Asymptotic dynamic modeling and response of hysteretic nanostructured beams

The nonlinear dynamic response of carbon nanotube (CNT)/polymer nanocomposite beams to harmonic base excitations is investigated asymptotically via the method of multiple scales. The hysteresis associated with the CNT/polymer interfacial frictional sliding is described by a 3D mesoscopic theory reduced via a uniaxial strain assumption for a beam in pure plane bending. Such reduction leads to a Bouc–Wen-like hysteretic moment–curvature relationship. The generalized memory-dependent constitutive law is developed asymptotically and, subsequently, introduced in two archetypal cases of nonlinear beam models. A beam model is tailored for axially restrained, extensible beams (e.g., hinged–hinged beams) for which the dominant geometric nonlinearity is associated with the multiplicative effect of the tension with the bending curvature. The second model is valid for inextensible beams (e.g., cantilever beams) dominated by inertia and curvature nonlinearities. The piece-wise integration of the moment–curvature relationship yields an exponential law which is treated asymptotically to obtain the quadratic and cubic curvature contributions. The ensuing asymptotic equations of motion in the unknown deflection field are discretized according to the Galerkin method employing the eigenmode directly excited near its primary resonance to thus obtain a piece-wise reduced-order model (ROM). The method of multiple scales applied to the ROM yields the asymptotic response together with the frequency response functions for the lowest mode. A parametric study unfolds rich nonlinear dynamic responses in terms of behavior charts highlighting regions of hardening and softening behavior, regions of single-valued stable behavior and regions of multi-valued multi-stable behavior. Such richness of responses is caused by the unusual and unique combination of material and geometric nonlinearities.