On the four-dimensional formalism in continuum mechanics

Remarkable hidden interrelations can be shown to exist between some fundamental principles of continuum mechanics if a four-dimensional formalism is used, e.g., if time is placed on the same footing as space. To exhibit these properties, the special theory of relativity supplies the proper framework. The fully relativistic energy-momentum tensor and the associated Euler-Lagrange equations are formulated for an elastic solid and for a perfect fluid as a limiting case. Approximate relativistic balance and conservation laws are then derived by expanding the exact relativistic relations in series of powers of the ratio of the velocity of motion to the velocity of light. Several orders of approximation are examined in detail, employing both the Eulerian and the Lagrangian description. This permits to gain new insight into the interrelated structure of the basic laws of continuum mechanics, such as the balance (or conservation) of mass, energy, physical (linear) momentum and material (linear) momentum. As a by-product, a hierarchy of approximate theories of continua is established whose velocity of motion is comparable to the velocity of light.