The Initial-boundary Value Problems For Two Classes Of Nonlinear Differential Equations

In this paper, by using the finite deformation theory of elasticity, problems of growth andmotion along radial direction of four spherical structures (namely, a solid sphere, a spherewith an initial micro-void, a spherical shell, a spherical membrane) are described as nonlineardifferential equations with the initial-boundary value. And then analytic solutions of equationsare obtained by using the inverse method, the incompressibility condition, etc. Next,qualitative analyses are performed for solutions and a series of some meaningful conclusionsare obtained. The main works are derived as follows:In Chapter2, the finite deformation statics problems are examined for four sphericalstructures composed of isotropic incompressible hyperelastic materials under tensile loads ontheir outer surfaces. The solid sphere is composed of a class of isotropic incompressibleOgden materials, it is proved that when the tensile load is less than the critical load, the solidsphere remains solid, if the tensile load exceeds the critical load, there is a cavity at the centerof the sphere. It is obtained that there exists a critical value for the sphere with an initialmicro-void composed of isotropic incompressible Ogden materials, if the tensile load issmaller than the critical value, the radius of micro-void grows slowly with the increasingtensile load. However, if the tensile load exceeds the critical value, the radius grows rapidlyall at once. The spherical shell and the spherical membrane are composed of a class ofisotropic incompressible Mooney-Rivlin materials, in theory, it can be concluded that theinner radius of the two structures will grow with the increasing tensile load.In Chapter3, the finite deformation dynamic problems are considered for four sphericalstructures composed of isotropic incompressible hyperelastic Ogden materials, where thespherical structures are subjected to radial tensile loads on their outer surfaces, throughqualitative analyses of solutions, it can be found that, for the solid sphere and the sphere withan initial micro-void, the structures would perform nonlinear periodic oscillations with time.For the spherical shell or the spherical membrane, it is proved that there exists a critical value,if the tensile load is smaller than the critical value, the radial motion of the inner surface is anonlinear periodic oscillation with the increasing time. If the tensile load exceeds the criticalvalue, the structures will be destroyed ultimately.