The Scale-Dependent Deformation Model of a Layered Rectangle

Abstract

We consider the problem of deformation of a layered rectangle whose lower side is rigidly clamped, a distributed normal load acts on the upper side, and the lateral sides are in conditions of sliding termination. One-parameter gradient elasticity theory is used to account for the scale effects. The boundary conditions on the lateral faces allow us to use separation of variables. The displacements and mechanical loads are expanded in Fourier series. To find the harmonics of displacements, we have a system of two fourth order differential equations. We seek a solution to the system of differential equations by using the elastic potential of displacements and find the unknown integration constants by satisfying the boundary and transmission conditions for the harmonics of displacements. Considering some particular examples, we calculate the horizontal and vertical distribution of displacements as well as the couple and total stresses of a layered rectangle. We exhibit the difference between the distributions of displacements and stresses which are found on using the solutions to the problem in the classical and gradient formulations. Also, we show that the total stresses have a small jump on the transmission line due to the fact that, in accord with the gradient elasticity theory, not the total stresses, but the components of the load vectors should be continuous on the transmission line. Furthermore, we reveal a significant influence of the increase of the scale parameter on the changes of the values of displacements and total and couple stresses.