Geometrically nonlinear gravito-elasticity: Hyperelastostatics coupled to Newtonian gravitation

A theory of nonlinear elasticity is presented that incorporates a spatially non-constant Newtonian gravitational field as is appropriate if deformable heavy masses of finite volume are considered. For Newtonian gravitation the mass density is a sink for the (scaled) gravitational field. This Gauss-type law for gravitation is incorporated into the mechanical balance equations of linear and angular momentum as well as into the thermomechanical balance equations of energy and entropy. To this end, a total energy density as well as total Piola and Cauchy stresses are introduced that directly capture the contribution of Newtonian gravitation. For the nonlinear elastic case the pertinent relations for the gravito-elastically coupled problem are recovered from a variational setting in terms of the nonlinear deformation map and a gravitational potential.