Modeling Residual Stress Development in Thermal Spray Coatings: Current Status and Way Forward

J Therm Spray Tech (2017) 26:1115–1145
DOI 10.1007/s11666-017-0590-1
REVIEW
Modeling Residual Stress Development in Thermal Spray

Coatings: Current Status and Way Forward
Abba A. Abubakar1 • Abul Fazal M. Arif1 • Khaled S. Al-Athel1 • S. Sohail Akhtar1 •
Javad Mostaghimi2
Submitted: 13 November 2016 / in revised form: 18 May 2017 / Published online: 17 July 2017
ASM International 2017
Abstract An overview of analytical and numerical meth- Introduction

ods for prediction of residual stresses in thermal spray
coatings is presented. The various sources and mechanisms Thermal spray coatings are advanced coatings that are
underlying residual stress development in thermal spray deposited on a substrate surface by heating a feedstock
coatings are discussed, then the various difficulties asso- material (usually in powder or wire form) to molten state
ciated with experimental residual stress measurement in (as demonstrated in Fig. 1) (Ref 1). The thermal energy
thermal spray coatings are highlighted. The various ana- supplied during the thermal spray process softens the
lytical and numerical models used for prediction of residual feedstock material and enables effective spreading and/or
stresses in thermal spray coatings are thoroughly discussed. deformation of the molten droplets to form a highly dense
While analytical models for prediction of postdeposition coating layer with high adhesive strength (Ref 2). Thermal
thermal mismatch stresses are fully developed, analytical spray processes are used extensively in highly advanced
quenching and peening stress models still require extensive industries, such as aerospace, automotive, electronics,
development. Various schemes for prediction of residual telecommunications, energy, consumer products, nuclear,
stresses using the finite element method are identified. The medical, and oil and gas industries. In most of their
results of the various numerical and analytical models are applications, thermal spray coatings are used to improve
critically analyzed, and their accuracy and validity, when the thermal, wear, cohesive, or corrosion/chemical resis-
compared with experiments, are discussed. Issues regard- tance of substrate surfaces. However, new types of coating
ing the accuracy and applicability of the models for pre- applications such as antibacterial coatings, osmotic filter-
dicting residual stresses in thermal spray coatings are ing, biocompatible coatings, superconductive coatings,
highlighted, and several suggestions for future develop- functionally graded coatings, etc. are rapidly emerging
ment of the models are given. (Ref 3).
Thermal spray coatings can be classified according to
Keywords analytical model birth–death technique finite how the feedstock material is heated, i.e., using either
element method numerical model residual stress combustion or direct (Joule) heating, as shown in Fig. 2
thermal spray coatings (Ref 2, 4). Among combustion-driven processes, high-ve-
locity oxyfuel (HVOF) is considered to be the most suit-
able nowadays, due to its numerous advantages over the
common flame spray process (Ref 5, 6). It utilizes com-
bustion to heat the feedstock material into molten or
& Abul Fazal M. Arif semimolten droplets, which are then accelerated to very
afmarif@kfupm.edu.sa
high velocities before impacting with the substrate surface.
1
Mechanical Engineering Department, King Fahd University HVOF is considered more powerful than other thermal
of Petroleum and Minerals, Dhahran 31261, Saudi Arabia spray processes (e.g., plasma and flame spray) because it
2
Mechanical Engineering Department, University of Toronto, results in highly dense coating microstructure, faster
Toronto, Canada deposition rate, less energy consumption, lower oxide
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1116 J Therm Spray Tech (2017) 26:1115–1145
Fig. 1 Schematic of typical

thermal spray deposition
process on flat substrate
Fig. 2 Overview of thermal

spray processes (Ref 2, 4)
content, and higher adhesive strength (Ref 7-9). The main processes such as high-velocity oxyfuel (HVOF), which
drawbacks of the HVOF process are that it is very costly can achieve particle velocities as high as 1000 m/s (Ref
for high-temperature materials (e.g., ceramics) and the 11). A drawback common to all thermal spray processes is
combustion gasses emitted may react with the coating that interactions between the various process (input)
material. On the other hand, plasma spray is the most parameters make it difficult to control coating quality and
widely used thermal spray process based on direct electric integrity (Ref 3). Consequently, high residual stresses are
(Joule) heating. It uses a high-energy thermal plasma as a developed and the coating microstructure is filled with
heat source to heat the feedstock (powder) material into various types of defect (e.g., pores, cracks, splat interfaces,
molten droplets and accelerate them towards the substrate second-phase particles, etc.) that significantly affect the
surface. The plasma spray process is widely used due to its properties and lifetime of the coating.
lower cost, lower overall environmental impact, and flex- Residual stresses are stresses that remain in a structure
ibility in terms of coating materials (Ref 2, 10). It is the or material after manufacture or removal of external loads.
most suitable process for deposition of high-temperature Mostly, residual stresses occur due to changes in the shape,
ceramics, since the deposition temperature can be as high size, or properties of solid materials. They can be useful or
as 12,000 C. However, due to its lower deposition rate (or harmful depending on the type of material or product
lower particle impact velocity, ranging from 150 to 400 m/ involved. Tensile residual stresses are known to be more
s) (Ref 11), the plasma spray process generally results in harmful than compressive residual stresses due to their
less dense (and more porous) coatings compared with other tendency to initiate and propagate cracks (Ref 12).
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J Therm Spray Tech (2017) 26:1115–1145 1117
Undoubtedly, residual stresses cannot be avoided in ther- only postdeposition residual stresses were included in that
mal spray coatings (Ref 13-16). Residual stresses affect the stress calculation. A detailed review of the effect of
adhesive strength, cohesive strength, thermal shock resis- residual stresses on the fracture toughness or bond
tance, thermal fatigue life, corrosion resistance, wear strength of coatings was given by Clyne and Gill (Ref
properties, and service life of coatings (Ref 13, 17-20). 13).
Residual stresses are known to evolve during and after Due to the limitations on experimental measurements of
coating deposition, thus their prediction is very compli- residual stresses in thermal spray coatings, several residual
cated due to their strong dependence on the process history. stress models have been developed. Such analytical models
Deposition stresses (occurring at micro- and mesoscale) have mostly been derived using the theory of elasticity,
can occur due to the sudden cooling and solidification of adopting several assumptions to simplify the steps required
splats, the peening action of droplets on a predeposited to derive closed-form solutions. With the recent advance-
layer, or high thermal gradients developed during deposi- ment in computation, numerical modeling has become
tion (Ref 21). Postdeposition stresses (occurring at mac- more popular for analysis of engineering systems. As the
roscale) develop after the coating cools to room finite element method (FEM) is more commonly applied
temperature, due to differential contraction or mismatch in than other numerical methods, it has been used by many
coefficient of thermal expansion (CTE) between the coat- researchers to predict residual stresses in thermal spray
ing and substrate. Similarly, substrate geometry and prior coatings (Ref 26). Most such research has used the element
surface treatment (e.g., grit-blasting or roller burnishing) birth–death FEM approach to include both deposition and
significantly influence the residual stress state of sprayed postdeposition stresses during the computation. Image-
coatings (Ref 22-24). During service, residual stresses may based finite element schemes have also been used recently
be severely affected by large loads, chemical reactions, to predict localized stresses that develop due to the
phase transformations, through-thickness thermal gradi- imperfections (e.g., cracks, pores, inclusions, etc.) that
ents, creep, etc. (Ref 25-28). Thus, the overall residual pervade the microstructure of coatings (Ref 31, 32). An
stress state is a superposition of the stresses that develop as attempt to predict residual stresses by coupling computa-
a result of various phenomena occurring on various length tional fluid dynamics (CFD) with the finite element method
and time scales. In fact, the residual stress state often (Ref 33) has also been made. However, all of the models
reverses (i.e., from tensile to compressive, or vice versa) used in literature to predict residual stresses in thermal
after the coating has cooled to room temperature. This spray coatings suffer from various issues. In some cases,
leads to complexity and uncertainty in determining the the results given by the models are not accurate, qualita-
nature and magnitude of the residual stress state of thermal tively or quantitatively. The last review on this topic was
spray coatings (Ref 10). Therefore, full tracking of the provided by Clyne and Gill (Ref 13) in 1996, with dis-
residual stress state can help with effective optimization of cussions about the development of residual stresses and
thermal spray processes for enhanced performance and their relationship with delamination failure of thermal
durability of coatings. spray coatings. To the best of the authors’ knowledge, no
The correlation between the residual stresses in and further reviews on this topic have been published, despite
lifetime of coatings has been investigated in many the fact that new research trends have been established.
research works; For instance, Khan and Liao (Ref 15) Therefore, there is a need to discuss the various issues,
investigated the effect of substrate surface roughness and challenges, and possibilities with regards to the available
coating deposition temperature on the adhesive strength models for prediction of residual stresses in coatings.
of coatings. They found that the adhesive strength (or Directions for future development of such models also
fracture toughness) of coatings is significantly affected by need to be highlighted.
the nature and magnitude of the induced residual stresses. We present herein an overview of the various models
Similarly, McGrann et al. (Ref 29) investigated the effect that have been used to predict residual stresses in thermal
of residual stresses on the fatigue life of thermal spray spray coatings. First, a brief discussion of the various
coatings. They also found that the fatigue life of coatings sources or components of residual stresses is presented.
is directly dependent on the sign and magnitude of the Then, the main difficulties associated with use of experi-
residual stresses. Araujo et al. (Ref 30) also found that mental residual stress measurements (for residual stress
the reduction of the bond strength of coatings with prediction in coatings) are identified. This is followed by a
increasing thickness is directly related to the influence of discussion about the various analytical and numerical
residual stresses. Ranjbar-Far et al. (Ref 29) also made models that are used for prediction of residual stresses in
the same observation when they numerically determined thermal spray coatings. The results given by those models
the effect of residual stresses on the lifetime of thermal are then critically analyzed, and their accuracy and valid-
spray coatings using the finite element method, although ity, compared with experiments, are discussed. Various
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Fig. 3 Quenching stress formation (at microscale) during single splat solidification
results obtained from the presented analytical and numer- (a) Quenching stress The term ‘‘quenching stress’’ is
ical methods are then analyzed and compared. Several commonly used in literature to describe stresses that
issues that must be considered in future research work are develop as a result of the sudden solidification of the
highlighted. Finally, possible future directions for research molten particles (or deposited splats) upon impact
in this field are discussed. with the substrate (or predeposited coating) surface.
During coating growth and deposition, quenching
stress evolves due to processes occurring on varying
Sources of Residual Stresses time and length scales. At the microscale, the sudden
solidification and shrinkage (occurring within
Thermal Mismatch Stresses 10-100 ls) of individual splats result in formation
of tensile stresses due to thermal mismatch and
During a thermal spray process, thermal mismatch (or constrained shrinkage of the splats due to the
misfit) strain occurs due to the differences in temperature interfacial bond (Ref 2, 3, 21) (as demonstrated in
and coefficient of thermal expansion (CTE) between the Fig. 3). This interfacial bonding is established by
coating and substrate material. With high CTE mismatch, various mechanisms including mechanical, diffu-
the misfit strain is appreciably high and results in thermal sional or chemical processes. With the arrival of new
mismatch stresses due to uneven contraction of the coating sets of droplets, the splat-level quenching stress field
and substrate material near the interface region. Due to the is significantly influenced by the high thermal flux
complicated nature of this process, the evolution of the and impact energy of the newly deposited droplets.
thermal mismatch stress is highly dependent on the process Consequently, at the mesoscale, steep through-
(parameters) and significantly affected by various pro- thickness thermal gradients develop (within hun-
cesses occurring on different time and length scales. dreds of splats), leading to slight bending or
Generally, a thermal spray coating process can be divided distortion of the substrate layer (depending on the
into two stages, i.e., deposition (involving layer-by-layer degree of thermal mismatch). The thermal gradient
coating growth to the required thickness) and postdeposi- is severe and more harmful in the case of ceramic
tion (involving the final cooldown of the coating to room coatings due to their low thermal conductivity (Ref
temperature). The thermal mismatch stresses are therefore 2, 13). The slight distortion/bending of the substrate
also classified according to these stages, i.e., thermal layer is necessary for the composite system to attain
deposition (or quenching) stress and postdeposition mis- equilibrium (by force and momentum balance) while
match stresses, as discussed below: accommodating the high quenching stresses
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Fig. 4 Bending of substrate

material (during deposition) due
to quenching stresses developed
after deposition of coating
sublayer
developed within the coating sublayers (as demon-

strated in Fig. 4).
(b) Postdeposition mismatch stress Likewise, the evo-
lution of thermal stresses extends to the postdepo-
sition stage, when the deposited coating is finally
cooled to room temperature. Upon its final cool-
down, uneven contraction occurs between the coat-
ing and substrate material near the interface region,
due to the difference/mismatch in their properties.
Being a macrolevel stress, the postdeposition mis-
match stress has the highest magnitude and signif-
icantly influences the final residual stress state, as
demonstrated in Fig. 5. Also, the mass or thickness
of the coating is strongly correlated with its postde-
position mismatch stress and bond strength. For Fig. 5 Bending of substrate/coating after (a) coating deposition,
(b) postdeposition cooling
coatings having lower CTE than the substrate (e.g.,
ceramic coatings), the postdeposition mismatch
stress is usually compressive, thus helping to reduce Peening Stress
the severity of the tensile quenching stresses at the
coating–substrate interface. On the other hand, In thermal spray literature, stresses that develop due to
coatings having a CTE higher than that of the particle impact on the substrate (or previous coating layer)
substrate (e.g., most metallic coatings) result in surface are called peening or impact stresses (shown in
tensile postdeposition mismatch stress state. How- Fig. 6) (Ref 2). Due to the type of reaction forces produced
ever, these tensile postdeposition mismatch and by impact, peening residual stresses are usually compres-
quenching stresses (in metallic coatings) are usually sive. Thus, peening stresses reduce the severity of
balanced by the high peening (compressive) stress quenching stress and result in a nonlinear compressive
associated with most metallic coatings. Therefore, residual stress state in coatings, as demonstrated in previ-
minimization of postdeposition mismatch stresses ous research works (Ref 35-37). Peening contributes
(through careful selection of the coating/substrate greatly to the overall residual stress profile, especially in
material combination, their thicknesses, and operat- metallic coatings deposited by high-speed thermal spray
ing temperature) is required for enhanced durability processes such as HVOF and D-Gun. Since metallic coat-
and performance of coatings (Ref 34). ings are susceptible to large plastic deformation, very high
123
Fig. 6 Peening stress developed during coating deposition
peening (compressive) stresses occur due to conversion of structuring/texturing processes based on laser treatment,
the droplet kinetic energy into sound, plastic work, and machining, grinding/brushing, degreasing, and chemical
heat energy. Meanwhile, in ceramic coatings, the resulting surface treatment also affect the stress state developed
peening (compressive) stresses often reduce the severity of within the coating and substrate layers.
the quenching (tensile) stress, initiate microcracks due to
stress wave propagation, and partially close the numerous Other Sources: Multiphase Systems, Discontinuities,
voids/cracks that pervade the microstructure of such Coating Posttreatment, and In-Service Loads
coatings.
In coatings, residual stresses can also be induced by other
Stresses Induced by Substrate Surface Pretreatment phenomena which are usually considered as minor sources
and occur only under certain loading or environmental
Before coating deposition, the usual practice is to conditions. The emergence of new crystals or phase
roughen the substrate surface and induce initial com- transformations in the coating can lead to the development
pressive (or peening) stresses in order to improve the of residual stresses due to volumetric expansion or
adhesive strength of the coating. This is mostly achieved inelastic deformation (Ref 2, 38, 39). Compositional
by grit-blasting, a process whereby hard particles are variation of second-phase particles through the coating
blasted at the surface of the substrate material to thickness significantly affects the residual stress formation
improve its surface properties (mainly surface rough- due to the gradual change in properties, e.g., in func-
ness, grain size, and hardness). The sharp changes due tionally graded thermal barrier coatings (TBCs) (Ref 40).
to the resulting asperities on the substrate surface help The presence of pores, cracks, and hard inclusions leads
to improve the bond strength at the coating–substrate to the development of higher residual stresses due to
interface. Furthermore, grit-blasting induces some initial localized phenomena and high stress gradients (Ref 41).
compressive (peening) stresses on the surface of the Shot peening and roller burnishing also significantly affect
substrate, and this greatly influences the eventual mag- the residual stresses developed within the coating, as
nitude and direction of the residual stresses developed in demonstrated by Luzin et al. (Ref 42) and Klusemann
the substrate at the end of the coating deposition pro- et al. (Ref 43). The residual stress state of coatings is also
cess, especially near the interface. Moreover, preheating affected by in-service loadings which induce inelastic
the substrate to higher temperatures is commonly car- deformations (e.g., creep, sintering, precipitation, chemi-
ried out to reduce the severity of quenching stresses, cal reactions, phase transformations, etc.) and reduce the
especially near the coating–substrate interface. Surface integrity of coatings.
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Difficulties in Measuring Residual Stresses: Why through-thickness stress profile did not provide good
Modeling? results, due to stress redistribution induced by the layer
removal process.
Over the years, various experimental setups for accurate The second nondestructive technique is the in situ cur-
estimation of residual stresses in coatings have been vature method, first proposed by Stoney (Ref 53) and later
developed. X-ray diffraction, in situ curvature measure- improved by Brenner and Senderoff (Ref 54). This method
ments, neutron diffraction, and incremental hole-drilling involves continuous monitoring of the substrate curvature
methods are frequently used. The first three test methods during both coating deposition and postdeposition cooling
are nondestructive, while the last is semidestructive. Using (Ref 13, 55). During coating deposition, the quenching
each technique, residual stresses developed in thermal stresses that develop within the deposited coating are
spray coatings have been measured and evaluated with a accommodated by the substrate material so that the com-
certain level of accuracy. However, each of these experi- posite system can reach stability/equilibrium. Conse-
mental methods suffers from its own limitations, as dis- quently, a slight change of substrate curvature occurs due
cussed in the following paragraphs (starting with the to the balancing of the overall forces and moment acting on
nondestructive tests). the system. Moreover, the substrate curvature undergoes
Starting with diffraction techniques, x-ray and neutron further changes after postdeposition cooling due to uneven
diffraction are commonly used. Due to the limited pene- contraction and the thermal mismatch between coating and
tration depth of x-rays, the x-ray diffraction method is substrate. In the in situ curvature method, the change in the
restricted to crystalline coatings or prediction of only curvature or displacement of the substrate during the
residual stresses developed near the coating surface. Thus, spraying process is monitored with the aid of an in situ
using x-ray diffraction analysis, it is almost impossible to coating properties (ICP) sensor, as described in a recent
measure the residual stresses developed near the substrate– study by Mutter et al. (Ref 55), which uses a laser to
coating interface, which are more critical than near-surface accurately estimate the central displacement of the sub-
residual stresses. Furthermore, x-ray diffraction analysis strate layer. The substrate curvature can be predicted to a
often gives inaccurate stress values due to uncertainties in high level of accuracy such that even vibrations resulting
determining the required elastic parameters (Ref 44, 45). from the sudden impacts of sprayed droplets or the plasma
Despite these limitations, x-ray diffraction has been used plume can be detected. The temperatures of the top and
by many researchers (Ref 46-50) not only to predict surface bottom of the substrate (measured continuously using
residual stresses in coatings but also to correlate surface thermocouples) help in predicting the associated thermal
stresses and spray process parameters. Neutron diffraction gradient developed during the process. Using these curva-
is another method for determining residual stresses in ture and temperature readings, the evolving residual
materials. In this technique, high-energy neutrons are stresses and coating properties (e.g., effective elastic
allowed to penetrate the sample while the scattering caused modulus and stress–strain response) are calculated using
by their interaction with atoms and nuclei (present in the the models of Stoney (Ref 53) or Brenner and Senderoff
sample) is recorded and analyzed. The high penetration (Ref 54) as functions of the process history. To date, in situ
power of neutrons and the nondestructive nature of such curvature measurement represents the only experimental
measurements are the main strengths of this method. technique that can track the evolution of residual stresses
However, this method is difficult to apply to coatings due (both deposition and postdeposition stresses separately)
its low resolution (*1-10 mm) compared with the thick- during the thermal spray process. The method is nonde-
ness range of thermal spray coatings (around 500 lm). For structive, real time, and widely used by many researchers,
this reason, the neutron diffraction technique is only used including Totemeier et al. (Ref 50), Sampath et al. (Ref
for experimental determination of residual stresses in thick 56), and Zhang et al. (Ref 57), to effectively predict the
coatings (Ref 36). Furthermore, neutron diffraction is very residual stresses in and effective properties of coatings. It
expensive. Thus, the cost of obtaining sufficient and can even be used to determine the quality of coatings (i.e.,
accurate data for residual stress analysis is very high (Ref whether cracking/delamination occurs or not) by compar-
51). Using an insufficient amount of neutrons usually ing the effective properties of the coating with those of
results in less accurate results due to low scattering inten- standard/acceptable specimens. Despite its numerous
sities. Therefore, it is not feasible to determine the varia- advantages, the in situ curvature method also suffers from
tion of the residual stress through the depth of a thermal important limitations. Firstly, this method is only applica-
spray coating using any common diffraction method. Even ble to model samples of specific thickness and shape. Thus,
an attempt by Kingswell et al. (Ref 52) to combine x-ray it is difficult to estimate residual stresses developed in
diffraction with a layer removal technique to predict the actual (coated) components that are commonly used in
123
industry. The stress state determined using such model (ANN) algorithm. Thus, the hole-drilling method has been
samples will not be the same as that of actual components widely used to estimate the residual stress state of thermal
due to their different thermal history. Similarly, it is usually spray coatings. This method can be used to measure
difficult to transform curvature data into the stress state for residual stresses with depth resolution up to 10 lm, as
thick coatings produced using high-energy processes (e.g., shown by Escribano and Gadow (Ref 61), Buchmann et al.
HVOF and D-Gun) due to the assumptions adopted by (Ref 62, 63), and Grant et al. (Ref 66). However, the
Stoney (Ref 53) and Brenner and Senderoff (Ref 54). accuracy of this method is often affected by several factors
Furthermore, the estimated residual stress is inaccurate for (Ref 59, 64), such as human error arising from the tedious
some coating systems due to the lack of consideration of nature of such measurements, high sensitivity and suscep-
nonlinear material behavior (e.g., plastic deformation, tibility to experimental errors, semidestructive nature, the
cracking, etc.) occurring during postdeposition cooling. complicated stress calculation procedure (especially for
Another disadvantage is that it is difficult and costly to low-yield and highly brittle coatings), and the difficult
optimize the coating process due to the need for repetitive interpretation of the results.
coating of several specimens to obtain sufficient data for A problem common to all experimental methods is that
optimization. Thus, development of an effective and cali- they give average stress values which cannot be used to
brated model is necessary for such optimization tasks. predict the microstresses (or localized stresses) that often
Among semidestructive test methods, the incremental occur near areas of stress concentration (such as cracks,
hole-drilling method is among the most commonly used for pores, splat boundaries, inclusions, etc.). Also, the long
measurement of residual stresses (Ref 58). This is partly time required to perform each experiment makes it difficult
due to its simplicity, portability, availability, flexibility to optimize the spray process using experimentally mea-
with regards to sample materials, and ability to track sured stress values. Furthermore, each experimental
residual stress variation with depth. The incremental hole- method suffers from one or more restrictions regarding the
drilling method has been used to determine the variation of size and shape of the specimen to be used, coating and
residual stresses through the thickness of thermal spray substrate material types, sensor tolerance, experimental
coatings. Due to the heterogeneous nature of such coating conditions, and cost. Thus, no available experimental
systems, there is a significant error in the final stress results methods are suitable for process optimization for enhanced
if standard calibration coefficients are used for the residual durability and quality of coatings. For this reason, various
stress calculation. Thus, Valente et al. (Ref 59) proposed a models have been developed for prediction of residual
methodology for accurate determination of residual stres- stresses in thermal spray coatings, as discussed in subse-
ses in thermal spray coatings by using the finite element quent sections.
method to determine the required calibration coefficients
numerically. Using this approach, Montay et al. (Ref 60)
predicted the residual stresses developed in zirconia, alu- Analytical Models for Prediction of Residual
mina, and tungsten carbide thermally sprayed coatings. Stresses
Escribano and Gadow (Ref 61) also proposed use of high-
speed drilling and milling processes to reduce the errors Theoretical Quenching Stress Models
resulting from temperature rise, plastic deformation, and
crack propagation (due to drill cutting forces). As done by Theoretically, quenching stresses are more difficult to
Valente et al., the correct calibration coefficients were calculate than the other residual stress components. This is
determined using finite element analysis. Buchmann et al. due to the phase-change phenomena (or dynamics) asso-
(Ref 62, 63) also used this hole-drilling and milling method ciated with particle impact, spread, and solidification. As
coupled with calibration using finite element analysis to explained above, the quenching stress is mainly tensile
investigate the residual stress state of thermally spray because most materials shrink upon solidification. The
coatings. Santana et al. (Ref 64) also determined the quenching stress is usually calculated from the following
residual stress state of HVOF-sprayed nickel coatings using relation (Ref 67-69):
the incremental hole-drilling method coupled with finite ZTsp
element analysis. The latest development was by Held and rqc ¼ Esp ac ðTÞdT: ðEq 1Þ
Gibmeier (Ref 65), who presented a new evaluation tech-
Ts
nique for accurate determination of residual stresses
developed in thick coatings using the incremental hole- Thus, the maximum value of the quenching stress can be
drilling method coupled with an artificial neural network analytically derived as (Ref 67-69)
123
Fig. 7 Comparison of theoretical and measured quenching stress developed in (a) alumina (ceramic) and (b) nickel (metallic) coatings as
functions of substrate temperature

rqc ¼ Esp asp Tsp Ts ; ðEq 2Þ various mechanisms, as demonstrated in previous work (Ref
67). In metallic coatings, the stresses relax by splat edge
where Esp is the elastic modulus of the splat, asp is the relaxation, plastic yielding, creep, and interfacial sliding. On
coefficient of thermal expansion of the splat, Tsp is the the other hand, residual stresses in ceramic coatings relax by
solidification temperature of the splat, and Ts is the sub- microcracking (due to thermal shock) and interfacial sliding.
strate temperature. A detailed discussion of such relaxation mechanisms was
The theoretical quenching stress Eq 2 gives very high values given in previous work (Ref 67). Valente et al. (Ref 72)
compared with the actual quenching stress; For instance, the suggested that the theoretical quenching stress (in Eq 1 and
theoretical value of quenching stress for alumina (ceramic) 2) should be reduced by multiplying Tsp with a reduction
coating (for 100 C cooling) is 100 MPa, far greater than the factor, n ¼ 0 ! 1. They suggested that the value of n should
actual value of 10 MPa (Ref 67). Similarly, the typical be 0.6 to accommodate stress relaxation by yielding and
observed quenching stress for nickel (metallic) coatings [de- creep. However, it was not clear whether the same reduction
posited by atmospheric plasma spraying (APS)] is less than factor applies to other stress relaxation mechanisms such as
100 MPa, far lower than the theoretical value of 1 GPa (Ref microcracking, interfacial sliding, and edge relaxation. We
67). This discrepancy has been attributed to the assumption that carried out investigations on the validity of the above sug-
the elastic modulus of the deposited splats is the same as that of gestions for correction of the theoretical quenching stress
the corresponding bulk material (Ref 67). Various researchers values of thermally sprayed alumina (ceramic) and nickel
have used three-point bending, nanoindentation, and acoustic (metallic) coatings. For the purpose of comparison, proper-
analyses to measure the effective modulus of thermal spray ties of alumina and nickel, as measured experimentally by
coatings. Typically, values lower than bulk are found, due to Kuroda et al., were used with melting temperatures of 2277
porosity, microcracks, and the lamellar microstructure (Ref and 1435 C, respectively. As shown in Fig. 7(a) and (b), the
67, 70, 71). Kuroda et al. (Ref 67) observed that the measured agreement of the theoretical and corrected maximum
elastic modulus of coating deposits was one-third and one-sixth quenching stress values (as given by Eq 2) with experiments
of the bulk value for metallic and ceramic coatings, respec- is not good. Moreover, there was no good qualitative com-
tively. Thus, they suggested that a good approximation is to use parison in the case of alumina. This is because quenching
effective values of elastic modulus for coatings (instead of those stress strongly depends not only on the substrate temperature
for bulk materials) as Esp in Eq 1 and 2. but also other factors such as substrate surface preparation,
Another reason for the differences between theoretical chemical compatibility, the thermal spray process, etc.
quenching stress values (given by Eq 1 and 2) and those For most ceramic coatings, the interfacial adhesive
observed experimentally is stress relaxation occurring due to strength is more dependent on diffusion or chemical
123
compatibility than mechanical bonding (as discussed in plane-strain conditions, the substrate elastic modulus can
ES
Sect. 13.4.2 of Ref 2). For this reason, the actual quenching be replaced by ð1v with vS being the Poisson’s ratio of the
SÞ
stress is commonly found to increase linearly with sub- substrate material.
strate temperature due to higher rates of diffusion (or It has been shown by many researchers (Ref 50, 74, 75)
chemical bond energy) at the coating–substrate interface, that Stoney’s model gives a good approximation for the
as shown in Fig. 7(a). On the contrary, metallic coatings quenching stresses developed in thin coatings such as those
(e.g., nickel) display a quenching stress which decreases deposited by thermal spray processes (with deviation of
with substrate temperature due to plastic deformation and/ less than 10%). What is interesting is that Stoney’s model
or creep. However, other metallic coatings (e.g., molyb- does not require the effective elastic modulus of the coating
denum) show a quenching stress that first increases with but still gives a good approximation to the deposition
substrate temperature then starts to decrease due to partial stresses. However, for thick coatings with elastic modulus
melting or softening of the coating material (Ref 67). Thus, comparable to that of the substrate, Soderberg (Ref 76)
for such coating materials, both diffusion and mechanical observed that Stoney’s model results in significant errors
bonding play a role in determining the adhesive strength. due to the lack of consideration of the coating elastic
Quenching stresses also depend on the type of thermal modulus. Thus, Brenner and Senderoff (Ref 54) corrected
spray process used; For example, quenching stresses this error by adding a term to Eq 3 to include the effect of
developed during the HVOF process were found to be the elastic modulus and thickness of the deposited coating
noticeably different from those resulting from the APS on the developed quenching stress (as shown in Eq 4).
process (Ref 51, 73), due to the lower deposition temper-
ature of HVOF coatings. Therefore, it can be seen that the ES tS ðtS þ v5=4 tc Þ
rqc ¼ ; ðEq 4Þ
quenching stress evolution is intricately dependent on 6 dr tc
many factors that can hardly be controlled during the
where v ¼ EESc is the coating–substrate elastic modulus ratio.
process, thus further theoretical understanding of the
It is important to note that both Eq 3 and 4 predict a
quenching stress is required.
quenching stress that is uniform and does not vary in the
thickness direction of the coating.
Semiempirical Quenching Stress Models
Due to the various challenges and limitations of using the Postdeposition Mismatch Stress Models
existing quenching stress model, several semiempirical
models have been used to predict the quenching stresses The first analytical model for prediction of the elastic thermal
developed during thermal spray coating. These are ana- mismatch stresses developed in a two-layer composite
lytical models that require experimental data for the change structure was developed by Timoshenko (Ref 77) based on
in the substrate curvature (as recorded using an ICP sensor classical beam theory. Since then, many researchers have
or other means) during the thermal spray process; hence, utilized his model to obtain the thermal stress distribution in
they are termed ‘‘semiempirical.’’ Stoney (Ref 53) first layered (composite) structures; For instance, Hans and Evans
derived a relationship between the in-plane stresses (Ref 78) predicted the thermal mismatch stresses developed
developed in a deposited thin film/coating and the radius of in metal–ceramic composite materials using the model of
curvature of the substrate. He arrived at a final relation for Timoshenko. Liu and Murarka (Ref 79) also used it to predict
the uniform in-plane stress developed in the deposit/coat- the thermal mismatch stresses developed in coated semi-
ing using laminate theory (based on only the linear elastic conductor devices. Many applications of Timoshenko’s
response) by balancing the forces and moments acting on model can be found in literature (Ref 80-84). However, it is
the entire coating system (as shown in Eq 3). difficult to predict mismatch residual stresses in multilayer
coatings using Timoshenko’s model. This is due to the many
ES tS2
rqc ¼ for plane stress; ðEq 3Þ unknowns, complex interfacial (compatibility) conditions,
6 dr tc
and thin nature of the coating as compared with the substrate
where rqc is the deposition (or quenching) stress developed material. Thus, it is necessary to establish an elegant way of
in the coating, ES is the elastic modulus of the substrate, tS deriving an analytical model for the thermal mismatch
is the thickness of the substrate, dr is the change in the stresses in coatings. Laminate theory, first developed by
radius of curvature due to deposition, and tc is the total Stoney (Ref 53), was recently used as an alternative, as it is
thickness of the deposit. The change in curvature of the suitable for very thin coatings. The only problem is that
substrate (usually obtained using an ICP sensor) is deter- laminate theory may result in complex formulations which
mined by taking the inverse of the radius, i.e., dk ¼ dr1 . For may be difficult to handle analytically, necessitating use of
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approximate solutions to predict the postdeposition (ther- different expressions for the postdeposition mismatch
mal) mismatch stresses developed in thermal spray coatings. stresses. Hsueh derived the following expressions for the
Hsueh (Ref 85) and Zhang et al. (Ref 86) developed an in-plane stresses developed in the coating (ri ) and substrate
approximate solution to Stoney’s model (for multilayered (rS ) (based on a first-order approximation):
composite systems) by additively decomposing the elastic !
strain tensor (e) into uniform and bending strain parts, as 2 2 X n X
n
shown in Eq 5. Using Eq 5, the continuity (or compati- rS ¼ 2 3z þ 2tS Ej t j E i t i ð ai aS Þ

tS ES j¼1 i¼1
bility) conditions are implicitly satisfied at the interface
DT ; for tS z 0;
between the various composite layers.
z tb Xn ðEq 6Þ
e¼cþ ; for tS z tn and tn ¼ ti ; ðEq 5Þ !!
r i¼1
Xn
Ej tj aj aS
ri ¼ Ei ð aS ai Þ þ 4 DT;
E S tS
where z is the coordinate through the thickness of the j¼1
coating (shown in Fig. 8), c is the uniform strain, z ¼ tb is for i ¼ 1; 2; 3; . . .; n;
the position of the bending axis, r is the radius of curvature, ðEq 7Þ
tS is the substrate thickness, tn is the free surface of the
topmost coating layer, and ti is the thickness of the ith where E is the effective elastic modulus (i.e., the elastic
coating layer. modulus divided by ð1 vÞ for the plane-strain case), v is
Thus, Hsueh and Zhang et al. used Eq 5 to derive exact Poisson’s ratio, a is the CTE, and DT is the change in
solutions for the uniform strain, bending axis position, and temperature due to cooling. Note that i denotes each
curvature of a multilayer coating system. Due to the use of coating layer, while ‘‘S’’ denotes the substrate layer.
slightly different expressions for the approximate position Meanwhile, Zhang et al. obtained the following
of the bending axis, Hsueh and Zhang et al. obtained expressions (based on a first-order approximation):
2 0 !13
P
i1
6X B Ei ti 2 tj þ ti C7
6 n Ei ti ðai aS Þ DT Xn
Ei ti ðai aS Þ DT B tS Xn
j¼0 C7
6
rS ¼ ES 6 þ6 Bzþ C7; for tS z 0; ðEq 8Þ
E t E t 2 B 2 E S tS
C7
4 i¼1 S S i¼1 S S @ i¼1 A5
2 0 i1 13
P
E t 2 t þ t
6 Xn
Ej t j a j aS DT Xn
Ei ti ðai aS Þ DT B Xn i i i i
C7
ri ¼ Ei 6ð a a Þ DT þ6 Bz þ t S 1 C7
4 S i
ES tS E S t 2 @ 2 E S tS
A5 ðEq 9Þ
j¼1 i¼1 S i¼1
for i ¼ 1; 2; 3; . . .; n:
Both Hsueh and Zhang et al. showed that, using a zeroth-

order approximation, their equations further reduced to a
previous solution obtained by Townsend et al. (Ref 87).
Adopting such a zeroth-order approximation results in the
following final expression, which can only be applied to
coatings of extremely small thickness:
2ð3z þ 2tS Þ Xn
rS ¼ 2
Ei ti ðai aS Þ DT; for tS z 0;
tS i¼1
ðEq 10Þ
Fig. 8 Variation of z values along the thickness direction (for ri ¼ Ei ðaS ai Þ DT; for i ¼ 1; 2; 3; . . .; n: ðEq 11Þ
multilayer coatings)
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However, the expression in Eq 11 shows that the stress due to a reduction in the CTE mismatch near the interface
induced in each coating layer is independent of the thick- region. However, they also observed that the stresses
ness, curvature, and properties of the other coating layers, induced on the top coat surface were constant and inde-
which obviously leads to inaccurate results, as demon- pendent of the compositional gradient. The main limitation
strated by Eq 9. Hsueh (Ref 85) used Eq 6 and 7 to predict of Hsueh’s and Zhang’s model is that they considered that
the postdeposition mismatch stresses developed in five- the materials of the individual coating and substrate layers
layered AlGa laser diodes. He observed that the zeroth- and are isotropic, temperature independent, and subjected only
first-order approximations led to conflicting predictions, to pure elastic strain. To date, their models have not been
due to the use of temperature-independent elastic moduli. extended to accommodate anisotropy or nonlinear behavior
Also, Zhang et al. (Ref 86) used Eq 8 and 9 to predict the of coatings due to the resulting mathematical complexities.
postdeposition mismatch stresses developed in a thermal To investigate the effect of substrate geometry and sur-
barrier coating (TBC). Unlike Hsueh, Zhang et al. found face asperities on the postdeposition mismatch stresses,
that the induced stresses are not uniform but decrease lin- Gong and Clarke (Ref 90) first developed a two-concentric
early along the thickness direction of the coating. They (two-layer) model to predict the effect of substrate surface
showed that the bending part of the strain tensor may be roughness on the induced thermal stresses. Then, Hsueh and
neglected in the closed-form expression for coatings of Fuller (Ref 91) and Zhang et al. (Ref 92) extended this fur-
very small thickness and low elastic modulus. ther to three-concentric (three-layer) and four-concentric
In other work, Hsueh and Lee (Ref 88) and Zhang et al. (four-layer) models and investigated the effect of substrate
(Ref 89) applied the model to predict the postdeposition surface roughness on the induced mismatch stresses. Then,
mismatch stresses in compositionally (or functionally) graded Song et al. (Ref 93) finally developed a four-concentric
coatings. In that case, the residual stresses are not only due to (four-layer) model to predict the effect of both substrate
the CTE mismatch between the respective layers but also due surface roughness and macrocurvature. All these concentric
to the variation of the composition through the coating models were derived using the theory of linear elasticity and
thickness. Thus, modified expressions for the effective elastic isotropic material behavior. After long derivations, Song
modulus and CTE are used as the properties of the graded et al. arrived at final expressions for the radial stresses
layers in Eq 6-9. For a graded coating layer composed of two developed near an undulation of the substrate. They found
distinct materials (a and b), the effective elastic modulus (Eg ) that increasing the undulation amplitude and decreasing the
and CTE (ag ) of the graded layer are given by undulation wavelength result in increased mismatch stresses
n near the interface. They also found that the macrocurvature
z of the substrate significantly affects the stress distributions,
Eg ¼ Ea þ ð Eb Ea Þ ; ðEq 12Þ
tg especially near regions of high stress concentration, e.g.,
n edges, interfaces, etc. Therefore, it can be concluded that
z
ag ¼ aa þ ðab aa Þ ; ðEq 13Þ postdeposition mismatch stresses developed in coatings are
tg
greatly affected by not only CTE mismatch but also substrate
where Ea and aa are the elastic modulus and CTE of mate- macrocurvature and surface roughness.
rial a (i.e., the material of major concentration at z ¼ 0), Eb
and ab are the elastic modulus and CTE of material b (i.e., the Overall Residual Stress Model
material of major concentration at z ¼ tg ), and n is the gra-
dient exponent, which determines how the mixture of the two To the best of the authors’ knowledge, analytical models
materials varies along the coating thickness; For instance, in which combine the various contributions to residual stres-
a compositionally graded TBC layer, n ¼ 0 when there is no ses are few in literature. Tsui and Clyne (Ref 94) were the
NiCrAlY composition and n ¼ 1 when there is 100% first to develop a simple and straightforward analytical
NiCrAlY composition. expression which combines both quenching and postde-
Using Eq 12 and 13, Hsueh and Lee (Ref 88) analyzed position mismatch stresses using Stoney’s model in com-
the postdeposition stresses developed in a TBC having a bination with thermoelasticity (not including heat transfer
functionally graded bond coat layer. Hsueh and Lee found in the analysis). Tsui and Clyne did not derive analytical
that, unlike in the top coat layer, the stress state of the bond equations for the quenching stress, but instead quenching
coat layer is significantly affected by the compositional stress (obtained from experiments or other models) serves
gradient (with minimum stress level developed at n = 6). directly as an input to the combined model (Ref 67-69).
Zhang et al. (Ref 89) also used Eq 12 and 13 to model a However, unlike the models presented above (in section
TBC system with a graded phase of NiCrAlY alloy within ‘‘Postdeposition Mismatch Stress Models’’), Tsui and
the ceramic top coat. They found that use of a graded top Clyne’s model represents the coating as an aggregate of
coat layer leads to lower values of postdeposition stresses individual sublayers deposited in discrete steps, while at
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each step, the force and momentum are balanced. Thus, the 6Al-4V (with deposition temperature of 500 C), and APS
model is more realistic as it takes into consideration the ZrO2 coating on PK33 (with deposition temperature of
layer-by-layer growth or evolution of the quenching stress 500 C), finding acceptable agreement between the ana-
while simultaneously integrating it with the postdeposition lytically and numerically predicted results. For the APS
mismatch stress to give the overall residual stress state. NiCrAlY coating, they found that ignoring the quenching
This model can predict the dependence of the residual stress would result in a large error since metallic coatings
stress on deposition temperature, material properties, and have low CTE mismatch and higher quenching stress. For
coating and substrate dimensions. However, its main lim- the VPS B4C coating, they found that the residual stress
itation is that it cannot use temperature-dependent material depends on both the quenching and thermal mismatch
properties due to difficulties associated with finding a stresses in equal proportions. For the APS ZrO2 coating,
closed-form solution for such a problem. The final the quenching stress is not very significant due to stress
expressions (obtained by Tsui and Clyne) for the in-plane relaxation via microcracking, thus the overall residual
residual stresses developed (after deposition of the ith stress state is mainly dependent on the thermal mismatch
sublayer) at the top and bottom of the substrate are (Ref 94) stress. Similarly, Bolelli et al. (Ref 95) experimentally
rsb ¼ rs jy¼H measured the peening stress and integrated it with the
Xn quenching stress in Tsui and Clyne’s model to achieve a
Es Fi
¼ þ Es ðji ji1 ÞðH þ di Þ good estimate of the residual stresses developed in func-
i¼1
bðHEs þ ði 1ÞwEd Þ tionally graded WC–Co/NiAl coatings deposited by
FðCTEÞ HVOF. In other work, Tsui and Clyne (Ref 96) developed a
þ Es ðjc jn ÞðH þ dn Þ;
bH similar analytical model for coatings sprayed on hollow
ðEq 14Þ cylindrical geometry (i.e., having some initial curvature).
In that case, stresses develop only due to quenching and
rst ¼ rs jy¼0
expansion rather than bending. They also tested the model
Xn
Es Fi using the same set of materials and process conditions as
¼ þ Es ðji ji1 Þdi
i¼1
bðHEs þ ði 1ÞwEd Þ presented in Ref 66. They found that curved substrates
FðCTEÞ having large radius of curvature result in lower residual
þ Es ðjc jn Þdn ; stresses than planar substrates.
bH
ðEq 15Þ
where H is the substrate thickness, b is the sub- Comparison of Various Analytical Models
strate/coating width, w is the deposited coating thickness, j
is the curvature, d is the neutral axis position (measured For the purpose of comparison, we evaluated and compared
from the coating–substrate interface, as demonstrated in the postdeposition mismatch stresses (as given by various
Fig. 8), F is the force developed in the coating layer (a analytical models) developed in several types of coating
function of the quenching stress), and FðCTEÞ is the force after cooling from deposition to room temperature, i.e.,
that arises due to misfit strain during final cooling. Note 25 C. Conventional TBC (deposited on 304 stainless
that n is the total number of deposited layers, while ‘‘s’’ steel), alumina (deposited on 316 stainless steel), and
denotes the substrate and ‘‘d’’ denotes the deposit (or NiCoCrAlY (deposited on aluminum AA1100 alloy) were
coating layer). More details regarding the derivation can be the coatings selected for this analysis. The properties of
obtained from Ref 96. TBC, alumina, and NiCoCrAlY were selected from the
The total stress developed at the midpoint of the ith work of Wu et al. (Ref 97), Lui et al. (Ref 98), and Gan
sublayer within the coating is (Ref 94) et al. (Ref 99), respectively (as presented in Table 1). From
Fig. 9(a), it can be observed that there is a linear variation
Fi 1
rdi ¼ rd jy¼ði1=2Þw ¼ Ed ðji ji1 Þ i w di of the mismatch stresses (developed in the substrate)
bw 2
Xn through the thickness direction. Moreover, only Zhang’s
Ed Fj 1
þ Ed ðjj jj1 Þ i w dj model predicts such linear variation of the developed
j¼iþ1
bðHEs þ ðj 1ÞwEd Þ 2
mismatch stresses through the thickness direction of the
FðCTEÞ 1
þ Ed ðjc jn Þ i w dn for 1 i n: coating, as is evident from Eq 7 and 9. It can be seen that
bh 2
the model of Hsueh (shown in Eq 7) does not depend on
ðEq 16Þ
the z-coordinate (i.e., through the coating thickness). Due
Tsui and Clyne tested their model on APS NiCrAlY to the use of the zeroth-order approximation, the model of
coating on PK33 substrate (with deposition temperature of Townsend et al. clearly overestimated the stresses devel-
420 C), vacuum plasma spray (VPS) B4C coating on Ti- oped in both the substrate and coating. Thus, Zhang’s
123
Table 1 Materials properties of coatings selected for analysis

S/N Material Properties used @ deposition temperature Source
1 TBC (top coat) E = 500 GPa, v = 0.2, a = 10.34 9 10-6/C @ 1025 C Ref 97
BC (bond coat) E = 109 GPa, v = 0.3, a = 16 9 10-6/C @ 1025 C
304 stainless steel (substrate) E = 152 GPa, v = 0.27, a = 1.96 9 10-6/C @ 1025 C
2 Alumina E = 84.7 GPa, v = 0.24, a = 9.68 9 10-6/C @ 865 C Ref 98
316 stainless steel (substrate) E = 152 GPa, v = 0.27, a = 1.96 9 10-6/C @ 865 C
3 NiCoCrAlY E = 225 GPa, v = 0.3, a = 14.0 9 10-6/C @ 645 C Ref 99
Al 1100 (substrate) E = 68.9 GPa, v = 0.33, a = 23.6 9 10-6/C @ 645 C
E elastic modulus, v Poisson’s ratio, a coefficient of thermal expansion
model seems to be more effective for predicting the increased. Figure 11(a) shows that the postdeposition
residual stresses in thermal spray coatings. The same mismatch stresses are severely affected by the composi-
observations were made when such analysis was carried tional variation of second-phase particles within a TBC
out on alumina and NiCoCrAlY coating, as shown in deposited on a bond coat (with properties obtained from
Fig. 9(b) and (c), respectively. All these analytical results Ref 97). Thus, using compositional variation, the lifetime
are reasonably similar, due to the adoption of similar of the coating may be increased by reducing the magnitude
assumptions during their derivation. However, their main of tensile residual stresses near the top coat–bond coat
limitation is that they all lead to unrealistically higher interface.
stresses that may not actually represent the true residual The effect of quenching stress on ceramic (alumina)
stress state in coatings. The residual stresses computed and metallic (NiCoCrAlY) coatings has also been inves-
using elastoplastic models by Wu et al. (Ref 97), Liu et al. tigated using the analytical model developed by Tsui and
(Ref 98), and Gan et al. (Ref 99) are far lower than the Clyne (Ref 94) (implemented in MATLAB). Based on
elastically predicted stresses. Thus, it is necessary to con- previous research by Liu et al. (Ref 98) and Gan et al.
sider nonlinear material behavior in future derivations. (Ref 99), the quenching stress in alumina and NiCoCrAlY
Otherwise, numerical methods should be used to compute was selected as 10 and 300 MPa, respectively. As shown
the induced residual stresses. A common observation is that in Fig. 11(b), the model correctly predicts that the
tensile stresses are developed in coatings that have higher quenching stress is more important in metallic coatings.
CTE than the substrate material, and vice versa. Thus, for As mentioned above, the quenching stress in ceramic
improved coating life, the coating/substrate material com- coatings is largely relieved by intense microcracking
bination should be carefully selected such that tensile (occurring as a result of thermal shock), which compli-
thermal mismatch stresses are minimized while maintain- cates modeling of quenching stresses in ceramic coatings.
ing optimum values of other properties required for good Since Liu et al. (Ref 98) and Gan et al. (Ref 99) used
coating performance. elastoplastic constitutive behavior for the top coat mate-
As expected, the postdeposition mismatch stresses rials, we could not compare their results with those pre-
increase with coating deposition temperature, as shown in sented in Fig. 11(b). However, the magnitude of the
Fig. 9(d), implying that coating deposition should be predicted residual stresses (as given in Fig. 11b) falls
optimized such that deposition occurs at the lowest tem- within the acceptable range. Thus, the model by Tsui and
perature possible while retaining other essential coating Clyne can be used as an effective tool in coating design as
properties. Figure 10(a) and (b) shows that the mismatch long as the correct values of quenching stress and effec-
stress distribution is strongly affected by the deposited tive elastic modulus are used. Further improvements could
coating thickness. It can be seen from Fig. 10(a) and be made by considering temperature-dependent material
(b) that the stress distribution for an alumina coating differs behavior and nonlinear constitutive behavior.
from that of a NiCoCrAlY coating. This is obvious since Based on the results presented above, it can be seen that
the elastic properties, substrate material, and CTE mis- analytical models can predict the residual stress profile in
match differ for the alumina and NiCoCrAlY coating coatings with a certain degree of accuracy. However, the
systems, thus the two results cannot be compared. How- effectiveness of such analytical models as tools for opti-
ever, in a general sense, one can say that a tensile stress mization is hindered by the various simplifying assump-
state becomes more tensile and a compressive stress state tions adopted. All the models were derived based on the
becomes more compressive as the coating thickness is theory of linear elasticity and therefore may predict very
123
Fig. 9 Comparison of various predictions of thermal mismatch stresses for (a) TBC system deposited at 1000 C, (b) alumina coating deposited
at 840 C, and (c) NiCoCrAlY coating deposited at 620 C; (d) variation of misfit stresses with temperature as predicted by Zhang et al.
high (inaccurate) stress values. Residual stresses in coat- ceramic coatings. Consequently, there is a need for
ings may develop as a result of nonlinear constitutive extension of the available models to handle nonlinear
behavior (Ref 36, 100), depending on the materials and constitutive behavior. Also, there are no analytical models
process conditions applied. In fact, recent findings have for prediction of peening stresses as well. Due to compu-
shown that the mechanical behavior of ceramic coatings is tational advancements, heat and momentum transfer
nonlinear elastic at low temperatures (Ref 101) but highly between impacting droplets and the substrate can be easily
inelastic at high temperatures (Ref 102, 103). Therefore, it modeled. Thus, simple empirical models for prediction of
can be said that the analytical models developed so far are peening stress can be developed using the results obtained
not very effective for predicting residual stresses in from single-splat analysis based on such numerical models.
123
Fig. 10 Effect of coating thickness on thermal mismatch stresses (using Zhang’s model and the same properties as in Table 1) for (a) alumina
coating and (b) NiCoCrAlY coating
Fig. 11 (a) Effect of gradient exponent on variation of misfit stress along coating thickness for TBC (interface is at z = 0 according to Eq 12).
(b) Effect of quenching stress on alumina and NiCoCrAlY coatings (using Tsui and Clyne’s model)
Additional empirical models that consider the effects of Numerical Models for Prediction of Residual
particle process parameters (such as particle size, temper- Stresses
ature, velocity, etc.) on the induced residual stresses would
be another interesting area of study. However, one must Due to recent advances in computational methods, pre-
bear in mind that analytical modeling is mostly limited by diction of residual stresses in thermal spray coatings is now
the mathematical complexities associated with obtaining largely carried out using numerical methods. The finite
closed-form solutions to engineering problems, which is difference method (FDM) is obsolete and has many limi-
why it is often necessary to use numerical methods to tations. Thus, predictions of residual stresses in thermal
model the residual stress evolution in coatings. spray coatings using FDM are now found in few research
123
works (Ref 104, 105). Rather, residual stress development a point of contention is whether such a pure elastic model
has largely been modeled using the finite element method can provide accurate predictions of the induced residual
recently. This is not surprising, as the finite element stresses, as coatings are susceptible to nonlinear deforma-
method is more popular than other numerical schemes due tions. Unlike Jiang et al., Yilbas and Arif (Ref 45) used an
its flexibility and suitability for handling complex geome- elastoplastic material model to predict the residual stresses
tries and transient boundary conditions. Modeling of developed in Inconel 625-type Ni-based alloy coatings
residual stresses with the finite element method is done using a conventional thermal stress formulation. They
using the conventional finite element formulation for found that the numerically predicted residual stress was
thermal stress analysis. This formulation couples the closer to experimental values than were results obtained
energy equation and local force balance (stress equilib- using a previous analytical (elastic) model (Ref 109). Thus,
rium) equation for the evolution of temperature and dis- they asserted that numerical models give more accurate
placements, respectively. Since the finite element estimates of induced residual stresses. Furthermore, Zhang
formulation for thermal stress analysis is very common, its et al. (Ref 110) and Khor and Gu (Ref 20) carried out
derivation is not presented here; more details can be found numerical investigation of the residual stress state in
in previous research work (Ref 106, 107). The constitutive functionally graded coatings (FGCs) using conventional
behavior of the material can be elastic or inelastic thermal stress analysis. They found that the bond strength
depending on the coating material and process conditions and lifetime of FGCs are significantly higher than those of
applied. Recently, modified finite element schemes such as conventional (duplex) coatings due to the gradual change
the element birth–death technique, explicit–implicit finite of properties near the coating–substrate interface. Even
element scheme, and image-based finite element schemes though the results presented by Zhang et al. agreed with
have been used to model residual stress formation in experiments qualitatively, there was no quantitative vali-
thermal spray coatings. In the course of the discussion dation of the model. Zhang et al. (Ref 110) also suggested
herein, we give more emphasis to recent results. that the various layers in an FGC should be deposited at
lower cooling rates to minimize residual stresses. Recently,
Modeling of Postdeposition Mismatch Stresses Using few papers using conventional thermal stress analysis to
Conventional Thermal Stress Analysis predict residual stresses in thermal spray coatings have
been published. Thus, use of other appropriate schemes for
For the sake of clarity, we start this discussion with the accurate prediction of residual stresses is required, as dis-
simplest numerical model, which involves use of the con- cussed in the next section.
ventional thermal stress formulation to model only postde-
position cooling and predict the resulting mismatch stresses. Modeling of Residual Stress Evolution Using
In this formulation, deposition stresses are neglected. Con- Element Birth–Death Approach
sequently, such finite element analysis often over- or
underestimates the residual stress field due to its lack of It is not possible to consider deposition stresses using the
consideration of the layer-by-layer evolution of residual methodology described in the preceding section, where
stresses. Also, perfect interfacial boundary conditions are only postdeposition cooling is modeled. As discussed
usually adopted at the interface. This may affect the residual above, the results obtained from such schemes are hardly
stress results due to the lack of consideration of important reliable and accurate. To consider deposition stresses (such
phenomena (e.g., thermal contact resistance) which should as quenching and peening), a popular approach known as
be considered when modeling the temperature distribution in the element birth–death technique is commonly applied.
a thermal spray coating. These inadequacies generally result This is a numerical procedure where elements are activated
in poor results. For this reason, this methodology has only or deactivated at a given time step according to a certain
been used by a few researchers, as discussed below. criterion, which can be set manually. The implementation
Jiang et al. (Ref 108) predicted the effect of holes on of such a technique in any finite element code is very
residual stresses developed in a TBC system using con- straightforward. To ‘‘kill’’ (or deactivate) an element, the
ventional linear elastic thermal stress analysis. They magnitudes of material properties such as stiffness, con-
observed that the specimen dimensions, spraying process, ductivity, heat capacity, etc. are reduced to nearly zero by
coating thickness, and hole radius all influence the stress multiplying them by a reduction factor, commonly taken as
distribution. They found that the radius of the hole had 1 9 10-6. On the other hand, all elements that should be
greater influence on the induced residual stress field than ‘‘born’’ (or reactivated) are multiplied by a unity reduction
the other factors investigated. Quantitatively, the numerical factor. This numerical activation/deactivation of the ele-
results obtained by Jiang et al. were not validated, possibly ments is usually done using a user-defined algorithm
due to a lack of experimental data for validation. However, designed based on the process history.
123
When applied to the coating process, the element birth– interactions with the environment. Another analysis by Liu
death approach is commonly used to model the layer-by- et al. (Ref 98) and Zhao et al. (Ref 114) (using a similar
layer evolution of the deposition (quenching or peening) approach) revealed that the residual stresses developed in
stresses based on the process time, gun location, gun speed, thermally sprayed alumina coatings are mainly compres-
spray rate, and deposited coating thickness. The thermal sive. In fact, Liu et al. tracked the evolution of the residual
deposition (or quenching) stresses are mainly predicted stress and found that it changes from tensile to compressive
using the thermal stress formulation applied only to the after cooling to room temperature, as shown in Fig. 12. As
activated elements, while peening deposition stresses are demonstrated in this figure, Liu et al. also found that lower
predicted using explicit finite element impact models. The substrate cooling rate resulted in higher compressive
heat influx due to heating of the substrate/coating sublayer residual stresses due to lower quenching stresses. As an
by the plasma plume and contact with the molten droplets improvement on the model used by Buchmann et al. and
is normally considered by inclusion of additional heat Wenzelburger et al., Liu et al. used an elastic-perfectly
source terms in the energy equation. To capture the gun plastic material model to restrict the induced tensile
movement adequately, the heat source term is defined to be (quenching) stresses to about 10 MPa (as experimentally
both space and time dependent (according to the process determined) in order to consider relaxation of quenching
history) using a Gaussian distribution (Ref 62, 98, 111). stress via microcracking. Though the model results look
Similarly, heat loss from the predeposited coating surface good qualitatively, there was a large discrepancy between
through convection and radiation is represented by another the modeled and experimentally measured residual stresses
heat source term. After correct prediction of the deposition obtained by the curvature method, which may be due to the
stresses using the element birth–death technique, the restriction of the coating yield strength. Due to the use of
postdeposition mismatch stresses are computed and added the more realistic material model, Zhao et al. found that the
to the preexisting deposition stresses to give the final overall residual stress values were reasonably comparable
residual stress field. Many researchers have used this to experimental results obtained by the XRD technique. In
technique to provide better estimates of residual stresses in another analysis, Valente et al. (Ref 72) found that the
coatings (as demonstrated in Table 2); detailed discussion contribution of quenching stress is negligible for alumina
and analysis of their findings follow. coatings (around 10-15 MPa), especially when the post-
deposition thermal mismatch stress is as high as 100 MPa.
Considering Thermal Stresses Only Nevertheless, addition of the quenching stress resulted in
numerical values that were closer to experimental results.
This discussion starts with alumina/titania coatings. Pre- They asserted that the areas of high stress concentration
dictions of residual stresses using the element birth–death (such as free edges of the specimen) are the potential sites
approach started with the research groups of Buchmann where failure might initiate.
et al. (Ref 62, 111) and Wenzelburger et al. (Ref 112, 113) For thermal barrier coatings (made of yttria-stabilized
from Germany, who investigated the effect of process zirconia top coat and NiCrAl bond coat), residual stresses
parameters on the residual stress field developed in alumina are also frequently modeled using the element birth–death
and titania coatings deposited inside engine cylinder liner approach. Starting with the research works carried out with
tubes. They found that the residual stress field developed in a pure elastic material model for the top coat, Fogarassy
such coatings was compressive due to relaxation of et al. (Ref 115) and Lugscheider and Nickel (Ref 116)
quenching (tensile) stress via microcracking. The results modeled the evolution of residual stresses in a TBC and
were found to be only qualitatively comparable to experi- observed that very high stresses (with large deviation from
mental ones (obtained by the incremental hole-drilling experiments) were predicted. Using a similar approach,
method) due to the modeling of the coating material Lee et al. (Ref 117) found that the TC–BC interface was
behavior using purely elastic material behavior. As the most critical region for coating failure due to formation
expected, their model predicts lower compressive residual of high tensile (quenching) stresses near the top coat–bond
stresses or even tensile stresses with higher substrate coat interface. Wang et al. (Ref 118) also found that
cooling rates (during deposition) due to higher quenching regions of high stress concentration (mostly near the free
stresses. On the other hand, they found that substrate pre- sample edges, interfaces, and defects) serve as weak points
heating resulted in more compressive stresses, as shown in where coating failure initiates at the microlevel. However,
Fig. 12. Thus, it is desirable to preheat the substrate they found that the overall effect of defects on the mac-
material before coating deposition to minimize develop- roscale is actually helpful, as they relax the postdeposition
ment of tensile residual stresses which might initiate mismatch stresses, which have a greater influence on the
interfacial cracks. However, substrate preheating should lifetime of ceramic coatings. Consequently, the presence of
not be done to the extent of mechanical failure or chemical defects limits the distortion or bending of coated samples
123
Table 2 Comparison of numerical results obtained using birth–death technique versus experiments
Coating material, material model, and model geometry Maximum residual stress and location Validation Deviation from experimental result Source
1
Titania, 3D cylindrical substrate r1;2 ¼ 50 MPa at S (a) [100% Ref 111
Alumina,1 3D cylindrical substrate r1;2 ¼ 5 MPa at I 0 ! [ 300% Ref 62
r1;2 ¼ 10 MPa at S and I
AT13,2 3D rr ¼ 244 MPa, ra ¼ 240 MPa at an edge near coating surface (b) -51.5%! 49.5% Ref 114
Alumina,2 3D r1;2 ¼ 55 MPa at S (a) -66.7%! -20% Ref 72
r1;2 ¼ 65 MPa at I
J Therm Spray Tech (2017) 26:1115–1145
2
Alumina, 3D rx ¼ 422 MPa at S (d) 15.6% Ref 98
rx ¼ 92 MPa at S
Nickel-cobalt,2 axisymmetric rr ¼ 2:6 GPa at I (d) Unpredictable Ref 99
TBC,2 axisymmetric rr ¼ 80 MPa at S (e) -300% Ref 119
rr ¼ 60 MPa at I
1
TBC, 2D curved geometry ra ¼ 190 MPa at I (c) -77.25%! 127.5% Ref 115
ra ¼ 10 MPa at S
TBC,1 3D turbine blade geometry rvm ¼ 3:37 GPa at blade profile edge … … Ref 116
TBC,1 axisymmetric rvm ¼ 145 MPa at I … … Ref 117
TBC,1 axisymmetric-on SCL and DCL DCL showed lower residual stresses in all cases … … Ref 118
TBC,2 axisymmetric rr ¼ 130 MPa at S (d) Good agreement on displacements Ref 97
rr ¼ 150 MPa at I
Titanium,3 axisymmetric rr ¼ 60 MPa at S (d) Fair agreement on displacements Ref 122 and 123
rr ¼ 60 MPa at I
for 1.45 lm thickness
3
IN718, single-particle model ra ¼ 400 MPa … … Ref 51
rr ¼ 1000 MPa
at contact region
SS 316,3 axisymmetric rr ¼ 250 MPa at S (b) -2.08%! 2.08% Ref 124 and 125
rr ¼ 500 MPa at I
for coatings deposited at 610 m/s and 1593 K
1-elastic, 2-elastoplastic, 3-elastoplastic with contact formulation, (a) incremental hole-drilling method, (b) XRD, (c) neutron diffraction, (d) curvature measurement, (e) modified layer removal
method, r1;2 -in-plane stress, rx -longitudinal stress, rr -radial stress, ra -axial stress, rvm -von Mises stress, S-coating surface, I-coating–substrate interface
SCL single ceramic layer, DCL double ceramic layer
1133
123
Fig. 12 Residual stress variation through the thickness of alumina coating: (a) effect of substrate preheating to 393 K on residual stress state of
alumina coatings (Ref 62), (b) effect of deposition/process cooling rate on residual stresses induced in alumina coatings (Ref 98)
due to lower mismatch stresses. Based on similar residual TBCs. Using this modified approach, a group of ‘‘soft/hy-
stress analysis, Wang et al. suggested using a double-layer pothetical elements’’ with suitable properties are added at
top coat for TBC applications in the future due to lower appropriate locations within the model. These elements can
residual stresses and longer lifetime compared with the undergo large deformation at low stress levels with little
conventional, single-layer top coat system. Most of the influence on other active elements. Unlike in previous
results presented in the cited works deviate largely from research works, Wu et al. used the Drucker-Prager criterion
experimental ones. This may be due to the use of elastic (Ref 120) to predict the stresses developed in the top coat,
constitutive behavior to model the top coat. Thus, we also since it captures brittle behavior better compared with the
studied and categorized the papers in which elastoplastic elastic-perfectly plastic model. They also modeled relax-
models were used. ation of quenching stress in the top coat (via microcrack-
Using the birth–death approach and an elastic-perfectly ing) by restricting the tensile stresses to low values. The
plastic material model, Bengtsson and Persson (Ref 119) analysis was carried out in axisymmetric geometry. They
modeled the residual stress evolution in a TBC. As done by found that, while the radial residual stress was compressive
Liu et al. (for alumina coatings), relaxation of quenching and increased linearly with depth, the axial residual stress
stress via microcracking was considered by limiting the was tensile and varied nonlinearly through the coating
tensile yield strength of the material to lower values. Even depth. They asserted that these stress states may lead to
though the qualitative validation of their results is not coating delamination. As expected, the radial stresses were
sound, they found qualitative agreement between the pre- found to be considerably higher than the axial stresses. The
dicted effective plastic strain and experimentally measured residual displacements obtained using the modified finite
vertical crack density. They also observed that the final element model were found to agree accurately with
residual stress state of the coating is compressive and high experimental results obtained by the hole-drilling method.
crack density was developed near the coating–substrate As the solidification time (or melting temperature) of
interface due to development of high quenching stresses metallic coatings is considerably lower than that of high-
near this region. In the conventional element birth–death temperature ceramics, modeling of their residual stress
technique, the nodes of dead elements do not take part in evolution becomes more complicated due to the need to
the calculation as long as they are inactive. Consequently, include peening stresses. For this reason, most metallic
severe distortion of some inactive elements occurs, since coatings are deposited using the HVOF process, which
they share nodes with the active elements. To avoid this results in improved bond strength due to higher peening
severe distortion, Wu et al. (Ref 97) developed the stresses. Thus, we found only few research works in which
‘‘modified element birth–death technique,’’ a significant residual stress evolution in metallic coatings was modeled
contribution to modeling of residual stress evolution in by considering thermal stresses only. Gan et al. (Ref 99)
123
Fig. 13 (a) Effect of coating thickness and substrate preheating to through thickness of SS 316 coating, demonstrating effect of particle
80 C on residual stress state of HVOF Ti coating (Ref 122, 123). velocity (Ref 124, 125)
(b) Variation of simulated thermal, peening, and final residual stresses
modeled residual stresses developed during deposition of nonlinearity associated with modeling impacts, explicit
nickel-cobalt (NiCoCrAlY) alloy on an aluminum disc finite element schemes are commonly used for modeling of
(AA1100) substrate using the element birth–death tech- peening stresses. Explicit schemes are very effective for
nique. They assumed that the coating undergoes elasto- solving nonlinear dynamic finite element problems such as
plastic deformation with bilinear kinematic hardening. The impact, crash analysis, large plastic deformations, etc.
displacements given by the model compared well with However, the main disadvantage of explicit schemes is
those obtained using a laser displacement sensor, except for their slowness due to the need for extremely short time
the fact that random fluctuations of displacement and stress steps. Furthermore, the analysis often becomes tedious due
values were observed in the experimental results. This is to the need to couple results from both explicit and implicit
likely due to slight vibrations caused by the spray gun finite element schemes. Nevertheless, coupled implicit–
movement, instantaneous impact of sprayed particles, and explicit schemes were used recently to predict the residual
heterogeneous nature of the coating microstructure. As stresses in metallic coatings. The explicit scheme is used to
found in ceramic coatings, they found that preheating the predict the peening stresses, while the implicit scheme is
substrate prior to coating deposition results in decreased used to model the thermal stress evolution using the ele-
residual stresses due to lower quenching stresses. In other ment birth–death approach.
research work, Gan et al. (Ref 121) applied the model to a Zimmerman (Ref 122, 123) predicted the residual
TBC and found that the displacements obtained by the stresses in a Ti coating (deposited on Al2O3 cylindrical disk
model compared well with experimental ones. sample by D-Gun) using this approach. They carried out
the computation in two stages using a two-dimensional
Considering Both Peening and Thermal Stresses (2D) axisymmetric model: the first stage modeled the stress
state and temperature distribution induced during the
As mentioned above, the contribution from peening stress impact of the coating onto the substrate using an explicit
is only significant when the particle (or droplet) impact finite element scheme; then, the second stage used the
velocity or spray rate is very high. Generally speaking, the results of the first stage to predict the overall residual
particle impact velocity is relatively lower in plasma spray stresses using an implicit thermomechanical model (de-
processes compared with HVOF and D-Gun. Thus, for veloped based on the element birth–death technique). They
HVOF and D-Gun processes, the peening stress is an assumed that, when sprayed particles are deposited on
essential component of the residual stress state. Metallic solidified splats, 80% of the kinetic energy of the sprayed
coatings are commonly sprayed using HVOF because of particles is converted into heat energy and plastic work.
the need for higher bond strength and density. Due to the They also assumed a perfect and uniform contact condition
123
at splat interfaces and that the temperature and stress fields state. Activation/deactivation of elements can be elegantly
developed by three particles are the same for all the layers. achieved using data from the thermal spray deposition
Even though a slight discrepancy was found when the history. However, certain problems are often encountered
numerically predicted in situ curvature results were com- when modeling residual stresses using this approach. First,
pared with the experimental values (measured by digital prediction of the actual value of quenching stresses is very
dial gauge), the sudden change in stress state within the challenging due to stress relaxation via microcracking,
thickness of the coating seems confusing (shown in plastic deformation, creep, interfacial sliding, and imper-
Fig. 13a). This may be due to high stresses developed near fect bonding. Microcracking is very common in ceramic
defects. The results (from Fig. 13a) also show that the coatings, while other stress relaxation mechanisms are
radial compressive residual stresses are higher in regions mostly encountered in metallic coatings. Several
near the coating–substrate interface. Moreover, greater researchers have considered stress relaxation by limiting
coating thickness was found to have a negligible effect on the maximum quenching stress to a low value using cer-
the stress state near the interface, as depicted in Fig. 13(a). tain elastoplastic material models (e.g., ideally plastic
It can also be seen from Fig. 13(a) that substrate preheating model, Drucker–Prager model, etc.). However, there is a
results in negligible change in the residual stresses within need for development of a properly calibrated model
both the substrate and coating layers. However, they based on more experimental findings of the stress relax-
observed that the residual stresses increased with coating ation mechanisms. This would not only relax the mac-
thickness in some regions. Similar analysis using slightly rostresses developed within the entire coating layer but
different assumptions was carried out for HVOF coatings also help to determine the localized residual stresses
by Lyphout et al. (Ref 51) and Bansal et al. (Ref 124, 125). developed near the defects or imperfections that pervade
Both Lyphout et al. and Bansal et al. assumed that the the coating microstructure. Using the localized stresses,
peening stress developed due to the impact of one particle regions where coating failure might initiate could be
is the same for the whole coating layer at a particle time correctly predicted, thus opening a new avenue for study
step or layer level. Lyphout et al. (Ref 51) found that the of the influence of residual stresses on the failure mech-
coating thickness did not have a significant effect on the anisms and lifetime of thermal spray coatings.
developed residual stresses and that the high compressive Furthermore, use of homogeneous, isotropic, and linear
residual stresses developed in HVOF coatings are the elastoplastic material models for predicting residual
reason for their good bond strength. Bansal et al. (Ref 124) stresses usually affects the quality of the stress results
made a somewhat different assumption, i.e., that 90% of obtained using the element birth–death technique. The
the kinetic energy was dissipated as heat. To incorporate mechanical behavior of thermal spray coatings is more
the effect of strain, strain rate, and temperature on the stress likely to be nonlinear and inelastic due to the presence of
fields developed during impact, they used the Johnson– many types of defects (such as voids, cracks, interfaces,
Cook model for predicting the resulting peening stress inclusions, etc.). Recent findings have shown that the
using an explicit finite element scheme. It can be seen from mechanical behavior of thermal spray coatings is usually
Fig. 13(b) that higher particle impact velocities result in nonlinear elastic and nonlinear inelastic, depending on the
higher compressive stresses. Elhoriny et al. (Ref 126) also temperature and extent of loading (Ref 101-103). More-
used a similar approach to predict the residual stresses in an over, the nature of the thermal spray process is such that
Al2O3 coating. However, instead of modeling the quench- coating properties are expected to vary with direction due
ing and peening stresses developed by a single droplet, they to the splat–splat growth of the coating. The properties in
considered large sets of material blocks consisting of a the thickness direction are expected to be remarkably
chunk of particles. They found the peening stresses to be different from those in the horizontal directions. Thus, use
considerably lower in ceramic coatings due to their high of isotropic material models is clearly wrong. Lack of
yield strength and porous nature. Nevertheless, inclusion of consideration of heterogeneity and anisotropy is the main
peening stress helped in predicting residual stresses closer reason why numerical models cannot predict the stress
to those measured by the hole-drilling method. fluctuations observed experimentally. Similarly, the poor
quantitative agreement of the numerical results with
Issues with Element Birth–Death Approach experimental ones is due to these and other issues (such as
not including proper convective and radiative heat transfer
Based on the reviewed papers, it can be seen that the sources).
birth–death technique is currently the most effective With regards to coupling the element birth–death
approach for modeling residual stresses in thermal spray approach with peening stresses (using explicit schemes),
coatings. The method combines both deposition and quantitative and qualitative comparisons of numerical
postdeposition stresses to give the overall residual stress with experimental results appear better with the addition
123
of peening stresses. However, several assumptions adop- image-based finite element method started with the work of
ted in such research work as presented herein lack a sound Hollister and Riemer (Ref 127) in 1993. Then, Tereda et al.
basis; For instance, Zimmerman (Ref 122) assumed that (Ref 128) demonstrated the potential of image-based finite
80% of the kinetic energy of sprayed particles is con- element modeling of materials using real (or complex)
verted into heat energy and plastic work during the impact microstructure and homogenization techniques. Later, an
of sprayed particles, whereas Bansal et al. (Ref 124) objected-oriented finite element method (OOF) for mod-
assumed that 90% of the kinetic energy is dissipated as eling materials using real images of their microstructure
heat. Since the sprayed particles hardly rebound after was developed by Langer et al. (Ref 129). Since then, OOF
impact, it is expected that almost 100% of the kinetic has been used for modeling of heterogeneous materials.
energy will be converted to plastic work, heat, elastic OOF uses adaptive meshing algorithms to convert an
wave propagation, and sound (with the energy dissipated image of the microstructure to a high-quality finite element
as elastic waves and sound being negligible). Thus, this mesh. Due to the fact that image-based finite element
issue needs to be clarified further and a standard needs to schemes are not very common in commercial packages,
be set for future research work. Also, perfect contact is only a few researchers have used this methodology for
assumed between the impacting particles and substrate modeling residual stresses in coatings.
surface. However, thermal contact resistance should play Hsueh et al. (Ref 130) analyzed the effect of substrate
a unique role in determining the amount of heat energy surface roughness or asperities on the residual stresses
transferred to the substrate, especially at such short length developed in TBCs using linear thermoelastic analysis with
scales. Thus, thorough investigation of the effects of these OOF. They considered only thermal mismatch stresses
assumptions should be carried out in the future. Despite developed as a result of postdeposition cooling, the pres-
the promising results obtained so far, the element birth– ence of localized regions (or defects), and surface undu-
death technique needs to be improved further. Indeed, the lation of the bond coat. Pores and cracks were the main
method predicts residual stress fields that deviate consid- microstructural defects considered in the model. As
erably from experimental ones, as demonstrated in expected, they found that very high localized stresses were
Table 2. Also, the method should be modified so as to induced in regions surrounding the defects. Therefore, they
capture the complex interactions between deposition found that substrate surface roughness (represented by a
process parameters for effective prediction of residual sinusoidal curve) resulted in high localized residual stres-
stresses. ses (in the GPa range), with the convex and concave por-
tions of the top coat and bond coat materials having tensile
Modeling of Localized Stresses Using Image-Based and compressive stresses, respectively. Due to the difficulty
FEM of experimental measurements of residual stresses at the
microlevel, Hsueh et al. could not provide quantitative
As discussed above, thermal spray coatings are full of validation of the presented stress results. Similarly,
heterogeneities or irregularly shaped regions which may Klusemann et al. (Ref 43) numerically predicted the
include cracks, pores, second-phase particles, splat inter- residual stresses developed in thermally sprayed WC–Co
faces, etc. However, modeling of residual stresses in ther- coating on steel substrate using OOF. They used images
mal spray coatings is often carried out using homogeneous obtained from both optical microscopy and scanning
material models, thus resulting in the deviations observed electron microscopy (SEM) to estimate the thermal mis-
between numerical and experimental results. Moreover, the match stresses developed after the deposited coating was
available damage models used for failure prediction in cooled to room temperature. Unlike Hsueh et al., they
thermal spray coatings are developed based on the modeled the resulting mismatch stresses using an elasto-
assumption of homogeneous material constitutive behav- plastic material model (for both the coating and substrate)
ior, which may not capture the true damage mechanisms of and considered initial the compressive stresses induced on
coatings. Apart from the difficulties associated with geo- the substrate during its surface preparation (by roller bur-
metric representation of such heterogeneities using com- nishing) process. They compared the thermal mismatch
puter-aided design (CAD) programs, numerical stress field developed in a homogeneous microstructure
computations become very intensive due to high geometric with that of a microstructure filled with numerous defects,
nonlinearity, poor mesh quality, and numerical disconti- such as dispersed Co particles within a WC matrix, voids,
nuities. This has led to the development of new numerical cracks, surface undulation, splat interfaces, etc. They found
schemes or tools for adequate handling of heterogeneous that the average stress distribution within the coating is the
material behavior. Recently, image-based (or object-ori- same for both the homogeneous and real microstructure
ented) finite element schemes have been commonly used model. However, they observed severe fluctuations of
for constitutive modeling of heterogeneous materials. The stress values within the coating due to the formation of
123
high localized stresses in regions near imperfections. Thus, group to have predicted the residual stresses developed in
image-based residual stress models could be used to thermal spray coatings using microstructure images
investigate the mechanism of coating failure via fracture obtained from a stochastic process model. However, the
initiation and propagation. The residual stresses were found residual stress model is not stochastic and can only capture
to reduce with coating depth, being mainly tensile on the postdeposition misfit stresses. It would be interesting to
surface and compressive in regions near the coating–sub- develop a stochastic residual stress model to capture the
strate interface. It is important to note that the stress state randomness associated with residual stress evolution in
near the coating interface is compressive in both the sub- thermal spray coatings effectively. With this, issues such as
strate and coating material. This is favorable for the life- deposition stress relaxation, inclusion of peening stresses,
time of the coating, since compressive stresses usually and stress concentration near imperfections could be con-
result in increased delamination fracture toughness. sidered properly. Consequently, numerical prediction of
Mostaghimi et al. (Ref 31) applied their previously residual stresses would be more realistic and may pave the
developed stochastic model to simulate the microstructure way for development of coating failure/damage and life-
of a Ni coating deposited on a stainless steel substrate by time prediction models.
HVOF. Using an image of the simulated microstructure,
they used a linear elastic thermomechanical model (im-
plemented in OOF) to predict the mismatch stresses Future Directions
developed after cooling of the deposited coating to room
temperature. Voids were the only source of discontinuities Analytical Models
considered in the analysis. They demonstrated that the
average mismatch stresses were mainly found to be tensile It can be seen that various analytical models have been
within the coating, with some fluctuating values of both developed and used for prediction of residual stresses in
tensile and compressive stresses developed in localized thermal spray coatings. The flaws of the various analytical
regions. They also found that the presence of pores/voids models have been thoroughly discussed. Due to the limi-
resulted in a reduced average stress level within the coating tations of such analytical modeling, we provide a few
due to local stress relaxation. However, there was no suggestions below on how to improve existing analytical
quantitative validation of either the average or localized models.
residual stresses, possibly due to a lack of experimental
1. Quenching stress Thorough investigation for correct
data for validation of both localized and macrostresses (the
estimation of the elastic modulus of a single splat (for
average size of voids and cracks being around 2 lm).
use in Eq 1) and that of the entire coating is required.
Another issue is that small stress values were observed at
There is insufficient justification for the assumption
image boundaries due to the nature of the boundary con-
that the elastic modulus of individual splats is the same
ditions used for this analysis. One must therefore question
as that of the entire coating layer (taking defects into
the validity of such results obtained from image-based
consideration). Otherwise, a modified expression for
finite element schemes, especially as the accuracy of finite
the theoretical quenching stress should be developed.
element results significantly depends on the quality of the
Similarly, empirical relations for correct estimation of
elements near the geometrical discontinuities within the
the stress relaxation reduction factor (n) (as given in
simulated microstructure.
Eq 1) should be developed. This would enable correct
Based on the discussion above, it can be seen that very
estimates of the stress relaxation factor as a function of
few research works have been carried out using this
the process parameters. Furthermore, the effect of
approach. This is mainly due to the lack of a clear
splat–splat interfacial bonding on the overall strength
methodology to model deposition stresses using
of the coating could be investigated using this reduc-
microstructural images. Microstructural images are usually
tion factor.
obtained after coating build-up, hence only postdeposition
2. Peening stress An analytical model for accurate
mismatch stresses can be estimated. Additionally, the
prediction of the peening stress is required. Closed-
presence of noise in such images poses difficulties during
form expressions can be derived based on the change
their conversion to a finite element mesh. Due to the
of energy state of sprayed particles during impact.
presence of noise and intricate defect shapes, formation of
Otherwise, effective empirical models should be
spurious finite elements (with large aspect ratio) leads to
developed using finite element analysis. Development
inaccurate results and prolonged simulation time. Thus,
of an effective peening stress model would greatly
better ways of eliminating noise (without compromising
improve the analytical model of Tsui and Clyne, which
the contents) and finite element meshing should be devel-
is currently used by many researchers.
oped. Mostaghimi et al. (Ref 31) are the only research
123
3. Postdeposition misfit stress Analytical expressions for improve the quality of stress results by minimizing
postdeposition thermal misfit stresses are mostly large distortions at nodes that are common to ‘‘born’’
derived using the theory of elasticity. However, the and ‘‘killed’’ elements.
mechanical behavior of thermal spray coatings is more 2. The coating elements should be activated block-by-
likely to be nonlinear elastic and/or inelastic due to the block (instead of layer-by-layer) in order to correctly
presence of many types of defect (such as voids, track the evolution of the residual stresses, as adopted
cracks, interfaces, inclusions, etc.), as revealed in in recent works by Elhoriny et al. (Ref 126) and
recent papers (Ref 101-103). Thus, new models that Berthelsen et al. (Ref 131).
can capture nonlinear elastic behavior should be 3. The main simulation parameters required to predict
developed to improve the accuracy of existing models. deposition stresses [such as effective modulus, con-
As heat transfer analysis is not included in the vective and radiative coefficients, thermal contact
derivation of mismatch stresses, the energy equation resistance, stress relaxation, the percentage of kinetic
should also be considered in the future. Furthermore, (peening) energy that is converted to plastic work and
the temperature dependence of material properties heat, etc.] should be calibrated using stress values
should be considered when deriving analytical models obtained by ICP sensor. It is currently not easy to
in the future. determine the values of these simulation parameters. In
this way, the calibrated model can easily be used for
optimization of the process without the need to
Finite Element Model
perform too many experiments. This is the essence of
modeling.
Various approaches have been used for modeling the
4. It is well known that finite element formulations give
residual stress state of coatings using the finite element
average stress results at the interfacial nodes (of two
method. Numerical predictions based on element birth–
dissimilar materials) when perfect (or glued/tied con-
death finite element schemes have been shown to be rea-
straint) bonding is assumed at the interface. This
sonably close to experimentally measured residual stresses.
ensures continuity of displacement fields across the
Image-based finite element schemes have been restricted to
interface. In reality, the stress values are not contin-
predictions of postdeposition thermal misfit stresses, but
uous across such interfaces, as demonstrated by the
they provide an elegant way of predicting localized stress
analytical models shown in Fig. 11 and 12. A common
regions which may act as potential failure initiation sites in
practice is to decompose the stress at interfacial nodes
thermally sprayed coatings. Therefore, the finite element
into individual values. However, this has not been done
method is a very powerful tool that can be used to predict
for the results presented in most research works. This
residual stresses in coatings. Various ways in which such
should be corrected in the future.
predictions of residual stresses could be made more
5. The mechanical behavior of the coating material
effective, realistic, and useful are highlighted in the fol-
should be simulated using realistic constitutive
lowing sections.
models including, e.g., nonlinear elastic and/or
inelastic behavior, as presented in recent papers
Element Birth–Death Approach: Points of Consideration
(Ref 101-103).
for Future Simulations
6. The element birth–death approach can be applied to an
image-based mesh in order to add deposition stresses.
As seen in many research works, the element birth–death
Even though the image/mesh is obtained after the
approach is currently the most widely used and accepted
coating has been cooled to room temperature, the
method for numerical estimation of residual stresses in
deposition stresses can still be predicted since the mesh
thermal spray coatings. When applying this approach, it is
nodes are allowed to deform. If the sharpness of the
necessary to adopt certain assumptions in order to mini-
defects is too small, mesh refinement near defects can
mize implementation difficulties. Consequently, the results
become challenging. In such a case, a smaller portion
are usually not as accurate as required, especially when
of the image [in the form of a representative volume
compared with experiment. We have identified a number
element (RVE)] should be modeled and integrated into
of corrections or suggestions to improve such stress
a multiscale modeling framework.
results in the future.
7. As commonly observed experimentally, cracking of
1. The ‘‘modified’’ element birth–death approach based ceramic coatings occurs after postdeposition cooling to
on an algorithm developed by Wu et al. (Ref 97) (as room temperature. Thus, it will be interesting to
discussed in ‘‘Considering Thermal Stresses Only’’ consider adding a damage model during the postdepo-
section) should be used in the future. It helps to sition stage in the future.
123
Process Modeling Using Meshless Lagrangian Solvers sprayed particles (less than hundred) due to the need for
implementation of several interfacial boundary conditions
Based on the review presented above, one can see that the during particle interactions. Even so, it is necessary to
main challenge in numerical modeling of residual stresses distribute the solution across parallel processors.
is the prediction of deposition (both quenching and peen- Using this approach, thorough investigation on the effect
ing) stresses, due to complications arising from stress of process parameters (such as particle temperature,
relaxation. Therefore, there is a need for the development velocity, and size, gun speed, gun path, deposition tem-
of an effective and computationally efficient model for perature, substrate temperature, spray rate, and cooling
prediction of deposition stresses during the thermal spray condition) and their interactions on residual stresses can be
process. Using Eulerian schemes (e.g., phase field, level- carried out. Several design charts can be developed by
set, volume of fluid, etc.), it is difficult to track splat conducting such analysis on only 1–6 particles. Such
interfaces due to the problem of coalescence (inherent to design charts will greatly help in material selection and
the formulation). Furthermore, Eulerian schemes require process optimization. In reality, particles are sprayed with
relatively higher mesh density or computational resources randomly (statistically) distributed parameters (such as
compared with mesh-free Lagrangian schemes. size, velocity, and temperature), thus the residual stress
In the future, mesh-free parallel graphics processing unit evolution is stochastic. Therefore, stochastic residual stress
(GPU)-compatible Lagrangian solvers can be used to prediction tools could be developed using established rules
effectively model the deposition process as well as stresses. for stress–parameter correlations as obtained from such
Each particle (or droplet) can be represented as a separate design charts. Furthermore, this approach could be used to
meshless Lagrangian domain consisting of only nodes or a generate several representative volume elements (RVEs)
point cloud having unique initial and boundary conditions. having localized residual stresses (at the microscale).
Recent meshless methods such as smooth particle hydro- Within a multiscale modeling framework, the correlation
dynamics (Ref 132-134) and the discrete gradient method between localized stresses (existing near discontinuities)
(Ref 135, 136) can be used to model large deformation and the macroscale residual stress field can be easily
problems on a point cloud. The process of splat formation established using computational homogenization (Ref
and solidification can be simulated on the point cloud using 140, 141), thus paving the way for investigation of the
a nonisothermal fluid–structure interaction model. Inter- effects of localized residual stresses on the constitutive
estingly, information on the solidification front during the behavior of coatings (e.g., property estimation and
phase change process (as given by the energy equation) can damage).
be directly used to compute the associated quenching Using the above suggestions, a robust model that may
stresses using the stress equilibrium equation. By con- serve as a versatile tool for investigating the relationship
verting the discrete points to a finite element mesh, residual between process parameters, coating microstructure, and
stress relaxation processes (such as thermal shock, inelastic coating properties may be developed. This could be used in
deformations, and damage) can be easily modeled using conjunction with fracture-based damage models to inves-
new numerical schemes, e.g., the extended finite element tigate all possible failure mechanisms of coatings. It may
method (Ref 137), discontinuous Galerkin’s method (Ref also serve as a tool for optimization of the thermal spray
138), cohesive zone models (Ref 139), etc., which are more process for enhanced coating quality and performance.
stable than meshless approaches. However, the stability of
meshless schemes can be improved such that they can be Acknowledgments The authors would like to acknowledge the
support provided by King Fahd University of Petroleum & Minerals
used for such modeling as well. For high-energy processes
(KFUPM) in funding this work through project FT161016.
(such as HVOF), the peening stress is implicitly repre-
sented at the fluid–fluid or fluid–structure interface. Thus,
there is no need for the ambiguous assumptions usually Appendix
adopted when solving for the peening stresses. The main
limitation of this approach is that it can only be used to See Table 3.
model stresses associated with the deposition of a few
123
Table 3 An overview of the previous models used for the prediction of residual stresses developed in thermal spray coatings
Source of residual stress Model type Experimental validation Coating material Reference
S1 S2 S3 S4
R A 7 Many Ref 67
R A 7 Many Ref 72
R A 7 AlGa Ref 85
R A & TBC Ref 86
R A & TBC Ref 88
R A & TBC Ref 89
R A & TBC Ref 90
R A & TBC Ref 91
R A & TBC Ref 92
R A & TBC Ref 93
R R A 4 Many Ref 94
R R R A 4 WC–Co Ref 95
R R A 4 Many Ref 96
R F1 & TBC Ref 108
R F1 4 Diamalloy Ref 45
R F1 & FG-TBC Ref 110
R R F1 & FG-TBC Ref 20
R R R F2 & Al2O3, TiO2 Ref 111 and 62
2
R R R F & Al2O3, TiO2 Ref 112 and 113
R R R F2 & Alumina Ref 98
R R R F2 4 Alumina Ref 114
R R R F2 & Alumina Ref 72
R R R F2 & TBC Ref 115
2
R R R F & TBC Ref 117
R R R F2 & DLC TBC Ref 118
R R R F2 4 TBC Ref 97
R R R F2 & NiCoCrAlY Ref 99
2
R R R R F & Ti Ref 122 and 123
R R R R F2 & … Ref 51
R R R R F2 & SS 316 Ref 124 and 125
R R R R F2 & Alumina Ref 126
R F3 & TBC Ref 130
3
R F & WC–Co Ref 43
R F3 & Ni Ref 31
R F3 & SS, WC–Co Ref 32
S1-quenching stress, S2-peening stress, S3-thermal gradient stress, S4-thermal misfit stress, A-analytical, F -conventional FEM, F2-birth–death
1
FEM, F3-image-based FEM, 4-quantitatively valid, &-qualitatively valid, 7-both quantitatively and qualitatively not valid
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Modeling Residual Stress Development in Thermal Spray Coatings: Current Status and Way Forward

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J Therm Spray Tech (2017) 26:1115–1145

Modeling Residual Stress Development in Thermal Spray

Abstract An overview of analytical and numerical meth- Introduction

Fig. 1 Schematic of typical

Fig. 2 Overview of thermal

Fig. 4 Bending of substrate

developed within the coating sublayers (as demon-

Fig. 6 Peening stress developed during coating deposition

shown in Eq 5. Using Eq 5, the continuity (or compati- rS ¼ 2 3z þ 2tS Ej t j E i t i ð ai aS Þ

Both Hsueh and Zhang et al. showed that, using a zeroth-

Table 1 Materials properties of coatings selected for analysis

References 4. Begriffe and Einteilung, DIN EN 657; Thermal Spray—Termi-

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