A loaded beam in full frictionless contact with a couple stress elastic half-plane: Effects of non-standard contact conditions. (1st December 2021)

• Record Type: Journal Article
• Authors: Radi, E.
• Is Part Of: International journal of solids and structures. Volume 232(2021)
• Publication Date: 2021-12-01
• Subjects: Contact problem -- Euler-Bernoulli beam -- Chebyshev polynomials -- Green's function -- Couple stress elasticity -- Size effects; Mechanics, Applied -- Periodicals; Structural analysis (Engineering) -- Periodicals; Elastic solids -- Periodicals; Mécanique appliquée -- Périodiques; Constructions, Théorie des -- Périodiques; Solides élastiques -- Périodiques; Elastic solids; Mechanics, Applied; Structural analysis (Engineering); Periodicals; 624.18
• DOI: 10.1016/j.ijsolstr.2021.111175
• Languages: English
• ISSNs: 0020-7683

Abstract

Highlights: Analytical solution is presented for the problem of a beam resting on couple stress elastic half-plane. The Green's functions for a point couple applied at the half-plane surface is given. The distributions of contact pressure and couple stress tractions are calculated analytically. Three different types of microstructural contact conditions are considered and discussed. The size effects due to the characteristic length of the couple stress half-plane are clearly illustrated. 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.