In
continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such m ...
, a compatible
deformation
Deformation can refer to:
* Deformation (engineering), changes in an object's shape or form due to the application of a force or forces.
** Deformation (physics), such changes considered and analyzed as displacements of continuum bodies.
* Defor ...
(or
strain
Strain may refer to:
Science and technology
* Strain (biology), variants of plants, viruses or bacteria; or an inbred animal used for experimental purposes
* Strain (chemistry), a chemical stress of a molecule
* Strain (injury), an injury to a mu ...
)
tensor field
In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis ...
in a body is that ''unique'' tensor field that is obtained when the body is subjected to a
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
,
single-valued,
displacement field. Compatibility is the study of the conditions under which such a displacement field can be guaranteed. Compatibility conditions are particular cases of
integrability condition In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms. The idea is to take advantage of the ...
s and were first derived for
linear elasticity
Linear elasticity is a mathematical model of how solid objects deform and become internally stressed due to prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mec ...
by
Barré de Saint-Venant in 1864 and proved rigorously by
Beltrami in 1886.
[C Amrouche, PG Ciarlet, L Gratie, S Kesavan, On Saint Venant's compatibility conditions and Poincaré's lemma, C. R. Acad. Sci. Paris, Ser. I, 342 (2006), 887-891. ]
In the continuum description of a solid body we imagine the body to be composed of a set of infinitesimal volumes or material points. Each volume is assumed to be connected to its neighbors without any gaps or overlaps. Certain mathematical conditions have to be satisfied to ensure that gaps/overlaps do not develop when a continuum body is deformed. A body that deforms without developing any gaps/overlaps is called a compatible body. Compatibility conditions are mathematical conditions that determine whether a particular deformation will leave a body in a compatible state.
[Barber, J. R., 2002, Elasticity - 2nd Ed., Kluwer Academic Publications.]
In the context of
infinitesimal strain theory
In continuum mechanics, the infinitesimal strain theory is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller (indeed, infinitesimal ...
, these conditions are equivalent to stating that the displacements in a body can be obtained by integrating the
strain
Strain may refer to:
Science and technology
* Strain (biology), variants of plants, viruses or bacteria; or an inbred animal used for experimental purposes
* Strain (chemistry), a chemical stress of a molecule
* Strain (injury), an injury to a mu ...
s. Such an integration is possible if the Saint-Venant's tensor (or incompatibility tensor)
vanishes in a
simply-connected body[N.I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity. Leyden: Noordhoff Intern. Publ., 1975.] where
is the
infinitesimal strain tensor and
:
For
finite deformations the compatibility conditions take the form
:
where
is the
deformation gradient
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 strai ...
.
Compatibility conditions for infinitesimal strains
The compatibility conditions in
linear elasticity
Linear elasticity is a mathematical model of how solid objects deform and become internally stressed due to prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mec ...
are obtained by observing that there are six strain-displacement relations that are functions of only three unknown displacements. This suggests that the three displacements may be removed from the system of equations without loss of information. The resulting expressions in terms of only the strains provide constraints on the possible forms of a strain field.
2-dimensions
For two-dimensional,
plane strain
In continuum mechanics, the infinitesimal strain theory is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller (indeed, infinitesimal ...
problems the strain-displacement relations are
:
Repeated differentiation of these relations, in order to remove the displacements
and
, gives us the two-dimensional compatibility condition for strains
:
The only displacement field that is allowed by a compatible plane strain field is a plane displacement field, i.e.,
.
3-dimensions
In three dimensions, in addition to two more equations of the form seen for two dimensions, there are
three more equations of the form
:
Therefore, there are 3
4=81 partial differential equations, however due to symmetry conditions, this number reduces to six different compatibility conditions. We can write these conditions in index notation as
[Slaughter, W. S., 2003, ''The linearized theory of elasticity'', Birkhauser]
:
where
is the
permutation symbol
In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol or Levi-Civita epsilon represents a collection of numbers; defined from the sign of a permutation of the natural numbers , for s ...
. In direct tensor notation
:
where the curl operator can be expressed in an orthonormal coordinate system as
.
The second-order tensor
:
is known as the incompatibility tensor, and is equivalent to the
Saint-Venant compatibility tensor
Compatibility conditions for finite strains
For solids in which the deformations are not required to be small, the compatibility conditions take the form
:
where
is the
deformation gradient
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 strai ...
. In terms of components with respect to a Cartesian coordinate system we can write these compatibility relations as
:
This condition is necessary if the deformation is to be continuous and derived from the mapping
(see
Finite strain theory
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 strai ...
). The same condition is also sufficient to ensure compatibility in a
simply connected body.
Compatibility condition for the right Cauchy-Green deformation tensor
The compatibility condition for the
right Cauchy-Green deformation tensor can be expressed as
:
where
is the
Christoffel symbol of the second kind. The quantity
represents the mixed components of the
Riemann-Christoffel curvature tensor
In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. I ...
.
The general compatibility problem
The problem of compatibility in continuum mechanics involves the determination of allowable single-valued continuous fields on simply connected bodies. More precisely, the problem may be stated in the following manner.
[Acharya, A., 1999, '' On Compatibility Conditions for the Left Cauchy–Green Deformation Field in Three Dimensions'', Journal of Elasticity, Volume 56, Number 2 , 95-105]
Consider the deformation of a body shown in Figure 1. If we express all vectors in terms of the reference coordinate system
, the displacement of a point in the body is given by
:
Also
:
What conditions on a given second-order tensor field
on a body are necessary and sufficient so that there exists a unique vector field
that satisfies
:
Necessary conditions
For the necessary conditions we assume that the field
exists and satisfies
. Then
:
Since changing the order of differentiation does not affect the result we have
:
Hence
:
From the well known identity for the
curl of a tensor we get the necessary condition
:
Sufficient conditions
To prove that this condition is sufficient to guarantee existence of a compatible second-order tensor field, we start with the assumption that a field
exists such that
. We will integrate this field to find the vector field
along a line between points
and
(see Figure 2), i.e.,
:
If the vector field
is to be single-valued then the value of the integral should be independent of the path taken to go from
to
.
From
Stokes' theorem
Stokes's theorem, also known as the Kelvin–Stokes theorem Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)" :ja:培風館, Bai-Fu-Kan( ...
, the integral of a second order tensor along a closed path is given by
:
Using the assumption that the curl of
is zero, we get
:
Hence the integral is path independent and the compatibility condition is sufficient to ensure a unique
field, provided that the body is simply connected.
Compatibility of the deformation gradient
The compatibility condition for the deformation gradient is obtained directly from the above proof by observing that
:
Then the necessary and sufficient conditions for the existence of a compatible
field over a simply connected body are
:
Compatibility of infinitesimal strains
The compatibility problem for small strains can be stated as follows.
Given a symmetric second order tensor field
when is it possible to construct a vector field
such that
:
Necessary conditions
Suppose that there exists
such that the expression for
holds. Now
:
where
:
Therefore, in index notation,
:
If
is continuously differentiable we have
. Hence,
:
In direct tensor notation
:
The above are necessary conditions. If
is the
infinitesimal rotation vector then
. Hence the necessary condition may also be written as
.
Sufficient conditions
Let us now assume that the condition
is satisfied in a portion of a body. Is this condition sufficient to guarantee the existence of a continuous, single-valued displacement field
?
The first step in the process is to show that this condition implies that the
infinitesimal rotation tensor is uniquely defined. To do that we integrate
along the path
to
, i.e.,
:
Note that we need to know a reference
to fix the rigid body rotation. The field
is uniquely determined only if the contour integral along a closed contour between
and
is zero, i.e.,
:
But from Stokes' theorem for a simply-connected body and the necessary condition for compatibility
:
Therefore, the field
is uniquely defined which implies that the infinitesimal rotation tensor
is also uniquely defined, provided the body is simply connected.
In the next step of the process we will consider the uniqueness of the displacement field
. As before we integrate the displacement gradient
:
From Stokes' theorem and using the relations
we have
:
Hence the displacement field
is also determined uniquely. Hence the compatibility conditions are sufficient to guarantee the existence of a unique displacement field
in a simply-connected body.
Compatibility for Right Cauchy-Green Deformation field
The compatibility problem for the Right Cauchy-Green deformation field can be posed as follows.
Problem: Let
be a positive definite symmetric tensor field defined on the reference configuration. Under what conditions on
does there exist a deformed configuration marked by the position field
such that
:
Necessary conditions
Suppose that a field
exists that satisfies condition (1). In terms of components with respect to a rectangular Cartesian basis
:
From
finite strain theory
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 strai ...
we know that
. Hence we can write
:
For two symmetric second-order tensor field that are mapped one-to-one we also have the
relation
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 ...
:
From the relation between of
and
that
, we have
:
Then, from the relation
:
we have
:
From
finite strain theory
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 strai ...
we also have
:
Therefore,
:
and we have
:
Again, using the commutative nature of the order of differentiation, we have
:
or
:
After collecting terms we get
:
From the definition of
we observe that it is invertible and hence cannot be zero. Therefore,
:
We can show these are the mixed components of the
Riemann-Christoffel curvature tensor
In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. I ...
. Therefore, the necessary conditions for
-compatibility are that the Riemann-Christoffel curvature of the deformation is zero.
Sufficient conditions
The proof of sufficiency is a bit more involved.
[Blume, J. A., 1989, "Compatibility conditions for a left Cauchy-Green strain field", J. Elasticity, v. 21, p. 271-308.] We start with the assumption that
:
We have to show that there exist
and
such that
:
From a theorem by T.Y.Thomas
[Thomas, T. Y., 1934, "Systems of total differential equations defined over simply connected domains", Annals of Mathematics, 35(4), p. 930-734] we know that the system of equations
:
has unique solutions
over simply connected domains if
:
The first of these is true from the defining of
and the second is assumed. Hence the assumed condition gives us a unique
that is
continuous.
Next consider the system of equations
:
Since
is
and the body is simply connected there exists some solution
to the above equations. We can show that the
also satisfy the property that
:
We can also show that the relation
:
implies that
:
If we associate these quantities with tensor fields we can show that
is invertible and the constructed tensor field satisfies the expression for
.
See also
*
Saint-Venant's compatibility condition In the mathematical theory of elasticity (mathematics), elasticity, Saint-Venant's compatibility condition defines the relationship between the Deformation (mechanics), strain \varepsilon and a displacement field (mechanics), displacement field \ u ...
*
Linear elasticity
Linear elasticity is a mathematical model of how solid objects deform and become internally stressed due to prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mec ...
*
Deformation (mechanics)
In physics, deformation is the continuum mechanics transformation of a body from a ''reference'' configuration to a ''current'' configuration. A configuration is a set containing the positions of all particles of the body.
A deformation can ...
*
Infinitesimal strain theory
In continuum mechanics, the infinitesimal strain theory is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller (indeed, infinitesimal ...
*
Finite strain theory
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 strai ...
*
Tensor derivative (continuum mechanics)
The derivatives of scalars, vectors, and second-order tensors with respect to second-order tensors are of considerable use in continuum mechanics. These derivatives are used in the theories of nonlinear elasticity and plasticity, particularly i ...
*
Curvilinear coordinates
In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally inve ...
References
External links
Amit Acharya's notes on compatibility on iMechanicaPlasticity by J. Lubliner, sec. 1.2.4 p. 35
Continuum mechanics
Elasticity (physics)