Mechanical Analysis And Numerical Simulation Of Finit Deformation Of Nonlinear Hyperelastic Toroidal Membrane

Inflatable membrane structures have received wide attention due to their widespread application in the field of civil engineering,and the membranes of inflatable membrane structures are usually modeled as hyperelastic materials.Therefore,the finite deformation of inflatable membrane structures under internal pressure loads has become a hot topic of current research.This paper mainly studies the finite expansion of nonlinear hyperelastic toroidal membrane structures under uniform internal pressure.The toroidal membrane structures are modeled as neo-Hookean and two-order Yeoh hyperelastic models with relaxed strain energy density functions,respectively.The changes in cross-section,principal stretches,and Cauchy stress during the expansion process of the toroidal membrane structure under different conditions are discussed,and the moment at which the membrane structure first undergoes bifurcation instability is also studied.The first part mainly studies the finite expansion of neo-Hookean hyperelastic toroidal membrane under uniform internal pressure.The neo-Hookean hyperelastic toroidal membrane with a circular initial cross-section undergoes finite deformation under uniform internal pressure.After obtaining the expression of the total potential energy of the system,the governing equations and boundary conditions determined using the variational method can be expressed as a nonlinear two-point boundary value problem.The boundary value problem can be transformed into a initial value problem.The shooting method is used for numerical solution,and the Nelder-Mead optimization algorithm is introduced to ensure that the numerical solution meets the accuracy requirements.By comparing the results with previous studies,the accuracy of the target method procedure was verified.The research results show that the magnitude of limit-point pressure of membrane structures is negatively correlated with the magnitude of geometric parameters.The membrane structure always maintains a circular cross-section shape and continuously increases during expansion.During the expansion process of membrane structures corresponding to all geometric parameters,the meridional stretch is always greater than 1.The smaller the geometric parameters,the smaller the difference between the principal stretches and Cauchy stress at different locations during the expansion process of the membrane structure.The second part mainly studies the finite expansion of two-order Yeoh hyperelastic toroidal membranes under uniform internal pressure based on the first part.Using the same numerical method as in the previous section,the research results show that compared to geometric parameters,changes in material parameters have less impact on the cross-sectional shape,principal stretches and Cauchy stress distribution at different locations in the expansion process of membrane structures.For membrane structures with the same geometric parameters,the magnitude of the material parameters is positively correlated with the magnitude of the limit-point pressure of the membrane structure,as well as with the magnitude of the principal stretches and Cauchy stress corresponding to the same pressure value after the membrane structure passes through the limit-point pressure.The third part mainly studies the influence of electric field on the expansion process of two-order Yeoh hyperelastic toroidal membrane and the analysis of bifurcation instability based on the previous two parts.Applying an electric field in the thickness direction of the membrane structure,the variational formula for coupling mechanical and electrical energy is obtained,and the numerical solution is performed using the shooting method.The results show that with the increase of the electrical load,the limit-point pressure of the membrane structure tends to increase substantially,and the membrane structure will have a larger cross-sectional area during the middle and late stages of expansion,leading to abnormal phenomena of the principal stretches and Cauchy stress of the membrane structure.Bifurcation instability first occurs before the limit-point pressure.Increasing geometric parameters will gradually approach the limit-point pressure,while increasing electrical load will gradually move it away from the limit-point pressure.