**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 *T x*ð Þ , *y*, *z*, *t* was attained by solving the heat conduction equation, Eq. (1), in the substrate, along with the initial and boundary conditions discussed in Section 2.2 [13]:

$$
\rho C \frac{\partial T}{\partial t} = \frac{\partial}{\partial \mathbf{x}} \left( k \frac{\partial T}{\partial \mathbf{x}} \right) + \frac{\partial}{\partial \mathbf{y}} \left( k \frac{\partial T}{\partial \mathbf{y}} \right) + \frac{\partial}{\partial \mathbf{z}} \left( k \frac{\partial T}{\partial \mathbf{z}} \right) + Q \tag{1}
$$

where *T* is the temperature, *ρ* is the density, *C* is the specific heat, *k* is the heat conductivity, and *Q* is the internal heat generation per unit volume. All material properties were considered temperature-dependent.

#### **2.2 Initial and boundary conditions**

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

$$T(\mathbf{x}, \mathbf{y}, \mathbf{z}, \mathbf{0}) = T\_0 \tag{2}$$

$$T(\mathbf{x}, \mathbf{y}, \mathbf{z}, \mathbf{ox}) = T\_0 \tag{3}$$

where *T*<sup>0</sup> is the ambient temperature. In this study, the room temperature of 298 *K* was used. The boundary conditions, including thermal convection and radiation, are governed by Newton's law of cooling and the Stefan-Boltzmann law, respectively. The heat source parameter, *Q* in Eq. (1), was considered in the boundary conditions as a surface heat source (moving laser beam). The boundary conditions then could be expressed as [13]

$$K(\Delta T \cdot \mathfrak{n})|\_{\Gamma} = \begin{cases} \left[ -h\_{\varepsilon}(T - T\_{0}) - \epsilon \sigma \left( T^{4} - T\_{0}^{4} \right) \right]|\_{\Gamma} & \Gamma \notin \Lambda \\\left[ Q - h\_{\varepsilon}(T - T\_{0}) - \epsilon \sigma \left( T^{4} - T\_{0}^{4} \right) \right]|\_{\Gamma} & \Gamma \in \Lambda \end{cases} \tag{4}$$

where *k*, *T*, *T*0, and *Q* bear their previous definitions; *n* is the normal vector of the surface; *hc* is the heat convection coefficient; *ε* is the emissivity which is 0.9; *σ* is the Stefan-Boltzmann constant which is 5*:*<sup>6704</sup> � <sup>10</sup>�<sup>8</sup> <sup>W</sup>*=*m<sup>2</sup> <sup>K</sup>4; <sup>Γ</sup> denotes the

*Residual Stress Modeling and Deformation Measurement in Laser Metal Deposition Process DOI: http://dx.doi.org/10.5772/intechopen.90539*

surfaces of the workpiece; and Λ denotes the surface area covered by the laser beam.

#### **2.3 Assumptions and adjustments**

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

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

The transient temperature domain *T x*ð Þ , *y*, *z*, *t* was attained by solving the heat conduction equation, Eq. (1), in the substrate, along with the initial and boundary

where *T* is the temperature, *ρ* is the density, *C* is the specific heat, *k* is the heat conductivity, and *Q* is the internal heat generation per unit volume. All material

where *T*<sup>0</sup> is the ambient temperature. In this study, the room temperature of 298 *K* was used. The boundary conditions, including thermal convection and radiation, are governed by Newton's law of cooling and the Stefan-Boltzmann law, respectively. The heat source parameter, *Q* in Eq. (1), was considered in the boundary conditions as a surface heat source (moving laser beam). The boundary

� � � � �

*<sup>Q</sup>* � *hc*ð Þ� *<sup>T</sup>* � *<sup>T</sup>*<sup>0</sup> *εσ <sup>T</sup>*<sup>4</sup> � *<sup>T</sup>*<sup>4</sup>

where *k*, *T*, *T*0, and *Q* bear their previous definitions; *n* is the normal vector of the surface; *hc* is the heat convection coefficient; *ε* is the emissivity which is 0.9; *σ* is the Stefan-Boltzmann constant which is 5*:*<sup>6704</sup> � <sup>10</sup>�<sup>8</sup> <sup>W</sup>*=*m<sup>2</sup> <sup>K</sup>4; <sup>Γ</sup> denotes the

� � � � �

*<sup>K</sup>*ð Þj *<sup>Δ</sup><sup>T</sup>* � *<sup>n</sup>* <sup>Γ</sup> <sup>¼</sup> �*hc*ð Þ� *<sup>T</sup>* � *<sup>T</sup>*<sup>0</sup> *εσ <sup>T</sup>*<sup>4</sup> � *<sup>T</sup>*<sup>4</sup>

þ *∂ ∂z k ∂T ∂z* � �

*T x*ð Þ¼ , *y*, *z*, 0 *T*<sup>0</sup> (2) *T x*ð Þ¼ , *y*, *z*, ∞ *T*<sup>0</sup> (3)

0

�

0

�

<sup>Γ</sup> Γ ∉ Λ

<sup>Γ</sup> Γ∈Λ

(4)

þ *Q* (1)

parameters and can be used for other materials.

*New Challenges in Residual Stress Measurements and Evaluation*

analysis not coupled with mechanical effect is considered.

properties were considered temperature-dependent.

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

*k ∂T ∂x* � � þ *∂ ∂y k ∂T ∂y* � �

conditions discussed in Section 2.2 [13]:

*<sup>ρ</sup>C∂<sup>T</sup> <sup>∂</sup><sup>t</sup>* <sup>¼</sup> *<sup>∂</sup> ∂x*

**2.2 Initial and boundary conditions**

conditions then could be expressed as [13]

**90**

(

**2. Thermal analysis**

**2.1 Governing equations**

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 *Q* in Eq. (1) was considered a constant and uniformly distributed surface heat flux defined as

$$Q = \frac{aP}{\pi r^2} \tag{5}$$

where *α* is the absorption coefficient, *P* is the power of the continuous laser, and *r* is the radius of the laser beam. *α* was set as 0.4 according to the previous experiments conducted, and *r* ¼ 1*:*25 mm.

#### *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 *R* with time *t* as follows

$$R = \left[ \left( x - \int\_{t\_0}^t u dt \right) + \left( y - \int\_{t\_0}^t v dt \right) + \left( z - \int\_{t\_0}^t w dt \right) \right]^{\frac{1}{2}} \tag{6}$$

where *x*, *y*, and *z* are the spatial coordinates and the laser beam centers, *u*, *v*, and *w*, are the continuous velocities the laser beam travels along *x*-, *y*-, and *z*-direction.

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 0*:*3 was assumed for the power feeding process to account for the powder that did not reach the melt pool.

#### *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 *c* <sup>∗</sup> *<sup>p</sup>* as [15]

$$\mathbf{C}\_p^\*\left(T\right) = \mathbf{C}\_p(T) + \frac{L}{T\_m - T\_0} \tag{7}$$

**2.4 Finite element modeling**

*DOI: http://dx.doi.org/10.5772/intechopen.90539*

*2.4.1 Dimension and parameter*

*2.4.2 Material properties*

*2.4.3 Element selection method*

written as [14]

**Figure 2.**

**Table 2.**

**93**

*DMD process parameters.*

*The dimensions of DMD specimen.*

These process parameters are detailed in **Table 2**.

The values of these properties appear in **Figure 3**.

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 50*:*8 12*:*7 3*:*175 mm (2 0*:*5 0*:*125 inch). Two cases were simulated with different process parameters including laser power, laser travel speed, and powder feed rate. These parameters were chosen according to the criterion that the final geometry of deposits and the total energy absorbed by the specimen be the same in each case.

*Residual Stress Modeling and Deformation Measurement in Laser Metal Deposition Process*

Temperature-dependent thermal physical properties of SS 304, including the density, specific heat, thermal conductivity, and latent heat, were used as inputs.

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

> *Δt*> 6*c ρk Δl*

where *ρ*, *c*, and *k* have been defined; *Δl* is the element dimension; and *Δt* is the time increment. If the time increment *Δt* is smaller than this value, nonphysical

**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

<sup>2</sup> (12)

where *C*<sup>∗</sup> *<sup>p</sup>* ð Þ *T* is the modified specific heat, *Cp*ð Þ *T* is the original temperaturedependent-specific heat, *L* is the latent heat of fusion, *Tm* is the melting temperature, and *T*<sup>0</sup> is the ambient temperature. The values of the latent heat of the fusion, solidus temperature, and liquidus temperature of SS 304 [16] appear in **Table 1**.

### *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:

$$k\_m(T) = \begin{cases} k(T) & T \le T\_{liq} \\ 2.5 \cdot k(T) & T > T\_{liq} \end{cases} \tag{8}$$

where *km*ð Þ *T* is the modified thermal conductivity, *Tliq* is the liquidus temperature, and *T* and *k T*ð Þ maintain their previous definitions.

#### *2.3.6 Combined boundary conditions*

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

$$K(\Delta T \cdot \mathfrak{n})|\_{\Gamma} = \begin{cases} [(-h\_c - h\_r)(T - T\_0)]|\_{\Gamma} & \Gamma \not\mathfrak{g} \not\subset \Lambda \\ [Q - (-h\_c - h\_r)(T - T\_0)]|\_{\Gamma} & \Gamma \in \Lambda \end{cases} \tag{9}$$

where *hr* is the radiation coefficient expressed as

$$h\_r = \varepsilon \sigma (T^2 + T\_0^2)(T + T\_0) \tag{10}$$

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 *T*, which is highly nonlinear. This would greatly increase the computational expense and time. Based on experimental data, an empirical formula combining convective and radiative heat transfer were given by [19] as

$$h = h\_{\varepsilon} + \varepsilon \sigma (T^2 + T\_0^2)(T + T\_0) \approx 2.41 \times 10^{-3} \varepsilon T^{1.61} \tag{11}$$

where *h* is the combined heat transfer coefficient which is a lower-order function of temperature *T* compared with *hr*. The associated loss in accuracy using this relationship is estimated to be less than 5% [20]. In ABAQUS, a user subroutine "FILM" is written to simulate heat loss.


**Table 1.** *Latent heat of fusion for stainless steel 304.* *Residual Stress Modeling and Deformation Measurement in Laser Metal Deposition Process DOI: http://dx.doi.org/10.5772/intechopen.90539*
