PSI - Issue 50

Alekseev D.I et al. / Procedia Structural Integrity 50 (2023) 17–26 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

21 5

According to the results of experiments, the Johnson-Cook model was verified in the range of strain rates up to 7500 1/s. 4. Magnetic-induction methods 4.1. Ring expansion The results of the experimental research of conducting rings deformation due to the interaction of current induced in them with the magnetic field of multi-turn solenoids are presented in the following works Morozov et al. (2014, 2016, 2018), Gourdin (1989). It is known that, the azimuth stresses σ φ of a thin cylindrical shell, the radius R of which is significantly greater than its thickness h under internal pressure P , can be determined by a simple equation:

R

P    

(3)

h

In case of the magnetic system consisting of a multi-turn solenoid and a coaxial conducting ring mounted on the outside of the winding, the current i c flows in the solenoid, the current i = k ‧ ic is induced in the ring, where k is the current coupling factor. If the distribution of the current density over the cross-section of the ring is uniform, then the azimuth stress σφ in the ring is determined by the expression:

2

0          2 R i h b

(4)





where b - the ring width. It is obvious that the current carrying leads to Joule heating of the ring, which can be determined in accordance with the action integral according to Knoepfel (1970):

   

   

   

   

t i t 

  

2

1

( ) bh



0 

( )

exp

1

T t

dt

(5)

0 

 



 

c





where β – temperature coefficient of resistance, c v – heat capacity, ρ 0 – electrical resistivity at room temperature, t – current carrying time. The calculation for one of the typical magnetic systems described in works of Morozov et al. (2014, 2016, 2018), which consists of the five-turn solenoid with 25 mm in diameter, made of copper wire 0.5 mm in diameter and a copper thin-walled ring in 28.6 mm diameter, 0.015 mm thickness and 2 mm height, see Fig. 4a. The simulation results shown in Fig. 2b, c demonstrate a significantly inhomogeneous pressure distribution over the width of the ring, which by the moment of the current maximum in the centre of the ring (blue curve in Fig. 2c) is around 2.5 times higher than the pressure on the corner (green curve in Fig. 2c). This effect is caused due to the heating of the conductor (red curve in Fig 3c) and nonlinear diffusion of the magnetic field according to Shneerson et al. (2014, 2019). The temperature of the ring is close to the estimates made by (5). The choice of loading parameters, taking into account the possible influence of induced currents on the heating of a deformable ring, can be performed by selecting parameters in the joint analysis of relations (4) and (5), and the analysis of the ring expansion mechanical process can be carried out by numerical simulation. Also, it should be noted, that induced current effect on the fracture process of the ring according to Adamyan (2018).