Analyses Of Finite Deformation Of Hyperelastic Rubber Structures With Axial Symmetry

Axisymmetric structures composed of rubber materials (such as rubber rings, rubber tubes, rubber blankets. etc.) are widely used in social production and life under certain circumstance and load, and so they often encounter the problems of deformation, instability, destroy and limited service life, and so on. The stability problems of finite deformation of relative materials and structures are always the focuses of experts and scholars at home and abroad. In this dissertation. the static and dynamical problems are examined for several axisymmetric structures composed of rubber materials based on the hyperelastic constitutive relations via using the basic viewpoints and conclusions of the nonlinear elasticity and the bifurcation theory, and so forth. Some new conclusions are obtained, as follows,1. The problems of finite deformation of several rectangular rubber rings composed of incompressible hyperelastic materials are studied, where both ends of the rings are subjected to axial compressive loads. Systems of implicit analytical solutions are derived respectively by using the incompressible constraint and the boundary conditions for the isotropic neo-Hookean material, the transversely isotropic neo-Hookean material and the isotropic Mooney-Rivlin material. Combining with numerical examples,the influences of axial compressive loads, radial thickness and axial height of the rubber rings on finite deformation of the rings are discussed in detail. It is proved theoretically that with the increasing axial compressive loads, the decreasing radial thickness and the increasing axial height, the lateral surfaces of these rings along the radial direction inflate and both ends along the axial direction shrink more and more. For the transversely isotropic neo-Hookean material, the larger the value of the anisotropy parameter is, the bigger the deformation of the ring is. For the isotropic Mooney-Rivlin material, the smaller the ratio of the shear moduli is, the bigger the deformation of the ring is. For the three kinds of material models, the ratios of axial compression are all the smallest at the central cross-sections of these rings and are the biggest at the ends. The ratios of axial compression and the axial displacements are also influenced by axial loads and structure parameters.2. The problems of axial compression and radial inflation are examined respectively for the composite rubber tubes composed of incompressible hyperelastic materials.For a finitely long composite rubber tube subjected to static axial compressive loads at its both ends, where the tube is composed of two classes of incompressible neo-Hookean materials, a system of implicit analytical solutions is derived by using the incompressible constraint, the continuous conditions of stress and strain, the boundary conditions. It is proved with numerical examples that the lateral surface and the interface of the tube along the radial direction inflate and both ends along the axial direction shrink more and more as the axial loads increase or as the ratio of the shear moduli of the outer and the inner materials decreases. Moreover, the deformation models of both the lateral surface and the interface of the composite rubber tube are nearly uniform at most of the middle portion of the tube, whereas, are obvious near the two ends. In the middle part, the ratio of axial compression maintains almost the same, while it changes very fast near the two ends and achieves the maximum at the ends. The absolute value of axial displacement increases gradually from the central cross-section and reaches the maximum at the two ends of the tube.For an infinitely long composite rubber tube subjected to a suddenly applied radial pressure at its inner surface, where it is composed of two classes of transversely isotropic incompressible power-law materials, the mechanisms of static inflation and dynamic inflation are given by qualitatively analyzing the equation describing the radial inflation of the tube. It is proved that if the power-law parameters of the strain energy functions associated with the two materials do not exceed1, then there exists a critical pressure such that the radial inflation mode of the tube with time is a nonlinearly periodic oscillation as the radial pressure does not exceed the critical pressure, otherwise, the tube will inflate infinitely. If at least one of the power-law parameters of the strain energy functions is larger than1, then for any given pressure, the radial inflation mode of the tube with time is always a nonlinearly periodic oscillation. Moreover, for some special material parameters, the oscillation amplitude will increase discontinuously.3. The bifurcation problems are studied for two classes of axisymmetric structures composed of isotropic incompressible hyperelastic materials. The mathematical models are reduced to two classes of second order nonlinear ordinary differential equations which are used to describe the radial motions of the structures with time, respectively. Through qualitatively analyzing the two equations, it yields the following conclusions:For an infinitely long cylinder composed of the isotropic incompressible Ogden material with a microvoid at its center, when it is subjected to a suddenly applied tensile load at the outer surface, the radius of the microvoid grows very slowly at the beginning with the increasing tensile load. However, there exists a critical load, when the tensile load exceeds the critical load the radius of the microvoid grows rapidly. Moreover, for the material parameters satisfying certain conditions, the motion of the microvoid is always a nonlinear periodic oscillation for any given load. As the material parameters take certain special values, the phase diagrams of the motion equation have homoclinic orbits, which means that the oscillation amplitude increases discontinuously with the increasing tensile load, such case is very rare for isotropic incompressible hyperelastic materials.For a pre-strained circular sheet composed of a class of isotropic incompressible hyperelastic materials, when it is subjected to a suddenly applied tensile load at the radial surface, it is shown that cavitation depends exactly on the power-law parameter. The larger the pre-strained value is prescribed, the earlier cavitation occurs; the larger the power-law parameter is, the later cavitation occurs. Dynamically, once a cavity forms at the axial line of the circular sheet, the motion of the formed cavity with time is a classical nonlinear periodic oscillation. Moreover, the oscillation amplitude increases with the increasing tensile load or with the increasing pre-strained value.