Boundary Value Problems Of High Order Differential Equations For High Temperature Superconducting Double Cantilever Beams And Their Fracture Mechanics Analysis

High-temperature superconducting materials have special properties such as zero resistance,Meissner effect,and macroscopic quantum effects.They also have lower operating costs than low-temperature superconducting materials,so they are widely used in the preparation of superconducting magnets and superconducting cables.High-temperature superconductors tend to crack under strong magnetic fields,on the one hand,fracture specimen of double cantilever beam is one of the most common and convenient ways to study the fracture characteristics of materials.On the other hand,higher order differential equations with boundary conditions are often used to describe the fracture mechanics of double cantilever beams.Therefore,studying the fracture mechanics characteristics of high-temperature superconducting double cantilever beams can be transformed into solving the boundary value problems of corresponding higher-order differential equations.This paper mainly uses the solution of higher order differential equations with boundary value conditions to study the fracture mechanical characteristics of macroscopic uniform functional gradients and micro-sized high-temperature superconducting double cantilever beams.The main research is as follows:The macroscopic uniform high-temperature superconducting double cantilever beam sample can be decomposed into uncracked and cracked parts.The governing equation of the cracking part is derived through the equilibrium equation of the force.The governing equation is a third-order non-homogeneous differential equation with boundary value conditions.By comparing the coefficient method and the eigen-root method,the analytical solutions of such third-order non-homogeneous differential equations with boundary value problems are obtained.Based on the theory of fracture mechanics,the analytical solution of the energy release rate and stress intensity factor of the double cantilever beam considering the root effect are obtained.The results show that the surface equivalence increases the energy release rate and stress intensity factor of the double cantilever beam;as the ratio of beam length to thickness increases,the effect of the root effect gradually decreases.On the other hand,in order to obtain better electro-magnetic-force performance,high-temperature superconducting materials are often prepared into functionally gradient structures in practical applications.Therefore,it is necessary to study the fracture characteristics of functionally graded high-temperature superconducting materials.According to the equilibrium equation of force,the governing equation of the functionally gradient double cantilever beam is derived.The governing equationis a fourth-order non-homogeneous differential equation with boundary values.An analytical solution of the equation is given by the comparison coefficient method.Based on the analytical solution,the effects of the material's Young's modulus,applied force,and surface effectiveness on the stress intensity factor and energy release rate of the double cantilever beam were studied.The results show that the increase of Young's modulus,surface efficiency,and applied force will increase the energy release rate of high-temperature superconducting materials.In addition,as the size of superconducting materials becomes more and more smaller,the effect of size effects on the fracture characteristics of high-temperature superconducting materials becomes increasingly significant.In the light of the couple stress theory,the fracture problem of micro-nano high temperature superconducting double cantilever beams can be described by the boundary value problem of non-homogeneous sixth-order differential equations.the analytical solution of the non-homogeneous sixth-order differential equation is given by combining the corresponding boundary conditions.Based on the analytical solution,we discussed the effects of varying magnetic fields and size effects on the fracture performance of nanobeams.The results show that the closer the size of the material is to its characteristic length,the smaller the stress intensity factor and the stronger the ability of the nanomaterial to prevent crack propagation.