On the stability of internally pressurized thick-walled spherical and cylindrical shells made of functionally graded incompressible hyperelastic material

Yavar Anani Gholamhossein Rahimi About the authors

Abstract

In this paper, stability analysis of thick-walled 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 non-monotonic 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 strain-energy functions, such as works of Mooney (1940) Mooney, M., (1940), A theory of large elastic deformation, Journal of Applied Physics, 11:582-592. , 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:223-252. , Yeoh (1993) Yeoh, O.H., (1993), Some forms of the strain energy function for rubber, Rubber Chem. Technology, 66:754-771. , 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): 1699-1735. . 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), Visco-hyperelastic constitutive law for modeling of foam’s behavior, J of Material& Design 32(5):2940-2948. , Bao et al. (2003) Bao R.H., Xue P., Yu T.X., Tao X.M. TAO (2003) Numerical simulation of large deformation of flat-topped 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 nearly-incompressible 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 two-parameter Mooney-Rivlin 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 rubber-like 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 pressure-volume coupling, International Journal of Mechanical Sciences, 82: 122-130. , 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, 93:166-180. , Tomita et al. (2008) Tomita, Y., Azuma, K. and Naito, M., (2008), Computational evaluation of strain-rate-dependent deformation behavior of rubber and carbon-black-filled rubber under monotonic and cyclic straining, International Journal of Mechanical Sciences, 50(6):856-868. and Barforooshi and Mohammadi (2016) Barforooshi S. and Mohammadi A., (2016), Study neo-Hookean and Yeoh hyper-elastic models in dielectric elastomer-based micro-beam 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 styrene-butadiene 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 Non-Linear 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: 1-7 , 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: 407-434 ).

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. Rubber-like 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 Non-Linear Mechanic, 40: 165-175. 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: 409-421. . 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: 615-630. 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: 492-512. study. Rudykh et al. (2012) Rudykh, S., Bhattacharya, K. and deBotton, G., (2012), Snap-through actuation of thickwalled electroactive balloons. Int. J. Nonlinear Mech., 47: 206-209. have researched about snap-through actuation of thick-walled electroactive balloons. Inflation and bifurcation of thick-walled compressible elastic spherical-shells has been studied by Haughton (1987) Haughton, D.M., (1987), Inflation and bifurcation of thick-walled compressible elastic spherical-shells. IMA J. Appl. Math., 39: 259-272. . 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 thick-walled hollow cylindrical shell made of isotropic FG rubber like materials which is shown in Fig.1 . A , B and Pi represent inner and outer radius of the shell and internal pressure, respectively. The cylinder is considered initially stress-free and presumed to be deformed statically. (R,Θ,Z) and (r,θ,z) represent reference and current configurations of cylindrical shell. The cylinder geometry in these configurations is described as follows:

Figure 1
Configuration of internally pressurized thick hollow cylinder
A R B ,0 Θ 2 π , 0 Z L (1)
ar b, 0θ2π, 0zl (2)

Deformation field of the cylinder can be expressed by Ericksen's proposed universal solutions (1954):

r = f ( R ) , θ = Θ , z = λ z Z (3)

The deformation gradient tensor F is presented by ( Fu and Ogden (2001) Fu,Y.B. and Ogden, R.W., (2001), Nonlinear Elasticity. Cambridge University Press. ):

F = d f ( R ) d R e r E R + f ( R ) R e θ E Θ + λ z e z E Z (4)

Radial, circumferential and axial stretches are defined by: λr=df(R)dR, λθ=f(R)R, λz=λz and J is determinant of the deformation gradient tensor F . As the subject of this paper is incompressible hyperelastic materials, incompressibility condition indicates: I3=J2= 1 and it leads to:

( λ r λ z λ θ ) = ( λ z f ( R ) d f ( R ) ) / ( R d R ) = 1 (5)

As a result, radial deformation of the cylinder is considered:

f ( R ) 2 = λ z 1 ( R 2 A 2 ) + a 2 (6)

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. ):

σ i = p + λ i W ^ λ i (7)

Where p is the undetermined scalar function that explains the incompressible internal constraint conditions and W^ is strain energy function which is function of the principal stretches. For thick walled hollow pressurized cylinder and in the absence of body forces only one of the equilibrium equation is not satisfied identically, which is ( Fu and Ogden (2001) Fu,Y.B. and Ogden, R.W., (2001), Nonlinear Elasticity. Cambridge University Press. ):

d σ r d r + σ r σ θ r = 0 (8)

Boundary conditions are expressed by:

σr(r=a)=Pi and σr(r=b)=0 (9)

Substitute σr and σθ from equation (7) to equation (8) and integrating in with respect to r yields:

σ r = a r ( 1 r [ W ^ λ r λ r W ^ λ θ λ θ ] ) d r P i (10)

By comparison σr from equations (7) and (10) , hydrostatic pressure is calculated as follows:

p = a r [ 1 r ( W ^ λ r λ r W ^ λ θ λ θ ) ] d r + W ^ λ r λ r + P i (11)

Equations (11) and (7) allow to calculate hoop stress, σθ , and axial stress, σz , by equations (12) and (13) as follows:

σ θ = σ r + W ^ λ θ λ θ W ^ λ r λ r (12)
σ z = σ r + W ^ λ z λ z W ^ λ r λ r (13)

b is determined by implementing second boundary condition of equation (9):

P i = a b ( 1 r [ W ^ λ r λ r W ^ λ θ λ θ ] ) d r (14)

Following parameters are introduced for simplicity:

β=BA , λ(θ)b=bB,λ(θ)a=aA,β2 (λzλ(θ)b21)=λzλ(θ)a21 (15)

With the differentiation of the strain energy function W^ in relation to λθ , denoted by W^λθ , and with the derivative of λθ , dλθ=drR(1λzλθ2) , the equation (14) can be rewritten as follows:

P i = ( λ θ ) b ( λ θ ) a W ^ λ θ d λ θ ( λ z λ θ 2 1 ) (16)

By differentiation of equation (16) with respect to (λθ)a and using equation (15) , it is found that:

( λ θ ) a 1 ( ( λ θ ) a 2 z 1 ) d P d ( λ θ ) a = W ^ λ θ ( ( λ θ ) a , λ z ) ( λ θ ) a W ^ λ θ ( ( λ θ ) b , λ z ) ( λ θ ) b (17)

Pressure turning-points at constant λz will exist, if W^λθ((λθ),λz)(λθ) is not monotonic in λθ and it leads to λθW^λθλθW^λθ=0 . Modified Ogden strain energy function for incompressible materials is used as follows ( Fu and Ogden (2001) Fu,Y.B. and Ogden, R.W., (2001), Nonlinear Elasticity. Cambridge University Press. ):

W ^ ( λ 1 , λ 2 , λ 3 ) = p = 1 N μ p ( R A ) n α p ( λ 1 α p + λ 2 α p + λ 3 α p 3 ) (18)

Where μp and αp are material parameter which varies by power law function in radial direction μp(R)=μp0(RA)n , N is Number of f Ogden strain energy function sentences and n is material inhomogeneity parameter. By considering p=1 in relation (18), the result is defined:

P i = μ p 0 ( A O G C N ( b , n , λ z ) A O G C N ( a , n , λ z ) ) (19)

Where:

A O G C N ( r , n , λ z ) = ( λ z r R ) α α 1 ( R A ) n ( 1 λ z r 2 λ z b 2 B 2 ) 1 2 ( n α ) ( ( R ) 2 α 2 F 1 ( n 2 α 2 , α 2 ; 1 α 2 ; λ z r 2 λ z b 2 B 2 ) + λ z α r 2 α ( 1 λ z r 2 λ z b 2 B 2 ) α 2 F 1 ( α 2 , α n 2 ; α + 2 2 ; λ z r 2 λ z b 2 B 2 ) ) (20)

Where 2F1(a,b;c;x) is the Gauss-hypergeometric function ( Arfken 1985 Arfken, G., (1985), Mathematical methods for physicists, 3rd ed., Academic Press, Orlando, FL. ).

3 Result and Discussion

Material constants “ μp0 ” and “ α ” are determined by using Levenberg–Marquardt nonlinear regression method for the rubber tested by Treloar (1944) Treloar, L.R.G., (1944), Stress-strain data for vulcanised rubber under various types of deformation. Trans. Faraday Soc., 40: 59-70. . By this method μp0=0.69 and α=1.3 are achieved. Figure 2 shows piμ0 versus (λθ)a for different λz in an approximately thin shell which its BA=1.1 . By analyzing this figure, it is found that critical pressure will increase by decreasing λz . Moreover related critical pressure happens in a higher (λθ)a when λz decreases. Figures 3 and 4 demonstrate piμ0 versus (λθ)a for different λz in shells with BA=2 and BA=3 , respectively. These figures show that critical pressure increases by increasing BA . For example piμ0 for BA=3 and λz=0.2 is 1.716 times of piμ0 for BA=2 and λz=0.2 . Δpμ0 is used to as dimensionless pressure. This dimensionless pressure can be used to predict behavior of the structure without knowing its material and after that it can be used very important factor in material tailoring. In addition, critical pressure happens in a higher (λθ)a when BA increases in a constant λz . Figure 5 shows piμ0 versus (λθ)a for different λz in a very thick shell with BA=20 . In this case, shell is very stable. For instance, instability does not occur for λz=0.2 even in (λθ)a7 .

Figure 2
piμ0 versus (λθ)a for different λz , constant material inhomogeneity parameter and BA=1.1
Figure 3
piμ0 versus (λθ)a for different λz , constant material inhomogeneity parameter a nd BA=2
Figure 4
piμ0 versus (λθ)a for different λz , constant material inhomogeneity parameter and BA=3
Figure 5
piμ0 versus (λθ)a for different λz , constant material inhomogeneity parameter and BA=20

Figures 6 , 7 and 8 show piμ0 versus (λθ)a by considering BA=2 for different inhomogeneity parameter and for λz=0.75 , 1, 1.25, respectively. These figures demonstrate critical pressure increases by increasing material inhomogeneity parameter (n) . For example, ratio of critical pressure for n=4 and n=0 is 4.78. Moreover, related critical pressure happens in a higher (λθ)a when n increases. For instance, critical pressure in figure 5 happens for n=4 and n=0 in (λθ)a=5.23 and (λθ)a=2.94 , respectively.

Figure 6
piμ0 versus (λθ)a by considering BA=2 for different inhomogeneity parameter and λz=0.75
Figure 7
piμ0 versus (λθ)a by considering BA=2 for different inhomogeneity parameter and λz=1
Figure 8
piμ0 versus (λθ)a by considering BA=2 for different inhomogeneity parameter and λz=1.25

Figures 9 , 10 and 11 show piμ0 versus (λθ)a for different BA . It is very interesting to observe that for very thick shells( BA=20 ), instability does not occur even at (λθ)a=7 . In contrast for a thin shells ( BA=1.05 and λz=0.75 ), instability occurs at (λθ)a=1.12 .

Figure 9
piμ0 versus (λθ)a for different BA , constant inhomogeneity parameter and λz=1.25
Figure 10
piμ0 versus (λθ)a for different BA , constant inhomogeneity parameter and λz=1
Figure 11
piμ0 versus (λθ)a for different BA , constant inhomogeneity parameter and λz=1 .25

4 Problem formulation for spherical shell

In this section, instability analysis of a pressurized thick-walled hollow spherical shell made of isotropic FG rubber like materials. A , B and Pi represent inner and outer radius of the shell and internal pressure, respectively. Condition s are the same of the cylindrical shell and reference and current configurations of spherical shell are presented by (R,Θ,Φ) and (r,θ,φ) . The sphere geometry in these configurations is described as follows:

A R B ,0 Θ 2 π ,0 Φ 2 π (21)
r b, 0θ2π, 0φ2π (22)

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. ):

r = f ( R ) , θ = Θ , φ = Φ (23)

Components of stretch in spherical coordinates are defined as follow: λr=df(R)dR, λθ=λφ=f(R)R The deformation gradient tensor F is presented by ( Fu and Ogden (2001) Fu,Y.B. and Ogden, R.W., (2001), Nonlinear Elasticity. Cambridge University Press. )::

F = d f ( R ) d R e r E R + f ( R ) R e θ E Θ + f ( R ) R e φ E Φ (24)

Method which is used for the cylinder is also applied for the sphere. As a result, internal pressure is found as follows:

P i = ( λ θ ) b ( λ θ ) a W ^ λ θ d λ θ ( λ θ 3 1 ) (25)

In this section, following parameters are also defined:

( λ θ ) a = a A , ( λ θ ) b = b B = ( ( λ θ ) a 3 + β 3 ) 1 3 , β , = B / A > 1, β 3 ( ( λ θ ) b 3 1 ) = ( λ θ ) a 3 1 (26)

By differentiation of equation (25) with respect to (λθ)a and some simplification, it is found that:

( ( λ θ ) a ( λ θ ) a 2 ) d P d ( λ θ ) a = W ^ λ θ ( ( λ θ ) a ) ( λ θ ) a 2 W ^ λ θ ( ( λ θ ) b ) ( λ θ ) b 2 (27)

Pressure turning-points will exist, if W^λθ(λθ)2 is not monotonic in λθ and it leads to λθW^λθλθ2W^λθ=0 . By using modified Ogden strain energy function for incompressible hyperelastic material and Eq. (25) , we have:

P i = μ 1 [ [ A O G S N ( b , m ) A O G S N ( a , m ) ] ] (28)

Where:

A O G S N ( r , m ) = 3 r 2 α 1 ( R ) m α 1 A m α 1 ( 1 r 3 b 3 B 3 ) 1 3 ( 2 α 1 m ) [ R 3 α 1 2 F 1 ( m 3 2 α 1 3 , 1 3 ( 2 α 1 ) ; 1 2 α 1 3 ; r 3 b 3 B 3 ) + 2 r 3 α 1 ( 1 r 3 b 3 B 3 ) α 2 F 1 ( α 1 3 , α 1 m 3 ; α 1 + 3 3 ; r 3 b 3 B 3 ) ] (29)

5 Result and Discussion

Figures 12 - 14 show piμ0 versus (λθ)a for different inhomogeneity parameter m . By analyzing these figures, it is found that critical pressure will increase by increasing m . Moreover related critical pressure happens in a higher (λθ)a when m increases. These figures show that critical pressure also increases by increasing BA . In addition, Figures 15 - 18 show piμ0 versus (λθ)a for different BA and specific material inhomogeneity parameter in each figure. These figures show that critical pressure increases by increasing BA . For example piμ0 for BA=4 and m=3 is 3.52 times of piμ0 for BA=1.1 and m=3 . In addition, critical pressure happens in a higher (λθ)a when BA increases.

Figure 12
piμ0 versus (λθ)a by considering BA=1.1 for different inhomogeneity parameter
Figure 13
piμ0 versus (λθ)a by considering BA=1.5 for different inhomogeneity parameter
Figure 14
piμ0 versus (λθ)a by considering BA=2 for different inhomogeneity parameter
Figure 15
piμ0 versus (λθ)a by considering m=6 for different BA
Figure 16
piμ0 versus (λθ)a by considering m=3 for different BA
Figure 17
piμ0 versus (λθ)a by considering m =0 for different BA
Figure 18
piμ0 versus (λθ)a by considering m =2 for different BA

6 Validation

For validating of obtained theoretical results, numerical method is used to find the accuracy of these result. In this order sphere with BA=2 and m=2 is considered. Comparison of numerical results and theoretical results is presented in Figure 19 . Comparison of numerical and theoretical results shows that, maximum differences between these results are about 6.3%; therefore it is concluded that there is good agreement between numerical and theoretical results, so theoretical solution can be applied for finding stability of the axisymmetric thick vessel composed of FG hyperelastic material.

Figure 19
Comparison between theoretical and numerical results of FG sphere

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 non-monotonic pressure-radius 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 (λθ)a increases by increasing material inhomogeneity parameter and shell thickness. This means that, shell can tolerate higher pressure and more radial deformation. Moreover, in cylindrical shells, critical pressure and its related hoop stretch increases by decreasing longitudinal stretch. This imply that in cylindrical shell when λz1 , longitudinal stretch has an opposite effect of internal pressure in shell behavior and instability point will be delayed by decreasing λz .

From an applied perspective, unstable conditions are absolutely unwelcome and should be evaded because the deformation becomes highly non-uniform, 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), Visco-hyperelastic constitutive law for modeling of foam’s behavior, J of Material& Design 32(5):2940-2948.
  • 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: 1-7
  • 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: 407-434
  • 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 flat-topped conical shells made of textile, Latin American Journal of Solid and Structures, 1:25–47.
  • Barforooshi S. and Mohammadi A., (2016), Study neo-Hookean and Yeoh hyper-elastic models in dielectric elastomer-based micro-beam 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 Non-Linear 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: 492-512.
  • Beatty, M.F., (1987) Topics in finite elasticity: hyperelasticity of rubber, elastomers, and biological tissues—with examples. Applied Mechanics Review 40(12): 1699-1735.
  • 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:223-252.
  • Coelho, M., Roehl. D. and Bletzinger. K. (2014), Numerical and analytical solutions with finite strains for circular inflated membranes considering pressure-volume coupling, International Journal of Mechanical Sciences, 82: 122-130.
  • 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 Non-Linear Mechanic, 40: 165-175.
  • Goriely, A., Destrade, M. and Ben Amar, M., (2006), Stability and bifurcation of compressed elastic cylindrical tubes, Q. J. Mech. Appl. Math., 59: 615-630.
  • 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 thick-walled compressible elastic spherical-shells. IMA J. Appl. Math., 39: 259-272.
  • Ikeda, Y, Kasai, Y., Murakami, S. and Kohjiya, S., (1998), Preparation and mechanical properties of graded styrene-butadiene rubber vulcanizates, J. Jpn. Inst. Metals, 62: 1013–1017.
  • Mooney, M., (1940), A theory of large elastic deformation, Journal of Applied Physics, 11:582-592.
  • Needleman, A., (1977), Inflation of spherical rubber balloons. Int. J, Solids Struct.,13: 409-421.
  • 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 rubber-like 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 two-parameter Mooney-Rivlin hyperelastic constitutive, Latin American Journal of Solid and Structures, 7:391–411.
  • Rudykh, S., Bhattacharya, K. and deBotton, G., (2012), Snap-through actuation of thickwalled electroactive balloons. Int. J. Nonlinear Mech., 47: 206-209.
  • 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, 93:166-180.
  • Silva C.A.C. and Bittencourt M.L., (2008), Structural shape optimization of 3D nearly-incompressible hyperelasticity problems, Latin American Journal of Solid and Structures, 5:129–156.
  • Tomita, Y., Azuma, K. and Naito, M., (2008), Computational evaluation of strain-rate-dependent deformation behavior of rubber and carbon-black-filled rubber under monotonic and cyclic straining, International Journal of Mechanical Sciences, 50(6):856-868.
  • Treloar, L.R.G., (1944), Stress-strain data for vulcanised rubber under various types of deformation. Trans. Faraday Soc., 40: 59-70.
  • Yeoh, O.H., (1993), Some forms of the strain energy function for rubber, Rubber Chem. Technology, 66:754-771.

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
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