Spatial description of the kinematics of continuum

When constructing the general theory of inelastic media there was used (2001) so called spatial description [1 - 4], where a certain fixed domain of a frame of reference contains different medium particles in different moments of time. Due to the use of the spatial description there was constructed a theory, where the concept of the smooth differential manifold is not used. Before such theories were developed only for fluids. For the first time such a theory is built for solids, where the stress deviator is non-zero. For the first time, the spatial description is applied to a medium consisting of particles with rotational degrees of freedom. A new definition of a material derivative, containing only objective operators, is given. This definition, including when using a moving co-ordinate system, does not contradict to the Galileo's Principle of Inertia [2]. It is shown that for the spatial description one can apply standard methods of the introduction of the stress tensor and other similar quantities [1]. The dynamic equations of the medium obtained basing upon the fundamental laws, formulated for the open systems. An error, which presents in the literature, appearing when integrating the differential equation expressing the law of conservation of particles, is eliminated.
  1. Zhilin P.A. Basic equations of the theory of non-elastic media // Proc. of the XXVIII Summer School “Actual Problems in Mechanics”. St. Petersburg. 2001. P. 14-58. (In Russian.)
  2. Zhilin P.A. Phase Transitions and General Theory of Elasto-Plastic Bodies // Proceedings of XXIX Summer School - Conference “Advanced Problems in Mechanics”, St. Petersburg, Russia, 2002. P. 36-48.
  3. Zhilin P.A. Mathematical theory of non-elastic media // Uspehi mechaniki (Advances in mechanics). Vol. 2. N 4. 2003. P. 3-36. (In Russian.)
  4. Zhilin P.A. On the general theory of non-elastic media // Mechanics of materials and strength of constructions. Proc. of St. Petersburg State Polytechnical University. N 489. 2004. P. 8-27. (In Russian.)