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

#### *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]

$$
\Delta t > \frac{\mathsf{G}\mathsf{c}}{\rho k} \Delta l^2 \tag{12}
$$

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



**Table 2.** *DMD process parameters.*

strain caused by transformation plasticity. Solid-state phase transformation does

*Trp*

*ij* <sup>þ</sup> *<sup>ε</sup><sup>T</sup>*

*ij* <sup>þ</sup> *<sup>ε</sup><sup>P</sup>*

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

where *Dijkl* is the elastic stiffness tensor calculated from Young's modulus *E* and

<sup>2</sup> *<sup>δ</sup>ikδjl* <sup>þ</sup> *<sup>δ</sup>ijδkl* <sup>þ</sup>

*<sup>δ</sup>ij* <sup>¼</sup> <sup>1</sup> *for i* <sup>¼</sup> *<sup>j</sup>*

*<sup>E</sup> <sup>σ</sup>ij* � *<sup>ν</sup>*

where *α* is the thermal expansion coefficient and *ΔT* is the temperature difference between two different material points. Rate-independent plasticity with the von Mises yield criterion and linear kinematic hardening rule [21] was utilized to

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 *t*. The plastic strain-stress relationship for isotropic material is governed by the

*dε<sup>P</sup>*

*sij* <sup>¼</sup> *<sup>σ</sup>ij* � <sup>1</sup>

the derivative with respect to time, the total strain rate can be described by [2]

3

By substituting Eq. (18), Eq. (19), Eq. (20), and Eq. (21) into Eq. (14) and taking

*ij* is the plastic strain increment, *λ* is the plastic multiplier, and *sij* is the

*εT*

0 *for i* 6¼ *j*

*ij* can be calculated from the thermal expansion constitutive

*ij* vanish. The total strain vector is then

*ij* ð Þ *i*, *j*, *k*, *l* ¼ 1, 2, 3 (16)

*ν* 1 � 2*ν <sup>δ</sup>ijδkl* (17)

*ij* (15)

*<sup>E</sup> <sup>σ</sup>kkδij* (19)

*ij* ¼ *αΔTδij* (20)

*ij* ¼ *λsij* (21)

*σkkδij* (22)

(18)

*ij* and *ε*

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

*<sup>ε</sup>ij* <sup>¼</sup> *<sup>ε</sup><sup>E</sup>*

*<sup>σ</sup>ij* <sup>¼</sup> *Dijklε<sup>E</sup>*

1

*Dijkl* <sup>¼</sup> *<sup>E</sup>*

1 þ *ν*

where *δij* is the Kronecker delta function defined as

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

*εE ij* <sup>¼</sup> <sup>1</sup> <sup>þ</sup> *<sup>ν</sup>*

not exist in stainless steel [21], so *ε<sup>Δ</sup><sup>V</sup>*

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

represented as

Poisson's ratio *ν* as [11]

Thermal strain *ε<sup>T</sup>*

model the plastic strain.

Prandtl-Reuss equation [22]:

deviatoric stress tensor defined by

where *dε<sup>P</sup>*

**95**

equation

**Figure 3.** *Temperature-dependent mechanical properties of SS 304.*

oscillations may appear in the solution. According to [14], the first-order elements can eliminate such oscillations but may lead to inaccurate solutions [13]. To take both the computational efficiency and accuracy into account, the first-order heat transfer elements, C3D8, with h-version mesh refinement, to refine the mesh by subdividing existing elements into more elements of the same order, were used for the whole domain. Fine meshes were used in zones close to the deposits, and the mesh size progressively increased with the distance from the deposits. In regions more away from the heat-affected zone, coarser meshes were implemented. A total of 14,496 elements and 17,509 nodes was created.

#### *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 5 K, and the time increments were selected automatically by ABAQUS to ensure that this value was not exceeded at any node during any increment of the analysis [14].

### **3. Mechanical analysis**

#### **3.1 Governing equations**

The total strain *εij* can be represented generally as

$$
\varepsilon\_{\vec{\imath}} = \varepsilon\_{\vec{\imath}}^{\mathcal{M}} + \varepsilon\_{\vec{\imath}}^{T} \tag{13}
$$

where *ε<sup>M</sup> ij* is the strain contributed by the mechanical forces and *ε<sup>T</sup> ij* is the strain from thermal loads. Eq. (12) can be decomposed further into five components as [6]

$$
\varepsilon\_{\vec{\imath}\vec{\jmath}} = \varepsilon\_{\vec{\imath}\vec{\jmath}}^E + \varepsilon\_{\vec{\imath}\vec{\jmath}}^P + \varepsilon\_{\vec{\imath}\vec{\jmath}}^T + \varepsilon\_{\vec{\imath}\vec{\jmath}}^{Trp} + \varepsilon\_{\vec{\imath}\vec{\jmath}}^{Trp} \tag{14}
$$

where *ε<sup>E</sup> ij* is the elastic strain, *ε<sup>P</sup> ij* is the plastic strain, *ε<sup>T</sup> ij* is the thermal strain, *ε<sup>Δ</sup><sup>V</sup> ij* is the strain due to the volumetric change in the phase transformation, and *ε Trp ij* is the

strain caused by transformation plasticity. Solid-state phase transformation does not exist in stainless steel [21], so *ε<sup>Δ</sup><sup>V</sup> ij* and *ε Trp ij* vanish. The total strain vector is then represented as

$$
\varepsilon\_{\vec{\imath}\vec{\jmath}} = \varepsilon\_{\vec{\imath}\vec{\jmath}}^{E} + \varepsilon\_{\vec{\imath}\vec{\jmath}}^{P} + \varepsilon\_{\vec{\imath}\vec{\jmath}}^{T} \tag{15}
$$

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

$$
\sigma\_{\vec{\imath}\vec{\jmath}} = D\_{\vec{\imath}\vec{\jmath}kl} \epsilon\_{\vec{\imath}\vec{\jmath}}^{E} \quad (i, j, k, l = 1, 2, 3) \tag{16}
$$

where *Dijkl* is the elastic stiffness tensor calculated from Young's modulus *E* and Poisson's ratio *ν* as [11]

$$D\_{ijkl} = \frac{E}{\mathbf{1} + \nu} \left[ \frac{\mathbf{1}}{\mathbf{2}} \left( \delta\_{ik} \delta\_{jl} + \delta\_{ij} \delta\_{kl} \right) + \frac{\nu}{\mathbf{1} - \mathbf{2}\nu} \delta\_{ij} \delta\_{kl} \right] \tag{17}$$

where *δij* is the Kronecker delta function defined as

$$
\delta\_{ij} = \begin{cases}
\mathbf{1} & \text{for} \quad i = j \\
\mathbf{0} & \text{for} \quad i \neq j
\end{cases}
\tag{18}
$$

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

$$
\sigma\_{i\dot{j}}^E = \frac{1+\nu}{E}\sigma\_{i\dot{j}} - \frac{\nu}{E}\sigma\_{kk}\delta\_{\dot{j}} \tag{19}
$$

Thermal strain *ε<sup>T</sup> ij* can be calculated from the thermal expansion constitutive equation

$$
\boldsymbol{\varepsilon}\_{\ddagger}^T = \boldsymbol{a} \boldsymbol{\Delta} \boldsymbol{T} \boldsymbol{\delta}\_{\ddagger} \tag{20}
$$

where *α* is the thermal expansion coefficient and *ΔT* is the temperature difference between two different material points. Rate-independent plasticity with the von Mises yield criterion and linear kinematic hardening rule [21] was utilized to model the plastic strain.

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 *t*. The plastic strain-stress relationship for isotropic material is governed by the Prandtl-Reuss equation [22]:

$$d\varepsilon\_{\vec{\text{ij}}}^{P} = \lambda \mathfrak{s}\_{\vec{\text{ij}}} \tag{21}$$

where *dε<sup>P</sup> ij* is the plastic strain increment, *λ* is the plastic multiplier, and *sij* is the deviatoric stress tensor defined by

$$
\sigma\_{\vec{v}\,\vec{j}} = \sigma\_{\vec{v}\,\vec{j}} - \frac{1}{3} \sigma\_{kk} \delta\_{\vec{v}\,\vec{j}} \tag{22}
$$

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]

oscillations may appear in the solution. According to [14], the first-order elements can eliminate such oscillations but may lead to inaccurate solutions [13]. To take both the computational efficiency and accuracy into account, the first-order heat transfer elements, C3D8, with h-version mesh refinement, to refine the mesh by subdividing existing elements into more elements of the same order, were used for the whole domain. Fine meshes were used in zones close to the deposits, and the mesh size progressively increased with the distance from the deposits. In regions more away from the heat-affected zone, coarser meshes were implemented. A total

In order to obtain reliable results from the mechanical analysis, the maximum nodal temperature change in each increment was set as 5 K, and the time increments were selected automatically by ABAQUS to ensure that this value was not

*<sup>ε</sup>ij* <sup>¼</sup> *<sup>ε</sup><sup>M</sup>*

strain from thermal loads. Eq. (12) can be decomposed further into five

*ij* <sup>þ</sup> *<sup>ε</sup><sup>P</sup>*

the strain due to the volumetric change in the phase transformation, and *ε*

*ij* is the strain contributed by the mechanical forces and *ε<sup>T</sup>*

*ij* <sup>þ</sup> *<sup>ε</sup><sup>T</sup>*

*ij* <sup>þ</sup> *<sup>ε</sup><sup>T</sup>*

*ij* <sup>þ</sup> *<sup>ε</sup><sup>Δ</sup><sup>V</sup> ij* þ *ε Trp*

*ij* is the plastic strain, *ε<sup>T</sup>*

*ij* (13)

*ij* (14)

*ij* is the thermal strain, *ε<sup>Δ</sup><sup>V</sup>*

*ij* is the

*Trp ij* is the

*ij* is

of 14,496 elements and 17,509 nodes was created.

*Temperature-dependent mechanical properties of SS 304.*

*New Challenges in Residual Stress Measurements and Evaluation*

exceeded at any node during any increment of the analysis [14].

The total strain *εij* can be represented generally as

*<sup>ε</sup>ij* <sup>¼</sup> *<sup>ε</sup><sup>E</sup>*

*ij* is the elastic strain, *ε<sup>P</sup>*

*2.4.4 Increment control*

**Figure 3.**

**3. Mechanical analysis**

**3.1 Governing equations**

where *ε<sup>M</sup>*

where *ε<sup>E</sup>*

**94**

components as [6]

$$\dot{\varepsilon}\_{\vec{\imath}\vec{\jmath}} = \frac{\mathbf{1} + \nu}{E} \dot{\sigma}\_{\vec{\imath}\vec{\jmath}} - \frac{\nu}{E} \dot{\sigma}\_{kk} \delta\_{\vec{\imath}\vec{\jmath}} + a \dot{T} \delta\_{\vec{\imath}\vec{\jmath}} + \lambda \left(\sigma\_{\vec{\imath}\vec{\jmath}} - \frac{\mathbf{1}}{3} \sigma\_{kk} \delta\_{\vec{\imath}\vec{\jmath}}\right) \tag{23}$$

#### **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**.

Therefore, when mapping the temperature data from the thermal analysis to the mechanical analysis, interpolation had to be conducted to obtain the temperature of

*Elements used in thermal and mechanical analysis. (a) 8-node brick element (b) 20-node brick element.*

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

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

**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 *t* ¼ 0*:*9 s, *t* ¼ 2*:*7 s, and *t* ¼ 4*:*5 s, laser beam is located in the center of the substrate. **Figure 7** shows the temperature field and isotherms of the substrate and deposits from the side view at *t* ¼ 4*:*5 s in Case 1. The peak temperature during the process was around 2350 K, while the lowest temperature was close to room temperature. The big temperature differences and small geometrical

The temperature gradient involved in the DMD process was quantitatively analyzed in details. The temperature of nodes along the *x*<sup>0</sup> and *y*<sup>0</sup> (shown in **Figure 8**)



, the temperature gradient reached a maximum of483 K*=*mm;





the 12 extra mid-side nodes (Nodes 9–20 in **Figure 5(b)**).

**4. Numerical results and experimental validation**

dimensions caused very large temperature gradients.

axes in simulation Case 1 at *t* ¼ 4*:*5 s is shown in **Figure 9**. The *x*<sup>0</sup>

were selected along the top surface of the substrate (bottom surface of the

deposits. The temperature of the substrate's top surface reached a maximum of 1069 K just below the center of the laser beam and decreased gradually along the *x*<sup>0</sup>

2220 K on the top surface of the deposits and decreased rapidly to1069 K . The slopes of the temperature curves represent the thermal gradients along the *x*<sup>0</sup>

deposits, reaching 1416 K*=*mm and then decreasing along the negative *y*<sup>0</sup>

, the maximum temperature gradient occurred near the top surface of the

**4.1 Temperature**

**Figure 5.**

*4.1.1 Temperature field*

*4.1.2 Temperature gradient*

deposits), while the *y*<sup>0</sup>


direction. In the *y*<sup>0</sup>

*y*0

**97**

along *y*<sup>0</sup>

#### *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 heataffected 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.

**Figure 4.** *Temperature-dependent mechanical properties of SS 304.*

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

**Figure 5.** *Elements used in thermal and mechanical analysis. (a) 8-node brick element (b) 20-node brick element.*

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)**).
