Residual stress and distortion modeling of electron beam direct manufacturing Ti-6Al-4V

Abstract

In this work, a finite element model is developed for predicting the thermo-mechanical response of Ti-6Al-4V during electron beam deposition. A three-dimensional thermo-elasto-plastic analysis is performed to model distortion and residual stress in the workpiece and experimental in situ temperature, and distortion measurements are performed during the deposition of a single-bead-wide, 16-layer-high wall built for model validation. Post-process blind hole–drilling residual stress measurements are also performed. Both the in situ distortion and post-process residual stress measurements suggest that stress relaxation occurs during the deposition of Ti-6Al-4V. A method of accounting for such stress relaxation in thermo-elasto-plastic simulations is proposed where both stress and plastic strain are reset to 0, when the temperature exceeds a prescribed stress relaxation temperature. Inverse simulation is used to determine the values of the absorption efficiency and the emissivity of electron beam–deposited, wire-fed Ti-6Al-4V, as well as the appropriate stress relaxation temperature.

Introduction

Additive manufacturing (AM) has seen increased attention in recent years due to the ability of the process to produce near-net shape parts directly from computer-aided design (CAD) files without the retooling cost associated with casting or forging. The electron beam deposition process is of particular interest to the aerospace industry due to its ability to deposit large amounts of feedstock material at rapid rates. The process involves melting metal wire onto a substrate and allowing for it to cool and form a fully dense geometry on a layer-by-layer basis. Unfortunately, large thermal gradients during the deposition process result in undesirable distortion and residual stress. Modifications to the build plan may reduce distortion and residual stress. To optimize the build plan without the expensive trial-and-error iterations, an accurate predictive model is needed.

Numerical modeling for the prediction of temperature, distortion, and residual stress caused by the AM process is similar to that of multi-pass welding. Weld modeling has been an active area of research for nearly four decades. Several weld models have been used to predict thermal and mechanical behaviors.^1–7 Other modeling work has focused on material phase change during welding.^8,9 Lindgren ¹⁰ has written detailed summaries on the development of weld model complexity, material modeling in welding, ¹¹ and improvements in computational efficiency for weld modeling. ¹² In more recent weld model work, Michaleris et al. ¹³ focused on predicting distortion modes caused by welding, as well as residual stress.

AM modeling adds significant computational cost when compared to multi-pass welding due to the increased amount of deposited material, passes, and process time. The addition of material into the simulation requires an element activation strategy. ¹⁴ Like weld models, AM models require accurate process parameters to yield acceptable results. The parameters of particular interest are absorption efficiency and surface emissivity, as these parameters determine the energy entering and exiting the system during the process. Yang et al. ¹⁵ experimentally determined the absorption efficiency and surface emissivity of deposited Ti-6Al-4V using a laser-assisted machining process. Shen and Chou ¹⁶ modeled the efficiency of a powder-based electron beam system as 0.90 and assumed the emissivity of Ti-6Al-4V to be a constant 0.70, resulting in close agreement with experimental results. No available literature provides values of efficiency and emissivity for a wire-fed electron beam system. Over the past decade, significant work has been performed to model AM.^17–20

Some researchers have focused on material phase change caused by AM such as predicting the resulting microstructure of deposited Ti-6Al-4V.^21–27 Additional work has focused on predicting stress and distortion in materials that undergo phase transformation. ²⁸ Ghosh and Choi ²⁹ concluded that simulated distortion results are significantly affected by failing to properly take into account the microstructural changes present in the deposited material. Griffith et al. ³⁰ showed that the high temperatures reached during the laser deposition of stainless steel 316 can cause the material to anneal, thus reducing the measured residual stress. Song et al. ³¹ accounted for the plastic strain relaxation in a deposited nickel-based alloy by setting a threshold temperature that, when surpassed, sets the plastic strain value to 0. Qiao et al. ³² performed microhardness measurements to demonstrate that the equivalent plastic strain needs to be dynamically adjusted at high temperatures in thermo-elasto-plastic models to simulate material annealing. An approach for managing the stress relaxation in AM-deposited Ti-6Al-4V has not yet been presented.

The objective of this work is to develop a finite element model for predicting the in situ thermo-mechanical response of a Ti-6Al-4V workpiece deposited using a wire-fed electron beam system. Workpiece distortion and residual stress is modeled using a three-dimensional (3D) thermo-elasto-plastic analysis. The model is validated using experimental in situ temperature and distortion measurements performed during deposition of a 16-layer-high, single-bead-wide wall build as well as post-process residual stress measurements taken using blind hole drilling. Both the in situ distortion and post-process residual stress measurements suggest that stress relaxation occurs in Ti-6Al-4V during deposition. The thermo-elasto-plastic model presented accounts for the observed stress relaxation by resetting both stress and plastic strain to 0 when the temperature exceeds a prescribed stress relaxation temperature. The absorption efficiency, emissivity, and stress relaxation temperature are determined by applying inverse simulation.

Electron beam deposition simulation

The thermal and mechanical histories are determined by performing a 3D transient thermal analysis and a 3D quasi-static incremental analysis, respectively. The thermal and mechanical analyses are performed independently and are weakly coupled, meaning that the mechanical response has no effect on the thermal history of the workpiece. ³³

Thermal analysis

The governing heat transfer energy balance is written as

$$\rho C_p\frac{\mathrm{dT}}{\mathrm{dt}}=-\nabla ·\mathit{q}(\mathit{r},t)+Q(\mathit{r},t)$$

(1)

where $\rho$ is the material density, $C_p$ is the specific heat capacity, $T$ is the temperature, $t$ is the time, $Q$ is the heat source, $\mathit{r}$ is the relative reference coordinate, and $\mathit{q}$ is the heat flux vector, calculated as

$$\mathit{q}=-k\nabla T$$

(2)

Table 1 lists the values of all temperature-dependent material properties (thermal conductivity $k$ and specific heat capacity $C_p$ ) for Ti-6Al-4V. Material properties at temperatures between those listed are obtained by linear interpolation over the temperature range. The material thermal properties above 800 °C are assumed to be constant. The density is a constant 4.43 × 10⁻⁶ g/mm³.

Table 1.

Temperature-dependent thermal properties of Ti-6Al-4V. ³⁴

$T$ (°C)	$k$ (W/m/°C)	$C_p$ (J/kg)
0	6.6	565
20	6.6	565
93	7.3	565
205	9.1	574
315	10.6	603
425	12.6	649
540	14.6	699
650	17.5	770
760	17.5	858
870	17.5	959

The electron beam heat source is modeled using the Goldak double-ellipsoid model as follows

$$Q=\frac{6\sqrt{3}p\eta f_s}{\mathrm{abc}\pi \sqrt{\pi }}{e}^{-[\frac{3x^2}{a^2}+\frac{3y^2}{b^2}+\frac{3{(z+v_{w}t)}^2}{c^2}]}$$

(3)

where $P$ is the power; $\eta$ is the absorption efficiency; $f_s$ is the process scaling factor; $x$ , $y$ , and $z$ are the local coordinates; $a$ , $b$ , and $c$ are the transverse, melt pool depth, and longitudinal dimensions of the ellipsoid, respectively; $v_w$ is the heat source travel speed; and $t$ is the time. The heat source is circular, making a = c = 6.35 mm and f_s  = 1. The penetration depth $b$ of the heat source is found to be 3.81 mm by cross-sectioning a deposited sample, as recommended by Goldak et al. ³⁵ Velocity is a constant 12.7 mm/s. Power is entered as the average power of each individual pass during the deposition. The efficiency $\eta$ is initially unknown and is calibrated based on experimental results.

Radiation is applied to all free surfaces, including those of the newly deposited material, using the Stefan–Boltzmann law

$$q_{\mathrm{rad}}=\epsilon \mathit{\sigma }(T_s^4-T_{\infty }^4)$$

(4)

where $\epsilon$ is the surface emissivity, $\mathit{\sigma }$ is the Stefan–Boltzmann constant, $T_s$ is the surface temperature of the workpiece, and $T_{\infty }$ is the ambient temperature (25 °C). The surface emissivity $\epsilon$ is unknown and determined by correlating computed and experimentally measured temperatures. Due to the deposition taking place in a vacuum, there is no surface convection.

When the electron beam is on, the magnitude of the time increments is calculated such that the beam travels a distance equal to its radius as follows

$$t^i=t^{i-1}+\frac{r_i}{v_w}$$

(5)

where $t$ is the time, $i$ is the current increment number, and $r$ is the modeled electron beam radius. When the electron beam is off, time steps are allowed to increase to magnitudes as large as 10s.

Mechanical analysis

A thermal history–dependent, quasi-static mechanical analysis is performed to obtain the mechanical response of the workpiece during deposition. The results of the thermal analysis are imported as a thermal load into the mechanical analysis. The governing stress equilibrium equation is

$$\nabla ·\mathit{\sigma }=0$$

(6)

where $\mathit{\sigma }$ is the stress. The mechanical constitutive law is

$$\mathit{\sigma }=\mathit{C}{\mathit{ϵ}}_e$$

(7)

$$\mathit{ϵ}={\mathit{ϵ}}_e+{\mathit{ϵ}}_p+{\mathit{ϵ}}_T$$

(8)

where $\mathit{C}$ is the fourth-order material stiffness tensor, and ${\mathit{ϵ}}_e$ , ${\mathit{ϵ}}_p$ , and ${\mathit{ϵ}}_T$ are the elastic strain, plastic strain, and thermal strain, respectively. The thermal strain is computed as

$${\mathit{ϵ}}_T={\mathit{ϵ}}_T\mathit{j}$$

(9)

$${\mathit{ϵ}}_T=\alpha (T-T^{\mathrm{ref}})$$

(10)

$$\mathit{j}={\left[\begin{array}{cccccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]}^T$$

(11)

where $\alpha$ is the thermal expansion coefficient and $T^{\mathrm{ref}}$ is reference temperature. The plastic strain is computed by enforcing the von Mises yield criterion and the Prandtl-Reuss flow rule

$$\begin{array}{c}\hfill f={\mathit{\sigma }}_m-{\mathit{\sigma }}_y({\mathit{ϵ}}_q,T)\le 0\hfill \end{array}$$

(12)

$${\stackrel{·}{\mathit{ϵ}}}_p={\stackrel{·}{\mathit{ϵ}}}_q\mathit{a}$$

(13)

$$\mathit{a}={\left(\frac{\partial f}{\partial \mathit{\sigma }}\right)}^T$$

(14)

where $f$ is the yield function, ${\mathit{\sigma }}_m$ is Mises’ stress, ${\mathit{\sigma }}_Y$ yield stress, ${\mathit{ϵ}}_q$ is the equivalent plastic strain, and $\mathit{a}$ is the flow vector.

For an incremental formulation, equation (7) is re-written as

$${}^n\mathit{\sigma }={}^{n-1}\mathit{\sigma }+\Delta \mathit{\sigma }$$

(15)

where ${}^{n-1}\mathit{\sigma }$ and ${}^n\mathit{\sigma }$ are the stress and the previous and current increment, and $\Delta \mathit{\sigma }$ is the stress increment computed as

$$\Delta \mathit{\sigma }=\Delta \mathit{C}({}^{n-1}\mathit{ϵ}-{}^{n-1}\mathit{ϵ}{}_p-{}^{n-1}\mathit{ϵ}{}_T)+\mathit{C}(\Delta \mathit{ϵ}-\Delta {\mathit{ϵ}}_p-\Delta {\mathit{ϵ}}_T)$$

(16)

where left superscripts denote the time increment where a quantity is computed and $\Delta \mathit{ϵ}$ is the total strain increment corresponding to the current displacement increment.

Incorporation of annealing in the constitutive system involves resetting the plastic strain ${\mathit{ϵ}}_p$ to 0 when the annealing temperature is reached. ³¹ Alternatively, Qiao et al. ³² proposed a gradual (dynamic) reduction in the equivalent plastic strain ${\mathit{ϵ}}_q$ based on the time duration that the material is exposed at high temperatures. As discussed in further detail in sections “Distortion history” and “Residual stress,” a relaxation mechanism is present during the deposition of Ti-6Al-4V manifested by the reduction in distortion recorded by in situ distortion measurements and reduced residual stress as measured by post-process blind hole drilling. Implementation of such annealing models did not result in significant improvement in the correlation of computed results with in situ displacement and post-process residual stress measurements. The deviation was in the order of 500%.

A stress relaxation is proposed in this work where instantaneous annealing and creep occurs when a stress relaxation temperature $T_{\mathrm{relax}}$ is reached. The relaxation is implemented by resetting the stress ${}^{n-1}\mathit{\sigma }$ , elastic strain ${}^{n-1}\mathit{ϵ}{}_e$ , thermal strain ${}^{n-1}\mathit{ϵ}{}_T$ , plastic strain ${}^{n-1}\mathit{ϵ}{}_p$ , and equivalent plastic strain ${}^{n-1}{\mathit{ϵ}}_q$ at the previous time increment to 0 when the norm of the temperature at all Gauss points of an element exceeds $T_{\mathrm{relax}}$ . The material response corresponding to this relaxation model for various relaxation temperatures are presented in section “Results and discussion.”

Table 2 displays the temperature-dependent mechanical properties of Ti-6Al-4V, including the elastic modulus $E$ , the yield strength ${\mathit{\sigma }}_y$ , and the coefficient of thermal expansion $\alpha$ . Material properties between the temperatures listed are linearly interpolated. Mechanical properties are assumed constant above 800 °C. The Poisson’s ratio $\nu$ is assumed to be a constant value of 0.34. The model assumes perfect plasticity, that is, no material hardening occurs.

Table 2.

Temperature-dependent mechanical properties of Ti-6Al-4V.^19,34

$T$ (°C)	$E$ (GPa)	${\mathit{\sigma }}_y$ (MPa)	$\alpha$ (μm/m °C)
0	105.00	777.15	8.60
20	103.95	768.15	8.64
250	91.81	664.65	9.20
500	78.63	552.15	9.70
800	62.80	417.15	9.70

Calibration and validation

Deposition process

Electron beam freeform fabrication is used to deposit Ti-6Al-4V in a vacuum chamber. A Ti-6Al-4V plate 254 mm long, 101.6 mm wide, and 12.7 mm thick is used as a substrate. The substrate is clamped at one end and cantilevered, allowing the unconstrained end to deflect freely, while monitored by a laser displacement sensor (LDS). Figure 1 shows the test fixture and constrained substrate after deposition.

Figure 1.

Sample mounted in the test fixture and a schematic of the approximated deposited geometry.

The AM system used is the Sciaky VX-300, which welds in the range of ${10}^{-4}$ – ${10}^{-5}$ Torr. The work envelope is approximately 5.8 × 1.2 × 1.2 m³ in volume. The deposition process begins by preheating the portion of the substrate where metal will be deposited. Figure 2 shows the preheating path. The electron beam power is set to 4.4 kW and scan speeds range between 35 and 42 mm/s.

Figure 2.

Scan pattern of the preheat performed on the top of the substrate.

After heating the top surface of the substrate, a one-bead-wide deposition is built 16 layers high and 203.2 mm long. The wire feed rate is set to 50.8 mm/s. The scan speed is a constant 12.7 mm/s. The electron beam operates at a power varying between 8 and 10 kW. The power is fluctuated to control the melt pool size. Figure 1 shows the resulting deposited geometry. The sloped wall is approximated as being 47.625 mm high near the clamped end of the substrate and 28.6 mm high near the free end for the succeeding finite element modeling work. Table 3 shows, for each deposition layer, the average power and start time, as well as the beginning and ending z-coordinates.

Table 3.

Varying electron beam process and path parameters.

Pass number	Power (kW)	Start time (s)	z start position (mm)	z end position (mm)
Preheat	4.4	0	0	0
1	8.8	115.72	2.9766	1.7875
2	8.7	216.02	5.9531	3.5750
3	9.3	297.03	8.9297	5.3625
4	8.7	436.69	11.9063	7.1500
5	8.6	535.33	14.8828	8.9375
6	8.6	624.49	17.8594	10.7250
7	8.6	736.94	20.8359	12.5125
8	8.4	833.99	23.8125	14.3000
9	8.3	941.90	26.7891	16.0875
10	8.4	1070.13	29.7656	17.8750
11	8.4	1195.32	32.7422	19.6625
12	8.3	1369.74	35.7188	21.4500
13	8.3	1524.81	38.6953	23.2375
14	8.4	1690.23	41.6719	25.0250
15	8.7	1990.64	44.6484	26.8125
16	8.4	2188.92	47.6250	28.6000

In situ distortion and temperature

In situ distortion measurements are taken using a Micro-Epsilon LDS, model LLT 28x0-100 and Micro-Epsilon controller scanCONTROL 28x0. The LDS is positioned to measure the longitudinal bowing distortion mode in the z-direction at the free end of the substrate, as shown in Figure 3(b). The LDS targets a point approximately 6.3 mm from the free end of the substrate. A National Instruments 9250 module reads the LDS analog voltage signal.

Figure 3.

Schematic showing the LDS measurement location and the thermocouple (T/C) locations on the (a) top of the substrate and (b) the bottom of the substrate.

Figure 3 shows the locations of the four thermocouples (0.25 mm diameter) used to monitor in situ temperature. Thermocouples 1–3 measure the temperature on the bottom of the substrate, parallel to the axis of deposition, and are located 63.5 mm from the clamped end of the substrate, at the center of the substrate, and 6.35 mm from the free end of the substrate, respectively. Thermocouple 4 measures temperature along the axis of deposition, on the top edge, and at the free end of the substrate. The thermocouples are placed to allow for temperature readings at various substrate locations, without interfering with the LDS. A National Instruments 9213 module reads the thermocouple analog voltage signals. Both National Instruments modules record data into LabVIEW at a frequency of 20 Hz.

Residual stress

Post-process residual stress is measured using the hole-drilling method. Eight residual stress measurements are taken, seven measurements on the bottom of the substrate along the axis of deposition (see Figure 4(a)) and one measurement on the deposited material (see Figure 4(b)). The majority of the measurements are taken on the substrate, as it provides a large smooth surface appropriate for applying strain gauges and placing the milling guide. Micro-Measurements^® strain gauges, model EA-06-062RE-120, are bonded to the bottom center of the substrate. Gauges are calibrated using the procedure described in manufacturer engineering data sheet U059-07 and technical note 503. The ASTM E837 drilling process is followed. Incremental drilling is done using RS-200 Milling Guide and high-speed drill from Micro-Measurements^®. A 1.52-mm-diameter, carbide-tipped, Type II Class 4A drill bit was used. Strain measurements are read by a Micro-Measurements^® P-3500 Strain Indicator. Bridges are balanced with a Micro-Measurements^® Switch and Balance Unit, model SB-1.

Figure 4.

Schematic showing the locations of the residual stress measurements (a) on the bottom of the substrate and (b) on the build portion.

Numerical implementation

Figure 5 displays the 3D finite element mesh used for both the thermal and mechanical analyses. The mesh contains 6848 Hex-8 elements and 9405 nodes. The mesh is generated using Patran 2012 by MSC. The mesh allots one element per deposition thickness and one element per heat source radius. A mesh convergence study was performed using two and then four elements per heat source radius resulting in 1.47% and 7.17% peak change, respectively, in the computed temperatures at the nodes corresponding to thermocouples and a 1.02% and 2.85% average change, respectively, at the nodes corresponding to thermocouples. However, the mesh with one element per heat source radius is used in this work because computational efficiency is critical in modeling electron beam deposition. The thermal and mechanical analyses are performed using the code CUBIC by Pan Computing LLC. ¹⁴ The quiet element activation approach is used in this work, where elements representing the metal deposition regions are present from the start of the analysis. However, they are assigned properties so that they do not affect the analysis. For the heat transfer analysis, the thermal conductivity $k$ is set to a lower value to minimize conduction into the quiet elements, and the specific heat $C_p$ is set to a lower value to adjust energy transfer to the quiet elements ¹⁴

$$k_{\mathrm{quiet}}=s_{k}k$$

(17)

$$C_{p_{\mathrm{quiet}}}=s_{C_p}{C}_p$$

(18)

Figure 5.

Mesh generated for the thermal and the mechanical analyses.

where $k_{\mathrm{quiet}}$ and $C_{p_{\mathrm{quiet}}}$ are the thermal conductivity and the specific heat used for quiet elements, respectively, and $s_k={10}^{-6}$ and $s_{C_p}={10}^{-2}$ are the respective scaling factors. When the heat source contacts any Gauss point of an element, the element is activated by switching $s_k$ and $s_{C_p}$ to 1. The initial temperature of the element is also reset to the ambient temperature to minimize erroneous heating of the element due to the activation. For each time increment, the continuously evolving surface between active and quiet elements is identified, and surface radiation is applied as a boundary condition.

For the mechanical analysis, the quiet elements are assigned a lower elastic modulus

$$\begin{array}{c}\hfill E_{\mathrm{quiet}}=s_{E}E\hfill \end{array}$$

(19)

where $s_E={10}^{-4}$ .

Figure 6 illustrates the mechanical constraints applied to the model. Three corner nodes on the clamped end of the substrate are constrained to model clamping the end of the substrate.

Figure 6.

Mechanical constraints used to fix the clamped end of the substrate.

The numerical model is calibrated, using inverse simulation, as in Aarbogh et al., ³⁶ to determine the unknown values of efficiency $\eta$ in equation (3), surface emissivity $\mathit{\epsilon }$ in equation (4), and stress relaxation temperature $T_{\mathrm{relax}}$ .

Efficiencies are varied from 0.90 ¹⁶ to 0.95, ³⁷ based on the available literature. The emissivity is varied from 0.44 to 0.69. ³⁸ The percent error is calculated as

$$\%\mathrm{Error}=\frac{100\sum _{i=1}^n\left|\frac{{(x_{\exp })}_i-{(x_{\mathrm{sim}})}_i}{{(x_{\exp })}_i}\right|}{n}$$

(20)

where $n$ is the total number of simulated time increments, $i$ is the current time increment, $x_{\mathrm{sim}}$ is the simulated value, and $x_{\exp }$ is the experimentally measured value. The combination of efficiency and emissivity yielding the lowest percent error is used for the model.

The results from the calibrated thermal model are imported as a thermal load into the mechanical simulation. The mechanical model is calibrated by adjusting the relaxation temperature $T_{\mathrm{relax}}$ . Temperatures between 600 °C (below the beginning of the alpha–beta phase transition ³⁹ ) and 980 °C (approaching the beta-transus of Ti-6Al-4V) are tested. The error is calculated using equation (20).

Results and discussion

Thermal history

Table 4 shows the results of the simulated cases used to calibrate the thermal model. The efficiency $\eta$ and emissivity $\mathit{\epsilon }$ values are found to be 0.90 and 0.54, respectively, as this combination results in the lowest percent error (7.7%) compared with experiment. The calibrated value for efficiency agrees with that used for the electron beam deposition in Shen and Chou. ¹⁶ The calibrated emissivity value falls within the range experimentally established by Coppa and Consorti. ³⁸

Table 4.

Cases examined for thermal model calibration.

Case	$\eta$	$\mathit{\epsilon }$	% Error
1	0.90	0.44	12.1
2	0.90	0.49	9.2
3	0.90	0.54	7.7
4	0.90	0.59	8.1
5	0.90	0.64	9.5
6	0.90	0.69	11.1
7	0.925	0.44	13.6
8	0.925	0.49	10.5
9	0.925	0.54	8.8
10	0.925	0.59	8.9
11	0.925	0.64	9.9
12	0.925	0.69	10.5
13	0.95	0.44	15.3
14	0.95	0.49	12.1
15	0.95	0.54	10.1
16	0.95	0.59	9.8
17	0.95	0.64	10.3
18	0.95	0.69	11.5

Figure 7 displays the thermal histories experimentally measured by the four thermocouples compared with the simulation results for the calibrated process efficiency of η = 0.90 and emissivity of ε = 0.54 at nodes corresponding to the locations of the thermocouples. A process time of 0 min corresponds to the start of the preheat. The ambient temperature is approximately 25 °C. Thermocouple 2 (located in the middle of the bottom of the substrate) records the highest temperature. Thermocouple 3 (located in the bottom of the substrate near the clamped end) records the lowest temperature, as it is nearest to the clamp which absorbs heat through conduction. Although the process efficiency $\eta$ and emissivity $\mathit{\epsilon }$ are calibrated to match the thermocouple measurements, a good correlation (7.7% error) is achieved for the entire duration of the process. In addition, Thermocouples 1, 3, and 4, which are not used for calibration, also display close correlation with simulated results.

Figure 7.

Computed thermal history with η = 0.90 and ε = 0.54 compared with the experimental measurements. (a) Thermocouple 1, (b) Thermocouple 2, (c) Thermocouple 3, and (d) Thermocouple 4.

Distortion history

Table 5 lists the results of the simulated cases run to investigate the effect of changing the relaxation temperature on the in situ distortion. A final error is also computed by comparing the post-process experimental distortion with the post-process simulation distortion. A stress relaxation temperature of 690 °C provides results in closest agreement with the experimental post-process distortion.

Table 5.

Cases examined for mechanical model calibration.

Case	$T_{\mathrm{relax}}$ (°C)	% Error	Final % error
19	600	43.4	45.3
20	650	16.9	23.2
21	670	6.6	12.9
22	680	4.7	8.0
23	690	7.4	2.2
24	700	14.3	4.4
25	980	144.3	157.5
26	None	431.0	520.3

Figure 8 illustrates the computed model distortion at the LDS point for selected relaxation temperatures. As seen in the figure, no relaxation (Case 26) results in a nearly linear increase in distortion with time up to 20 min. The increase rate becomes lower afterward. The distortion trend is quite different from that of the LDS measurement which shows a decrease in distortion after 7 min of processing. Also, the final distortion is over-predicted by 520.3%. A high relaxation temperature (case 25, 980 °C) exhibits a similar distortion trend when compared to that with no relaxation, but with lower magnitudes. The distortion is still over-predicted by 157.5%. Case 23 with $T_{\mathrm{relax}}$ of 690 °C exhibits the same distortion behavior as the LDS measurement. The distortion decreases after 7 min of processing and starts to increase again at 20 min of processing time. The error compared to the LDS measurement is 7.4% over the entire process duration. An even lower relaxation temperature (Case 19, 600 °C) exhibits a decrease of distortion after 5 min and an increase again after 25 min. This case under-predicts the final distortion, by 45.3%. These results illustrate the importance of accounting for stress relaxation in the model when predicting in situ and post-process distortion.

Figure 8.

Computed distortion histories at various stress relaxation temperatures compared with the experimental measurement.

Figure 9 shows the post-process distorted shape of the workpiece as predicted by the calibrated model. The maximum distortion occurs at the free end of the substrate. Figure 10 displays a close-up of the in situ measured distortion and the computed results for the calibrated relaxation temperature of 690 °C. It is noted that although the relaxation temperature is calibrated to match the in situ measured distortion, a very good correlation (7.4% error) is achieved for the entire duration of the process.

Figure 9.

Final post-process distortion (1× magnification) and longitudinal $(\mathit{\sigma }{}_{\mathrm{yy}})$ residual stress (MPa).

Figure 10.

Computed in situ distortion at the LDS point compared with the experimental results.

Residual stress

Figure 11 displays the computed post-process longitudinal component of the residual stress at nodes on the bottom of the substrate along the axis of deposition as well as experimental values at corresponding locations. Computed results for no stress relaxation (case 26) and stress relaxation temperature of $T_{\mathrm{relax}}=690\circ C$ (Case 23) are shown in the figure. The results with no relaxation are over-predicted by more than 500% and reach values as high as 1040 MPa. The computed residual stress also exhibits a shift to negative values close to the edges which is not present in the measured residual stress. The computed results with a stress relaxation temperature of $T_{\mathrm{relax}}=690\circ C$ do not show this shift to negative values and are in close agreement (within 25%) with the measured residual stress. The maximum stress predicted using stress relaxation is 41.3 MPa. Figure 9 also illustrates the computed longitudinal residual stress on the entire part. A high stress concentration is seen along the interface between the substrate and the build, and lower values (less than 250 MPa) elsewhere. The stress concentration located at the interface is artificially high in the model due to the sudden transition from the substrate to the deposition. On the actual workpiece, the transition is filleted, meaning the stress concentration will not occur. The measured residual stress on the deposition is 20.0 MPa, which is in good agreement with the prediction from the model of 3.15 MPa considering that the measurement error of blind hole drilling can be ±50 MPa. ⁴⁰ The residual stress results on the deposition are lower than those on the substrate. This is because the deposited material reaches a temperature exceeding the stress relaxation temperature more frequently, leading to a reduction in residual stress. These results show that, in addition to causing errors in the simulated distortion, failure to account for stress relaxation present in Ti-6Al-4V during electron beam deposition results in erroneous residual stress predictions.

Figure 11.

Computed residual stress results with and without stress relaxation, compared with the experimental measurements along the axis of deposition.

Conclusion

A finite element model is developed for predicting the thermo-mechanical response of Ti-6Al-4V during electron beam deposition. A 3D thermo-elasto-plastic analysis is performed, and experimental in situ temperature and distortion measurements are performed during the deposition of a single-bead-wide, 16-layer-high wall build for model validation. Post-process blind hole–drilling residual stress measurements are also performed.

Both the in situ distortion and post-process residual stress measurements suggest that stress relaxation occurs during the deposition of Ti-6Al-4V. A method of accounting for such stress relaxation in thermo-elasto-plastic simulations is proposed where both stress and plastic strain is reset to 0, when the temperature exceeds a prescribed stress relaxation temperature.

Inverse simulation is performed to determine the values of the absorption efficiency and the emissivity of electron beam–deposited, wire-fed Ti-6Al-4V, as well as the appropriate stress relaxation temperature. An efficiency of 0.90 and an emissivity of 0.54 result into the best correlation between measured and computed temperature history. Both values are in agreement with those published for laser-assisted machining and powder-based AM. A stress relaxation value of 690 °C is found to provide the best correlation between in situ measured and computed distortion. The results show that failure to implement stress relaxation in the constitutive model leads to errors in the residual stress and the in situ distortion predications of over 500% when compared with the experimental measurements.

Suggested future work includes sectioning and microstructural analysis of the test piece and establishing a correlation with the computed results. In addition, the possibility of a gradual stress relaxation model should be investigated to compare with the instantaneous model used in this work. This would require further testing to establish the suitable rate of relaxation.