Study On Cavitation And Shape Bifurcation Of Hyperelastic Membranes

Due to their ability to withstand large deformation,hyerelastic membranes are widely used in many fields such as space engineering,medical science,biological engineering,construction and civil engineering.The deformation,instability,and failure process of hyperelastic membranes are more complicated than the common behaviors in linear elastic materials when subjected to certain load due to its inherent material and geometrical non-linearity.Among many nonlinear problems involving hyperelastic membranes,the instability of materials and structures as the important factors that induce failure,damage and destruction has been the focus of attention.The problem of cavitation and shape bifurcation of hyperelastic membranes is studied in this thesis.Firstly,a uniformly stretched hollow circular membrane was adopted to investigate the characteristics of the stress and deformation at the edge of cavity.Analytic results show that,if the hole is small enough,the critical load for the defect model is almost the same as one for the defect-free model,and the distribution of stress and deformation around the cavity undergo significant transitions when the load changes across the critical value.Taking the initial radius?of the pre-existing cavity as a perturbation parameter,asymptotic analyses for the stress and deformation at the edge of the cavity were also carried out by dividing the load range into three cases.By focusing on the behavior near the critical value when using different strain energy functions,the sensitivity coefficient of initial defect in the cavitation of plane membrane was obtained.Asymptotic results show that,when the load is much smaller than the critical value,the deformed radius r???is of order?,and the leading orders of circumferential stress,radial stretch and circumferential stretch are all bounded constants,which are independent of the initial radius of cavity.When the load reaches a small neighborhood of the critical value,the deformed radius r???is of order?^?1+??/?3+??,and the leading orders of the three quantities are ln?,?^2?/?3+??,and?^-2/?3+??,respectively,where?is Poisson's ratio.And once the load significantly exceeds its critical value,the leading order term in r???is independent of?,while the leading orders of circumferential stress,radial stretch and circumferential stretch at the edge of cavity are of orders ln?,?^?,and 1/?.This indicates that when the pre-existing cavity is small enough,the circumferential stress and stretch at the edge of cavity would be quite large,which may help gain general insight into the damage of the material.Secondly,the sudden growth of cavitation for a pressurized spherical balloon with pre-existing void was studied.Considering half of the model,the shooting method was employed to solve the governing equations under symmetric boundary conditions.By comparing the curves of the deformed radius with respect to the pressure,it is found that the pre-exiting cavity in spherical balloon would also undergo sudden expansion.The asymptotic characteristics of the stress and deformation at the edge of cavity are similar to the ones for plane membrane.Therefore cavitation occurs in the hyperelastic balloons considered.Moreover,the same method was applied for the research on mechanical properties of the incomplete balloons.The effects of size,shape,anisotropy and the internal pressure load on the shape,stress and principal stretch of the spherical balloon were studied.And then,for both pressurized rugby-shaped and pumpkin-shaped balloons,shape bifurcation were studied by using the shooting method.The results show that there are different types bifurcation in the two shaped ellipsodal balloons.For a rugby-shaped balloon,there exists a threshold axes ratio below which the slender ellipsoidal balloon behaves more like a tube and bifurcation into a pear shape becomes impossible,whereas for a pumpkin-shaped balloon bifurcation into a pear shape is always possible.In addition,by using an energy criterion,we determine the stability properties of the primary and bifurcated solutions under pressure control and volume control,respectively.The total energy of the equilibrium state and its disturbed state are calculated,and the difference between these two states is used to evaluate the stability of current state.Our analyses indicate that under pressure control,both primary and bifurcated solutions that exist on the descending branch of the pressure versus volume curve are unstable,but under volume control,the bifurcated solution is always stable whenever it appears while the primary solution is only stable when there does not exist any bifurcated solution.However,the primary solutions that exist on the two ascending branches are always stable.Finally,our reserch on the shape bifurcation phenomenon of the hyperelastic balloon was further expanded to dielectric elastomers,and shape bifurcation of rugby-shaped and pumpkin-shaped balloons were investigated under both internal pressure and voltage.Results reveal that there exists an obvious pear deformation corresponding to the bifurcated solution and large stress and electric field concentration can be observed in the pear-shaped mode.In addition,the voltage has an influence on the range of bifurcation interval and the critical size of the short axis of rugby-shaped bifurcation.And when subjected to a certain load,multiple bifurcation occurs.In summary,cavitation and shape bifurcation were studied in this paper.An asymptotic method combined with numerical results was proposed to study the asymptotic characteristics of the stress and deformation at the edge of cavity for plane membranes,which may help gain general insight into the damage of the material.Cavitation for a pressurized spherical balloon containing a pre-existing cavity was investigated,and it is shown that cavitation may also occur in hyperelastic spherical balloons.For cases that subjected to pressure only and cases that also subjected to electric actuation,shape bifurcation was studied for both rugby-shaped and pumpkin-shaped balloons with different geometry configurations,and the results can provide reference for the design of membrane sensors and smart actuators.