Analytical Models

In service, residual stresses gradually generate and accumulate in the TBC due to the mismatch of material properties, which may eventually cause interface crack propagation and coating spallation (310). Thus, the evolution of residual stress plays an important role in predicting the life of TBCs. In the past decade, many engineering methods have been developed to evaluate the evolution of residual stress in different types of TBCs, such as X-ray diffraction (311), Raman spectroscopy(312), substrate removal (313), curvature measurement (314),

photoluminescence piezo-spectroscopy (PLPS) (315) and indentation methods (316).

At the same time, researchers have proposed the different forms of analytical solutions to evaluate the stress distribution of multilayer systems.

The simplest analytical model involves assuming that the TBC system is only composed of the ceramic coating and the metallic substrate. The coating is considered to be significantly thinner than the underlying substrate and all interfaces are flat, so the normal stress to the coating surface is zero. At the beginning of the cooling stage, the entire TBC system is stress-free due to creep relaxation. These assumptions lead to the coating being under biaxial stress 𝜎_B. In the elastic regime, the magnitude of the stress after cooling is proportional to the temperature drop Δ𝑇 (317):

𝜎_B =𝐸_TBC𝛼_TBC 1 − 𝜈_TBC Δ𝑇

Equation 41

𝐸TBC: elastic modulus of the coating

𝛼TBC: thermal expansion coefficient of the coating 𝜈TBC: Poisson’s ratio of the coating

For the case of a plasma-sprayed zirconia, the calculated stress in the coating associated with cooling from 1150 ºC to room temperature is:

𝐸TBC= 50 GPa

𝛼_TBC= 10 × 10⁻⁶ C^{o −1} 𝜈TBC= 0.25

𝜎_B = −750 MPa

On cooling from high temperature, the continued contraction of the underlying metallic substrate puts the coating under a significant amount of compression. In the areas of the system with cracks at the interface between the coating and the substrate, the residual compressive stresses will lead to buckling delamination and spalling of the coating (318), which may result in the associated problems of

structural integrity and stability, reduction in load-bearing capacity, stiffness degradation and global fracture of TBCs.

The prediction of TBC failure can be based on a simplified analytical relationship that correlates TBC delamination with TGO development. The TBC failure model starts with the assumption that delamination occurs when the critical out-of-plane residual stress intensity (𝐾res) is exceeded at the TGO/bond coat interface.

In the absence of additionally applied out-of-plane stresses only the thermal mismatch and oxidation induced stresses contribute to the stress intensity. The modeling approach, therefore, reduces to an evaluation of the residual out-of-plane stress (𝜎_resout−of−plane).

The out-of-plane stress of the multilayer thermal barrier system is a function of different thermo-mechanical properties and geometrical features (e.g. thermal expansion coefficients, Young’s moduli, thickness of the layers, maximum temperature) (319):

𝜎_resout−of−plane= 𝑓(𝑑_i, 𝐸_i, 𝛼_i, 𝑇_max)

Equation 42

𝐸_i: elastic modulus

𝛼_i: thermal expansion coefficient 𝑑_i: thickness

𝑇_max: maximum temperature

𝑖: refers to the layers of the TBC system

The thickness of the thermally grown oxide 𝑑TGO changes with time. In the above relationship the parameters which do not significantly change during the oxidation tests can be joined to a constant parameter 𝐴 to obtain a simplified equation

leaving only the TGO thickness as variable. Considering the normal residual stress for a convex asperity at the TGO/bond coat interface increases linearly with TGO thickness after an initial non-linear increase for small TGO thicknesses (320), the residual out-of-plane stress can be expressed as:

𝜎_resout−of−plane= 𝐴 ∙ 𝑑_TGO

Equation 43

The growth of the TGO thickness follows the parabolic law:

𝑑_TGO= 𝑘 ∙ 𝑡ⁿ= 𝐶 ∙ exp �− 𝑄 𝑅𝑇� ∙ 𝑡ⁿ

Equation 44

𝑡: oxidation time

𝑘: oxidation rate constant 𝑛: oxidation rate exponent

𝑄: activation energy of the TGO growth 𝑅: universal gas constant

𝑇: temperature 𝐶: pre-factor

Yielding for the critical out-of-plane stress:

𝜎_cout−of−plane= 𝐴 ∙ 𝐶 ∙ exp �− 𝑄 𝑅𝑇� ∙ 𝑡ⁿ

Equation 45

Logarithmic notation and re-arrangement with respect to the accumulated oxidation time yields:

ln(𝑡) = 𝑄 𝑛 ∙ 𝑅 ∙

1 𝑇 +

1

𝑛 �ln �𝜎^cout−of−plane� − ln(𝐶) − ln(𝐴)�

Equation 46

In case of TBC failure the parameter 𝑡 provides a direct measure of the lifetime, as this model is based on the critical out-of-plane stress concept, which states that lifetime is terminated when a critical stress is reached. In graphical

representation, the calculated lifetime of the TBC can be plotted as a function of inverse temperature (Figure 94). The slope in the ln(𝑡) − 1 𝑇⁄ diagram is 𝑄 𝑛 ∙ 𝑅⁄ , if the second term in the above equation is constant. It can be seen that the lower the oxidation temperature the longer is the expected lifetime.

Figure 94 - Lifetime of TBCs as a function of temperature

The presented lifetime analytical model does not consider the slow crack growth at room temperature, which constitutes its biggest limitation. Nevertheless, it

provides a good basis for the understanding the effect of temperature on delamination failure.

When spalling is deemed to occur by edge delamination at the TGO/BC

interface (as for smooth bond coat surfaces), the process is governed by the steady-state energy release rate. The Mumm, Evans and Spitsberg model (290) is based on linear-elastic fracture mechanics and shows that for a TGO/TBC bilayer coating, the critical steady-state energy release rate before decohesion from the substrate is given by:

𝐺ss=(𝐸1′ℎ₁(1 + 𝜐1)𝜀1r+ 𝐸₂^′ℎ₂(1 + 𝜐2)𝜀2r)²

2(𝐸₁^′ℎ₁+ 𝐸₂^′ℎ₂) −𝑀Δ𝛫 2

The quantity M is the net moment acting on the bilayer coating, given by:

𝑀 =𝐸1′𝐸2′ℎ1ℎ2(ℎ1+ ℎ2)[(1 + 𝜐2)𝜀2r− (1 + 𝜐1)𝜀1r] 2(𝐸₁^′ℎ₁+ 𝐸₂^′ℎ₂)

while Δ𝛫 is the net curvature change of the bilayer upon decohesion:

Δ𝛫 =6[(1 + 𝜐₂)𝜀_2r− (1 + 𝜐₁)𝜀_1r]

ℎ1ℎ2 �ℎ₁+ ℎ₂ ξ �

where the non-dimensional function ξ is given by:

ξ =𝐸₁^′

The subscripts 1 and 2 refer to the TGO and TBC layers, respectively. The residual strain is dominated by the thermal expansion mismatch, Δ𝛼, between the coating and substrate and by the temperature drop from the peak temperature, Δ𝑇. The critical TGO thickness can be calculated by equating 𝐺ss to the mode II interfacial toughness. If this toughness is assumed to be 80 J/m² (321) and

𝜐₁= 0.2

the relationship between the critical thickness and TBC elastic modulus can be plotted, Figure 95.

Figure 95 - Predicted TGO thickness for TBC buckling-delamination failure as a function of (a) YSZ Young's modulus (b) TBC thickness (c) interfacial toughness

The result in Figure 95 indicates that the critical TGO thickness (and therefore the spallation lifetime) has a maximum value of ~6 µm at a ceramic coating modulus of

~25 GPa. A large in-plane elastic modulus is responsible for higher misfit strains between the TGO and the TBC, whereas small coating modulus mean the residual strains in the TGO/TBC bilayer induced by the substrate are essentially reflected in the TGO layer.

The analysis also highlights the importance of the top coat thickness and interface toughness. It should be noted that even modest increases of toughness (from 80 to 90 J/m²) are predicted to increase the critical thickness of a TBC with a 30 GPa Young’s modulus by about 1 µm and therefore increase the lifetime of the coating.

While changes to the coating thickness are predicted to have little effect upon life, they do affect the optimum elastic modulus for maximum life, where thicker coatings require the use of a lower coating modulus. This result means that larger in-plane modulus or greater coating thickness increase the energy release rate G and thus reduce the durability.

The analytical studies, however, have some limitations which can result in erroneous results and may lead to incorrect conclusions regarding the stress

distribution in TBCs after prolonged thermal cycling exposure. Some of these limitations are outlined below:

1. The coating is considered homogenous and the effect of the columnar microstructure in EB-PVD systems is not taken into consideration. The formation of these columns significantly alters the elastic modulus of the coating, causing an anisotropic behaviour;

2. The effect of sintering is neglected. Sintering is a temperature dependent process that increases the thermal conductivity of the coating, redistributes stresses and more importantly increases the elastic modulus of the coating.

Hence, the elastic modulus of the coating will increase during high temperature service.

3. Close to the end of life, cracks are observed in the coating microstructure.

The formation and propagation of these cracks has two major consequences:

a. Cracking releases some of the stress in the coating;

b. Cracking also alters the elastic modulus of the coating, making this property dependent on the location of the cracks within the ceramic top coat.

The competition detailed above between elastic modulus decrease due to cracking and increase due to sintering is rather complex and beyond the scope of typical analytical analysis.

4.3. Finite Element Model

Analytical simulations are generally conducted for the case of thermo-elastic deformation. However, in practical situation the inelastic deformations such as plastic and creep deformations occur, which can produce different stress states in TBCs, possibly leading to other critical conditions of failure of TBCs. The details of the effect of the inelastic deformations on thermal stress states in TBCs can be numerically investigated. In addition, the method can include other phenomena responsible for time-dependent failure of TBCs, like growth of the TGO, sintering, temperature-dependent material properties and their evolution with time.

A new thermo-mechanical model has been formulated using the commercial finite element code Abaqus to investigate the residual stresses arising during service of the thermal barrier coatings system. The numerical simulations take into account all the parameters that crucially affect the stress state of the coatings during service, such as thermal mismatch and elastic-viscoplastic behaviour of the material constituents, oxidation of the bond coat, complex shape of the TBC/TGO/BC interface and redistribution of stresses via creep.