Multi-Scale Thermo-Mechanical Model Simulation of Residual Stress in Atmospheric Plasma Spray Process

Abstract

This work presents a multi-scale one-way-coupled thermo-mechanical method to determine the residual stress in an Atmospheric Plasma Spray (APS) process. The model uses three submodelling scale levels that range from the entire component (macroscopic) to a splat coating layer (microscopic) dimension. The three-level scale temperature and stress evolutions of an Al₂O₃ coating material on a flat aluminium substrate were analysed. The quenching stress for different substrate preheating temperatures up to 600 K at the end of the APS coating process was discussed and validated through an experimental in situ curvature method and Stoney’s quenching stress equation.

1. Introduction

Thermal spray coating technology is a widely used technique for producing advanced surfaces adapted to industrial requirements, such as wear resistance, corrosion resistance, abrasion resistance, thermal/electrical conductivity, or thermal/electrical insulation, among others, at low cost [1,2]. Different thermal spraying processes have been developed depending on the energy source used, such as plasma spraying, flame spraying, high-speed oxyfuel spraying, and arc spraying. Atmospheric Plasma Spray (APS) allows for the processing of a wide range of materials, such as metals and their alloys, metal oxides, ceramics, and others, due to the high temperatures reached in the deposition flow up to 15,000 K. It also allows for coating thicknesses in the range of micrometres to several millimetres [3].

Complex dynamic heat and mass-transfer processes occur between the high-energy process gases, the coating material particles, and the substrate, which determine the final properties of the coating [4]. Each thermal spray process is characterised by its temperature range, enthalpy, and velocity, resulting in coatings with unique properties such as adhesion, porosity, inclusions, hardness, and residual stress [5,6].

Residual stress is one of the most critical factors influencing coating properties and behaviour, as it can lead to cracks that reduce coating life when facing adhesion wear. Differences in the material properties of the coating and the substrate, such as the coefficient of thermal expansion and the elastic modulus, generate residual stress at various stages of the coating process [7,8,9]. Residual stresses are classified according to the phase of the process in which they are generated. In the first phase during the deposition process, quenching stress, σ_q, is generated by the sudden solidification of the deposited splats at the impact with the substrate surface. Quenching stress is always tensile and causes bending or distortion of the substrate layer at mesoscopic scales. The coating material, the substrate temperature, and the stress relaxation mechanisms determine the σ_q value [9]. During the cooling phase, so-called thermal stress, σ_t, is due to the unequal shrinkage of the substrate and the coating due to the mismatch in the Coefficient of Thermal Expansion (CTE) between the two materials [10,11]. The stress is usually compressive for ceramic coatings with a lower CTE than that of the substrate. The final residual stress is the sum of the quenching and thermal stress described above.

Several works have presented simulations to study the evolution of temperatures and residual stress during the thermal spray process [12,13,14,15,16]. They all focused on obtaining the residual stresses at the end of the ceramic coating process. None of them studied the deposition phenomena at the splat level in detail. The novelty of the multi-scale thermomechanical approach is that it provides a more complete understanding by incorporating advanced sub-modelling techniques that have not been explored in previous studies. Three scale levels are defined, from the complete component (macroscopic) to a splat coating layer (microscopic). This allows for the study of the interaction between the splat layers and how it affects the temperature and stress distribution within the coating and on the substrate’s surface.

The main objective of this work is to implement a multi-scale thermal–mechanical model that predicts the temperature and stress evolution at different coating and substrate depths during the APS deposition process on a flat substrate. Section 2 introduces the Finite Element Model (FEM) based on the submodelling technique of one-way coupled thermal–mechanical analysis. The temperature and residual stress evolution at the three modelling levels during the APS coating process are analysed in detail in Section 3. The simulated quenching stress at different substrate preheating temperatures up to 600 K is discussed. Section 4 proposes an experimental in situ curvature method to calculate the residual stress from the curvature produced in the substrate. The quenching stress is experimentally measured and determined by the Stoney equation, which validates the simulation results from the model.

2. Numerical Model

This section presents the FEM proposed to obtain the evolution of temperature and stress during the APS coating process. Section 2.1 describes one-way coupled thermo-mechanical analysis based on the submodelling method. This allows for temperature and stress analysis at different scale levels, from the entire component (macroscopic) to the splat coating layer (microscopic). In such modelling, it is necessary to define the operating parameters used in the APS process, such as torch speed and trajectory, as well as the plasma plume profile and the sequence of heat transfer mechanisms followed during the process. These are described in Section 2.2 and Section 2.3, respectively. Finally, Section 2.4 summarises the thermal and mechanical properties of the materials used in the model.

2.1. One-Way Coupled Thermo-Mechanical Analysis Based on Submodelling Method

A one-way coupled thermomechanical analysis has been proposed to study the residual stress generated in an APS coating process. First, a thermal analysis of the coating process is performed at each time step. Then, the temperature distribution obtained at each time step is used as a boundary condition in the mechanical analysis to obtain the residual stress at each time step [17,18,19]. Analysis was performed in ANSYS Mechanical APDL 2024 R1 software (ANSYS Inc., Canonsburg, PA, USA) [18].

The difference in the scale between the coating thickness in micrometres and the substrate thickness, usually in centimetres, requires adaptive meshing to increase the accuracy in the region close to the coating layer. Two techniques are usually used. The first technique is to create a full adaptive mesh of the sample, varying the size and density of the mesh [12,14]. This method has the disadvantage of a large number of elements with a high computational cost.

A second method is called submodelling [17,20,21,22]. A global sample mesh is analysed. From the results obtained in the global mesh, the region of interest is separated into a submodel, which is analysed independently with a more refined mesh. This technique is based on the Saint Venant Principle, which states that the stress, strain, and displacement fields caused by two different, but statically equivalent, force distributions on a body are approximately the same distance from the loading points. The distribution of stress and strains only changes near the load application regions. It is possible to select an area of the global model and analyse that area, defining the results of the global model in the study area as the boundary conditions for the submodel study.

On the other hand, additional material layers are deposited on the substrate in the coating process. The proposed model considers a sequential creation of additional layers. For this purpose, the element birth–death technique is used [9,12,14]. This technique activates or deactivates the mesh elements in a time step according to a specific criterion. Initially, all mesh elements are defined (substrate and coating), but the coating elements are deactivated and do not interfere with the thermo-mechanical analysis. In the time step where the plasma plume passes over the coating element, it is activated (birth) and participates in the thermo-mechanical analysis. This allows for the residual stress in the coating to be better estimated.

The model developed in this study is based on the submodelling technique, where three scale levels were defined from the full dimension of the sample (macroscopic model) to a splat coating layer (microscopic model) [17], as shown in Figure 1a. In the macroscopic model, the entire component is modelled with overall dimensions of 101.6 × 25.4 × 4 mm³. The first layer of the mesh was discretised into a total of 192 × 48 elements. Each element in the top layer was 0.5292 × 0.5292 × 0.365 mm³. One element of the top layer of the macroscopic level, Kth, was selected and discretised into 10 × 10 × 10 elements to create the mesoscopic model. Each element of this mesh had a size of 52.92 × 52.92 × 36.50 µm³. In addition, a layer of 10 × 10 × 1 elements was added to form the cover layer. To generate the microscopic model, a set of four elements of the coating layer and the four adjacent elements of the first layer of the substrate were selected. The total size of this set was 105.8 × 0.105.8 × 73.0 µm³. The set of eight elements was discretised into 24 × 24 × 20 elements. Each element had a size of 4.41 × 4.41 × 3.65 µm³. A layer thickness of 3.65 µm was considered for each coating layer [23]. All simulation parameters are summarised in

Table 1. Definition of the submodelling simulation parameters.

Parameter	Value
Substrate dimensions (DX × DY × DZ), mm3	101.6 × 25.4 × 4
Number elements: first layer (MTX × MTY)	192 × 48
Number of node layers	5
Surface element dimensions (dx × dy × dz), µm3	529.2 × 529.2 × 365
Analyzed elements (Ne)	21,312
Torch speed (vtorch), mm/s	500
Torch line change time (timetor), s	1
Ambient temperature (TAmb), K	300
Number elements of the line offset (ΔLZ)	5
Total number of spray lines (TL)	15

.

Figure 1b summarises the simulation sequence using the proposed one-way coupled thermal–mechanical model. First, the geometry, mesh, and time step were defined for each modelling level. Second, thermal analysis was performed to obtain the temperature field. Third, mechanical analysis was performed to obtain the displacements and residual stress generated at each time step. The thermal results were the initial conditions for the mechanical analysis. The following submodels introduce the resulting temperatures and displacements as boundary conditions. The above simulation sequence is repeated in each submodel.

Three different types of mesh elements in ANSYS were used [18]. The eight-node hexahedral thermal solid element called SOLID70 was used for the thermal analysis. For the mechanical analysis, the SOLID185 element was used for the substrate and the SOLID65 element for the coating, in accordance with the material model utilised in this work.

2.2. Plasma Spray Process Parameters

An APS process was simulated on an Al₂O₃ coating on a flat aluminium plate with dimensions of 101.6 × 25.4 × 4 mm³. An F6 APS torch (GTV, Luckenbach, Germany) and a six-axis robot (Type RX 130 B, Stäubli Tec-Systems GmbH, Bayreuth, Germany) were considered as references for the simulation and used for the experimental validation. The torch followed the path over the surface on the real process, as shown in Figure 2 [24]. The path was divided into a sequence of linear trajectories of length L_x, with a vertical distance between successive trajectories, D_Ly, to ensure uniform coating thickness. The operating parameters were defined as shown in

Table 2. Plasma torch parameters.

Parameter	Value
Plasma gun	GTV F6
Torch distance to the plate, mm	120
Torch Speed (vtorch), mm/s	500
Hydrogen, slpm	10
Argon, slpm	44
Current, A	500
Voltage, V	75
Power, kW	37.5
Powder carrier gas mass flow rate, slpm	5
Powder feed rate, g/min	4.2
Trajectory length (Lx), mm	200
Line offset (DLy), mm	2.65

.

2.3. Thermal Model

For the thermal analysis, the heat flux of the plasma plume on the substrate surface was modelled as a forced convection heat flux. Three parameters are required to define the heat fluxes: the flame footprint, the convective coefficient, and the flame temperature. The flame footprint was defined as an area of 36 × 16 elements, which was 19.05 × 8.46 mm². A temperature profile and a convective coefficient within the flame footprint were defined based on a Gaussian heat flux distribution [9,10,19]. The Gaussian plasma plume temperature profile parameters used in this work are summarised in

Table 3. Definition of the Gaussian plasma plume temperature profile parameters [24].

Parameter	Value
Number of elements for the X-axis dimension of the plasma area or footprint (FLx)	36
Number of elements for the Y-axis dimension of the plasma area or footprint (FLy)	16
Maximum plasma plume temperature (THigh), K	1476
Minimum plasma plume temperature (TLow), K	634
X-Axis standard deviation of the Gaussian plasma plume temperature profile, σx2	49
Y-Axis standard deviation of the Gaussian plasma plume temperature profile, σy2	14.5
Maximum convective coefficient (FilmHigh), W/m2·K	2500
Minimum convective coefficient (FilmLow), W/m2·K	2000

.

Before the plasma plume reaches the Kth substrate element, step K, the heat transfer is modelled as a convective and radiative heat flow from the substrate to the surroundings. When the plasma plume reaches the Kth substrate element, the heat flow is modelled as a forced convection flow [24]. At this step time, the coating element is activated at mesoscopic level. In the case of microscopic submodelling, the coating layer is divided into 10 successive splat layers that are activated during the plasma plume passage. The activation sequence of each of the ten layers is shown in Figure 3. Once the plasma plume leaves the deposited coating element, step K + F_Lx + 1, the heat transfer is modelled by the convective and radiative heat flow between the coating layer and the environment.

The ratio between the torch velocity, $v_{torch}$, and the element dimension in the forward direction of the flame, $d_x$, defines the time step, $t_{step}=d_x/v_{torch}$. At each time step of the simulation, a row of 16 elements perpendicular to the front line of the torch begins to receive the heat flux, while a row of 16 elements leaves the area covered by the plasma plume from the rear.

The heat transfer through the plasma plume, $q_{torch}$, is defined by Equation (1) [25]. The convective heat transfer through the surface to the surroundings, $q_{conv}$, is given by Equation (2) [25]. The radiative heat transfer through the surface to the surroundings, $q_{rad}$, is given by Equation (3) [26]. The Fourier–Biot equation determines the heat balance between the heat flow on the surface and inside the material and is based on Equation (4) [26].

$$q_{torch}=h_{flame}·\left({\mathrm{T}}_{flame}-{\mathrm{T}}_{surf}\right)$$

(1)

$$q_{conv}=h_{conv}·\left({\mathrm{T}}_{surf}-{\mathrm{T}}_0\right)$$

(2)

$$q_{rad}=\mathsf{\sigma }·\mathsf{\epsilon }·\left({\mathrm{T}}_{surf}^4-{\mathrm{T}}_0^4\right)$$

(3)

$$-{\displaystyle \frac{\partial }{\partial x}}\left(k_x·{\displaystyle \frac{\partial T}{\partial x}}\right)-{\displaystyle \frac{\partial }{\partial y}}\left(k_y·{\displaystyle \frac{\partial T}{\partial y}}\right)-{\displaystyle \frac{\partial }{\partial z}}\left(k_z·{\displaystyle \frac{\partial T}{\partial z}}\right)+\mathrm{q}=\rho ·C·{\displaystyle \frac{\partial T}{\partial t}}$$

(4)

where h_flame is the plasma plume convection coefficient, in W/(m²·K); h_conv is the natural convection coefficient, in W/(m²·K); T_flame is the plasma plume temperature, in K; T_surf is the surface temperature, in K; T₀ is the ambient temperature, in K; σ represents the Stefan–Boltzmann constant, 5.67 × 10⁻⁸ J/(s·m²·K⁴); ε is the emissivity coefficient of the coating/substrate surface; k_x, k_y, and k_z are the material’s conductivity, in W/(m²·K); $\rho$ is the material density, in kg/m³; C is the specific heat capacity, in J/kg·K; and $q$ is the heat transfer by convection and radiation through surfaces, in W/m³.

2.4. Thermal and Mechanical Material Properties

In the proposed model, the substrate material is aluminium AL99 AW 1050A H14, and the coating material is Al₂O₃ 99.7%, melted and acid washed. ANSYS Mechanical APDL provides many material models for different material behaviours [18]. This work uses the ANSYS material model called Multilinear Isotropic Hardening Plasticity (MISO) for the ceramic coating. The MISO material model describes the multilinear isotropic hardening process of a material during the application of a force. The thermal and mechanical properties of the implemented materials are summarised in

Table 4. Thermal and mechanical properties of aluminium.

Temp (K)	Density (kg/m3) [27]	Thermal Conductivity (W/(m·K)) [28]	Specific Heat (J/(kg·K)) [27]	Emissivity Coefficient [25]	Elastic Modulus (Pa) [29]	Poisson’s Ratio [30]	Thermal Expansion Coefficient (10−6 K−1) [31]
300	2700	237	871	0.07	83.80 × 109	0.334	23.0
600		232	1072		56.16 × 109		30.0
900		213	1190		48.00 × 109		34.0
1200		99	1273		100		36.9
1500		102	1338		100		39.1
1800		103	1391		100		41.0
2100		103	1436		100		42.5

and

Table 5. Thermal and mechanical properties of Al₂O₃.

Temp (K)	Density (kg/m3) [32]	Thermal Conductivity (W/(m·K)) [30]	Specific Heat (J/(kg·K)) [30]	Emissivity Coefficient [25]	Elastic Modulus (Pa) [32]	Poisson’s Ratio [30]	Thermal Expansion Coefficient (10−6 K−1) [33]
300	3950	36.96	789	0.8	3.85 × 1011	0.254	4.3
600		16.01	1114		3.63 × 1011		7.1
900		8.87	1220		3.41 × 1011		8.7
1200		6.66	1270		3.19 × 1011		9.9
1500		5.78	1308		2.97 × 1011		10.8
1800		6.13	1304		2.75 × 1011		11.5
2100		6.09	1310		2.53 × 1011		12.1
2400		6.05	1316		100		12.6
2700		6.01	1322		100		13.1

.

3. Simulation Results

This section presents the simulation results for the temperature and residual stress evolution at the three submodelling levels (macroscopic, mesoscopic, and microscopic) during the APS process of Al₂O₃ coating on an aluminium substrate plate preheated at 300 K.

In the macroscopic model, a path of 15 trajectories was simulated. An element in the centre of the top layer of the mesh was selected as a reference (point under test). Five vertical nodes through the thickness of the substrate were selected, as shown in Figure 1. These vertical nodes correspond to the substrate’s different depths at the point under test in the substrate. The simulated temperature evolution with the macroscopic submodel at different substrate depths at the point under test during the coating deposition is shown in Figure 4a. An increase in the temperature of the homogeneous substrate was due to the accumulation of heat during the 15 trajectories of the coating phase. A temperature peak pattern was observed when the plasma plume passed near the point under test in the central trajectories (sixth to tenth). The maximum temperature peak occurred when the plasma plume passed over the analysed element in the eighth trajectory, as shown in Figure 4b. Figure 4b describes the temperature evolution at different substrate depths of the point under test in the eighth trajectory in detail. In particular, the plasma plume comes into contact with the plate at point A at the beginning of the eighth trajectory, but does not affect the selected element until it arrives at point B. Heat transfer with the environment takes place during this time slot t_B−t_A. The temperature rises rapidly because of the heat input from the plasma plume at instant t_B. The centre of the plasma plume reaches the point under test at instant t_C. The maximum temperature peak showed a delay due to the thermal inertia of the plate. The temperature decreases from the maximum temperature peak until the plasma plume leaves the selected element at instant t_D. From that moment, the selected element is cooled by heat transfer with the environment. The plasma plume leaves the surface of the plate at instant t_E, and the plasma plume movement changes to the ninth trajectory, which takes place until instant t_F, when the plasma plume starts at point A of the ninth trajectory.

The plasma plume on the selected element at 0 mm substrate depth produced a maximum increase in the substrate surface temperature of 27 K and an increase in the global plate temperature of 7 K at the end of the trajectory. It is observed that the largest thermal gradient occurred on the substrate’s surface (0 mm depth) during the passage of the plasma plume and had a much smaller effect with the simulated depth of the point under test in the substrate, where temperature increased by only 14 K and 13 K at half the substrate thickness (2.10 mm depth) and total substrate thickness (4 mm depth) respectively. This justifies the use of mesh refinement by concentrating this study on the first layers with a greater temperature gradient.

Figure 5a shows the stress evolution at different substrate depths of the point under test during the deposition and cooling phases of the entire APS coating process. There was a correlation between the temperature peaks of the plasma plume pass and the maximum peaks of the stress. The stress was due to the effect of the contraction of the substrate due to the heat provided by the plasma plume over an area of the plate. This generated compressive stresses in that area and tensile stresses on the rest of the plate. In the time interval from 0 s to 4 s, the plasma plume came into contact with the plate, producing tensile stresses at the point under test placed in the centre of the plate. The plasma plume passed over the centre area of the plate in the time interval from 4 s to 9 s and produced compressive stresses at the point under test. Finally, the plasma plume moved away from the upper edge of the plate in the time interval from 9 s to 14 s, generating tensile stresses at the point under test.

Figure 5b shows the details of stress evolution at the different depths of the point under test during the eighth trajectory. As for the simulated temperature, stress showed a low variation during the approach plasma plume movement at time slot t_B−t_A. The compressive stress rose rapidly due to the effect of the high plasma plume temperature from time t_B up to a −43 MPa maximum stress value at the substrate’s surface (0 mm depth), corresponding to the maximum temperature in Figure 4b at the substrate’s surface. After that, the compressive stress decreases slowly when the torch trajectory moves away from the centre of the substrate plate.

The mesoscopic model mesh was generated from the element Kth selected in the macroscopic model, as shown as the point under test in Figure 1. The following discussion focuses on the eighth trajectory, where the plasma plume passes over the point under test and the deposition of the coating material occurs. In the mesoscopic model, the coating material is defined as an additional layer on the substrate that is activated when the plasma plume passes over the substrate surface. Figure 6a shows details of the mesoscopic model mesh and the depths of the nodes considered. The microscopic model was generated from the four elements in the centre of the substrate mesh’s top layer and the coating layer’s four adjacent elements, as shown in Figure 1a. Figure 6b shows details of the microscopic mesh and the depths of the nodes considered.

The thermal analysis results obtained in the macroscopic, mesoscopic, and microscopic models in the central time period of the eighth trajectory is shown in Figure 7. The substrate was heated progressively as the plasma plume approached. When the centre of the plasma plume arrived over element Kth, the coating layer (0 µm coating—mesoscopic and −35.6 µm coating—mesoscopic) was activated in the mesoscopic model at 1900 K, point A, and it cooled rapidly in contact with the substrate (0 µm substrate—mesoscopic). Concurrently, the substrate temperature rose to 375 K, point C. This temperature peak was not observed in the macroscopic model (0 µm coating—macroscopic) because the deposition process was not modelled at this level. The temperature gradient between the coating and substrate layers was due to the imperfect contact between the two surfaces, and thermal conduction occurred over a small fraction of the nominal contact area. This effect was defined in the model by the thermal contact conductivity [34]. Once deposited, the coating layer cooled down to point B, where the plasma plume left the substrate element and the temperatures of the coating and the substrate layer equalised.

In the microscopic model a more detailed effect was observed at this time. At this level, 10 splat layers were deposited progressively in six intervals, as explained in Section 2.1. Each splat layer caused an increase in temperature in the preceding coated splat layers. Six consecutive temperature peaks were observed for the first deposited layer (0 µm coating—microscopic). The P₁ point corresponded to the deposition of the first layer at 1900 K, which cooled rapidly in contact with the substrate. As the second and third layers were deposited, the temperature of the first layer rose to P_2, and the three layers cooled together. As the fourth and fifth layers were deposited, the temperature of the previous layers increased up to point P_3, and all the coating layers cooled down together. This phenomenon was repeated at points P₄, P₅, and P₆. These temperature peaks decreased as the volume of coating material increased. It is concluded that the thermal analysis results of the three models agree. At the substrate surface, the global temperatures were at the same level. The effect of coating deposition in the mesoscopic and microscopic models produced the same temperature peak at the substrate surface, point C. The coating temperature in the mesoscopic and microscopic models showed the same trend. The temperature curve of the coating in the mesoscopic model was centred with respect to the temperature curves of the different coating layers in the microscopic model.

Figure 8 compares the stresses obtained in the three submodels. The stress in the different layers showed similar trends at the three submodelling levels. Compressive stress was generated in the substrate layers during the coating phase, and tensile stress was generated in the coating layers, as expected [9]. The stress generated by the heat flux of the plasma plume was only considered in the macroscopic model, with a maximum compressive stress of −40 MPa. On the other hand, in the mesoscopic and microscopic models, a compressive stress peak of −80 MPa was observed in the substrate when the coating layers were activated. This compressive stress increment was due to the additional heat from the rapid solidification of the coating, not considered at the macroscopic level.

Figure 8b compares the stress of the coating layers in the mesoscopic and microscopic models. The single layer in the mesoscopic model produced a quenching stress of 17.07 MPa at point A. However, when the initial coating layer in the microscopic model was subdivided into 10 splats, stress progressively decreased as the number of layers increased. This effect was because each successive layer was deposited at a higher temperature than the previous one, so the stress was lower during the coating solidification. On the other hand, it was observed that the generation of splat layers in the microscopic model generated 70% higher residual stress in the surface layer of the substrate than in the mesoscopic model for the stress accumulated accumulative effect.

Quenching stress was generated by the sudden solidification of the deposited splats on the substrate surface, and is defined by point A in Figure 8a. The relationship between the simulated quenching stress and the substrate preheating temperature was analysed in the mesoscopic model for a 36.5 µm coating layer. Figure 9 illustrates the quenching stress for substrate preheating temperatures from 300 K to 600 K. The simulated quenching stress increased slightly with increasing substrate temperature, which agreed with the trend of the results from the literature [35]. It was also observed that the dependence is lower at high substrate preheating temperatures [14,35].

4. Experimental Quenching Stress

This section describes the experimental measurement of the quenching stress to validate the results obtained from the mesoscopic model for the APS coating process of Al₂O₃ coating on an aluminium substrate plate preheated to 300 K, point A in Figure 8a. An in situ curvature method based on continuously measuring the substrate curvature during the deposition process is proposed to measure the instant quenching stress. This non-destructive method allows for the direct calculation of residual stress from the curvature generated by a laser triangulation displacement sensor, and the substrate temperature measurement during the APS process through an infrared pyrometer. Several previous studies used the in situ curvature method to measure residual stress, and the thermocouple attached to the substrate to measure substrate temperature [36,37,38,39,40]. In contrast with previous works, the method proposed in this work allows for higher accuracy in non-contact temperature measurement, so it does not interfere with the curvature of the sample.

The quenching stress during the deposition phase could be determined using the Stoney Equation (5) [41,42].

$${\sigma }_q={\displaystyle \frac{∆k·E_s^{\prime }·t_s^2}{6·t_c}}$$

(5)

$$E_s^{\prime }={\displaystyle \frac{E_s}{\left(1-{\nu }_s\right)}}$$

(6)

where Δk is the radius of curvature, in m⁻¹; $E_s^{\prime }$ is the corrected modulus of elasticity of the substrate, in Pa; $t_s$ is the thickness of the substrate, in m; $t_c$ is the thickness of the coating, in m; and ${\nu }_s$ is the Poisson’s coefficient of the substrate. For the experiment in this work, these parameters are summarised in

Table 6. Parameters used in the calculation of Stoney’s quenching stress equation for aluminium and Al₂O₃ materials.

Parameter	Value
Young’s modulus of aluminium, GPa	69
Young’s modulus of Al2O3, GPa	306
Poisson’s coefficient for aluminium	0.334
Poisson’s coefficient for Al2O3	0.254
Specimen width, mm	25.4
Substrate thickness, mm	4
Coating thickness, µm	35

.

The thermal stress due to CTE mismatch during cooling down to room temperature could be calculated using the Tsui and Clyne Equation (7) [43,44].

$${\sigma }_{y=0}={\displaystyle \frac{F_{CTE}}{b·h}}+E_c^{\prime }·\left(k_c-k_n\right)·\delta$$

(7)

where, σ_y=0 is the residual stress due to cooling in the joint layer, Pa; F_CTE is the force set up due to differential thermal contraction, N; δ is the neutral position to the link-layer, m; b is the sample width, m; h is the coating thickness, m; k_c is the after-cooling curvature, m⁻¹; and k_n is the after-deposition coating process curvature, m⁻¹.

An APS coating process was carried out on a flat aluminium plate (AL99 AW 1050A H14), with dimensions of 101.6 × 25.4 × 4 mm³ and Al₂O₃ as coating material (Al₂O₃ 99.7%, melted and acid washed provided by Ceram GmbH Ingenieurkeramik, Albbruck, Germany). The F6 APS torch (GTV, Germany) was used, and a six-axis robot (Type RX 130 B, Stäubli Tec-Systems GmbH, Germany) was used to describe the movement of the torch. The substrate was degreased using acetone. A coating path was applied with the parameters summarised in Table 2. The experimental setup used is shown in Figure 10. The flat aluminium plate was placed on the supports fixed with springs. A laser triangulation displacement sensor (Micro-Epsilon ILD1420-10 OptoNCDT (Micro-Epsilon Messtechnik GmbH & Co. KG, Ortenburg, Germany)) was used to measure the central deflection. An infrared pyrometer (model Optris 3MH1-CF3-CB3 (Optris GmbH, Berlin, Germany)) was used to measure the temperature of the back surface of the plate.

The arc height, A_h, and temperature were measured during the APS process, as shown in Figure 11. Two phases can be clearly identified. The first phase corresponds to the coating process and the second phase corresponds to the cooling down phase. During the coating process, a global substrate temperature increase from 304 K to 413 K was observed as the torch progressed. Additionally, a pattern of temperature peaks was shown. Each temperature peak was due to the torch approaching the measurement point during each trajectory. It was also observed that as the torch began to pass over the surface of the plate, the peaks were lower, but broader. This was because, in the approaching torch trajectories, a gas reflux passed along the edge of the plate to the back part and affected the sensor measurement. This effect disappeared when the torch passed over the plate and reappeared when the torch left the plate.

On the other hand, the arc height also followed a peak pattern. It was observed that each maximum arc height peak corresponded to the maximum temperature peak of the trajectory. In addition, a change in the tendency of the arc height was observed when the torch reached the plate. The plate expanded due to the heat effect as the torch approached the plate. As the coating layer began to form on the plate, the tendency was inverted due to the tensile stress effect generated by the coating layer on the substrate surface. The maximum relative arc height occurred in the path where the plasma column passed through the centre of the plate. The deflection variations in each trajectory were attributed to the displacement of the plasma column, the pressure effect of the gas flow on the flat plate, and the vibration generated by the impact of the particles on the substrate surface.

The thermal analysis validation of the macroscopic model was discussed in a previous paper [24]. Figure 12 compares the measured and simulated temperatures in the macroscopic model. The experimental test traced a total path of 21 trajectories to ensure complete coating on the aluminium plate. Only 15 trajectories were simulated, corresponding to the trajectories where the torch passed directly over the substrate. It was verified that the model predicted the measured temperature curve quite accurately. The difference between the experimental and simulated temperature was because the model did not consider the indirect heat flow from the plasma torch in the approach trajectories, and trajectories m₁ to m₃ and m₁₉ to m₂₁, as shown in Figure 11b. On the other hand, the temperature peaks when the torch passed over the plate were lower in intensity but longer in duration than those in the simulation. This was because the heat contributed by the gas flow parallel to the plate surface was not considered in the flame profile model.

As shown in Figure 11b, from an initial arc height equal to zero, P_a, the relative arc height at the end of the deposition phase, P_b, was −1.60 µm. The relative arc height at the end of the cooling phase, P_c, was 36.05 µm. The curvature radius was calculated with the arc height and strip support distance, L, by the equation in Figure 10b [36]. The strip support distance was 7.68 cm in this experiment. Finally, a 16.45 MPa experimental quenching stress at the end of the deposition phase was calculated using Equation (5), the curvature radius and arc height relation in Figure 10b, and the relative arc height at the end of the deposition phase. This value fits perfectly with the simulated 17.07 MPa quenching stress at 300 K in Figure 9.

5. Conclusions

This work presents a multi-scale thermal and stress transient analysis based on a one-way-coupled thermal–mechanical method and submodelling technique during the APS process. This model enables an in-depth understanding of the deposition process not only at the mesoscopic level of a total single coating layer and how the coating interacts with the substrate, but also at the microscopic level of the splat layers. The temperature and stress results from the microscopic model show the behaviour of the splat layers produced during the deposition phase and how they interact with each other, and that they are involved in the APS process optimisation, such as flame temperature, substrate preheating temperature, torch speed, coating thickness, etc. An exhaustive thermal and stress analysis from the macroscopic level to microscopic or splat level was conducted to study the APS process of Al₂O₃ coating on a flat aluminium plate with dimensions of 101.6 × 25.4 × 4 mm³, which showed the consistency of the thermal and stress results between the three scaled submodels (macro, meso, and microscopic models). A study of the quenching stress at the end of the deposition phase of the APS process with respect to the substrate preheating temperature up to 600 K was conducted at the mesoscopic level. The results of this study show that the increment in the substrate preheating temperature caused a quenching stress increment in the layer–substrate interface from 17.07 MPa at 300 K to 28.6 MPa at 600 K. In conclusion, the quenching stress showed little dependence on the substrate preheating temperature for the Al₂O₃ coating on aluminium substrate. An APS coating process was carried out on a flat aluminium plate with Al₂O₃ coating material to validate the simulation results. An in situ curvature technique was developed to obtain the experimental arc height and temperature during one 35 µm Al₂O₃ coating deposition on 4 mm thick aluminium substrate at 300 K preheating temperature. The correlation between the two experimental parameters was justified. An arc height of −1.60 µm was obtained at the end of the deposition phase of the coating process. A quenching stress of 16.45 MPa was calculated using the Stoney equation. Therefore, the simulated quenching stress was 17.07 MPa at 300 K, which showed great fitting between the experimental and simulation results.