Buckling of elastic structures in unilateral contact with or without friction

The objective of this paper is to present a general study of the stability and bifurcation of elastic structures in the presence of unilateral contact, with or without friction. The stability in the dynamic, sense of an equilibrium position is first considered without friction. It is shown that a local strict minimum of the potential energy represents stable equilibrium. The positivity of the second variation of the Lagrangian with respect to some compatible virtual displacement rates also leads to a criterion for dynamic stability. Stability and bifurcation analysis are performed in the second part of the paper by the classical method of asymptotic development. The study of the rate problem leads to a non-bifurcation criterion and a stability criterion. The obtained stability criterion is identical to the aforementioned criterion of the second variation of the Lagrangian. In the last section the effect of dry friction following Coulomb's law is taken into consideration. The formulation of the rate problem in this case leads again to stability and non-bifurcation criteria. Some simple examples of buckling of elastic structures with unilateral contacts are also given in order to illustrate its dominant features.