Non-standard contact conditions between a beam and a couple stress elastic half-plane

In the present work, the problem of a deformable Euler-Bernoulli beam of length 2a in bilateral frictionless contact with a couple stress elastic half-plane, whose constitutive parameters are the shear modulus , the Poisson coefficient  and the material characteristic length l, is investigated by assuming that both contact pressure and couple stress tractions are transmitted across the contact zone. The present 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. It may be considered an extension of the works on beams in contact with an elastic half-plane performed by Shield and Kim (1992), Lanzoni and Radi (2016), and on rigid indenters in contact with an elastic couple-stress half-plane developed by Guorgiotis and Zisis (2016) and Zisis et al. (2018). The couple stress theory of elasticity requires boundary conditions on the microrotation and couple stress tractions in addition to the usual boundary conditions of the classic non-polar continuum on displacements and stress tractions. A challenging problem is thus how to extend the classic contact conditions to include the effects of the microrotation and couple stress tractions. In the proposed approach, the classical strain compatibility condition between the slope of the beam and that of the half-plane surface is imposed along the contact region. Moreover, three alternative kinds of microstructural contact conditions are considered and discussed, namely, vanishing of couple stress tractions, vanishing of microrotations and compatibility between microrotatons of the half-plane surface and slope of the beam. The first two types of boundary conditions are usually assumed in the technical literature on micropolar materials, although the third boundary condition seems the most correct one. 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 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 of the first kind displaying the classical 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 Chebyshev series expansion adopted for the contact pressure and couple stress tractions. 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 EI of the beam and the characteristic length l of the elastic half-plane. The three alternative conditions lead to significantly different results in term of bending moment along the beam. The size effects due to the characteristic length of the half-plane and the implications of the generalized contact conditions are illustrated and discussed. The classical elastic solution is recovered as the characteristic length becomes vanishing small. Generally, the magnitude of the couple stress tractions is found to increase with the characteristic length. Although its contribution is usually smaller than that of the contact pressure and mainly restricted to the edges of the beam, it may provide a significant influence on the shear force and bending moment along the beam. Therefore, the obtained results show that the couple stress tractions exhibit a large influence on the beam internal forces and moments and display size dependent behavior when the beam length is comparable to the intrinsic characteristic length scale of the ground. Moreover, we show that accounting for the micropolar behavior of the ground, but neglecting the moment tractions in the contact region may lead to a substantial underestimation of the bending moment in the beam, in particular for the intermediate range of values of the material characterisic length (Fig. 1). The most interesting applications concern the case of beam length comparable with the microstructural characteristic length, namely for the ratio of l/a equal 0.5 and 1 considered in the plots. These results are expected to be significant and useful for engineering applications, specially in the field of micromechanics. We aspire indeed that the provided results may serve as a reference for the design of structural components in contact with heterogeneous and complex materials, not only at the macroscale, but also at the micro and nanoscale, providing a fundamental basis for the assessment of the proper microstructural contact conditions. References 1. Gourgiotis, P.A., Zisis, Th., 2016. Two-dimensional indentation of microstructure solids characterized by couple-stress elasticity. Journal of Strain Analysis and Enginering Design, 51, 1-14. 2. Lanzoni, L., Radi, E., 2016. A loaded Timoshenko beam bonded to an elastic half plane. International Journal of Solids and Structures, 92(1), 76-90. 3. Shield, T.W., Kim, K.S., 1992. Beam theory models for thin film segments cohesively bonded to an elastic half space, International Journal of Solids and Structures, 29, 1085-1103. 4. Zisis, Th., Gourgiotis, P.A., Georgiadis, H.G., 2018. Contact mechanics in the framework of couple stress elasticity. In H. Altenbach et al. (eds.), Generalized models and non-classical approaches in complex