Abstract
In this paper, stability analysis of thickwalled spherical and cylindrical shells made of functionally graded incompressible hyperelastic material subjected to internal pressure is presented. Instability point happens in the inflation of above mentioned shells and in this paper effect of material inhomogeneity and shell thickness has been investigated. Extended Ogden strain energy function with variable material parameter is used to model the material behavior. To model inhomogeneity, we assume that material parameter varies by a power law function in the radial direction and inhomogeneity factor is a power in the power law function. Analytical method is used to find the internal pressure versus hoop extension ratio relations in explicit form for both of cylindrical and spherical shells and the nonmonotonic behavior of the inflation curves is studied. Following this, profile of inflation pressure versus hoop stretch is presented and effect of the inhomogeneity and shell thickness in the onset of instability is studied. The obtained results show that the material inhomogeneity parameter and shell thickness have a significant influence on the stability of above mentioned shells. Thus with selecting a proper material inhomogeneity parameter and shell thickness, engineers can design a specific FGM hollow cylinder that can meet some special requirements.
Keywords
Hyperelasticity; Instability; functionally graded material; thick shells
1 INTRODUCTION
Rubber like materials under external loading possibly display different reactions dependent on their material properties, forces and geometry. Finite deformation theory is used to describe large deformation responses of these materials. A hyperelastic material shows a nonlinear behavior which imply that its answer to the load is not directly proportional to the deformation. Different strain energy functions are used to model hyperelastic behavior of these materials and there are numerous efforts in the literature on the derivation and/or fitting of various forms of strainenergy functions, such as works of Mooney (1940) Mooney, M., (1940), A theory of large elastic deformation, Journal of Applied Physics, 11:582592. , Blatz and Ko (1962) Blatz, P.J. and Ko, W.L., (1962), Application of finite elastic theory to the deformation of rubbery materials. Transactions of the Society of Rheology, VI:223252. , Yeoh (1993) Yeoh, O.H., (1993), Some forms of the strain energy function for rubber, Rubber Chem. Technology, 66:754771. , Ogden (1972) Ogden, R.W., (1972), Large deformation isotropic elasticity  on the correlation of theory and experiment for incompressible rubberlike solids, Proc. R.Soc. Lond. A. , Beatty (1987) Beatty, M.F., (1987) Topics in finite elasticity: hyperelasticity of rubber, elastomers, and biological tissues—with examples. Applied Mechanics Review 40(12): 16991735. . Presenting precise constitutive model to describe hyperelastic behavior of rubber like material is the subject of a lot of researches in the recent years such as works by Anani and Alizadeh (2011) Anani, Y. and Alizadeh, Y., (2011), Viscohyperelastic constitutive law for modeling of foam’s behavior, J of Material& Design 32(5):29402948. , Bao et al. (2003) Bao R.H., Xue P., Yu T.X., Tao X.M. TAO (2003) Numerical simulation of large deformation of flattopped conical shells made of textile, Latin American Journal of Solid and Structures, 1:25–47. , Silva and Bittencourt (2008) Silva C.A.C. and Bittencourt M.L., (2008), Structural shape optimization of 3D nearlyincompressible hyperelasticity problems, Latin American Journal of Solid and Structures, 5:129–156. , Pereira and Bittencourt (2010) Pereira C.E.L. and Bittencourt M.L., (2010), Topological sensitivity analysis for a twoparameter MooneyRivlin hyperelastic constitutive, Latin American Journal of Solid and Structures, 7:391–411. , Pascon and Coda (2013) Pascon J.P. and Coda H.B., (2013), Large deformation analysis of homogeneous rubberlike materials via shell finite elements, Latin American Journal of Solid and Structures, 10:1177–1210. , Coelho et al. (2014) Coelho, M., Roehl. D. and Bletzinger. K. (2014), Numerical and analytical solutions with finite strains for circular inflated membranes considering pressurevolume coupling, International Journal of Mechanical Sciences, 82: 122130. , Santos et al. (2015) Santos, T., Alves, M.K. and Rossi, R. (2015), A constitutive formulation and numerical procedure to model rate effects on porous materials at finite strains, International Journal of Mechanical Sciences, 93:166180. , Tomita et al. (2008) Tomita, Y., Azuma, K. and Naito, M., (2008), Computational evaluation of strainratedependent deformation behavior of rubber and carbonblackfilled rubber under monotonic and cyclic straining, International Journal of Mechanical Sciences, 50(6):856868. and Barforooshi and Mohammadi (2016) Barforooshi S. and Mohammadi A., (2016), Study neoHookean and Yeoh hyperelastic models in dielectric elastomerbased microbeam resonators, Latin American Journal of Solid and Structures, 13:1823–1837.
As functionally graded rubber is the subject of this study it should be noted that graded rubber like materials were created by Ikeda et al. (1998) Ikeda, Y, Kasai, Y., Murakami, S. and Kohjiya, S., (1998), Preparation and mechanical properties of graded styrenebutadiene rubber vulcanizates, J. Jpn. Inst. Metals, 62: 1013–1017. in the laboratory for the first time, a while after these materials have attracted the attention of investigators for modeling these materials behavior under mechanical and geometrical boundary conditions. Some important and novel researches about analysis of inhomogeneous rubber like materials structures are presented in details by Bilgili (2003 Bilgili, E., (2003), Controlling the stress–strain inhomogeneities in axially sheared and radially heated hollow rubber tubes via functional grading, Mech. Res. Commun. 30:257–66. , ^{2004} Bilgili, E., (2004), Modelling mechanical behaviour of continuously graded vulcanized rubbers., Plast Rubbers Compos 33(4):163–169. ), Batra (2006) Batra, R.C., (2006), Torsion of a functionally graded cylinder. AIAA J, 44:1363–1365. , Batra and Bahrami (2009) Batra, R.C. and Bahrami, A. (2009), Inflation and eversion of functionally graded nonlinear elastic incompressible cylinders, Int. J. of NonLinear Mechanics, 44: 311–323. , Anani and Rahimi (2015 Anani, Y. and Rahimi G.H., (2015), Stress analysis of thick pressure vessel composed of functionally graded incompressible hyperelastic materials, International Journal of Mechanical Sciences, 104: 17 , ^{2016} Anani, Y. and Rahimi G.H., (2016), Stress analysis of thick spherical pressure vessel composed of transversely isotropic functionally graded incompressible hyperelastic materials, Latin American Journal of Solid and Structures, 13: 407434 ).
This paper concern is about instability analysis of internally pressurized spherical and cylindrical thick shells made of isotropic functionally graded incompressible hyperelastic materials. Moreover, finding effect of material inhomogeneity and shell thickness in the onset of instability of above mentioned shells are also analyzed in this paper. Rubberlike materials experiencing large deformations, display a diversity of amazing instabilities. A review of some of the more interesting instances has been presented in a recent paper by Gent (2005) Gent, A.N., (2005), Elastic instabilities in rubber, International Journal of NonLinear Mechanic, 40: 165175. where background literature can be found. In the context of stability of spherical and cylindrical shells, several aspects of this problem have been studied and cited here. Inflation of spherical rubber balloons has been studied by Needleman (1977) Needleman, A., (1977), Inflation of spherical rubber balloons. Int. J, Solids Struct.,13: 409421. . Goriely et al. (2006) Goriely, A., Destrade, M. and Ben Amar, M., (2006), Stability and bifurcation of compressed elastic cylindrical tubes, Q. J. Mech. Appl. Math., 59: 615630. have investigated stability and bifurcation of compressed elastic cylindrical tubes. Small amplitude radial oscillations of an incompressible, isotropic elastic spherical shell has been a subject of Beatty (2011) Beatty, M.F., (2011), Small amplitude radial oscillations of an incompressible, isotropic elastic spherical Shell, Math. Mech. Solids.,16: 492512. study. Rudykh et al. (2012) Rudykh, S., Bhattacharya, K. and deBotton, G., (2012), Snapthrough actuation of thickwalled electroactive balloons. Int. J. Nonlinear Mech., 47: 206209. have researched about snapthrough actuation of thickwalled electroactive balloons. Inflation and bifurcation of thickwalled compressible elastic sphericalshells has been studied by Haughton (1987) Haughton, D.M., (1987), Inflation and bifurcation of thickwalled compressible elastic sphericalshells. IMA J. Appl. Math., 39: 259272. . Comparison of stability and bifurcation criteria for inflated spherical elastic shells has been done by Haughton and Kirkinis (2003) Haughton, D.M. and Kirkinis. E., (2003), A comparison of stability and bifurcation criteria for inflated spherical elastic shells. Math. Mech. Solids., 8: 561–572. .
A large number of works has been done on to analyzing instability of spherical and cylindrical shell but we found that there is a gap in the literature about stability analyzing of functionally graded incompressible hyperelastic cylindrical/spherical thick shells. Therefore, in this paper we focus on instability of these shells and effect of material parameters and structural parameters are investigated, carefully.
2 Problem formulation for cylindrical shell
In this section, instability analysis of a pressurized thickwalled hollow cylindrical shell made of isotropic FG rubber like materials which is shown in Fig.1 .
Deformation field of the cylinder can be expressed by Ericksen's proposed universal solutions (1954):
The deformation gradient tensor
Radial, circumferential and axial stretches are defined by:
As a result, radial deformation of the cylinder is considered:
Cauchy stress for incompressible hyperelastic materials is stated by ( Fu and Ogden (2001) Fu,Y.B. and Ogden, R.W., (2001), Nonlinear Elasticity. Cambridge University Press. ):
Where
Boundary conditions are expressed by:
Substitute
By comparison
Equations (11) and (7) allow to calculate hoop stress,
Following parameters are introduced for simplicity:
With the differentiation of the strain energy function
By differentiation of equation (16) with respect to
Pressure turningpoints at constant
Where
Where:
Where
3 Result and Discussion
Material constants “
Figures 6 , 7
and 8 show
Figures 9 , 10
and 11 show
4 Problem formulation for spherical shell
In this section, instability analysis of a pressurized thickwalled hollow spherical shell made of isotropic FG rubber like materials.
Ericksen's universal solutions is used to find deformation of spherical shell ( Ericksen, 1954 Ericksen, J.L., (1954), Deformations possible in every isotropic incompressible perfectly elastic body, Z. Angew. Math. Phys. 5:466–486. ):
Components of stretch in spherical coordinates are defined as follow:
Method which is used for the cylinder is also applied for the sphere. As a result, internal pressure is found as follows:
In this section, following parameters are also defined:
By differentiation of equation (25) with respect to
Pressure turningpoints will exist, if
Where:
5 Result and Discussion
Figures 12  14 show
6 Validation
For validating of obtained theoretical results, numerical method is used to find the accuracy of these result. In this order sphere with
6 Conclusion
One of the most important and amazing problems in inflated hyperelastic bodies from a hypothetical viewpoint is instability analysis and finding instability point onset of these bodies, because they happen unexpectedly. Universal solution of Ericksen's family is used to find expansion of thick spherical/cylindrical shells made of inhomogeneous, isotropic incompressible hyperelastic material. Material inhomogeneity is assumed to model by functionally graded material. Modified Ogden strain energy function with power law variable material property is used to model grading of material properties. It should be noted, Ogden material shows nonmonotonic pressureradius relationship and behaviour for cylindrical and spherical shells. Therefore, onset of instability with the predictions of effect of material inhomogeneity parameter and shell thickness is investigated.
Results show that in the thick spherical/cylindrical shell, critical internal pressure and its related hoop stretch
From an applied perspective, unstable conditions are absolutely unwelcome and should be evaded because the deformation becomes highly nonuniform, leading to early failure. Above results in investigating mechanical behavior of these shells reveal great effect of shell thickness and material inhomogeneity to delay instability and should be noted in design of these shells.
References
 Anani, Y. and Alizadeh, Y., (2011), Viscohyperelastic constitutive law for modeling of foam’s behavior, J of Material& Design 32(5):29402948.
 Anani, Y. and Rahimi G.H., (2015), Stress analysis of thick pressure vessel composed of functionally graded incompressible hyperelastic materials, International Journal of Mechanical Sciences, 104: 17
 Anani, Y. and Rahimi G.H., (2016), Stress analysis of thick spherical pressure vessel composed of transversely isotropic functionally graded incompressible hyperelastic materials, Latin American Journal of Solid and Structures, 13: 407434
 Arfken, G., (1985), Mathematical methods for physicists, 3rd ed., Academic Press, Orlando, FL.
 Bao R.H., Xue P., Yu T.X., Tao X.M. TAO (2003) Numerical simulation of large deformation of flattopped conical shells made of textile, Latin American Journal of Solid and Structures, 1:25–47.
 Barforooshi S. and Mohammadi A., (2016), Study neoHookean and Yeoh hyperelastic models in dielectric elastomerbased microbeam resonators, Latin American Journal of Solid and Structures, 13:1823–1837.
 Batra, R.C. and Bahrami, A. (2009), Inflation and eversion of functionally graded nonlinear elastic incompressible cylinders, Int. J. of NonLinear Mechanics, 44: 311–323.
 Batra, R.C., (2006), Torsion of a functionally graded cylinder. AIAA J, 44:1363–1365.
 Beatty, M.F., (2011), Small amplitude radial oscillations of an incompressible, isotropic elastic spherical Shell, Math. Mech. Solids.,16: 492512.
 Beatty, M.F., (1987) Topics in finite elasticity: hyperelasticity of rubber, elastomers, and biological tissues—with examples. Applied Mechanics Review 40(12): 16991735.
 Bilgili, E., (2003), Controlling the stress–strain inhomogeneities in axially sheared and radially heated hollow rubber tubes via functional grading, Mech. Res. Commun. 30:257–66.
 Bilgili, E., (2004), Modelling mechanical behaviour of continuously graded vulcanized rubbers., Plast Rubbers Compos 33(4):163–169.
 Blatz, P.J. and Ko, W.L., (1962), Application of finite elastic theory to the deformation of rubbery materials. Transactions of the Society of Rheology, VI:223252.
 Coelho, M., Roehl. D. and Bletzinger. K. (2014), Numerical and analytical solutions with finite strains for circular inflated membranes considering pressurevolume coupling, International Journal of Mechanical Sciences, 82: 122130.
 Ericksen, J.L., (1954), Deformations possible in every isotropic incompressible perfectly elastic body, Z. Angew. Math. Phys. 5:466–486.
 Fu,Y.B. and Ogden, R.W., (2001), Nonlinear Elasticity. Cambridge University Press.
 Gent, A.N., (2005), Elastic instabilities in rubber, International Journal of NonLinear Mechanic, 40: 165175.
 Goriely, A., Destrade, M. and Ben Amar, M., (2006), Stability and bifurcation of compressed elastic cylindrical tubes, Q. J. Mech. Appl. Math., 59: 615630.
 Haughton, D.M. and Kirkinis. E., (2003), A comparison of stability and bifurcation criteria for inflated spherical elastic shells. Math. Mech. Solids., 8: 561–572.
 Haughton, D.M., (1987), Inflation and bifurcation of thickwalled compressible elastic sphericalshells. IMA J. Appl. Math., 39: 259272.
 Ikeda, Y, Kasai, Y., Murakami, S. and Kohjiya, S., (1998), Preparation and mechanical properties of graded styrenebutadiene rubber vulcanizates, J. Jpn. Inst. Metals, 62: 1013–1017.
 Mooney, M., (1940), A theory of large elastic deformation, Journal of Applied Physics, 11:582592.
 Needleman, A., (1977), Inflation of spherical rubber balloons. Int. J, Solids Struct.,13: 409421.
 Ogden, R.W., (1972), Large deformation isotropic elasticity  on the correlation of theory and experiment for incompressible rubberlike solids, Proc. R.Soc. Lond. A.
 Pascon J.P. and Coda H.B., (2013), Large deformation analysis of homogeneous rubberlike materials via shell finite elements, Latin American Journal of Solid and Structures, 10:1177–1210.
 Pereira C.E.L. and Bittencourt M.L., (2010), Topological sensitivity analysis for a twoparameter MooneyRivlin hyperelastic constitutive, Latin American Journal of Solid and Structures, 7:391–411.
 Rudykh, S., Bhattacharya, K. and deBotton, G., (2012), Snapthrough actuation of thickwalled electroactive balloons. Int. J. Nonlinear Mech., 47: 206209.
 Santos, T., Alves, M.K. and Rossi, R. (2015), A constitutive formulation and numerical procedure to model rate effects on porous materials at finite strains, International Journal of Mechanical Sciences, 93:166180.
 Silva C.A.C. and Bittencourt M.L., (2008), Structural shape optimization of 3D nearlyincompressible hyperelasticity problems, Latin American Journal of Solid and Structures, 5:129–156.
 Tomita, Y., Azuma, K. and Naito, M., (2008), Computational evaluation of strainratedependent deformation behavior of rubber and carbonblackfilled rubber under monotonic and cyclic straining, International Journal of Mechanical Sciences, 50(6):856868.
 Treloar, L.R.G., (1944), Stressstrain data for vulcanised rubber under various types of deformation. Trans. Faraday Soc., 40: 5970.
 Yeoh, O.H., (1993), Some forms of the strain energy function for rubber, Rubber Chem. Technology, 66:754771.
Publication Dates

Publication in this collection
04 June 2018 
Date of issue
2018
History

Received
27 July 2017 
Reviewed
13 Oct 2017 
Accepted
16 Oct 2018