Variation statement of related dissipative tasks of the mechanics of a continuous media | Mekhanika | kompozitsionnykh | materialov i konstruktsii
> Volume 25 > №3 / 2019 / Pages: 434-447

Variation statement of related dissipative tasks of the mechanics of a continuous media



The effectiveness of using the variational approach, formulated in the general case for the spacetime transversely-isotropic four-dimensional continuum together with the generalized Sedov’s variational equation is shown for the formulation of reversible and irreversible processes of deformation and heat transfer in both linear and nonlinear formulations. We use the procedure proposed by the authors for constructing a non-integrable variational form, which allows us to extend the formal variational models of the mechanics of deformable media to irreversible processes of deformation. Non-integrable variational forms determine the possible channels of dissipation depending on the list of generalized variables for a particular model of the medium. It is proved that if the reversible part of the processes under consideration is physically linear, then it can be distinguished in the Sedov’s equation as a variation of a separate functional. In this case, the Sedov’s variational equation can always be represented as the sum of the variation of the Lagrangian of the reversible part and the linear combination of dissipation channels of irreversible physically nonlinear processes. The importance of introducing of transverse isotropy properties in describing the connected processes of thermodynamics of deformation is noted. Such a generalization not only expands the physical models of the media under consideration, but is also necessary to obtain a consistent statement of the problem in generalized voltages for the generalized vector of momenta and the heat flux vector. The possibility of generalizing the variational approach to for formulating non-linear models of dissipative processes, including non-linear Navier-Stokes equations, is shown. In particular, examples of using the variational approach to describe hydrodynamic models in the absence of heat transfer in the medium are considered. The corresponding variational models are constructed: Darcy hydrodynamics, Navier-Stokes linear hydrodynamics, Brinkman hydrodynamics, gradient hydrodynamics and some generalization of the classical non-linear Navier-Stokes hydrodynamics.