Systematic description of imperfect bifurcation behavior of symmetric systems

A systematic method is presented for describing experimental curves of force vs strain of a system with regular polygonal "dihedral group# symmetry subject to bifurcation behavior\ with an aim toward over! coming the following problems ] "0# it is di.cult to judge whether the system is undergoing bifurcation or not ^ "1# the perfect behavior of the system cannot be known due to the presence of initial imperfections ^ "2# those curves are often qualitatively di}erent from bifurcation diagrams predicted by mathematics[ The tools employed are ] the asymptotic theory for imperfect bifurcation\ such as the Koiter law\ and the stochastic theory of initial imperfections[ The former theory is extended in this paper to the system with regular!polygonal symmetry to present asymptotic laws for recovering perfect curves with reference to the experimental ones[ These laws are formulated for physically observable displacements\ instead of the variables in the mathematical bifurcation diagrams\ in order to make them readily applicable to the experimental curves[ The stochastic theory is combined with an asymptotic law to develop a means to identify the multiplicity of the bifurcation point[ The systematic method for describing the experimental curves developed in this manner is applied to the bifurcation analysis of regular!polygonal truss domes to testify its validity[ Furthermore\ this method is applied to the shear behavior of cylindrical sand specimens to show that they\ in fact\ are undergoing bifurcation\ and\ in turn\ to demonstrate the importance of a viewpoint of bifurcation in the study of shear behavior of materials[ The need of a dual viewpoint of bifurcation and plasticity in the study of constitutive relationship of materials is emphasized to conclude the paper