On Boussinesq's problem for Maxwell continua subject to an external gravity field

Boussinesq's problem is solved for a uniform and incompressible Maxwell half-space subject to an external gravity field. The solution is based on momentum equations which account for stress advection in the hydrostatically pre-stressed continuum during its deformation. The analysis shows that disregarding the pre-stress term renders the theoretical stress distribution incorrect and the deformation singular in the inviscid limit of the Maxwell continuum. Our solution is contrasted with a recently published alternative solution of the same problem, where regularity in the inviscid limit was forced by modified boundary conditions.