Bifurcation of response of a nonlinear viscoelastic spherical membrane

The quasistatic inflation of a nonlinear viscoelastic spherical membrane by monotonically increasing pressure is considered. The deformation is assumed to be spherically symmetric. For the constitutive equation assumed, circumstances are shown to exist when the radius history must either have a jump discontinuity or bifurcate. A necessary condition for bifurcation and its dependence on material properties and radius history is analysed. Examples of bifurcation for various pressure histories are presented. Post-bifurcation branches are constructed and the possibility of secondary bifurcation is discussed. A fascinating phenomenon in nonlinear elasticity arises in the problem of the inflation of a spherical membrane by internal pressure. Using the Mooney form of strain energy function and assuming spherically symmetric deformation Green and Shield(l) showed that there could be more than one equilibrium state corresponding to each internal pressure. This paper considers the analogous situation when the membrane material is viscoelastic. In particular, there may be a pressure history which generates several possible radius histories. An example is presented in which such pressure histories are constructed. The corresponding radius histories arise by bifurcation from a common history. Consider first how bifurcation occurs in the elastic case. Starting from the undeformed state let the membrane be subjected to a continuous monotonically increasing pressure history. The deformation is considered quasi-static and can therefore be determined from the pressure- radius relation obtained by Green and Shield. This relation is shown schematically in Fig. 2(a). As the internal pressure increases from zero, a continuous monotonically increasing radius history is found from the portion of the f-x relation to the left of the local maximum. If the internal pressure. increases beyond this local maximum, the radius history has a jump dis- continuity and then is found from the monotonically increasing portion of the f - x relation to the right of the local minimum. Thus a jump discontinuity in radius history is produced even though the pressure history is continuous. In order to construct a continuous radius history. the pressure must be reduced after reaching the local maximum. The radius history then bifurcates into two continuous branches which are found by using the portions of the f-x relation to either side of the local maximum. There are several reasons for avoiding a jump discontinuity in the radius history. The internal pressure being greater than the local maximum of f implies that the membrane wall force and pressure resultant are no longer in balance and inertial effects must be considered. The apparent size of the jump discontinuity suggests that inertial effects could be quite considerable. If the occurance of this phenomenon in polymer processing, such as blow forming, is considered, the motion is no longer under control. This is an undesirable situation since, for example, the sudden increase in radius could be so large that the corresponding reduction in wall thickness could lead to bursting. When the membrane is viscoelastic, the above bifurcation analysis is complicated by history effects. The example presented here shows that the requirement of a continuous radius history can still lead to bifurcation. The question of restrictions on constitutive equations as related to the possibility of bifurcation is not considered. It is assumed that just as in the elastic case, there are material models for which bifurcation is possible. The example also includes the determination of the post-bifurcation solution branches corresponding to a given pressure history. The stability of such branches is not considered. The governing equations are presented in Section 2. The material is modeled by a nonlinear single integral constitutive equation which displays Mooney elasticity in its instantaneous and long time equilibrium response limits. Consequently, the response of the spherical membrane is