Signorini problem

The Signorini problem is an elastostatics problem in linear elasticity: it consists in finding the elastic equilibrium configuration of an anisotropic non-homogeneous elastic body, resting on a rigid frictionless surface and subject only to its mass forces.

The name was coined by Gaetano Fichera to honour his teacher, Antonio Signorini: the original name coined by him is problem with ambiguous boundary conditions.

The problem was posed by Antonio Signorini during a course taught at the Istituto Nazionale di Alta Matematica in 1959, later published as the article (Signorini 1959), expanding a previous short exposition he gave in a note published in 1933.

Signorini (1959, p. 128) himself called it problem with ambiguous boundary conditions,[2] since there are two alternative sets of boundary conditions the solution must satisfy on any given contact point.

The statement of the problem involves not only equalities but also inequalities, and it is not a priori known what of the two sets of boundary conditions is satisfied at each point.

Signorini asked to determine if the problem is well-posed or not in a physical sense, i.e. if its solution exists and is unique or not: he explicitly invited young analysts to study the problem.

[3] Gaetano Fichera and Mauro Picone attended the course, and Fichera started to investigate the problem: since he found no references to similar problems in the theory of boundary value problems,[4] he decided to approach it by starting from first principles, specifically from the virtual work principle.

During Fichera's researches on the problem, Signorini began to suffer serious health problems: nevertheless, he desired to know the answer to his question before his death.

Picone, being tied by a strong friendship with Signorini, began to chase Fichera to find a solution: Fichera himself, being tied as well to Signorini by similar feelings, perceived the last months of 1962 as worrying days.

[5] Finally, on the first days of January 1963, Fichera was able to give a complete proof of the existence of a unique solution for the problem with ambiguous boundary condition, which he called the "Signorini problem" to honour his teacher.

A preliminary research announcement, later published as (Fichera 1963), was written up and submitted to Signorini exactly a week before his death.

Signorini expressed great satisfaction to see a solution to his question.

A few days later, Signorini had with his family Doctor, Damiano Aprile, the conversation quoted above.

[6] The solution of the Signorini problem coincides with the birth of the field of variational inequalities.

[8] The problem consists in finding the displacement vector from the natural configuration

of an anisotropic non-homogeneous elastic body that lies in a subset

characterize the natural configuration of the body and are known a priori.

Therefore, the body has to satisfy the general equilibrium equations written using the Einstein notation as all in the following development, the ordinary boundary conditions on

Obviously, the body forces and surface forces cannot be given in arbitrary way but they must satisfy a condition in order for the body to reach an equilibrium configuration: this condition will be deduced and analyzed in the following development.

, then the ambiguous boundary condition in each point of this set are expressed by the following two systems of inequalities Let's analyze their meaning: Knowing these facts, the set of conditions (3) applies to points of the boundary of the body which do not leave the contact set

in the equilibrium configuration, since, according to the first relation, the displacement vector

In an analogous way, the set of conditions (4) applies to points of the boundary of the body which leave that set in the equilibrium configuration, since displacement vector

For both sets of conditions, the tension vector has no tangent component to the contact set, according to the hypothesis that the body rests on a rigid frictionless surface.

Each system expresses a unilateral constraint, in the sense that they express the physical impossibility of the elastic body to penetrate into the surface where it rests: the ambiguity is not only in the unknown values non-zero quantities must satisfy on the contact set but also in the fact that it is not a priori known if a point belonging to that set satisfies the system of boundary conditions (3) or (4).

The set of points where (3) is satisfied is called the area of support of the elastic body on

The form assumed by Signorini and Fichera for the elastic potential energy is the following one (as in the previous developments, the Einstein notation is adopted) where The Cauchy stress tensor has therefore the following form and it is linear with respect to the components of the infinitesimal strain tensor; however, it is not homogeneous nor isotropic.

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 Signorini (1959, pp.

is a solution of the problem with ambiguous boundary conditions (1), (2), (3), (4) and (5), provided it is a

619–620) showing that in general, admissible displacements are not smooth functions of these class.