Markov fracture models of inhomogeneous media and design of layered structures synthesized by additive technology

We consider the problem of studying the fracture of a composite macrolayer synthesized by additive technology on a solid substrate, to which the boundary of the lower microlayer is rigidly bonded. The materials of the microlayers are homogeneous and isotropic. The thicknesses of microlayers and the total thickness of the macrolayer are the control parameters of synthesis. The physical and mechanical characteristics of the microlayers and their combination in the macrolayer are also controlling parameters. The layer is fractured by a crack, which may appear in the top layer (or any other layer) and move toward the substrate or free surface, perpendicular to the boundaries of the microlayers. The crack can stop at the boundaries between the layers (then it is treated as a separate state) or pass through the boundaries without stopping. Based on the theory of Markov chains, the average lifetime of a stratified layer synthesized on a substrate using additive technology and the variance of the number of cycles (lifetime) characterizing the quality of the synthesized structure are calculated. On the basis of the materials available for the synthesis of the layered coating, the problems of optimal design of the composite body with maximum average resource, minimum resource dispersion, maximum specific strength for given geometric and physical-mathematical constraints on the composite parameters are solved. To find optimal solutions, dynamic programming algorithms are used, implemented on circuits (graphs) of sequential selection of a combination of layer materials according to their synergistic properties.