PSI - Issue 36

Odarka Prokhorenko et al. / Procedia Structural Integrity 36 (2022) 290–297 4 Odarka Prokhorenko, Serhii Hainutdinov, Volodymyr Prokhorenko et al. / Structural Integrity Procedia 00 (2021) 000 – 000 Solution of equation (1) by analytical or numerical methods makes it possible to determine the three-dimensional unsteady temperature field T(x, y, z, t) and thermal cycles for any point in the welded joint. However, the existing analytical methods make it possible to obtain solutions of equation (1) only for processes described by linear differential equations under linear boundary conditions, i.e. for those cases when the thermophysical properties can be considered independent on temperature. At the same time, numerical methods, in comparison with analytical ones, allow solving the problem of heat transfer in a complex formulation, i.e. taking into account the real geometry of the welded structure, the temperature dependence of the thermophysical properties, the distribution of the heating source. Currently, the most widespread in scientific research practice is the numerical finite element method (FEM). The solution of the heat conduction problem with the use of FEM is reduced to the minimization of the functional that describes the boundary value problem (1) – (3). For the finite element group this leads to the following matrix equation as presented in the work of Zienkiewicz et al. (2014):           T C K T F + =  , (4) where [C], [K] – global matrices of heat capacity and thermal conductivity; {T} – vector-column of temperatures in the nodes of the finite element mesh; {F} – vector-column of the heat load in the nodes. To solve the thermal conductivity differential equation by finite element method a SYSWELD (2015) software package, developed for modeling thermal deformation processes occurring during welding, was used. For arc welding methods a volumetric heat source proposed by Goldak (2005) in the form of a double ellipsoid (Fig. 2a) with an independent normal (Gaussian) power density distribution of the source in accordance with equations (5) and (6) for the front (index f) and rear (index r) parts of the ellipsoid is used in the software package. In the works of modern researchers Slyvinskyy et al. (2014) and Khudyakov et al. (2019) it is shown that for fusion arc welding the use of the Goldak (2005) heat source model allows prediction of the thermal cycle in the welded joints with high accuracy. 293

(a)

(b)

Z

a f

a r

X

b

Y

c

Fig. 2. The J. Goldak volumetric welding heat source model: (a) analytical model; (b) used in the SYSWELD software package.

2

   

  

2

2

   

( )  + −     + +      f x v t a y z b c     

2

   

   

2

2

  

( )  + −     + +          r x v t a y z b c 

3

−

3

 

−

6 3

Q e

(5)

6 3

Q e

q

f

=

q

f

=

, vol f

f

3/2

,r

vol

r

a bc



3/ 2

r a bc



f

where Q – effective thermal power of the heating source (for arc welding Q = η · I · U, W), τ – time elapsed since the beginning of the source action, s; t - current time, s; v – the heating source travel speed (welding velocity), mm/s; x, y, z – semi-axes of the ellipsoid in coordinate directions OX, OY, OZ, mm; f f and f r – coefficients determining the relations for the heat introduced into the front and rear parts of the ellipsoid; a f , a r , b, c - the corresponding radiuses