Mechanical Properties of Polymer Rubber Materials Based on a New Constitutive Model

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1 Mechanical Properties of Polymer Rubber Materials Based on a New Constitutive Model Mechanical Properties of Polymer Rubber Materials Based on a New Constitutive Model J.B. Sang*, L.F. Sun, S.F. Xing, B.H. Liu and Y.L. Sun School of Mechanical Engineering, Hebei University of Technology, Tianjin, , China Received: 7 April 2014, Accepted: 4 May 2014 Summary A new constitutive model, which is a modification from Gao s second constitutive model, has been introduced by using the finite deformation theory. The new model fulfils the condition that strain energy becomes zero under no deformation. For incompressible polymer rubber materials, when n=1 and a=0, the new constitutive relation may be simplified to the Mooney-Rivlin model; when n=1 and a=1, the new constitutive relation may be simplified to the Neo-Hookean model. The discussions of basic finite deformation examples illustrate how the new constitutive model reasonably describes the deformation properties of polymer rubber materials, and how the applied range of polymer rubber materials have been broadened by using the new constitutive model. Keywords: Polymer rubber materials; Constitutive model; Finite deformation theory 1. Introduction The finite deformation theory has developed greatly since 1940s. The theoretical achievements and experimental results based on this theory are extensively applied to engineering field. However, the mathematic problems are difficult in both theory and application. Researches have been conducted in recent years on the nonlinear problems of the finite elasticity theory by means of mathematics and numeric analysis. In the research, people gradually understand that the key is to construct a constitutive model that can reflect the polymer rubber materials, which determine the precision of the numerical analysis and the availability of the results 1-4. At present, people pay more attention to the construction of the constitutive model of polymer rubber materials. Many constitutive models have Smithers Information Ltd., 2014 been constructed to describe the physical characteristics of polymer rubber materials based on continuum mechanics, thermodynamics, phenomenological theory of the phase transition and statistic physics. In 1948, Rivlin proposed the strain energy function model to the isotropic hyperelastic materials from an phenomenological perspective as follows 5 : W = *Corresponding author, sangjianbing@hebut.edu.cn i, j=0 C ij (I 1 3) i (I 2 3) j In which C ij stands for material constant; I 1 and I 2 are respectively the first and second invariants of the left Cuachy-Green deformation tensor. The constitutive model is complicated just because there are many physical parameters. According to the research, introducing more physical parameters may clarify the physical meanings of the material deformation in theory, but it is difficult to determine these parameters by experiments. To be simplified, taking the first item of the Rivlin model, it is the Neo-Hookean model; while, taking the linear combination of the Rivlin model is the Mooney-Rivlin Model. From the theory, if taking C 10 = nkt/2 for Neo-Hookean model, the result is same as that of Gaussian statistics model by Treloar in 1943, and the Mooney-Rivlin model is same as the rubber constitutive model by Mooney in The above two models provide a solid foundation for the new constitutive model because they are easy in form and easy to verify by experiments. what s more, they are extensively applied to the engineering and have accumulated large quantities of theoretical and experimental achievements. These two models can fit the material mechanics features of the incompressible polymer rubber materials on small and medium deformation, but can t precisely describe the mechanical features of the polymer rubber materials in the large deformation. Polymers & Polymer Composites, Vol. 22, No. 8,

2 J.B. Sang, L.F. Sun, S.F. Xing, B.H. Liu and Y.L. Sun To solve these problems, people construct logarithmic type and power law type constitutive models for polymer rubber materials 7-9. Gao Y.C. proposed a strain energy function from the perspectives of tensile and compression of the materials in 1997 and applied it to the research of rubber like material s fracture 10. However, the strain energy function cannot be simplified to Neo-Hookean material or to Mooney-Rivlin material, which influences the experimental basis of the strain energy function application. Based on Gao s constitutive model, this paper constructs a new constitutive model that can describe the incompressible polymer rubber materials. That is, when n=1 and a=0, the new model transforms to Neo- Hookean model; However, when n=1 and a=1, the new model transforms to Mooney-Rivlin model. This paper verifies the rationality and applicability of the new model by analyzing and calculating the basic homogeneous deformation. What s more, the two constitutive parameters influences are discussed with the example of the spherical membrane expansion. 2. The Construction of New Constitutive Model Gao proposed the following strain energy function 10 in 1997: W = A(I n 1 + I n 1 ) (1) Based on the elastic finite deformation theory, the Cauchy stress tensor can be expressed as follows: σ = 2AnJ 1 (I n 1 1 B I n 1 1 B 1 ) (2) In which, B is left Cauchy-Green deformation tensor, I 1, I 2 and I 3 are the three invariants of B; A and n are material parameters and I -1 = I 2 /I 3. From which it indicates that the greater the tensile deform, the larger the I 1 ; the greater the compressive deform, the larger the I -1, which can describe the finite deformation features of the rubber like materials. Based on Gao s constitutive model, we proposed a strain energy function for the incompressible rubber like materials: ( ) +α ( I n 2 3 n ) W = A I n 1 3 n (3) In which a is the material parameter reflecting I 2 s influence on stress distribution and I 3 =1. From the new constitutive model 1.3, we can see that when n=1 and a=0, it transforms to Neo-Hookean model; when n=1 and a=1, it transforms to Mooney-Rivlin model. We can get the expression of Cauchy stress tensor to the isotropic incompressible polymer rubber materials based on the finite deformation theory as follows: σ = p * I + 2W 1 B 2W 2 B 1 (4) In which, W i = W/ I i, I is unit tensor. p * is the undetermined scalar function that justifies the incompressible internal constraint conditions. In the spontaneous configuration p * =0. and: W 1 = β 1 =2AnI 1 n 1 ; W 2 = β 2 = 2AαnI 2 n 1 (5) 2.1 Homogeneous Deformation of Uniaxial Tension As Figure 1 illustrates, a cube unit of a finite deformation material under uniaxial tension. The deformation meets the following expressions: x 1 = λ X 1 ; x 2 = µx 2 ; x 3 = µx 3 (6) The principal stretch is l 1 =l, l 2 =l 3 =m. The deformation gradient and left Cuachy-Green deformation tensor are as follows: F = λe 1 e 1 + µe 2 e 2 + µe 3 e 3 B = λ 2 e 1 e 1 + µ 2 e 2 e 2 + µ 2 e 3 e 3 (7) The strain invariants are: I 1 = λ 2 + 2µ 2 ; I 2 = 2λ 2 µ 2 + µ 4 ; I 3 = λ 2 µ 4 (8) The stress component can be achieved from the constitutive relation 1.4 as: σ 11 = p + 2λ 2 W 1 2λ 2 W 2 σ 22 = σ 33 = p + 2µ 2 W 1 2µ 2 W 2 (9) Because s 22 =s 33 =0 and the incompressible condition J=lm 2 =1, with Equation (9), the tension stress s 11 can be expressed as: σ 11 = 2An (λ 2 λ 1 ) λ 2 + 2λ 1 α(λ 2 λ) 2λ + λ 1 (10) The relation of s 11 l has been established from above equation. Under the small deformation, that is, l=1+e (e is the small deformation strain), Equation (10) can be simplified to: σ 11 = 2An(1+α)3 n ε; E = 2An(1+α)3 n Figure 1. The uniaxial deformation diagram (11) 694 Polymers & Polymer Composites, Vol. 22, No. 8, 2014

3 Mechanical Properties of Polymer Rubber Materials Based on a New Constitutive Model E stands for the Young s modulus under the small deformation. We can get E=3G, which shows the relationship between Young s modulus and shear modulus under the small deformation. In order to discuss the effect of constitutive parameters a and n on the mechanical properties of material, non-dimensional stress is introduced. From the Equation (10), we can get: F / A 0 = σ 11 µ 2 / A 0 = 2n (λ λ 2 ) λ 2 + 2λ 1 α(λ 3 λ 1 )( 2λ + µ 2 ) n 1 (12) R 0 H 2 R R 0 + H 2 ; 0 Θ π; 0 Φ 2π R 0 stands for the radius of middle plane of the spherical membrane; H is thickness of spherical membrane. The deformation of the spherical membrane can be expressed as: r = r(r); θ = Θ; ϕ = Φ (13) Figure 2 and Figure 3 have shown the computed result of Equation (12). Figure 2. The relation between F/A 0 l with effect of n (a=0.1) When the parameter a is given (a=0.1), as illustrated in Figure 2. if the constitutive parameter n increases, the stress becomes greater and it has the reinforcement feature apparently. Therefore, n is considered as the material s reinforcement parameter. The stress-strain curve becomes sharp lifting in large deformation when n=1.6, therefore a determines I 2 s influence on material s deformation and stress. As is shown in Figure 3, for the given n=1.2, the larger the a is, the greater the stress is, however, its influence is smaller than that of reinforcement parameter n. 3. The Finite Deformation Analysis on the Spherical Membrane Expansion Figure 3. The relation between F/A 0 l with effect of a (n=1.2) The stability of internal pressure on the spherical shell has attracted more attention both domestic and international scholars due to its theoretical significance and engineering application background With the constitutive model given by this paper, we study the internal pressure p on the incompressible hyper elastic spherical membrane. The spherical membrane model is illustrated as Figure 4. (R, Q, F) and (r, q, j) stand for the coordinates system before and after deformation respectively. Before the deformation: Polymers & Polymer Composites, Vol. 22, No. 8,

4 J.B. Sang, L.F. Sun, S.F. Xing, B.H. Liu and Y.L. Sun If e i (i=r, q, j) is the unit basis vector of coordinate (r, q, j), then the deformation tensor of left Cuachy- Green is shown as: B = λ 2 e r e r + µ 2 (e θ e θ + e ϕ e ϕ ) (14) In which, λ = dr dr = r ; µ = r R. The invariant of tensor B is: I 1 = λ 2 + 2µ 2 ; I 2 = µ 4 + 2µ 2 λ 2 ; I 3 = λ 2 µ 4 (15) As spherical membrane can be considered that the deformation Figure 4. The deformation of spherical membrane. (a) The deformation pattern of spherical membrane, (b) The reference coordinate system (a) Figure 5. The relation between P # m (a=0.1) distributes evenly along the thickness direction and radial stress is much less than circumferential stress. Then the balance equation of spherical membrane can be deduced as: (b) h(σ θθ +σ ϕϕ ) rp = 0 (16) h is the current spherical membrane thickness; and p is internal pressure. From the incompressible condition, we can get: h = µ 2 H; λ = µ 2 (17) If s rr =0, Substituting the Equations (14), (15) and (17) to Equation (9), the following expressions can be achieved: p = 2λ 2 W 1 2λ 2 W 2 σ θθ = σ ϕϕ = 2An (µ2 µ 4 )( µ 4 + 2µ 2 ) n 1 α(µ 2 µ 4 ) 2µ 2 + µ 4 (18) Introduce the non-dimensional pressure p # =pr 0 /4AH, with Equations (18) and (16), we can get: p # = n (µ 1 µ 7 ) µ 4 + 2µ 2 α(µ 5 µ) 2µ 2 + µ 4 (19) In order to discuss the influences of constitutive parameters a and n on the mechanical properties of material, the Equation (19) is calculated. The result is shown in Figure 5 and Figure 6. For the given a(a=0.1) illustrated in Figure 5, if the constitutive parameter n increases, the stress becomes greater and apparently it has the reinforcement feature. Therefore, n is considered as the material s reinforcement parameter. As is shown in Figure 6, a determines I 2 s influence on material deformation and stress. For the given n=1.2, the larger the a is, the greater the stress is. However, when a=0.1, n=0.8 and n=1.2, a=0, P # m becomes softening and the material becomes unstable. In fact, for the model (4), the material s stability not only depends on load, but also on the material parameters a and n. When a=0, n 1.5 and a 0, n > 0.75, the material can be stable. For the special case a=0: p # = n(2µ 2 + µ 4 ) n 1 (µ µ 7 ) (20) The result is shown by Figure 7. It is shown from Figure 7 that when n<1.5, there exists instability. The maximum of internal pressure becomes greater gradually with the increase of n as the value of P # change. When n>1.5, P # monotonically increases with the effect of m, which is consensus 696 Polymers & Polymer Composites, Vol. 22, No. 8, 2014

5 Mechanical Properties of Polymer Rubber Materials Based on a New Constitutive Model Figure 6. The relation between P # m (n=1.2) problems of the finite deformation analysis due to the introduction of a and n. Acknowledgement This paper is supported by Tianjin National Nature Science Foundation (grant No. 12JCYBJC19600) and Scientific Research Key Project of Hebei Province Education Department (grand No. ZD ) References Figure 7. The relation between P # m with effect of n with the conclusions of document 15. Further research is to be carried out about the effect on spherical membrane stability by material parameters a and n when a Conclusions In summary, the constitutive model put forward by this paper is verified to be reasonable and the parameters are required as A>0, n>0, a 0. The models describing the material are in greater use due to the introduction of a and n. When n=1 and a=0, the new constitutive model transforms to Neo- Hookean model; When n=1 and a=1, it transforms to Mooney-Rivlin model. Therefore, we conduct theoretical and experimental researches with the new constitutive model so that the results can benefit from the previous ones. Further research is to be conducted about the theoretical and experimental 1. Millard F. Beatty, Topics in finite elasticity: hyperelasticity of rubber, elastomer, and biological tissueswith example, Appl. Mech. Rev. 1987, 40: Boyce M.C. and Arruda E.M., Constitutive models of rubber elasticity: a review, Rubber Chem. Technol., 73 (2000) Bede T., Hyperelastic behavior of rubber: a novel invariant-based and review of constitutive models, J. Polym. Sci. Part B: Polym. Phy., 45 (2007) Ali A., Hosseini M. and Sahari B.B., A review of constitutive models for rubber-like materials, J. Eng. Appl. Sci., 3(1) (2010) Rivlin R.S., Large elastic deformations of isotropic materials: I. Fundamental concepts, II. Some uniqueness theories for homogenous deformation, Philos. Trans. Roy. Soc., Lond: Ser A., 240 (1948) Huang Z.P., Fundamentals of Continuum Mechanics, Beijing, 2003, Higher Education Press. 7. Gent A.N., A new constitutive relation for rubber, Rubber Chem. Technol., 69(1) (1996) Horgan C.O. and Saccomandi G.A., A molecular statistical basis for the Gent constitutive model of rubber elasticity, J. Elasticity, 68 (2002) Knowles J.K., The finite anti-plane field near the tip of a crack of incompressible elastic solids, Int. J. Fract., 13(4) (1977) Gao Y.C., Large deformation field near a crack tip in rubber-like Polymers & Polymer Composites, Vol. 22, No. 8,

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