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.
Highlights
Two-dimensional analyses of the contact problem of beams and plates resting on a deformable ground with microstructure are usually required in the field of micromechanics as well as in civil and biomechanical engineering, e.g. in the design of building foundations resting on granular materials, railway trucks laying on railway ballast and bone implants
The works on the problem of indentation of a couple stress elastic materials usually assume that the moment tractions exchanged between the rigid indenter and the halfplane are equal to zero, even if these interactions can in principle be transmitted along the contact region
The contact problem was solved by imposing the classical compatibility condition between the slope of the beam and that of the half-plane along the contact region together with an additional microstructural contact condition of three different types
Summary
Two-dimensional analyses of the contact problem of beams and plates resting on a deformable ground with microstructure are usually required in the field of micromechanics as well as in civil and biomechanical engineering, e.g. in the design of building foundations resting on granular materials, railway trucks laying on railway ballast and bone implants. The works on the problem of indentation of a couple stress elastic materials usually assume that the moment tractions exchanged between the rigid indenter and the halfplane are equal to zero, even if these interactions can in principle be transmitted along the contact region This choice is not necessarily justified from the physical point of view, but it is commonly assumed in order to recover the classical elastic solution as the characteristic length tends to zero. By using the results of the asymptotic analysis performed by Gourgiotis and Georgiadis (2011) for Mode I crack-tip fields, both the contact pressure and couple stress tractions are assumed to display a square root singularity at the edges of the contact region They are expanded in series of orthogonal Chebyshev polynomials of the first kind. The associated unilateral contact problem will be considered in a forthcoming paper
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