  The three-dimensional axisymmetric Boussinesq problem of an isotropic
half-space subjected to a concentrated normal quasi-static load is studied
within the framework of linear dipolar gradient elasticity. Our main concern is
to determine possible deviations from the predictions of classical linear
elastostatics when a more refined theory is employed to attack the problem. Of
special importance is the behavior of the new solution near to the point of
application of the load where pathological singularities exist in the classical
solution. The use of the theory of gradient elasticity is intended here to
model the response of materials with microstructure in a manner that the
classical theory cannot afford. A linear version of this theory results by
considering a linear isotropic expression for the strain-energy density that
depends on strain-gradient terms, in addition to the standard strain terms
appearing in classical elasticity. Through this formulation, a microstructural
material constant is introduced, in addition to the standard Lam\'e constants.
The solution method is based on integral transforms and is exact. The present
results show significant departure from the predictions of classical
elasticity. Indeed, continuous and bounded displacements are predicted at the
points of application of the concentrated load. Such a behavior of the
displacement field is, of course, more natural than the singular behavior
exhibited in the classical solution.
