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A micromechanics model for predicting the stress–strain relations of filled elastomers
Computational Materials Science 67 (2013) 104–108
Contents lists available at SciVerse ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
A micromechanics model for predicting the stress–strain relations of filled elastomers Yunpeng Jiang ⇑, Hualin Fan Department of Engineering Mechanics, Hohai University, Nanjing 210098, China
a r t i c l e
i n f o
Article history: Received 14 June 2012 Received in revised form 16 August 2012 Accepted 23 August 2012 Available online 4 October 2012 Keywords: Polymer–matrix composites (PMCs) Stress/strain curves Modeling Non-linear behavior Micromechanics
a b s t r a c t In this paper, a simple micromechanics model has been developed for predicting the effective stress–strain relations of filled elastomers. The present model constitutes of the instantaneous Young’s modulus and Poisson’s ratio characterizing rubber-like material, a double-inclusion configuration taking account of the absorption of rubber chains onto carbon-black (CB) particles, and the incremental Mori–Tanaka method to compute the effective stress–strain curves. The subsequent predictions are capable of reflecting the well-known S-shape of hyper-elastic composites, and verified via the comparison to the available experiments and analytical models. Parametric analysis is fatherly conducted on the microstructure effect. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction Filled elastomers have been widely applied in various industrial fields due to their large-deformability. In order to know their performance, the equivalent properties at mesoscopic scale should be exactly determined from their microstructures and constituents. Therefore, an accurate but simple homogenization method is very necessary and impending for their widely application. The available researches are grossly classified intomicromechanics method and molecular chain network models. Firstly, micromechanics approaches are briefly reviewed. Mullins and Tobin [1], Qi et al. [2] systematically measured the stress–strain relations of vulcanized rubbers containing CB powders, and introduced a strain amplification factor v ¼ E=EM ¼ 1 þ 2:5f P þ 14:1fP2 (EM – matrix modulus, fP – particle volume fraction) to describe the enhanced elastic property. Although a good agreement was achieved between the predictions and experiments, the constraint effect of the matrix on the CB powders is not accounted for by such a factor. Bergstrom and Boyce [3] investigated the influence of hard particles on the mechanical response of rubber matrix composites with experiment and FEM. A new concept was proposed based on the first strain invariant instead of strain amplification, and could predict the experiments very well. To the best of the authors’ knowledge, the finite deformation of composites was not really solved until Ponte Castaneda [4,5] developed a second-order homogenization method, and adopted the optimization computation to solve the ⇑ Corresponding author. Tel./fax: +86 25 83786046. E-mail address: [email protected] (Y. Jiang). 0927-0256/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.commatsci.2012.08.036
energy minimum problem to determine the appropriate deformation compatible conditions between two phases. But their method cannot but to solve many partial differential equations by a special code programming. Moreover, such method is so complicated and only confined to the plane strain case. Bouchart et al. [6,7] proposed another alternative formula to study compressible hyper-elastic composites. Omnès et al. [8] developed a generalized self-consistent scheme, containing the occluded rubber, the bound rubber and a percolating network, to predict the elasticity of filled rubber. Yin [9] developed a constitutive model for particle reinforced elastomers based on Eshelby’s tensor [10]. Although the interaction between particles and matrix is fully considered, some deductions are arguable. For instance, small strain Eshelby’s tensor was adopted in dealing with finite strain deformation. NematNasser [11,12] first generalized the Eshelby’s tensor for finite strain deformation, and obtained the corresponding formula. Yang and Xu [13] investigated the macroscopic hyper-elastic behavior of fiber reinforced polymers using a micromechanical model and finite strain theory based on the hyper-elastic constitutive law. deBotton [14] proposed a new strain energy function in terms of the behaviors of the individual phases and volume fractions for a finitely extensible solid with a preferred direction. Huang et al. [15] presented a micromechanical analysis for predicting the stress/strain behavior of the composite made of weft-knit polyester fiber interlock fabric and a polyurethane elastomer matrix. Li et al. [16] experimentally studied CB filled rubber and numerically analyzed the local strain filed, concluded that the local strain distribution in a rubber matrix approximately obeys the statistical Gaussian distribution. Abadi [17] presented a homogenization
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Y. Jiang, H. Fan / Computational Materials Science 67 (2013) 104–108
procedure to predict the shear response of heterogeneous materials at large deformation. Many molecular chain network models are commonly adopted to understand the inherent morphology evolution. Drozdov and Dorfmann [18] emphasized the mechanical energy instead of entropy theory of polymer chains at finite strain deformation, and developed a micromechanics model for polymer and filled polymer as well. Based on the additive network configurations, Govindjee and Simo [19] developed a micromechanics model and analyzed the Mullins’ effect and debonding damage in CB filled rubbers. Furthermore, Dargazany and Itskov [20] proposed a network evolution model to study the deformation-induced anisotropy and damage. In summary, it is challenging to establish the deformation compatible conditions between particle and matrix in finite strain case, but which is a necessary prerequisite to interpret the effective mechanical behaviors of filled elastomers. A constitutive model for filled elastomers should be accurate in predicting the stress–strain relations. While for its practical application, the model should be comparatively simple to be readily implemented in a general purpose finite element code for analyzing the macroscopic composite structures. To meet these requirements, this paper presents a simple but accurate micromechanics model for the nonlinear stress–strain relations of filled elastomers. The predicted stress–strain curves are compared with published experiments, and parametric analysis is also performed. 2. The micromechanics model 2.1. The tangent elasticity of rubber matrix Rubber is supposed to be isotropic, hyper-elastic, nearly incompressible and described by the following Ogden’s strain energy function [21] for its high accuracy: n X 2ai bi b b W¼ k1 þ k2i þ k3i 3 bi i¼1
ð1Þ
here ai and bi are unknown parameters, which are determined by fitting to the uniaxial stress–strain curves. ki are the material principal stretches which related to the principal strains via ki = 1 + ei. The equivalent strain is given by:
ee ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 e:e¼ e þ e222 þ e233 þ 2 e223 þ e231 þ e212 3 3 11
ð2Þ
Due to the incompressibility condition k1k2k3 = 1. The strain components should satisfy the following relation:
e22 ¼ e33 ¼ ð1 þ e11 Þ1=2 1
2.2. The double-inclusion (DI) model Fig. 1 sketches the homogenization procedure of CB reinforced rubber. As for the actual composites, CB nano-powders easily agglomerate to form a network structure in the rubber matrix. These networks surround a part of polymer chains whose deformation is greatly impeded during the applied stretch. CB phase deforms linear elastically due to its high stiffness in relation to the rubber phase. Therefore, CB particles and the surrounded rubber form a new kind of inclusions dispersed in the matrix. What follows is the homogenization procedure. CB particles are equivalent to a homogeneous layer, and its volume fraction just equals to the given CB concentration as a known parameter. The surrounded rubber is assumed to deform linear elastically, and its material properties are given as the already known parameters. The volume fraction of DI is defined by:
fDI ¼ fP
1 þ 0:02139DBP abs 1:46
ð6Þ
where DBPabs denote DBP (dibuty1 phthalate) absorption and already measured for different rubbers [22]. A DI configuration is subsequently constructed as shown in Fig. 1, and its equivalent stiffness can be computed by our previous research [24]. In order to realize this purpose, a new configuration including a particle and its surrounding layer is firstly constructed. During the DI construction, the boundaries of particle and its surrounding layer are supposed to be parallel for simplicity. In the DI model, a centered particle is surrounded by an inhomogeneous layer, in turn embedded in an infinite matrix CM, and the layer and particle constructs a DI. The DI includes a layer of ellipsoid XC with a uniform stiffness tensor CM. The layer volume C fraction is fDI ¼ V C =V DI . The equivalent stiffness of DI is determined by [23]:
h i 1 1 P CDI ¼ fDI CP ðI þ S K1 þ fDIC CC ðI þ S K1 P Þ C Þ 1 1 1 1 P C ðI þ S K1 Þ þ f I þ S K fDI DI C P
ð7Þ
where KC ¼ ðCC CM Þ1 ðCM Þ. After giving the equivalent stiffness of a DI, the next step is to predict the effective properties of filled elastomers reinforced by the DI instead of CB particles. During the computation of the effective composite behaviors, the stiffness tensor CP should be replaced by CDI, which determined by Eq. (7), with DI fraction fDI instead of particle fraction fP. 2.3. The incremental Mori–Tanaka method
ð3Þ
In order to linearly handle the nonlinear elastomer, the whole deformation procedure is divided into n loading step, i.e., the applied stretch is imposed on the specimen step by step up to the final deformation. The tangent modulus at each step is different, and the instantaneous modulus and Poisson’s ratio at the nth step are defined by:
The stress–strain relations would be predicted for the fictitious composites with DI inclusions. Based on Benveniste’s formula of Mori–Tanaka method [24], and after a simple deduction, the incremental constitutive relation of composite is given as:
de ¼ ðI þ fP AÞC1 M dr
ð8Þ 1
En ¼
rn11 rn1 11 en1 en1 1
ð4Þ
mn ¼
1=2 ð1 þ en1 ð1 þ en1 Þ1=2 1 Þ n e1 en1 1
ð5Þ
where r11 and e11 denotes the nominal stress and strain in the loading direction, respectively. In this paper, all the terms for the reinforcement and matrix are represented by symbols with subscripts ‘P’ and ‘M’, respectively, and those of the composite are denoted by symbols without any script. All the tensors and vectors are written in boldface letters.
with A ¼ ½ðCP CM ÞS þ CM þ fP ðCP CM ÞðI SÞ ðCP CM Þ The incremental first P–K stress tensors (nominal stress) of constituents are respectively given by:
drM ¼ CM ½I þ ð1 fP ÞðI SÞAC1 M dr
ð9:aÞ
drM ¼ CM ½I þ fP ðI SÞAC1 M dr
ð9:bÞ
Similarly, the incremental strain tensors of constituents are respectively written as:
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Carbon-Black CB layer
Rubber
Rubber
ΩΓ
homogeneous material
ΩP
deM ¼ ½I þ fP ðI
ð10:aÞ
SÞAC1 M d
r
ð10:bÞ
3. Results and discussion 3.1. Verification of the analytical model Fig. 2 shows the comparisons between theoretic predictions and experiments [1] for rubber containing different concentrations of MT-CB under tension. Unless indicated otherwise, the constituent properties are EM ¼ 10 MPa; v M ¼ 0:5, EP ¼ 10 GPa; v P ¼ 0:3, and 0.02139DBPabs = 2.5 [22]. It is clearly shown that all the predictions are in fair agreement with experiments except at the higher stretch stage. The main reason is that the debonding damage between particle and rubber matrix would occur with increasing the applied stretch. A better prediction is undoubtedly expected by considering the debonding damage in the model. Fig. 3 shows the comparisons between theoretic predictions and experiments [3] for natural rubbers filled with CB N351 under compression. Here, 0:02139 DBP abs ¼ 2:05 [23]. Unlike uniaxial tension, particle–matrix interface does not separate under compression, and thus all the predicted stress–stretch curves agree with the test data very well in the whole stretch range, which again confirms the accuracy of the developed model. The present model is fatherly compared with published analytical models at fP = 21% as shown in Fig. 4. Compared to other models, the present method gives the best predictions. What’s the most important is that only the present model captures the S-shape of the stress–stretch relations. 3.2. Verification of the model for fiber filled elastomers
Nominal Stress σ11 (MPa)
Two analytical models are recently developed for fiber reinforced composites. One is developed by Lopez-Pamies as [4]:
fP=21%
2
13.8% 8.4%
1 matrix
0
0
0.5
1.0
Experiment [3] Predictions EP=10GPa, vP=0.3
-6 -5
fP=25% 15%
-4 -3 -2
7%
-1 matrix
0
-0.1 -0.2 -0.3 -0.4 -0.5 -0.6
Stretch λ1 Fig. 3. Theoretical predictions & experiments [3] for natural rubbers filled with of CB N351 under compression.
Experiment [1] Bergstrom-Boyce Govindjee-Simo Yin-Sun The present model
3
2
1
0 0
0.5
1.0
1.5%
5.3%
1.5
2
Engineering Strain (λ1−1) Fig. 2. Theoretical predictions & experiments [1] for rubber filled with different concentrations of MT-CB under tension.
1.5
Engineering Strain (λ1−1) Fig. 4. Comparisons between different analytical models and experiments [1] for fP = 21%.
l ð1 þ fP Þ 2 1 WðkÞ ¼ M k þ 22 2 ð1 fP Þ k
ð11Þ
where k ¼ k1 ¼ 1=k2 . The other is proposed by deBotton as [14]:
l ðk 1Þ2 ½ðk þ 1Þ2 þ 2f P ðk2 þ 1Þ WðkÞ ¼ M 2 2 k ð1 fP Þ
Experiment [1] Predictions EP=10GPa, vP=0.3
3
-7
0
Nominal Stress σ11 (MPa)
1 deP ¼ C1 P CM ½I ð1 fP ÞðI SÞACM dr
Nominal Stress σ11 (MPa)
Fig. 1. Homogenization procedure of carbon black (CB) reinforced rubber.
ð12Þ
Fig. 5 shows the comparison between the predicted stress– strain relations by different analytical models for rigid fiber reinforced composites, where 1-axis is in the transverse plane vertical to the fiber direction. The deBotton’s model [25] gives much better predictions with FEM results [26] than the other models. The predictions of the present model and Lopez’s model superpose each other, and slightly deviate from FEM results. Figs. 6 and 7 compared the stress–strain relations of rigid fiber reinforced composites predicted by the present model with FEM [26]. The matrix materials in the two diagrams are very different, and the matrix in Fig. 7 reaches the lock-up stage at the lower stretch as compared to that in Fig. 6. The present model is
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Y. Jiang, H. Fan / Computational Materials Science 67 (2013) 104–108
Nominal Stress σ11 (MPa)
Nominal Stress σ11 (MPa)
5 FEM Lopez's model deBotton's model the present model
4 3 2
matrix
1 0
fP=0.2
1
1.5
2.0
2.5
4 EP=10GPa vP=0.3 fP=0.5
3
0.3 0.2
2
0.1 0
1 0
3
0
3 2
matrix fP=0.1 fP=0.2 fP=0.3 fP=0.4
1 0 1
1.5
2.0
4
1.5
2.0
2.5
EP= EP=10000MPa EP=1000MPa EP=1MPa EP=0
3 2 1
x matri increasing the modulus
0 0
0.5
1.0
1.5
2.0
2.5
Engineering Strain (λ1−1)
2.5
Stretch λ1
1.0
Fig. 8. Dependence of the stress–strain relations on particle volume fraction fP.
Nominal Stress σ11 (MPa)
Nominal Stress σ11 (MPa)
4
0.5
Engineering Strain (λ1−1)
Stretch λ1 Fig. 5. The predicted stress–strain relations by different analytical models for fiber reinforced composites.
0.4
Fig. 9. Dependence of the stress–strain relations on particle elastic modulus.
Fig. 6. Theoretical predictions & FEM [26] for rigid fiber reinforced composites.
Stress σ11 or τ12 (MPa)
Nominal Stress σ11 (MPa)
3
20
fP=0.1 0.2 0.3 0.4
15 10 5
matrix
EP=10GPa, vP=0.3 fP=0.3
2
1
1.5
2.0
2.5
Stretch λ1
biaxial shear
1 matrix
0
0
uniaxial
0
0.5
1.0
1.5
2
Engineering Strain (λ1−1) Fig. 10. The effect of loading path on the composite stress–strain relations.
Fig. 7. Theoretical predictions & FEM [26] for rigid fiber reinforced composites.
essentially based on the volumetric average of the stress/strain fields, which is very hard to account for the deformation localization induced by the lock-up of rubber matrix. The analytical model provides good estimates of the deformation response at a low particle volume fraction. But the predictions deviate rapidly from FEM results with increasing particle volume fraction and applied stretches. The discrepancy between the predictions and FEM data in Fig. 6 is a little lower as compared to those in Fig. 7, which is induced by the early lock-up of the matrix at the first stretch stage. In order to better predict the deformation response of filled elastomers, high-order homogenization method should be developed to fully reflect the microstructural effect on the soft-hard heterogeneous systems. Furthermore, the microstructures, e.g., particle distribution, matrix properties and etc., maybe changes a lot during the large deformation, and the corresponding induced influence should be considered in improving the analytical models.
3.3. Effect of particle stiffness and particle concentration Fig. 8 illustrates the dependence of the composite stress–strain relations on particle volume fractions fP in the range of 0–50%. As expected, increasing the particle volume fraction also increases the overall stress at the same applied stretch. Fig. 9 shows the effect of particle stiffness on the deformation response of the composites, where EP is in a wide range of 0–1. EP = 0 denotes void and 1 rigid inclusion. It is clearly shown that the effective stress increases with increasing particle stiffness at the very beginning, but cannot enhance fatherly after exceeding a certain value, which is also confirmed by Bergstrom’s simulations [3] and experiments [27]. 3.4. Effect of loading path In the graph of Fig. 10, the evolution of the nominal stresses r11 or s12 are reported as a function of the principle elongation k1 for
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Y. Jiang, H. Fan / Computational Materials Science 67 (2013) 104–108
tensile, pure shear and biaxial tension). Stress levels obtained for the biaxial tension is higher than that computed in uniaxial tension, and that of shear is the lowest. These results highlight the strong influence of the loading path on stress–strain relations. 4. Conclusions A micromechanics model is proposed to predicate the finite strain stress–strain relations of filled elastomers. The double-inclusion model is employed to consider the constraint effect of CB network on the rubber matrix. The nonlinear constitutive relation of rubber-like materials is linearized by the instantaneous Young’s modulus and Poisson’s ratio. The stress–strain curves of filled elastomers are predicted by the incremental Mori–Tanaka method. The developed model is verified via published experiments and analytical models. Additionally, parametric analysis is performed on the microstructure effect. Several important conclusions are reached: (1) The CB network configuration should be taken into account in predicting the effective behaviors of filled elastomers. The deformation of rubber matrix surrounded by CB network is greatly impeded, and in turn enhances the load-carrying capability of composites. (2) The volumetric-average based homogenization model cannot account for the deformation location of rubber matrix, and thus overestimate the stretch response of filled elastomers. (3) Compared to the existing analytical models, the present model is very accurate and very simple in interpreting the effective performance of filled elastomers. Acknowledgements This work was supported by the Fundamental Research Funds for the Central Universities (No. B11020079), Jiangsu Provincial
Natural Science Foundation (No. BK2012407) and National Natural Science Foundation of China (No. 11202064). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]
L. Mullins, N.R. Tobin, J. Appl. Polymer. Sci. 9 (1965) 2993–3009. H.J. Qi, M.C. Boyce, J. Mech. Phys. Solids 52 (2004) 2187–2205. J.S. Bergstrom, M.C. Boyce, Rubber Chem. Tech. 72 (1999) 633–656. O. Lopez-Pamies, P. Ponte-Castaneda, J. Mech. Phys. Solids 54 (2006) 807–830. O. Lopez-Pamies, P. Ponte-Castañeda, J. Mech. Phys. Solids 55 (2007) 1677– 1701. V. Bouchart, M. Brieu, D. Kondo, M. Naït Abdelaziz, Comput. Mater. Sci. 43 (2008) 670–680. V. Bouchart, M. Brieu, N. Bhatnagar, D. Kondo, Int. J. Solids Struct. 47 (2010) 1737–1750. B. Omnès, S. Thuillier, P. Pilvin, Y. Grohens, S. Gillet, Composites: Part A 39 (2008) 1141–1149. H.M. Yin, L.Z. Sun, J.S. Chen, Mech. Mater. 34 (2002) 505–516. J.D. Eshelby, Proc. Roy. Soc. London A 241 (1957) 376–396. S. Nemat-Nasser, Mater. Sci. Eng. A 285 (2000) 239–245. S. Nemat-Nasser, Mech. Mater. 31 (1999) 493–523. Q.S. Yang, F. Xu, J. Mater. Sci. Eng. 2 (2008) 56–61. G. deBotton, J. Mech. Phys. Solids 53 (2005) 1334–1361. Z.M. Huang, S. Ramakrishna, A.A.O. Tay, Compos. Sci. Technol. 60 (2000) 671– 691. X. Li, Y. Xia, Z.R. Li, Y.M. Xia, Comput. Mater. Sci. 55 (2012) 157–165. M.T. Abadi, Compos. Struct. 92 (2010) 578–584. A.D. Drozdov, A. Dorfmann, Int. J. Plasticity 19 (2003) 1037–1067. S. Govindjee, J. Simo, J. Mech. Phys. Solids 39 (1991) 87–112. R. Dargazany, M. Itskov, Int. J. Solids Struct. 46 (2009) 2967–2977. R.W. Ogden, Proc. Roy. Soc. London A 326 (1972) 565–584. A.I. Medalia, J. Colloids Interface Sci. 32 (1970) 115–131. Y.P. Jiang, K. Tohgo, Y. Shimamura, Comput. Mater. Sci. 46 (2009) 507–515. Y. Benveniste, Mech. Mater. 6 (1987) 147–157. G. deBotton, G. Shmuel, J. Mech. Phys. Solids 57 (2009) 1165–1181. J. Moraleda, J. Segurado, J. Llorca, J. Mech. Phys. Solids 57 (2009) 1596– 1613. B. Bittmann, F. Haupert, A.K. Schlarb, J. Appl. Polym. Sci. 124 (2012) 1906–1911.
Contents lists available at SciVerse ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
A micromechanics model for predicting the stress–strain relations of filled elastomers Yunpeng Jiang ⇑, Hualin Fan Department of Engineering Mechanics, Hohai University, Nanjing 210098, China
a r t i c l e
i n f o
Article history: Received 14 June 2012 Received in revised form 16 August 2012 Accepted 23 August 2012 Available online 4 October 2012 Keywords: Polymer–matrix composites (PMCs) Stress/strain curves Modeling Non-linear behavior Micromechanics
a b s t r a c t In this paper, a simple micromechanics model has been developed for predicting the effective stress–strain relations of filled elastomers. The present model constitutes of the instantaneous Young’s modulus and Poisson’s ratio characterizing rubber-like material, a double-inclusion configuration taking account of the absorption of rubber chains onto carbon-black (CB) particles, and the incremental Mori–Tanaka method to compute the effective stress–strain curves. The subsequent predictions are capable of reflecting the well-known S-shape of hyper-elastic composites, and verified via the comparison to the available experiments and analytical models. Parametric analysis is fatherly conducted on the microstructure effect. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction Filled elastomers have been widely applied in various industrial fields due to their large-deformability. In order to know their performance, the equivalent properties at mesoscopic scale should be exactly determined from their microstructures and constituents. Therefore, an accurate but simple homogenization method is very necessary and impending for their widely application. The available researches are grossly classified intomicromechanics method and molecular chain network models. Firstly, micromechanics approaches are briefly reviewed. Mullins and Tobin [1], Qi et al. [2] systematically measured the stress–strain relations of vulcanized rubbers containing CB powders, and introduced a strain amplification factor v ¼ E=EM ¼ 1 þ 2:5f P þ 14:1fP2 (EM – matrix modulus, fP – particle volume fraction) to describe the enhanced elastic property. Although a good agreement was achieved between the predictions and experiments, the constraint effect of the matrix on the CB powders is not accounted for by such a factor. Bergstrom and Boyce [3] investigated the influence of hard particles on the mechanical response of rubber matrix composites with experiment and FEM. A new concept was proposed based on the first strain invariant instead of strain amplification, and could predict the experiments very well. To the best of the authors’ knowledge, the finite deformation of composites was not really solved until Ponte Castaneda [4,5] developed a second-order homogenization method, and adopted the optimization computation to solve the ⇑ Corresponding author. Tel./fax: +86 25 83786046. E-mail address: [email protected] (Y. Jiang). 0927-0256/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.commatsci.2012.08.036
energy minimum problem to determine the appropriate deformation compatible conditions between two phases. But their method cannot but to solve many partial differential equations by a special code programming. Moreover, such method is so complicated and only confined to the plane strain case. Bouchart et al. [6,7] proposed another alternative formula to study compressible hyper-elastic composites. Omnès et al. [8] developed a generalized self-consistent scheme, containing the occluded rubber, the bound rubber and a percolating network, to predict the elasticity of filled rubber. Yin [9] developed a constitutive model for particle reinforced elastomers based on Eshelby’s tensor [10]. Although the interaction between particles and matrix is fully considered, some deductions are arguable. For instance, small strain Eshelby’s tensor was adopted in dealing with finite strain deformation. NematNasser [11,12] first generalized the Eshelby’s tensor for finite strain deformation, and obtained the corresponding formula. Yang and Xu [13] investigated the macroscopic hyper-elastic behavior of fiber reinforced polymers using a micromechanical model and finite strain theory based on the hyper-elastic constitutive law. deBotton [14] proposed a new strain energy function in terms of the behaviors of the individual phases and volume fractions for a finitely extensible solid with a preferred direction. Huang et al. [15] presented a micromechanical analysis for predicting the stress/strain behavior of the composite made of weft-knit polyester fiber interlock fabric and a polyurethane elastomer matrix. Li et al. [16] experimentally studied CB filled rubber and numerically analyzed the local strain filed, concluded that the local strain distribution in a rubber matrix approximately obeys the statistical Gaussian distribution. Abadi [17] presented a homogenization
105
Y. Jiang, H. Fan / Computational Materials Science 67 (2013) 104–108
procedure to predict the shear response of heterogeneous materials at large deformation. Many molecular chain network models are commonly adopted to understand the inherent morphology evolution. Drozdov and Dorfmann [18] emphasized the mechanical energy instead of entropy theory of polymer chains at finite strain deformation, and developed a micromechanics model for polymer and filled polymer as well. Based on the additive network configurations, Govindjee and Simo [19] developed a micromechanics model and analyzed the Mullins’ effect and debonding damage in CB filled rubbers. Furthermore, Dargazany and Itskov [20] proposed a network evolution model to study the deformation-induced anisotropy and damage. In summary, it is challenging to establish the deformation compatible conditions between particle and matrix in finite strain case, but which is a necessary prerequisite to interpret the effective mechanical behaviors of filled elastomers. A constitutive model for filled elastomers should be accurate in predicting the stress–strain relations. While for its practical application, the model should be comparatively simple to be readily implemented in a general purpose finite element code for analyzing the macroscopic composite structures. To meet these requirements, this paper presents a simple but accurate micromechanics model for the nonlinear stress–strain relations of filled elastomers. The predicted stress–strain curves are compared with published experiments, and parametric analysis is also performed. 2. The micromechanics model 2.1. The tangent elasticity of rubber matrix Rubber is supposed to be isotropic, hyper-elastic, nearly incompressible and described by the following Ogden’s strain energy function [21] for its high accuracy: n X 2ai bi b b W¼ k1 þ k2i þ k3i 3 bi i¼1
ð1Þ
here ai and bi are unknown parameters, which are determined by fitting to the uniaxial stress–strain curves. ki are the material principal stretches which related to the principal strains via ki = 1 + ei. The equivalent strain is given by:
ee ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 e:e¼ e þ e222 þ e233 þ 2 e223 þ e231 þ e212 3 3 11
ð2Þ
Due to the incompressibility condition k1k2k3 = 1. The strain components should satisfy the following relation:
e22 ¼ e33 ¼ ð1 þ e11 Þ1=2 1
2.2. The double-inclusion (DI) model Fig. 1 sketches the homogenization procedure of CB reinforced rubber. As for the actual composites, CB nano-powders easily agglomerate to form a network structure in the rubber matrix. These networks surround a part of polymer chains whose deformation is greatly impeded during the applied stretch. CB phase deforms linear elastically due to its high stiffness in relation to the rubber phase. Therefore, CB particles and the surrounded rubber form a new kind of inclusions dispersed in the matrix. What follows is the homogenization procedure. CB particles are equivalent to a homogeneous layer, and its volume fraction just equals to the given CB concentration as a known parameter. The surrounded rubber is assumed to deform linear elastically, and its material properties are given as the already known parameters. The volume fraction of DI is defined by:
fDI ¼ fP
1 þ 0:02139DBP abs 1:46
ð6Þ
where DBPabs denote DBP (dibuty1 phthalate) absorption and already measured for different rubbers [22]. A DI configuration is subsequently constructed as shown in Fig. 1, and its equivalent stiffness can be computed by our previous research [24]. In order to realize this purpose, a new configuration including a particle and its surrounding layer is firstly constructed. During the DI construction, the boundaries of particle and its surrounding layer are supposed to be parallel for simplicity. In the DI model, a centered particle is surrounded by an inhomogeneous layer, in turn embedded in an infinite matrix CM, and the layer and particle constructs a DI. The DI includes a layer of ellipsoid XC with a uniform stiffness tensor CM. The layer volume C fraction is fDI ¼ V C =V DI . The equivalent stiffness of DI is determined by [23]:
h i 1 1 P CDI ¼ fDI CP ðI þ S K1 þ fDIC CC ðI þ S K1 P Þ C Þ 1 1 1 1 P C ðI þ S K1 Þ þ f I þ S K fDI DI C P
ð7Þ
where KC ¼ ðCC CM Þ1 ðCM Þ. After giving the equivalent stiffness of a DI, the next step is to predict the effective properties of filled elastomers reinforced by the DI instead of CB particles. During the computation of the effective composite behaviors, the stiffness tensor CP should be replaced by CDI, which determined by Eq. (7), with DI fraction fDI instead of particle fraction fP. 2.3. The incremental Mori–Tanaka method
ð3Þ
In order to linearly handle the nonlinear elastomer, the whole deformation procedure is divided into n loading step, i.e., the applied stretch is imposed on the specimen step by step up to the final deformation. The tangent modulus at each step is different, and the instantaneous modulus and Poisson’s ratio at the nth step are defined by:
The stress–strain relations would be predicted for the fictitious composites with DI inclusions. Based on Benveniste’s formula of Mori–Tanaka method [24], and after a simple deduction, the incremental constitutive relation of composite is given as:
de ¼ ðI þ fP AÞC1 M dr
ð8Þ 1
En ¼
rn11 rn1 11 en1 en1 1
ð4Þ
mn ¼
1=2 ð1 þ en1 ð1 þ en1 Þ1=2 1 Þ n e1 en1 1
ð5Þ
where r11 and e11 denotes the nominal stress and strain in the loading direction, respectively. In this paper, all the terms for the reinforcement and matrix are represented by symbols with subscripts ‘P’ and ‘M’, respectively, and those of the composite are denoted by symbols without any script. All the tensors and vectors are written in boldface letters.
with A ¼ ½ðCP CM ÞS þ CM þ fP ðCP CM ÞðI SÞ ðCP CM Þ The incremental first P–K stress tensors (nominal stress) of constituents are respectively given by:
drM ¼ CM ½I þ ð1 fP ÞðI SÞAC1 M dr
ð9:aÞ
drM ¼ CM ½I þ fP ðI SÞAC1 M dr
ð9:bÞ
Similarly, the incremental strain tensors of constituents are respectively written as:
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Y. Jiang, H. Fan / Computational Materials Science 67 (2013) 104–108
Carbon-Black CB layer
Rubber
Rubber
ΩΓ
homogeneous material
ΩP
deM ¼ ½I þ fP ðI
ð10:aÞ
SÞAC1 M d
r
ð10:bÞ
3. Results and discussion 3.1. Verification of the analytical model Fig. 2 shows the comparisons between theoretic predictions and experiments [1] for rubber containing different concentrations of MT-CB under tension. Unless indicated otherwise, the constituent properties are EM ¼ 10 MPa; v M ¼ 0:5, EP ¼ 10 GPa; v P ¼ 0:3, and 0.02139DBPabs = 2.5 [22]. It is clearly shown that all the predictions are in fair agreement with experiments except at the higher stretch stage. The main reason is that the debonding damage between particle and rubber matrix would occur with increasing the applied stretch. A better prediction is undoubtedly expected by considering the debonding damage in the model. Fig. 3 shows the comparisons between theoretic predictions and experiments [3] for natural rubbers filled with CB N351 under compression. Here, 0:02139 DBP abs ¼ 2:05 [23]. Unlike uniaxial tension, particle–matrix interface does not separate under compression, and thus all the predicted stress–stretch curves agree with the test data very well in the whole stretch range, which again confirms the accuracy of the developed model. The present model is fatherly compared with published analytical models at fP = 21% as shown in Fig. 4. Compared to other models, the present method gives the best predictions. What’s the most important is that only the present model captures the S-shape of the stress–stretch relations. 3.2. Verification of the model for fiber filled elastomers
Nominal Stress σ11 (MPa)
Two analytical models are recently developed for fiber reinforced composites. One is developed by Lopez-Pamies as [4]:
fP=21%
2
13.8% 8.4%
1 matrix
0
0
0.5
1.0
Experiment [3] Predictions EP=10GPa, vP=0.3
-6 -5
fP=25% 15%
-4 -3 -2
7%
-1 matrix
0
-0.1 -0.2 -0.3 -0.4 -0.5 -0.6
Stretch λ1 Fig. 3. Theoretical predictions & experiments [3] for natural rubbers filled with of CB N351 under compression.
Experiment [1] Bergstrom-Boyce Govindjee-Simo Yin-Sun The present model
3
2
1
0 0
0.5
1.0
1.5%
5.3%
1.5
2
Engineering Strain (λ1−1) Fig. 2. Theoretical predictions & experiments [1] for rubber filled with different concentrations of MT-CB under tension.
1.5
Engineering Strain (λ1−1) Fig. 4. Comparisons between different analytical models and experiments [1] for fP = 21%.
l ð1 þ fP Þ 2 1 WðkÞ ¼ M k þ 22 2 ð1 fP Þ k
ð11Þ
where k ¼ k1 ¼ 1=k2 . The other is proposed by deBotton as [14]:
l ðk 1Þ2 ½ðk þ 1Þ2 þ 2f P ðk2 þ 1Þ WðkÞ ¼ M 2 2 k ð1 fP Þ
Experiment [1] Predictions EP=10GPa, vP=0.3
3
-7
0
Nominal Stress σ11 (MPa)
1 deP ¼ C1 P CM ½I ð1 fP ÞðI SÞACM dr
Nominal Stress σ11 (MPa)
Fig. 1. Homogenization procedure of carbon black (CB) reinforced rubber.
ð12Þ
Fig. 5 shows the comparison between the predicted stress– strain relations by different analytical models for rigid fiber reinforced composites, where 1-axis is in the transverse plane vertical to the fiber direction. The deBotton’s model [25] gives much better predictions with FEM results [26] than the other models. The predictions of the present model and Lopez’s model superpose each other, and slightly deviate from FEM results. Figs. 6 and 7 compared the stress–strain relations of rigid fiber reinforced composites predicted by the present model with FEM [26]. The matrix materials in the two diagrams are very different, and the matrix in Fig. 7 reaches the lock-up stage at the lower stretch as compared to that in Fig. 6. The present model is
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Y. Jiang, H. Fan / Computational Materials Science 67 (2013) 104–108
Nominal Stress σ11 (MPa)
Nominal Stress σ11 (MPa)
5 FEM Lopez's model deBotton's model the present model
4 3 2
matrix
1 0
fP=0.2
1
1.5
2.0
2.5
4 EP=10GPa vP=0.3 fP=0.5
3
0.3 0.2
2
0.1 0
1 0
3
0
3 2
matrix fP=0.1 fP=0.2 fP=0.3 fP=0.4
1 0 1
1.5
2.0
4
1.5
2.0
2.5
EP= EP=10000MPa EP=1000MPa EP=1MPa EP=0
3 2 1
x matri increasing the modulus
0 0
0.5
1.0
1.5
2.0
2.5
Engineering Strain (λ1−1)
2.5
Stretch λ1
1.0
Fig. 8. Dependence of the stress–strain relations on particle volume fraction fP.
Nominal Stress σ11 (MPa)
Nominal Stress σ11 (MPa)
4
0.5
Engineering Strain (λ1−1)
Stretch λ1 Fig. 5. The predicted stress–strain relations by different analytical models for fiber reinforced composites.
0.4
Fig. 9. Dependence of the stress–strain relations on particle elastic modulus.
Fig. 6. Theoretical predictions & FEM [26] for rigid fiber reinforced composites.
Stress σ11 or τ12 (MPa)
Nominal Stress σ11 (MPa)
3
20
fP=0.1 0.2 0.3 0.4
15 10 5
matrix
EP=10GPa, vP=0.3 fP=0.3
2
1
1.5
2.0
2.5
Stretch λ1
biaxial shear
1 matrix
0
0
uniaxial
0
0.5
1.0
1.5
2
Engineering Strain (λ1−1) Fig. 10. The effect of loading path on the composite stress–strain relations.
Fig. 7. Theoretical predictions & FEM [26] for rigid fiber reinforced composites.
essentially based on the volumetric average of the stress/strain fields, which is very hard to account for the deformation localization induced by the lock-up of rubber matrix. The analytical model provides good estimates of the deformation response at a low particle volume fraction. But the predictions deviate rapidly from FEM results with increasing particle volume fraction and applied stretches. The discrepancy between the predictions and FEM data in Fig. 6 is a little lower as compared to those in Fig. 7, which is induced by the early lock-up of the matrix at the first stretch stage. In order to better predict the deformation response of filled elastomers, high-order homogenization method should be developed to fully reflect the microstructural effect on the soft-hard heterogeneous systems. Furthermore, the microstructures, e.g., particle distribution, matrix properties and etc., maybe changes a lot during the large deformation, and the corresponding induced influence should be considered in improving the analytical models.
3.3. Effect of particle stiffness and particle concentration Fig. 8 illustrates the dependence of the composite stress–strain relations on particle volume fractions fP in the range of 0–50%. As expected, increasing the particle volume fraction also increases the overall stress at the same applied stretch. Fig. 9 shows the effect of particle stiffness on the deformation response of the composites, where EP is in a wide range of 0–1. EP = 0 denotes void and 1 rigid inclusion. It is clearly shown that the effective stress increases with increasing particle stiffness at the very beginning, but cannot enhance fatherly after exceeding a certain value, which is also confirmed by Bergstrom’s simulations [3] and experiments [27]. 3.4. Effect of loading path In the graph of Fig. 10, the evolution of the nominal stresses r11 or s12 are reported as a function of the principle elongation k1 for
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Y. Jiang, H. Fan / Computational Materials Science 67 (2013) 104–108
tensile, pure shear and biaxial tension). Stress levels obtained for the biaxial tension is higher than that computed in uniaxial tension, and that of shear is the lowest. These results highlight the strong influence of the loading path on stress–strain relations. 4. Conclusions A micromechanics model is proposed to predicate the finite strain stress–strain relations of filled elastomers. The double-inclusion model is employed to consider the constraint effect of CB network on the rubber matrix. The nonlinear constitutive relation of rubber-like materials is linearized by the instantaneous Young’s modulus and Poisson’s ratio. The stress–strain curves of filled elastomers are predicted by the incremental Mori–Tanaka method. The developed model is verified via published experiments and analytical models. Additionally, parametric analysis is performed on the microstructure effect. Several important conclusions are reached: (1) The CB network configuration should be taken into account in predicting the effective behaviors of filled elastomers. The deformation of rubber matrix surrounded by CB network is greatly impeded, and in turn enhances the load-carrying capability of composites. (2) The volumetric-average based homogenization model cannot account for the deformation location of rubber matrix, and thus overestimate the stretch response of filled elastomers. (3) Compared to the existing analytical models, the present model is very accurate and very simple in interpreting the effective performance of filled elastomers. Acknowledgements This work was supported by the Fundamental Research Funds for the Central Universities (No. B11020079), Jiangsu Provincial
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