Classical and relativistic balance of configurational forces
This article develops a unified variational framework for configurational (or material) forces in both Classical (3D, non-relativistic) and Relativistic (4D) Continuum Mechanics. Configurational forces describe the evolution of material defects-such as cracks, dislocations, and interfaces-which move relative to the material rather than through physical space. In the classical setting of hyperelasticity, the authors revisit the balance of configurational forces using an intrinsic Lagrangian formulation, where the material body is modeled as an abstract three-dimensional manifold. By treating the reference configuration as a variable and performing a Lagrangian variation with respect to it, they show that the configurational forces balance naturally emerges. Importantly, this balance equation is not independent: it is equivalent to the standard balance of linear momentum combined with constitutive relations, and it is expressed through the Eshelby stress tensor on the reference configuration. The framework is then extended to Relativistic Hyperelasticity within General Relativity. Matter is described by a matter field, a vector valued function, defined on the four-dimensional Universe, and the Lagrangian (i.e., Action) includes both matter and gravitational contributions. Two stress-energy tensors arise: the Noether stress-energy tensor (from variations with respect to the matter field) and the Hilbert stress-energy tensor (from variations with respect to the Universe metric). Assuming General Covariance, the authors prove that these tensors and their associated balance laws are equivalent. By introducing the notion of an observer and specializing to static spacetimes, the authors define a relativistic generalization of the deformation and derive a four-dimensional Eshelby tensor. They show that in Special Relativity, as in Classical Continuum Mechanics, the relativistic configurational forces balance is not a new equation but follows from the conservation laws of the Noether stress-energy tensor. Finally, they recover the classical configurational forces balance as the non-relativistic limit of the relativistic theory. Overall, the paper provides a rigorous geometric and variational interpretation of configurational forces, unifying classical and relativistic formulations and clarifying their deep connection with standard equilibrium equations.