Latent heat of fusion for stainless steel 304.

## Abstract

Direct metal deposition (DMD) has become very popular within the space of rapid manufacturing and repair. Its capability of producing fully dense metal parts with complex internal geometries, which could not be easily achieved by traditional manufacturing approaches, has been well demonstrated. However, the DMD process usually comes with high thermal gradients and high heating and cooling rates, leading to residual stresses and the associated deformation, which can have negative effect on product integrity. This paper studies the features of thermal stress and deformation involved in the DMD process by constructing a 3-D, sequentially coupled, thermomechanical, finite element model to predict both the thermal and mechanical behaviors of the DMD process of Stainless Steel 304 (SS 304). A set of experiments were then conducted to validate deformation using a laser displacement sensor. Comparisons between the simulated and experimental results show good agreement. This model can be used to predict the mechanical behavior of products fabricated by the DMD process and to help with the optimization of design and manufacturing parameters.

### Keywords

- additive manufacturing
- residual stress
- deformation
- finite element analysis
- experimental validation

## 1. Introduction

### 1.1 Laser-aided direct metal deposition

Laser-aided direct metal deposition (DMD) is an advanced additive manufacturing (AM) technology which can produce fully dense, functional metal parts directly from CAD model. In its operation, the laser beam is focused onto a metallic substrate to create a melt pool, and a powder stream is continuously transported into the melt pool by the powder delivery system. The substrate is attached to a computer numerical control (CNC) multi-axis system, and by moving the substrate according to the desired route pattern, a 2-D layer can be deposited. By building successive layers on top of one another (layer by layer), a 3-D object can be formed. The DMD process has demonstrated its ability in the area of rapid manufacture, repair, and modification of metallic components. Practically, this process is most suitable for components with complex internal geometries that cannot be fabricated by traditional manufacturing methods such as casting. Furthermore, this process is very cost-effective compared with traditional subtractive manufacturing techniques because it can produce near net shape parts with little or no machining.

### 1.2 Residual stress and distortion

Residual stresses exist in a part while no external loads were applied. When a part is heated evenly from its previous residual stress-free state, it expands evenly, and no thermal stress is generated. However, when a part is heated unevenly, thermal stress is produced [1].

During DMD processing, highly localized heating and cooling lead to nonuniform thermal expansion and contraction, which further results in a complicated distribution of residual stresses in the heat-affected zone (HAZ) and distortion across the entire structure. These residual stresses could be detrimental—they may cause fractures, promote fatigue, and induce unpredictable buckling during the service of deposited parts; the deformation often is harmful to the dimensional accuracies of structures. Therefore, it becomes critical to predict the two behaviors of materials after the DMD process and to optimize the manufacturing parameters to reduce the residual stresses and deformation.

### 1.3 Literature review

The thermal behavior of the DMD process has been investigated numerically by many scholars. In [2], a 2-D finite element model was created to simulate the temperature field during the laser cladding process. The results indicated that quasi-steady thermal field could not be reached in a short time. Other scholars have chosen to experimentally investigate thermal behavior. In [3], Griffith et al. employed radiation pyrometers and thermocouples to monitor the thermal signature during laser engineered net shaping (LENS) processing. The results showed that the integrated temperature reheat had a significant effect on the microstructural evolution during the fabrication of hollow H13 tool steel parts. Utilizing a two-wavelength imaging pyrometer, Wang and Felicelli [4] measured the temperature distribution in the melt pool and the area surrounding it during the LENS deposition process. It was found that the maximum temperature in the molten pool is approximately 1600. Only thermal behaviors were investigated in these papers, while no residual stresses were modeled and analyzed.

Some researchers have focused on the modeling and simulation of traditional welding processes, which share many similarities with DMD processes. Using a double-ellipsoid heat source, Gery et al. [5] generated the transient temperature distributions of the welded plates. The results demonstrated that the welding speed, energy input, and heat source distributions had important effects on the shape and boundaries of heat-affected zone. Deng [6] investigated the effects of solid-state phase transformation on the residual stress and distortion caused by welding in low carbon and medium steels. The model discovered that the solid-state phase transformation did not have a noteworthy impact on the final residual stress and the welding deformation in low carbon steel. However, the final residual stresses and the welding deformation appear to be significantly affected by the martensitic transformation in medium carbon steels. Feli et al. [7] analyzed the temperature history and the residual stress field in multi-pass, butt-welded, stainless steel pipes. It was found that in the weld zone and its vicinity, a tensile axial residual stress is produced on the inside surface and compressive axial stress at the outside surface.

Other researchers have attempted to obtain the distribution of residual stress caused by the DMD process through experiments. For example, Moat et al. [8] measured strain in three directions using a neutron diffraction beam line to calculate the stress in DMD manufactured Waspaloy blocks. They found that large tensile residual stresses exist in the longitudinal direction near the top of the structure. Zheng et al. [9] measured residual stress in PZT thin films fabricated by a pulsed laser using X-ray diffraction. Although experiments can provide relatively accurate results, their flexibility and high cost limit their ability to serve as a general method by which to solve residual stress problems.

In recent years, analyses of the residual stress involved in laser deposition processes using the FE model have been well documented in the literary. In [10], 2-D FE model was built to observe the impact of process parameters on the melt pool size, growth-direction residual stress, and material properties in laser-based deposition processes. They observed that after the deposition was completed and the wall was cooled to room temperature, large tensile stresses exist in the vertical direction at vertical free edges, which is a contrast to the observations in this study. Wang et al. [11] utilized commercial welding software SYSWELD to characterize the residual stress in LENS-deposited AISI 410 stainless steel thin-wall plates. Tensile longitudinal stresses were found near the mid-height, and compressive stresses were found near the top and bottom of the walls. In [12], Kamara et al. investigated the residual stress characteristics of a laser-deposited, multiple-layer wall of Waspaloy on an Inconel 718 substrate. The results indicated that along the length of the wall, residual stresses were almost zero at the bottom and top of the wall. Along the height of the wall, tensile stress with large magnitudes existed at the bottom of the wall while close to the top surface, near stress-free condition, seem to prevail. This matches well with the results presented in this thesis.

### 1.4 Simulation and experiment approach

Based on the finite element (FE) analysis package ABAQUS, a 3-D, sequentially coupled, thermomechanical model was developed to simulate the transient temperature field, residual stress, and final deformation involved in the DMD process of Stainless Steel 304 (SS 304). The numerical modeling consisted of two major steps shown in Figure 1. A transient thermal analysis was firstly carried out to produce the temperature history of the entire workpiece. Then, in the second step, using the temperature field file generated in the previous step as load, a mechanical analysis was carried out to calculate the residual stress and deformation of workpiece.

A laser displacement sensor was used in the experiment to record the vertical deflection of the workpiece resulted from thermal stresses during the deposition process. The accuracy of the numerical model was checked and validated by comparing the experimental results with the simulation results. This validated model can be applied to a multilayer DMD process of stainless steel under different process parameters and can be used for other materials.

## 2. Thermal analysis

### 2.1 Governing equations

During DMD processing, the stress and deformation field in the workpiece largely depend on the temperature history; however, the effect of the stress and deformation field on the temperature field is insignificant. Thus, a heat transfer analysis not coupled with mechanical effect is considered.

The transient temperature domain

where

### 2.2 Initial and boundary conditions

To solve Eq. (1), the initial conditions were set as

where

where

### 2.3 Assumptions and adjustments

Accurate modeling of the thermal process yields highly nonlinear coupled equations, which is time-consuming and expensive to solve. To speed up the solution process and reduce the computational time without sacrificing accuracy, the following assumptions and adjustments were considered.

#### 2.3.1 Energy distribution of the deposition process

In this study, a circular-shaped laser beam with a constant and uniform power density was used. Thus, to match the experiment setup, the heat source parameter

where

#### 2.3.2 Movement of laser beam

The motion of the laser beam was taken into account by updating the position of the beam’s center

where * x*,

*, and*y

*are the spatial coordinates and the laser beam centers,*z

*,*u

*, and*v

*, are the continuous velocities the laser beam travels along*w

*-,*x

*-, and*y

*-direction.*z

In ABAQUS, a user subroutine “DFLUX” [14] was written to simulate the motion of the laser beam.

#### 2.3.3 Powder projection

When modeling, the continuous powder injection process is broken into many small discrete time steps. Using the model change method provided by ABAQUS [14], in each time step, a set of finite elements was added onto the substrate to form deposits along the center line of the substrate. The width of the deposits was assumed to be the same as the diameter of the laser beam, and the thickness of the deposits was calculated from the laser or table travel speed and the powder feed rate. An efficiency of

#### 2.3.4 Modeling the latent heat of fusion

To account for the effect of the latent heat of fusion during the melting and solidification process, the specific heat capacity is modified to generate an equivalent specific heat capacity

where

Latent heat of fusion (J/kg) | Solidus temperature (K) | Liquidus temperature (K) |
---|---|---|

273,790 | 1703 | 1733 |

#### 2.3.5 Marangoni effect

As discussed in [17], the temperature distribution is significantly impacted by the effect of Marangoni flow, which is caused by the thermocapillary phenomenon. To obtain an accurate thermal field solution, based on the method proposed by [18], an artificial thermal conductivity was used to account for the Marangoni effect:

where

#### 2.3.6 Combined boundary conditions

The boundary conditions shown in Eq. (4) can be rewritten as

where

Eq. (9) shows that when the temperature is low, convection is dominant in heat loss and when temperature is high, radiation becomes dominant. As shown in Eq. (10), radiation coefficient is the third-order function of temperature

where

### 2.4 Finite element modeling

#### 2.4.1 Dimension and parameter

As shown in Figure 2, a finite element model for a one-pass, three-layer DMD process was built. The dimension of the substrate under consideration is

Case number | Laser power (W) | Laser travel speed (mm/min) | Powder feed rate (g/min) |
---|---|---|---|

1 | 607 | 250 | 6.3 |

2 | 910 | 375 | 9.4 |

#### 2.4.2 Material properties

Temperature-dependent thermal physical properties of SS 304, including the density, specific heat, thermal conductivity, and latent heat, were used as inputs. The values of these properties appear in Figure 3.

#### 2.4.3 Element selection method

Based on the computational accuracy and cost, the type and size of the finite elements used to simulate the substrate and deposits were determined. In transient heat transfer analysis, second-order elements generally produce more accurate results; however, there is a minimum time increment. A simple guideline can be written as [14]

where * ρ*,

#### 2.4.4 Increment control

In order to obtain reliable results from the mechanical analysis, the maximum nodal temperature change in each increment was set as

## 3. Mechanical analysis

### 3.1 Governing equations

The total strain

where

where

The elastic stress-strain relationship is governed by isotropic Hooke’s law as

where

where

For isotropic elastic solids, Eq. (15) can be simplified as

Thermal strain

where

Unlike the elastic and thermal strain, no unique relationship exists between the total plastic strain and stress; when a material is subjected to a certain stress state, there exist many possible strain states. So strain increments, instead of the total accumulated strain, were considered when examining the strain-stress relationships. The total strain then was obtained by integrating the strain increments over time

where

By substituting Eq. (18), Eq. (19), Eq. (20), and Eq. (21) into Eq. (14) and taking the derivative with respect to time, the total strain rate can be described by [2]

### 3.2 Initial and boundary conditions

The temperature history of all the nodes generated in the thermal analysis was imported as a predefined field into the mechanical analysis. The only boundary condition applied to the domain was that the substrate was fixed on one side to prevent rigid body motion. In ABAQUS, the node displacements on the left side of the substrate were set as 0.

### 3.3 Finite element modeling

#### 3.3.1 Material properties

Temperature-dependent mechanical properties including the thermal expansion coefficient [23], Young’s modulus, Poisson’s ratio [21], and yield stress [16] were used to model the thermomechanical behavior of SS 304. The values of these properties appear in Figure 4.

#### 3.3.2 Element selection

The order of element and integration method used in the mechanical analysis differed from those used in the thermal analysis, while the element dimension and meshing scheme remained unchanged. To ensure the computational accuracy of the residual stress and deformation, second-order elements were utilized in the heat-affected zone, while first-order elements were used in other regions to reduce the computation time. Preventing shear and volumetric locking [14] requires the selection of reduced-integration elements. Therefore, elements “C3D20R” and “C3D8R” in ABAQUS were combined in use to represent the domain.

As shown in Figure 5, the 3-D 20-node element used in the mechanical analysis had 12 more nodes than the 3-D 8-node element used in the thermal analysis. Therefore, when mapping the temperature data from the thermal analysis to the mechanical analysis, interpolation had to be conducted to obtain the temperature of the 12 extra mid-side nodes (Nodes 9–20 in Figure 5(b)).

## 4. Numerical results and experimental validation

### 4.1 Temperature

#### 4.1.1 Temperature field

Figure 6 shows the temperature field of the melt pool and surrounding areas from top view at different times in Case 1 (laser power 607 W, laser travel speed 250 mm/min, powder feed rate 6.3 g/min). Laser beam cyclically moves along +z and –z-direction. At

#### 4.1.2 Temperature gradient

The temperature gradient involved in the DMD process was quantitatively analyzed in details. The temperature of nodes along the * x*′ and

*′ (shown in Figure 8) axes in simulation Case 1 at*y

*′-direction nodes were selected along the top surface of the substrate (bottom surface of the deposits), while the*x

*′-direction nodes were selected along the height of the deposits. The temperature of the substrate’s top surface reached a maximum of*y

*′-direction. In the*x

*′-direction, the temperature of the deposits reached a maximum of*y

*′- and*x

*′-directions. Along*y

*′, the temperature gradient reached a maximum of*x

*′, the maximum temperature gradient occurred near the top surface of the deposits, reaching*y

*′-direction. These steep thermal gradients induced large compressive strains within the deposits and substrates [24].*y

### 4.2 Residual stress

The nature and magnitude of residual stresses existing in final deposits would affect the integrity of the entire structure. In general conditions, compressive residual stresses are advantageous since they increase the load resistance and prevent crack growth, while tensile residual stresses are detrimental that they reduce the load resistance and accelerate crack growth.

The residual stress (in Pa) distribution within the final deposits is shown in Figures 10 and 11, where Figure 10 shows the whole substrate and deposit, while Figure 11 shows a * y*–

*cross section view with half of the deposits hidden to show the internal residual stress. Normal stresses*y

Various experimental methods for measuring residual stress have been developed, such as destructive methods, including incremental hole drilling, layer removal, and crack compliance, and nondestructive methods including X-ray diffraction and neutron diffraction [8, 9, 24, 25, 26, 27]. These methods could be used to measure the residual stress directly with relatively good accuracy; however, they usually are not cost-effective or easy to set up. Therefore, instead of measuring the residual stress directly, a flexible indirect method has been developed for residual stress validation. A one-one relationship exists between the deflection of the substrate and residual stress; therefore, by validating the deflection of the substrate, the residual stress results can be validated indirectly.

### 4.3 Deformation

During the DMD process, the substrate will continuously expand and shrink, finally maintaining a deformed shape (Figure 12). In this study, deflection along y was the main deformation under consideration and is shown in Figure 13.

#### 4.3.1 Experiment setup

As shown in Figure 14, in the experiment, the substrate was clamped at the left end to prevent rigid body motion. Keyence’s LK-G5000 series laser displacement sensor shown in Figure 15 was placed just below the right end of the substrate to record the displacement of the free end along the y-direction with a frequency of 25 Hz during the process. The experimental results appear in Figure 16. The entire DMD process was controlled by the “Laser Aided Material Deposition System”.

#### 4.3.2 Experimental and simulation results

Figure 16 illustrates the comparisons of the substrate deflection between the experimental and simulation results for both cases. It is obvious that the simulated deflection matched well with the experimental results. During each deposition layer, the substrate firstly bent down due to thermal expansion on the top surface and then bent up due to thermal shrinkage from the cooling process. After cooling down, the substrate still maintained its distorted shape.

The deflection values from simulation were 28.5 and 24.6% higher than the deflection measured from experiments for Cases 1 and 2, respectively. There are several potential factors that could cause these differences. Firstly, the experimental setup could not perfectly match the setup in the simulation. For example, in the simulation, the laser beam traveled exactly along the centerline of the substrate. While in experiments (Figure 12), this cannot be perfectly achieved. These offsets may greatly affect the deflection value since it is very sensitive to the positions of heated zone (where expansion and shrinkage mainly happen) and measuring point. Moreover, the laser displacement sensor did not track the displacement of one particular point on the workpiece. Instead, it sensed the signal reflected by an obstacle, so the positions it tracked were always changing as the substrate continuing to deform. Last but not the least, the simplifications and assumptions considered in both thermal and mechanical analyses could also contribute to the differences between the simulation and experiment. For example, although the substrate material is originally isotropic elastic solid, it may become orthotropic after the DMD process.

It is also worth noting that for Cases 1 and 2, even the total amount of energy applied to the substrate is the same and Case 2 has significantly higher distortion than Case 1, which is caused by higher laser power.

## 5. Conclusion

To investigate the features of thermal and mechanical behavior of deposited materials involved in the DMD process, a sequentially coupled, thermomechanical finite element model was developed for multilayer DMD process of Stainless Steel 304. The results revealed the characteristics of temperature distribution, residual stress, and deformation within the formed deposits and substrates. A set of experiments were conducted to validate the mechanical effects using a laser displacement sensor. This FEA model can be used to predict the mechanical behavior of products fabricated by the DMD process or similar processes with localized heat sources such as laser sintering, laser cladding, and welding.

## Acknowledgments

The support from NASA’s grant, under NRA NNX11AI73A, is appreciated. The authors would like to acknowledge William Seufzer and Karen Taminger of NASA Langley Research Center for their mentorship. Support from Missouri S&T’s Material Research Center, Manufacturing Engineering program, and Intelligent Systems Center is also greatly appreciated.

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