Solutions by Potentials

Solutions by Potentials

Not too surprisingly since the field is steady state, the deviatoric vector field, like the isotropic scalar field, can be derived from potential functions. These poten-tials may, under certain conditions, be harmonic themselves or, more generally, can be related to harmonic functions. Most of the classic analytic solutions were obtained by inverse methods where one guesses a part of the solution to the field equations leaving enough freedom in terms of unspecified coefficients or func-tions so that both the differential equations and specific boundary conditions can be satisfied. However, a direct approach is possible for the linear fields of classic strength-of-materials or certain half-space situations, which is very important. This direct approach is the subject of the next pages.

For more general problems, various potential functions have been devel-oped that lead to solutions. Related to displacements are the scalar and vector potentials, the Galukin vectors, and the Papkovich-Neuber functions. How-ever, as we have noted, boundary conditions in this case are seldom simple. Potentials that generate solutions for stress fields are, in three dimensions, those of Maxwell and Morera and for two dimensions the Airy stress func-tion. Both Maxwell's and Morera's stress functions can be reduced to Prandtl's stress function for the torsion problem. Airy's stress function and the solution of plane problems will be presented in next pages. Other impor-tant solutions, which can be obtained by direct solution of the field equations using semi-inverse reasoning, will be given as we go along or in the home-work problems.