The Scale-Dependent Deformation Model of a Layered Rectangle

We consider the problem ofdeformation of a layered rectangle whose lower side is rigidly clamped, adistributed normal load acts on the upper side, and the lateral sides are in conditions of slidingtermination. One-parameter gradient elasticity theory is used to account for thescale effects. The boundary conditions on the lateral faces allow us to useseparation of variables. The displacements and mechanical loads areexpanded in Fourier series. To find the harmonics ofdisplacements, we have a system of two fourth order differential equations.We seek a solution to the system of differential equationsby using the elastic potential ofdisplacements and find the unknown integration constants bysatisfying the boundary and transmission conditionsfor the harmonics of displacements. Considering some particular examples,we calculate the horizontal and vertical distribution ofdisplacements as well as the couple and total stresses of a layered rectangle.We exhibit the difference between the distributions ofdisplacements and stresses which are found on using the solutions to theproblem in the classical and gradient formulations.Also, we show that the total stresses have a smalljump on the transmission line due to the fact that, in accord with thegradient elasticity theory, not the total stresses, but thecomponents of the load vectors should be continuous on thetransmission line.Furthermore, we reveala significant influence of the increase of the scale parameter on thechanges of the values of displacements and total andcouple stresses.