An investigation of the stability of a linear viscoelastic bar

The intent of this study is to investigate the stability of an initially straight, simply supported, linear viscoelastic bar subjected to a harmonically varying axial load. Infinitesimal viscoelasticity is assumed through the study. The study is based upon the concepts of continuum mechanics developed by W. Noll. A Volterra integral of the second kind is derived and solved for the deflection of the bar as a function of time. The solution of the equation is based on the assumption that the stress relaxation function can be approximated as a series of monotonically decreasing exponentials. The study proceeds by defining the concept of the systems stability. Based on the general solution, two- lemmas and a theorem are proved with respect to stability. The theorem proves that two conditions are necessary and sufficient for stability. A numerical example is presented for which four cases are investigated. The stress relaxation function, c(t), is assumed to be a single exponential. The lemmas and theorem are applied to determine the system's stability.