Submodels of the model of dynamic deformation of a transversally isotropic thermoelastic medium for solving the problems of horizontal crack formation at 3D printing

• Published in: Acta mechanica 2023-09, Vol.234 (9), p.4315-4321
• Main Authors: Chirkunov, Yu. A., Pikmullina, E. O., Molodin, V. V., Lapidus, A. A., Topchiy, D. V.
• Format: Article
• Language: English
• Subjects: 3-D printers; 3D printing; Analysis; Classical and Continuum Physics; Control; Cracks; Differential equations; Dynamic models; Dynamical Systems; Engineering; Engineering Fluid Dynamics; Engineering Thermodynamics; Group dynamics; Heat and Mass Transfer; Linear equations; Original Paper; Plane waves; Solid Mechanics; Theoretical and Applied Mechanics; Three dimensional models; Three dimensional printing; Vibration
• ISSN: 0001-5970; 1619-6937
• DOI: 10.1007/s00707-023-03587-2

A three-dimensional dynamic model of a transversally isotropic thermoelastic medium satisfying the Gassman conditions is used to study the problem of the formation of horizontal cracks in a printed array arising during 3D printing during its cooling and maturation. For a system of second-order differential equations defining this model, a group foliation is performed with respect to the pseudogroup admitted by this system. As a result, a system of first-order differential equations equivalent to the equations of the original model is obtained. This system consists of an automorphic system and a resolving system. With a help of a resolving system, a system (R) of first order equations for the components of the displacement vector and temperature is obtained. The system (R) contains fewer additional functions than the union of the automorphic and resolving systems of the performed group stratification. Two submodels are found, which are determined by exact invariant solutions of the system (R). The first submodel describes a wave traveling inside of the layer along one of the coordinate axes. The second submodel describes a plane wave traveling inside of the layer. For these submodels, heating modes are indicated that do not lead to the formation of horizontal cracks in the product, and modes in which horizontal cracks will necessarily appear.