Half‐space contact problem considering strain gradient and surface effects: An analytical approach

Abstract

AbstractIn this article, an analytical approach is proposed for the problem of an elastic half‐space under symmetrically distributed normal force with arbitrary profile including uniform pressure, Boussinesq flat‐ended punch, conical punch and Hertz's spherical punch. The formulation of the work is on the basis of Mindlin's first strain gradient theory with five material length scale parameters. Also, the surface effects are taken into account using the Gurtin‐Murdoch surface elasticity theory. The developed mathematical formulation is general so that it can be simply reduced to different strain gradient‐based theories such as the modified versions of strain gradient and modified couple stress theories (MSGT and MCST), the simplified strain gradient theory (A‐SGT), as well as the classical theory (CT). In the solution procedure, an analytical method is applied using the Fourier and Hankel transforms. In the numerical results, the effects of material length scale parameters and kind of punch on the surface displacements and stresses of the half‐space are analyzed. The surface influences are also comprehensively studied. Moreover, comparisons are made between the predictions of MSGT, MCST, A‐SGT and CT. This work shows that the material length scale parameters of Mindlin's first SGT play important roles in the response of half‐space. Also, the presented results provide a comparison between the intensity of effects of strain gradient parameters and surface parameters.