PSI - Issue 50

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Dmitry Parshin et al. / Procedia Structural Integrity 50 (2023) 314–319 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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3. Material and methods In the paper by Parshin (2020), a non-classical mathematical model of solid mechanics is constructed to quantitatively describe the above delineated additive process. This model takes into account the following three factors affecting the character of deformation of the coated product being created:  possible rheological deformation properties of the material used, which are manifested in the strain response delay and the aging of the material;  inertial force effects on the formed product during its creation, which are caused by intensive rotation of the substrate body surface around its longitudinal symmetry axis;  arbitrary initial stretching-compression of the material in the hoop direction, which occurs at the time of its entry into the composition of the created solid ply. For the purposes of the research undertaken in this paper, we will use this model, leaving the latter only taking into account the third factor which will be determined by setting the program for changing the initial hoop stress ,0   in the applied material depending on the current radius  of the elementary material layer being added in the process of the ply creation, assuming that the modeled process allows of this stress regulated variation. Note that the neglect of the first factor is permissible at a relatively low rate of change during the additive process in the variable internal radius ( ) a t of the created product and can be implemented in the model from the paper by Parshin (2020) if we take in it 0    C C and 0   G . (1) And neglecting the second factor will be correct in the case of not only a sufficiently surgeless but also a quite slow rotation of the considered body during the process of creating the coating ply on its surface. The latter fact will be adopted in the discussed model if we accept there ( ) 0  t  . (2) By solving the non-classical initial boundary value problem given in the paper by Parshin (2020), we can strictly show that with made simplifications (1) and (2), the used model will give the following accurate analytical integral expressions describing the evolvement of internal technological stresses in the cylindrical coating in question being additively created under the above made assumptions (we skip the corresponding calculations here because of their ponderousness):                                 2 0 2 0 [ ( ) ] ,0 ,0 [ ( ) ] ,0 1 ( ) 2 1 ( ) , 1 ( ) 2 1 a t a a t a d d (3)   is the dimensionless material constant depending only on the Poisson ’s ratio and being positive for all compressible materials; the constant  is defined in the paper by Parshin (2020). The role of the spatial variable plays here the dimensionless quantity 2 0 ) ( a    . In (3) and further, we consider that the program for changing the initial hoop stress is set with respect to this variable: ( ) ,0 ,0       . Based on result (3), it is possible to state and analytically solve a number of problems on controlling the stress state of the additively manufactured cylindrical coatings, what the further recital is devoted to. 4. Relevant calculations Below we will consider the following two problems as particular examples: where 1  