Computers and Structures 89 (2011) 977–985

Contents lists available at ScienceDirect

Computers and Structures

journal homepage: www.elsevier .com/locate /compstruc

Numerical simulation of the electron beam welding process

Piotr Lacki a,⇑, Konrad Adamus b

a Faculty of Civil Engineering, Czestochowa University of Technology, Akademicka 3, Czestochowa 42-218, Polandb Faculty of Mechanical Engineering and Computer Science, Armii Krajowej 21, Czestochowa 42-201, Poland

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

Article history:Received 31 May 2010Accepted 20 January 2011Available online 4 March 2011

Keywords:Electron beam welding (EBW)Heat-affected zoneNumerical simulation3D conical heat sourceThermomechanical coupling analysis

0045-7949/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.compstruc.2011.01.016

⇑ Corresponding author. Fax: +48 34 3250609.E-mail address: piotr@lacki.com.pl (P. Lacki).

Electron beam welding is a highly efficient and precise welding method that is being increasingly used inindustrial manufacturing and is of growing importance in industry. Compared to other welding processesit offers the advantage of very low heat input to the weld, resulting in low distortion in components.Modeling and simulation of the laser beam welding process has proven to be highly efficient for research,design development and production engineering. In comparison with experimental studies, a modelingstudy can give detailed information concerning the characteristics of weld pool and their relationshipwith the welding process parameters (welding speed, electron beam power, workpiece thickness, etc.)and can be used to reduce the costs of experiments. A simulation of the electron beam welding processenables estimation of weld pool geometry, transient temperature, stresses, residual stresses and distor-tion. However this simulation is not an easy task since it involves the interaction of thermal, mechanicaland metallurgical phenomena. Understanding the heat process of welding is important for the analysis ofwelding structure, mechanics, microstructure and controlling weld quality.

In this paper the results of numerical simulation of electron beam welding of tubes were presented. Thetubes were made of 30HGSA steel. The numerical calculation takes into consideration thermomechanicalcoupling (TMC). The simulation aims at: analysis of the thermal field, which is generated in welding pro-cess, determination of the heat-affected zone and residual stresses in the joint. The obtained results allowfor determination both the material properties, and stress and strain state in the joint. Furthermore,numerical simulation allows for optimization of the process parameters (welding speed, power of theheat source) and shape of the joint before welding. The numerical simulation of electron beam weldingprocess was carried out with the ADINA System v. 8.6. using finite element method.

� 2011 Elsevier Ltd. All rights reserved.

1. EBW characteristics

Electron beam welding, EBW, is a fusion welding process thatutilizes electrons as a source of energy which is used to join ele-ments. Successful applications of electron in welding processes fol-low from several of its traits [1]:

– Electrons occur in external atom shells, thus they can be easilydetached from atom and beam can be created.

– Electrons have negative electric charge (�1.6 � 10�19 C) andthey can be accelerated using electric field, the higher the elec-tron speed the higher its kinetic energy which will be used tomelt metal.

EBW is performed in vacuum. The transport of electrons in vac-uum is intended to eliminate electron collisions with much heaviergas particles. The collisions would cause beam defocus and loss of

ll rights reserved.

electron kinetic energy [2]. Additionally collisions would create airionization which in turn would destroy cathode [3]. There are alsoEBW units that operate in atmospheric gases and in partial vac-uum. However, due to beam defocus they achieve poorer perfor-mance and can be treated as a supplementary method and not asa replacement [4].

The weld creation mechanism during EBW process is not en-tirely explained. Electron beam allows for achieving high powerdensity (5 � 108 W/cm2) at a small area (10�7 cm2). Dependingon accelerating voltage electrons themselves can penetratethrough external layer of material at depth of about 10�2 mm.Although electrons themselves can penetrate only such small dis-tance known fusion zones are much deeper. Possible mechanismbehind creation of deep welds is described by Schultz [2]. Afterelectrons penetrate the external layer of element they start to heatthe metal. The metal melts and subsequently changes into vapor.The high pressure vapor bubble bursts and destroys the externallayer of element. Once the vapor is released the beam is refocusedand starts to penetrate through next layers. The cycle repeats andthin deep weld is created.

Fig. 2. EBW welded specimen selected for comparison with numerical analysisresults.

Fig. 3. Weld microstructure in the selected specimen.

978 P. Lacki, K. Adamus / Computers and Structures 89 (2011) 977–985

Chattopadhyay [5] lists advantages of EBW. EBW is the pre-ferred welding method because of the following reasons:

– Application of vacuum allows for welding of materials that reactwith atmospheric gases, for instance welding of titanium.

– Does not require preheating of materials that are characterizedby high melting point.

– Capability of welding materials that have high thermal conduc-tivity and tend to reflect laser beam.

– Capability of welding both large and small elements since elec-tron beam parameters can be easily controlled and modified.

The main drawback of EBW is the size of the welded elementsthat must fit into the vacuum chamber. Another factor is the highcost of vacuum generation.

1.1. EBW unit used during experiments

Experimental work was performed using polish SE10/60 EBWunit which is presented in Fig. 1. The main components of EBW unitare electron beam gun, working chamber and vacuum pumps. Elec-tron beam gun comprises cathode emitting electrons, anode that at-tracts electrons and through which beam is transported, and thesystem focusing electron beam. Working chamber has the shapeof cuboid with dimensions 1200 � 710 � 850 mm. The chamber ismade of acid resistant steel. It is equipped with system that allowsfor moving welded objects in XY plane and also for their rotation.One of the vacuum pumps is used for empting working chamberand the other for empting chamber containing electron beam gun.Pumps are controlled by system dedicated for measuring pressure.

The following parameters can be controlled in the specifiedranges:

– accelerating voltage: 10–60 kV;– beam current: 0–250 mA;– cathode heating current: up to 30 A;– beam power: up to 15000 W,– vacuum in working chamber: 610�5 h Pa;– vacuum in chamber containing electron beam gun: 610�5 h Pa;– chamber empting time: ca. 30 min.

1.2. Experiment

During experimental research a series of welded joints was per-formed using different input parameters. One of the weldmentswas selected for the purpose of comparison with numerical calcu-lation results. Fig. 2 presents the selected specimen. The specimencomprises two tubes joined together. Tubes are made of 30HGSA

Fig. 1. EBW unit used for experiments.

steel. Specimen dimensions used during numerical simulationare the same as in the actual experiment.

The following parameters were used for the selected specimen:accelerating voltage was set to 60 kV, beam current was set to165 mA, specimen revolution time was set to 10 s. Pulsed modewas selected with pulse width set to 27 ms and pulse pause timeset to 111 ms. Sample application of pulsed mode is presented in[6].

In order to determine the size and shape of heat affected zone,HAZ, microsection was created at weld cross section. Fig. 3 pre-sents microsection of the weld joint. The changes resulting frommaterial heating can be seen. Three zones corresponding to differ-ent heat change rates during welding process can be distinguished.The parting line can be clearly determined between differentzones. The zone in the upper part of weld corresponds to additionalpost-weld cosmetic pass which was performed with lower valuesof beam power in order to smooth out the external surface of joint.The two other zones correspond to main pass during which jointwas created. The knowledge of size and shape of particular zonesand the corresponding zone development conditions can be usedto validate numerical model.

2. Heat source models

Goldak and Akhlaghi [7] described several basic heat sourcemodels that can be used during weld process simulations. Threebasic heat source models were presented in Fig. 4.

Point heat source is placed on the top surface of the welded ob-ject. It is used to model shallow welds. Line heat source which isperpendicular to the top surface of the welded object and occupiesspace inside object. It is used to model deep welds.

Disk heat source is the extension of point heat source. Heat fluxis assigned to the object top surface represented by a disk. It hasuniform or Gaussian distribution. Gaussian distribution is de-scribed using the following function which corresponds to Fig. 4a:

qðrÞ ¼ qð0Þe�Cr2 ð1Þ

where q(r) is the heat flux for radius r, q(0) the maximal value ofheat flux in the center of disk, C the heat flux distribution coefficient

a1

a2x

x

x

z

z(a)

(b)

(c) z

y

y

y

b

c

Power density

Power density

Power density

Fig. 4. Heat source models: (a) disk source, (b) double ellipsoid source and (c)conical source [6].

P. Lacki, K. Adamus / Computers and Structures 89 (2011) 977–985 979

and r is the distance from disk center. Disk heat source is extendedby 3D hemi-spherical heat source which is placed beneath the topsurface of the welded object.

Disk heat source and hemi-spherical heat source assume thesymmetry of weld pool with regard to axis going through the mid-dle of heat source. Thus they fail to reflect the shape of weld poolcreated by a moving heat source. This problem is solved by doubleellipsoid heat source. Initially single ellipsoid was taken into ac-count as the moving heat source produces oval weld pool at thesurface of the welded object. Whole power was assigned to the halfof ellipsoid beneath the top surface and the half of ellipsoid abovethe top surface was ignored. Since temperature gradient in thefront part of weld pool is lower than in the rear part two differentellipsoids were used to represent heat source. The quarter of oneellipsoid represents heat distribution in front part and the quarterof the other ellipsoid represents heat distribution in rear part. Heatdistribution is described by the following equation which corre-sponds to Fig. 4b:

qðx; y; z; tÞ ¼ 6ffiffiffi3p

fQabc

ffiffiffiffipp

pe�3x2

a2 e�3y2

b2 e�3ðzþvtÞ2

c2 ð2Þ

where Q is the overall input power, f the fraction of power assignedto ellipsoid quarter, a,b,c the ellipsoid semi-axes, v the heat sourcespeed and t is the time from the beginning of weld process.

Double ellipsoid is relatively accurate for description of heatdistribution in shallow welds that are produced by electric arcwelding process. On the other hand conical heat source is used

for the purpose of heat distribution description in deep weldsoccurring during laser welding and electron beam welding. Conicalheat source assumes Gaussian heat distribution in radial directionand linear heat distribution in axial direction. Conical heat sourcemodel is presented in Fig. 4c.

2.1. Analytical temperature distribution model for EBW processes

Ho et al. [8] suggested analytical model describing temperaturedistribution in cavity created during electron beam welding. Thesolution presented by authors offers results accuracy similar tothe existing analytical models. What distinguishes this model fromother is that the predicted temperature at the bottom of the cavitydoes not approach infinity. The modified parabolic coordinate sys-tem is used to describe heat source:

x ¼ 2ffiffiffiffiffiffing

pcos /

Pe; y ¼ 2

ffiffiffiffiffiffing

psin /

Pe; z ¼ ðn� gÞ

PeffiffiffiSp ð3Þ

where Pe is the Peclet number and S is the convection coefficientdefined as S = a/az. Parameters a, az denote liquid diffusivity and en-hanced diffusivity in vertical direction z. Dimensionless tempera-ture H was introduced:

H ¼ T � T1T � Tm

expffiffiffiffiffiffing

pcos /

h ið4Þ

where Tm is the melting point temperature and T1 is the ambienttemperature.

At the walls of cavity the balance between incident flux andconduction was assumed:

@H@g�H

2

ffiffiffing

scos /

�����g¼g0

¼ �3QffiffiffiSp

pPeexp

ffiffiffiffiffiffiffiffing0

pcos /� 12ng0

Pe2

� �ð5Þ

where Q is the dimensionless electron beam power and g0 is the va-lue of g corresponding to the cavity walls.

The temperature predicted by model is the highest at the bot-tom of cavity, 2200 �C, and decreases linearly toward upper partof the cavity, 1500 �C. At the upper part of cavity temperature ishigher than experimental values measured by Schauer and Giedt[9,10]. The discrepancy is probably caused by heat dissipation toambient air. In the model heat dissipation was ignored as it was as-sumed that it is negligibly small compared to heat flux carried byelectrons. Temperature predicted by model at the bottom of cavityis too high and temperature in the middle part of cavity is too lowcompared to empirical data. The possible explanation of these dif-ferences is that the model does not take into account plasmaabsorption and multiple reflections.

Most of beam power focuses at the bottom of cavity as powerhas Gaussian distribution. This is the reason for high temperatureat cavity bottom.

High value of convection parameter S, and thus low value of li-quid diffusivity in vertical direction, corresponds to higher temper-ature at the bottom of cavity and greater cavity depth. Similareffect can be produced by increasing value of Peclet number,increasing beam power or using material that has greater heatabsorption. In case of low beam power temperature tends to de-crease more rapidly toward the upper part of cavity than in caseof high beam power.

2.2. Heat source model generations

Goldak and Akhlaghi [7] suggested division of heat source mod-els into five generations. Consecutive generations extend theirpredecessors.

First generation of models uses simple geometric objects likepoint, line and disk to describe heat sources. Specified amount of

980 P. Lacki, K. Adamus / Computers and Structures 89 (2011) 977–985

heat is assigned to geometric object and it has uniform distributionand constant value. This approach gives good results for predictionof temperatures that are far from weld pool. Since these models as-sume constant operating conditions they cannot describe initiali-zation and finalization of welding process.

Second generation extends heat source description with func-tion representing power density within geometric object. Morecomplex objects are introduced such as double ellipsoid for arcwelding and cone for electron beam welding. In case of ellipsoidpower is assigned to volume and appropriate function describespower distribution in such a way that power value is highest inthe ellipsoid center and decreases to a certain value at ellipsoidborder. Shape of weld pool and its temperature distribution areapproximation of actual values. If calculated shape of weld poolis significantly different from the actual shape then temperaturesin the area of actual weld pool have fictitious values. Despite thesedisadvantages second generation models can relatively accuratelypredict temperature distribution outside weld pool.

In third generation shape of weld pool is a result of calculationsand not as it is in case of second generation input data. In order todetermine the shape of liquid–solid interface Stefan problem mustbe solved. The curvature and velocity of liquid–solid interface mustbe taken into account during definition of melting temperature.Additionally, these models include hydrostatic stress in weld pool,pressure on the weld pool surface from the arc, surface tensionforces and flow of mass into and out of weld pool. From the numer-ical point of view these methods are moderately more computa-tionally complex than first and second generation models. On theother hand they offer more reliable results.

Fourth generation of models extends third generation withequations describing fluid dynamics inside weld pool. Macroscopicfluid dynamics is described with Navier–Stokes equations. Theseequations take into account buoyancy and Lorentz forces actingon the weld pool. Additionally fourth generation models take intoaccount Marangoni effect, arc pressure and shear forces impact onthe surface of weld pool. Some of the models take into account alsothe transport of molten material from electrode but only for cur-rents below 100/150 A. For higher values of current fluid motiondescription problems occur.

Fifth generation combines the model of electric arc and themodel of heat source. This is achieved by introduction of mag-neto-hydrodynamics equations. Due to high complexity of fourthand fifth generation models they require complicated mathemati-cal apparatus. Currently it often cannot be proofed that solutionexist or it is unique. For these reasons the application of thesemodels to prediction of weld pool geometry in industry is limited.

2.3. Deep penetration welding model

Goldak and Kazemi [11] present 3D FEM model describing laserbeam welding process. From numerical point of view laser beamwelding is similar to electron beam welding. The suggested modelproduces temperature distribution and shape of weld pool. Heatsource comprises surface source and inner source. Surface sourceis represented by disk model using Gaussian power distribution.Inner source is represented by a line divided into segments. Withinone segment the value of produced heat is constant. The amount ofheat is the function of segment depth relative to top surface ofwelded object.

Authors used empirical equation suggested by Lankalapalli et al.[12] to determine the amount of heat produced by laser beam atspecified depth. This equation defines power as a function of ther-mal conductivity, temperature and Peclet number:

Pz ¼ kðTv � T0Þð2:1995þ 6:2962Pe� 0:4994Pe2 þ 0:0461Pe3Þð6Þ

where Pz is the power produced at depth z, k the thermal conductiv-ity, Tv the heat source temperature, T0 the initial temperature and Peis the Peclet number.

It was assumed that cavity created by laser beam has the shapeof a cone. Peclet number at a specified depth was defined as a func-tion of Peclet number at the top surface of workpiece, penetrationdepth and distance from top surface:

Pe ¼ Peð0Þ 1� zd

� �ð7Þ

where Pe(0) is the Peclet number at top surface, z the distance fromtop surface and d is the penetration depth. Based on Lampa andKaplan work [13] Peclet number at top surface was defined as afunction of weld width, welding speed and thermal diffusivity:

Peð0Þ ¼ va2km

ð8Þ

where v is the welding speed, a the radius of surface source, j thethermal diffusivity of liquid material and m is the user defined coef-ficient representing multiple of thermal diffusivity.

Integrating Eq. (6) and using Eq. (7) to change the variable ofintegration from z to Pe the authors defined the amount of powerabsorbed by inner source Pl as:

Pl ¼ tðakliqÞðTv � T0Þð2:1995þ 3:1481Peð0Þ� 0:16647Peð0Þ2 þ 0:01152Peð0Þ3Þ ð9Þ

where t is the plate thickness, kliq the liquid metal conductivity, athe user defined parameter taking into account impact of weldingvelocity on thermal conductivity and Pe(0) is the value of Pecletnumber at top surface. The amount of absorbed power by surfaceheat source Pc is calculated as product of laser beam power andabsorption coefficient g specific to laser welding which was definedusing Bramson’s formula [14]:

gðTÞ ¼ 0:365Rl

� �12

� 0:0667Rl

� �þ 0:006

Rl

� �32

ð10Þ

where R is the electrical resistivity of material and l is the is the la-ser beam wavelength. Whole power absorbed by welded object isdefined as Pt = Pc + Pl.

Thermal conductivity and specific heat were assumed to be afunction of temperature.

The model allows for prediction of weld pool shape. Good re-sults were obtained for prediction of weld width at the top surfaceand at the bottom of weld. Model fails to predict accurately weldwidth in depth near the top surface. From calculations it followsthat three parameters have the largest impact on the results: Pecletnumber, thermal conductivity and absorption coefficient of mate-rial used for surface heat source.

3. EBW numerical model

EBW numerical model was built using ADINA System v8.6.1[15,16]. The program utilizes Finite Element Method. The basic fi-nite element procedures used in ADINA System were described in[17].

Fourier-Kirchoff equation was used to describe heat propa-gation:

@T@t¼ ar2T þ qv

qcpð11Þ

where a is the thermal diffusivity, q the density, cp the specific heatand qv is the efficiency of inner volume heat source.

The conical heat source model with uniform power distributionwas assumed. The assumed shape of heat source follows from pen-etrative motion of electron beam. As the electron beam penetrates

Tube 1Fixed SurfaceSingle heat source

Tube 2 X

YZ

Fig. 5. Tubes cross-section with conical heat source.

0

2e+11

4e+11

6e+11

8e+11

1e+12

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

Hea

t, W

/m3

Time, s

Fig. 6. Amount of heat generated during pulse EBW for welding speed 10 mm/s.

P. Lacki, K. Adamus / Computers and Structures 89 (2011) 977–985 981

through material it creates cavity called keyhole. Heat is being gen-erated inside keyhole thus in the model it was assumed that heat isgenerated in the volume elements. Electron beam produces key-holes of several millimeters to several centimeters deep and heatis generated inside the weld.

In order to map the conical heat source to FEM mesh the powerassigned to conical heat source was assigned to elements in theshape of prism. The triangle cross-section of prism reflects the ac-tual cross-section of HAZ in the analyzed specimen. The elongatedshape of prism takes into account the movement of heat sourcealong welding trajectory. Geometry of prism is defined by itsdepth, length and flare angle. The length is the quotient of tube cir-cumference and number of prism elements along tube circumfer-ence. The movement of heat source is represented by productionof power in the consecutive prism elements. At any time poweris produced inside only a single prism element.

Two tubes of different thickness were joined using EBW. Buttjoint was created. One of the tubes has welding collar which facil-itates fitting tubes to each other. Each of tubes is 50 mm long. Tubeexternal diameter is 31.8 mm which corresponds to circumferenceof about 100 mm. One of the tubes has walls 5 mm thick the otherhas walls 3 mm thick. In the vicinity of weld, mesh was thickenedand mesh elements were modified so that they reflect the shape ofheat source. Fig. 5 shows mesh representing tubes. It can be seenthat mesh elements are thickened near conical heat source.

As initial condition temperature of tubes was set to 20 �C. Onthe tube walls convection coefficient was set to 0 as welding is per-formed in vacuum in short time. Calculations were done for threedifferent welding speeds: 6.7, 10 and 20 mm/s which correspondto overall welding time of: 5, 10 and 15 s. Welding time of 10 s cor-responds to actual experiment and welding times of 5 and 15 swere used for comparative purposes.

Heat is generated in the series of pulses. Compared to continu-ous beam pulse welding offers higher ratio of fusion zone depth towidth and thus enables creating thin welds as explained in [2].Modeling of pulse welding requires considerably higher numberof time steps than modeling of continuous welding. For each weld-ing speed welding time was divided into 72 periods. Each of peri-ods corresponds to 5� rotation of heat source which gives togetherfull 360� rotation. Single pulse lasts 20% of period time and idletime between pulses corresponds to 80% of period time. Thereare 72 pulses and 72 prism elements. Each of pulses is assignedto the appropriate prism element. For welding speeds 6.7, 10 and20 mm/s single pulse lasts 0.042, 0.028, and 0.014 s, respectively.

One EB pulse is represented by 20 time steps and idle period be-tween pulses is represented by 80 time steps. Power density ofheat source during pulse welding is shown in Fig. 6. During activeperiod power density equals 9 � 1011 W/m3. The actual beampower was 9900 W. Power absorption coefficient equal to 60%was assumed in the simulation.

Tubes were made of 30HGSA steel according to PN-89/H-84030/04 standard. This is chromium–manganese–silicon steelwith chemical constitution presented in Table 1. 30HGSA steel isused for mills, average machines and high-strength parts. 30HGSAsteel has limited weldability especially for thick cross-sections.During welding processes allowable hardness is often exceeded.For large elements it is advisable to apply intermediate annealing.Immediately after end of welding process it is advisable to applysoft annealing or toughening. In case of materials that had under-gone heat treatment prior to welding it is possible that within HAZstrength properties will deteriorate. In order to restore these prop-erties in HAZ appropriate heat treatment should be applied.

The following material properties were assumed for 30HGSAsteel:

– thermal conductivity

50 W/mK – specific heat 472 J/kgK – density 7850 kg/m3

– Young’s modulus

210 GPa – Poisson ratio 0.3 – yield stress 850 MPa for 20 �C

150 MPa for 1350 �C

– strain hardening modulus 85 MPa for 20 �C

0 MPa for 1350 oC,

– coefficient of thermal expansion 11 � 10�6 K�1

4. Modeling results

Finite element method program ADINA allows for convenientmodeling of pulse EBW process. Theoretical model representingphysical process allows us to estimate changes occurring inwelded tubes. Modification of welding speed enables us predicthow this change will impact temperature distribution. Tempera-ture range will depend on welding speed and the assumed material

Table 1Chemical constitution of 30HGSA steel according to PN-89/H-84030/04.

Fe C Mn Si p S Cr

Min Max Min Max Min Max Min Max Min Max Min Max Min Max

Residue 0.28 0.34 0.8 1.1 0.9 1.2 0 0.025 0 0.025 0.8 1.1

Ni Mo W V Ti Cu Al

Min Max Min Max Min Max Min Max Min Max Min Max Min Max

0 0.3 0 0.1 0 0.2 0 0.05 0 0.05 0 0.3 0 0.1

TEMPERATURE

1350.1260.1170.1080.990.900.810.720.630.540.450.360.270.180. 90.

TIME 0.3056

a) b) c) d)

TIME 0.58333 TIME 0.86111 TIME 1.4167MAXIMUM MAXIMUM MAXIMUM MAXIMUM

4833. 4947. 5011. 5042.NODE 25803 NODE 26991 NODE 28179 NODE 30555

Fig. 7. Development of HAZ in numerical model of tube welding process: (a) time 0.3056 s, (b) time 0.58333 s, (c) time 0.86111 s and (d) time 1.4167 s.

982 P. Lacki, K. Adamus / Computers and Structures 89 (2011) 977–985

properties. According to [18] temperature of 25,000 �C can beachieved during EBW. Hai-Xing and Chen [19] analyzed plasmatemperature during laser beam welding, which offers beam powersimilar to EBW, show that keyhole temperature can reach about13000 �C. During simulation the maximal temperature of 6216 �Cwas reached within 2.125 s for welding speed of 6.7 mm/s.

Changes of temperature distribution in HAZ were presented inFig. 7. Within about 2 s from beginning of the process HAZ devel-ops in non-stationary way. Subsequently stationary stage isachieved for HAZ. During stationary stage HAZ does not changeits shapes and it merely changes its position relative to its initialposition moving along welding trajectory.

Fig. 8 presents temperature values as a function of time forpoints at the same distance from heat source in normal directionto welding trajectory and at different depths, for welding speedof 10 mm/s. Initially as the depth increases temperature signifi-cantly decreases. After period of 1 s for all points temperature ap-proaches 250 �C and starts to linearly decrease. It can be seen thatduring first second temperature values have tendency to createwave which reflect pulse character of electron beam.

Fig. 9 presents temperature values as a function of time forpoints at the same depth and different distances from heat sourcein normal direction to welding trajectory, for welding speed of10 mm/s. Only for two points closest to the heat source pulse char-acter of heat source can be seen. After the period of about 1 s tem-perature for all points stabilizes.

Fig. 10 presents changes of temperature values as a function oftime for point P2 and different welding speeds. Plot analysis showsthat as welding speed increases maximal temperature decreasesfor point P2. Also temperature amplitude of consecutive pulsesdecreases.

As the welding speed increases the amount of power generatedin weld decreases. Smaller amount of power results in lower key-hole temperature and smaller size of HAZ. Fig. 11 presents threematerial volumes for temperatures above 500 �C and for differentwelding speeds. It can be seen that as welding speed increasesthe volume of material above 500 �C decreases and temperaturein center of volume also decreases. HAZ for welding speeds of 10and 6.7 mm/s has shape of solid of elliptic base that decreases inone direction. For welding speed of 20 mm/s HAZ has the shapeof half-ellipsoid with growing cone that represents heat generatedby EB pulse.

The assumed heat source model allows for distinction betweensolid and liquid phase based on temperature calculated for 3D solidelements. Due to the assumed kind of finite elements the processesoccurring in HAZ cannot be analyzed for temperatures above melt-ing point.

4.1. Thermomechanical analysis

Thermomechanical coupled, TMC, analysis was used to deter-mine the magnitude of thermal stresses. Similar approach was ap-plied in [20]. Thermal elasto-plastic material model was assumedin the numerical model. Martensitic phase transformation thatmay occur during cooling period was neglected. The face of thickertube shown in Fig. 5 was fixed. All degrees of freedom were takenaway for the mesh nodes in the surface corresponding to the tubeface.

Fig. 12 shows results of TMC numerical analysis. Temperaturedistribution and the corresponding effective stress distributionare presented in Fig. 12. Fig. 12a presents distribution corre-sponding to half of pulse duration, Fig. 12b presents distribution

4 mm 1.55 mm

P1

P5 P6 P7 P8 P9 P10 P11 P12 P13 P14P2

P3

P4

P0

Tem

pera

ture

[°C

]

0

100

200

300

400

500

600

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6Time [s]

Fig. 8. Temperature values as a function of time for points P0–P4 at constantdistance from heat source axis. Welding speed of 10 mm/s.

0

100

200

300

400

500

600

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6Time [s]

1 mm 15.18 mm

Tem

pera

ture

[C

]°

P1

P5P2

P3

P4

P0

P12 P13 P14P6 P7 P8 P9 P10 P11

Fig. 9. Temperature values as a function of time for points P2, P5–P14 at the samedepth. Welding speed of 10 mm/s.

0

100

200

300

400

500

600

700

800

Tem

pera

ture

,[°C

]

Time,[s]

welding speed = 6.7 mm/swelding speed = 10 mm/s

welding speed = 20 mm/s

Fig. 10. Temperature value changes for point P2 and different welding speeds.

TIME 0.70833(a)

(b)

(c)

MAXIMUM3448.

NODE 30555

TIME 1.4167MAXIMUM

5042.NODE 30555

TIME 2.1250MAXIMUM

NODE 30555

Fig. 11. Volume of material above 500 �C for welding speeds: (a) 20 mm/s,(b) 10 mm/s and (c) 6.7 mm/s.

P. Lacki, K. Adamus / Computers and Structures 89 (2011) 977–985 983

corresponding to the end of pulse and Fig. 12c presents distribu-tion corresponding to the end of idle period between pulses.

The impact of temperature on effective stress field can be in-ferred from the plots. The lowest values of effective stress occurin the area of molten pool. High values of effective stress surroundwelding pool. The highest values equal to 370 MPa occur beneathwelding pool at the end of idle period.

As it was shown in [21] the yield stress is the key mechanicalproperty in welding simulation. The yield stress dependency ontemperature must be considered in a welding process simulationto obtain correct result. Young’s modulus and thermal expansioncoefficient have small impact on the residual stress and distortionin welding deformation simulation.

4.2. Model assessment

In order to asses numerical model isotherm calculated by themodel were put onto macrostructure picture of weld (Fig. 13).For temperatures above 1200 �C isotherms are close to each other

which indicates that high temperature gradient occurs. Isotherm inbottom part of weld is in 95% consistent with lines observed inmacrostructure.

a) TIME 0.01389 s

b) TIME 0.02778 s

c) TIME 0.1389 s

EFFECTIVESTRESS, MPa

373320266213160106530

X

Y

ZMAX TEMPERATURE2552 Co

MAX TEMPERATURE4110 Co

MAX TEMPERATURE2435 Co

TEMPERATURE, Co

1500

1300

1100

900

700

500

300

100

Fig. 12. Temperature and effective stress distribution for: (a) half of pulse duration, (b) end of pulse and (c) end of idle period between pulses.

0 5mm

TEMPERATURETIME 1.4167

3000.2400.1800.1200.600. 0.

MAXIMUM5042.

NODE 30555

Fig. 13. Comparison of numerical calculation results and actual results.

984 P. Lacki, K. Adamus / Computers and Structures 89 (2011) 977–985

In upper part of weld isotherms are consistent with actual mac-rostructure areas to a lesser extent. This is caused by the fact thatafter welding finished specimen underwent additional heat treat-ment with low power electron beam in order to smooth weld face.The impact of additional low power heat source can be seen inmacrostructure. Second heat source slightly extends HAZ in upperpart and causes discrepancy between calculated isotherms andmacrostructure lines. Further research will focus on including sec-ond low power heat source in numerical model in order to predictHAZ shape more accurately.

5. Conclusions

Numerical calculation results and empirical results allowed forvalidation of assumed model of EBW process. The model was cre-ated using ADINA System. The following conclusions can be drawnfrom the analysis of results:

– After period of 1 s for points P0–P4 that are 1 mm from heatsource axis the temperature oscillations due to pulse weldingtend to stabilize.

– As welding speed increases maximal temperature decreases forpoint P2. Also temperature amplitude due to pulse weldingdecreases.

– Temperature gradient in the area corresponding to heat sourcecauses increase of effective stresses. Maximal values of thermalstresses occur in specimen area beneath heat source. The mag-nitude of thermal stresses is dependent on yield stress ofwelded material at given temperature.

– Comparison of calculated isotherms and weld macrostructureallowed for validation of the model accuracy. The suggestednumerical model produced results that are satisfyingly consis-tent with empirical data.

Acknowledgments

Financial support of Structural Funds in the Operational Pro-gramme – Innovative Economy (IE OP) financed from the European

P. Lacki, K. Adamus / Computers and Structures 89 (2011) 977–985 985

Regional Development Fund – Project ‘‘Modern material technolo-gies in aerospace industry’’, No. POIG.01.01.02-00-015/08-00 isgratefully acknowledged.

References

[1] Nikolaev G, Olshansky N. Advanced welding processes. Moscow: Mir; 1977.[2] Schultz H. Electron beam welding. Abington: Woodhead Publishing; 1994.[3] Molokovsky SI, Sushkov AD. Intense electron and ion beams. Berlin: Springer;

2005.[4] Jeffus L. Welding: principles and applications. Delmar Cengage Learning; 1997.[5] Chattopadhyay R. Advanced thermally assisted surface engineering

processes. Springer; 2004.[6] Cui HC, Lu FG, Peng K, Tang XH, Yao S. Research on electron beam welding of

in situ TiB2p/ZL101 composite. J Shanghai Jiaotong Univ 2010;15:479–83.[7] Goldak JA, Akhlaghi M. Computational welding mechanics. Springer; 2005.[8] Ho CY, Wen MY, Lee YC. Analytical solution for three-dimensional model

predicting temperature in the welding cavity of electron beam. Vacuum2007;82:316–20.

[9] Schauer DA, Giedt WH. Prediction of electron beam welding spiking tendency.Welding J 1978;57:189–95.

[10] Schauer DA, Giedt WH, Schintaku SM. Electron beam welding cavitytemperature distributions in pure metals and alloys. Welding J 1978;57:127–33.

[11] Goldak JA, Kazemi K. Numerical simulation of laser full penetration welding.Comput Mater Sci 2009;44:841–9.

[12] Lankalapalli KN, Tu JF, Gartner M. A model for estimating penetration depth oflaser welding processes. J Phys D 1996;29:1831–41.

[13] Lampa C, Kaplan AFH, Powell J, Magnusson C. An analytical thermodynamicmodel of laser welding. J Phys D: Appl Phys 1997;30:1293–9.

[14] Bramson MA. Infrared radiation: a handbook for applications. NewYork: Plenum Press; 1968.

[15] ADINA theory and modeling guide, vol. I: ADINA, report ARD 09-7, May 2009.ADINA R&D, Inc.

[16] ADINA-T theory and modeling guide, vol. II: ADINA heat transfer, report ARD09-7, May 2009. ADINA R&D, Inc.

[17] Bathe KJ. Finite element procedures. Upper Saddle River, New Jersey: Prentice-Hall Inc.; 1996.

[18] Cary HB, Helzer SC. Modern welding technology. New Jersey: PearsonEducation; 2005.

[19] Hai-Xing W, Chen X. Three-dimensional modelling of laser-induced plasmaplume characteristics in laser welding. J Phys D: Appl Phys 2003;36:628–39.

[20] Zhu XK, Chao YJ. Effect of temperature-dependent material properties onwelding simulation. Comput Struct 2002;80:967–76.

[21] Teng TL, Chang PH, Tseng WC. Effect of welding sequence on residual stresses.Comput Struct 2003;81:273–86.