Issue 53

P. Ferro et alii, Frattura ed Integrità Strutturale, 53 (2020) 252-284; DOI: 10.3221/IGF-ESIS.53.21 259   T m ( ) ( ) p [( )] [1 (T T )] K(T) s t                         u u u u u g u F (8) (p is the static pressure, μ is the viscosity, ρ g is the gravitational body force, ρ g[1- β (T-T m )] is the buoyancy force term induced by the temperature dependency of density, K(T) = A mesh (1-f l ) 2 /(f l 3 + ε ) is a Darcy condition that suppress the motion of un-melted metal, where f l is the liquid fraction, ε = 0.01 [34] is a small number to avoid zero denominators and A mesh = 10 14 is the mesh zone constant [34]). The last term in Eqn. (8) is the additional momentum source term that in the work by Zhang and Zhang [34] is applied at the metal-air interface [47,49]: (9) P v  0.54 P a e  K B 1 T  1 T B           (10)    2  2  (1   2 )  1 (11) In Eqs 9-11 σ is the surface tension, κ is the surface curvature, is the surface normal, is the tangential gradient, is the interface delta function, and is the average density at the interface. P v is the recoil pressure that induces a surface depression over the melt pool. P a is the ambient pressure, λ is the evaporation energy, K B is the Boltzmann constant and T b is the boiling temperature. It is observed from Eqn. (10) that P v is temperature dependent. The temperature values are obtained by solving the time-dependent energy equation in fluid [50]: p p C T ( C T) k T Q t           u (12) where  (  u C p T) is convection within the fluid due to fluid flow, k is the thermal conductivity, C p is the specific heat and Q is the energy loss because of various reasons (such as convection, radiation, evaporation) and laser heat source [34] represented by the following Gaussian distributed power per unit area [51]: S laser   P  r 0 2 exp  2[x  v laser t  x i  ( y  y i ) 2 r 0 2         (13) where P is the laser power, α is the absorbance, r 0 is the laser radius at 1/e 2 , v laser is the scan speed of the laser, x i , y i is the initial position of the laser focal center. To simplify the RT approach, in [34] the laser beam power is projected to the top surface of the metal supposing all the heat source is absorbed by the top surface at the first interaction (no optical reflection occurs). Despite the complexity of thermo-fluid dynamic formulation of the problem, interesting phenomena can be captured and studied in detail. For example, Zhang and Zhang [34] showed how the surface morphology of the laser melted power bed is formed by a competition between the Marangoni and recoil pressure effect (Fig. 7). The Marangoni effect is due to the variation of the surface tension ( γ ) with the temperature. The higher the temperature the lower the surface tension (Fig. 7a). To reduce the surface energy, the hottest fluid is moved to the lower temperature regions. On the other hand, the recoil pressure tends to move the fluid downwards as shown in Fig. 7b. Since the magnitude of the recoil pressure is an exponential function of temperature, whereas the Marangoni convection is caused by a linear temperature dependence of the surface tension, a depression zone is formed in the melted pool. A sufficient good agreement was obtained by Zhang and Zhang [34] between the predicted and observed size of the fusion zone. The higher the scan speed, the lower the melted region depth but with a non-linear trend (Fig. 8). As a matter of fact, it is observed that both the Marangoni effect and the recoil pressure are prone to increase the penetration depth. However,