Certain issues in the analytical integration of the Boussinesq problem (Fracture and Structural Integrity, 2025)
The Boussinesq solution, one of the fundamental problems in the theory of elasticity, enables an analysis of stresses and strains (displacements) in a semi-elastic space subject to surface loads. This solution has a form of formulas for displacements evoked by a concentrated force; these formulas can be treated as Green functions for calculation of displacements (and then – stresses) in a half-space loaded in any way at its surface z = 0. The study presents difficulties met during the analytical integration of the Green functions in the Mathematica environment as well as methods of coping with these difficulties. The authors are going to present particular issues which can be quite surprising and confusing, for example a failure to obtain a close result for definite integrals in Wolfram Mathematica or differences between results of calculations of the sum of integrals and the integral of the sums. The results of the study can help in establishing more exact benchmarks for the numerical methods applied in the analysis of settlement under foundations as well as other contact issues of the theory of elasticity based on the Boussinesq solution.
Keywords: Boussinesq solution; Green functions; numerical integration; symbolic integration