The ambiguous boundary conditions

If ${\displaystyle \scriptstyle {\boldsymbol {\tau }}=(\tau _{1},\tau _{2},\tau _{3})}$ is any tangent vector to the contact set ${\displaystyle \Sigma }$, then the ambiguous boundary condition in each point of this set are expressed by the following two systems of inequalities

(3)     ${\displaystyle \quad {\begin{cases}u_{i}n_{i}&=0\\\sigma _{ik}n_{i}n_{k}&\geq 0\\\sigma _{ik}n_{i}\tau _{k}&=0\end{cases}}}$     or     (4)     ${\displaystyle {\begin{cases}u_{i}n_{i}&>0\\\sigma _{ik}n_{i}n_{k}&=0\\\sigma _{ik}n_{i}\tau _{k}&=0\end{cases}}}$

Let's analyze their meaning:

• Each set of conditions consists of three relations, equalities or inequalities, and all the second members are the zero function.
• The quantities at first member of each first relation are proportional to the norm of the component of the displacement vector directed along the normal vector ${\displaystyle n}$.
• The quantities at first member of each second relation are proportional to the norm of the component of the tension vector directed along the normal vector ${\displaystyle n}$,
• The quantities at the first member of each third relation are proportional to the norm of the component of the tension vector along any vector ${\displaystyle \tau }$ tangent in the given point to the contact set ${\displaystyle \Sigma }$.
• The quantities at the first member of each of the three relations are positive if they have the same sense of the vector they are proportional to, while they are negative if not, therefore the constants of proportionality are respectively ${\displaystyle \scriptstyle +1}$ and ${\displaystyle \scriptstyle -1}$.

Knowing these facts, the set of conditions (3) applies to points of the boundary of the body which do not leave the contact set ${\displaystyle \Sigma }$ in the equilibrium configuration, since, according to the first relation, the displacement vector ${\displaystyle u}$ has no components directed as the normal vector ${\displaystyle n}$, while, according to the second relation, the tension vector may have a component directed as the normal vector ${\displaystyle n}$ and having the same sense. 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 ${\displaystyle u}$ has a component directed as the normal vector ${\displaystyle n}$, while the tension vector has no components directed as the normal vector ${\displaystyle n}$. 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 ${\displaystyle \Sigma }$, while its complement respect to ${\displaystyle \Sigma }$ is called the area of separation.

The above formulation is general since the Cauchy stress tensor i.e. the constitutive equation of the elastic body has not been made explicit: it is equally valid assuming the hypothesis of linear elasticity or the ones of nonlinear elasticity. However, as it would be clear from the following developments, the problem is inherently nonlinear, therefore assuming a linear stress tensor does not simplify the problem.

The form of the stress tensor in the formulation of Signorini and Fichera

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)

${\displaystyle W({\boldsymbol {\varepsilon }})=a_{ik,jh}({\boldsymbol {x}})\varepsilon _{ik}\varepsilon _{jh}}$

where

• ${\displaystyle \scriptstyle {\boldsymbol {a}}({\boldsymbol {x}})=\left(a_{ik,jh}({\boldsymbol {x}})\right)}$ is the elasticity tensor
• ${\displaystyle \scriptstyle {\boldsymbol {\varepsilon }}={\boldsymbol {\varepsilon }}({\boldsymbol {u}})=\left(\varepsilon _{ik}({\boldsymbol {u}})\right)=\left({\frac {1}{2}}\left({\frac {\partial u_{i}}{\partial x_{k}}}+{\frac {\partial u_{k}}{\partial x_{i}}}\right)\right)}$ is the infinitesimal strain tensor

The Cauchy stress tensor has therefore the following form

(5)     ${\displaystyle \sigma _{ik}=-{\frac {\partial W}{\partial \varepsilon _{ik}}}\qquad {\text{for }}i,k=1,2,3}$

and it is linear with respect to the components of the infinitesimal strain tensor; however, it is not homogeneous nor isotropic.

The potential energy

The first step of the analysis of Fichera as well as the first step of the analysis of Antonio Signorini in Signorini 1959 is the analysis of the potential energy, i.e. the following functional

(6)      ${\displaystyle I({\boldsymbol {u}})=\int _{A}W({\boldsymbol {x}},{\boldsymbol {\varepsilon }})\mathrm {d} x-\int _{A}u_{i}f_{i}\mathrm {d} x-\int _{\partial A\setminus \Sigma }u_{i}g_{i}\mathrm {d} \sigma }$

where ${\displaystyle u}$ belongs to the set of admissible displacements ${\displaystyle \scriptstyle {\mathcal {U}}_{\Sigma }}$ 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 ${\displaystyle u}$ which minimize the integral ${\displaystyle I(u)}$ is a solution of the problem with ambiguous boundary conditions (1), (2), (3), (4) and (5), provided it is a ${\displaystyle C^{1}}$ function supported on the closure ${\displaystyle \scriptstyle {\bar {A}}}$ of the set ${\displaystyle A}$: 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 ${\displaystyle \scriptstyle {\boldsymbol {u}}\in {\mathcal {U}}_{\Sigma }}$, and then requires it to be greater than or equal to zero

${\displaystyle \left.{\frac {\mathrm {d} }{\mathrm {d} t}}I({\boldsymbol {u}}+t{\boldsymbol {v}})\right\vert _{t=0}=-\int _{A}\sigma _{ik}({\boldsymbol {u}})\varepsilon _{ik}({\boldsymbol {v}})\mathrm {d} x-\int _{A}v_{i}f_{i}\mathrm {d} x-\int _{\partial A\setminus \Sigma }\!\!\!\!\!v_{i}g_{i}\mathrm {d} \sigma \geq 0\qquad \forall {\boldsymbol {v}}\in {\mathcal {U}}_{\Sigma }}$

Defining the following functionals

${\displaystyle B({\boldsymbol {u}},{\boldsymbol {v}})=-\int _{A}\sigma _{ik}({\boldsymbol {u}})\varepsilon _{ik}({\boldsymbol {v}})\mathrm {d} x\qquad {\boldsymbol {u}},{\boldsymbol {v}}\in {\mathcal {U}}_{\Sigma }}$

and

${\displaystyle F({\boldsymbol {v}})=\int _{A}v_{i}f_{i}\mathrm {d} x+\int _{\partial A\setminus \Sigma }\!\!\!\!\!v_{i}g_{i}\mathrm {d} \sigma \qquad {\boldsymbol {v}}\in {\mathcal {U}}_{\Sigma }}$

the preceding inequality is can be written as

(7)      ${\displaystyle B({\boldsymbol {u}},{\boldsymbol {v}})-F({\boldsymbol {v}})\geq 0\qquad \forall {\boldsymbol {v}}\in {\mathcal {U}}_{\Sigma }}$

This inequality is the variational inequality for the Signorini problem.

Historical references

• Antman, Stuart (1983), "The influence of elasticity in analysis: modern developments", Bulletin of the American Mathematical Society, 9 (3): 267–291, doi:10.1090/S0273-0979-1983-15185-6, MR 0714990, Zbl 0533.73001.
• Duvaut, Georges (1971), "Problèmes unilatéraux en mécanique des milieux continus" (PDF), Actes du Congrès international des mathématiciens, 1970, ICM Proceedings, vol. Mathématiques appliquées (E), Histoire et Enseignement (F) – Volume 3, Paris: Gauthier-Villars, pp. 71–78. A brief research survey describing the field of variational inequalities.
• Fichera, Gaetano (1972), "Boundary value problems of elasticity with unilateral constraints", in Flügge, Siegfried; Truesdell, Clifford A. (eds.), Festkörpermechanik/Mechanics of Solids, Handbuch der Physik (Encyclopedia of Physics), vol. VIa/2 (paperback 1984 ed.), Berlin–Heidelberg–New York: Springer-Verlag, pp. 391–424, ISBN 0-387-13161-2, Zbl 0277.73001. The encyclopedia entry about problems with unilateral constraints (the class of boundary value problems the Signorini problem belongs to) he wrote for the Handbuch der Physik on invitation by Clifford Truesdell.
• Fichera, Gaetano (1995), "La nascita della teoria delle disequazioni variazionali ricordata dopo trent'anni", Incontro scientifico italo-spagnolo. Roma, 21 ottobre 1993, Atti dei Convegni Lincei (in Italian), vol. 114, Roma: Accademia Nazionale dei Lincei, pp. 47–53. The birth of the theory of variational inequalities remembered thirty years later (English translation of the contribution title) is an historical paper describing the beginning of the theory of variational inequalities from the point of view of its founder.
• Fichera, Gaetano (2002), Opere storiche biografiche, divulgative [Historical, biographical, divulgative works] (in Italian), Napoli: Giannini, p. 491. A volume collecting almost all works of Gaetano Fichera in the fields of history of mathematics and scientific divulgation.
• Fichera, Gaetano (2004), Opere scelte [Selected works], Firenze: Edizioni Cremonese (distributed by Unione Matematica Italiana), pp. XXIX+432 (vol. 1), pp. VI+570 (vol. 2), pp. VI+583 (vol. 3), archived from the original on 2009-12-28, ISBN 88-7083-811-0 (vol. 1), ISBN 88-7083-812-9 (vol. 2), ISBN 88-7083-813-7 (vol. 3). Three volumes collecting Gaetano Fichera's most important mathematical papers, with a biographical sketch of Olga A. Oleinik.
• Signorini, Antonio (1991), Opere scelte [Selected works], Firenze: Edizioni Cremonese (distributed by Unione Matematica Italiana), pp. XXXI + 695, archived from the original on 2009-12-28. A volume collecting Antonio Signorini's most important works with an introduction and a commentary of Giuseppe Grioli.

Research works

• Andersson, John (2016), "Optimal regularity for the Signorini problem and its free boundary", Inventiones Mathematicae, 1 (1): 1–82, arXiv:1310.2511, Bibcode:2016InMat.204....1A, doi:10.1007/s00222-015-0608-6, MR 3480553, S2CID 118934322, Zbl 1339.35345.
• Fichera, Gaetano (1963), "Sul problema elastostatico di Signorini con ambigue condizioni al contorno" [On the elastostatic problem of Signorini with ambiguous boundary conditions], Rendiconti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, 8 (in Italian), 34 (2): 138–142, MR 0176661, Zbl 0128.18305. A short research note announcing and describing (without proofs) the solution of the Signorini problem.
• Fichera, Gaetano (1964a), "Problemi elastostatici con vincoli unilaterali: il problema di Signorini con ambigue condizioni al contorno" [Elastostatic problems with unilateral constraints: the Signorini problem with ambiguous boundary conditions], Memorie della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, 8 (in Italian), 7 (2): 91–140, Zbl 0146.21204. The first paper where aa existence and uniqueness theorem for the Signorini problem is proved.
• Fichera, Gaetano (1964b), "Elastostatic problems with unilateral constraints: the Signorini problem with ambiguous boundary conditions", Seminari dell'istituto Nazionale di Alta Matematica 1962–1963, Rome: Edizioni Cremonese, pp. 613–679. An English translation of the previous paper.
• Petrosyan, Arshak; Shahgholian, Henrik; Uraltseva, Nina (2012), Regularity of Free Boundaries in Obstacle-Type Problems, Graduate Studies in Mathematics, vol. 136, Providence, RI: American Mathematical Society, pp. x+221, ISBN 978-0-8218-8794-3, MR 2962060, Zbl 1254.35001.
• Signorini, Antonio (1959), "Questioni di elasticità non linearizzata e semilinearizzata" [Topics in non linear and semilinear elasticity], Rendiconti di Matematica e delle sue Applicazioni, 5 (in Italian), 18: 95–139, Zbl 0091.38006.