The Potential Energy
where belongs to the set of admissible displacements i.e. the set of displacement vectors satisfying the system of boundary conditions (3) or (4). The meaning of each of the three terms is the following
- the first one is the total elastic potential energy of the elastic body
- the second one is the total potential energy due to the body forces, for example the gravitational force
- the third one is the potential energy due to surface forces, for example the forces exerted by the atmospheric pressure
Signorini (1959, pp. 129–133) was able to prove that the admissible displacement which minimize the integral is a solution of the problem with ambiguous boundary conditions (1), (2), (3), (4) and (5), provided it is a function supported on the closure of the set : however Gaetano Fichera gave a class of counterexamples in (Fichera 1964b, pp. 619–620) showing that in general, admissible displacements are not smooth functions of these class. Therefore Fichera tries to minimize the functional (6) in a wider function space: in doing so, he first calculates the first variation (or functional derivative) of the given functional in the neighbourhood of the sought minimizing admissible displacement, and then requires it to be greater than or equal to zero
Defining the following functionals
the preceding inequality is can be written as
This inequality is the variational inequality for the Signorini problem.
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