Nonlinear Dyn https://doi.org/10.1007/s11071-019-05386-8 ORIGINAL PAPER Asymptotic dynamic modeling and response of hysteretic nanostructured beams Giovanni Formica · Walter Lacarbonara Received: 17 August 2019 / Accepted: 19 November 2019 © Springer Nature B.V. 2019 Abstract The nonlinear dynamic response of car- bon nanotube (CNT)/polymer nanocomposite beams to harmonic base excitations is investigated asymptot- ically via the method of multiple scales. The hystere- sis associated with the CNT/polymer interfacial fric- tional sliding is described by a 3D mesoscopic the- ory reduced via a uniaxial strain assumption for a beam in pure plane bending. Such reduction leads to a Bouc–Wen-like hysteretic moment–curvature rela- tionship. The generalized memory-dependent consti- tutive law is developed asymptotically and, subse- quently, introduced in two archetypal cases of nonlin- ear beam models. A beam model is tailored for axi- ally restrained, extensible beams (e.g., hinged–hinged beams) for which the dominant geometric nonlinear- ity 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 rela- tionship yields an exponential law which is treated asymptotically to obtain the quadratic and cubic cur- G. Formica (B) Dipartimento di Architettura, Roma Tre University, via Madonna dei Monti 40, Roma, Italy e-mail: giovanni.formica@uniroma3.it W. Lacarbonara Department of Structural and Geotechnical Engineering, Sapienza University of Rome, via Eudossiana 18, Roma, Italy e-mail: walter.lacarbonara@uniroma1.it vature contributions. The ensuing asymptotic equa- tions of motion in the unknown deflection field are discretized according to the Galerkin method employ- ing 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 low- est mode. A parametric study unfolds rich nonlinear dynamic responses in terms of behavior charts high- lighting 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 combi- nation of material and geometric nonlinearities. Keywords Nanocomposite beam · Carbon nanotube/polymer · Nanostructured beam · Method of multiple scales · Hysteretic moment–curvature law · Nonlinear frequency response 1 Introduction Nanocomposite materials are regarded as high- performance structural materials for demanding appli- cations in dynamic environments. Lightweight nano- composites, made of engineering (thermoplastic or thermosetting) polymers integrated with various 0D/1D/ 2D carbon nanofillers, are being employed to realize dynamic devices, such as microresonators, microac- 123