A computationally efficient thermo-mechanical model with temporal acceleration for prediction of residual stresses and deformations in WAAM

Figures & data

Figure 1. Illustration of the semi-analytical thermal approach.

Figure 2. (a) The geometry of the simulated structure [42]. (b) Half of the geometry modelled by taking advantage of the symmetry of the problem.

Figure 3. (a, b) The analytical temperature $\tilde{T}$ distribution along x₂ axis. In (a), various acceleration factor κ with the original diffusion time t_r = 0.01 s are employed, and in (b), the acceleration factor κ is set as 200 with various diffusion time t_r. (c) The real temperature T along path 1 (see Figure 2(a)).

Figure 4. The temperature as a function of time for point (a) G1 and (b) G2 (see Figure 2(a)). For the simulated results, only the cooling period is displayed, and the correspondingly starting moment of the simulated cooling period is shifted to match that of the experiment.

Figure 5. The von Mises stress along path 2 (a) before and (b) after removal from the backing plate.

Figure 6. The displacement component U₃ after building the fourth layer and removing the clamps.

Figure 7. (a) Geometry of the WAAM component. (b) Experiment setup of the WAAM process.

Figure 8. Moving path of the heat source for a certain layer.

Figure 9. The deformation after the part is fabricated. A scale factor of 5 is applied.

Figure 10. The von Mises stresses of the component (a) before and after (b) the annealing process. The clamps are removed for both situations.

Figure 11. The displacement component U₃ along path 3 before and after the annealing. The clamps are removed for both situations.

Table C1. Thermal properties of mild steel S355 employed in the simulation [27].

A (−)	ρcp	k
0.78	3.9 MJ/m^3K	48.6 W/mK

Table C2. Mechanical properties of mild steel S355 employed in the simulation.

T (°C)	E (GPa)	βth (°C^−1)	ν (−)	${\sigma }_Y$ $\left({ϵ}_{\mathrm{eq}}^P=0\right)$ Part (MPa)	${\sigma }_Y$ $\left({ϵ}_{\mathrm{eq}}^P=0.01\right)$ Part (MPa)	${\sigma }_Y$ $\left({ϵ}_{\mathrm{eq}}^P=0\right)$ Baseplate (MPa)	${\sigma }_Y$ $\left({ϵ}_{\mathrm{eq}}^P=0.01\right)$ Baseplate (MPa)
20	206	1.2 × 10^−5	0.29	450	520	350	420
100	203	1.2 × 10^−5	0.39	450	520	330	400
200	201	1.2 × 10^−5	0.29	420	500	305	380
300	200	1.2 × 10^−5	0.29	390	450	270	350
400	165	1.2 × 10^−5	0.30	320	370	230	290
500	100	1.2 × 10^−5	0.30	260	300	180	230
600	60	1.2 × 10^−5	0.32	170	215	125	160
700	40	1.2 × 10^−5	0.32	60	100	60	100
800	30	1.2 × 10^−5	0.35	50	80	60	100
900	20	1.2 × 10^−5	0.35	50	50	60	60
1000	10	1.5 × 10^−5	0.39	50	50	60	60

The E is the elastic modulus, υ is the Poisson’s ratio, and ${ϵ}_{\mathrm{eq}}^P$ is the equivalent plastic strain [32].

Table C3. Thermal properties of TC11 employed in the simulation.

A (−)	ρcp	k
0.8	3.8 MJ/m^3K	17.2 W/mK

Table C4. Mechanical properties of TC11 employed in the simulation.

T (°C)	E (GPa)	ν (−)	βth (°C^−1)	${\sigma }_Y$ (MPa) $\left({ϵ}_{\mathrm{eq}}^P=0\right)$	${\sigma }_Y$ (MPa) $\left({ϵ}_{\mathrm{eq}}^P=0.004\right)$
20	123.6	0.3	9.3 × 10^−6	1004	1004
200	105.6	0.3	9.3 × 10^−6	736.2	764.6
300	99	0.3	9.5 × 10^−6	676.7	719
400	94.3	0.3	9.7 × 10^−6	656.6	693.6
500	86.1	0.3	10 × 10^−6	628.4	664.4
600	73	0.3	10.2 × 10^−6	552.2	567.6

Table C5. Creep behaviour of TC11 [36].

T (°C)	${ϵ}_{\mathrm{eq}}^{\mathrm{creep}}$ (%)	${\dot{ϵ}}_{\mathrm{eq}}^{\mathrm{creep}}$ (×10^−7s^−1)	${\sigma }_{\mathrm{VM}}$ (MPa)
450	0.335	1.31	350 MPa
450	0.445	1.40	400 MPa
450	0.538	1.67	450 MPa
500	0.410	2.70	350
500	0.461	5.06	400
500	0.526	6.94	450
550	0.428	7.81	300
550	0.449	17.3	350
550	0.704	29.8	400

Table C6. Creep parameter employed in the simulation (in MPa-mm unit).

T (°C)	B	λ	m
450	5.20e−9	1.57	−0.66
500	1.06e−7	1.29	−0.89
550	1.61e−15	3.89	−0.26

Ding J, Colegrove P, Mehnen J, et al. Thermo-mechanical analysis of wire and arc additive layer manufacturing process on large multi-layer parts. Comput Mater Sci. 2011;50:3315–3322. doi: 10.1016/j.commatsci.2011.06.023

 Web of Science ®Google Scholar

Sahoo S. Prediction of residual stress and deformation of build part with variation of hatch spacing in direct metal laser sintering of AlSi10Mg built part: thermo-mechanical modeling. J Laser Appl. 2021;33(3). doi: 10.2351/7.0000393

 Web of Science ®Google Scholar

Ning J, Mirkoohi E, Dong Y, et al. Analytical modeling of 3d temperature distribution in selective laser melting of Ti-6Al-4V considering part boundary conditions. J Manuf Process. 2019b;44:319–326. doi: 10.1016/j.jmapro.2019.06.013

 Web of Science ®Google Scholar

Yang Y, van Keulen F, Ayas C. A computationally efficient thermal model for selective laser melting. Addit Manuf. 2020;31:100955. doi: 10.1016/j.addma.2019.100955

 Web of Science ®Google Scholar

Data availability

The raw/processed data required to reproduce these findings is available from the corresponding authors on reasonable request.