Fig. 1 Bouc–Wen hysteretic loop under sine excitation: ( a ) γ = 0.5, ν = − 0.5, A = 1 and ( b ) γ = 0.9, ν = 0.1, A = 1 More

Fig. 2 Bilinear hysteretic loops: ( a ) hysteretic restoring force z with z y = 1 and ( b ) the overall restoring force g ( t , x , x ˙ ) with α = 0.5 More

Fig. 3 Iwan–Jenkins hysteretic loop ( K J = 4, f max = 25, f min = 0) More

Fig. 4 Response of the Bouc–Wen oscillator ( λ = 0.5, α = 0.6): ( a ) variance of the displacement and ( b ) variance of the fractional derivative term More

Fig. 5 Response of the Bouc–Wen oscillator ( λ = 0.5, α = 0.4): ( a ) variance of the displacement and ( b ) variance of the fractional derivative term More

Fig. 6 Response of the bilinear oscillator ( λ = 0.5, α = 0.6): ( a ) variance of the displacement and ( b ) variance of the fractional derivative term More

Fig. 7 Response of the bilinear oscillator ( λ = 0.5, α = 0.4): ( a ) variance of the displacement and ( b ) variance of the fractional derivative term More

Fig. 8 Response of the Preisach oscillator ( λ = 0.5): ( a ) variance of the displacement and ( b ) variance of the fractional derivative term More

Fig. 1 Schematic of a typical water-triggered shape memory process. F s and F d denote the elastic deformation gradient of sparse and dense phases, respectively; F f denotes the frozen deformation gradient. More

Fig. 2 Relationship between the dense volume fraction ϕ d and normalized chemical potential μ ^ ; μ ^ r denotes the reference normalized chemical potential at the phase transformation center More

Fig. 3 Schematics of three types of shape memory cycles of water-trigger shape memory effects, including free-recovery process, fixed-recovery process 1, and fixed-recovery process 2 More

Fig. 4 General schematics of the stress-stretch-normalized chemical potential ( σ − λ − μ ^ ) relationship for ( a ) free-recovery process, ( b ) fixed-recovery process 1, and ( c ) fixed-recovery process 2 More

Fig. 5 Validation of the accuracy of the FEM implementation by comparing with analytical results in different processes: ( a ) stress–chemical potential relationship in the constraint-dehydrating process; ( b ) stretch–chemical potential relationship in the free-recovery process; ( c ) stress–chem... More