Size-dependent non-linear dynamics of curvilinear flexible beams in a temperature field

• Published in: Applied Mathematical Modelling Vol. 67; pp. 283 - 296
• Main Authors: Krysko, V.A., Awrejcewicz, J., Kutepov, I.E., Babenkova, T.V., Krysko, A.V.
• Format: Journal Article
• Language: English
• Published: New York Elsevier Inc 01.03.2019; Elsevier BV
• Subjects: Beams (structural); Beams and columns; Dynamic stability; Dynamics; Finite difference method; Finite differences; Hamilton's principle; Isotropic material; Liapunov exponents; Linearity; Load distribution (forces); Mathematical models; Microstructures; Nanoelectromechanical systems; Nonlinear dynamics; Nonlinearity; Numerical methods; Parameters; Runge-Kutta method; Stress concentration; Temperature distribution; Thermal stress; Transverse loads; Thermal stress; Microstructures; Dynamics; Beams and columns; Finite differences
• ISSN: 0307-904X; 1088-8691; 0307-904X
• DOI: 10.1016/j.apm.2018.10.026

•Dynamical stability loss of a curvilinear size-dependent beam is studied.•A combination of the couple stress theory and the Euler–Bernoulli and Duhamel–Neumann hypotheses is proposed.•Reliability of solution to non-linear thermomechanical PDEs is validated.•Numerous engineering-oriented problems are investigated. A mathematical model of the loss of dynamic stability of curvilinear size-dependent MEMS and NEMS elements embedded in a temperature field and subjected to large deflections was derived and studied. The fundamental governing dynamical equations of MEMS/NEMS members were yielded by Hamilton's principle. The investigations were based on combining the modified couple stress theory, the first-order approximation kinematic (Euler–Bernoulli) model, the von Kármán geometric non-linearity, and the Duhamel–Neumann law regarding the temperature input (the beam material is elastic, isotropic and there are no constraints imposed on the temperature distribution). The temperature field was defined by solving a heat transfer equation. The computational algorithm was based on the finite difference method and the Runge–Kutta method. The numerical methods were validated by estimating the temporal and spatial convergence and reliability of the obtained solution was validated with the Lyapunov exponents obtained by qualitatively different methods. A few case studies related to the loss of stability, the magnitude of the size-dependent parameter, the type and intensity of the temperature input, and the parameters of uniformly distributed transverse load were investigated.