Not too surprisingly since the field is steady state, the deviatoric vector field, like the isotropic scalar field, can be derived from potential functions.

**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.

Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail

Civil : Principles of Solid Mechanics : Strategies for Elastic Analysis and Design : Solutions by Potentials |

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