Compatibility (mechanics)
Introduction In continuum mechanics, a '''compatible''' deformation (or strain) '''tensor field''' in a body is that ''unique'' tensor 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.
Compatibility conditions for infinitesimal strains The compatibility conditions in linear elasticity 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 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 arethree more equations of the form:Therefore, there are 34=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:where  is the permutation symbol. 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 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.=== 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.
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.Figure 1.Motion of a continuum body.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===Figure 2.Integration paths used in proving the sufficiency conditions for compatibility.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 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 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.===Sufficient conditions===The proof of sufficiency is a bit more involved. We start with the assumption that: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 .
See also * Saint-Venant's compatibility condition* Linear elasticity* Deformation (mechanics)* Infinitesimal strain theory* Finite strain theory* Tensor derivative (continuum mechanics)* Curvilinear coordinates
References
External links * Amit Acharya's notes on compatibility on iMechanica* Plasticity by J. Lubliner, sec. 1.2.4 p. 35