On the compatibility condition of displacement field for finite deformation of continuum

The vanishing of Riemann-Christoffel tensor is usually adopted as the compatibility condition of finite deformation. However, we prove in this paper by the method of Cesaro that this condition is necessary but not sufficient for guarantee of a single-valued, continuous displacement field. A new general compatibility condition, based on the theorem of strain-rotation decomposition(Chen4]) is derived. The displacement compatible condition reduces to Saint-Venant's condition when strain and rotation are infinitesimal.