PSI - Issue 50

4

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

317

 it is required to provide a necessary law of change in the contact stress that is peeling the coating ply being created from the body surface or, on the contrary, pressing the coating ply to it;  it is required to obtain in the finished product a necessary final distribution of hoop stresses throughout the coating ply, which will determine the strength of this product during its operation under a certain internal or external pressure. 4.1. Result and brief discussion for the contact stress controlling problem The law of change in the contact stress taking place in a particular process realization can be found from the first formula (3) by setting 1   in it:       1 [ ( ) ] ,0 1 2 0 1 ( ) 2 1 ( ) a t a d p t          . (4) By differentiating (4) by time t , we find the required program for changing the initial hoop stresses that provides the given contact pressure law ( ) p t : Remark that the figuring here dependence ( ) a t is assumed to be known in virtue of the realized coating ply additive creation program. In order to discuss obtained result (5), we may note that one will never be able to achieve the contact stress that will press the ply being created against the body surface during the ply additive growth process if the initial hoop stresses created in the added material are not compressive. Indeed, the internal radius ( ) a t of the coating ply growing on the body surface monotonically decreases over time, so ( ) 0   a t . Therefore, for non-compressive initial stresses 0 ,0    it will be ( ) 0   p t in accordance with (5). This means that the contact pressure cannot increase during the coating growth process. But at the initial moment of the growth process, the pressure is zero on the body surface to be coated with the layerwise created ply. It turns out that the pressure will not be able to increase from the zero point and will never become positive. 4.2. Result and brief discussion for the hoop stresses distribution controlling problem Let it be required to achieve a necessary final distribution ( ) ,end    of the hoop stresses in the finished coating, where end t is the moment when the coating creation ends. According to the second formula (3), we should write the following expression for the desired distribution:                        2 0 end end [ ( ) ] ,0 ,0 ,end 1 ( ) 2 1 ( ) ( ) a t a t t d . We denote for brevity the resulting internal radius ( ) end a t of the created coating ply by end a . That is, we have to find the proper program ( ) ,0    for setting the initial stresses in the added material by solving the integral equation ( ) ( 1) ( ) ( ) ( ) 2 0 2 a t a t a t a      ( ) ,0   p t  a t    . (5)

1 ( )

 

1

   







,0



( )

( )

   .

(6)

d



,0

,end





2







2

(

)

a a

end 0

We can construct the strict closed form analytical solution of equation (6), that has the following representation: