Signorini problem Information & Signorini problem Links at HealthHaven.com

The Signorini problem is an elastostatics problem in linear elasticity: it consists in finding the elastic equilibrium configuration of a 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 Signorini: the original name coined by him is problem with ambiguous boundary conditions.

[edit] History

The classical Signorini problem: what will be the equilibrium configuration of the orange spherically shaped elastic body resting on the blue rigid frictionless plane?

The problem was posed by Antonio Signorini during a course of lessons taught at the Istituto Nazionale di Alta Matematica in 1959, later published as the paper (Signorini 1959), expanding a previous short exposition he gave in a note published in 1933. As stated in (Signorini 1959, p. 128), he called it problem with ambiguous boundary conditions i.e. problema con ambigue condizioni al contorno in Italian, since there are two alternative sets of boundary conditions involving not only equalities but also inequalities the solution must satisfy on any given contact point, but it is not a priori known what of the two sets is satisfied for each point: he posed the question if the problem was or not well posed in a physical sense, i.e. if its solution exist and is unique or not, inviting young analysts to study the problem (Signorini 1959, p. 129). Gaetano Fichera and Mauro Picone attended the lessons and Fichera started to investigate the existence and uniqueness of the solutions: as he remembers in (Fichera 1995, p. 49), there were no references to a similar problem in the theory of boundary value problems, so he decided to deal with it by starting from the virtual work principle. While the problem was under investigation, Signorini began to suffer serious health problems: nevertheless, he desidered 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, who, being himself tied to Signorini by similar feelings, perceived the last months of 1962 as worrying days (Fichera 1995, p. 51). But 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. The preliminary note later published as (Fichera 1963) was written up and submitted to Signorini exactly a week before his death: He was very satisfied to see a positive answer to his question. A few days later, he told his family Doctor Damiano Aprile (Fichera 1995, p. 53):-"Il mio discepolo Fichera mi ha dato una grande soddisfazione (My disciple Fichera gave me a great contentment)."-"Ma Lei ne ha avute tante, Professore, durante la Sua vita (But you had many, Professor, during your life)"- replied Doctor Aprile, but also Signorini replied:-"Ma questa è la più grande (But this is the greatest one)"-. And those were his last words. According to Antman (1983, p. 282) the solution of the Signorini problem coincides with the birth of the field of variational inequalities.

[edit] Formal statement of the problem

The content of this section and of the included 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 (see Signorini 1959, p. 127). The problem consist in finding the displacement vector from the natural configuration $\scriptstyle\boldsymbol{u}(\boldsymbol{x})=\left(u_1(\boldsymbol{x}),u_2(\boldsymbol{x}),u_3(\boldsymbol{x})\right)$ of a anisotropic non-homogeneous elastic body that lies in a subset $A$ of the three-dimensional euclidean space whose boundary is $\scriptstyle\partial A$ and whose interior normal is the vector $n$ , resting on a rigid frictionless surface whose contact surface (or more generally contact set) is $Σ$ and subject only to its body forces $\scriptstyle\boldsymbol{f}(\boldsymbol{x})=\left(f_1(\boldsymbol{x}),f_2(\boldsymbol{x}),f_3(\boldsymbol{x})\right)$ , and surface forces $\scriptstyle\boldsymbol{g}(\boldsymbol{x})=\left(g_1(\boldsymbol{x}),g_2(\boldsymbol{x}),g_3(\boldsymbol{x})\right)$ applied on the free (i.e. not in contact with the rest surface) surface $\scriptstyle\partial A\setminus\Sigma$ : the set $A$ and the contact surface $Σ$ characterize the natural configuration of the body and are known a priori. Therefore the body has to satisfy the general equilibrium equations

(1) $\qquad\frac{\partial\sigma_{ik}}{\partial x_k}- f_i= 0\qquad\forall i=1,2,3$

written using the Einstein notation as all in the following development, the ordinary boundary conditions on $\scriptstyle\partial A\setminus\Sigma$

(2) $\qquad\sigma_{ik}n_k-g_i=0\qquad\forall i=1,2,3$

and the following two sets of boundary conditions on $Σ$ , where $\scriptstyle\boldsymbol{\sigma} = \boldsymbol{\sigma}(\boldsymbol{u})$ 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.

[edit] The ambiguous boundary conditions

If $\scriptstyle\boldsymbol{\tau}=(\tau_1,\tau_2,\tau_3)$ is any tangent vector to the contact set $Σ$ , then the ambiguous boundary condition in each point of this set are expressed by the following two systems of inequalities

(3) $\quad \left\{ \begin{array}{l} u_i n_i=0\\ \sigma_{ik} n_i n_k \geq 0\\ \sigma_{ik} n_i \tau_k = 0 \end{array} \right.$ or (4) $\left\{ \begin{array}{l} u_i n_i > 0\\ \sigma_{ik} n_i n_k = 0\\ \sigma_{ik} n_i \tau_k = 0 \end{array} \right.$

Let's analyze their meaning:

Each set of conditions const 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 $n$ .
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 $n$ ,
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 $τ$ tangent in the given point to the contact set $Σ$ .
The quantities at the first member of each of the three relation 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 $\scriptstyle +1$ and $\scriptstyle -1$ .

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

The above formulation is general since the 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.

[edit] 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 previus developments, the Einstein notation is adopted)

$W(\boldsymbol{\varepsilon})=a_{ikjh}(\boldsymbol{x})\varepsilon_{ik}\varepsilon_{ik}$

where

$\scriptstyle\boldsymbol{a}(\boldsymbol{x})=\left(a_{ikjh}(\boldsymbol{x})\right)$ is the elasticity tensor
$\scriptstyle\boldsymbol{\varepsilon}=\boldsymbol{\varepsilon}(\boldsymbol{u})=\left(\varepsilon_{ik}(\boldsymbol{u})\right)=\left(\frac{1}{2} \left( \frac{\partial u_i}{\partial x_k} + \frac{\partial u_k}{\partial x_i} \right)\right)$ is the infinitesimal strain tensor

The Cauchy stress tensor has therefore the following form

(5) $\sigma_{ik}= - \frac{\partial W}{\partial \varepsilon_{ik}} \qquad\forall i,k=1,2,3$

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

[edit] Solution of the problem

As for the section on the formal statement of the Signorini problem, the contents of this section and the included subsections follow closely the treatment of Gaetano Fichera in (Fichera 1963), (Fichera 1964b), (Fichera 1972) and also (Fichera 1995): obviously, the exposition focuses on the basics steps of the proof of the existence and uniqueness for the solution of problem (1), (2), (3), (4) and (5), rather than the technical details.

[edit] 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) $I(\boldsymbol{u})=\int_A W(\boldsymbol{x},\boldsymbol{\varepsilon})\mathrm{d}x - \int_A u_i f_i\mathrm{d}x - \int_{\partial A\setminus\Sigma}\!\!\!\!\! u_i g_i \mathrm{d}\sigma$

where $u$ belongs to the set of admissible displacements $\scriptstyle\mathcal{U}_\Sigma$ 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 ammissible displacement $u$ which minimize the integral $I (u)$ is a solution of the problem with ambiguous boundary conditions (1), (2), (3), (4) and (5), provided it is a $C 1$ function supported on the closure $\scriptstyle \bar A$ of the set $A$ : however Gaetano Fichera gave a class counterexamples in (Fichera 1964b, pp. 619-620) showing that in general ammissible displacements are not smooth functions of those 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 functonal in the neighborhoodof the sought minimizing admissible displacement $\scriptstyle\boldsymbol{u} \in \mathcal{U}_\Sigma$ , and then requires it to be greather or equal to zero

$\left. \frac{\mathrm{d}}{\mathrm{d}t} I( \boldsymbol{u} + t \boldsymbol{v}) \right\vert_{t=0} = -\int_A \sigma_{ik}(\boldsymbol{u})\varepsilon_{ik}(\boldsymbol{v})\mathrm{d}x - \int_A v_i f_i\mathrm{d}x - \int_{\partial A\setminus\Sigma}\!\!\!\!\! v_i g_i \mathrm{d}\sigma \geq 0 \qquad \forall \boldsymbol{v} \in \mathcal{U}_\Sigma$

Defining the following functionals

$B(\boldsymbol{u},\boldsymbol{v}) = -\int_A \sigma_{ik}(\boldsymbol{u})\varepsilon_{ik}(\boldsymbol{v})\mathrm{d}x \qquad \boldsymbol{u},\boldsymbol{v} \in \mathcal{U}_\Sigma$

and

$F(\boldsymbol{v}) = \int_A v_i f_i\mathrm{d}x + \int_{\partial A\setminus\Sigma}\!\!\!\!\! v_i g_i \mathrm{d}\sigma\qquad \boldsymbol{v} \in \mathcal{U}_\Sigma$

the preceding inequality is can be written as

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

This inequality is the variational inequality for the Signorini problem.

[edit] See also

[edit] References

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 34 (2): 138–142, Zbl 0128.18305 (in Italian). A short paper describing briefly the approach to 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 7 (2): 91–140, Zbl 0146.21204 (in Italian). The paper containing the existence and uniqueness theorem for the Signorini problem.
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.
Signorini, Antonio (1959), "Questioni di elasticità non linearizzata e semilinearizzata (Issues in non linear and semilinear elasticity)", Rendiconti di Matematica e delle sue applicazioni, 5 18: 95–139, Zbl 0091.38006 (in Italian).

[edit] Bibliography

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 714990, Zbl 0533.73001, http://www.ams.org/bull/1983-09-03/S0273-0979-1983-15185-6/home.html?pagingLink=%3Ca+href%3D%22%2Fjoursearch%2Fservlet%2FDoSearch%3Fco1%3Dand%26co2%3Dand%26co3%3Dand%26cperpage%3D50%26csort%3Dd%26endmo%3D00%26f1%3Dmsc%26f2%3Dtitle%26f3%3Danywhere%26f4%3Dauthor%26format%3Dstandard%26jrnl%3Done%26sendit22%3DSearch%26sperpage%3D30%26ssort%3Dd%26startmo%3D00%26timingString%3DQuery%2Btook%2B71%2Bmilliseconds.%26v4%3DAntman%26onejrnl%3Dbull%26startRec%3D1%22%3E .
Fichera, Gaetano (1972), "Boundary value problems of elasticity with unilateral constraints", in Flügge, Siegfried; Truesdell, Clifford A., Festkörpermechanik/Mechanics of Sol, Handbuch der Physik (Encyclopedia of Physics), VIa/2 (paperback 1984 ed.), Berlin-Heidelberg-New York: Springer-Verlag, pp. 391–424, Zbl 0277.73001, ISBN 3-540-13161-2, ISBN 0-387-13161-2 . 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 (The birth of the theory of variational inequalities remembered thirty years later)", Incontro scientifico italo-spagnolo. Roma, 21 ottobre 1993, Atti dei Convegni Lincei, 114, Roma: Accademia Nazionale dei Lincei, pp. 47–53 (in Italian).
Fichera, Gaetano (2002), Opere storiche biografiche, divulgative (Historical, biographical, divulgative works), Napoli: Giannini (distribuited by the Società Nazionale di Scienze, Lettere e Arti in Napoli), pp. 491, http://www.socnazsla.unina.it/italiano/fismat/pubbl.htm . A volume collecting works of Gaetano Fichera in the fields of history of mathematics and scientific divulgation.
Fichera, Gaetano (2004), Opere scelte (Selected works), Firenze: Edizioni Cremonese (distribuited by Unione Matematica Italiana), pp. XXIX+432 (vol. 1), pp. VI+570 (vol. 2), pp. VI+583 (vol. 3), ISBN 88-7083-811-0, ISBN 88-7083-812-9, ISBN 88-7083-813-7 . Three volumes collecting the most important mathematical papers of Gaetano Fichera, with a biographical sketch of Olga A. Oleinik.
Signorini, Antonio (1991), Opere scelte (Selected works), Firenze: Edizioni Cremonese (distribuited by Unione Matematica Italiana), pp. XXXI + 695 . A volume collecting the most important works of Antonio Signorini with an introduction and a commentary of Giuseppe Grioli.

[edit] External links

Barbu, V. (2001), "Signorini problem", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104, http://eom.springer.de/S/s110130.htm

Contents