The Lavrentiev–Ishlinsky problem at the initial stage of motion

The paper addresses the dynamic buckling of thin rod subjected to discontinuous constant longitudinal load at the initial stage of motion. The time interval equal to the twice the time path of the longitudinal wave along the rod length is understood as the initial stage of the rod motion. Two types of the boundary conditions are analyzed in detail. The transverse deflection of the rod is sought for in the form of a Fourier series in terms of the normal modes of bending vibration. The problem is solved in a linear approximation. The evolution of the beam deflection is studied. The deflection magnitude due to dynamic buckling is compared with the initial deflection and its ratio is taken as a measure of instability. At the initial stage of motion the additional deflection of the buckled rod is shown to be much greater than the amplitude of the initial curvature only for a very flexible rod.