elastostatics In physics and materials science, elasticity is the ability of a body to resist a distorting influence and to return to its original size and shape when that influence or force is removed. Solid objects will deform when adequate loads are ap ...

configuration Configuration or configurations may refer to: Computing * Computer configuration or system configuration * Configuration file, a software file used to configure the initial settings for a computer program * Configurator, also known as choice boar ...

anisotropic Anisotropy () is the property of a material which allows it to change or assume different properties in different directions, as opposed to isotropy. It can be defined as a difference, when measured along different axes, in a material's phys ...

elastic body In physics and materials science, elasticity is the ability of a body to resist a distorting influence and to return to its original size and shape when that influence or force is removed. Solid objects will deform when adequate loads are ap ...

friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding (motion), sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative la ...

Antonio Signorini Antonio Signorini may refer to: * Antonio Signorini (physicist) Antonio Signorini (2 April 1888 – 23 February 1963) was an influential Italian mathematical physicist and civil engineer of the 20th century.boundary conditions.

History

The problem was posed by

Antonio Signorini Antonio Signorini may refer to: * Antonio Signorini (physicist) Antonio Signorini (2 April 1888 – 23 February 1963) was an influential Italian mathematical physicist and civil engineer of the 20th century.Istituto Nazionale di Alta Matematica'' in 1959, later published as the article , expanding a previous short exposition he gave in a note published in 1933. 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 In mathematics, equality is a relationship between two quantities or, more generally two mathematical expressions, asserting that the quantities have the same value, or that the expressions represent the same mathematical object. The equality b ...

''but also

inequalities Inequality may refer to: Economics * Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy * Economic inequality, difference in economic well-being between population groups * ...

'', and ''it is not

a priori ("from the earlier") and ("from the later") are Latin phrases used in philosophy to distinguish types of knowledge, justification, or argument by their reliance on empirical evidence or experience. knowledge is independent from current ex ...

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 problem In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to ...

s, he decided to approach it by starting from first principles, specifically from the

virtual work principle D'Alembert's principle, also known as the Lagrange–d'Alembert principle, is a statement of the fundamental classical laws of motion. It is named after its discoverer, the French physicist and mathematician Jean le Rond d'Alembert. D'Alember ...

. 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 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 , 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. The solution of the Signorini problem coincides with the birth of the field of variational inequalities.

Formal statement of the problem

The content of this section and the following subsections follows closely the treatment of Gaetano Fichera in , and also : 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 ) for the original approach. The problem consist in finding the displacement vector from the natural configuration

\scriptstyle\boldsymbol(\boldsymbol)=\left(u_1(\boldsymbol),u_2(\boldsymbol),u_3(\boldsymbol)\right)

of an

anisotropic Anisotropy () is the property of a material which allows it to change or assume different properties in different directions, as opposed to isotropy. It can be defined as a difference, when measured along different axes, in a material's phys ...

non-homogeneous

elastic body In physics and materials science, elasticity is the ability of a body to resist a distorting influence and to return to its original size and shape when that influence or force is removed. Solid objects will deform when adequate loads are ap ...

that lies in a

subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...

A

of the three-

dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...

al

euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...

whose boundary is

\scriptstyle\partial A

and whose

interior normal In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at ...

is the

vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...

''

n

'', resting on a rigid frictionless surface whose contact surface (or more generally contact set) is

\Sigma

and subject only to its

body force In physics, a body force is a force that acts throughout the volume of a body. Springer site - Book 'Solid mechanics'preview paragraph 'Body forces'./ref> Forces due to gravity, electric fields and magnetic fields are examples of body forces. ...

s

\scriptstyle\boldsymbol(\boldsymbol)=\left(f_1(\boldsymbol),f_2(\boldsymbol),f_3(\boldsymbol)\right)

, and surface forces

\scriptstyle\boldsymbol(\boldsymbol)=\left(g_1(\boldsymbol),g_2(\boldsymbol),g_3(\boldsymbol)\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

\Sigma

characterize the natural configuration of the body and are known a priori. Therefore, the body has to satisfy the general equilibrium equations :

\qquad\frac- f_i= 0\qquad\text 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

:

\qquad\sigma_n_k-g_i=0\qquad\text i=1,2,3

and the following two sets of boundary conditions on

\Sigma

, where

\scriptstyle\boldsymbol = \boldsymbol(\boldsymbol)

is the

Cauchy stress tensor In continuum mechanics, the Cauchy stress tensor \boldsymbol\sigma, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components \sigma_ that completely ...

. 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.

The ambiguous boundary conditions

If ''

\scriptstyle\boldsymbol=(\tau_1,\tau_2,\tau_3)

'' is any tangent vector to the contact set

\Sigma

, then the ambiguous boundary condition in each point of this set are expressed by the following two systems of

inequalities Inequality may refer to: Economics * Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy * Economic inequality, difference in economic well-being between population groups * ...

:

\quad
\begin
u_i n_i & = 0 \\
\sigma_ n_i n_k & \geq 0\\
\sigma_ n_i \tau_k & = 0
\end

or

\begin
u_i n_i & > 0 \\
\sigma_ n_i n_k & = 0 \\
\sigma_ n_i \tau_k & = 0
\end

Let's analyze their meaning: *Each set of conditions consists of three

relations Relation or relations may refer to: General uses *International relations, the study of interconnection of politics, economics, and law on a global level *Interpersonal relationship, association or acquaintance between two or more people *Public ...

,

equalities In mathematics, equality is a relationship between two quantities or, more generally two mathematical expressions, asserting that the quantities have the same value, or that the expressions represent the same mathematical object. The equality b ...

or

inequalities Inequality may refer to: Economics * Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy * Economic inequality, difference in economic well-being between population groups * ...

, and all the second members are the zero function. *The quantities at first member of each first relation are

proportional Proportionality, proportion or proportional may refer to: Mathematics * Proportionality (mathematics), the property of two variables being in a multiplicative relation to a constant * Ratio, of one quantity to another, especially of a part compare ...

to the norm of the component of the displacement vector directed along the

normal vector In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve ...

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 In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve ...

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 Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...

\tau

tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...

in the given point to the contact set

\Sigma

. *The quantities at the first member of each of the three relations are positive if they have the same

sense A sense is a biological system used by an organism for sensation, the process of gathering information about the world through the detection of stimuli. (For example, in the human body, the brain which is part of the central nervous system rec ...

of the

vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...

they are

proportional Proportionality, proportion or proportional may refer to: Mathematics * Proportionality (mathematics), the property of two variables being in a multiplicative relation to a constant * Ratio, of one quantity to another, especially of a part compare ...

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 applies to points of the boundary of the body which ''do not'' leave the contact set

\Sigma

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

u

''has no components'' directed as the

normal vector In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve ...

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 A sense is a biological system used by an organism for sensation, the process of gathering information about the world through the detection of stimuli. (For example, in the human body, the brain which is part of the central nervous system rec ...

. In an analogous way, the set of conditions 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 A hypothesis (plural hypotheses) is a proposed explanation for a phenomenon. For a hypothesis to be a scientific hypothesis, the scientific method requires that one can testable, test it. Scientists generally base scientific hypotheses on prev ...

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 In physics and materials science, elasticity is the ability of a body to resist a distorting influence and to return to its original size and shape when that influence or force is removed. Solid objects will deform when adequate loads are ap ...

to penetrate into the surface where it rests: the ambiguity is not only in the unknown values non-

zero 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usu ...

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 or . The set of points where is satisfied is called the area of support of the elastic body on

\Sigma

, while its complement respect to

\Sigma

is called the area of separation. The above formulation is ''general'' since the

Cauchy stress tensor In continuum mechanics, the Cauchy stress tensor \boldsymbol\sigma, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components \sigma_ that completely ...

i.e. the

constitutive equation In physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities (especially kinetic quantities as related to kinematic quantities) that is specific to a material or substance, and appr ...

of the

elastic body In physics and materials science, elasticity is the ability of a body to resist a distorting influence and to return to its original size and shape when that influence or force is removed. Solid objects will deform when adequate loads are ap ...

has not been made explicit: it is equally valid assuming the

hypothesis A hypothesis (plural hypotheses) is a proposed explanation for a phenomenon. For a hypothesis to be a scientific hypothesis, the scientific method requires that one can testable, test it. Scientists generally base scientific hypotheses on prev ...

of linear elasticity or the ones of

nonlinear elasticity In continuum mechanics, the finite strain theory—also called large strain theory, or large deformation theory—deals with deformations in which strains and/or rotations are large enough to invalidate assumptions inherent in infinitesimal stra ...

. However, as it would be clear from the following developments, the problem is inherently

nonlinear In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...

, 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 Elastic energy is the mechanical potential energy stored in the configuration of a material or physical system as it is subjected to elastic deformation by work performed upon it. Elastic energy occurs when objects are impermanently compressed, ...

is the following one (as in the previous developments, the Einstein notation is adopted) :

W(\boldsymbol)=a_(\boldsymbol)\varepsilon_\varepsilon_

where *

\scriptstyle\boldsymbol(\boldsymbol)=\left(a_(\boldsymbol)\right)

is the elasticity tensor *

\scriptstyle\boldsymbol=\boldsymbol(\boldsymbol)=\left(\varepsilon_(\boldsymbol)\right)=\left(\frac \left( \frac + \frac \right)\right)

is the infinitesimal strain tensor The

Cauchy stress tensor In continuum mechanics, the Cauchy stress tensor \boldsymbol\sigma, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components \sigma_ that completely ...

has therefore the following form :

\sigma_= - \frac \qquad\text i,k=1,2,3

and it is ''

linear Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...

'' with respect to the components of the infinitesimal strain tensor; however, it is not

homogeneous Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...

nor isotropic.

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 , , and also : obviously, the exposition focuses on the basics steps of the proof of the existence and uniqueness for the solution of problem , , , and , rather than the technical details.

The potential energy

The first step of the analysis of Fichera as well as the first step of the analysis of

Antonio Signorini Antonio Signorini may refer to: * Antonio Signorini (physicist) Antonio Signorini (2 April 1888 – 23 February 1963) was an influential Italian mathematical physicist and civil engineer of the 20th century.functional :

I(\boldsymbol)=\int_A W(\boldsymbol,\boldsymbol)\mathrmx - \int_A u_i f_i\mathrmx - \int_u_i g_i \mathrm\sigma

where

u

belongs to the set of admissible displacements

\scriptstyle\mathcal_\Sigma

i.e. the set of displacement vectors satisfying the system of boundary conditions or . The meaning of each of the three terms is the following *the first one is the total

elastic potential energy Elastic energy is the mechanical potential energy stored in the configuration of a material or physical system as it is subjected to elastic deformation by work performed upon it. Elastic energy occurs when objects are impermanently compressed, ...

of the

elastic body In physics and materials science, elasticity is the ability of a body to resist a distorting influence and to return to its original size and shape when that influence or force is removed. Solid objects will deform when adequate loads are ap ...

*the second one is the total

potential energy In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potentia ...

due to the

body force In physics, a body force is a force that acts throughout the volume of a body. Springer site - Book 'Solid mechanics'preview paragraph 'Body forces'./ref> Forces due to gravity, electric fields and magnetic fields are examples of body forces. ...

s, for example the

gravitational force In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...

*the third one is the potential energy due to surface forces, for example the

force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a ...

s exerted by the

atmospheric pressure Atmospheric pressure, also known as barometric pressure (after the barometer), is the pressure within the atmosphere of Earth. The standard atmosphere (symbol: atm) is a unit of pressure defined as , which is equivalent to 1013.25 millibar ...

was able to prove that the admissible displacement

u

which minimize the integral

I(u)

is a solution of the problem with ambiguous boundary conditions , , , and , provided it is a

C^1

function supported on the closure

\scriptstyle \bar A

of the set

A

: however Gaetano Fichera gave a class of

counterexample A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "John Smith is not a lazy student" is ...

s in showing that in general, admissible displacements are not

smooth function In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...

s of these class. Therefore, Fichera tries to minimize the functional in a wider function space: in doing so, he first calculates the

first variation In applied mathematics and the calculus of variations, the first variation of a functional ''J''(''y'') is defined as the linear functional \delta J(y) mapping the function ''h'' to :\delta J(y,h) = \lim_ \frac = \left.\frac J(y + \varepsilon ...

(or

functional derivative In the calculus of variations, a field of mathematical analysis, the functional derivative (or variational derivative) relates a change in a functional (a functional in this sense is a function that acts on functions) to a change in a function on ...

) of the given functional in the

neighbourhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; American and British English spelling differences, see spelling differences) is a geographically localised community ...

of the sought minimizing admissible displacement

\scriptstyle\boldsymbol \in \mathcal_\Sigma

, and then requires it to be greater than or equal to

zero 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usu ...

:

\left. \frac I( \boldsymbol + t \boldsymbol) \right\vert_ = -\int_A \sigma_(\boldsymbol)\varepsilon_(\boldsymbol)\mathrmx - \int_A v_i f_i\mathrmx - \int_\!\!\!\!\! v_i g_i \mathrm\sigma \geq 0 \qquad \forall \boldsymbol \in \mathcal_\Sigma

Defining the following functionals :

B(\boldsymbol,\boldsymbol) = -\int_A \sigma_(\boldsymbol)\varepsilon_(\boldsymbol)\mathrmx \qquad \boldsymbol,\boldsymbol \in \mathcal_\Sigma

and :

F(\boldsymbol) = \int_A v_i f_i\mathrmx + \int_\!\!\!\!\! v_i g_i \mathrm\sigma\qquad \boldsymbol \in \mathcal_\Sigma

the preceding inequality is can be written as :

B(\boldsymbol,\boldsymbol) - F(\boldsymbol) \geq 0 \qquad \forall \boldsymbol \in \mathcal_\Sigma

This inequality is the variational inequality for the Signorini problem.

Notes

References

Historical references

*. *. A brief research survey describing the field of variational inequalities. *. The encyclopedia entry about problems with unilateral constraints (the class of

boundary value problem In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to ...

s the Signorini problem belongs to) he wrote for the ''Handbuch der Physik'' on invitation by

Clifford Truesdell Clifford Ambrose Truesdell III (February 18, 1919 – January 14, 2000) was an American mathematician, natural philosopher, and historian of science. Life Truesdell was born in Los Angeles, California. After high school, he spent two years in Eur ...

. *. ''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. * . A volume collecting almost all works of Gaetano Fichera in the fields of history of mathematics and scientific divulgation. * , (vol. 1), (vol. 2), (vol. 3). Three volumes collecting Gaetano Fichera's most important mathematical papers, with a biographical sketch of Olga A. Oleinik. * . A volume collecting Antonio Signorini's most important works with an introduction and a commentary of

Giuseppe Grioli Giuseppe is the Italian form of the given name Joseph, from Latin Iōsēphus from Ancient Greek Ἰωσήφ (Iōsḗph), from Hebrew יוסף. It is the most common name in Italy and is unique (97%) to it. The feminine form of the name is Giuse ...

.

Research works

*. *. A short research note announcing and describing (without proofs) the solution of the Signorini problem. *. The first paper where aa

existence Existence is the ability of an entity to interact with reality. In philosophy, it refers to the ontological property of being. Etymology The term ''existence'' comes from Old French ''existence'', from Medieval Latin ''existentia/exsistenti ...

and uniqueness theorem for the Signorini problem is proved. * . An English translation of the previous paper. *. *.

External links

*
Alessio Figalli, On global homogeneous solutions to the Signorini problem
{{DEFAULTSORT:Signorini Problem Calculus of variations Continuum mechanics Elasticity (physics) Partial differential equations

History

Formal statement of the problem

The ambiguous boundary conditions

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

Solution of the problem

The potential energy

See also

Notes

References

Historical references

Research works

External links