Compatibility conditions

Remarks

Necessity is the symmetry of second partial derivatives of $\varphi$. For sufficiency, the condition says that each row $(F_{i1}, F_{i2}, F_{i3})$ has zero curl. On a simply connected set such a field is the gradient of a function $\varphi_i$, obtained by integrating along paths. So a candidate deformation gradient field comes from an actual deformation exactly when it is curl-free row by row. Simple connectivity is needed. Take $\Omega = \set{X : 1 < X_1^2 + X_2^2 < 4, \ 0 < X_3 < 1}$, and let $F$ have first row $\left(-X_2, X_1, 0\right)/(X_1^2 + X_2^2)$, the gradient of the polar angle, and other rows $e_2^T$ and $e_3^T$. This $F$ satisfies the condition but is not a gradient, since the angle is not single-valued on $\Omega$. The analogous condition for the infinitesimal strain tensor is the Saint-Venant compatibility condition.

Going deeper

For a student

In this form the statement is the Poincaré lemma for closed $1$-forms, applied to each component of $\varphi$. In solid mechanics compatibility is usually imposed on strains rather than on $F$. For small strains this leads to the Saint-Venant equations, and for finite strains to the vanishing of the Riemann curvature of the metric $C$. Failure of compatibility in a multiply connected body is the continuum picture of dislocations.