J.  Awrejcewicz, I.  Kutepov, S.P.  Pavlov, I.V.  Papkova, A.V.  Krysko, 
    Non-Linear Dynamics of Flexible Curvilinear Bernoulli-Euler Nano-Beams in a Stationary Temperature Field,
    Journal of Engineering and Applied Sciences,
    Volume 11,Issue 9,
    2016,
    Pages 2079-2084,
    ISSN 1816-949x,
    jeasci.2016.2079.2084,
    (https://makhillpublications.co/view-article.php?doi=jeasci.2016.2079.2084)
    Abstract: In this study the mathematical model of non-linear dynamics of flexible curvilinear beams in a stationary temperature field is proposed. On a basis of the variation principles the PDEs governing nonlinear dynamics of curvilinear nano-beams are derived. The proposed mathematical model does not include any requirements for the temperature distribution along the beam thickness and it is defined via solution to the 2D Laplace equation for the corresponding boundary conditions. The governing PDEs are reduced to ODEs employing the finite difference method of a second order and then the counterpart Cauchy problem has been solved using the 4th order Runge-Kutta method. The convergence of reduction from PDEs to ODEs is validated by the Runge principle. In particular, it has been shown that the solutions obtained taking into account the material nano-structural features are more stable in comparison to the case where the micro-effects are neglected.
    Keywords: Curvilinear Bernoulli-Euler beam;micro and nano-beams;temperature field;non-linear dynamics;PDEs