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.

我们考虑层状矩形的变形问题，该矩形的下边被刚性夹紧，上边作用有分布法向载荷，并且侧面处于滑动终止状态。单参数梯度弹性理论用于解释尺度效应。侧面的边界条件允许我们使用变量分离。位移和机械载荷以傅里叶级数展开。为了找到位移的谐波，我们有一个由两个四阶微分方程组组成的系统。我们利用位移的弹性势来寻求微分方程组的解，并通过满足位移谐波的边界和传输条件来找到未知的积分常数。考虑到一些特定的例子，我们计算了位移的水平和垂直分布以及层状矩形的力偶应力和总应力。我们展示了在使用经典公式和梯度公式中的问题解决方案时发现的位移和应力分布之间的差异。此外，我们还表明，传输线上的总应力有一个小的跳跃，因为根据梯度弹性理论，在传输线上，负载矢量的分量而不是总应力应该是连续的。此外，我们揭示了尺度参数的增加对位移以及总应力和力偶应力值的变化的显着影响。