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A two-dimensional mathematical model describing a combination of the processes of self-propagating hightemperature synthesis and selective laser sintering of powder compositions (it is controlled by the laser-radiation energy) has been proposed. The model makes it possible to evaluate the correspondence of the geometric characteristics of the system to the values of the velocities of a moving laser spot at which the layer poured completely reacts in the vicinity of the spot.Introduction. Selective laser sintering (SLS) is one of the most developed (commercially) procedures of realization of the technology of fast prototyping. The SLS process represents the creation, layer by layer, of material copies of three-dimensional objects using spatially selective laser action on the surface of a freely poured powder composition [1]. A distinguishing feature of the SLS method is the possibility of creating functional composite products, not just their physical models, directly.In our opinion, the development of the SLS method can be the most promising when the SLS process is combined with the well-known process of self-propagating high-temperature synthesis (SHS) [2]. Three-dimensional products created due to the SHS reaction in this case can possess a set of physicomechanical properties that cannot be obtained by other methods. We have shown experimentally the possibility of realizing a controlled SHS combustion reaction in a diffusion regime to form intermetallide and piezo-and ferroelectric phases in the systems Ni-Ti, Ni-Al, Ti-Al [3-5], Pb(Zr x Ti 1−x )O 3 , Li 0.5 Fe 2.5−x Cr x O 4 , and BaFe 12x Cr x O 19 [6,7]. We note that the combination of the SHS and SLS processes is difficult to control. The beginning of the SHS reaction can "anticipate" the process of sintering itself or can "lag" behind it; also, we have the combustion (incomplete burning) of the starting components. The prime objective of laser sintering is precision selection of laser-action regimes in which both processes -SHS and SLSwould be in dynamic equilibrium. Therefore, the large number of controlling parameters in the above processes makes the problem of theoretical modeling topical.The SLS of multicomponent powder compositions is traditionally associated with the process of liquid-phase sintering; this process is well known in powder metallurgy. We note that, even in a one-component powder medium, we always have a variation in the particle size and the early melting of the finely divided part of the fraction is actually analogous to liquid-phase sintering. We know of numerous attempts at modeling the SLS process by the finiteelement method and within the framework of the semiclassical approach in the literature [8][9][10].The present work seeks to construct and investigate a two-dimensional model theoretical description of the SLS of such SHS layers. It should be emphasized that, unlike the SHS macrokinetics, our interest was predominantly with the conditions of preservation of the controllability of the process in question due to the correspondi...

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