A Note on the Balance Laws of Nonlinear Hyperelasticity

It is known that the balance laws of hyperelasticity (Green elasticity), i.e., conservation of mass and balance of linear and angular momenta, can be derived using the first law of thermodynamics by postulating its invariance under rigid body motions of the Euclidean ambient space---the Green-Naghdi-Rivlin theorem. In the case of a non-Euclidean ambient space, covariance of the energy balance---its invariance under arbitrary diffeomorphisms of the ambient space---gives all the balance laws and the Doyle-Ericksen formula---the Marsden-Hughes theorem. In this note, we show that the constitutive equations as well as the balance laws of hyperelasticity can be derived using the first and second laws of thermodynamics without assuming any invariance.