PSI - Issue 42

5

P. Ferro et al. / Procedia Structural Integrity 42 (2022) 259–269 P. Ferro et al./ Structural Integrity Procedia 00 (2022) 000 – 000

263

ì í ï ï ï î ï ï ï

¶ T ¶ x ¶ T ¶ y ¶ T ¶ z

R x = − k x

R x = − k y

(4)

R x = − k z

where k x , k y and k y are the alloy thermal conductivity in the three directions x, y and z, respectively. Due to temperature-dependent material properties (k and c), the associated material behavior was nonlinear. Now, by substituting Eqs. (4) into Eq. (3), the differential governing heat conduction equation can be rewritten as:

é ë ê

ù û ú +

¶ T ¶ t

¶ ¶ x

¶ T ¶ x

¶ ¶ y

¶ T ¶ y

¶ ¶ z

¶ T ¶ z

é ë ê

ù û ú +

é ë ê

ù û ú + q

ρ c

=

k x

k y

k z

(5)

The solution of Eq. (5) requires the definition of initial conditions:

T(x, y, z,0) = T 0 (x, y, z)

(6)

as well as boundaries conditions:

æ èç

ö ø÷ + q s + h c (T − T a ) + h r (T − T r ) = 0

¶ T ¶ x

¶ T ¶ y

¶ T ¶ z

N x + k y

N y + k z

k x

N z

(7)

In Eq. (6), T 0 (x,y,z) is taken equal to the ambient temperature (T a = 20 °C) or eventually the pre-heating temperature. In Eq. (7) N x , N y and N z are the direction cosine of the outward projected normal to the boundary; h c (25 W/m 2 K (Solomon et al., 2018)) and h r are the heat transfer coefficients of convection and radiation, respectively; T is the surface temperature of the model and T r is the temperature of the heat source instigating radiation. The boundary heat flux is designated by q s . The heat transfer coefficient of radiation can be written as:

2 − T r

2 )(T + T

h r = σε F(T

r )

(8)

where  is the Stefan Boltzmann’s constant,  is the emissivity (0.7) and F is the configuration factor. The heat generation due to the heat source moving over the weld line and quantified by the term q in Eq. (5) is described by the power distribution function proposed by Goldak et al. (1984) for arc welding processes:

(9)