From Ziegler to Beck's column: a nonlocal approach

This paper is concerned with the dynamic stability of a microstructured elastic column loaded by circulatory forces. This nonconservative lattice (or discrete) problem is shown to be equivalent to the finite difference formulation of Beck's problem (cantilever column loaded by follower axial force). The lattice problem can be exactly solved from the resolution of a linear difference eigenvalue problem. The first part of the paper deals with the theoretical and numerical analyses of this discrete Beck's problem, with a particular emphasis on the flutter load sensitivity with respect to the discretization parameters, such as the number of links of the lattice. The second part of the paper is devoted to the elaboration of a nonlocal equivalent continuum that possesses similar mathematical or physical properties as compared to the original lattice model. A continualized nonlocal model is introduced first by expanding the difference operators present in the lattice equations in terms of differential operators. The length scale of the continualized nonlocal model is size independent. Next, Eringen's nonlocal phenomenological stress gradient is considered and applied at the beam scale in allowance for scale effects of the microstructured Beck column. The nonlocal Euler-Bernoulli beam model is able to capture the softening scale effect of the lattice model, even if the length scale of Eringen's model appears to be size dependent in this case. The continualized nonlocal continuum slightly differs from the Eringen's one, in the sense that the length scale affecting the static and the inertia terms differs in the deflection equation. A general parametric study illustrates the capability of each nonlocal model, the phenomenological and the continualized one, with respect to the reference lattice model. Nonlocal Beck's column is shown to be a transient medium from Ziegler's column (two-degree-of-freedom system) to the local continuous Beck's column (with an infinite degree of freedom).