Dynamic Snap-Through of Imperfect Viscoelastic Shallow Arches

Abstract

Dynamic snap-through or dynamic buckling of imperfect viscoelastic shallow arches with hinged ends is considered under step loads of infinite duration. Attention is principally devoted to the influence both of small imperfections and of small amounts of damping, acting together, on the critical loads. For the problem considered, the Voigt model is used for viscoelasticity, the deflection is represented by the first two harmonic modes, and imperfections have the shape of the second (antisymmetric) mode. Results obtained by numerical integration of the differential equations show that the critical load exhibits a jump discontinuity in the limit for vanishing imperfection. Critical loads for slightly imperfect and elastic (inviscid) arches are slightly higher than those from the saddle-point formula of Hoff and Bruce [1, p. 276], confirming that the formula gives a lower bound on the critical load. However, critical loads for arches with slight imperfection and slight viscosity are considerably higher than for the slightly imperfect elastic arches. Another closed-form expression is shown to be in good agreement with these results. For finite amounts of viscosity, the critical loads tend rapidly to the values obtained for infinite viscosity, which are the same as the critical loads for quasi-static buckling. Apart from the jump discontinuity at zero, the critical load for any viscosity decreases continuously and monotonically with imperfection.