Static and Dynamic Behavior of Elastic Circular Non-Local Plate Dynamics on Unilateral Two-Parameter Foundation by Using Chebyshev Polynomial Expansion

Abstract This paper investigates the static and dynamic response of an elastic circular plate, focusing on forced vibrations. The plate is modeled as resting on a unilateral two-parameter foundation and subjected to both rotationally symmetric distributed and concentrically applied concentrated loads, all of which are time-dependent. The two-parameter foundation model incorporates non-local properties, and the plate is assumed to exhibit non-local elastic behavior. Additionally, the foundation is considered unilateral, accounting for potential separation between the foundation and the plate. The present study incorporates two characteristic parameters of the nonlocal foundation $$g_{o}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>g</mml:mi> <mml:mi>o</mml:mi> </mml:msub> </mml:math> and the plate $$\mu$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>μ</mml:mi> </mml:math> to capture size effects of the foundation and plate. Within this framework of these assumptions, the static and dynamic behavior of the plate is studied by deriving the lift-off condition of the plate from the foundation. The governing equation of the system is formulated by accounting for the foundation beyond the contact region. An approximate solution is obtained using the Galerkin method. Given the rotational symmetry of the plate geometry and loading, the displacement function is assumed to have the same symmetry and is expressed using a symmetric Chebyshev polynomial expansion, which includes the contact region. To address a broad range of foundation, plate and load parameters, special attention is given to adopting non-dimensional parameters. Because of the nonlinear response of the unilateral foundation, numerical treatment is performed using an iterative solution procedure in the static case. Detailed numerical solutions are presented to investigate both static and dynamic behaviors of the plate. For dynamic loads, the second-order differential equation of motion is solved numerically using constant acceleration procedures. The effects of the plate and the foundation parameters are studied under various loading scenarios for both static and dynamic cases. Although arbitrary time variations are assumed for the loading, a stepwise time increment is employed in the numerical analysis for both concentrated and uniformly distributed loads, with forced vibration studied comparatively. Numerical results are presented to offer insight into the time variations of the contact region and the displacements at both the center and edge of the plate, for various plate and foundation parameters. Plate lift-off which induces nonlinear behavior is evidenced by the results presented. The effect of unilateral foundation characteristics on the static and dynamic behavior of the circular plate is illustrated through a comprehensive parameter analysis.