dv. Theor. ppl. Mech., Vol. 5, 2012, no. 6, 257-262 Radial Growth of a Micro-Void in a Class of Compressible Hyperelastic Cylinder Under an xial Pre-Strain * Yuxia Song, Datian Niu and Xuegang Yuan College of Science, Dalian Nationalities University, 116600, Dalian, China bstract In this paper, the mathematical model that describes the radial growth of a pre-existing micro-void in a compressible hyperelastic circular cylinder under an axial pre-strain is formulated as the boundary valued problem of a class of second order nonlinear differential equations. The analytic solutions of parametric type that describe the growth of the micro-void are obtained and some corresponding simulations are given. The physical interpretations are well shown by the numerical results. Keywords: compressible hyperelastic circular cylinder; nonlinear second order differential equation; stress distribution; radial displacement 1 Introduction The nonlinear compressible hyperelastic materials have been widely used in industrial and medical applications. The deformation problems of them have been investigated by many researchers for a long time. Several works [1-] studied the cavitation problem of compressible solid cylinder and sphere composed of different hyperelastic materials. Liao, et al [5], Niu, et al [6] studied the compression and inflation of the compressible spherical shell composed of class * This work was supported by the National Natural Science Foundation of China (No.1087205, 1100109), the Program for New Century Excellent Talents in University (NCET-09-0096) and the Fundamental research Funds for the Central Universities (No. DC1201011121, DC1201011122). Corresponding author, Email: yuanxg@dlnu.edu.cn
258 Yuxia Song, Datian Niu and Xuegang Yuan of single and composite hyperelastic materials, respectively. Zhang, et al [7] studied the radial growth of the micro-void centered at a compressible hyperelastic cylinder with infinite length. In this paper, based on the theory of finite deformation for hyperelastic materials, we study the radial growth of a pre-existing micro-void centered at a compressible hyperelastic cylinder under an axial pre-strain. We first formulate the mathematical model of the problem as a nonlinear differential equation with boundary conditions. Then we obtain the analytic solutions of parametric type of the model by using the dimensionless transformation. Finally, we analyze the radial growth of the micro-void by numerical simulations. 2 Formulation of Mathematical Model xially, the cylinder with a micro-void we studied was under a given pre-strain η, that is, the cylinder is stretched axially if η > 1 while is compressed axially if 0 <η < 1. Furthermore, the outer surface of the shell is subjected to a prescribed radial stretch λ > 1, and the surface of the micro-void is traction-free. 2.1 Configurations Under the assumption of radically symmetric deformation, the undeformed and deformed configurations of the cylinder are respectively given by D0 = {( R, Θ, Z) 0 < R <,0 Θ 2π, l Z l} (1) D = {( r, θ, z ) r = r ( R ) 0, R ; θ = Θ,0 Θ 2 π ; z = λ Z } (2) where r = r(r) is the radial deformation function to be determined. and are the inner and outer radii of the undeformed cylinder. Moreover, λ 1 = dr / dr = r&, λ2 = r / R, λ = η () are the principal stretches of the deformation gradient tensor, and λ i > 0, i = 1,2. The corresponding principal components of the Cauchy stress tensor are 1 W 1 W 1 W σ rr =, σ θθ =, σ ϕϕ ( R ) = = 0 () λ2λ λ1 λ1λ λ2 λ1λ2 λ where W is the strain energy function of the hyperelastic material. 2.2 Strain energy function s is well known, the constructive relation of the hyperelastic materials can be completely described by its strain energy function W. In this paper, the strain energy function we considered is given by [] 1 1 1 W = C1( λ1 + λ2 + λ ) + C2 ( λ1 + λ2 + λ ) + C ( λ1λ2λ 1), (5) with v v 2v C1 = μ, C2 = μ, C = μ, 2v 2v 2v where μ and v are the infinitesimal shear modules and the Poisson s ratio of the material, respectively, μ > 0 and 0 < v <1/.
Radial growth of a micro-void 259 2. Governing equations In the absence of the body force, the differential equation governing the radial motion of the cylinder is given by the following nonlinear second order differential equation 2 2 W 1 W r( R) W W & r + ( ) = 0 2 r& R + (6) λ1 R λ1λ 2 R λ1 λ2 2. oundary conditions Since the outer surface of the cylinder is subjected to a prescribed radial stretch λ and the inner surface of the cylinder is traction-free, therefore r( ) = λ, σ rr ( ) = 0, (7) Obviously, Eqs.(5)~(7) form the mathematical model of the problem to be discussed. We can solve Eqs. (5)~(7) to analyze the radial growth of the inner surface of the micro-void. Solution and nalysis Substituting the strain energy function (5) into Eq. (6), we obtain that 2& r& 1 1 1 + = 0 2 2 (8) λ1 R λ2 λ1 Using the following transformation λ1 r& R t = t( R) = =, λ2 r( R) Eq.(8) can be transformed to the following system of the first order differential equations dr 2dt dr 2dt, r = = (1 t)( + t) R t(1 t)( + t). (9) On integration of Eq.(9) yields 6 Ct 2 D( + t) R =, r = (10) (1 t) ( + t) t where C > 0, D > 0 are integral constants to be determined. Thus we obtain the general solution (10) with parameter t of Eq.(6). The parameter t satisfies that 0 < t < t < t < 1, where t and t are the values of t corresponding to R = and R =, respectively. Let R = and R = in the first equation of Eq.(10), we have 6 Ct 6 Ct =, = (1 t) ( + t) (1 t) ( + t). (11) On substitution the boundary condition (7) 1 into the second equation in Eq.(10) yields
260 Yuxia Song, Datian Niu and Xuegang Yuan 2 2 D( + t ) λ = (12) (1 t ) Substituting the strain energy function (5) into the inner boundary condition (7) 2, we can obtain 1/ 1/ 1/ 6 2 / C 1 + D t C1 C2 = 0 + 2 ( ) C λ D t + t C t (1) Therefore, for given radial stretch λ > 1, the ratio of the inner and the outer radii /, and the pre-strain λ = η, we can determine the parameters t, t, C and D from Eqs. (11)~(1). s a result, the analytic solutions with parametric type of mathematical model, Eqs.(5)~(7), are also in hand. Furthermore, we obtain the relation between r c / and λ by combining Eqs. (11)~(12), namely, 1/ 6 2 / 1/ 6 2 / C + t rc C + t λ = and = D t D t where r c is the radius of micro-void. Fig.1 Radii of micro-void versus radial stretches under axial pre-strains η = 1.1,1.2 Fig.2 Radii of micro-void versus radial stretches under axial pre-strains η = 0.8,0.9
Radial growth of a micro-void 261 Figs. 1~2 plot the relation curves between the radii of micro-void and the radial stretches for different values of η. From Figs. 1~2, we see that, for certain relative small values of /, the radius of the micro-void increases very slow for small λ. However, if λ exceeds certain value, the growth of the micro-void will grow very quickly. Meanwhile, when the pre-strain is extension, the larger η is, the faster the radius of the micro-void increases. Similarly, when the pre-strain is compression, the larger η is, the faster the radius of the micro-void increases. Conclusions In this paper, we investigate the growth of a micro-void centered at the axial line of the circular cylinder composed of a compressible hyperelastic material under axial pre-strain. Through analyzing the relation between the growth of the micro-void and the prescribed stretch, we have the following results: (1) The radius of the micro-void increases with the increasing of the radial stretch. The radius of the micro-void increases slowly for larger /. When / is relative small, the radius of the micro-void increases very slow first, and increases faster suddenly as λ exceeds certain value. Therefore, the case of smaller / is similar to the case of the cavitation of the solid circular cylinder. (2) For the same values of / and for different pre-strains, the radius of the micro-void grows faster for larger pre-extension, while grows slower for larger pre-compression. References [1] C. O. Horgan, R. beyaratne, bifurcation problem for a compressible nonlinear elastic medium: growth of a micro-void, Journal of Elasticity, 16 (1986), 189-200. [2] X. C. Shang, C. J. Cheng, Exact solution for cavitated bifurcation for compressible hyper-elastic materials, International Journal Engin. Sci., 9 (2001), 1101-1117. [] J. S. Ren, C. J. Cheng, Cavity bifurcation for incompressible hyper-wlastic material, pplied Mathematics and Mechanics, 2 (2002), 881-888. [] X. G. Yuan, Z. Y. Zhu and C. J. Cheng, Study on cavitated bifurcation problem for spheres composed of hyper-elastic materials, Journal of Shanghai University, 51 (2005), 1-. [5]. F. Liao, P. Liu and X. G. Yuan, Radially symmetric deformation of a class of compressible hyper-elastic spherical shells, Journal of Jilin Normal University, (2006), 7-9. (Chinese Edition)
262 Yuxia Song, Datian Niu and Xuegang Yuan [6] D. T. Niu, X. G. Yuan, H. M. Deng, Radially symmetric deformation of spherical shell composed of composite compressible hyperelastic materials, dvanced Material Research, 181-182 (2011), 50-5. [7] W. Z. Zhang, C. J. Cheng, X. G. Yuan, Radial growth of a micro-void in a class of compressible hyperelastic circular cylinders, Chinese Quarterly of Mechanics, 29 (2008), 86-90. (Chinese Edition) Received: May, 2012