Finite Element Analysis of Residual Stress in Plasma‐Sprayed Ceramic

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T Valente, C Bartuli, M Sebastiani and F CasadeiFinite element analysis of residual stress in plasma-sprayed ceramic coatings

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Finite element analysis of residual stress inplasma-sprayed ceramic coatings

T Valente1, C Bartuli1�, M Sebastiani1 and F Casadei2

1Department of Chemical and Materials Engineering, University of Rome ‘La Sapienza’, Rome, Italy2Centro Sviluppo Materiali Spa, Rome, Italy

Abstract: A numerical study was carried out by finite element analysis (FEA) for the calculation of

absolute values and through-thickness variation of residual stress originating in thermal spray coatings.

The investigated deposit was an air plasma sprayed alumina coating sprayed on a carbon steel substrate pre-

viously coated with Ni20Al bond coat. Results show compressive residual in-plane stresses with linear

through-thickness variation and tensile normal and shear stresses having a peak at the coating–substrate

interface. The influence of deposition temperature on residual stress was also investigated. The experimental

validation of the FEA model was carried out using a high-speed hole drilling technique, suitably adapted

for the analysis of a multimaterial structure through FEA calculation of the required calibration coefficients.

A very good agreement between simulated and measured stresses was obtained, inspite of the adopted

simplification hypothesis.

Keywords: residual stress, plasma spray, ceramic coating, finite element, hole drilling

1 INTRODUCTION

Residual stresses originating within thermal sprayed coat-

ings in different stages of their production process can

strongly affect their mechanical properties and therefore

play a key role in their functionality and in the lifetime of

coated components. Thus, it is important to understand the

evolution of the stresses during the coating process and to

investigate the influence of processing parameters on their

absolute value and spatial distribution.

In particular, residual stresses are known to depend firstly

on the deposition temperature and secondly on particle

temperature and velocity, on deposition rate and, most

importantly, on coating thickness [1–3]. They also have a

fundamental influence on coating structure and substrate–

coating adhesion, thus strongly affecting deposit perform-

ance in working conditions.

Different experimental techniques are available nowa-

days for their direct measurement or indirect evaluation

[4–10], such as X-ray or neutron diffraction methods, cur-

vature measurements, high-speed hole drilling techniques

and metallurgical and layer removal methods. However,

numerical modelling tools [11, 12] offering reliable

predictions of the mechanical properties of a selected

substrate–coating system on the basis of the operating con-

ditions of the manufacturing process are of even greater

importance, assisting processing parameter optimization

procedures, saving time and reducing the economic impact

of the trial and error approach.

Residual stress evaluation for plasma-sprayed coatings by

finite element analysis (FEA) is a complex problem owing

to the relevant number of processing parameters involved

which can result in non-independent sources of stress con-

tributing to the final state. Such stresses are generally classi-

fied as ‘quenching stresses’ and ‘DTC stresses’, originating

from differential thermal contractions.

Quenching stresses [13–15] arise from impact, spreading

and solidification of each molten particle (at temperature

Tm) on the substrate surface. They are always tensile

and can be, in principle, approximately estimated by the

following equation

s0 ¼ E0

ðTm

Ts

ac(T) dT (1)

where Tm is the coating melting temperature, Ts is the sub-

strate temperature and ac is the coating thermal expansion

coefficient.

Several sources of error, however, can affect equation (1),

usually giving results that do not match experimental data

The MS was received on 6 February 2004 and was accepted after revisionfor publication on 15 June 2004.�Corresponding author: Department of Chemical and Materials Engineer-ing, University of Rome ‘La Sapienza’, Via Eudossiana 18, 00184 Rome,Italy.

321

L00604 # IMechE 2004 Proc. Instn Mech. Engrs Vol. 218 Part L: J. Materials: Design and Applications



with good accuracy [13]; among the most important, two

need to be mentioned:

1. Young’s modulus of sprayed materials is very different

from the elastic modulus of the corresponding bulk

materials (with values of the Ec/Eb ratio as low as 1/

10), mainly as a consequence of coating porosity and

lamellar microstructure.

2. Stress relaxation phenomena [13] (creep and yielding for

metals, microcracking for ceramics) can be responsible for

a remarkable decrease in the final value of quenching stress.

In the case of metallic materials, equation (1) can thus be

modified as follows

sq ¼ E�c

ðbTm

Ts

ar(T) dT (2)

where E�c is the actual elastic modulus of the coating, as

determined by experimental measurements, and b is a temp-

erature reduction coefficient (approximately equal to about

0.6) adopted to take into account stress relaxation phenom-

ena such as yielding or creep.

On the other hand, for ceramic-based coatings, a very

marked reduction in quenching stress has also been

observed [1, 16, 17], produced as a consequence of exten-

sive microcracking of individual lamellae after droplet

impact. As an example, a residual tensile stress of about

10 MPa can be experimentally evaluated by in situ curva-

ture measurements for plasma-sprayed alumina [13], while

quenching stresses of the order of GPa would be obtained

simply by applying equation (1).

This aspect simplifies the numerical evaluation of the

total stress generated within ceramic coatings, and the con-

tribution of quenching stresses can be taken into account

simply by superimposing a uniform tensile stress, typically

in the range 10–15 MPa, to the thermally induced stress

fields [13]. This last source of stress results from misfit

strains generated by differential thermal contractions

(DTCs) [15, 18] caused by thermal expansion coefficient

(CTE) mismatch between coating and substrate; in particu-

lar, DTC stresses are induced during the final cooling stage

from processing temperature, Ts, to room temperature, Tr.

The equation correlating these misfit strains to the ther-

mal properties of the substrate and coating is as follows

D1 ¼ðTs

Tr

½ac(T)� as(T)� dT (3)

whereas is the coefficient of thermal expansion of the substrate.

Starting from equation (3), residual stress can be evalu-

ated, in the case of linear elastic material behaviour,

simply by applying the well-known Hooke’s law

sth ¼ Er

1� nc

ðTs

Tr

½ac(T)� as(T)� dT (4)

where nc is Poisson’s ratio of the coating. Equation (4)

shows that, for ac , as compressive thermal stresses arise

in the coating owing to final cooling after spraying.

Moreover, the CTE is a function of temperature, so that

thermal stresses can also arise within individual layers if

through-thickness thermal gradients are originated. This is

generally the case when the heat flux generated by the

plasma torch is repeatedly transferred to the substrate and

to the deposited layers during coating build-up.

Although complex interactions between coexisting resi-

dual stresses of various sources can be generally expected,

a sufficiently reliable prediction model can be developed

for qualitative stress estimation and preliminary process

optimization by adopting realistic simplification hypoth-

eses. The following assumptions are proposed:

1. The coating thickness is negligible compared with the

substrate thickness.

2. Spraying is carried out in constant substrate temperature

conditions (Ts).

3. The sprayed deposit is instantly quenched to Ts.

4. The substrate–deposit couple remains isothermal as it

cools down to room temperature.

5. The mechanical behaviour of the coating material is

linear elastic in the cooling range.

6. Interface bonding between the deposit and the substrate

is perfect.

On the basis of the above simplifications, the final residual

stress state can be expressed as the sum of quenching stres-

ses, sq and DTC stresses, sth

sr ¼ sq þ sth (5)

The aim of the present study is to evaluate, by using finite

element analysis (FEA) [11, 14, 19, 20], average values

and spatial distributions of DTC stresses in thermal sprayed

ceramic coatings. Experimental validation of the model was

carried out using a suitable implementation of the incremen-

tal hole drilling method [21], adapted to the analysis of

plasma-sprayed deposits by developing a further FEA

model for the calculation of calibration coefficients [22–24]. Both FEA simulation and experimental measurements

were carried out for commercialy pure alumina coatings

deposited by air plasma spraying (APS) onto AISI 4037

carbon steel substrates previously coated with a Ni20Al

bonding layer.

2 FINITE ELEMENT MODELLING

The developed model was implemented using ANSYSw 7.0

software. The complexity of the analytical process made

it expedient to carry out the modelling procedure in two

consecutive steps:

1. In the first analytical step the aim was to simulate the ther-

mal history of the coating during spraying, in order to cal-

culate the effective deposition temperature of each layer.

322 T VALENTE, C BARTULI, M SEBASTIANI AND F CASADEI

Proc. Instn Mech. Engrs Vol. 218 Part L: J. Materials: Design and Applications L00604 # IMechE 2004



2. In the second step a structural analysis was performed to

estimate the residual stresses by applying the boundary

thermal conditions obtained from the previous step, and

by further implementing the final cooling phase.

2.1 Thermal history

In a thermal spray process a source of thermal and kinetic

energy is used to melt and accelerate the starting powders

towards a solid substrate where successive layers are

deposited. In the case of the APS deposition technique, a

high energy density plasma is generated electrically within

the spraying torch. The heat flux generated by each pass of

the torch was estimated by a numerical analysis procedure

based on a proprietary one-dimensional Fortran code.

Among other features, the analysis requires the calcu-

lation of the heat exchange coefficient between the plasma

jet and the substrate, starting from the plasma processing

parameters. To this end, the Holger approach [25] was

adopted, based on the equation

Nu ¼ D

r

1� 1:1(D=r)

1þ 0:1((H=D)� 6)(D=r)Pr0:42F(Re) ¼ hD

K

ð6Þ

where

D ¼ nozzle diameter

H ¼ torch–substrate distance

Pr ¼ cpg � mg

kg(Prandtl number)

cpg ¼ specific heat of the plasma gas

mg ¼ viscosity of the plasma gas

kg ¼ thermal conductivity of the plasma gas

Re ¼ rgjvg � vpjdpmg

(Reynolds number)

rg ¼ density of the plasma gas

vg ¼ velocity of the plasma gas

dp ¼ average diameter of the particles in the plasma jet

Vp ¼ velocity of the particles in the plasma jet

r ¼ 2:5D

K ¼ thermal conductivity of the substrate

h ¼ heat exchange coefficient

The function F(Re) is defined as:

F(Re) ¼ 1:36Re0:574 if 2000 , Re , 30 000

F(Re) ¼ 0:54Re0:667 if 30 000 , Re , 120 000

F(Re) ¼ 0:151Re0:775 if 120 000 , Re , 400 000

The thermal flux, _qq, transferred to the substrate (or to the

coating/substrate system) can be expressed as a function

of the heat exchange coefficient, h (obtained for calculated

Reynolds numbers in the range 1500–40 500) by the fol-

lowing equation

_qq ¼ h(Tg � Ts) ¼ K@T

@x(7)

where Ts is the temperature of the substrate or the substrate/

coating system and x is the distance from the substrate.

As an example, in the case of air plasma spray with

a nozzle diameter of 6 mm, for a power input of 40 kW,

a plasma gas mixture flowrate of 50 standard liters per

minute (slpm) Arþ 12 slpm H2 and a spraying distance of

115 mm, a heat flux of 283 kW/m2 (Fig. 1) with a spot

area of 25 � 25 mm2 was estimated.

The total heat flux released by the substrate, _qqout, in the

ordinary temperature range for a spraying process can be

considered as the sum of a convection and a radiation term

_qqout ¼ k1(T � T0)þ k2(T4 � T4

0 ) (8)

For the simulation of the mass flow, the simultaneous

deposition of a whole layer of material was assumed at

each torch pass.

Mass flow and heat fluxes were then fitted into a two-

dimensional finite element model (Fig. 2), built with plane

thermal elements (of the type ‘plane 55’). The adopted

algorithm involved non-static analyses: the movement of

the torch was simulated by the movement of the previously

evaluated heat flux, and coating built-up was simulated with

the controlled activation (birth) of preselected elements.

Fig. 1 Calculated heat flux on a 50 � 50 � 4 mm substrate as

a function of spraying distance for a nozzle diameter

of 6 mm, a power input of 40 kW and a plasma gas

mixture flowrate of 50 slpm Arþ 12 slpm H2

FINITE ELEMENT ANALYSIS OF RESIDUAL STRESS IN PLASMA-SPRAYED CERAMIC COATINGS 323




The thickness of each layer was kept constant and calculated

by dividing the coating final thickness by the number of

torch passes.

Thermal properties of sintered a-Al2O3, available in the

literature [26] and given in Table 1, were used for the simu-

lation of the coating thermal behaviour. This assumption

does not introduce significant errors [13] in the case of

alumina coatings.

The variation in the generic property P with temperature

was expressed in the model according to a polynomial

function

P(T) ¼ C0 þ C1T þ C2T2 þ C3T

3 þ C4T4 (9)

where the coefficients Ci were estimated from the literature

values with regression models (least-squares method).

A different approach was used for the selection of the

value of Young’s modulus, for which the difference between

the coating and the bulk (sintered) material cannot be neg-

lected. Four-point bending tests of as-sprayed coatings

were performed at room temperature, and they gave an

average E value of 55 GPa with a standard deviation of

5 MPa, as calculated on the basis of the beam theory. Ten

3 � 4 � 50 mm specimens (AISI4037 substrates coated

with Ni20Al/Al2O3, total coating thickness 0.5 mm) were

tested according to ASTM C1161-02c standard by means

of a MTS Bionix universal testing machine with a load

cell of 25 kN. A totally different value of 410 GPa charac-

terizes sintered a-Al2O3 [26].

2.2 Residual stress

The selected shape for structural analysis was a three-

dimensional plane disc (Fig. 3) built with eight-node solid

finite elements (‘solid 45’). In consideration of the axial

symmetry of the geometrical system, a two-dimensional

model (‘plane 42’) was also tested, with a remarkable

reduction in the solution time. By implementing the

general algorithm illustrated in Fig. 4 for the three- and

two-dimensional model, results characterized by the same

accuracy were obtained: the two-dimensional model was

therefore used for all simulations with consistent simplifica-

tion of the calculation procedure.

The spraying process was simulated using the ANSYSw

EKILL/EALIVE function in order to set the birth or death

of elements and to build up the coating layer by layer. For

each layer, the calculated deposition temperaturewas imposed

as a boundary condition and was also assumed as a reference

temperature for the calculation of residual stress according to

equation (4). In terms of constitutive relationship models, a

bilinear hardening and a pure elastic behaviour were assumed

for carbon steel and alumina-sprayed coating respectively.

3 EXPERIMENTAL PROCEDURES

3.1 Coating deposition

Coatings simulated in FEA models were fabricated by air

plasma spraying starting from alumina powders (Metco

105NS, 2325 meshþ 15 mm), using the standard spraying

parameters reported in Table 2. An average thickness of

320 mm was obtained with 46 torch passes. Metallic bond

Fig. 2 Geometrical model for the simulation of thermal

history

Table 1 Thermo-mechanical properties for sintered alumina at various temperatures (from [26])

Property 20 8C 500 8C 1000 8C 1200 8C 1400 8C 1500 8C

Poisson’s ratio 0.231(+0.001) 0.237 0.244 0.247 0.250 0.252Specific heat (J/kg K) 755(+15) 1165 1255 1285 1315 1330Thermal conductivity (W/m K) 33(+2) 11.4 7.22 6.67 6.34 6.23Thermal expansion coefficient (1026 K21) 4.6(+0.2) 7.1 8.1 8.3 8.5 8.6

Fig. 3 Three-dimensional finite element model: disc





coats with an average thickness of 180 mm were obtained

from Ni20Al powders (Metco 404NS, 2170þ 270 mesh).

X-ray diffraction patterns of as-sprayed coatings revealed

a dual-phase microstructure composed of a-Al2O3 and

g-Al2O3. A coating cross-section is shown in Fig. 5.

3.2 Residual stress measurement

Residual stresses of as-sprayed coatings were experimen-

tally evaluated using the hole drilling technique on the

basis of an incremental method [21, 27]. The procedure con-

sists of the removal of stressed material by drilling a small

blind hole in the top surface of the coating, and of the

measurement of the strain relaxation occurring in the adja-

cent material by means of a proper set of strain gauges.

By means of suitable stress–strain relationships, residual

stresses can be evaluated provided that correct values for

calibration coefficients [22, 28] are adopted. The most

widely used method for calculating residual stresses from

measured relaxed strains is the ‘integral method’ [22]:

stress values are obtained from the strain relaxation

measured at various depths with a rosette strain gauge on

the basis of the following set of equations

Xj¼i

j¼1

�AAijPj ¼ pi, 1 4 j 4 i 4 nc

Xj¼i

j¼1

�BBijQj ¼ qi, 1 4 j 4 i 4 nc

Xj¼i

j¼1

�BBijTj ¼ ti, 1 4 j 4 i 4 nc (10)

where

Pj ¼ s(1) j þ s(3) j

2, Qj ¼ s(1) j � s(3) j

2,

T j ¼ t(13) j (11)

pi ¼ 1(1)i þ 1(3)i2

, q j ¼ 1(1)i � 1(3)i2

,

ti ¼ 1(3)i þ 1(1)i � 21(2)i2

(12)

represent stress and strain components in the direction of the

three strain gauges (Fig. 6) in accordance with the nc steps in

to which the calculation is divided.

The terms Pj and pi represent the hydrostatic component

of residual stresses and the corresponding volumetric strain

relaxation. Similarly, the other variables (Qj, Tj, qi, ti) rep-

resent the shear stresses and the shear strain components.

The problem can only be solved if the two matrices of

calibration constants Aij and Bij appearing in equation (10)

(the first relating to the hydrostatic stress component, the

second to the deviatoric one) are known. Since this is

not the case for blind holes, for which closed analytical

Fig. 4 Algorithm adopted for the simulation of residual

stresses

Table 2 Spraying parameters for the alumina

coating

APS spraying parameters for Al2O3

Substrate AISI 4037 steelSpraying distance 115 mmGas plasma (slpm) 50Ar, 12H2

Plasma current 560 APlasma voltage 73 VNumber of torch passes 46Cooling gas Ar, 4.5 barCoating thickness 320 mm

Fig. 5 Microstructure of an air plasma sprayed Al2O3 coating

(cross-section, optical microscope, �200)





solutions are not available, calibration constants were calcu-

lated by FEA. The adopted procedure to evaluate the general

term Aij requires the application of a uniform pressure at the

jth step for a hole i steps deep (Fig. 7), and the calculation of

the corresponding deformation on the gauge area [29].

Furthermore, the number and distribution of selected

steps has to be selected with the maximum care: it has

been demonstrated [30] that this number should not be

higher than 10. A step distribution that makes it possible

to minimize experimental errors is achieved if the following

conditions are met

Ann ¼ cos t, 1 4 n 4 nc

Bnn ¼ cos t, 1 4 n 4 nc (13)

These conditions are substantially subjected by adopting an

increasing distribution of calculation steps. In general, the

total number of calculation steps, nc, is different from the

number of experimental steps, ne. The experimental number

is commonly greater, which allows for a better interpolation

of strain data. In this case a continuous strain–depth (1–h)curve is available for the optimization of the calculation

step process. For better precision of the developed model,

the eccentricity of the actual drilled hole was also taken into

account using appropriate geometric relations [31] describing

the real strain distribution created around the hole.

Experimental measurements were carried out using a

high-velocity drilling machine (Restan, 350 000 r/min),

which allows for minimization of the stresses induced by

drilling and accurate measurement of the hole size and

eccentricity. To avoid induced thermal strains during

the drilling process, possibly generated owing to the low

thermal conductivity of the ceramic coatings, a stop time

of 100 s between each drilling step was adopted.

4 RESULTS AND DISCUSSION

4.1 Results of thermal analyses

The substrate temperature was monitored during spraying

by three thermocouples placed on the back of the substrate,

Fig. 7 FEM model for the calculation of calibration

coefficient A32

Fig. 6 Rosette strain gauge for hole drilling measurements

Fig. 8 Measured substrate temperature during spraying

Fig. 9 Simulation of the variation in substrate temperature

during spraying





as reported in Fig. 8. A very good agreement is observed

between the measured and the numerically predicted

temperature values (Fig. 9) with maximum differences of

2 K over the whole temperature profile. The developed

thermal model can thus be considered, for the investigated

case, as an appropriate simulation model. The substrate

temperature was maintained at a maximum value of 415 K

by appropriate gas front cooling, with the aim of reducing

the amount of residual stress after final cooling to room

temperature.

4.2 Results of structural analysis

The first qualitative results in terms of stress evaluation can

be obtained by the analysis of the deformed shape of the

coating element, as illustrated in Fig. 10: the curvature of

the coated specimen indicates the presence of compressive

stresses on the top of the coating and tensile stresses on

the back of the substrate. This is in agreement with equation

(4) since the thermal expansion coefficient of alumina is

lower than that of the carbon steel substrate.

The results of the simulation of through-thickness vari-

ation of in-plane stress are reported in Fig. 11. The graph

refers to a middle section of the system and indicates a com-

pressive in-plane stress linearly varying through the coating

thickness. At the two interfaces (coating/bond coat and bond

coat/substrate) the model predicts discontinuities leading to

tensile stresses in the bond coat and in the substrate. Look-

ing at the free edge of the coated substrate (Fig. 12), the

model predicts a peak in the axial stress as well as in the

shear stress, in accordance with the interface. This result

is consistent with problems observed in practice that lead

to coating detachment and delamination in this area, thus

confirming the strong influence, predicted by the model,

of residual stresses on coating adhesion [32].

Coating thickness is a fundamental parameter for control-

ling the occurrence of the above-mentioned problem:

according to the model, the value of compressive in-plane

stresses decreases with coating thickness. With the same

model, the effect of variation in the deposition tempera-

ture on residual stress can also be evaluated. Simulated

in-plane maximum stresses corresponding to spraying temp-

eratures varying from 150 to 400 8C are reported in Fig. 13.

It appears that an eccessively high deposition temperature,

while responsible for improved coating cohesion and

reduced coating porosity, can lead to residual stresses able

to promote coating failure. Appropriate upper limits in the

Fig. 10 Simulation of the deformed shape of a coating element after spraying and cooling at room temperature

Fig. 11 Simulation of through-thickness variation of residual

in-plane stress





deposition temperature, selected as a function of the mech-

anical resistance of the coating material, should therefore be

respected in the optimization of processing parameters. For

the purpose of model validation, experimental measurement

of the residual stress in plasma-sprayed ceramic coatings

was carried out by the hole drilling method. Results

obtained with seven calculation steps and for a hole eccen-

tricity lower than 50 mm are reported in Fig. 14. Very low

stresses were measured in the direction perpendicular to

the torch scanning direction (smax, almost null as an aver-

age), whereas an average stress of about 244 MPa was cal-

culated from the measurements of the smin component of

the stress, parallel to the torch direction. The measured

smin stress is then the quantity that should be compared

with the result of in-plane stress calculation obtained from

FEA analysis.

The low absolute value of the residual stresses in the

ceramic coating is to be ascribed to the low deposition

temperature (415 K), selected with the aim of evaluating

whether the numerical model and the hole drilling measur-

ing technique could also give satisfactory and consistent

results in the case of low-stress coatings endowed with

improved mechanical properties and prolonged lifetime.

The comparison between predicted and measured values

of residual stress is graphically illustrated in Fig. 15 as a

function of coating thickness. Numerical and experimental

values are both compressive and of the same order of mag-

nitude. Nevertheless, Fig. 15a, representing the simulated

values of the DTC stress as directly compared with the

measured total stress, indicates a slight overestimation of

FEA results (in the range from255 to265 MPa as opposed

to the measured range from 230 to 250 MPa). Taking into

Fig. 14 Main residual stresses calculated from strain gauge

measurements for different depths: smin

perpendicular and smax parallel to the torch

scanning direction

Fig. 12 Simulation of axial stresses: (a) contour plot at the

edge of the plate; (b) through-thickness variation

Fig. 13 Simulation of maximum in-plane stress as a function

of deposition temperature





account the presence of tensile quenching stresses of about

10–15 MPa, results of numerical calculation appear to be in

very good agreement with measured values, as illustrated in

Fig. 15b.

5 CONCLUSIONS

A finite element model to simulate residual stresses for air

plasma sprayed ceramic coatings was developed and

validated. The model makes it possible to predict the

thermal history of the coating during the spraying process,

and the through-thickness variation of in-plane normal and

shear stresses. An agreement between the predicted and

experimental results in the coating was observed.

Numerical simulation at a low residual stress level in the

case of brittle ceramic coatings should be corrected to con-

sider effects of tensile quenching stress, in order to repro-

duce the experimental data with the best accuracy. When

residual stresses, expected or predicted, exceed 100 MPa,

quenching stresses can be neglected since their influence

on the final stress state is lower than about 10 per cent of

the total.

The developed model predicts stress concentrations at the

free edge of the specimen as well as stress increase with

deposition temperature.

The high-speed hole drilling method implemented with

FEA calibration of the required coefficients was adopted

for the validation of the model with an Al2O3 coating

sprayed with standard plasma parameters in air. This

method has thus been shown to offer an expert tool for

measuring residual stresses and stress profiles over the

drilling depth.

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Fig. 15 Comparison between predicted and measured

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Finite Element Analysis of Residual Stress in Plasma‐Sprayed Ceramic

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