In this work we introduce the so-called dissipative function of internal volumetric friction in the Lagrange equations. Unlike the Rayleigh dissipation function, which has been used to describe dissipation due to velocity-dependent friction on surfaces of discrete (rigid) bodies within the Lagrange formalism for a long time, this paper presents a new rheological model—the so-called dissipative gyrostat—which allows to study dissipation within the volume of (rigid) bodies by means of the aforementioned dissipation function. In a series of two papers we will, first, introduce the new concept and outline the theory. To this end, we will initially explain the physical ideas behind the new rheological model in words and formulae as illustrative as possible. Since our intention is also to build a bridge between the world of Lagrange and Truesdellian rational mechanics we will then compare and relate terms in the equations pertinent to both points of view. This will culminate in a rational explanation of the volumetric dissipation function. In the second of this series of papers we shall apply the theory to a study of various beam pendulum problems and also use it for an investigation of the locking phenomenon of planetary bodies.