Compatible Strains

An exact solution to a problem in continuum mechanics requires that the components of strain be compatible with each other. If this condition is satisfied then there exists a displacement field corresponding to the strain field. The compatibility conditions are a set of differential equations relating the strain components. In the case of plane stress/strain, for example, where there are three strains and only two displacement components, there is a single equation of strain compatibility. If, as in assumed displacement finite elements, the strains are calculated from displacements then the strains are, by definition, compatible. Continuity of displacements across interelement boundaries leads to continuity in some, but not all components of the strain. In the plane stress/strain case, the direct strain parallel to the interface is continuous whilst the other two components are generally not continuous unless the FE model has recovered the exact solution.

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