Compatibility (mechanics)

Compatibility is the study of the conditions under which such a displacement field can be guaranteed.

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.

is the infinitesimal strain tensor and For finite deformations the compatibility conditions take the form where

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.

In direct tensor notation where the curl operator can be expressed in an orthonormal coordinate system as

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

The same condition is also sufficient to ensure compatibility in a simply connected body.

represents the mixed components of the Riemann-Christoffel curvature tensor.

The problem of compatibility in continuum mechanics involves the determination of allowable single-valued continuous fields on simply connected bodies.

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

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 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

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 over a simply connected body are The compatibility problem for small strains can be stated as follows.

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 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.

Hence the compatibility conditions are sufficient to guarantee the existence of a unique displacement field

be a positive definite symmetric tensor field defined on the reference configuration.

does there exist a deformed configuration marked by the position field

In terms of components with respect to a rectangular Cartesian basis From finite strain theory we know 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

Therefore, We can show these are the mixed components of the Riemann-Christoffel curvature tensor.

is invertible and the constructed tensor field satisfies the expression for