Continuum mechanics in manifolds: Kinematics

Motion f_t is diffeomorphic mapping, which preserve the topology of the initial and final state, from manifold B to S=f(B)

                                                 (1)

The tangent bundle homeomorphism

                                                          (2)

where two point tensor F, which involving points in two distinct configurations, is called the deformation gradient, tangent map or the differential of f. Or push forward

From (2)

The metric tensor G can be written as $dS^2=G_{AB}(X)dX^A⊗dX^B$  . We may also write this tensor g, at the point x as $ds^2=g_{ab}(x)dx^a⊗dx^b$ . For the pull-back under f we have

which is right Cauchy-Green tensor C

One measure of the deformation taking place is given by the Green-Lagrange deformation tensor

The small strain tensor defines as the Lie derivative of metric tensor with repsect to the displacement vector field u

$\varepsilon:=L_ug$