Study On Axisymmetric Deformation Of Circular Pre-stressed Membrane Under Gas Pressure

Membranes,as structures or structural members,have been widely used in the engineering fields of construction,aerospace,MEMS devices,biology and so on.They are often used as key components in many technical applications,such as the conductive membrane that acts as a deformable plate electrode in the contact(or non-contact)capacitive pressure transducer,as well as the blister membrane in the blister test and the bulge membrane in the bulge test.By analyzing the mechanical behavior of membranes,the mechanical properties of an independent membrane or membrane/substrate system can be determined.In these technical applications,membranes are usually in the form of circular membrane structures that are subjected to surface loads,primarily gas pressure.In addition,membranes usually carry a uniform in-plane stress,which is usually called residual stress,initial stress,or pre-stress,before the membranes are subjected to surface loads.Therefore,for these technical applications,a more accurate analytical understanding of the mechanical behavior of circular membranes with or without pre-stress under gas pressure is required,i.e.,a more accurate analytical solution is required.However,due to the lack of such an analytical solution in these fields,the analytical solution for lateral uniformly-distributed loads has to be adopted.As a result,only the effect of gas pressure on the lateral direction is considered,while the effect on the longitudinal direction is ignored,which inevitably leads to the generation of errors.only the transverse component of the gas pressure acting on the membrane is considered,but the longitudinal component is ignored,which inevitably leads to errors.This thesis is devoted to the theoretical study of circular membrane problems,and presents the analytical solution for the problem of axisymmetric deformation of a circular pre-stressed membrane under gas pressure.First of all,the research status of circular membrane problem is reviewed,and the inadequacies of the relevant existing analytical theories are pointed out.Then,the circular membrane problem studied in this thesis was mechanically simplified,and based on the simplification,the mechanical model was established,the coordinate system is established,the governing equations describing the mechanical behavior of circular membrane(including the out-of-plane equilibrium equation,the in-plane equilibrium equation,geometric equations,physical equations,and the compatibility equation)are established,and the boundary conditions for solving the governing equations are determined.Finally,the governing equations and boundary conditions are nondimensionalized,the power series method is used to solve the problem analytically,and the power series expressions of stress and deflection are given.Thus,a new analytical solution for axisymmetric deformation of circular pre-stressed membrane under gas pressure is obtained.The obtained analytical solution is comprehensively discussed in this thesis.The validity of the obtained analytical solution is theoretically analyzed,which proves that the obtained analytical solution meets the boundary condition which has not been used during the derivation.It is verified by numerical examples that the obtained solution has good convergence.The differences among the obtained analytical solution and the existing analytical solutions are compared,which shows that the outcome given by the obtained analytical solution is more reliable and more accurate than the relevant existing analytical solutions(the extended Hencky solution and Campbell solution)for the problem studied in this thesis,and the obtained analytical solution can take the influence of pre-stress(which Fichter solution does not considered)into consideration.At last,the parameter analysis is carried out with finite element simulation,which gives the curves of deflection and radial stress of membrane with the radius under different values of loads and pre-stresses.