Dynamical responses, such as motion and destruction of hyper-elastic cylindrical shells subject to periodic or suddenly applied constant load on the inner surface, are studied within a framework of finite elasto-dynamics. By numerical computation and dynamic qualitative analysis of the nonlinear differential equation, it is shown that there exists a certain critical value for the internal load describing motion of the inner surface of the shell. Motion of the shell is nonlinear periodic or quasi-periodic oscillation when the average load of the periodic load or the constant load is less than its critical value. However, the shell will be destroyed when the load exceeds the critical value. Solution to the static equilibrium problem is a fixed point for the dynamical response of the corresponding system under a suddenly applied constant load. The property of fixed point is related to the property of the dynamical solution and motion of the shell. The effects of thickness and load parameters on the critical value and oscillation of the shell are discussed.
The mechanical response of the human arterial wall under the combined loading of inflation, axial extension, and torsion is examined within the framework of the large deformation hyper-elastic theory. The probability of the aneurysm formation is explained with the instability theory of structure, and the probability of its rupture is explained with the strength theory of material. Taking account of the residual stress and the smooth muscle activities, a two layer thick-walled circular cylindrical tube model with fiber-reinforced composite-based incompressible anisotropic hyper-elastic materials is employed to model the mechanical behavior of the arterial wall. The deformation curves and the stress distributions of the arterial wall are given under normal and abnormal conditions. The results of the deformation and the structure instability analysis show that the model can describe the uniform inflation deformation of the arterial wall under normal conditions, as well as formation and growth of an aneurysm under abnormal conditions such as the decreased stiffness of the elastic and collagen fibers. From the analysis of the stresses and the material strength, the rupture of an aneurysm may also be described by this model if the wall stress is larger than its strength.
Mechanical properties, such as the deformation and stress distributions for venous walls under the combined load of transmural pressure and axial stretch, are examined within the framework of nonlinear elasticity with one kind of hyper-elastic strain energy functions. The negative pressure instability problem of the venous wall is explained through energy comparison. First, the deformation equation of the venous wall under the combined loads is obtained with a thin-walled circular cylindrical tube. The deformation curves and the stress distributions for the venous wall are given under the normal transmural pressure, and the regulations are discussed. Then, the deformation curves of the venous wall under the negative transmural pressure or the internal pressure less than the external pressure are given. Finally, the negative pressure instability problem is discussed through energy comparison.
Dynamical cavitation and oscillation of an anisotropic two-family fiber-reinforced incompressible hyper-elastic sphere subjected to a suddenly applied constant boundary dead load are examined within the framework of finite elasto-dynamics.An exact differential equation between the radius of the cavity and the applied load is obtained.The curves for the variation of the maximum radius of the cavity with the load and the phase diagrams are obtained by vibration theories and numerical computation.It is shown that there exists a critical value for the applied load.When the applied load is larger than the critical value,a spherical cavity will suddenly form at the center of the sphere.It is proved that the evolution of the cavity radius with time follows that of nonlinear periodic oscillation,and oscillation of the anisotropic sphere is not the same as that of the isotropic sphere.
The growth of a prolate or oblate elliptic micro-void in a fiber reinforced anisotropic incompressible hyper-elastic rectangular thin plate subjected to uniaxial extensions is studied within the framework of finite elasticity. Coupling effects of void shape and void size on the growth of the void are paid special attention to. The deformation function of the plate with an isolated elliptic void is given, which is expressed by two parameters to solve the differential equation. The solution is approximately obtained from the minimum potential energy principle. Deformation curves for the void with a wide range of void aspect ratios and the stress distributions on the surface of the void have been obtained by numerical computation. The growth behavior of the void and the characteristics of stress distributions on the surface of the void are captured. The combined effects of void size and void shape on the growth of the void in the thin plate are discussed. The maximum stresses for the void with different sizes and different void aspect ratios are compared.
