Stress and displacement analysis of an auxetic quarter-plane under a concentrated force

The problem of a quarter-space under distributed normal and shear loads is considered.A mathematical model is formulated for the plane strain state.Theoretical background of the Mellin integral transform and calculation of residues is outlined.An analytical procedure involving the Mellin transform is presented for the general reduced problem of a quarter-plane.Numerical computation of residues allows for evaluation of the inverse transforms for the displacements and stresses.Simulation results are obtained for a special load case: a concentrated force.The deformation of the loaded boundary is analyzed for various values of Poisson's ratio.It turns out that auxetics exhibit locally negative stiffness, which leads to an anomalous behavior of the system.A simple explanation of the unusual deformation mechanism is proposed.The semianalytical solutions are compared with the results obtained by means of the finite element method.