| Circular membranes are especially welcomed as structures or structural members in many technical applications,because they are axisymmetric and therefore convenient to be dealt with analytically.For instance,they can be used to develop normal mode or touch mode capacitive pressure sensors,and can also be used to design bulge or blister tests or constrained blister tests,to characterize mechanical properties of free-standing thin film or thin film/substrate systems.In order to ensure the reliability and accuracy of technical application,it is usually necessary to have a more accurate analytical understanding for the mechanical behavior of membranes,or in other words,a more accurate analytical solution is usually needed.However,the problem of axisymmetric deformation of a circular membrane under gas pressure loading and rigid constraint of maximum deflection,whose analytical solution is essential to the development of touch mode capacitive pressure sensors and the design of constrained blister tests,is usually difficult to be solved analytically.Therefore,at present,in these development and design fields,the analytical solution for the problem of axisymmetric deformation of a circular membrane under transverse uniform loading and rigid constraint of maximum deflection is widely used as a rough analytical solution,due to the lack of the correct analytical solution suitable for gas pressure loading.Such doing,in fact,considers only the transverse(vertical)component of the applied gas pressure,and neglects its longitudinal(horizontal)component,which inevitably leads to the introduction of errors and loss of precision.This thesis is devoted to the analytical study of the problem of axisymmetric deformation of a circular membrane under gas pressure loading and rigid constraint of maximum deflection.In the research process,a mechanical model of this problem was established.The deformed circular membrane is divided into two parts: the central contact part and the annular non-contact part.The mechanical problem of the central contact part is the in-plane radial stretching and compression problem of a circular membrane,and the mechanical problem of the annular non-contact part is the out-of-plane axisymmetric deformation problem of a circular membrane.The two local problems are connected by the continuous condition of stress and strain.Then the governing equations of the problem of axisymmetric deformation of a circular membrane under gas pressure loading and rigid constraint of maximum deflection are established based on the large deflection theory of membranes.The boundary conditions for solving the governing equations are determined,then the governing equations and boundary conditions are simultaneously transformed into dimensionless forms,and the power series method is used to solve the dimensionless governing equations.Finally,the power series expressions of the stress and deflection are given.After presenting an analytical solution,the effectiveness and convergence of the analytical solution are discussed.The analytical solution presented in this thesis is compared with the analytical solution for the problem of axisymmetric deformation of a circular membrane under transverse uniform loading and rigid constraint of maximum deflection.The results show that the difference between the two analytical solutions will become larger with the increase of load value,which means that it is necessary to select the appropriate analytical solution according to the actual load,otherwise there will be large errors.At the same time,through the contact experiment between the circular membrane and the frictionless rigid plate under gas pressure,it is shown that the analytical solution presented in this thesis has higher calculation accuracy than the existing analytical solution suitable for transverse uniform loading when solving the problem of axisymmetric deformation of a circular membrane under gas pressure loading and rigid constraint of maximum deflection.Besides,for the convenience of practical applications,the variation laws of parameters,critical contact pressure and membrane stress are discussed. |
