Mathematical Modeling of Bending of a Thin Orthotropic Rectangular Plate Based on Couple Stress Theory

A two-parameter mathematical model is developed for studying the strain state of a thin orthotropic nanoplate of constant thickness subjected to an arbitrary transverse load. Assuming that the deformation conditions are isothermal and the displacements of the plate points are small compared to the plate thickness, a differential equation for the nanoplate deflection is derived from the minimum total free energy condition, taking into account the curvature tensor components under small rotation at the microlevel. A solution to this equation for a rectangular nanoplate is constructed with Chebyshev polynomials of the first kind used as a basis in a Hilbert function space. The coefficients in the expansion of the approximating function in terms of these polynomials are found using the collocation method. The roots of Chebyshev polynomials of the first kind are used as collocation points. The error of the constructed solution is estimated in the norm of the Banach space of essentially bounded functions. The plate deflection under constant and distributed loading for which there is an analytical solution is computed and analyzed depending on nonlocal length scale parameters.