Fractal basin of elastic shakedown attractors under cyclic rectilinear generalized strain paths

Abstract

For perfectly elastoplastic model under rectilinear generalized strain paths, the response subspace technique is used to reduce the k‐dimensional problem to a more economical two‐dimensional problem, in which two coordinates (x, y) suffice to determine the generalized stress. Furthermore, in this subspace a Minkowski space can be endowed, on which the group action is found to be a proper orthochronous Lorentz group. In order to further understand the model behavior we subject it to a cyclic rectilinear generalized strain path and perform a critical analysis, establishing the theorems about relaxation of mean stresses and elastic shakedown. The concept of elastic shakedown attractor is introduced; several critical values of amplitude ratio are derived and then the phenomenon of fractal basin is explored when the amplitude ratio is smaller than the elastic shakedown limit and larger than one of the critical values.

Key words:

• perfectly elastoplastic model
• elastic shakedown attractor
• fractal basin
• Lorentz group

Notes

C. ‐S. Liu is with the Department of Mechanical and Mechatronic Engineering, National Taiwan Ocean University, Keelung, Taiwan 202, R.O.C. (Tel: [phone] ext. 3252; Fax: [phone]; Email: [email protected])