A generalized Coleman–Noll procedure and the balance laws of hyper-anelasticity

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 and by postulating its invariance under superposed 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 time-dependent diffeomorphisms of the ambient space—gives all the balance laws and the Doyle–Ericksen formula—the Marsden–Hughes theorem. It is also known that, by assuming the balance laws, and positing the first and second laws of thermodynamics, the Doyle–Ericksen formula can be derived—the Coleman–Noll procedure. Traditionally, the first law of thermodynamics combined with an invariance assumption has been used to derive the balance laws, while the second law has served to constrain constitutive equations. In this paper, we explore how the balance laws themselves can be derived directly from thermodynamic principles. We achieve this via a generalization of the Coleman–Noll procedure: it is shown that the Doyle–Ericksen formula as well as the balance laws for both hyperelasticity and hyper-anelasticity can be derived using the first and second laws of thermodynamics without assuming any (observer) invariance.