On a viscoplastic Shanley-like model under constant load

Motivated by applications devoted to study the behavior of steel and aluminum alloys columns, inelastic Shanley-like models have been extensively studied in the literature, mainly to investigate buckling and post buckling problems (see Sewell, 1971; Hutchinson, 1974 for a complete review). On the other hand, recent papers discussing geotechnical problems point out that those models may be useful for the study of the essential features of the equilibrium of towers. In this case, the structure's proper weight (which is a conservative load with constant magnitude), and the verticality imperfection, appear to be responsible for the leaning evolution, as well as the time variation of the mechanical property of the soil . Throughout this paper, a 'T' shaped rigid rod on two no-tension viscoplastic springs under constant load with initial imperfection is considered. Under fairly general assumptions, a viscoplastic constitutive law is derived as a particular case of the theory developed in (Gurtin et al., 1980), studying its behavior under loading processes. By virtue of a time rescaling procedure, extreme retardation leads to determine a yielding parameter, which allows to distinguish between viscoelastic and viscoplastic ranges. For all the states attained by the rod, explicit expressions for the two displacement parameters characterizing its evolution are given. Noting that failure may occur if the reaction of one spring goes to zero, sufficient conditions under which no bifurcation and no failure occur are given for all the phases, leading so to determine the minimum upper bound for the load parameter. This new result turns out to depend only on the relaxation surface parameters at equilibrium, irrespective of the behavior under non-zero finite deformation velocities.