Finite rotations in the description of continuum deformation

Finite rotations in continuum mechanics are described by means of either a proper orthogonal tensor or finite rotation vectors. Some algebraic relations concerning the finite rotations are reviewed. Formulae expressing them in terms of displacements are given. Along each of the curvilinear coordinate lines the finite rotations are shown to satisfy some systems of the linear first-order differential equations. Each system of the equations is presented in four different but equivalent forms associated with an intermediate stretched basis or with an intermediate rotated basis. Integrability conditions of the system of equations provide various alternative forms of compatibility conditions in continuum mechanics. The displacement field is expressed through the stretch and rotation fields in the form of three successive line integrals. The formula describes the displacements to within a constant finite translation and a constant finite rotation. The procedure proposed here generalizes the formula derived by Cesàro (1906) within the classical linear theory of elasticity.