Computational Materials Science 117 (2016) 221–232

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Computational Materials Science

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Finite element simulation and experimental investigation of residualstresses in selective laser melted Ti–Ni shape memory alloy

http://dx.doi.org/10.1016/j.commatsci.2016.01.0440927-0256/� 2016 Elsevier B.V. All rights reserved.

⇑ Corresponding author at: College of Materials Science and Technology, NanjingUniversity of Aeronautics and Astronautics, Yudao Street 29, 210016 Nanjing, PRChina. Tel./fax: +86 25 52112626.

E-mail address: dongdonggu@nuaa.edu.cn (D. Gu).

Dongdong Gu ⇑, Beibei HeCollege of Materials Science and Technology, Nanjing University of Aeronautics and Astronautics, Yudao Street 29, 210016 Nanjing, PR ChinaInstitute of Additive Manufacturing (3D Printing), Nanjing University of Aeronautics and Astronautics, Yudao Street 29, 210016 Nanjing, PR China

a r t i c l e i n f o a b s t r a c t

Article history:Received 17 September 2015Received in revised form 25 January 2016Accepted 29 January 2016

Keywords:Selective laser meltingFinite element methodStress distributionTemperature gradientCrack

A three-dimensional transient finite element method (FEM) model was established to predict the stressdistribution of parts shaped by selective laser melting (SLM) technology, using Ti–Ni two-componentpowders as the raw materials. The moving heat source with a Gaussian distribution was applied in thesimulation process. By simulating the laser beam scanning process, the peak values of the thermal stres-ses were first recorded at the onset of the first track where the first heating–cooling cycle occurred. Afterthe whole part was cooled down, the largest residual stresses were found at the end of the first track andthe last track. To verify the simulation results, the experimental investigation with the same parameterswas conducted. The initial area of the first track of the fabricated part was fully dense where the residualstresses were small. The larger residual stresses obtained at the following track resulted in the formationof the cracks at the end edge of the parts, which testified that the results were in good agreement with thesimulation predictions.

� 2016 Elsevier B.V. All rights reserved.

1. Introduction

Titanium–nickel shape memory alloys (Ti–Ni SMA) are widelyused in engineering and biomedicine fields because of their typicalcharacteristics, such as shape memory effect, good elastic modulus,damping capacity, pseudo-elasticity and good biological compati-bility [1–5]. Typically, Ti–Ni alloys exhibit shape memory andsuper-elasticity effects at different temperatures [6]. During thetransformation process, the stress and temperature distributionshave significant effect on the martensitic transformation [1,7,8].However, the difficulties of alloy processing and high costs posea challenge to further applications of Ti–Ni alloys [9].

In recent years, due to the high flexibility in feedstock andshapes, selective laser melting (SLM) technique provides an alter-native way of fabricating the Ti–Ni alloys parts [10]. As one ofthe typical technology of Rapid Manufacturing (RM), the laserinduced SLM freeform fabrication can produce the customizedthree-dimensional functional components with complex configu-rations directly from metals, alloys, or composites powders[11–13]. In virtue of its flexibility in feedstock, solid freeformfabrication, highly material efficiency and near net shape produc-

tion of parts, SLM shows great superiority for the production ofthree-dimensional (3D) parts with complex configurations in com-parison with conventional methods [14–18]. Moreover, in the SLMprocess, the highly localized heat input leads to a very fast meltingand solidification within a micron-sized molten pool, resulting inthe unique microstructure of the fabricated parts which is aremarkable advantage of the technology [19–22].

However, during the rapid manufacturing process, the depos-ited material undergoes consecutive thermal cycles and complexphysical–chemical reactions, which results in the complex thermaldistribution within the solidified materials. Due to the high tem-perature gradients, non-uniform thermal expansions and contrac-tion caused by the inhomogeneous thermal distribution in theHeat Affected Zone (HAZ), the high residual stresses are inevitablyformed accompanied by the SLM process [23]. In general, a highresidual stress region is formed around the molten zone whichmay result in the formation of shrinkage, promotion of cracking,delamination, fatigue-failure and thermal distortion in the finishedpart [24–27]. As a result, the dimensional precision, shape, andmechanical properties of the fabricated parts would be signifi-cantly affected. Thus, it is quite important to understand the distri-bution and evolution of formed residual stress within the SLMshaped parts. Considering the complexity of the process, experi-mental measurements of the temperature and thermal stressesare difficult because of the insufficient time, high cost for

Fig. 1. Schematic of SLM processing of Ti–Ni two-component powder.

Fig. 2. (a) Gaussian laser energy density; (b) three-dimensional thermal finite element model; (c) three-dimensional thermal stress finite element model (Point 1: at the startof the first track on the top layer; Point 2: in the center of the first track on the top layer; Point 3: at the end of the first track on the top layer; Point 4: in the center of thesecond track on the top layer; Point 5: in the center of the third track on the top layer).

222 D. Gu, B. He / Computational Materials Science 117 (2016) 221–232

measurement and the highly transient nature of the SLM process[28]. Instead of the experimental trial-and-error approaches,three-dimensional finite element modeling is proved to be themost commonly used numerical method to investigate the temper-ature variation, residual stresses distribution and crack formationsin the final component fabrication during the SLM process [29].

Up to now, a series of researches have been conducted to inves-tigate the temperature and residual stresses distribution duringthe SLM process. Some heat conduction models have been estab-lished to predict the density, temperature distribution and molten

pool dimensions taking account of the thermo-physical propertiesand Gaussian distribution by Childs and his workers [30,31]. Chenand Zhang [32] simulated two-dimensional melting accompaniedby shrinkage and resolidification of the metal powder layer usinga temperature transforming model. The effects of the moving heatsource intensity, the parameters applied in the process and thetemperature distribution were discussed. Matsumoto et al. [33]calculated and analyzed the distribution of stress formed on thepowder bed in rapid prototyping. The final results of deformationand residual stresses distribution predicted the possibility and

Fig. 3. The different stress distributions of three different points on the first track: (a) X-component of stress along the scan direction; (b) Y-component of stress transversal tothe scan direction; (c) Z-component of stress through the thickness of the layer; (d) Von Mises stress.

D. Gu, B. He / Computational Materials Science 117 (2016) 221–232 223

positions of cracking during forming process. Due to temperaturechanges and phase transformations during the selective laser sin-tering (SLS) process, an element thermal model allowing for thenon-liner behavior of thermal conductivity and specific heat wasdeveloped by Kolossov et al. [34]. Their study showed that sinter-ing had a strong influence on thermal evolution through changingthe thermal properties of the material. Dai and Shaw [29,35,36]established a 3D thermo-mechanical finite element model toinvestigate the transient temperature, transient stresses, residualstresses and warpage of the component made of multiple materialsin a layer-by-layer fabrication approach. In their study, it wasfound that the transformation from the initial powder elementsto solid elements resulted in higher temperature gradients, largertransient and residual stresses, and increased warpage.

Although many numerical simulations of SLS/SLM process havebeen conducted to investigate the temperature and residual stres-ses distribution, there are few simulation researches on the ther-mal behavior and stresses field during the SLM process of theTi–Ni shape memory alloys. In the previous work, the residualstress of the fabricated part with different parameters was investi-gated through the finite element method and experimental study.However, the evolution law of thermal stress and the distributionof the residual stress in different positions have been seldomlystudied. In the present study, a three-dimensional thermal modelbased on sequentially coupled thermo-mechanical field analysiswas developed to simulate SLM of a powder bed containing a mix-ture of Ti and Ni two-component powders with different melting

points. The model was used to investigate the thermal behavior,temperature gradient of the powder bed, characteristics of themolten pool and the residual stresses distribution during SLM pro-cess of the two-component Ti–Ni metal powders, using ANSYS 13.0commercial finite element method software. Five typical points indifferent positions were selected to comprehensively investigatethe thermal stress variation and the residual stress distribution.A user-written subroutine is implemented in ANSYS ParametricDesign Language (APDL) for the simulation. The correspondingexperiment was conducted to verify the morphologies and micro-strucutres of SLM parts under the optical microscope.

2. Physical principle and modeling approach of SLM

2.1. Assumptions

Owing to the complex nature of SLM process, the followingassumptions are made to make the problem mathematicallytractable:

1. The heat flux from laser beam is considered as Gaussian distri-bution and irradiates straightforwardly to the surface of thepowder layer.

2. The whole powder bed is considered to be isotropic and contin-uous material. Powders melted by the laser beam are consid-ered to be spherical due to the surface tension, and thevolume is changed as the relative density increases to unity.

Fig. 4. The different stress distributions of three points in the Y-direction of three different tracks: (a) X-component of stress along the scan direction; (b) Y-component ofstress transversal to the scan direction; (c) Z-component of stress through the thickness of the layer; (d) Von Mises stress.

224 D. Gu, B. He / Computational Materials Science 117 (2016) 221–232

3. The coefficient of convection between powder bed and the envi-ronment is assumed to be constant. The powder bed is assumedto be subjected to plane stress type of temperature variationbecause of the thin thickness of the powder layer.

4. During SLM process, the heat generated by phase transition isignored during simulation process.

2.2. Physical principle statement

The schematic of the physical model of SLM process under con-sideration is shown in Fig. 1. A laser beam scans the powder sur-face in a back and forth manner layer by layer at a constantscanning velocity along the X-direction. As a fraction of the laserenergy is absorbed by the powders, the temperature of the pow-ders is brought up to Tm and the melting of a certain thickness ofthe powder bed occurs. A molten pool is formed under the highenergy laser beam, and the melted metal infiltrates into the un-melted powder layer driven by capillary and gravitational forces.The temperature gradient of the molten pool is formed regularlyas shown in Fig. 1. The HAZ is formed after the laser beam movesaway, the molten pool cools and solidifies to form the fully densepart. The melting depth pierces into the previously solidified layerand leads to the formation of the remolten zone. Consequently, thenewly solidified layer can bond together with the existing layer toform an integrated resolidified region. During SLM process, theoverlap area between the adjacent scan tracks is determined bythe scanning spacing [37]. In order to get optimum bondingbetween the neighboring tracks, the value of scan spacing is

assumed to be 52.5 lm in this work and the diameter of the laserbeam is settled as 70 lm. Then, the overlap width between the twoneighboring tracks is 17.5 lm. Since the powder bed is sufficientlylarger compared to the diameter of the laser beam, SLM process isconsidered to be a quasi-steady state [32,38,39].

2.3. Physical model

A sequentially coupled 3D thermo-mechanical model is estab-lished to calculate the temperature distribution using ANSYS soft-ware [25]. In the thermal simulation, the laser beam was modeledas aheat flux,q(x, y, z, t),with aGaussianpowerdistributionas aheatsource as shown in Fig. 2a. The thermal behavior results wereobtained firstly through the transient thermal calculation duringthe laser melting. Then the element type was exchanged from ther-mal to structure toperform the transient stress analysis, and thepre-viously obtained temperatures which were regarded as thermalloadings were applied to the mechanical analysis. The 3D finiteelement thermo-model is shown in Fig. 2b. The top Ti–Ni two-component powder layer had the size of 1.19 � 0.315 �0.0375 mm3 and was meshed with the fine size of 0.025 � 0.025 �0.0125 mm3. As the enormous differences between the powderbed and the substrate in mesh elements could lead to the non-convergence result, the interlayer was established as a transitionlayer to connect the two layers. The interlayer had the size of1.19 � 0.315 � 0.02 mm3andwasmeshedwith the tetrahedronele-ment structure. The bottom layer had the size of 1.19 � 0.315 �0.16 mm3 and was meshed with the coarse size of hexahedron

Fig. 5. The residual stress distributions when the powder bed was cooled down at 100 s: (a) X-component of stress along the scan direction; (b) Y-component of stresstransversal to the scan direction; (c) Z-component of stress through the thickness of the layer; (d) Von Mises stress.

D. Gu, B. He / Computational Materials Science 117 (2016) 221–232 225

element structure. The laser beam struck a number of elementsequal to the laser spot size of 70 lm and scanned with a constantscanning velocity of 100 mm/s along the X-direction on the powderbed.

As not all the elements were calculated in the stress field, theedged elements outside the scanning area were not presented inthe model. Thus, the new model was established as shown inFig. 2c. There were three different parts in this model. The topTi–Ni two-component powder layer which was calculated in stressfield had the size of 1.05 � 0.25 � 0.0375 mm3. Both the interlayerand the bottom layer were taken as the substrate and had the samesize as the thermal model described in Fig. 2b. In order to obtainthe accurate calculation, fine mesh element of hexahedron elementstructure with the size of 0.025 � 0.025 � 0.0125 mm3 was used inthe powder bed as the temperature gradients of the powder bedwere very high. Considering the calculation efficiency, the sub-strate was meshed with the coarse element using hexahedron ele-ment structure. In this simulation, the heat source was loaded asthree tracks scanning back and forth. The converged temperatureresults were required to be transferred to the elements used forthe stress analysis [40]. As the moving laser beam was used inSLM process, the thermal stress of the different points at the sametime varied significantly. Five typical points were selected to studythe process of SLM fabrication. P1, P2, P3 was the start point, inter-mediate point and end point of the first track, respectively. Thesethree points were selected to reflect the stress distribution ofthe same track, which could indicate the appropriate length ofthe track. P4, P5 was the intermediate point of the second and thirdtrack, respectively. The combination between the two neighboringtracks could be indicated through these intermediate points. Thus,the evolution of the thermal stress and the distribution of theresidual stress during the SLM process could be investigatedcomprehensively.

2.4. Governing equations

2.4.1. Thermal modelingThe transient spatial distribution T (x, y, z, t) satisfies the follow-

ing differential equation for 3D heat conduction in the component[41]:

q@ðCpTÞ

@t¼ @

@xK@T@x

� �þ @

@yK@T@y

� �þ @

@zK@T@z

� �ð1Þ

where q is the density of the powder; t is the time; Cp is the specificheat capacity; T is the surface temperature of the part being fabri-cated; K is the thermal conductivity of the material; x, y and z arethe coordinates. As the specific heat capacity Cp is the thermal–physical properties of the material and the temperature varied indifferent time, Cp varies with the time and the temperature.

Considering the latent heat of fusion during the phase changes,their relationship can be described as:

H ¼ZqcdT ð2Þ

where q is the density of the material, c is the specific heat capacity.

2.4.2. Boundary conditionsThe thermal initial and boundary conditions are defined by

Tðx; y; z; 0Þ ¼ T0 ð3Þand

K@T@n

þ hcðT � T0Þ þ qr ¼ 0 ðx; y; zÞ 2 Sn ð4Þ

where n is the normal vector of the top surface Sn; hc is the heattransfer coefficient of natural thermal convection; T0 is the ambienttemperature and is considered as 25 �C; and qr is the heat losseswhich can be defined as:

qr ¼ reðT4 � T40Þ ð5Þ

where e is the emissivity; and r is the Stefan–Boltzmann constantwhich has the value of 5.67 � 10�8 W/m2 K4.

The boundary of pressure can be determined by the Laplace–Young equation [37].

p ¼ p1; z ¼ s0: ð6Þ

pl ¼ pv �2rreff

¼ p1 � 2rreff

; z ¼ s: ð7Þ

Fig. 6. The residual stress distributions along the X-distance of three points in the Y-direction on three different tracks when the powder bed was cooled down at 100 s: (a) X-component of stress along the scan direction; (b) Y-component of stress transversal to the scan direction; (c) Z-component of stress through the thickness of the layer; (d) VonMises stress.

226 D. Gu, B. He / Computational Materials Science 117 (2016) 221–232

2.4.3. Heat source modelingThe moving laser heat source can be considered as the Gaussian

heat source distribution which is shown in Fig. 2a [42,43]:

q ¼ 2APpR2 exp �2r2

R2

� �ð8Þ

where P is the laser power, r is the radial distance of a point to thebeam center, A is the laser absorptivity of powder material whichwas taken as 0.75 in the present work, R is the laser beam radiusat which the heat flux density falls to its 1/e2 at the center of thelaser beam. When the center of the laser beam scans the surfaceof the powder bed from the starting point (x0, y0, z0) to (x, y0, z0)point along positive X-direction with a constant velocity v for aperiod time t, R can be described by:

R2 ¼ ðx� x0 � vtÞ2 þ ðz� z0Þ2 ð9ÞIn the previous study, the appropriate scan velocity is optimized

as 100 mm/s.

2.5. Thermal physical parameters of powder

The two-component powder used in this investigation was Tiand Ni mixed powder with atom proportion of Ti:Ni = 60:40. Theporosity of the powder bed can be defined by:

/ ¼ qs � qp

qsð10Þ

where u is the porosity of the powder, qs and qp are the densities ofthe solid and powder materials, respectively. In this finite element

model, the porosity is assumed to vary from u0 = 0.4 for initial pow-der state to u1 = 0 for solid state.

The effective thermal conductivity of the powder bed can bedescribed as follows by [37]:

kkf

¼ 1�ffiffiffiffiffiffiffiffiffiffiffiffi1� /

p� �1þ /kr

kf

� �

þffiffiffiffiffiffiffiffiffiffiffiffi1� /

p 2

1� kfks

1

1� kfks

lnkskf

� �� 1

24

35þ kr

kf

8<:

9=; ð11Þ

where ks is the thermal conductivity of the solid, kf is the thermalconductivity of the fluid, B is deformation parameter of the particle,kr is the thermal conductivity which can be given by:

kr ¼ 4FrT3Dr ð12Þ

where F is a view factor with the value of 1/3, Dr is the averagediameter of the powder particles, and T is the particle temperature.As the thermal conductivity is the function of the temperature, itvaries with the temperature during the SLM process.

The applied SLM conditions are as follows: laser powerP = 200 W, scan velocity m = 100 mm/s, layer thicknessx = 37.5 lm, laser spot diameter D = 70 lm, overlap widthw = 17.5 lm, hatch spacing s = 52.5 lm.

Two steps were conducted during the stress analysis simula-tion. The first step was the heating process which lasts about0.12 s. During this process, the powder is heated by the movinglaser beam. The second step was simulated as the cooling processwhich lasts 100 s. The simulation temperature results obtained

Fig. 7. Stress distributions along the Y-distance of different points on the first track: (a) X-component of stress along the scan direction; (b) Y-component of stress transversalto the scan direction; (c) Z-component of stress through the thickness of the layer; (d) Von Mises stress.

D. Gu, B. He / Computational Materials Science 117 (2016) 221–232 227

from thermal analysis are used as the input to the thermal stressanalysis.

3. Results and discussion

3.1. Thermal stress behavior analysis

Three typical points were selected, which located at the start(P1), middle (P2) and end (P3) of the first track, respectively. Thestress variation and history in the X, Y, Z directions and von Misesare shown in Fig. 3. Obviously, the similar variation trend of differ-ent stress directions was presented. When the part was cooleddown, the residual stresses elevated rapidly in a short time andthen maintained at a constant value eventually. The final residualstress was found to be variant in different points. The highest val-ues of final residual stress in different directions (X-componentstress 86.3 MPa, Y-component stress 95.7 MPa, Z-component stress23.2 MPa and von Mises stress 92.5 MPa) were found at P3 whichlocated at the end of the first track. The minimum residual stressvalues (X-component stress 65.1 MPa, Y-component stress85.7 MPa, Z-component stress 1.2 MPa and von Mises stress83.5 MPa) were obtained at P1 which located at the start of the firsttrack. The thermal stress distributions in different directions dur-ing heating period (0–0.12 s) were also shown in this figure. Thevariations of the stresses were complicated as the laser heat sourcemoved rapidly. It was worth noting that the value of the thermalstress of P1 was negative at the beginning of the heating process,which meant that the compressive stress was obtained. As theheating process proceeded, the compressive stress decreased

rapidly and turned into tensile stress [44]. Afterwards, the peakthermal stresses were observed at P2 and P3 sequentially. Allpoints presented one obvious peak except for P2 which had twopeaks during heating period. The variations of all the thermal stres-ses near the peaks were found to be sharp.

The minimum value of the ultimate residual stress is obtainedat P1 when the part is totally cooled down. As P1 is the first pointthat the laser beam passes by, the first heating–cooling cycle is alsoobtained in this point. When the laser beam passes away andmoves to the other position, the fabricated part in P1 is re-melted. As a result, the residual stress in this position is releasedpartially. In comparison, the highest value of the ultimate residualstress is obtained at P3 when the part is totally cooled down. As P3locates at the end of the first track, the temperature of the previ-ously solidified parts is high enough to prevent the release of theresidual stress [25]. It is worth noting that the values of the resid-ual stresses arise up as the track length increases [33].

At the heating stage, P1 is scanned by the laser beam firstly,which results in the higher temperature compared to the ambientpart. As the area of the higher temperature is restrained by theareas of the lower temperature, the compressive stresses areobtained at P1. Afterwards, the laser beam passes away, the hightensile stresses occur at P1 which results from the fact that thispart is cooled down and is stretched by the vicinity areas of hightemperature [33]. When the laser beam moves on and passes overthe end of the second track which is the nearest position onanother track to P1, P1 is reheated at the highest degree. Conse-quently, the second peak values of P1 residual stresses areobtained. There are also two peak values of P2 which experiences

Fig. 8. The different stress and temperature gradient distributions at P2 (the center of the first track where the laser beam passes by at 0.0167 s): (a) X-component of stressalong the scan direction; (b) Y-component of stress transversal to the scan direction; (c) Z-component of stress through the thickness of the layer; (d) Von Mises stress.

228 D. Gu, B. He / Computational Materials Science 117 (2016) 221–232

the same process as P1. It is worth noting that the first peak valuesare lower while the second peak values are higher in comparison toP1. For the first peak values, the metal plate corresponded to P2 isalso heated before the laser beam arrives. As a result, the metalplate is equivalent to be preheated which decreases the stressesof P2. For the second peak values, P2 is reheated in the shorterinterval time than P1 as the laser beam moves back and forth con-stantly which results in the higher stresses of P2. In comparison tothe other two points, the laser beam draws near and passes awayfrom P3 constantly, resulting in the two closer variations of stresspeaks which are presented as a long peak. After the part is cooleddown, all the stresses develop quickly. In general, the stresses ondifferent points vary with certain limits and the final values arisealong the positive X-direction. It thus can be inferred that it is aneffective strategy to reduce residual stresses through shorteningthe individual tracks by adjusting the scan pattern to a named‘‘island scanning” method [45–48].

Three different points, named as P2, P4 and P5, which werelocated along the center Y-axis of three different tracks, as shownin Fig. 2c, were selected. Fig. 4 illustrates the stress variation andhistory of the three different points in the X, Y, Z directions andvon Mises. As shown in Fig. 4, the similar variation trend of differ-ent stress directions was presented. The residual stresses of everydirection were fluctuated strongly at the heating stage andascended rapidly until the values turned to be constant finally afterthe fabricated part was cooled down. For the stresses of all thedirections, the maximum values of final residual stresses werefound at P5 (X-component stress 112.3 MPa, Y-component stress

113.5 MPa, Z-component stress �4.8 MPa and von Mises stress104.3 MPa). In comparison, the minimum values of all the finalstresses except for Z-component stress were obtained at P2(X-component stress 85.5 MPa, Y-component stress 71.8 MPa andvon Mises stress 88.6 MPa). During the heating process(0–0.12 s), there was obvious variation of residual stresses. Forall the directions, the first peak values occurred at P2 which locatedat the first track. It was noting that P2 was the only point whichowned two peak values in each direction. Noticeably, the first peakvalues appeared at P2 at the time of 0.02 s after the laser beampasses away from P2. P2 had the second peak at the time of0.056 s, which was coincide with the first peak of P4. Later, lessoscillated peaks were found at the time of 0.08 s. For stress ofZ-component displayed in Fig. 6c, the stress value of P4 duringheating process and P5 during the whole shaping process werenegative, i.e., the compress stress.

As P2 locates at the first scan track on the powder bed, it is re-melted when the laser beam scans the other track. The long-timere-melting of P2 leads to its release of the residual stresses to thegreatest extend, which results in its minimum values of the finalresidual stress after the fabricated part is cooled down totally.Since P5 locates at the edge of the part and it is the last point tobe heated, it thus can be pre-heated twice as the laser beam scansthe first and the second tracks. The ambient temperature aroundP5 is relatively lower which brings about high cooling rate andsolidification. Thus, highest values of residual stresses areobserved. It is also worth noting that the negative value ofZ-component stress is obtained at P5. Since the existence of

Fig. 9. The different residual stress distributions with different sections at P2 (the center of the first track) when the laser beam passed by P2 at 0.0167 s: (a) X-component ofstress along the scan direction; (b) Y-component of stress transversal to the scan direction; (c) Z-component of stress through the thickness of the layer; (d) Von Mises stress.

D. Gu, B. He / Computational Materials Science 117 (2016) 221–232 229

enormous temperature difference between the ambient environ-ment and P5, the inflation of P5 resulted from the high tempera-ture is restrained severely. Hence, the transition from tensilestresses to compressive stresses is possible to occur.

On the other hand, the overlap area between two adjacenttracks is formed by partly scan of the previous track in order toguarantee the continuity of the formed parts. At the point of P2,the temperature ascends above its melting point again, leading toits re-melting when the laser beam scans over the second track.Therefore, more than one peak of residual stress is obtained owingto the multiple occurrence of temperature gradient. However, theresidual temperature resulted from the first scanning process playsa pre-heating impact on P4, which in turn partially relieves stressand makes it lower than the first peak of P2.

3.2. Residual stress distribution

Fig. 5 depicts the residual stress distributions in different com-ponents after the part was cooled down totally. The quantitativedescription of stresses variations along the X-direction andY-direction are shown in Figs. 6 and 7, respectively. As shown inFig. 5, the larger stresses were obtained near the edge of the part,especially at the end of the scanning track. As can be seen inFig. 5, the stresses distributions in the powder bed were variousfor different oriented components. It was worth noting that thevalues of the substrate stresses were generally negative. For thestresses distribution along X-direction depicted in Fig. 6, the largeststresses of all components were all in the third scan which could bereflected on P5. All the values were positive except for the stress inthe Z-component. As shown in Fig. 7, the variation of stressesdistribution along Y-direction was more complicated as the laserbeam scanned over the points located at the same track in a short

time. The minimum values of all components were obtained in P1located at the starting of the track, while the higher stress valueswere obtained in P3 located at the end of the track.

In the scanning process of the first track, the lower stressesvalue in P1 results from the fact that it experiences an early heatingand cooling cycle. As a result, the higher temperature gradients andthus the related thermal stresses are obtained [35]. As the laserbeam moves away, the thermal stresses can be released partly.However, at the end of the scanning track, it has less time torelease its thermal stresses, which results in the larger residualstresses. As the laser beam scans back and forth, the first trackshrinks and deposits during cooling cycle. Since the laser beamscans the same position at the next track, the first track is reheatedand then a part of the residual stresses can be released. As moretracks are deposited, the already solidified parts also constrainand prevent further shrinking of the subsequent tracks. As theabove mechanism occurs at each track of the SLM process, theresidual stresses at the start of the tracks is smaller than that atthe end of track, while the residual stress of later scanning trackis larger than that of the previous scanning tracks. It thus can beexplained that the cracks and fractures normally occur at the edgeof the SLM shaped parts. Normally, the residual stress can berelieved though reheated process by increasing the amount ofscanning tracks. As the residual stresses are closely related to themechanical performance of the fabricated parts, more scanningtracks are necessary to advance the performance.

3.3. Mechanism of residual stress formation

Fig. 8 depicts the relationship between the residual stresses andthe temperature gradient by taking an example of P2. As previouslymentioned, P2 was a typical point located in the middle of the first

Fig. 10. Optical microscope images showing morphology on different points of SLM processed Ti–Ni shape memory alloy parts: (a) P1; (b) P2; (c) P3; (d) P4; (e) P5.

230 D. Gu, B. He / Computational Materials Science 117 (2016) 221–232

track. The laser beam passed by P2 at the time of 0.0167 s. Asshown in Fig. 8, for all the stress directions, the peak values ofthe residual stresses occurred behind the peak values of the tem-perature gradients. Furthermore, the quantitative values of theresidual stresses were tightly depended on the temperature gradi-ents values. For the first peak values of the residual stresses in alldirections, the peaks were obtained after the laser beam passedby P2. In a similar way, the second peak values of the residualstresses occurred after the laser beam passed by the midpoint ofthe second track. Afterwards, both the temperature gradients andthe residual stresses tended to be constant.

Fig. 9 shows the different residual stress distributions with dif-ferent sections at P2 when the laser beam passes by P2 at the timeof 0.0167 s. The residual stresses contours were tightly corre-sponded to the curve graphs as shown in Fig. 8. As the heat fluxpassed by P2, the high temperature gradients were formed whilethe low residual stresses were obtained. On the other band, theresidual stresses in different sections presented the similar distri-bution trend, i.e. the residual stresses developed from the centerto the edge of the part gradually.

When the laser beam passes by P2, the high temperature gradi-ents in the vicinity of the laser beam on the powder bed can beclearly seen due to the applied Gaussian heat source. The temper-ature of the powder particles is elevated rapidly under the action of

the absorbed energy, producing a molten pool when the tempera-ture exceeds the melting point temperature. However, the residualstresses are low around the molten pool due to the elastic modu-lus, which reduces with increase in the temperature [40]. Afterthe laser beam passes away from P2, the region is cooled downgradually. As a result, the first heating–cooling cycle is formedwhich is associated with the change of the thermal residual stres-ses. The residual stresses drop again when the laser beam passesby the midpoint of the second track, because the previous trackis reheated due to the overlap of the scan tracks.

3.4. Experimental validation

The preparation of the powder was conducted before the SLMprocess. The 99.5 pct purity Ti powder with an average size of30 lm and the 99.5 pct purity Ni powder with a mean particle sizeof 50 lm were used as the starting materials. According to the Ti:Ni atom ration of 60:40 (i.e., the equivalent weight ratio of55.4:44.6), the two components were milled using high-energymill. Stainless steel grinding balls and the Ti and Ni powder mix-ture were blended in a Pulverisette 4 vario-planetary mill, withthe ball-to-powder weight ratio of 4:1. High-purity argon was usedas the protective gas. The milling time and the rotation speed of themain disk were settled at 4 h and 250 rpm, respectively. The ball

D. Gu, B. He / Computational Materials Science 117 (2016) 221–232 231

milling duration of 20 min was followed by an interval of 10 min inorder to avoid the excessive temperature increase within thegrinding bowl. The as-milled composite powder which was calledTi–Ni two-component powder was obtained with a near-spherical shape and a refined particle size of 25–35 lm.

The SLM system developed by NUAA was used in the SLM pro-cess experiment. The system consisted a YLR-500-SM fiber laserwith a power of �500W and a spot size of 70 lm, a computer sys-tem for process control, an automatic powder feeding apparatus,and a gas protection system. The Ti–Ni alloy was fabricated underthe same processing parameters as numerical simulation using theTi–Ni two-component powders. The polished longitudinal-sectionand cross-section morphologies of the Ti–Ni alloy parts areprovided in Fig. 10. Generally, the overlap features between twoadjacent tracks can be observed. Fig. 10a–c showed thelongitudinal-section morphological characteristics of the areasaround the points of P1, P2 and P3. As can be seen in Fig. 10a,the fabricated part was fully dense at the vicinity around P1. Smallcracks around P2 can be seen in Fig. 10b, which extended from thetop of the powder bed to the bottom. As shown in Fig. 10c, obviouscracks were obtained around P3 which located at the end of thefirst track. The general direction was similar with cracks roundP2, meanwhile, the cracks also extended to the other areas ran-domly. The cross-section morphologies corresponding to thepoints of P4 and P5 in different tracks were provided inFig. 10d and e. The morphologies were fully dense around P4and the cracks formed in the vicinity of P5 which located at theedge of the fabricated part.

As for the longitudinal-section, the obvious cracks can be seenat the end of the track which is corresponding to the larger residualstresses point of P3. The main residual stresses in the longitudinal-section are generally along the X-direction and Y-direction, whilethe stresses perpendicular to the XY substrate are negligible [49].As shown in Fig. 7, the stresses along the direction of depositionare smaller compressive stresses compared with the stresses onthe XY plane in a tensile stresses state. Thus the cracks extendalong the direction perpendicular to the XY plane, which is in goodagreement with the larger stresses region predicted in the previousnumerical simulation. For the cross-section morphologies, the sim-ilar trend of the stress distributions is shown in Fig. 6. The largerresidual stresses obtained at the third track lead to the cracks atthe end edge of the parts. Thus, the experimental results can bewell verified with the numerical simulation.

4. Conclusions

Amethod for calculating the distribution of the residual stressesduring the SLM process was proposed and the effects on the mor-phologies of the fabricated parts formed on the powder bed weredisclosed. Three-dimensional finite element model was establishedto obtain the simulation results which have good agreement withthe experimental results. The results can be summarized in thefollowing points.

1. During the process of the laser beam scanning, the peak valuesof the thermal stresses were first recorded at P1 which locatedat the start of the first track.

2. After cooled down, the largest residual stresses were found atP3 which located at the end of the first track. Thus, in orderto prevent the cracks of the fabricated part, it is necessary toshorten the scanning track.

3. According to the description of different points along the Y-axisin different tracks, the distribution of the stresses was compli-cated. At the heating process, the point at the first track demon-strated the larger thermal stress. While, after the part was

cooled down totally, the larger residual stresses were found atthe last track. The results explained the appearance of the crackat the edge of the part.

4. It was considered that when the neighboring track began tosolidify, a large residual stress was introduced at the edge endof the solid part and then cracking might be caused. To reducethe possibility of cracking at the edge of the part, strong framesof the areas should be formed first.

5. The final experimental results, which had good agreement withthe quantitatively calculation using the simulation method,indicated the relative positions of the cracks. On the other band,the directions of the cracks demonstrated that the longitudinalstresses were larger compared to the normal and transversestresses.

Acknowledgments

The authors gratefully acknowledge the financial support fromthe National Natural Science Foundation of China (Nos. 51575267and 51322509), the Top-Notch Young Talents Program of China,the Outstanding Youth Foundation of Jiangsu Province of China(No. BK20130035), the Program for New Century Excellent Talentsin University (No. NCET-13-0854), the Science and TechnologySupport Program (The Industrial Part), Jiangsu Provincial Depart-ment of Science and Technology of China (No. BE2014009-2), the333 Project (No. BRA2015368), Science and Technology Foundationfor Selected Overseas Chinese Scholar, Ministry of HumanResources and Social Security of China, the Aeronautical ScienceFoundation of China (No. 2015ZE52051), the Shanghai AerospaceScience and Technology Innovation Fund (No. SAST2015053), theFundamental Research Funds for the Central Universities (Nos.NE2013103 and NP2015206), and the Priority Academic ProgramDevelopment of Jiangsu Higher Education Institutions.

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