Nonlinear vibration of buckled nanowires on a compliant substrate

• Published in: Applied Mathematical Modelling 2020-03, Vol.79, p.230-242
• Main Authors: Wang, Bo, Bi, Haohao, Wang, Yan, Ouyang, Huajiang, Deng, Zichen
• Format: Article
• Language: English
• Subjects: Amplitudes; Beam theory (structures); Compliant substrate; Electronics; Equations of motion; Equilibrium; Euler-Bernoulli beams; Euler-Lagrange equation; Kinetic energy; Mathematical analysis; Modulus of elasticity; Nanowire; Nanowires; Nonlinear vibration; Nonlinearity; Residual stress; Resonant frequencies; Runge-Kutta method; Stretchable electronics; Substrates; Surface effect; Symplectic (Partitioned) Runge–Kutta method; Unstable equilibrium point; Vibration analysis
• ISSN: 0307-904X; 1088-8691; 0307-904X
• DOI: 10.1016/j.apm.2019.10.033

•A nonlinear vibration model for a nanowire-substrate structure as a basic stretchable electronics system is established.•The surface effect is found to affect the number of equilibrium points; positive surface constants stiffen the structure.•Negative surface elastic modulus shift the stable and unstable equilibrium points.•An increasing pre-strain will expand the homoclinic orbits and move the locations of the equilibrium points.•Symplectic (Partitioned) Runge–Kutta method is adopted to solve the equation of motion. Buckling of thin nanowires on a pre-strained compliant substrate has been widely used to make nanowire-based stretchable electronics. On nanometer scale, surface effect plays an important role on a buckled nanowire structure. In addition, as the amplitude of the deflection of the buckled nanowire is larger than its thickness, geometrical nonlinearity should be taken into account. Taking the kinetic energy caused by the out-of-plane motion into account, and on the basis of Euler beam theory, a theoretical model for a nanowire-substrate structure is established, combined with the influences of the nano-scale surface effect and geometrical nonlinearity. By means of Lagrange's equation, the equation of motion is derived and then solved by the Symplectic (Partitioned) Runge–Kutta method (PRK). Several numerical examples are analysed to study the nonlinear vibration of the structure. The analytical expressions of stable and unstable equilibrium points, and the relationship between the vibration amplitude and the natural frequency are obtained. The influences of surface effect and pre-strain on the dynamic behaviour are analysed. Through these numerical results, one can find that when the surface elastic modulus and surface residual stress are considered, the number of unstable equilibrium points would increase to three. The frequency obtained with positive surface elastic modulus is greater than that obtained with negative surface elastic modulus, implying that the positive surface elastic modulus can make the nanowire-substrate structure stiffer. Furthermore, when the pre-strain increases, the locations of stable and unstable equilibrium points move further away from the initial displacement, and the homoclinic orbits become expanded. The results presented in this paper should be useful to guide the design of nanowire-based stretchable electronics.