Fig. 1 Schematic diagram of the spectrum of a QP solution from the double-step MMS More

Fig. 2 Amplitudes of type I QP solutions of the damped nonlinear QP Mathieu equation versus the parametric excitation magnitude β with ω = 1.2075, ω d = 3 / 20 , c = 0.001, ω ~ 0 = 1.1 , α = 0.08, and ξ = 0.5 from the three methods, i.e., the IHB method with ... More

Fig. 3 IHB method with two time scales, NI, and the double-step MMS for type I QP solutions of the damped nonlinear QP Mathieu equation with ω = 1.2075, ω d = 3 / 20 , c = 0.001, ω ~ 0 = 1.1 , α = 0.08, and ξ = 0.5 at the parametric excitation magnitude β = 0... More

Fig. 4 QP responses of the damped nonlinear QP Mathieu equation with ω = 1.2075, ω d = 3 / 20 , c = 0.001, ω ~ 0 = 1.1 , α = 0.08, and ξ = 0.5: ( a ) β ∼ A 1,0 and ( b ) enlargement of a zone highlighted in ( a ) More

Fig. 5 Poincaré sections of the QP responses of the damped nonlinear QP Mathieu equation with ω = 1.2075, ω d = 3 / 20 , c = 0.001, ω ~ 0 = 1.1 , α = 0.08, and ξ = 0.5 from NI and the IHB method with two time scales with different numbers of harmonic terms: ( a... More

Fig. 6 Amplitudes of type II QP solutions versus the parametric excitation magnitude β of the damped nonlinear QP Mathieu equation with ω = 1.2075, ω d = 3 / 20 , c = 0.001, ω ~ 0 = 1.1 , α = 0.08, and ξ = 0.5 from the IHB method with two time scales and NI; ... More

Fig. 7 IHB method with two time scales and NI for type II QP solutions of the damped nonlinear QP Mathieu equation with ω = 1.2075, ω d = 3 / 20 , c = 0.001, ω ~ 0 = 1.1 , α = 0.08, and ξ = 0.5 at the parametric excitation magnitude β = 0.6: ( a ) time histor... More

Fig. 8 Amplitudes of type III QP solutions versus the parametric excitation magnitude β of the damped nonlinear QP Mathieu equation with ω = 1.2075, ω d = 3 / 20 , c = 0.001, ω ~ 0 = 1.1 , α = 0.08, and ξ = 0.5 from the IHB method with two time scales and NI:... More

Fig. 9 The IHB method with two time scales and NI for type III QP solutions of the damped nonlinear QP Mathieu equation with ω = 1.2075, ω d = 3 / 20 , c = 0.001, ω ~ 0 = 1.1 , α = 0.08, and ξ = 0.5 at the parametric excitation magnitude β = 0.6: ( a ) time h... More

Fig. 10 Time histories of the damped nonlinear QP Mathieu equation with ω = 1.2075, ω d = 3 / 20 , c = 0.001, ω ~ 0 = 1.1 , α = 0.08, and ξ = 0.5 at the parametric excitation magnitude β = 0.4 More

Fig. 1 (Sketch not to scale) ( a ) A strained nanostructured layer of material A and thickness h L is embedded in an infinite-size matrix of material B. A nanowire of material A, width 2 h x and height h y , is considered at the upper layer–matrix interface. The misfit strain between ... More

Fig. 2 Dimensionless force F ~ G + versus the dislocation position x ~ d for different values of the dipole vertical position, with h ~ y = 0.4 and K ~ m = 1 : ( a ) in the upper part of the nanostructure ( y ~ d ... More

Fig. 3 Different regions in the ( h ~ y , y ~ d ) plane of the nanostructure where the dislocation dipole can reach or not equilibrium positions, with K ~ m = 1 More

Fig. 4 Critical misfit parameter ( δa / a eq ) c versus h y / h x for h x = 20 b , 50 b , and 100 b More