  Atomistic simulations are used to test the equations of continuum contact
mechanics in nanometer scale contacts. Nominally spherical tips, made by
bending crystals or cutting crystalline or amorphous solids, are pressed into a
flat, elastic substrate. The normal displacement, contact radius, stress
distribution, friction and lateral stiffness are examined as a function of load
and adhesion. The atomic scale roughness present on any tip made of discrete
atoms is shown to have profound effects on the results. Contact areas, local
stresses, and the work of adhesion change by factors of two to four, and the
friction and lateral stiffness vary by orders of magnitude. The microscopic
factors responsible for these changes are discussed. The results are also used
to test methods for analyzing experimental data with continuum theory to
determine information, such as contact area, that can not be measured directly
in nanometer scale contacts. Even when the data appear to be fit by continuum
theory, extracted quantities can differ substantially from their true values.
