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 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, 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. 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, he decided to attack it by starting from first principles, precisely 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. Finally, on the first days of January 1963, Fichera was able to give a complete proof of the existence and uniqueness of a solution for the problem with ambiguous boundary condition, which he called '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: and He was very satisfied to see a positive answer to his question. A few days later, during a conversation with his family Doctor Damiano Aprile, Signorini told him: According to Antman (1983, p. 282) the solution of the Signorini problem coincides with the birth of the field of variational inequalities. The content of this section and the following subsections follows closely the treatment of Gaetano Fichera in Fichera 1963, Fichera 1964b and also Fichera 1995: his derivation of the problem is different from Signorini's one in that he does not consider only incompressible bodies and a plane rest surface, as Signorini does. The problem consist in finding the displacement vector from the natural configuration u ( x ) = ( u 1 ( x ) , u 2 ( x ) , u 3 ( x ) ) {displaystyle scriptstyle {oldsymbol {u}}({oldsymbol {x}})=left(u_{1}({oldsymbol {x}}),u_{2}({oldsymbol {x}}),u_{3}({oldsymbol {x}}) ight)} of an anisotropic non-homogeneous elastic body that lies in a subset A {displaystyle A} of the three-dimensional euclidean space whose boundary is ∂ A {displaystyle scriptstyle partial A} and whose interior normal is the vector n {displaystyle n} , resting on a rigid frictionless surface whose contact surface (or more generally contact set) is Σ {displaystyle Sigma } and subject only to its body forces f ( x ) = ( f 1 ( x ) , f 2 ( x ) , f 3 ( x ) ) {displaystyle scriptstyle {oldsymbol {f}}({oldsymbol {x}})=left(f_{1}({oldsymbol {x}}),f_{2}({oldsymbol {x}}),f_{3}({oldsymbol {x}}) ight)} , and surface forces g ( x ) = ( g 1 ( x ) , g 2 ( x ) , g 3 ( x ) ) {displaystyle scriptstyle {oldsymbol {g}}({oldsymbol {x}})=left(g_{1}({oldsymbol {x}}),g_{2}({oldsymbol {x}}),g_{3}({oldsymbol {x}}) ight)} applied on the free (i.e. not in contact with the rest surface) surface ∂ A ∖ Σ {displaystyle scriptstyle partial Asetminus Sigma } : the set A {displaystyle A} and the contact surface Σ {displaystyle Sigma } 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 ∂ A ∖ Σ {displaystyle scriptstyle partial Asetminus Sigma } and the following two sets of boundary conditions on Σ {displaystyle Sigma } , where σ = σ ( u ) {displaystyle scriptstyle {oldsymbol {sigma }}={oldsymbol {sigma }}({oldsymbol {u}})} is the Cauchy stress tensor. 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 analized in the following development.