Dislocation mechanics simulations of the bilinear behavior in micro- and nanoindentation

The hardnesses of electropolished, polycrystalline alpha-brass, and aluminum were measured as functions of load using Vickers microindentation, and Berkovich nanoindentation. Data were compared with literature data for silver, copper, and tungsten. In all cases, the hardness was observed to increase with decreasing size. Theories in the literature based on strain gradient plasticity and the addition of statistically stored and geometrically stored dislocation densities predict that the square of the hardness should increase linearly with inverse size of the indent. Our data and the literature data agree with this prediction over a limited range of indent diameters represented by microhardness and deep nanohardness data, whereas for the shallow nanohardness data, a second linear behavior is observed. The linear behavior of the microhardness and deep nanohardness data together with the second linear behavior generated by the shallow nanohardness data constituted what we designate a bilinear behavior. An algorithm is developed to calculate the induced shear stresses in a circular Volterra dislocation loop from a line integral of the Peach-Koehler equation using dislocation mechanics and isotropic elasticity. The computed induced shear stresses when plotted versus depth of indentation produced curves that exhibited a bilinear behavior identical to the bilinear behavior resulted from experiments, and the curves collapsed to a single curve for the different materials simulated.