Skip to main content

An analytic derivation of the bifurcation conditions for localization in hyperelastic tubes and sheets


We provide an analytic derivation of the bifurcation conditions for localized bulging in an inflated hyperelastic tube of arbitrary wall thickness and axisymmetric necking in a hyperelastic sheet under equibiaxial stretching. It has previously been shown numerically that the bifurcation condition for the former problem is equivalent to the vanishing of the Jacobian determinant of the internal pressure P and resultant axial force N, with each of them viewed as a function of the azimuthal stretch on the inner surface and the axial stretch. This equivalence is established here analytically. For the latter problem for which it has recently been shown that the bifurcation condition is not given by a Jacobian determinant equal to zero, we explain why this is the case and provide an alternative interpretation.

This is a preview of subscription content, access via your institution.

Fig. 1


  1. Mallock, A.: Note on the instability of India-rubber tubes and balloons when distended by fluid pressure. Proc. Roy. Soc. A 49, 458–463 (1891).

    Article  MATH  Google Scholar 

  2. Yin, W.-L.: Non-uniform inflation of a cylindrical elastic membrane and direct determination of the strain energy function. J. Elast. 7, 265–282 (1977).

    Article  MATH  Google Scholar 

  3. Chater, E., Hutchinson, J.W.: On the propagation of bulges and buckles. J. Appl. Mech. 51, 269–277 (1984).

    Article  Google Scholar 

  4. Kyriakides, S., Chang, Y.-C.: On the inflation of a long elastic tube in the presence of axial load. Int. J. Solids Struct. 26(9–10), 975–991 (1990).

    Article  Google Scholar 

  5. Kyriakides, S., Chang, Y.-C.: The initiation and propagation of a localized instability in an inflated elastic tube. Int. J. Solids Struct. 27(9), 1085–1111 (1991).

    Article  Google Scholar 

  6. Pamplona, D.C., Goncalves, P.B., Lopes, S.R.X.: Finite deformations of cylindrical membrane under internal pressure. Int. J. Mech. Sci. 48(6), 683–696 (2006).

    Article  Google Scholar 

  7. Goncalves, P.B., Pamplona, D., Lopes, S.R.X.: Finite deformations of an initially stressed cylindrical shell under internal pressure. Int. J. Mech. Sci. 50(1), 92–103 (2008).

    Article  MATH  Google Scholar 

  8. Rodríguez, J., Merodio, J.: A new derivation of the bifurcation conditions of inflated cylindrical membranes of elastic material under axial loading. application to aneurysm formation. Mech. Re. Commun. 38(3), 203–210 (2011).

    Article  MATH  Google Scholar 

  9. Alhayani, A.A., Rodríguez, J., Merodio, J.: Competition between radial expansion and axial propagation in bulging of inflated cylinders with application to aneurysms propagation in arterial wall tissue. Int. J. Eng. Sci. 85, 74–89 (2014).

    Article  Google Scholar 

  10. Alexander, H.: Tensile instability of initially spherical balloons. Int. J. Eng. Sci. 9, 151–160 (1971).

    Article  Google Scholar 

  11. Kanner, L.M., Horgan, C.O.: Elastic instabilities for strain-stiffening rubber-like spherical and cylindrical thin shells under inflation. Int. J. Non-linear Mech. 42, 204–215 (2007).

    Article  Google Scholar 

  12. Horny, L., Netusil, M., Horak, Z.: Limit point instability in pressurization of anisotropic finitely extensible hyperelastic thin-walled tube. Int. J. Non-linear Mech. 77, 107–114 (2015).

    Article  Google Scholar 

  13. Fu, Y.B., Pearce, S.P., Liu, K.K.: Post-bifurcation analysis of a thin-walled hyperelastic tube under inflation. Int. J. Non-linear Mech. 43(8), 697–706 (2008).

    Article  MATH  Google Scholar 

  14. Fu, Y.B., Liu, J.L., Francisco, G.S.: Localized bulging in an inflated cylindrical tube of arbitrary thickness-the effect of bending stiffness. J. Mech. Phys. Solids 90, 45–60 (2016).

    MathSciNet  Article  MATH  Google Scholar 

  15. Ye, Y., Liu, Y., Fu, Y.B.: Weakly nonlinear analysis of localized bulging of an inflated hyperelastic tube of arbitrary wall thickness. J. Mech. Phys. Solids 135, 103804 (2020).

    MathSciNet  Article  Google Scholar 

  16. Lin, Z., Li, L., Ye, Y.: Numerical simulation of localized bulging in an inflated hyperelastic tube with fixed ends. Int. J. Appl. Mech. 12(10), 2050118 (2020).

    Article  Google Scholar 

  17. Wang, S., Guo, Z., Zhou, L., Li, L., Fu, Y.B.: An experimental study of localized bulging in inflated cylindrical tubes guided by newly emerged analytical results. J. Mech. Phys. Solids 124, 536–554 (2019).

    MathSciNet  Article  Google Scholar 

  18. Varatharajan, N., DasGupta, A.: Study of bifurcation in a pressurized hyperelastic membrane tube enclosed by a soft substrate. Int. J. Non-linear Mech. 95, 233–241 (2017).

    Article  Google Scholar 

  19. Wang, J., Althobaiti, A., Fu, Y.B.: Localized bulging of rotating elastic cylinders and tubes. J. Mech. Mater. Struct. 12(4), 545–561 (2017).

    MathSciNet  Article  Google Scholar 

  20. Wang, J., Fu, Y.B. (2018).: Effect of double-fibre reinforcement on localized bulging of an inflated cylindrical tube of arbitrary thickness. J. Eng. Math. 109(1), 21–30.

  21. Liu, Y., Ye, Y., Althobaiti, A., Xie, Y.X.: Prevention of localized bulging in an inflated bilayer tube. Int. J. Mech. Sci. 153, 359–368 (2019).

    Article  Google Scholar 

  22. Ye, Y., Liu, Y., Althobaiti, A., Xie, Y.X.: Localized bulging in an inflated bilayer tube of arbitrary thickness: effects of the stiffness ratio and constitutive model. Int. J. Solids Struct. 176, 173–184 (2019).

    Article  Google Scholar 

  23. Hejazi, M., Hsiang, Y., Srikantha Phani, A.: Fate of a bulge in an inflated hyperelastic tube: theory and experiment. Proc. Roy. Soc. A 477(2247), 20200837 (2021).

    MathSciNet  Article  Google Scholar 

  24. Fu, Y.B., Jin, L., Goriely, A.: Necking, beading, and bulging in soft elastic cylinders. J. Mech. Phys. Solids 147, 104250 (2021).

    MathSciNet  Article  Google Scholar 

  25. Emery, D., Fu, Y.B.: localized bifurcation in soft cylindrical tubes under axial stretching and surface tension. Int. J. Solids Struct. 219, 23–33 (2021).

    Article  Google Scholar 

  26. Emery, D., Fu, Y.B.: Post-bifurcation behaviour of elasto-capillary necking and bulging in soft tubes. Proc. R. Soc. A 477, 20210311 (2021).

    MathSciNet  Article  Google Scholar 

  27. Wang, M., Jin, L.S., Fu, Y.B.: Axi-symmetric necking versus Treloar-Kearsley instability in a hyperelastic sheet under equibiaxial stretching. Math. Mech. Solids (February 2022).

  28. Ogden, R.W.: Non-linear elastic deformations. Ellis Horwood, New York (1984)

    MATH  Google Scholar 

  29. Haughton, D.M., Ogden, R.W.: Bifurcation of inflated circular cylinders of elastic material under axial loading—I. Exact theory for thick-walled tubes. J. Mech. Phys. Solids 27(5-6), 489–512 (1979).

  30. Kirchgässner, K.: Wave-solutions of reversible systems and applications. J. Diff. Eq. 45(1), 113–127 (1982).

    MathSciNet  Article  MATH  Google Scholar 

  31. Mielke, A.: Hamiltonian and Lagrangian flows on center manifolds, with applications to elliptic variational problems. Springer-Verlag, Berlin Lecture Notes in Mathematics (1991)

  32. Haragus, M., Iooss, G.: Local bifurcations, center manifolds, and normal forms in infinite-dimensional dynamical systems. Springer, London (2011)

    Book  Google Scholar 

  33. Wang, M., Fu, Y.B.: Necking of a hyperelastic solid cylinder under axial stretching: Evaluation of the infinite-length approximation. Int. J. Eng. Sci. 159, 103432 (2021).

    MathSciNet  Article  MATH  Google Scholar 

Download references


This work was supported by the National Natural Science Foundation of China (Grant No. 12072224).

Author information

Authors and Affiliations


Corresponding author

Correspondence to Xiang Yu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix: A Proof of equality (2.58)

Appendix: A Proof of equality (2.58)

To prove (2.58), we start by analyzing the linearized forms of (2.57). Assuming that the perturbed configuration is also uniform, we show below that the linearized forms of the following two equations

$$\begin{aligned}&-P^*(\lambda _a,\lambda _z)+P=0,\end{aligned}$$
$$\begin{aligned}&\frac{1}{2}b^2 (P^*(\lambda _a,\lambda _z)-P)+\frac{1}{2\pi }(N^*(\lambda _a,\lambda _z)-N)=0 \end{aligned}$$

agree with (2.42) and (2.43).

Let us denote by \(\overline{\varvec{\sigma }}\) and \(\varvec{\sigma }\) the Cauchy stresses associated with the uniformly inflated configuration and perturbed configuration, respectively. Then it follows from the definition of the incremental stress tensor \(\varvec{\chi }\) that

$$\begin{aligned} \varvec{\sigma }=(\varvec{I}+\varvec{\eta })(\overline{\varvec{\sigma }}+\varvec{\chi }). \end{aligned}$$

Note that the incremental deformation gradient \(\varvec{\eta }\) is diagonal since the perturbed configuration is uniform.

Specifying (A.1) to the perturbed configuration, we see from the definition of \(P^*\) that the resulted equation takes the form

$$\begin{aligned} \int _{{\tilde{a}}}^{{\tilde{b}}}\frac{\sigma _{33}-\sigma _{11}}{{\tilde{r}}}\,d{\tilde{r}}+P=0. \end{aligned}$$

where \({\tilde{r}}=r+u\) denotes the radius of the tube after perturbation, and \({\tilde{a}}={\tilde{r}}(A)\) and \({\tilde{b}}={\tilde{r}}(B)\). Substituting (A.3) into (A.4) and making a change of variables (integration by substitution) by applying the incompressibility equality

$$\begin{aligned} {\tilde{\lambda }}_z({\tilde{r}}^2-{\tilde{a}}^2)=\lambda _z (r^2-a^2), \end{aligned}$$

where \({\tilde{\lambda }}_z\) is the axial stretch of the tube in the perturbed configuration, we obtain

$$\begin{aligned} \int _{a}^b \frac{(1+\eta _{33})({\overline{\sigma }}_{33}+\chi _{33})-(1+\eta _{11})({\overline{\sigma }}_{11}+\chi _{11})}{r}\frac{\lambda _z r^2}{{\tilde{\lambda }}_z {\tilde{r}}^2}\,dr+P=0. \end{aligned}$$

When expanded to linear order, we have

$$\begin{aligned} \frac{\lambda _z r^2}{{\tilde{\lambda }}_z {\tilde{r}}^2}=1-2\frac{{\tilde{r}}-r}{r}-\frac{{\tilde{\lambda }}_z-\lambda _z}{\lambda _z}=1-2\eta _{11}-\eta _{22}. \end{aligned}$$

Thus by ignoring nonlinear terms, one can simplify (A.6) as

$$\begin{aligned} \int _a^b \frac{\chi _{33}-\chi _{11}}{r}\,dr+\int _a^b ({\overline{\sigma }}_{33}\frac{\eta _{33}-\eta _{11}}{r}+ \frac{{\overline{\sigma }}_{33}-{\overline{\sigma }}_{11}}{r}\eta _{33})\,dr=0, \end{aligned}$$

where we have used incompressibility constraint \( \eta _{11}+\eta _{22}+\eta _{33}=0. \) The radial equilibrium for the unperturbed deformation implies that

$$\begin{aligned} \frac{d{\overline{\sigma }}_{33}}{dr}+\frac{{\overline{\sigma }}_{33}-{\overline{\sigma }}_{11}}{r}=0. \end{aligned}$$

Using integration by parts, one can write the second integral in (A.8) as

$$\begin{aligned} \begin{aligned} \int _a^b \left( {\overline{\sigma }}_{33}\frac{\eta _{33}-\eta _{11}}{r}+ \frac{{\overline{\sigma }}_{33}-{\overline{\sigma }}_{11}}{r}\eta _{33}\right) \,dr&=\int _a^b \left( {\overline{\sigma }}_{33}\frac{\eta _{33}-\eta _{11}}{r}-\frac{d\overline{\sigma }_{33}}{dr}\eta _{33}\right) \,dr\\ {}&=-\overline{\sigma } _{33}\eta _{33}|_{r=a}^{r=b}+\int _a^b {\overline{\sigma }}_{33}\left( \frac{\eta _{33}-\eta _{11}}{r}+\frac{d\eta _{33}}{dr}\right) \,dr\\ {}&=-Pu_{r}|_{r=a}+\int _a^b {\overline{\sigma }}_{33}\left( \frac{\eta _{33}-\eta _{11}}{r}+\frac{d\eta _{33}}{dr}\right) \,dr, \end{aligned} \end{aligned}$$

where use has been made of the boundary conditions \({\overline{\sigma }}_{33}|_{r=a}=-P\) and \({\overline{\sigma }}_{33}|_{r=b}=0\). In view of incompressibility constraint and the fact that \(\eta _{22}\) is constant since the perturbed configuration is uniform, we deduce that

$$\begin{aligned} \begin{aligned} \frac{\eta _{33}-\eta _{11}}{r}+\frac{d\eta _{33}}{dr}&=\frac{\eta _{33}-\eta _{11}}{r}-\frac{d\eta _{11}}{dr}=\frac{u_r-u/r}{r}-\frac{d}{dr}\left( \frac{u}{r}\right) =0. \end{aligned} \end{aligned}$$

Putting these together, we see that the linearized form of (A.1) can be written as

$$\begin{aligned} \int _a^b \frac{\chi _{33}-\chi _{11}}{r}\,dr-Pu_r|_{r=a}=0, \end{aligned}$$

which agrees with (2.42) (note that \(\chi _{32}=0\) for uniform inflations).

Equation (A.2) applied to the perturbed configuration can be written as

$$\begin{aligned} \int _{{\tilde{a}}}^{{\tilde{b}}}\sigma _{22}{\tilde{r}}\,d{\tilde{r}}-\frac{1}{2}{\tilde{a}}^2P-\frac{N}{2\pi }=0. \end{aligned}$$

Using (A.3) and (A.5), we can rewrite the above equation as

$$\begin{aligned} \int _a^b ({\overline{\sigma }}_{22}+\chi _{22})r\,dr-\frac{1}{2}{\tilde{a}}^2P-\frac{N}{2\pi }=0. \end{aligned}$$

Its linearized form is

$$\begin{aligned} \int _a^b \chi _{22}r\,dr-Pau|_{r=a}=0, \end{aligned}$$

which is the same as (2.43).

Now let \({\tilde{\lambda }}_a\) and \({\tilde{\lambda }}_z\) be the two principal stretches of the perturbed configuration, thus

$$\begin{aligned} {\tilde{\lambda }}_a&=\lambda _a+d\lambda _a=\lambda _a+\alpha ^2\left( c_1\lambda _a+\frac{c_2}{\lambda _a A^2}\right) ,\end{aligned}$$
$$\begin{aligned} {\tilde{\lambda }}_z&=\lambda _z+d\lambda _z=\lambda _z-2\alpha ^2 \lambda _z c_1. \end{aligned}$$

Then equation (A.1) applied to the unperturbed and perturbed configurations takes the form \( -P^*(\lambda _a,\lambda _z)+P=0\) and \(-P^*({\tilde{\lambda }}_a,{\tilde{\lambda }}_z)+P=0\), respectively. Subtraction of these two equalities yields

$$\begin{aligned} -P^*({\tilde{\lambda }}_a,{\tilde{\lambda }}_z)+P^*(\lambda _a,\lambda _z)=0. \end{aligned}$$

In a similar way, we can deduce from (A.2) that

$$\begin{aligned} \frac{1}{2}{\tilde{b}}^2(P^*({\tilde{\lambda }}_a,{\tilde{\lambda }}_z)-P^*(\lambda _a,\lambda _z))+\frac{1}{2\pi }(N^*({\tilde{\lambda }}_a,{\tilde{\lambda }}_z)-N^*(\lambda _a,\lambda _z))=0, \end{aligned}$$

where \({\tilde{b}}={\tilde{r}}(B)\) is the outer radius of the tube after perturbation. Linearizing the above two equations at \((\lambda _a,\lambda _z)\) followed by the use of (A.16) and (A.17), we obtain

$$\begin{aligned}&\left( 2\lambda _z\frac{\partial P^*}{\partial \lambda _z}-\lambda _a\frac{\partial P^*}{\partial \lambda _a}\right) c_1-\frac{1}{\lambda _aA^2}\frac{\partial P^*}{\partial \lambda _a}c_2=0,\end{aligned}$$
$$\begin{aligned}&\Big (\frac{\lambda _a}{2\pi }\frac{\partial N^*}{\partial \lambda _a}+\frac{\lambda _ab^2}{2}\frac{\partial P^*}{\partial \lambda _a}-\frac{\lambda _z}{\pi }\frac{\partial N^*}{\partial \lambda _z}-\lambda _zb^2\frac{\partial P^*}{\partial \lambda _z}\Big )c_1+\Big (\frac{1}{2\pi \lambda _a A^2}\frac{\partial N^*}{\partial \lambda _a}+\frac{b^2}{2\lambda _a A^2}\frac{\partial P^*}{\partial \lambda _a}\Big )c_2=0, \end{aligned}$$

Comparing them with (2.39) and (2.44), we conclude that

$$\begin{aligned} \begin{aligned} m_{11}&=2\lambda _z\frac{\partial P^*}{\partial \lambda _z}-\lambda _a\frac{\partial P^*}{\partial \lambda _a},\\ m_{12}&=-\frac{1}{\lambda _aA^2}\frac{\partial P^*}{\partial \lambda _a},\\ m_{21}&=\frac{\lambda _a}{2\pi }\frac{\partial N^*}{\partial \lambda _a}+\frac{\lambda _ab^2}{2}\frac{\partial P^*}{\partial \lambda _a}-\frac{\lambda _z}{\pi }\frac{\partial N^*}{\partial \lambda _z}-\lambda _zb^2\frac{\partial P^*}{\partial \lambda _z},\\ m_{22}&=\frac{1}{2\pi \lambda _a A^2}\frac{\partial N^*}{\partial \lambda _a}+\frac{b^2}{2\lambda _a A^2}\frac{\partial P^*}{\partial \lambda _a}. \end{aligned} \end{aligned}$$

In view of (2.49), it follows from (A.22) that \(\Omega (\lambda _a,\lambda _z)\) can be expressed as

$$\begin{aligned} \Omega (\lambda _a,\lambda _z)=-\frac{\lambda _z}{\pi \lambda _a A^2}\left( \frac{\partial P^*}{\partial \lambda _a}\frac{\partial N^*}{\partial \lambda _z}-\frac{\partial P^*}{\partial \lambda _z}\frac{\partial N^*}{\partial \lambda _a}\right) , \end{aligned}$$

which completes the proof.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Yu, X., Fu, Y. An analytic derivation of the bifurcation conditions for localization in hyperelastic tubes and sheets. Z. Angew. Math. Phys. 73, 116 (2022).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI:


  • Localized bulging
  • Axisymmetric necking
  • Bifurcation
  • Nonlinear elasticity

Mathematics Subject Classification

  • 74B20
  • 74G10
  • 74G60
  • 35A20