechanical Engineering, University of Washington, Seattle, WA, 98195, USA b
A R T I C L E I N F O
A B S T R A C T
Keywords: Failure Rubber DIC Finite element modeling
Rubbers have broad applications in various areas. The engineering application requires accurately capturing the response of these materials under large deformation. In this work, we investigate the deformation behavior of two engineering rubbers (EPDM and Silicone rubbers) until failure occurs. The digital image correlation tech nique is used to capture the strain field of rubber specimens with different types of geometry defects. The results show that EPDM rubbers are more ductile than Silicone rubbers. For both rubbers a homogenous deformation behavior can be observed for specimens without defects, while strain localization behaviors occur for specimens with geometry defects. Finite element analysis (FEA) is also performed. The Arruda-Boyce eight chain model is used to represent the constitutive relationship of both rubbers with the parameters determined from the stressstrain response of specimens without geometry defects. The strain field obtained from FEA shows good agree ment with the experimental data even in the large strain region close to failure. However, the simulated forcedisplacement is consistent with the experimental results only for specimens without geometry defects. For specimens with geometry defects, discrepancies exist in all the cases, which indiates that more mechanisms need to be incorporated into the constitutive relationship.
1. Introduction Rubbers, which can sustain large deformations, have been widely applied in various areas, including tires, sealing materials and shock absorbers. However, natural rubbers are relatively soft, which are not suitable for applications requiring a high strength material. To overcome this limitation, the most common method is adding nanoparticles into rubber matrix to form rubber composites [1]. For example, carbon black filled rubbers are widely used in tires. For practical applications, rubbers are also required to have a high tolerance to geometry defects. Thus, various works have been done to investigate deformation behaviors of engineering rubbers in presence of different geometry imperfections [2–5]. The geometry defects can induce an inhomogeneous deformation state, which is a challenge for mechanical characterization. The recent development of digital image correlation (DIC) technique has provided several advantages to obtain the strain field of materials compared with traditional methods [6–11]. The traditional stiff strain gage may
influence strain distributions when attached to a soft rubbery material. In contrast, the DIC technique is a non-contact method, which eliminates this drawback. Also the DIC technique can measure strain distributions over a wide strain region, while the traditional methods typically can only measure the response of materials in a small strain region. More over, the measurement length-scale of the DIC technique ranges from a size from nanometer to megameter [12,13]. For all these advantages, the DIC technique has been widely used to characterize the deformation behaviors of materials. Due to abundant works in the literature, in the following we will only review the appli cation of the DIC technique to rubbery materials [2–4]. Mzabi et al. [14] applied DIC to measure the strain field around a crack tip of filled rubbers in the cyclic loading tests, while Major and Lang [15] and Martinez et al. [16] combined the DIC technique and infrared ther mography to characterize the fracture behaviors of rubbers in mono tonic and cyclic loading conditions respectively. Brighenti et al. [5] analyzed the defect tolerance of silicone rubbers with a central crack while the DIC technique was used to measure the strain distribution of
* Corresponding author. Department of Engineering Mechanics, College of Mechanics and Materials, Hohai University, Nanjing, Jiangsu, 210098, China. E-mail address: [email protected] (R. Xiao). https://doi.org/10.1016/j.polymertesting.2019.106111 Received 24 June 2019; Received in revised form 21 August 2019; Accepted 19 September 2019 Available online 25 September 2019 0142-9418/© 2019 Elsevier Ltd. All rights reserved.
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engineering rubbers, EPDM and Silicone rubbers. The DIC technique is used to measure the strain distribution of specimens with different types of geometry defects. The FEA is also performed to analyze the effects of geometry imperfections on the mechanical response. The paper is ar ranged as follows. The experimental methods are firstly described. Section 3 presents some representative experimental results. The constitutive relationship of the model and the methods of parameter determinations are then shown in Section 4. The followed section compares the experimental and simulated response of two rubbers. The conclusion part summarizes the main findings in this paper. 2. Experimental methods Fig. 1. The geometry of the specimens used in uniaxial tension tests. The unit of the dimension is mm. The marked area in the first subfigure is the measured area using the DIC technique.
The EPDM and Silicone rubbers used in this work were directly ob tained from the McMaster-Carr Supply Company and used as the inreceived condition. The filler for both rubbers is carbon black. Howev er, the content of the filler is not provided by the supplier. The film with a thickness of 1.59 mm (1/16 inch) was then manufactured to the experimental shape using laser cutting. Since Silicone rubbers can withstand a high temperature (250 � C), the laser cutting seems to have a limited influence on the property of the specimens. However, for EPDM rubbers the power of laser needs to be carefully controlled to minimize the influence of laser cutting. As shown in Fig. 1, three different ge ometry defects were also introduced to the dog-bone shaped specimens: a circular defect in the middle, a symmetric distributed half-circular defect on each edge, an antisymmetric distributed half-circular defect on each edge. Before tests, the surfaces of the EPDM specimens were sprayed with white speckles while the black speckles were used for Silicone speci mens. All experiments were performed on an Instron 3367 Series Testing Machine at room temperature with a 1 kN load cell using a uniaxial tension mode. The specimens were deformed at a rate of 15 mm/min until failure. A preliminary study showed that the performance of both rubbers had negligible dependence on the loading rate, which indicated the viscoelastic effects could be neglected for EPDM and Silicone rubbers at room temperature. During the tests, a CCD camera was used to record the surface of the specimens. The strain distribution was calculated using the commercial software VIC 2D.
specimens with different crack sizes and loading conditions. Schwan et al. [17] used the DIC technique to characterize the strain localization behaviors of aerogels in uniaxial compression tests, which were mainly caused by the local fracture of gels. Meunier et al. [18] combined ex periments and numerical simulation to investigate the deformation be haviors of unfilled silicone rubbers in tensile, compression, shear and bulge tests, which provided an insight on the choice of the constitutive models for rubbers. To obtain the strain field in large deformation re gion, the incremental DIC method is often used to guarantee a good correlation of images. Goh et al. [19] developed a single-step image correlation method and demonstrated this method could also provide an accurate calculation of strain field. In addition to experiments, numerical models are also widely employed to characterize the mechanical response of rubbers and soft materials [20,21]. For rubbers, the hyperelasticity should be first considered. It is now well accepted that the hyperelastic behaviors of rubbers origin from the entropy change during the deformation. Based on statistical thermodynamics, various models have been developed to describe the stress response of rubbers in the finite deformation region, such as three chain model, four chain model, eight chain models and full chain models [22–24]. Alternatively, phenomenological models have also been proposed based on the requirement of continuum mechanics [25,26]. Some rubbers exhibit a rate-dependent viscoelastic response and a deformation history dependent softening behavior. In these cases, the viscoelastic and damage mechanisms need to be incorporated into the constitutive relationships [27]. To capture the response of rubbers with complex geometry and loading conditions, the constitutive model needs to be implemented into a finite element suite. The finite element analysis (FEA) is a standard method and has already been widely used to characterize rubbery materials [27–31]. In this paper, we investigate the deformation behaviors of two
3. Experimental results In this section, we briefly summarize the main experimental obser vations. Fig. 2 plots the force-displacement relationships for EPDM and Silicone rubbers. Both types of rubbers exhibits good ductility, though EPDM rubbers are more ductile in all the conditions. The failure displacement of EPDM rubbers specimens without geometry defects is around 110 mm, which is larger than the failure displacement of Silicone rubbers with a value of less than 75 mm. Geometry defects reduce the
Fig. 2. The force as a function of the displacement for: (a) EPDM rubbers and (b) Silicone rubbers. 2
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Fig. 3. Representative strain contours of Eyy for EPDM rubber specimens: (a) without geometry defects and (b) with a symmetric distributed half-circular defect.
Fig. 4. The Eyy distribution along the middle line of the specimens for (a) EPDM rubbers without geometry defects and (b) EPDM rubbers with a symmetric distributed half-circular defect.
failure displacement for both rubbers. For EPDM rubbers, the specimens with a circular defect in the middle exhibits the lowest failure displacement with a value of around 75 mm. However, compared with plastics, geometry imperfections have a less pronounced effect on the failure displacement in rubbers [32]. Shi et al. [32] found that the failure displacement of specimens with a geometry defect for thermoplastics was reduced by 4–10 times compared with specimens without geometry imperfections. For EPDM rubbers, the force shows a nearly linear in crease with the displacement. In contrast, the force-displacement of Silicone rubbers exhibites a linear region till a displacement of around 5 mm. The slope then decreases to a lower value. The geometry defects do not change the shape of the force-displacement curves, though the force is smaller for specimens with geometry defects. For all the condi tions, the tests were repeated at least twice. For EPDM rubbers, good repeatability has been observed. However, for Silicone rubbers, small variances can be observed for the repeated tests, as shown in Fig. S1 in the supplementary material. The lowest failure displacement may occur for the specimens with different geometry defects. Fig. 2 and Fig. S1 also shows that different locations of geometry defects seem to have an insignificant influence on the force response, though small variances can be observed for Silicone rubbers. The similar force response is probably because the same size of geometry defects are adopted in all the con ditions. To further validate the influence of the geometry size on the force response, the size of the circular defects was changed to a radius of
3 mm. As shown in Fig. S2 in the supplementary material, the specimens with the same size of geometry defects again exhibit a nearly identical force response. The DIC technique was employed to obtain the strain distribution. For the uniaxial deformation case, we show the engineering strain in the tension direction (Fig. 1), defined as Eyy ¼ ∂u=∂y, where u is the displacement in the tension direction. Fig. 3 plots two representative strain contours for EPDM rubbers. For specimens without geometry defects, a nearly homogenous strain field is observed as shown in Fig. 3a, though inhomogeneous strain appears at the two ends of the measured area. This is because the shoulders influence the strain distribution and can induce an inhomogeneous strain field for the middle part close to the shoulders. The strain contour before failure, which is the last subfigure in Fig. 3-a, shows that the strain distribution falls into a narrow region. This is different from glassy polymers, which exhibit a strain localization behavior for specimens without geometry defects [33,34]. To quanti tatively investigate the deformation behaviors, we also plot the strain along the middle line of the specimens in Fig. 4-a, which clearly shows a nearly constant strain value along the middle line for specimens without defects. Geometry defects can significantly affect the strain distribution. Fig. 3-b shows the strain contours of EPDM rubbers with a symmetric distributed half-circular defect on each edge. The strain around the defect is larger than the other parts and the specimens also fail at the 3
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Fig. 5. Comparison between measured and predicted stress-strain relationship of two rubbers: (a) EPDM rubbers and (b) Silicone rubbers.
with strain. A clear strain stiffening response can be observed under large deformation. As a consequence, the strain difference of rubber specimens with defects decreases with deformation. All the other experimental results of strain contours and strain evolution are showed in Figs. S3–S12 in the supplementary material.
defect region eventually. However, the strain in other regions is not negligible. For example, before failure the minimum engineering strain is 1.54 occurring in the undisturbed area and the maximum engineering strain is 2.29 around the geometry defects. This can also been observed from the strain profile as shown in Fig. 4-b. The strain reaches a maximum value in the middle point and gradually decreases to a steadystate value. This phenomenon significantly differs from the glassy polymers [32]. For glassy polymers, the deformation mainly occurs around the defect region. The strain far from the defect region is much smaller and remains in the elastic region. Thus, the strain field exhibits a very sharp peak. The different response to geometry defects arises from the distinct stress-strain relationship of rubbers and glassy polymers. For glassy polymers, the stress-strain relationship has a stiff elastic region followed by a strain softening region to reach a steady-state flow stress. The region around the defect first reaches the strain softening region and results in a strong localization region. As a consequence, the other parts are still in the small strain elastic region. Rubbers do not exhibit the stiff elastic and strain softening behaviors. The stress continuously increases
4. Numerical methods We also applied finite element simulation to investigate the effects of geometry defects on the deformation behaviors. To obtain the stressstrain relationship, we tracked the displacement of two points sepa rated by 5 mm in the middle part of the specimens without geometry defects. The correspondingly true strain was then calculated. Thus, this true strain is actually an averaging strain. From Fig. 4, it can be seen that the true strain obtained using this method can well represent the actual deformation state due to a nearly homogenous deformation behavior in the middle region. The true stress can be calculated by assuming volu metric incompressible. The response of both rubbers shows negligible
Fig. 6. Comparison between the measured and model predicted force-displacement relationship for EPDM rubber with different geometry defects: (a) no defect, (b) a circular defect in the middle, (c) a symmetric distributed half-circular defect, and (d) an antisymmetric distributed half-circular defect. 4
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Fig. 7. Comparison between the measured and model predicted force-displacement relationship for Silicone rubbers with different geometry defects: (a) no defect, (b) a circular defect in the middle, (c) a symmetric distributed half-circular defect, and (d) an antisymmetric distributed half-circular defect.
dependence on the loading rate. So the Arruda-Boyce eight chain hyperelastic model was used to fit the stress response of both rubbers. For finite deformation, we first define the deformation gradient as F ¼ ∂x=∂X, which maps the point X in the reference configuration to the point x in the current configuration. The left Cauchy deformation tensor b ¼ FFT and the right Cauchy-Green deformation tensor C ¼ FT F are also defined. The corresponding deviatoric part of Cauchy-Green deformation tensors can be calculated as C ¼ J 2=3 C and b ¼ J 2=3 b, where J ¼ detðFÞ. The Helmholtz free energy density of a quasiincompressible Arruda-Boyce model has the following form: � � � �� � x �� � λeff 1 y κ Ψ ¼ μN λ2L þ J 2 2lnJ 1 ; x þ ln μN λ2L y þ ln sinhx λL sinhy 2 λL (1)
are μN ¼ 1:194 MPa, λL ¼ 5:3 and κ ¼ 119:4 MPa. The values of the bulk modulus κ were chosen as two orders larger of the shear modulus to ensure the volumetric quasi-incompressible condition. As shown in Fig. 5, the Arruda-Boyce model can accurately describe the experi mentally measured stress for EPDM rubbers. For Silicone rubbers, the Arruda-Boyce model fails to capture the stress response in the small strain region, where Silicone rubbers exhibits a slightly stiffer response within a true strain of 20%. A more robust approach to obtain the hyperelastic constitutive relationships of rubbers can be found in Li and Liu [35] and Han et al. [36]. 4.1. Comparison between experiments and simulation The commercial finite element package ABAQUS was employed to simulate the response of two rubbers with or without geometry defects. The mesh was discretized using the C3D8H element. A convergence study was also applied until the simulation results showed negligible change with increasing the mesh density. We first compare the experi mentally measured and simulated global force-displacement relation ships under various geometry conditions. As shown in Fig. 6, for EPDM rubbers the simulated force-displacement shows a good agreement with the measured response for specimens without geometry defects. For the other cases, the simulation results overestimates the measured force response. For Silicone rubbers, discrepancies can be observed between experiments and simulations. For specimens without geometry defects, the simulation underestimates the experimentally measured force response. While for the specimens with geometry defects, compared with the measured response, the predicted force is initially smaller in the small region and larger in the large strain region. This discrepancy may be caused by the Arruda-Boyce model not fully capturing the constitu tive relationship of Silicone rubbers. For each rubber, the simulated force-displacement relationships with different geometry conditions are shown in Fig. S13 of the supplementary material. The general trend is consistent with the experimental data in Fig. 2. We then compare the strain contours from DIC and FEA. Fig. 8 plots
where μN is the characteristic chain stiffness of polymer chain, λL is the qffiffiffiffiffiffi 1 stretch limit, κ is the bulk modulus, λeff ¼ 3I1 , I 1 ¼ C : I, x ¼ � � � � L 1 λλeffL and y ¼ L 1 λ1L . The L ðuÞ ¼ cothu u 1 is the Langevin function. The Cauchy stress can which gives � �� 1μ λ λ σ ¼ N L L 1 eff b J 3 λeff λL
then be calculated through σ ¼ � 1 1 I 1 1 þ κðJ 3 J
1Þ1:
2 ∂Ψ T J F ∂CF ,
(2)
The stress response for uniaxial deformation tests can be obtained by assuming the deformation gradient only has three principal components F ¼ λ1 e1 � e1 þ λ2 e2 � e2 þ λ3 e3 � e3 . Given the stretch ratio λ1 in the tension direction, λ2 and λ3 can then be calculated based on σ2 ¼ 0 and σ 3 ¼ 0, which gives the corresponding stress σ1 . To obtain the model parameters, an iterative method was used to achieve the minimum value P of Ni ðσexp σsimu Þ2 . In total, 50 pairs of experimentally measured stressstrain data were used. The obtained μN , λL and κ for EPDM rubbers are 1.047 MPa, 3.47 and 104.7 MPa, while these values for Silicone rubbers 5
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Fig. 8. Comparison of the strain contour from DIC and finite element simulation of specimens without geometry defects: (a) EPDM rubbers and (b) Silicone rubbers.
Fig. 9. Comparison between measured and simulated Eyy distribution along the middle line of the specimen for (a) EPDM rubber without geometry defects and (b) Silicone rubber without geometry defects.
the strain contours of specimens without geometry defects. The results from both DIC and FEA show a homogenous strain field for both rubbers. To quantitatively evaluate the performance of finite element model, we also show the strain distribution along the middle line of the specimen in Fig. 9. The predicted engineering strain exhibits a good agreement with the measured value. The largest discrepancy occurrs for Silicone speci mens, showing that the simulated strain is smaller than the measured strain in the middle region. This is because the Arruda-Boyce model predicts a softer response in the small region, which leads to an over estimate of the strain on the shoulders of specimens and an underesti mate of the strain in the middle region. We also investigate the effects of geometry defects on the deforma tion behaviors. Fig. 10 directly compares the strain field of specimens at different deformation stages. In general, the simulation can well repro duce the main features observed in experiments. Specifically, simulation captures the shape of circular defects changes to an elliptical shape with deformation. The finite element simulation can also describe the for mation of different necked shapes of specimens with a half circular defect distributed symmetrically or antisymmetrically on each edge. The strain distribution around and far from the defect is quantitatively captured. The strain region from DIC and FEA is in general consistent. However, several inconsistencies also exist between experiments and simulation. For example, at the same displacement the elliptical shape predicted by finite element method is smaller than the measured shape. The strain field from DIC results does not show the small localized region
at the top and bottom of the elliptical shape. The strain localization zones in Fig. 10-c and Fig. 10-d is also smaller in the simulation than the zones measured in the experiment. The force-displacement relations in Figs. 6 and 7 show that when failure occurs the simulated force is larger than the measured force for specimens with an antisymmetric distrib uted half-circular defect. However, Fig. 10-c and d show that the maximum Eyy from the simulated contours is slightly smaller than the experimental data. This is probably because measuring the maximum strain for specimens with antisymmetric distributed defects is more challenge, which is more sensitive to the geometry inhomogeneous. To quantitatively evaluate the performance of numerical methods, we also compare the strain distribution of specimens with geometry defects from DIC and FEA. As shown in Fig. 11, the simulation results exhibit a good agreement with experimental results, which captures the maximum strain as well as the strain profile with locations. For speci mens with geometry defects, the crosssection is reduced by 40%. Due to strain stiffening, the average strain in the undisturbed crosssection should be more than 60% of the strain in the reduced crosssection, which is consistent with the observation in Fig. 11. However, the prin cipal strain around the defect is not always in the tension direction. Thus, the minimum strain Eyy from the contours is often smaller than 60% of the maximum Eyy from the contours as shown in Fig. 10. Fig. 11 also demonstrates some discrepancies between experiments and simu lation. For example, the predicted maximum strain at a displacement of 15 mm is smaller than the measured value for Silicone rubbers. This is 6
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Fig. 10. Comparison of the strain contours from DIC and finite element simulation: (a) EPDM rubbers with a circular defect, (b) Silicone rubbers with a circular defect, (c) EPDM rubbers with an antisymmetric distributed half-circular defect, (d) Silicone rubbers with an antisymmetric distributed half-circular defect, (e) EPDM rubbers with a symmetric distributed half-circular defect, and (f) Silicone rubbers with a symmetric distributed half-circular defect.
caused by the inaccuracy of the hyperelastic model to describe the stress response of Silicone rubbers in the small strain region. Among the three different defect positions, both experimental and simulation results show that the maximum Eyy for specimens with a circular defect in the middle is the highest at the same displacement, while the maximum Eyy for specimens with an antisymmetric distributed half-circular defect on each edge is the lowest at the same dispalcement. Both rubbers exhibit an abrupt failure when the local strain reaches a critical value. For EPDM rubbers, the failure strain is close for specimens with different geometry defects. However, for Silicone rubbers, the specimen with a symmetric distributed half-circular defect exhibits the lowest failure strain since the corresponding failure displacement is also the lowest in this set of experiments. Due to the variance in specimens, the lowest failure strain can also occur in specimens with other geometry defects (Fig. S1 in the supplementary material). The macroscopic response of materials is determined by the geom etry, loading conditions and constitutive relationship of the material. In this work, we investigate the mechanical response of two filled rubbers with different geometry defects. Both experiments and simulation were
performed to fully understand the mechanical response of these mate rials. The DIC method provides an accurate measurement of the strain distribution under large deformation, while the finite element analysis can provide a mechanism to explain the experimental observations. The presence of the geometry defects can induce strain localizations. The monotonic stress response of two rubbers has a restriction on the degree of localization. Thus, in general no severe localization occurs, which leads to a ductile response even for specimens with geometry defects. However, we only investigate the circular defects. In fact, the shape of defects has a significant influence on the deformation behaviors. The crack-type defects are most commonly used in the literature to charac terize the deformation behaviors of rubbers [3,4,31]. A strong strain localization zone can be observed around the crack tip. When beyond a certain threshold, the crack propagates, which further results in the failure of rubbers. For a circular defect, we do not observe a clearly damage formation and propagation around the defect. Instead, rubbers exhibit an abrupt failure when the local strain reaches a critical value. The polymer types also determine the deformation behaviors. In our previous works, we mainly focused on characterization the mechanical 7
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Fig. 11. Comparison between measured and simulated Eyy distribution along the middle line of the specimen for: (a) EPDM rubbers with an antisymmetric distributed half-circular defect, (b) Silicone rubbers with an antisymmetric distributed half-circular defect, (c) EPDM rubbers with an symmetric distributed halfcircular defect, (d) Silicone rubbers with a symmetric distributed half-circular defect.
localization behaviors and the final failure strain. We also applied FEA to investigate the experimental observations. The Arruda-Boyce eight chain model was adopted for both rubbers. The measured and simulated strain distributions show good agreement especially in the large defor mation region. However, discrepancies still exist between the predicted and measured force-displacement relationship, which indicates that other mechanisms need to be incorporated into the hyperelastic model. We should also emphasize that the characterization methods in this work are limited. We only investigated the influence of circular defects on the deformation behaviors. Also only uniaxial tension tests were performed. More tests need to be carried out to fully understand the effects of geometry imperfections on the deformation behaviors of en gineering rubbers.
behaviors of glassy polymer, such as amorphous thermoplastic poly (ethylene terephthalate)-glycol (PETG). The deformation behavior of PETG is inherently different from the two rubbers investigated in this work. Strain localization always occurs even for specimens without geometry defects due to strain softening [34]. The geometry defects can cause a strong localization zone around the defect. The strain remains at a low value for positions away from the defects. Thus, the failure displacement of specimens with defects is much smaller [32]. 5. Conclusions The deformation behaviors of rubbers play an essential role in their applications. The main challenge of characterization the behaviors of rubbers is that large deformation has to be considered. The recent development of the DIC technique has provided a powerful tool to measure the strain field of rubbers in the large deformation condition. In this work, we employed the DIC technique to obtain the uniaxial deformation behavior of two engineering rubbers, EPDM and Silicone rubbers. The results show that homogenous deformation occurs for specimens without geometry defects. In contrast, the geometry imper fections induce a localization phenomenon and a smaller failure displacement. All the specimens exhibit an abrupt failure when the local strain reaches a critical value. EPDM rubbers exhibit similar critical failure strains. However, variances can be observed in Silicone rubbers. In the repeated tests, the lowest critical failure strain may occur for specimens with different geometry defects. Compared with glassy polymers, the geometry defects have a less pronounced effect on the
Data availability The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study. Acknowledgements The authors acknowledge the Fundamental Research Funds for Central Universities, Hohai University (Grant No. 2019B85814), the project (No: 19HNKLE01) supported the Hunan Province Key Labora tory of Engineering Rheology, Central South University of Forestry and Technology.
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Appendix A. Supplementary data
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