The plane problem of a loaded Euler-Bernoulli beam of finite length in frictionless bilateral contact with a microstructured half-plane modelled by the couple stress theory of elasticity is considered here. The study is aimed to investigate the size effects induced on the beam internal forces and moments by the contact pressure and couple stress tractions transmitted across the contact region. Use is made of the Green’s functions for point force and point couple applied at the surface of the couple stress elastic half-plane. The problem is formulated by imposing compatibility of strain between the beam and the half-plane along the contact region and three alternative types of microstructural contact conditions, namely vanishing of couple stress tractions, vanishing of microrotations and compatibility between rotations of the beam cross sections and microrotations of the half-plane surface. The first two types of boundary conditions are usually assumed in the technical literature on micropolar materials, without any sound motivation, although the third boundary condition seems the most correct one. The problem is thus reduced to one or two (singular) integral equations for the unknown distributions of contact pressure and couple stress tractions, which are expanded in series of Chebyshev orthogonal polynomials displaying square-root singularity at the beam ends. By using a collocation method, the integral equations are reduced to a linear algebraic system of equations for the unknown coefficients of the series. The contact pressure and couple stress along the contact region and the shear force and bending moment along the beam are then calculated under various loading conditions applied to the beam, varying the flexural stiffness of the beam and the characteristic length of the elastic half-plane. The size effects due to the characteristic length of the half-plane and the implications of the generalized contact conditions are illustrated and discussed.

此处考虑了有限长度的受载欧拉-伯努利梁在与微结构半平面无摩擦双边接触时的平面问题，该问题由弹性耦合应力理论建模。该研究旨在研究由接触压力和跨接触区域传递的耦合应力牵引对梁内力和力矩引起的尺寸效应。对施加在耦合应力弹性半平面表面上的点力和点耦合使用格林函数。该问题是通过在梁和沿接触区域的半平面之间施加应变的兼容性以及三种不同类型的微观结构接触条件，即耦合应力牵引的消失，微旋转的消失和光束横截面旋转与半平面表面微旋转之间的兼容性。前两种边界条件通常在微极材料的技术文献中假设，​​没有任何合理的动机，尽管第三种边界条件似乎是最正确的。因此，问题被简化为一个或两个（奇异）积分方程，用于接触压力和耦合应力牵引的未知分布，这些方程在一系列 Chebyshev 正交多项式中展开，在梁端显示平方根奇异性。通过使用搭配方法，积分方程被简化为该级数未知系数的线性代数方程组。然后在施加到梁的各种载荷条件下计算接触区域的接触压力和耦合应力以及沿着梁的剪切力和弯矩，改变梁的弯曲刚度和弹性半平面的特征长度。说明和讨论了由于半平面的特征长度和广义接触条件的影响而引起的尺寸效应。