Compatibility (mechanics)
Encyclopedia
In continuum mechanics
, a compatible deformation (or strain) tensor field in a body is that unique field that is obtained when the body is subjected to a 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 conditions and were first derived for linear elasticity
by Barré de Saint-Venant
in 1864 and proved rigorously by Beltrami
in 1886.
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.
In the context of infinitesimal strain theory, these conditions are equivalent to stating that the displacements in a body can be obtained by integrating the strains. Such an integration is possible if the Saint-Venant's tensor (or incompatibility tensor)
vanishes in a simply-connected body where 
is the infinitesimal strain tensor and
For finite deformations the compatibility conditions take the form
where
is the deformation gradient.
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.
Combining these relations 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.,
.
three more equations of the form
Therefore there are six different compatibility conditions. We can write these conditions in index notation as
where
is the permutation symbol. In direct tensor notation
where the curl operator can be expressed in a orthonormal coordinate system as
.
The second-order tensor
is known as the incompatibility tensor.
where
is the deformation gradient. 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). The same condition is also sufficient to ensure compatibility in a simply connected body.
where
is the Christoffel symbol of the second kind
. The quantity
represents the mixed components of the Riemann-Christoffel curvature tensor.
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
exists and satisfies
. Then
Since changing the order of differentiation does not affect the result we have
Hence
From the well known identify for the curl of a tensor
we get the necessary condition
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, 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.
Then the necessary and sufficient conditions for the existence of a compatible
field over a simply connected body are
Given a symmetric second order tensor field
when is it possible to construct a vector field 
such that
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 
.
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.
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
exists that satisfies condition (1). In terms of components with respect to a rectangular Cartesian basis
From finite strain theory 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
From the relation between of
and 
that 
, we have
Then, from the relation
we have
From finite strain theory 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. Therefore the necessary conditions for
-compatibility are that the Riemann-Christoffel curvature of the deformation is zero.
We have to show that there exist
and 
such that
From a theorem by T.Y.Thomas 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 
.
Continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and the mechanical behavior of materials modelled as a continuous mass rather than as discrete particles...
, a compatible deformation (or strain) tensor field in a body is that unique field that is obtained when the body is subjected to a continuous
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
, single-valued, displacement field
Displacement field (mechanics)
A displacement field is an assignment of displacement vectors for all points in a region or body that is displaced from one state to another. A displacement vector specifies the position of a point or a particle in reference to an origin or to a previous position...
. Compatibility is the study of the conditions under which such a displacement field can be guaranteed. Compatibility conditions are particular cases of integrability conditions and were first derived for linear elasticity
Linear elasticity
Linear elasticity is the mathematical study of how solid objects deform and become internally stressed due to prescribed loading conditions. Linear elasticity models materials as continua. Linear elasticity is a simplification of the more general nonlinear theory of elasticity and is a branch of...
by Barré de Saint-Venant
Adhémar Jean Claude Barré de Saint-Venant
Adhémar Jean Claude Barré de Saint-Venant was a mechanician and mathematician who contributed to early stress analysis and also developed the one-dimensional unsteady open channel flow shallow water equations or Saint-Venant equations that are a fundamental set of equations used in modern...
in 1864 and proved rigorously by Beltrami
Eugenio Beltrami
Eugenio Beltrami was an Italian mathematician notable for his work concerning differential geometry and mathematical physics...
in 1886.
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.
In the context of infinitesimal strain theory, these conditions are equivalent to stating that the displacements in a body can be obtained by integrating the strains. Such an integration is possible if the Saint-Venant's tensor (or incompatibility tensor)
vanishes in a simply-connected body where is the infinitesimal strain tensor andFor finite deformations the compatibility conditions take the form
where
is the deformation gradient.Compatibility conditions for infinitesimal strains
The compatibility conditions in linear elasticityLinear elasticity
Linear elasticity is the mathematical study of how solid objects deform and become internally stressed due to prescribed loading conditions. Linear elasticity models materials as continua. Linear elasticity is a simplification of the more general nonlinear theory of elasticity and is a branch of...
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 problems the strain-displacement relations areCombining these relations 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 arethree more equations of the form
Therefore there are six different compatibility conditions. We can write these conditions in index notation as
where
is the permutation symbol. In direct tensor notationwhere the curl operator can be expressed in a orthonormal coordinate system as
.The second-order tensor
is known as the incompatibility tensor.
Compatibility conditions for finite strains
For solids in which the deformations are not required to be small, the compatibility conditions take the formwhere
is the deformation gradient. In terms of components with respect to a Cartesian coordinate system we can write these compatibility relations asThis condition is necessary if the deformation is to be continuous and derived from the mapping
(see Finite strain theory). 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 aswhere
is the Christoffel symbol of the second kindCurvilinear coordinates
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 invertible at each point. This means that one can convert a point given...
. The quantity
represents the mixed components of the Riemann-Christoffel curvature tensor.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 .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 byAlso
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 satisfiesNecessary conditions
For the necessary conditions we assume that the field exists and satisfies. ThenSince changing the order of differentiation does not affect the result we have
Hence
From the well known identify for the curl of a tensor
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 in the design of algorithms for numerical...
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, the integral of a second order tensor along a closed path is given by
Using the assumption that the curl of
is zero, we getHence 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 thatThen the necessary and sufficient conditions for the existence of a compatible
field over a simply connected body areCompatibility 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 thatNecessary conditions
Suppose that there exists such that the expression for holds. Nowwhere
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 gradientFrom Stokes' theorem and using the relations
we haveHence 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 thatNecessary conditions
Suppose that a field exists that satisfies condition (1). In terms of components with respect to a rectangular Cartesian basisFrom finite strain theory we know that
. Hence we can writeFor two symmetric second-order tensor field that are mapped one-to-one we also have the relation
From the relation between of
and that , we haveThen, from the relation
we have
From finite strain theory 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. 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. We start with the assumption thatWe have to show that there exist
and such thatFrom a theorem by T.Y.Thomas we know that the system of equations
has unique solutions
over simply connected domains ifThe 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 thatWe 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
- Linear elasticityLinear elasticityLinear elasticity is the mathematical study of how solid objects deform and become internally stressed due to prescribed loading conditions. Linear elasticity models materials as continua. Linear elasticity is a simplification of the more general nonlinear theory of elasticity and is a branch of...
- Deformation (mechanics)Deformation (mechanics)Deformation in continuum mechanics is the 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...
- Infinitesimal strain theory
- Finite strain theory
- Tensor derivative (continuum mechanics)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 in the design of algorithms for numerical...
- Curvilinear coordinatesCurvilinear coordinatesCurvilinear 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 invertible at each point. This means that one can convert a point given...