On concentrated surface loads and Green's functions in the Toupin–Mindlin theory of strain-gradient elasticity

Abstract

The two-dimensional Green's functions are derived for the half-plane in the context of the complete Toupin–Mindlin theory of isotropic strain-gradient elasticity. Two types of Green's functions exist for a concentrated force and a concentrated force dipole acting upon the surface of a traction-free half-plane. Our purpose is to examine the possible deviations from the predictions of classical theory of elasticity as well as from the simplified strain-gradient theory, which is frequently utilized in the last decade for the solution of boundary value problems. Of special importance is the behavior of the new solutions near to the point of application of the loads where pathological singularities and discontinuities exist in the classical solutions. The boundary value problems are attacked with the aid of the Fourier transform and exact full-field solutions are provided. Our results indicate that in all cases the displacement field is bounded and continuous at the point of application of the concentrated loads. The new solutions show therefore a more natural material response. For the concentrated force problem, both displacements and strains are found to be bounded, whereas the strain-gradients exhibit a logarithmic singularity. Thus, in marked contrast with the classical elasticity solution, a finite strain energy is contained within any finite portion of the body. On the other hand, in the case of the concentrated dipole force, the strains are logarithmically singular and the strain gradients exhibit a Cauchy type singularity. The nature of the boundary conditions in strain-gradient elasticity is highlighted through the solution of the pertinent boundary value problems. Finally, based on our analytical solution, the role of edge forces in strain-gradient elasticity is elucidated employing simple equilibrium considerations.