We consider the key systems describing steady state of a locally inhomogeneous electroconductive non-ferromagnetic solid within framework of the local gradient approach in thermomechanics. An arbitrarily chosen subdomain of the solid is regarded as a thermodynamically open system that can exchange by mass with environment. It is assumed that this exchange occurs suddenly at the initial time when the body structure is instantly set. The mass sources are introduced into the model to conform the actual and reference body states. The sources are associated with method of body surface forming.

- Shaofan Li, Xin-Lin Gao. Handbook of Micromechanics and Nanomechanics. CRC Press, 1256 p. (2013).
- Gurtin M. E., Murdoch A. I. A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal.
**57**, 291–323 (1975). - Aifantis E. C. Strain gradient interpretation of size effects. Int. J. Fract.
**95**, 299–314 (1999). - Eringen A. C. Nonlocal Continuum Field Theories. Springer, New York (2002).
- Burak Y. I., Nagirnyi T. S. Mathematical modeling of local gradient processes in inertial thermomechanical systems. Int. Appl.Mech.
**28**, 775 (1992). - Burak Y., Nahirnyj T., Tchervinka K. Local Gradient Thermomechanics. In: Richard B. Hetnarski (eds) Encyclopedia of Thermal Stresses, Springer Reference: 2794–2801 (2014).
- Nagirnyi T. S., Tchervinka K. A. Interface phenomena and interaction energy at the surface of electroconductive solids. Comput. Meth. Sci. Technol.
**14**, 105 (2008). - Nagirnyi T. S., Tchervinka K. A. Thermodynamical models and methods of thermomechanics taking into account near-surface and structural nonhomogeneity. Bases of nanomechanics I. Lviv, SPOLOM, (2012) (in ukrainian).
- Nagirnyi T. S., Tchervinka K. A. Basics of mechanics of local non-homogeneous elastic bodies. Bases of nanomechanics II. Lviv, Rastr–7, 167 p. (2014) (in ukrainian).
- Nahirnyj T. S., Chervinka K. A., Boiko Z. V. On the choice of boundary conditions in problems of the local gradient approach in thermomechanics. J. Math. Sci.
**186**, 130 (2012). - Nahirnyj T. S., Tchervinka K. A. On steady state description for electroconductive non-ferromagnetic local non-homogeneous solid I Int. XX Ukr. Conf. “Modern Problems of Applied mathematics and informatics”, Lviv, 7–9 April 2014, 110–111 (2014).
- Burak Y. I., Halapats B. P., Gnidets’ B. M. Physical and mechanical processes in conductive bodies. Kyiv, Nauk. Dumka, 229 (1978) (in ukrainian).
- Glansdorff P., Prigogine I. Thermodynamic Theory of Structure, Stability and Fluctuations. Wiley, New York (1971).

Math. Model. Comput. Vol. 1, No. 2, pp. 214-223 (2014)