A loaded beam in full frictionless contact with a couple stress elastic half-plane: Effects of non-standard contact conditions

Abstract

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.

Indentation of a free beam resting on an elastic substrate with an internal lengthscale

“…The normal displacement, slope and microrotation at the surface of a couple stress homogenenous isotropic elastic half-plane were derived by Radi (2021) for a clockwise couple C acting at the origin of the coordinate system…”

“…Lately, the indeterminate couple stress theory of elasticity was successfully applied to fracture mechanics (Radi, 2007;Radi, 2008;Morini et al, 2013;Gourgiotis and Georgiadis, 2011;Nobili et al, 2019) and contact mechanics (Zisis et al, 2015;Karuriya and Bhandakkar, 2017). Moreover, the effects of friction and adhesion on contact problems, within the framework of couple-stress elasticity, have been included in the works of Song et al (2017), Wang et al (2018) and Li and Liu (2020). There are several assumptions, especially concerning the new boundary conditions relating rotations and couple tractions, that were introduced in the solution of the above-mentioned problems in order to retrieve the classical elastic solution as the characteristic length of the microstruture tends to zero.…”

“…Surprisingly, this assumption is generally introduced also when the substrate is a micropolar continuum (Zisis et al, 2014(Zisis et al, , 2018Song et al, 2017) and therefore it is capable of supporting couple tractions. Similarly, Euler-Bernoulli (EB) beams in contact with a micropolar material are capable of exchanging distributed couples.…”

“…The plane problem of the bonded frictionless contact between a finite EB beam and a couple stress elastic half-plane has been recently investigated by Radi (2021) assuming that both normal stresses and couple tractions are transmitted along the contact zone. The interface conditions considered in that work assumes that both the slope and the rotation of the contacting bodies correspond.…”

A Method to Calculate the Support Length of Beams Resting on Masonry Walls

“…In current design practice, a maximum allowable contact pressure is usually assumed and checked against the pressure transmitted by the supported beam. Although accurate models were proposed in the literature, they are too complex to be adopted in common practice [ 19 ]. For this reason, it is crucial to have a fast and reliable method to estimate the contact pressure between the masonry wall and the beam resting on it.…”