Elastoplastic analysis of process induced residual stresses in thermally sprayedcoatingsYongxiong Chen, Xiubing Liang, Yan Liu, and Binshi Xu Citation: Journal of Applied Physics 108, 013517 (2010); doi: 10.1063/1.3374710 View online: http://dx.doi.org/10.1063/1.3374710 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/108/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Reduction of NOx emission on NiCrAl-Titanium Oxide coated direct injection diesel engine fuelled with radish(Raphanus sativus) biodiesel J. Renewable Sustainable Energy 5, 063121 (2013); 10.1063/1.4843915 Effects of alloying elements and temperature on the elastic properties of dilute Ni-base superalloys from first-principles calculations J. Appl. Phys. 112, 053515 (2012); 10.1063/1.4749406 Residual stress analysis in the film/substrate system with the effect of creep deformation J. Appl. Phys. 106, 033512 (2009); 10.1063/1.3191684 Residual stress relaxation in the film/substrate system due to creep deformation J. Appl. Phys. 101, 083530 (2007); 10.1063/1.2717551 Response of a Zr-based bulk amorphous alloy to shock wave compression J. Appl. Phys. 100, 063522 (2006); 10.1063/1.2345606

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Elastoplastic analysis of process induced residual stresses in thermallysprayed coatings

Yongxiong Chen (陈永雄�,a� Xiubing Liang (梁秀兵�, Yan Liu (刘燕�, and Binshi Xu (徐滨士�National Key Laboratory for Remanufacturing, Academy of Armored Forces Engineering, Beijing100072, People’s Republic of China

�Received 8 July 2009; accepted 8 March 2010; published online 9 July 2010�

The residual stresses induced from thermal spraying process have been extensively investigated inprevious studies. However, most of such works were focused on the elastic deformation range. Inthis paper, an elastoplastic model for predicting the residual stresses in thermally sprayed coatingswas developed, in which two main contributions were considered, namely the deposition inducedstress and that due to differential thermal contraction between the substrate and coating duringcooling. The deposition induced stress was analyzed based on the assumption that the coating isformed layer-by-layer, and then a misfit strain is accommodated within the multilayer structure afterthe addition of each layer �plastic deformation is induced consequently�. From a knowledge ofspecimen dimensions, processing temperatures, and material properties, residual stress distributionswithin the structure can be determined by implementing the model with a simple computer program.A case study for the plasma sprayed NiCoCrAlY on Inconel 718 system was performed finally.Besides some similar phenomena observed from the present study as compared with previous elasticmodel reported in literature, the elastoplastic model also provides some interesting features forprediction of the residual stresses. © 2010 American Institute of Physics. �doi:10.1063/1.3374710�

I. INTRODUCTION

In the process of coating production by thermal spray-ing, the material to be deposited is introduced as the form ofpowder or wire into the core of a high velocity gas jet, whereit is melted by the heat sources such as plasma, electric arc,and high velocity oxygen flame �HVOF�. The molten par-ticles are atomized and accelerated toward the substrate, re-sulting in flattening, solidification �within a very short time,approximately few milliseconds�, and then coating forma-tion. Due to the large temperature difference during the pro-cess, residual stress is unavoidably generated in the coating/substrate structure, which strongly affects the coatingproperties and therefore the service life. Generally, the re-sidual stress can be divided into two main sources,1,2

“quenching” �primary� and “thermal” �secondary� stresses.During the deposition, the temperature of molten particlesstriking on the substrate drops instantaneously, thermal con-traction is constrained by the underlying solid, which resultsin tensile stress being developed in the splat layer, known as“quenching,” “intrinsic” or “deposition” stress. After spray-ing, the deposit and substrate cool down to room tempera-ture, accompanied with thermal cooling stress �tensile orcompressive� due to mismatch of the coefficient thermal ex-pansion �CTE� between the materials. As a result, the re-sidual stress is locked into the coating at the room tempera-ture and any subsequent usage of the coating components. Inmost cases, residual stresses have adverse effects on coatingproperties, such as bonding strength, resistance to thermalshock, fatigue life under bending, and erosion resistance. Un-derstanding of the development of stresses during spraying is

therefore of utmost importance, in order to devise countermeasures and to have desirable coating properties.

As an important factor controlling the coating properties,the residual stress has been widely investigated in previousworks. Among them the Stoney equation3 can be consideredas one of the earliest achievements, which can be conve-niently used to determine the residual stress in very thinfilms from the knowledge of curvature. Based on the Stoneyequation,4,5 some ex situ and in situ curvature experimentalmethods were developed to measure stresses. However, thisequation may result in considerable error if applied in thickfilms, and only an average stress can be obtained in thewhole film structure. Besides the curvature methods, otherexperimental methods to determine stresses in coatings werealso developed and modified, e.g., x-ray and neutron diffrac-tions, material removal methods.6,7 Each method has specificrequirements on the specimens, instruments, measurement,and evaluation procedures, and they also differ in the “rich-ness” and nature of provided information.4 For the x-ray andneutron diffraction stress measurement methods, as an ex-ample, one main limitation is that only on the coating surfacethe stress can be detected. This is a significant problem inmany applications, because the stress status within thecoating/substrate structure may be more important than thaton the coating surface if the maximum stress locates withinthe structure.

Besides the experimental methods for prediction of coat-ing residual stresses, numerical and analytical modeling canalso be considered as useful candidates,8–14 because both thestress on the coating surface and that within the structure canbe determined by the models. For example, the finite element�FE� modeling technique has been widely used to simulatethe residual stress generation during the thermal spraying

a�Author to whom correspondence should be addressed. FAX: �86� 1066717144. Electronic mail: yongxiongchen@yahoo.cn.

JOURNAL OF APPLIED PHYSICS 108, 013517 �2010�

0021-8979/2010/108�1�/013517/9/$30.00 © 2010 American Institute of Physics108, 013517-1

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process. But some models considered the stresses only due todifferential thermal contraction and neglect the quenchingcontribution, and others assumed that deformation is limitedin elastic range. These assumptions are often totally unjusti-fied. Recently some FE models were developed based on theelastoplastic assumption for thermally sprayed coatings, es-pecially for the graded or multilayered compositecoatings.15–18 The numerical results obtained from the FEmodels presented detailed stress distributions in the wholecoating and substrate. Some of them can be used to predictthe coating residual stress distributions induced from thethermal spraying process. But it is always unaffordable forthe computer ability when a thick coating is to be considered.Analytical modeling provides another way in understandingthe stress distributions in thermally sprayed coatings, whichis more cost-and-time efficient for calculation as comparedwith the FE method. Tsui and Clyne developed analyticalmodels to estimate the generation of stresses in coatings de-posited onto planar6,19,20 and cylindrical21 specimens, assum-ing that the coating is formed with a large amount of thinlayers continuously deposited one by one, and the quenchingstresses are taken into account. In their studies, the modelswere limited in the elastic range and the intrinsic �quenching�stress should be predetermined before calculation of the finalresidual stress was performed. In reality, the thermal contrac-tion of a thermal-sprayed splat during quenching is usuallylarge, which may initiate plastic deformation in the structure,and it’s difficult to obtain the real value of intrinsic stress forthe thin layers. Therefore, the present study is aimed at de-veloping an elastoplastic analytical model to investigate theresidual stresses in thermally sprayed coatings in the wholeelastic-plastic range, where the intrinsic stress need not beprespecified. Section II presents the quenching stress forma-tion, in which the coating system with planar geometry isconsidered as a multilayer composite beam which is formedwith progressively deposited thin layers. Section III de-scribes the formation of thermal stress in the coating systemduring cooling down to the room temperature. Section IVgives the result of the final residual stresses. Finally a casestudy is presented in Sec. V.

II. FORMATION OF DEPOSITION INDUCED STRESS

The geometry of the multilayered material consideredhere is such that the problem can be reduced to one dimen-sion, and that analytically tractable solutions can be used.For analytical simplicity, attention is confined to those caseswhere only thermal, elastic, and plastic deformations areconsidered, while other factors affecting the stresses such ascreeping, microcracks, and micropores in the material can beignored. The interfaces between the layers �including thecoating/substrate interface� are assumed to be perfectlybonded at all times. For most thermal spraying processes, thefeedstock can be heated upper its melting point, and depos-ited on the substrate in the form of small molten droplets. Sothe initial temperature of each depositing layer is assumed tobe the melting point of the material, and the layered materialis consequently considered to be initially stress-free. Itshould be noted that in some cases the spraying particles

may have an initial temperature lower than the melting tem-perature �e.g., for the process of HVOF�, where significantpeening stresses are generated during the impact of the semi-molten particles on the substrate.22 Prediction of the residualstresses for these cases is therefore become more complex,which is not taken into account in the present study.

A. Deposition of the first layer „bilayer structures…

Figure 1 schematically illustrates the geometry of thelayered structure. Consider that the sample is long enough ascompared with its total thickness. The analysis developed forthe plane stress problem can be easily extended to the biaxialcase by simply replacing Young’s modulus E by E / �1–v�where v is Poisson’s ratio; the equal biaxial stress state rep-resents the most realistic geometrical condition in the layeredmaterial.

Assume that the first layer impinging on the substrate isquenched from its initial temperature Tin to the depositiontemperature Td, and that the temperature change of the sub-strate induced by the deposition is so little that can be ig-nored, namely the temperature of the substrate is equal to Td

which is assumed to be fixed during the process. The misfitstrain �� between the first layer and substrate is given by

�� = �d�Tin − Td� = �d�T , �1�

where �d is the deposit CTE. Imposition of this misfit strainsets up a tensile force acting on the deposit and a compres-sive force acting on the substrate. Because of this pair ofequal and opposite forces, a bending moment is generated inthe structure. The bending strain �K can be defined as

�K = �K1 − K0��y − ye1� �− ts � y � td� , �2�

where K1, K0, and ye1 is the curvature due to the depositionof the first layer, the initial curvature, and the position of theneutral axial, respectively. Normally the initial curvature K0

would be equal to zero. The curvature is defined to be nega-tive in this case, as shown in Fig. 1. For the deposit andsubstrate, the total strain is therefore composed of two parts,one is correlated with the planar force, and the other is cor-related with the bending, which can be expressed as

�d1 = �dp1 + �K1 − K0��y − ye1� �0 � y � td� , �3�

FIG. 1. Schematics depiction of the generation of strains due to the depo-sition of the first layer on the substrate surface: �a� stress-free condition; �b�quenching induced misfit strain �unconstrained� ��; �c� planar strain com-ponent of the layer �dp1 and the substrate �sp1; and �d� asymmetric stressesinduced bending.

013517-2 Chen et al. J. Appl. Phys. 108, 013517 �2010�

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�s1 = �sp1 + �K1 − K0��y − ye1� �− ts � y � 0� . �4�

Note that K1−K0 and ye1 are independent of y, and that�dp1−�sp1=��=�d�T. Equations �3� and �4� can be rede-fined as

�d1 = �K1 − K0�y + �1 + �d�T �0 � y � td� , �5�

�s1 = �K1 − K0�y + �1 �− ts � y � 0� , �6�

where �1=�sp1− �K1−K0�ye1. For convenience in the discus-sion followed, it is assumed that the material constants aretemperature independent and that the substrate is always inthe elastic deformation range, the stress versus strain rela-tionship of the substrate is therefore given by

�s1 = Es�s1. �7�

In most cases, the temperature difference Tin−Td is signifi-cant and the contraction of the layer is usually large �at leastseveral millistrain�, which results in that the deposited layermay be beyond its elastic deformation range. The stress ver-sus strain relationship of most coating materials in a rangeslightly over the elastic limit may be estimated using a bilin-ear model as illustrated in Fig. 2, where Ed, Hd, and �Y arethe young’s modulus, strain hardening �in the tensile direc-tion�, and initial yield stress, respectively. Here, providingthe temperature change of the deposit is higher than the criti-cal value �this critical value, denoted by �T1, will be solvedlater�, which leads to the whole layer reaches the plasticstate, the stress versus strain relationship of the deposit canbe expressed as23,24

�d1 = Hd��d1 −�Y

Ed� + �Y . �8�

For the deposit/substrate structure, the following equilibriumconditions should be met:

�−ts

0

�s1dy + �0

td

�d1dy = 0, �9a�

�−ts

0

�s1�y − ye1�dy + �0

td

�d1�y − ye1�dy = 0. �9b�

Rearranging Eq. �9b� results in

�−ts

0

�s1ydy + �0

td

�d1ydy − ye1��−ts

0

�s1dy + �0

td

�d1dy�= 0. �9c�

Referring to Eq. �9a�, Eq. �9c� can be reduced to

�−ts

0

�s1ydy + �0

td

�d1ydy = 0. �9d�

Combining Eqs. �5�, �9a�, and �9d�, the explicit expressionsfor K1, �1, �d1 and �s1 can be conveniently obtained as

K1 = −6Eststd�ts + td�Hd��d�T − �Y/Ed� + �YHd

2td4 + Es

2ts4 + 2HdEstdts�2td

2 + 3tdts + 2ts2�

, �10�

�1 = K1Hdtd

3 + 4Ests3 + 3Ests

2td

6Ests�ts + td�, �11�

�d1 = K1Hdtd�6�ts + td�y − 3tdts − 4td

2� − Ests3

6td�ts + td�, �12�

�s1 = K1Hdtd

3 − 2Ests3 − 3Ests

2td + 6Ests�ts + td��y + ts�6ts�ts + td�

.

�13�

It should be noted that Eqs. �10�–�13� are identical tothat reported in literature,23,25 but there are some differencesbetween them. The solution in literature23,25 is based on thelinear strain assumption while the present study is based onthe generation of force and moment. Moreover, the final ob-jective of the present study is to obtain an elastoplastic solu-tion for thermal sprayed multilayer structures, while the dis-cussions in previous literature23,25 are only limited in bilayeror trilayer structures.

If the above solutions are compared with that performedin literature,19 it can be found that, since there are somesimilarities of the solution procedure used between the twomodels, the present study deduced the generation of stressesin the structure based on the assumption of temperaturevariation induced misfit strain, where using the fixed quench-ing stress is avoided �while it is used in literature19�, more-over, plastic deformation of the layer during deposition istaken into account in the present study.

B. The critical temperature variation required for fullplastic deformation

From the above discussion, it can be found that with theincrease in temperature difference �i.e., �T�, the stress in thecoating layer will increase accordingly. If the temperaturedifference is increased to a critical value, plastic deformationwill generate in the coating. According to Eq. �12�, it can befound that the maximum and minimum stresses in the coat-ing layer appear at the coating/substrate interface and the topsurface of the coating layer, respectively. The minimumstress can be expressed as

FIG. 2. Schematic representation of the stress-strain curve for the coatingand substrate.

013517-3 Chen et al. J. Appl. Phys. 108, 013517 �2010�

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�d1min = K1

Hdtd�3tdts + 2td2� − Ests

3

6td�ts + td�=

Ests�Hd�d�T − �YHd/Ed + �Y��Ests3 − Hdtd�3tdts + 2td

2��Hd

2td4 + Es

2ts4 + 2HdEstdts�2td

2 + 3tdts + 2ts2�

. �14�

When plastic deformation reaches the whole coating layer, the minimum stress in the layer must be equal to the initial yieldstress. As defined previously, �T1 is the critical temperature at which plastic deformation just reaches the whole layer.Therefore, substituting �d1

min=�Y into Eq. �14�, �T1 can be obtained as

�T1 =�Y

Ed�d+

�Y

Es�d

Hd�td/ts�4 + 6Es�td/ts�3 + 9Es�td/ts�2 + 4Es�td/ts�Es − 2Hd�td/ts�3 − 3Hd�td/ts�2 . �15�

In most thermal spraying cases, the thickness of the depos-ited layer �td� is about 2–3 levels lower than that of thesubstrate �ts�. Therefore, Eq. �15� can be estimated by

�T1 ��Y

Ed�d. �16�

Since �Y /Ed is equal to the elastic strain limit of the coatingmaterial, as shown in Fig. 2, Eq. �16� manifests that thecritical temperature �T1 corresponds to the condition thatmisfit strain initiated from temperature variation reaches thematerial elastic limit. For the spraying process, as mentionedbefore, the temperature difference Tin−Td is significant, andthe misfit strain ��=�d�T is much higher than the elasticstrain limit, therefore the condition of Eq. �15� can be easilysatisfied for most thermal sprayed materials. The followingstudy will only focus on the case in which this condition issatisfied. Since the minimum stress is higher than the initialyield stress and plastic deformation is generated in the wholelayer, discussion about the maximum stress is somewhat un-necessary here.

C. Deposition of the second layer „trilayer structures…

Consider the second layer impinging on the coated sub-strate �as shown in Fig. 3�. The initial temperature of the

second layer and the deposition temperature of the substrate�together with the first layer� are the same as before, so themisfit strain between the second layer and the coated sub-strate can also be expressed by Eq. �1�. The strains in thisstage for the two coating layers and the substrate are given as

�d2 = �dp2 + �K2 − K1��y − ye2� �td � y � 2td� , �17�

�d1 = K1y + �1 + �d�T + �sp2

+ �K2 − K1��y − ye2� �0 � y � td� , �18�

�s2 = K1y + �1 + �sp2

+ �K2 − K1��y − ye2� �− ts � y � 0� , �19�

where �dp2 and �sp2 is the strain component correlated withthe planar force of the second layer and the coated substrate�together with the first layer�, respectively. Their relationshipis

�dp2 − �sp2 = �� = �d�T . �20�

If it is assumed that �2=�sp2− �K2−K1�ye2, Eqs. �17�–�19�can be rearranged as

�d2 = �K2 − K1�y + �2 + �d�T �td � y � 2td� , �21�

�d1 = K1y + �1 + �d�T + �K2 − K1�y + �2 �0 � y � td� ,

�22�

�s2 = K1y + �1 + �K2 − K1�y + �2 �− ts � y � 0� . �23�

Consider the elastic deformation for the substrate and plasticdeformation for the whole second layer. Their stress versusstrain relationships are given by

�s2 = Es�s2, �24�

�d2 = Hd��d2 −�Y

Ed� + �Y . �25�

For the first layer, consider that the stress component resultedfrom deposition of the second layer is always compressive�the tensile possibility will be discussed later�, and that themodulus in the compressive direction of the plastically de-formed coating material can be set equal to its elastic modu-lus, the stress versus strain relationship for the first layer isthen given as

FIG. 3. Schematics depiction of the generation of strains due to the depo-sition of the second layer on the coated substrate surface: �a� stress-freecondition; �b� quenching induced misfit strain �unconstrained� between thesecond layer and the coated substrate ��; �c� planar strain component of thelayer �dp2 and the substrate �sp2; and �d� asymmetric stresses inducedbending.

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�d1 = Hd�K1y + �1 + �d�T −�Y

Ed� + �Y + Ed

���K2 − K1�y + �2� . �26�

The stress and momentum equilibrium conditions are

�−ts

0

�s2dy + �0

td

�d1dy + �td

2td

�d2dy = 0, �27�

�−ts

0

�s2ydy + �0

td

�d1ydy + �td

2td

�d2ydy = 0. �28�

Combining Eqs. �21�–�28�, the explicit expressions of �2 andK2 can be obtained as

�2 = − td

�4ts3Es + 9ts

2Estd − 4Hdtd3��− Hd�Y + �YEd + �d�THdEd�

Ed�32tstd3HdEs + 24ts

2td2HdEs + 8ts

3tdHdEs + Es2ts

4 + 16Hd2td

4�, �29�

K2 = 6�2�2td

6Hd3 + 2ts

6Es3 + 4ts

5Es3td + tstdHdEs�12ts

4Es + 52ts3tdEs + 8ts

2td2Hd + 78ts

2td2Es + 44tstd

3Es + 29tstd3Hd + 27td

4Hd���Es

2ts4 + 6ts

2td2HdEs + Hd

2td4 + 4tstd

3HdEs + 4ts3tdHdEs��4ts

3Es + 9ts2Estd − 4Hdtd

3�. �30�

With the expressions of �2 and K2, the solutions for �d2, �d1,and �s2 can be conveniently obtained, while they are notpresented here due to the expressions are very long.

D. Deposition of the nth layer „multilayer structures…

The above procedure can be extended to analysis depo-sition of the nth layer. The strain for n layers of the coatingas well as the substrate can be expressed as follows

�dn = �Kn − Kn−1�y + �n + �d�T ��n − 1�td � y � ntd� ,

�31�

�dj = �Kj − Kj−1�y + � j + �d�T

+ �i=j+1

n

��Ki − Ki−1�y + �i� ��j − 1�td � y � jtd� ,

�32�

�sns = Kny + �

i=1

n

�i �− ts � y � 0� , �33�

where 1� j�n−1. Assume that, for all previously depositedn−1 layers, the stress component resulted from deposition ofthe nth layer is still compressive. The strain versus stressrelationship for the coating/substrate structure can be ob-tained as

�dn = Hd��Kn − Kn−1�y + �n + �d�T − �Y/Ed� + �Y

��n − 1�td � y � ntd� , �34�

�dj = Hd��Kj − Kj−1�y + � j + �d�T − �Y/Ed� + �Y

+ Ed �i=j+1

n

��Ki − Ki−1�y + �i�� ��j − 1�td � y � jtd� ,

�35�

�sn = Es�Kny + �i=1

n

�i� �− ts � y � 0� . �36�

The equilibrium conditions become

�−ts

0

�sndy + �j=1

n−1 ��j−1�td

jtd

�djdy + ��n−1�td

ntd

�dndy = 0, �37�

�−ts

0

�snydy + �j=1

n−1 ��j−1�td

jtd

�djydy + ��n−1�td

ntd

�dnydy = 0.

�38�

With the above equations, expressions of K1 ,K2 . . .Kn and�1 ,�2 . . .�n as well as the stresses set up due to deposition aredetermined. This can be done by writing a simple computerprogram, which is straightforward and time-effective.

It should be noted that it is possible that, for some pre-viously deposited layers, the stress �strain� component re-sulted from the deposition of nth layer becomes tensile insome cases. But it is clear that the resulted planar strain �dueto the thermal contraction of nth layer� acting on the previ-ously deposited layers and substrate is still compressive. Thesign of the total strain component �the sum of planar strainand the bending strain� is therefore determined by the bend-ing component, namely only when the bending strain be-comes positive �tensile� and its absolute value is higher thanthat of the planar strain, the total strain may become tensile.As defined in Eq. �2�, the bending strain for the materiallocated lower than the neutral axis is positive and that higherthan the neutral axis is negative. As a result, if the bendingstrain component for some deposited layers changes fromcompressive to tensile, it must starts from the layer/substrateinterface �i.e., at point y=0�. For a coating/substrate structurein pure elastic range, the neutral axis position yen� can begiven as19

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yen� =�ntd�2Ed − ts

2Es

2�ntdEd + tsEs�. �39�

If yen� 0, the neutral axis will located in the deposit, and thelayers located lower than yen� will become tensile, and it ispossible that some of them have a tensile total strain compo-nent mentioned above. While for a coating/substrate struc-ture in an elastic-plastic range, if plastic deformation is ini-tiated in the layers, the neutral axis position will lower thanthat of pure elastic coating/substrate structure under the sameconditions �e.g., the same dimensions, load, and material�.Therefore, an approximately maximum thickness of ntd canobtained if Eq. �39� is equal to zero, which can be used topredict the critical status in which the total strain componentchanges from compressive to tensile. This value is very con-servative because when the neutral axis position reaches tothe substrate/coating interface, only the bending strain be-comes positive �tensile� in the layers below the neutral axis,but the total stain �sum of the bending and planar straincomponents� in these layers is still negative. In another side,this prediction method is enough because few industrial coat-ings have the thickness beyond this critical value.

III. FORMATION OF CTE MISMATCH INDUCEDSTRESS DURING COOLING

When the whole coating/substrate structure cools downto room temperature, stresses due to CTE mismatch will gen-erate in the structure. Therefore, a misfit strain between thetwo bonded plates, ��= ��d−�s��Td−Tr�, is created. Similarto that described in Sec. II, a planar strain/stress and a bend-ing strain/stress are acted on the whole coating and substratedue to the misfit strain between the coating and substrate.The final curvature after cooling is Kc, while Kn is the cur-vature adopted after the last layer has been deposited. Forconvenience in expressions followed, the coating thickness isdefined as h �i.e., h=ntd�. The total strain of the system isgiven as

�dc = �dp + �Kc − Kn��y − yec� �0 � y � h� , �40�

�sc = �sp + �Kc − Kn��y − yec� �− ts � y � 0� . �41�

Consider �c=�sp− �Kc−Kn�yec. Equations �40� and �41� canbe rearranged as

�dc = �Kc − Kn�y + �c + ��d − �s��Td − Tr� �0 � y � h� ,

�42�

�sc = �Kc − Kn�y + �c �− ts � y � 0� . �43�

As discussed previously, in the progressive depositionprocess, the tensile component of stress in each layer is pro-gressively reduced by deposition of successive layers on topof it �see Eq. �35��. Therefore, a rebound process to somedegrees has been generated in the deposited layers, whichmay result in that the stress acting on some layers is belowthe material yielding strength again. Consider the tensile de-formation ��dc0, or �sc0� as a loading process and thecompressive deformation ��dc0, or �sc0� as an unload-ing process for the plastically deformed deposit. The stress

status will become very complex in the deposit during cool-ing down because the sign of the deformation is determinedby the parameters such as the deposition temperature, CTEdifference between the coating and substrate, and sample di-mensions as well as the pre-existed stress magnitude duringthe deposition. In other words, besides the deposition in-duced stresses generated in the layers, there are several pos-sibilities that an additional loading or unloading process actson the coating, therefore, it is difficult to list out every pos-sibility in the analytical model, and to list out whether elasticmodulus or strain hardening modulus should be used to de-scribe the stress versus strain relationship for the coating. Forconvenience in discussion followed, the CTE mismatch in-duced strain in the coating is only limited in the case that�dc0, the relationship of stress versus strain generated inthe cooling process for the substrate and coating is given as

�sc = Es�sc, �44�

�dc = Ed�dc. �45�

Consider the equilibrium conditions below

�−ts

0

�scdy + �0

h

�dcdy = 0, �46�

�−ts

0

�scydy + �0

h

�dcydy = 0. �47�

Combining Eqs. �42�–�47�, the solutions for Kc−Kn, �c, �dc,and �sc can be obtained as

Kc − Kn = −6htsEsEd�ts + h���d − �s��Td − Tr�

Ed2h4 + Es

2ts4 + 2htsEdEs�2h2 + 3hts + 2ts

2�,

�48�

�c = �Kc − Kn�Edh3 − 2Ests

3 − 3Ests2h

6Ests�ts + h�, �49�

�dc = �Kc − Kn�Edh�6�ts + h�y − 3hts − 4h2� − Ests

3

6h�ts + h�, �50�

�sc = �Kc − Kn�Edtd

3 − 2Ests3 − 3Ests

2td + 6Ests�ts + td��y + ts�6ts�ts + td�

.

�51�

IV. FINAL RESIDUAL STRESSES

From the above knowledge, the final residual stressesdistribution in the substrate and coating layer can be conve-niently obtained by adding the contributions represented byEq. �51� to Eq. �36�, and adding Eq. �50� to Eqs. �34� and�35�, respectively.

�snres = �Kc − Kn�

Edtd3 − 2Ests

3 − 3Ests2td + 6Ests�ts + td��y + ts�

6ts�ts + td�

+ Es�Kny + �i=1

n

�i� �− ts � y � 0� , �52�

013517-6 Chen et al. J. Appl. Phys. 108, 013517 �2010�

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�dnres = �Kc − Kn�

Edh�6�ts + h�y − 3hts − 4h2� − Ests3

6h�ts + h�

+ Hd��Kn − Kn−1�y + �n + �d�T − �Y/Ed� + �Y

��n − 1�td � y � ntd� , �53�

�djres = �Kc − Kn�

Edh�6�ts + h�y − 3hts − 4h2� − Ests3

6h�ts + h�+ Hd��Kj

− Kj−1�y + � j + �d�T − �Y/Ed� + �Y + Ed �i=j+1

n

��Ki

− Ki−1�y + �i�� ��j − 1�td � y � jtd� . �54�

Therefore, the solution of the residual stresses is just asimple addition of the deposition induced quenching stressesand the cooling induced thermal stresses solved in the abovesections.

V. A CASE OF ELASTOPLASTIC ANALYSIS

Once the dimensions, material properties of the substrateand coating layer are determined, implementation of themodel is fairly straightforward. In order to illustrate the useof the present model, typically atmospheric plasma sprayed�APS� NiCoCrAlY coating on the Ni alloy substrate is con-sidered, because this coating system is often used in industryand widely investigated in previous literature. The propertiesrequired for the model are listed in Table I. The flow chartgiving the sequence of the residual stress calculation isshown in Fig. 4. It should be noted that the current modeldoes not consider the variation in material properties withtemperature, which would affect the accuracy of the predic-tions if the deposition temperature is very high and the prop-erties varies considerably with temperature.

When using the model, selection of n �hence h� is arbi-trary. Figure 5 shows the effect of vary n on the predictedresidual stress distribution due to quenching only for the APSNiCoCrAlY on Inconel 718 system. For the case when n isequal to 1, i.e., block deposition, the stress level and thestress gradient in the structure �especially in the coating� aresignificantly different to those when n is equal to 5 or 100.This is due to the fact that, in the progressive depositionprocess, the tensile stress in each layer, as mentioned previ-ously, is progressively reduced by deposition of successive

layers on top of it, because the deposition induced stresscomponent in the successive layers is always compressive.Therefore, the stress at the coating top surface should bemore tensile as compared to that at the interface �i.e., a posi-tive stress gradient�. But for the case that the process is con-sidered as a block deposition, a negative stress gradient isobtained. This is because the stress component originatedfrom the curvature in each layer is compressive. In fact, sucha negative stress gradient is generated within every layer,which is very apparent for the n=5 plot. However, as n be-comes large, this effect becomes insignificant because thethickness of each layer is too small to be identified from theplot, and the real trend is that of the stress rising from thebottom to the top surface of the coating. Therefore, using asuitable large value for n will be more identical to the realityand will give a fairly accurate prediction of the stress distri-bution. It is important to be noted that the above resultsobtained from the present model are similar to that reportedby Tsui and Clyne,19 which is solved based on the elasticdeformation assumption. However, details of the plots indi-

TABLE I. Material properties used in the analytical model �Refs. 1, 26, and27�.

Property

Material

Inconel 718 NiCoCrAlY

CTE �MK−1� 14.4 11.6Young’s modulus �GPa� 200 204Poisson’s ratio 0.3 0.3Yield start stress �MPa� 1185 270Strain hardening �GPa� ¯ 5Melting point �K� ¯ 1673

FIG. 4. Flow chart showing the main calculation processes for the presentmodel.

FIG. 5. Plots of deposition induced stress for APS NiCoCrAlY on Inconel718, by treating the deposition as a block process �n=1� or as a progressiveprocess with n �n=5,100�.

013517-7 Chen et al. J. Appl. Phys. 108, 013517 �2010�

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cate some remarkable differences. For example, for the casewhen n is equal to 1, the negative gradient of the stress in thepresent model is less obvious than that in the literature, whilefor the case when n is equal to 100, the positive gradient inthe former is much higher than the later. This is mainly dueto the different assumptions of the deformation range andthat of the intrinsic stress. In the previous literature,19 asmentioned before, the material properties are assumed to beof pure elastic deformation, and a fixed quenching �intrinsic�stress has to be prespecified before calculation. In practice, itis difficult to obtain the real value of intrinsic stress for thethin layers, even if by some experimental apparatuses or nu-merical methods. In the present model, all the material prop-erties �see Table I�, including the parameters correlated withthe plastic deformation, can be directly obtained from experi-ments or referring previous works.

Figure 6 shows the stress distributions for the APS NiC-oCrAlY on Inconel 718 system with and without the finalcool-down process. It is found that the final residual stress isclose to that due to quenching only, which manifests that thestresses arising from quenching of splats account for most ofthe final stress levels. For the case shown in Fig. 6, a con-stant deposition temperature of 700 K was assumed androom temperature is assumed to be 300 K. Whereas in factthe deposition temperature can be varied somewhat. So it isworthwhile to investigate the effect of assuming differentdeposition temperatures on the final residual stresses. Figure7 shows the residual stress plots with three different deposi-tion temperatures �covered the actual range during spraying�,i.e., 473, 623, and 773 K, respectively. It can be found that,the residual stress in the coating decreases with the increasein deposition temperature. For the substrate, as the deposi-tion temperature increases, the curvature slope �absolutevalue� of the residual stress decreases correspondingly.Therefore, it can be concluded that using a relatively higherdeposition temperature is helpful to improve the residualstress distribution in the coating system.

The effect of coating thickness on the residual stressdistribution was also investigated. As shown in Fig. 8, withthe increase in coating thickness, the residual stress on thecoating surface �tensile� increases correspondingly, and themagnitude of residual stress �compressive stress in most

cases� at the interface also increases. For the substrate, ahigher curvature slope �absolute value� of the residual stressis obtained when the coating thickness increased. In addition,it can be found that, in the coating systems with three differ-ent thicknesses, the coating residual stresses at the same dis-tance from the interface are different from each other, andthat, at the same position, the stress value in the structurewith higher coating thickness is always lower than that withlower thickness.

VI. CONCLUSIONS

This paper presents an elastoplastic analysis of the re-sidual stress in thermally spraying coating systems. Twomain residual stress generation mechanisms are taken intoaccount in the developed elastoplastic model, i.e., thequenching stress during deposition and thermal mismatch be-tween coating and substrate during cooling. The depositioninduced stress was analyzed based on the layer-by-layerdeposition assumption which clearly illustrated how stressesbuild up during the spraying process. Although only one pos-sibility for the thermal mismatch stress generation was dis-cussed in the present study, the approach presented here canbe extended to other possibilities. As compared with othernumerical or analytical models, the main advantage of thepresent model lies in that, without predetermining the intrin-

FIG. 6. Predicted stress distributions from the present model �n=100� forAPS NiCoCrAlY on Inconel 718 with and without the final cool-down pro-cess �from 700 K�.

FIG. 7. Predicted residual stress distributions from the present model �n=100� for APS NiCoCrAlY on Inconel 718 with different deposition tem-peratures, 473, 623, and 773 K.

FIG. 8. Predicted residual stress distributions from the present model �n=100� for APS NiCoCrAlY on Inconel 718 with different coating thick-nesses, 0.5, 1.5, and 2.5 mm.

013517-8 Chen et al. J. Appl. Phys. 108, 013517 �2010�

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sic stress, the stress distribution in the coating can be easilydetermined by knowing the specimen dimensions and mate-rial properties. The case study for the APS NiCoCrAlY onInconel 718 system shows that the stresses resulting fromquenching of splats account for most of the final stress lev-els, and that controlling the residual stress levels can be eas-ily performed based on the model by optimizing the processparameters, e.g., deposition temperature and coating thick-ness.

Although the material properties considered in thepresent model are temperature independent, in most cases,this does not lead to large errors, at least in the applicationsof qualitative prediction of the residual stress level in thecoating system. In addition, the thermally sprayed coatingconcerned in the elastoplastic model is considered as amultilayer structure, so the model can be easily applied intoother multilayer thin films, such as chemical vapor deposi-tion and physical vapor deposition films.

ACKNOWLEDGMENTS

The authors are grateful to the priority support by ChinaNatural Science Foundation �Grant Nos. 50735006 and50905185�, National Science and Technology Support Pro-gram �Grant No. 2006BAF02A19�, and National “863”Project �Grant No. 2009AA03Z342�.

NOMENCLATURE

� coefficient of thermal expansion �K−1��� misfit strain

E Young’s modulus� strain

Hd strain hardening �Pa�h thickness of the all deposited layers �m�K curvature �m−1�n number of layersT temperature �K�

�T difference between the droplet initial tempera-ture and deposition temperature �K�

t thickness �m�� strain component� stress �Pa�v Poisson’s ratioy displacement, relative to the interface �m�

ye neutral axial position, relative to the interface�m�

Subscriptsd ,s deposit and substrate, respectively

dp ,sp planar strain component �for deposit and sub-strate, respectively�

in initial �for temperature�K bending �for strain�

i , j ,n ,c for the deposition of ith, jth, and nth layer andthe final cooling, respectively

r room temperatureY yielding �for stress�

Superscriptsmin minimum �for stress�

ns for substrate after deposition of n layersres residual �for stress�

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