In this work, two kinds of silicon nitride-based composites, namely, those with titanium nitride or silicon carbide additives, were sintered using the hot pressing technique (HP). The phase composition, microstructure, and mechanical and elastic properties of the materials were characterized. Three-dimensional geometric models of the composites were created on the basis of microstructure parameters. Using these models, bulk residual thermal stresses were calculated by the finite element method (FEM). Surface stresses were determined using the XRD method of sin²ψ.