Longitudinal Free Vibrations of a Fixed-Free Two-Phase Elastic Bar

摘要

In this paper, the longitudinal free vibrations of a fixed-free bar are studied.It is assumed that the bar initially consists of two phases, one of which was obtained from the other by a martensitic phase transformation. It is also assumed that both phases of the bar have elastic constitutive behavior. For a bar that consists entirely of one phase that behaves elastically, it is well known that during the free vibrations of the bar the displacement and stress at each point of the bar oscillate as time progresses. If there is damping present, these oscillations will decay and go to zero as time goes to infinity, otherwise their amplitudes will remain constant in time. Considering this, for a bar that initially consists of two different phases that both behave elastically, one might expect that the displacement and stress at each point of the bar will also have oscillatory-type behavior during the free vibrations of the bar. If this is the case, the driving traction at the interface separating the two phases will oscillate. As a result of this, if the nominal phase boundary velocity is related to the driving traction through a kinetic relation that does not have an interval of the driving traction corresponding to a zero nominal phase boundary velocity, the nominal phase boundary velocity will also oscillate. Since energy is dissipated when the phase boundary moves and passes over particles of material of one phase converting them into particles of material of the other phase, one might conclude that the oscillatory-type response of the two-phase bar during the free vibrations of the bar should decay as time increases. This damping behavior of the two-phase bar is discussed.