Consistent hypo-elastic behavior using the four-dimensional formalism of differential geometry

The covariance principle of the theory of relativity within a four-dimensional framework ensures the validity of any equations and physical relations through any changes of frame of reference, due to the definition of the 4D space–time and the use of 4D tensors, operations and operators. This 4D formalism enables also to clearly distinguish between the covariance principle (i.e., frame-indifference) and the material objectivity principle (i.e., indifference to any rigid body motion superposition). We propose and apply here a method to build a constitutive relation for elastic materials using such a 4D formalism. The present article is specifically devoted in the application of this methodology to construct hypo-elastic materials with the use of the 4D Lie derivative. It enables thus to obtain consistent non-dissipative models equivalent to (hyper)elastic ones.