PSI - Issue 34

J. Gil et al. / Procedia Structural Integrity 34 (2021) 6–12

8

J. Gil et al. / Structural Integrity Procedia 00 (2019) 000–000

3

c p Specific energy D Constitutive matrix f

Goldak’s weighting fraction

k

Thermal conductivity

K P Q

Sti ff ness matrix

Laser power

Body heat

T Temperature u zz Vertical displacement

2. Simulating distortions and residual stresses

The additive process is inherently complex, due to the interactions between the heat source and the powder, the successive addition of material in the component, melt pool thermodynamics and multiple heating and cooling cycles. Therefore, approximations are employed depending on the particular phenomena that the simulation is trying to cap ture. In the case of this work, where macroscopic distortions and stresses are to be analysed, there are three common routes employed to compute residual stresses: • The inherent strain approach: inherent strain (IS) is a technique developed by Y. Ueda et al. (1975) initially developed for the calculation of strains in welded joints which computes the plastic deformation a component is subjected to through an inherent strain vector whose components can be either experimentally obtained or calculated through the finite element method in an object with a smaller domain than the intended analysis domain, as shown by X. Liang et al. (2018); • Weakly-coupled thermomechanical analysis: in this formulation, the thermal history of nodal points is calcu lated firstly in its entirety, and subsequently used as the foundation for the structural calculations, meaning that the thermal response is independent from the mechanical outputs. The governing di ff erential equations are shown in Equations 1; • Strongly-coupled thermomechanical analysis: this method employs the same underlying logic as the weakly coupled analysis, with the di ff erence being the computation of the mechanical response of the component at each simulation step, thus implying a feedback loop between the thermal history and mechanical response. As expected, this method is computationally more intensive.

∂ T ( x , y , z , t ) ∂ t

= ∇ ( k ( T ) · ∇ T ( x , y , z , t )) + Q ( x , y , z , t )

ρ ( T ) c p ( T )

= 0

∇ · σ

(1)

= D ε el

σ

ε tot ε th

= ε el + ε pl + ε th = α ( T ) ( T − T 0 )

Certain aspects of AM simulations are left out of the previous introduction, such as nodal activation method and boundary conditions, but further reading may be of interest M. Gouge et al. (2018). Most of the available software packages base their codes on the weakly-coupled thermomechanical analysis, with varying degrees of complexity. In this paper, two di ff erent approaches are used: approach #1, which structures the problem by activating the nodes respective to the complete, newly deposited layer, and assuming its temperature at its melting temperature, and applies several cooling steps whose duration depends on the idle time between layers, a user-defined parameter. Afterwards, a new layer is introduced and allowed to exchange heat with the already present layers, powder and environment through coe ffi cients that are also user-defined. The process is repeated until the component is completely built, hence undergoing a final cooldown stage until a cooldown temperature is achieved. The absence of a scanning strategy is of note, as this approach relinquishes calculations associated with the interaction between the laser and the powder.