Asymptotic model equations and bifurcations analysis of compressible hyperelastic layer
Student thesis: Doctoral Thesis
Related Research Unit(s)
In this dissertation, we study the equilibriumstates of a compressible hyperelastic layer under compression after the primary and secondary bifurcations. This type of problem is an old one and has been studied from different points of view. It is very difficult to find analytical post-bifurcation solutions, especially for the secondary bifurcation solutions. Bifurcation studies using two or three-dimensional continuum mechanics formulations have already been presented for the case of axially load elastic material. But the general post-bifurcation analysis of this problem for arbitrary hyperelastic material is very few in the literature due to the complexity of the required calculations, thus motivating the present work. It is worth noticing that for the more complicated case of elastic material, numerical as well as asymptotic post-bifurcation analysis have been presented in the literature. Of interesting here is to find the asymptotic analytical bifurcation solutions for the compressible hyperelastic layer. Starting from the two-dimensional field equations for a compressible hyperelastic material, we use a methodology of coupled series-asymptotic expansions developed earlier to derive two coupled nonlinear ordinary differential equations (ODEs) as the model equations. The critical buckling stresses are determined by a linear bifurcation analysis, which are in agreement with the results in literature. The method of multiple scales is used to solve the model equations to obtain the second-order asymptotic solutions after the primary bifurcations. An analytical formula for the post-buckling amplitudes is derived. Two kinds of numerical solutions are also obtained, the numerical solutions of themodel equations by a differencemethod and those of the two-dimensional field equations by the finite elements method. Comparisons among the analytical solutions, numerical solutions and solutions obtained by the Lyapunov-Schmidt-Koiter (LSK) method in literature are made and good agreements for the displacements are found. It is also found that at some places the axial strain is tensile, although the layer is under compression. To consider the secondary bifurcation, we superimpose a small deformation on the state after the primary bifurcation. With the analytical solution of the primary bifurcation, we manage to reduce the problem of the secondary bifurcation to one of the first bifurcation governed by a second order variable-coefficient ODE. Our analysis identifies an explicit function (4.16) and from the existence/nonexistence of its zero points one can immediately judge whether a secondary bifurcation can take place or not. The zero corresponds to a turning point of the governing ODE, which leads to nontrivial solutions. Further, by theWKBmethod, the equation (in a very simple form) for determining the critical stress for the secondary bifurcation is derived. We further use AUTO to compute the secondary bifurcation point numerically, which confirms the validity of our analytical results. The numerical results in the secondary bifurcation branch computed by AUTO indicate that the secondary bifurcation induces a "wave number doubling" phenomenon and also the shape of the layer has a convexity change along the axial direction. Under the general three-dimensional pre-stress condition, for a new hyperelastic material subjected to axially load, using the same method to derive a similar five order asymptoticmodel equation. The impact of the pre-stretchm3 on the principal stretches of the uniform pre-bifurcation state and the primary and secondary bifurcation of the model equation have been studied. The results show that the pre-stretch m3 play a key role in determining the solutions and bifurcation of model equation.
- Elastic analysis (Engineering), Asymptotic expansions, Bifurcation theory, Mathematical models