Thermally grown oxide (TGO) is a critical factor for the service life of thermal barrier coatings (TBC). Numerical simulations of the growth process of TGO have become an effective means of comprehensively understanding the progressive damage of the TBC system. At present, technologies of numerical simulation to TGO growth include two categories: coupled chemical-mechanical methods and mechanical equivalent methods. The former is based on the diffusion analysis of oxidizing elements, which can describe the influence of bond coat (BC) consumption and phase transformation in the growth process of TGO on the mechanical behavior of each layer of TBC, and has high accuracy for the thickness evolution of TGO, but they cannot describe the lateral growth of TGO and the rumpling phenomenon induced. The latter focuses on describing the final stress and strain state after the growth of a specific TGO rather than the complete growth processes of TGO. Based on the measured TGO thickness growth curve, simulations of thickening and lateral growth can be achieved by directly applying anisotropic volumetric strain to oxidized elements and switching elements properties from the BC to the TGO.

Thermal Barrier Coating (TBC) is effective thermal protection widely used in the gas turbine to protect air-cooled blades or vanes exposed to high temperatures environment [

A typical TBC system consists of a ceramic topcoat (TC) with low thermal conductivity and an Al-rich and oxidation-resistant metallic bond coat (BC) [_{2}O_{3}, and its thickness is much small, generally less than 10

TGO plays the role of oxygen barrier due to very dense that is difficult for oxygen to penetrate further, preventing the substrate from being oxidized and corroded. However, the coefficient of thermal expansion (CTE) of TGO is less than that of others, which is detrimental to the internal mechanical compatibility of the TBC system [_{2}O_{3} at the ceramic-metal interface is one of the dominant damage mechanisms that affect the life of the TBC system, especially for aero engines and industrial gas turbines [

The growth of TGO is essentially a complex dynamic process coupled with chemistry and mechanics, generally including internal oxidation, grain boundary oxidation, and external oxidation. Internal oxidation is that oxygen anions diffuse inward through the TC and react with Al cations from the BC at the TGO-BC interface. Grain boundary oxidation is that Al cations react with oxygen at the grain boundaries within the TGO, and external oxidation is that Al cations in TGO subsequently diffuse outward and react with oxygen anions at the TC-TGO interface [

In 2001, Busso et al. [

For the BC made of MCrAlY material, the main oxidation reaction is the internal oxidation reaction in

Thus, the average volumetric strain caused by the internal oxidation is

Once the material at some point starts to be oxidized, the total expansion strain of TGO inward growth is controlled by the volume fraction of Al_{2}O_{3} generated by the main oxidation reaction. Suppose BC is composed of an oxidation-resistant phase (Phase 1) and an oxidation-prone phase (Phase 2). With the start of oxidation, a very thin layer of internal oxidation zone (IOZ) is formed at the ceramic-metal interface, where an internal growth oxide co-exists with Phase 1 and Phase 2. The IOZ is a thin transition zone between BC and TGO that moves inward as the oxidation progresses. Due to the inter-diffusion process driven by the concentration gradient from one side of the IOZ to the other, the local oxide volume fraction is increased from approximately

Based on the known volumetric fractions of each material phase before, during and after oxidation, a constitutive framework based on a self-consistent scheme was established to describe the mechanical behavior of an equivalent homogeneous solid in the IOZ. The inelastic deformation rate tensor can be decomposed into the average inelastic deformation rate term caused by slip and the irreversible phase transformation deformation rate term characterizing the metal phase oxidation, assuming that which is controlled by the rate of change of the internal oxidation volume fraction. If the local oxygen concentration of each material point is greater than or equal to the critical value, then this point starts to be oxidized.

For the external oxidation, to improve the calculation efficiency, it is assumed that the oxide is generated on the original TGO instantaneously and its constitutive behavior is thermo-elastic, and that the volumetric strain related to the external oxidation is zero in the plane of the TGO-ceramic interface and is maximum in the out-of-plane direction. In fact, Al cations will also react with oxygen at the internal interface of TGO, causing TGO to grow laterally at a low local defect density. However, Busso's model did not consider this situation to simplify the calculation.

The oxidation kinetics equation for the growth of the TGO thickness under isothermal oxidation was established based on the Arrhenius's equation [

Furthermore, Busso et al. [

Then the average volumetric strain associated with the primary oxidation reaction can be obtained,

In Busso's research,

Hille et al. [_{2}O_{3}, and

The model describes the diffusion of oxygen to Al in BC, and the sink term on the right side of the equation relates the consumption of free oxygen to the formation of TGO, i.e., Al_{2}O_{3}. Both the reaction rate and the growth kinetics depend on the availability of reactants. By reasonably selecting the oxidation rate parameters, it can be ensured the calculated TGO morphology evolution can be consistent with the experimental observation results. In essential, this model considers that there is an oxidation zone, that is, a thin conversion layer (mixed TGO and BC) where the volumetric fraction of the oxide is between the TGO volume fraction from

Assuming that the initial TGO thickness is

In fact, the predicted results by the above models reflect the average thicknesses of the growth of the TGO. Shen et al. [

However, experimental observations show that there is a clear boundary between the BC and the TGO, which is inconsistent with the assumption in the coupled diffusion-constitutive framework that in the IOZ or BC-TGO mixed zone the initial phase of the alloy co-exists with the oxide. To handle this problem, Caliez et al. [

To calculate the oxidation kinetic, global diffusivities have to be fitted by an inverse method, considering the stoichiometric mass concentration in the critical chemical activity assessment. The experimental observation results show that the anion growth mechanism is dominant, and the mesh size has no effect on the results of computed kinetic if the thickness of TGO grows at least one row within a time increment step.

It should be noted that the inverse method or iterative process is required to obtain some parameters to maintain consistency with the experimental results when the local shape of the TGO is established using a sine wave to assess the growth stress in the above methods. Gupta et al. [

• TGO growth is occurring only during the dwell phase of the thermal cycle test;

• BC is mainly composed of aluminum, TGO is pure Al_{2}O_{3}, and other oxides are neglected;

• The rate of Al_{2}O_{3} formation is much higher than the rate of the diffusion of the Al and oxygen;

• The consumption of Al in BC cannot be considered during exposure;

• Only the inward growth of TGO is considered in the model;

• The oxygen diffusion rate in TC is very high because ZrO_{2} is transparent to oxygen flow.

Based on these assumptions, a TGO growth model based on diffusion is established using the computational fluid dynamics (CFD) method in ANSYS Fluent. Two solid domains are defined in the TBC system, namely a stationary gas area containing oxygen (representing TC) and a solid aluminum region (representing BC). The system includes two initial solid scalar quantities, namely oxygen and aluminum diffused in the solid region, and the third solid scalar quantity is Al_{2}O_{3} formed during the simulation. Each of these scalar concentrations _{2}O_{3} formation.

Assuming that the element concentration threshold value in BC is _{2}O_{3} in an element is greater than _{2}O_{3}. The choice of this value is mainly to achieve a better convergence, and the parameter sensitivity analysis shows that this value does not affect the final result. The aluminum diffusion coefficient in the BC is based on the composition of the BC and calculated using the commercial software DICTRA. The initial concentrations of aluminum and oxygen scalar are set to 1 and 0 in the BC, and 0 and 1 in the TC, respectively. The profiles of TGO indicated by solid lines based on the growth model at different stages are plotted over the microstructure image of a TBC sample at failure, as shown in

Hermosilla et al. [_{2}O_{3}, once a given element is oxidized, the alloy phase will instantaneously change to _{2}O_{3} and volumetric strain is simultaneously applied.

The PBRs of

The growth equation of TGO thickness is as follows:

The coupled microstructure-mechanics analysis process is shown in _{2}O_{3} in TGO. The field variable file is read by the ABAQUS solver and called in the user subroutine UMAT defining the mechanical behavior of BC and TGO.

The material model of each layer is defined as follows:

The substrate is IN-713LC, which is defined via temperature-related tables of constitutive properties in the ABAQUS input file. Although the microstructure evolution simulation of the substrate-coating interface is carried out in the 1D finite-difference model, the influence of the local interdiffusion between the BC and the substrate on the performance of the substrate is ignored in the FE model.

BC and TGO are treated as a single region in the FE model whose performances are described by the UMAT subroutine as a self-consistent model. The constitutive model simulates the performance evolving of BC with the microstructure changes, or a single-phase Al_{2}O_{3} representing the TGO. This method can describe the moving BC/TGO interface when the performance of elements changes from the multiphase BC material to the TGO. For a given BC element, oxidation occurs instantaneously, and the BC and TGO phases do not exist at the same time in an element. TGO growth simulation is performed by changing the BC element adjacent to the TGO to pure Al_{2}O_{3} and applying volume expansion based on the PBR of the oxidation reaction.

The properties of the TC are defined via temperature-related tables in the ABAQUS input file, like the substrate.

Based on a similar method, Kyaw et al. [

In 2017, Kyaw et al. [

To realize the transformation of the one-dimensional phase proportion data into the three-dimensional FE model, BC is modeled with a multi-layer structure, as shown in

Totally, coupled chemical-mechanical methods, including the coupled oxidation-constitutive framework and the coupled microstructure-mechanical framework, can thoroughly describe the TGO thickening process, but the influence of the lateral growth of TGO forming on the internal grain boundaries cannot be considered. The obtained interface roughness basically does not change with thermal cycles, so these methods cannot simulate the TGO rumpling [

Numerical analysis and experiments have confirmed that the BC surface morphology changes during cyclic oxidation, driven by the mismatch strain caused by the thermal expansion misfit between the TGO and the BC during cooling and by the strain associated with the oxide growth at the high temperature. To deal with the above-mentioned situation, Karlsson et al. [

Four-node axisymmetric bilinear elements were used for the large deformation analysis of BC and TGO in ABAQUS. The in-plane strain component,

The morphological changes of BC and TGO with

Rosler et al. [

Based on a method similar to that of Karlsson et al. [

Similarly, considering the thickening and lateral growth of TGO at the same time, Seiler et al. [

Ebrahimi et al. [

The FE model for simulating TGO growth under the thermal fatigue load in ABAQUS is shown in

Jiang et al. [

The thickening of TGO is simulated in ABAQUS, but the influence of the lateral growth of TGO is not considered. The FE model is shown in

Cen et al. [

The oxidation time is from

Then, the oxidation strain is as follows:

Therefore, the volumetric strain rate of the TGO can be calculated. It should be noted that the growth of the TGO is anisotropic, and the strain rate in the thickness direction is about 10 times that in the lateral direction, which is consistent with the previous assumption [

Nayebashaee et al. [

Ranjbar-Far et al. [

A FE model with the quadratic six-node triangular elements with generalized plane strain approximation and reduced integration is used for analysis. The results of stress analysis for the sine shape of TGO are shown in

It can be seen from the above results of FE models that elements volumetric swelling is only suitable for the small deformation of the thickness of TGO. These methods do not involve the BC consumption process and simplify the TGO growth as the anisotropic swelling of the volume of oxidized elements. The equivalent thickness direction strain rate is calculated through the measured TGO thickness growth equation. Assuming the lateral growth strain rate is proportional to the strain rate in the thickness direction, the thickening and lateral growth of the TGO can be achieved.

Based on Karlsson's model, He et al. [

Due to that

Jinnestrand et al. [

The growth of TGO is caused by the diffusion of anions or cations. Above

Shen et al. [_{2}O_{3}, it produces isotropic volume swelling, and PBR is 1.28. Therefore, TGO is compressed by

Busso et al. [

Aiming at the chemical failure caused by the growth of non-protective oxides such as NiCr_{2}O_{4}, Busso et al. [

In fact, the protective oxide is still produced in the TGO valley region during the breakaway oxidation. To speed up the computation process, an equation similar to the wave peak can be used to describe wave valleys as follows:

The TGO thickness in the zone between apexes and valleys can be determined by the linear interpolation. Here, the PBR of protective _{2}O_{3} (primary TGO) is _{2}O_{4} (secondary TGO) is

Wei et al. [

The TGO thickness increment for each thermal cycle can be calculated according to

The TGO lateral strain rate

It can be seen that methods of element conversion transform properties of elements in the BC to Al_{2}O_{3} to simulate the TGO growth, which replaces the complex diffusion model with the measured TGO thickness growth curve or the distance between the material point and the TGO interface as the oxidation criterion of BC elements. Unlike methods of element volumetric swelling, which is only suitable for small changes in the thickness of TGO, the larger-scale thickening of TGO can be achieved by converting the material properties of oxidized elements and simultaneously applying the anisotropic volume expansion strain. However, the achievement of the lateral growth of TGO is as same as that in methods of element volumetric swelling, that is, the lateral growth of TGO is modeled by imposing the lateral volumetric swelling to the whole TGO layer.

The chemical-mechanical coupling method can directly or indirectly describe the element diffusion, BC consumption, and TGO thickening during the growth of TGO. The oxidation-constitutive coupling framework integrated the diffusion equation into the material constitutive model to achieve the dynamic simulation of the growth process of TGO and the consumption of oxidation in the BC, while the microstructure-mechanics coupling framework can consider the influence of microstructure degradation caused by BC oxidation and phase transformation on the mechanical behavior of materials at high temperatures by treating the BC as a mixture. However, the chemical-mechanical coupling methods are all aimed at the TGO thickening process and do not consider the influence of the lateral growth of TGO caused by grain boundary oxidation. The interface roughness obtained basically cannot evolve with the number of thermal cycles, and cannot simulate the TGO wrinkling and the stress redistribution phenomenon occurring in the thermal fatigue test.

The mechanical equivalent method cannot describe the complete process of TGO growth, but focuses on the final stress and strain state after the specific TGO growth. The volumetric swelling of TGO elements is to treat TGO growth equivalently as the anisotropic volume expansion of TGO elements, in which the equivalent thickness direction strain rate is calculated through the measured TGO thickness growth oxidation kinetic equation, and assume the transverse growth strain rate is proportional to the strain rate in the thickness direction, and then the description of TGO thickening and lateral growth is achieved. This method is only suitable for the case where the thickness of TGO does not change much. Another equivalent method is to realize the TGO growth simulation through elements conversion in the BC, replace the complex diffusion model with the measured TGO thickness growth curve and the distance between the material point and the TGO interface as the oxidation criterion of the BC element, and then convert the material properties of the oxidation element and apply the anisotropic volume expansion strain to simulate the large-scale thickening of TGO. The simulation of the lateral growth of TGO is realized by applying lateral volumetric strain to the entire TGO layer.

Based on the current research progress, the subsequent numerical simulation of TBC should focus on the following issues. Firstly, some numerical methods, such as adaptive meshing or meshfree, should be introduced to solve the moving interface of TGO growth and the large-deformation question due to TGO rumpling in FE models. Secondly, the effects of structural features in turbine blades or vanes on the evolution of TGO morphology and stress distribution should be further studied, such as the curvature of substrates, film holes, edge effects, etc. Thirdly, advanced microscopic measurement techniques should be developed to obtain accurate mechanical/chemical properties of each layer, considering the influence of BC phase transformation and TC sintering. Lastly, the correlation between the progressive damage of TBC and some factors, including the TGO growth, the interface morphology, and the mechanical behavior of each layer, should be established to identify the driving force of damage and provide support for the lifetime prediction of TBC system.

The authors wish to express their appreciation to the reviewers for their helpful suggestions which greatly improved the presentation of this paper.

_{2}O

_{3}-ZrO

_{2}thermal barrier coating after in-phase thermo-mechanical test

^{3+}piezospectroscopy

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_{3}scales during the oxidation of Fe-22Cr-4.8Al-0.3Y alloy