Another approach of WKB method for the stability analysis of the bending of an elastic rubber block

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Abstract

Our concern in this paper is to study the bifurcation of the pure bending of a rubber block made of elastic Varga material which has a ratio of thickness A to length L. The governing equations are reduced to an eigenvalue system.

We then apply the WKB method with repeated roots for describing the angle of bending and also the azimuthal shear. In this process, very interesting solvability conditions are derived and solved. We obtained sufficient conditions for different kinds of L/A. Our findings shows that by increasing the value of L/A the bending angle also increases.

Introduction

The most rubbers which are used in engineering and industrial devices are often subjected to large bending deformation and rely on the excellent bending characteristics of such blocks. The importance of bending of blocks in applications is the motivation for study of their stability. Large bending is the favorable mode of deformation of rubber blocks. When an elastic block is subjected to severe bending, it is expected that its inner curved face will eventually become unstable. This phenomenon is well captured by the theory of incremental nonlinear elasticity.

The stability analysis of the problem of pure bending in different tubular structures (specially in the thin-walled) has a long history and there is a wide mechanical engineering literature dealing with different aspects; some are summarized by Kyriakides and Corona (2007). Some studies in the present context which has been done mathematically are the works of Seide and Weingarten (1961) and those by Tovstik and Smirnov (2000).

New and better methods introduced for the first time on pure bending by Triantafyllidis (1980), who examined incremental bifurcation equations for a couple of piecewise power law constitutive models, including a hypo-elastic one. He discovered that the underlying instability mechanism is a surface instability similar to that happened in the plane-strain half-space problems discussed by Hill and Hutchinson (1975) or Young (1976).

Haughton (1999) did a similar analysis for the hyper-elastic materials (mostly for neo-Hookean) in a three-dimensional context, and also discussed the vertical compression. Dryburgh and Ogden (1999) introduced thin coatings on the curved boundaries of the bent block and made comparisons with the uncoated case. Their findings show that, relative to the latter case, bifurcation is generally promoted by the presence of surface coating, on either or both curved boundaries, that is the bifurcation occurs at smaller strains. The relative sizes of the shear moduli for the coating and, respectively the bulk material was found to play an important role in describing this phenomenon. For several models of materials, Destrade, Annaidh, and Coman (2009) described critical stretch ratio of a bent block. For models used to describe arteries it is found, somewhat surprisingly, that the strain-stiffening effect promotes instability.

The WKB method is an asymptotic method for studying singular perturbation problems. An excellent review or its application on solving the eigenvalue problem has been made by Fu (1998) to the bifurcation analysis of a spherical shell of arbitrary thickness. Sanjaranipour (Sanjaranipour, 2005, Sanjaranipour, 2010) have continued the work of Fu (1998) and used this method not only to the buckling analysis of Varga but also for the neo-Hookean cylindrical shell of arbitrary thickness subjected to an external hydrostatic pressure. Coman and Destrade (2008) made an extension of the investigation of the previous studies about bending of a rubber blocks and used WKB method to the bifurcation of an incompressible neo-Hookean thick hyper-elastic plate and reduced the system of two incremental equilibrium equations into a fourth-order linear eigen-problem that displays a multiple turning point. In the above mentioned and similar cases the incremental equations could be simplified into a single eigen-value problem (i.e. a simple fourth order O.D.E with variable coefficient). While using WKB method on solving a system of differential equations instead of a single fourth order O.D.E, we have to employ a new form with special techniques of WKB method, which generally is not easily possible. For the first time Fu and Sanjaranipour (2002) applied this form of WKB method to the stability analysis of an everted cylindrical tube. They indicated that for obtaining a proper asymptotic expansion, the solvability condition should by apply and in order to get a correct approximation the higher order WKB analysis should be employed. We simplified the incremental equilibrium equations in section three and finally reduced the problem to a system of three simple partial differential equation with variable coefficients. Then, the system of the eigen-value problem is solved by the aid of the higher order WKB method in section four. By applying this method on solving such a system, some interesting points arises, i.e. the solvability condition. The details related to such interesting situation will be analyzed on a separate paper. The paper concludes with a discussion of the results obtained, together with the suggestions for the further study. Comparison between the results obtained by using this method with the data of the numerical compound matrix method, shows an excellent agreement.

Section snippets

Deformation

The reference configuration of the initially undeformed rectangular of the hyperelastic occupies the regionB0={(X,Y,Z)R3|-AXA,-LYL,0ZH},where A, L and H are the thickness, length and the height respectively. By applying the surface traction on the faces at Y = ±L, the following deformation will appearB1={(r,θ,z)R×(-π,π]×R|arb,-ψ0θψ0,0zH},where a, b are the radii of the inner and outer faces and ψ0 = ψL, (ψ is a prescribed constant). For an incompressible elastic material we will have

Stability of the bending configuration

In the absence of the body forces the incremental equilibrium equations for the incompressible elastic materials as fully described in Haughton and Ogden (1979) isdivs˙=0,where s˙ denotes the incremental nominal stress tensor and the incremental boundary conditions ares˙Tn=0,where n is the unit outward normal to the surface. The linearized form of s˙ is written bys˙=AΓ+pΓ-p˙I,where Γ and I are the incremental deformation gradient of F and the identity tensor respectively, and A is the fourth

WKB approach for the stress concentration

The WKB method is a simple efficient tool for dealing with the variable-coefficient linear differential equations containing small or large parameters. We shall study the eigen-value problem for the case of ξ =   1 and α = O(1). Our aim is to describe the dependence of λ on this large parameter. We may use the following form of WKB method for the solution of the eigen-problem (20), (21), (22) and the relevant boundary conditions,{f,g,k}T={Φ,Ψ,ξΩ}Texpξρaρs(τ)dτ,whereΦ=Φ0+Φ1ξ+Φ2ξ2+,Ψ=Ψ0+Ψ1ξ+Ψ2ξ2+,

Conclusion

We are studying the bifurcation of a bending cylindrical rubber block made of elastic Varga material. Our aim on this study is two-fold. First deriving the system of differential equations with related boundary conditions and second, solving this system by using the so-called WKB method. We have to mention that by using WKB method on the leading order analysis, repeated roots arises and the process of obtaining the equations of the solvability conditions is very interesting. By solving this

Acknowledgements

The authors would like to express their thanks to Prof. Yibin Fu regarding to his valuable comments on this work.

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