Analytical Modeling of Residual Stress in Selective Laser Melting Considering Volume Conservation in Plastic Deformation

: Residual stress (RS) is the most challenging problem in metal additive manufacturing (AM) since the build-up of high tensile RS may influence the fatigue life, corrosion resistance, crack initiation, and failure of the additively manufactured components. While tensile RS is inherent in all the AM processes, fast and accurate prediction of stress state within the part is extremely valuable and would result in optimization of the process parameters in achieving a desired RS and control of the build process. This paper proposes a physics-based analytical model to rapidly and accurately predict the RS within the additively manufactured part. In this model, a transient moving point heat source (HS) is utilized to determine the temperature field. Due to the high temperature gradient within the proximity of the melt pool area, material experience high thermal stress. Thermal stress is calculated by combining three sources of stresses known as stresses due to the body forces, normal tension, and hydrostatic stress in a homogeneous semi-infinite medium. The thermal stress determines the RS state within the part. Consequently, by taking the thermal stress history as an input, both the in-plane and out of plane RS distributions are found from incremental plasticity and kinematic hardening behavior of the metal by considering volume conservation in plastic deformation in coupling with the equilibrium and compatibility conditions. In this modeling, material properties are temperature-sensitive since the steep temperature gradient varies the properties significantly. Moreover, the energy needed for the solid-state phase transition is reflected by modifying the specific heat employing the latent heat of fusion. Furthermore, the multi-layer and multi-scan aspects of metal AM are considered by including the temperature history from previous layers and scans. Results from the analytical RS model presented excellent agreement with XRD measurements employed to determine the RS in the Ti-6Al-4V specimens.

biomedical, aerospace, automotive, and marine as stated by Herderick [1]. Namatollahi et al. [2] stated that "AM is the process of joining metallic powders layer by layer to produce complex threedimensional parts". AM has several advantageous over conventional manufacturing including lower density, reduction in scrap rate, complex nearly net-shape parts, elimination of multi-step manufacturing, and many more as expressed by Camacho et al. [3]. In contrast, there are still some limitations that hinder the applicability of AM such as steep temperature gradient, high thermal stress, and tensile residual stress (RS) as reported by Ngo et al. [4], anisotropy and heterogeneity in microstructure as explained in the work of Ji et al. [5] and mechanical properties as explained in the work of Tabei et.al [6]. Bartlett et al. [7] described that steep temperature gradient induced by high laser power and low conduction may cause high thermal stress in the component. Since the yield strength of the material depends on temperature, and at elevated temperatures around the melting point, materials are usually soft, they can easily undergo plastic deformation. Consequently, due to the repeated heating and cooling, material experience a high magnitude of RS. Residual stress is inherent in all the manufactured components, and high RS is reported by many researchers. Roehling et al. [8] reported high tensile RS in parts built via powder bed fusion (LPBF) systems. Wang et al. [9] measured the RS using neutron diffraction, they observed high tensile RS in IN625 parts. An et al. [10] reported high RS in curved thin-walled structure manufactured via LPBF. Denlinger and Heigel [11] simulated the RS during the additively manufactured Ti-6Al-4V samples, they stated that the predicted RS was considerably high. Zhao et al. [12] predicted high RS in titanium alloys build via direct metal laser sintering (DMLS) using FEM. Romano et al. [13] reported that RS has a crucial impact on fatigue life of the component, corrosion resistance, crack initiation and growth, and also microstructure and mechanical properties of the materials.
There is a considerable amount of research on literature explaining different methods for the prediction of RS including experimentation, numerical modeling, and analytical modeling.
Experimental procedures to measure the RS in the components could be categorized into destructive and non-destructive methods. The non-destructive methods can be classified into X-ray diffraction-which is cable of near surface RS measurements-and neutron diffraction-which is capable of volumetric measurements. Other non-destructive methods include ultra-sonics as explained by Noronha and Wert [14], electrical resistivity as explained by Chung [15] magnetic behavior as explained by Krause et al. [16] and piezo-spectroscopy in thin films as explained by Ager III and Drory [17]; these are material and geometry specific. Destructive methods such as hole drilling, sectioning, crack compliance, digital image correlation, and electronic speckle pattern interferometry (ESPI) essentially create a free surface in the part and correlate resultant deformation to RS as stated by Prime [18].
Numerical modeling is another approach for the forecast of RS. Aggarangsi and Beuth [19] used a finite element method (FEM) to simulate the RS. In their modeling, they defined a temperature gradient from the melt pool and correlated it to the maximum RS. They have shown that localized preheating could reduce the maximum RS in the additively manufactured 304 stainless steel. Panda and Sahoo [20] used FEM to predict the RS in the direct metal deposition (DMD) of AlSi10Mg. They have predicted temperature distribution using transient temperature and coupled the results to a structural model to predict the RS. In their modeling, the thermal and mechanical material properties are considered constant. Also, the effects of scan strategies are not considered in this modeling. Chen et al. [21] used FEM to simulate the RS in additively manufactured parts using inherent strain method. They have utilized the temperature results obtained from thermocouples to calibrate the process parameters. Then, they have used Goldak's HS model to obtain the thermal gradients and the inherent strains. Ganeriwala et al. [22] also used FEM to simulate the stress state in laser powder bed fusion of Ti-6Al-4V. They have used lumping approach to speed up the computations. They indicated that the stress is higher near the boundaries for the island scan strategies. Ding and Shin [23] proposed a 3D thermo-elastoplastic finite element model to simulated the RS. They have validated the results using neutron diffraction strain scanner.
Physics-based closed-form analytical solutions are the other approach for the prediction of RS. Prediction of RS should be started by modeling of the temperature field induced by laser in AM.
There are various analytical methods to model the temperature field in metal AM. Mirkoohi et al. [24] introduced five distinct HS models to analytically simulate the three-dimensional temperature profile. They have investigated the viability of all the models under various process conditions. They have considered the material properties temperature-dependent; phase change is also considered through modification of specific heat. Furthermore, the effect of scan strategy is also considered in their modeling. Mirkoohi et al. [25] have proposed a model to analytically simulate the temperature field considering the effect of layer thickness. Fergani et al. [26] presented a model for the prediction of stress. In their modeling, properties are considered constant. Also, the effect of scan strategy is not considered in their modeling. Moreover, the stress in the build direction is considered to be elastic.
Experimentation is a different approach which facilitates the perception of RSin metal AM. But, measuring the stress state of the entire part is challenging, time-consuming, and expensive. Physicsbased numerical models are an assuring approach for the calculation of RS when the results are compared to experiments. However, the simulation of the entire process considering the multiphysics aspects of metal AM cannot be achieved in a reasonable amount of time. On the other hand, physics-based closed-form analytical solutions validated by physical experiments enable prediction of the stress state within the additively manufactured part much faster than FEM and experimentation. Furthermore, the multi-physics aspect of metal AM such as multi-layer and multiscan aspect of AM-which is challenging to consider by any other methods-can be considered in the process modeling of AM using closed-form analytical solutions. This accurate and reliable model can be adopted to optimize the AM parameters and guide the build process in achieving the desired part quality. Consequently, having one model to accurately and rapidly predict the stress state within the part is extremely valuable.
The physics governing the AM is considerably complicated, thus, capturing the entire physical phenomena involved in this process currently is not possible. In this work, a physics-based closedform analytical model is proposed to predict the RS within the additively manufactured part. Herein, the proposed thermomechanical model is capable of the prediction of RS accurately in a few seconds. Prediction of RS cannot be achieved within this amount of time by any numerical methods or experimentation, this shows the computational efficiency of the proposed model. In this work, first, a transient moving point HS is employed to analytically simulate the in-process temperature field within the components. Second, temperature gradient along the scan direction (x-axis) and build direction (z-axis) is calculated from temperature field. Third, thermal stress is obtained by combining three different sources of stress known as stresses due to body forces, normal tension, and hydrostatic stress. Last, the RS distributions along the build direction and also along the scan direction are found from incremental plasticity and kinematic hardening behavior of the metal based upon the premises of plane strain condition in the build of isotropic and homogeneous properties. Moreover, the RS is calculated according to the conservation of volume in plastic deformation, in coupling with the equilibrium and compatibility conditions. In this modeling, the thermal and mechanical material properties are temperature-sensitive, since the steep temperature gradient causes the material properties to vary substantially. Moreover, due to the cyclic melting and solidification, the build part experiences phase change. The heat capacity is modified to consider the effect of latent heat of melting during this process. Furthermore, the impacts of multi-layer and multi-scan aspects of metal AM is analyzed by including the effect of temperature history on former layers and scans.
X-ray diffraction experimental measurements are conducted to validate the proposed analytical model. The Ti-6Al-4V samples are built using direct metal deposition (DMD) process under different process condition. Good agreement is obtained between measured and predicted RS.

Process Modeling
The process modeling of residual stress is divided into three main parts; (1) prediction of temperature distribution, (2) prediction of thermal stress, and (3) prediction of residual stress. To predict the temperature distribution in the presence of a laser, different approaches are proposed by Carslaw and Jaeger [27] and Goldak et al. [28]. The complete review of the available heat source modeling can be found in [24].

Thermal Analysis
In this section, a transient three-dimensional heat source model is used to predict the temperature field as; This equation is obtained by solving the differential equation of heat conduction as explained in [24].
where is laser power, is absorption coefficient, is density, is the modified heat capacity to consider the latent heat of fusion for solid-state phase change as explained in the work of Mirkoohi et al. [24].
is scan speed, is thermal diffusivity, and 0 is the initial temperature.

Mechanical Analysis
Steep temperature gradient and fast solidification may induce thermal stress. The resulting stress is calculated by the combination of the different sources of stresses due to the body forces, normal stress tension, and hydrostatic stress as described by Saif et al. [29]: where, is the coefficient of the thermal expansion, E represents the elastic modulus, is the temperature gradient and ( ) is expressed by: At elevated temperatures, the yield strength of the Ti-6Al-4V approaches zero. Thus, the AM part experiences a high magnitude of plastic deformation due to heating cycles and cooling cycles. If the stress passes the yield strength, upon unloading (cooling in this situation) some amount of stress remains in the body which is known as RS. The RS is inherent in all the AM parts and can be reduced or eliminated with proper control and optimization of process parameters or post-processing techniques such as pitting, and heat treatment since it has an elastic nature.
Both the in-plane and out of plane RS distributions are obtained from incremental plasticity and kinematic hardening behavior of metal according to the conservation of volume in plastic deformation in coupling with equilibrium and compatibility conditions. McDowell et al. [30] proposed an algorithm to predict the RS in rolling process. This algorithm has several advantages including high computational efficiency, rapid and accurate prediction of RS; however, this algorithm has several limitations and should be modified for the AM applications; First, the stress along the build direction ( ) is assumed to be elastic. In AM processes, the stress along the build direction has a high magnitude. Thus, the stress along the build direction should be solved simultaneously with , and . In this case, there are two equations with three unknowns, which cannot be solved using McDowell model. Second, this algorithm does not consider the conservation of volume in plastic deformation. Qi et al. [31] explained that these limitations may reduce the precision of the predicted RS.
The Johnson Cook flow stress model is used to determine the yield surface.
where is material yield stress, represents the effective plastic strain, ̇ is the effective plastic strain rate, is the temperature of material, is the melting point of material, and 0 is the initial temperature. The terms , , , , and ̇0 are the material constant which are listed in Table  1 for Ti-6Al-4V.
If the obtained yield stress passes the yield threshold of the material, the total plastic strain can be calculated incrementally. The plastic strain rate can is introduced by Khan and Huang [32] as , and ℎ is the plastic modulus.
As mentioned before, the previous state of the art algorithm assumed the stress along the build direction (z-direction) is elastic and ignored the volume conservation in plastic deformation. Below we explaine how these two limitations can be solved by considering the elastic and plastic components of strain in all three directions. In the elastoplastic loading the total strain is a combination of elastic part and plastic part as According to the conservation of volume in plastic deformation; ̇+̇+̇= 0 (7) Based on plain strain assumption Taking plastic flow rule (Equation (5)) into Equation (9); Based on plastic loading condition; Therefore; ( The incremental form of Equation (13) can be written as; ̇= Then, by solving three equations with three unknowns, ( , , ) can be obtained as; where, ̇ * ,̇ * ,̇ * are the elastic thermal stresses calculated from Equations (2 and 3).
is the hybrid function which depends on the modulus ratio (ℎ/ ) as; where = 0.15 is the algorithm constant, and = /(2(1 + ))is the elastic shear modulus. Three systems of equations are solved simultaneously for ̇, ̇, and ̇ for each elastic-plastic increment of strain.  Figure 4. These data are gathered from the work of Welsch et al. [34], Heigel et al. [35] Jamshidinia et al. [36], Li et al. [37] Mills [38] , and Murgau [39]. The melting temperature range of Ti-6Al-4V is from 1600℃ to 1660℃. In this modeling melting temperature is 1600℃. Density of Ti-6Al-4V drops for about 6.7% from room temperature to 2000℃; thermal conductivity increases as the temperature increases; specific heat increases gradually with the increase in temperature; coefficient of thermal expansion increases gradually up to 1000℃, and reaches a constant value; elastic modulus decreases with the increase in temperature and approaches zero at around melting temperature. This shows the material at liquid phase has negligible elasticity; Poisson's ratio increases with the increase in temperature; yield strength decreases rapidly from room temperature up to 1000℃ and slows the slope above1000℃. This signifies that the Ti-6Al-4V is extremely soft and it is quite easy to undergo plastic deformation at elevated temperatures which causes the medium to experience high residual stress.  Yield strength The equations of temperature sensitive material properties are also derived from the data points as listed in Table 2.

Experimental Residual Stress Analysis
Two blocks of Ti-6Al-4V specimens with the size of 20×10×3 mm is produced via DMD process using LENS CS 1500 SYSTEMS under different process conditions as shown in Table 3. The selected laser powers are 206 W, and 385 W, and the scan speeds are 25 mm/s, 40 mm/s, respectively. The deposited layer thickness for both samples is 250 , and hatch spacing is 105 . A bi-directional continuous scan strategy is used to build the parts as shown in Figure 2. PANalytical Empyrean multipurpose X-ray diffractometer is used to measure the RS of the specimens using the where, and 0 are the stressed and unstressed lattice parameter, respectively. The generalized Hook's law for isotropic material is used to calculate stress as = (1+ )(1−2 ) ((1 − ) + ( + ) where , , ∈ , , .

(g)
In Equation (35), an elastic modulus ( ), and Poisson's ratio of 114 GPa and 0.33 are used, respectively. Samples are polished using liquid abrasive of 1 and 0.05 at a very slow speed to eliminate macroscopic residual stresses. Measurements are collected every 0.5 mm along the build direction (z-axis) of the samples. Figure 5 illustrates the melt pool area and heat affected zone induced by highly localized heat input and low thermal conductivity. Non-uniform heating induces non-uniform thermal expansion. During heating cycle, the melt pool area and the heat affected zone are trying to expand; however, they are surrounded by solidified metal. This would generate the compressive stress state within the melt pool and heat-affected zones as shown in Figure 6(a). During the cooling cycle, the shrinkage of the material would develop tensile stress state as illustrated in Figure 6(b).

Results and Discussion
The proposed analytical model is extremely valuable since it provides fast (less than 45 seconds with 4-processor laptop) and accurate prediction of stress state within the build. In this modeling, first, a transient moving heat source approach is used to predict the temperature field in AM. Second, the thermal stress induced is calculated by combining three stresses known as stresses due to body forces, normal tension, and hydrostatic stress. Last, the stresses may exceed the yield point and material would experience plastic deformation. As a consequence of repeated loading and unloading (heating and cooling), material experiences high RS. Therefore, both the in plane and out of plane RS distributions are calculated from incremental plasticity and kinematic hardening behavior of the metal based upon the premises of plane strain condition in the build of isotropic and homogeneous properties, in coupling with the equilibrium and compatibility conditions.  process, material experience cyclic melting and solidification. The energy needed for the phase change is taken into account by incorporating the latent heat of fusion into specific heat. Furthermore, the multi-layer and multi-scan aspect of metal AM are considered by including the temperature histories from the previous layers and scans. Experimental measurements are conducted via X-ray diffraction to measure the RS in Ti-6Al-4V specimens built via DMD process at the middle of the specimens (X=10 mm, Y=1.5 mm) along the build direction (z-axis) at every 0.5 mm. The scan strategy in both experimentation and analytical modeling is bi-directional. Moreover, the hatching space and layer thickness are 105 , 250 , respectively. Comparison of the results from the proposed analytical model and experimentation of Ti-6Al-4V specimens built via DMD process showed good qualitative and quantitative agreement. Figure 7 illustrates the predicted temperature field for Ti-6Al-4V specimens. Figure 7(a) demonstrates the predicted melt pool area and heat-affected zone (HAZ) for the first specimen in Table 3, which has a laser power of 206 W, the scan speed of 25 mm/s, with the layer height and hatch spacing of 250 and 105 , respectively. It should be noted that the absorption ratio is 30% for Ti-6Al-4V samples as explained in the work of Selvan et al. [42]. Since the evaporation of the metallic powders is not considered in the modeling, the maximum temperature does not go beyond the evaporation temperature which is around 3000℃ for Ti-6Al-4V as reported by Selvan et al. [42]. In this figure, the melt pool depth is around 0.1 mm based on the melting point of 1600℃, and the depth of heat affected zone is around 0.15 mm. Below this depth, the material is not affected by laser. The rapid temperature change at the border of HAZ and complete solid material-which is not affected by the laser-is the region where the stress state within the build part changes from tensile to compressive. This phenomenon will be explained in detail in the following sections of this manuscript. Figure 7  As explained before material properties vary significantly since the temperature gradient is quite high in AM. The variation of thermal and mechanical material properties for the second specimen with the laser power of 385 W, and scan speed of 40 mm/s illustrated in Figure 8. Density of the solid material is around 4400 / 3 . As the temperature increase the density decreases to 4050 / 3 in the liquid zone; in the melt pool zone, thermal conductivity reaches its maximum value due to the high magnitude of temperatures, and decreases to around 5 / ℃ in the solid zone; specific heat has the highest magnitude of 530 / ℃ at the melt pool area and drops to 480 / ℃ when the material is solidified; thermal expansion is more expanded in the liquid zone compare to other properties which shows the thermal expansion is more sensitive to temperature, and has the highest magnitude in the liquid zone; elastic modulus in the liquid zone is almost zero, in the HAZ reaches Residual stress along the scan direction and transverse direction is predicted using the proposed model and validated experimentally. Figure 9 illustrates predicted RS along the scan direction and transverse direction as a function of depth into the build part for the first sample with the laser power of 206 W, scan speed of 25 mm/s, layer height of 250 , and hatch spacing of and 105 . Since the samples have the rough surface, the top surface of the samples is polished up to 100 with the electromechanical polishing to be able to accurately measure the RS on the surface.
Both the in plane and out of plane residual stresses are highly tensile in the melt pool zone and heat affected zone in coherence with most of reported results in literature as explained in introduction section. As shown in Figures 9 and 10, in both scan direction and transverse direction, there is a gradual change in stress state from tensile to compressive. This change occurs at around the depth where the medium is solidified and is not affected by the laser. The results confirm that upon cooling the melt pool and heat affected zones are under tension due to the shrinkage of the material, and when the below material is completely solidified and is not affected by the laser anymore, material experience compression stress state. The oscillations in the melt pool and heat affected zone is due to the fact that the thermal and mechanical material properties vary significantly in these regions. Therefore, an abrupt change in one of the material properties would results in oscillations. This could be better improved by having more material properties data points at different temperatures. Figure 10 shows the predicted RS in the scan and transverse directions for the second sample in Table 2 with the laser power of 385 W, scan speed of 25 mm/s, and layer height and hatch spacing of 250 and 105 , respectively. The stress state is tensile in both the scan and transverse directions and changes to compressive at the depth around 0.9 mm. Around this region the material is completely solidified and is not affected by the laser. Comparison of predicted and measured RS follows each other closely.

Conclusion
A physics-based analytical model is proposed to rapidly and accurately calculate the stress state within the additively manufactured part. As the explicit computation is executed without meshing or iteration, it can be completed at a speed several orders of magnitude higher than that of conventional FEM's. The high computational efficiency of the proposed model affects a wide range of applications becoming a powerful tool for design and also fatigue assessment when component undergoes cyclic loading. It also enables efficient control and optimization of process parameters to achieve a high-quality part.
In this modeling, a transient moving point heat source is employed to predict the in-process temperature field within the build part. Thermal stress induced by steep temperature gradient is then obtained using the Green's functions of stresses due to the point body load in a homogeneous semiinfinite medium. Last, the stress state is obtained by importing the thermal history as an input for residual stress prediction using modified McDowell algorithm, in which both the in plane and out of plane residual stress distributions are obtained from incremental plasticity and kinematic hardening behavior of the metal, in coupling with the equilibrium and compatibility conditions. In this modeling, the relationship between three normal stresses are considered according to the property of volume invariance in plastic deformation.
Due to the fast irradiation of the laser and quite low thermal conductivity of the material, the material experience high temperature gradient. High temperature gradient induced variation of the material properties in the medium. Consequently, material properties are considered temperature sensitive. Moreover, the energy needed for the solid-state phase change is considered by modifying the heat capacity using latent heat of fusion. Furthermore, the effect of layer thickness, hatching space and scan strategy is considered in this modeling by incorporating the temperature history from the previous scans.
In this work, first, the temperature field is predicted for Ti-6Al-4V specimens. Distribution of thermal and mechanical material properties including density, specific heat, thermal conductivity, elastic modulus, coefficient of thermal expansion, yield strength, and Poisson's ratio for the laser power of 385 W and scan speed of 40 mm/s showed that high temperature gradient in the AM process changes the material properties within the medium notably.
The predicted temperature fields for both samples showed that at the vicinity of melt pool area the temperature and the magnitude of temperature gradient is high enough to cause the thermal stress to exceed the yield strength of material. Upon cyclic loading and unloading of material during the heating and cooling cycles, material experience high residual stress. The predicted residual stress showed that the stress state is highly tensile in the melt pool and heat affected zones, and compressive