Shang Xin-chun

Exact solution for cavitated bifurcation for compressible hyperelastic materials

Abstract In this paper, a new exact analytic solution for spherical cavitated bifurcation is presented for a class of compressible hyperelastic materials. The strain energy density of the materials is assumed to be a linear function of three strain invariants, which may be regarded as a first-order approximation to the general strain energy density near the reference configuration, and also may satisfy certain constitutive inequalities of hyperelastic materials. An explicit formula for the critical stretch for the cavity nucleation and a simple bifurcation solution for the deformed cavity radius which describes the cavity growth are obtained. The potential energy associated with the cavitated deformation is examined. It is always lower than that associated with the homogeneous deformation, thus the state of cavitated deformation is relatively stable. On the basis of the presented analytic solutions for the stretches and stresses, the catastrophic transition of deformation and the jumping of stresses for the cavitation are discussed in detail. The boundary layers of the displacements, the strain energy distribution and stresses near the formed cavity wall are observed. These investigations illustrate that cavitation reflects a local behaviour of materials.

An exact analysis for free vibration of a composite shell structure-hermetic capsule

An exact analytical solution was presented for free vibration of composite shell structure-hermetic capsule. The basic equations on axisymmetric vibration were based on the Love classical thin shell theory and derived for shells of revolution with arbitrary meridian shape. The conditions of the junction between the spherical and the cylindrical shell segments are given by the continuity of deformation and the equilibrium relations near the junction point. The mathematical model of problem is reduced to as an eigenvalue problem for a system of ordinary differential equations in two separate domains corresponding to the spherical and the cylindrical shell segments. By using Legendre and trigonometric functions, exact and explicitly analytical solutions of the mode functions were constructed and the exact frequency equation were obtained. The implementation of Maple programme indicates that all calculations are simple and efficient in both the exact symbolic calculation and the numerical results of natural frequencies compare with the results using finite element methods and other numerical methdos. As a benchmark, the exactly analytical solutions presented in this paper is valuable to examine the accuracy of various approximate methods.

Growth of a void in a rubber rectangular plate under uniaxial extension

The finite deformation and stress analyses for a rectangular plate with a center void and made of rubber with the Yeoh elastic strain energy function under uniaxial extension were studied in this paper. An approximation solution was obtained from the minimum potential energy principle. The numerical results for the growth of the cavitation and stresses along the edge of the cavitation were discussed. In addition, the stress concentration phenomenon was considered.

Numerical computation for nonlinear unsteady percolation of shale gas and prediction of production

According to the reality of the marine shale reservoirs in South China, multi-scale gas flow in the shale reservoirs with special pore structure characteristics is considered. By using the Knudsen number, it is judged that the dominant pattern of flow in shale porous media is slip flow, and it is pointed out that the process of gas transport is an isothermal process. On the basis of multiscale Beskok-Karniadakis flow model, the corresponding second-order approximation correction is given and the mathematic model of nonlinear unsteady radial percolation is established for the tight shale gas wells. The problem is simplified by applying the Boltzmann transformation and an explicit iteration scheme to solve the pressure is presented by numerical discrete technique. The distribution curves of pressure with time and space are obtained for the case of the constant pressure at the inner boundary. The variation of output over time is predicted and the influence of desorption, slippage and diffusion on the production is analyzed.