Modeling of long, thin elastic structures with periodic geometry

The author considers the problem of modeling a class of elastic structures characterized by having one dimension large relative to the other two and being periodic along the length. Since a direct approach to modeling such structures leads to intractable numerical difficulties, a simpler model is modeled by a linearized elastic system on a 3-D domain which is not simply connected. By letting a small parameter (the width of the cross section) tend to zero, two uncoupled (1-D) equations having the same form as the usual Euler-Bernoulli beam equation but with periodic coefficients are obtained. By a second limiting process (letting the period tend to zero), equations with constant coefficients (the homogenized equations) are obtained. Numerical results comparing the eigenvalues of the first limit equation with those of the homogenized equation are presented