Qualitative analysis for solutions of a class of nonlinear ordinary differential equations
The boundary value problem (BVP) for a class of nonlinear ordinary differential equations is examined. It can be used to describe the eversion problem for a spherical shell composed of a class of transversely isotropic incompressible Mooney-Rivlin materials. The solutions of the nonlinear equation describing the relation among deformed thickness, initial thickness and material parameters of the spherical shell are obtained. The effects of structure parameter and material parameters on the thickness of the everted spherical shell are discussed by numerical examples.
Inflation and eversion of spherical shells of a special compressible material
A study is made of the qualitative features of the inflation of a shell of the so-called material of type II. This material has a strain energy function which is linear in i1, i3 and arbitrary in i2, where i1, i2 and i3 are the three principal invariants of the stretch tensors. For a specific form of the arbitrary function, a closed-form solution to the equations of equilibrium describing spherical eversion can also be found. By using this solution, a comparison can be made between the qualitative features of the regular and everted shells under internal pressure. It is found that the stress distribution in both cases is different and that a larger pressure can be sustained by the everted shell. However eversion has no effect on the other features studied.
Instability of Finite Amplitude Elastic Waves
The equations of motion for hyper-elastic material form a hyperbolic system of second order quasi-linear partial differential equations 
 
$$ \frac{\partial^{2}u_{i}}{\partial t^{2}}=\sum_{k,r,s}c_{ikrs}\frac{\partial^{2}u_{r}}{\partial x_{k}\partial x_{s}} $$ 
 
for the components u i of the displacement vector u = u(x, t). The coefficients c ikrs are functions of the matrix u ‵ whose elements are the displacement gradients \( u_{i,k}=\partial u_{i}/\partial x_{k} \). In order to avoid assumptions about the behavior of the material for large stresses or strains we require the c ikrs (u ‵) to be defined only for u ‵ in the neighborhood of the origin. More generally we consider hyperbolic systems (1) of N equations for a vector u(x, t) with N components u i defined for \( x\in \mathbb{R}^{n},t\geqq0 \), the coefficients c ikrs depending on u ‵ and u t. The initial value problem consists in finding a solution of (1) for prescribed values of u (x, 0), u t (x, 0).