To begin with, we identify the equations of elastostatics in a Riemannian
manifold, which generalize those of classical elasticity in the
three-dimensional Euclidean space. Our approach relies on the principle of
least energy, which asserts that the deformation of the elastic body arising in
response to given loads minimizes over a specific set of admissible
deformations the total energy of the elastic body, defined as the difference
between the strain energy and the potential of the loads. Assuming that the
strain energy is a function of the metric tensor field induced by the
deformation, we first derive the principle of virtual work and the associated
nonlinear boundary value problem of nonlinear elasticity from the expression of
the total energy of the elastic body. We then show that this boundary value
problem possesses a solution if the loads are sufficiently small (in a sense we
specify).Comment: 43 page
