Finite Element Modelling

In 80s, in the early time of TBC research work, the finite element analysis has been found with experimental results to develop life prediction models for the plasma-sprayed TBC. As an example, a two-dimensional, axisymmetric finite element analysis with cyclic loading and material nonlinearities has been developed at company GE [117]. This finite element analysis was carried out to examine the stresses that occur during the operation cycle showing that a considerable residual stress exists in the specimen. The plastic deformation of substrate, bond coat and the top coating creep are considered in the FE analysis. At that time, this FE Model

82 is limited to a maximum of three materials, it was not possible to directly model the substrate, bond coat, oxide scale, and topcoat with separate materials; the substrate properties were substituted for the bond coat.

The next FE model to be mentioned is that one developed by J. Cheng et al. [118] in 1998 with elastic and plastic material properties to determine the thermal/residual stress in the bond coat of an EB deposited TBC. Based on the test data from the cyclic furnace testing performed on TBC specimens and the observation of progressive failure [103], they found that purely elastic analysis failed to show some important tensile regions associated with the observed failure. The stresses computed in the elastic analysis were higher than those from the elastic/plastic calculations. The elastic calculations fail to show tensile stresses that occur on reheating; the stresses upon reheating that are missed in elastic analysis were responsible for some of the observed cracking. In addition, J. Cheng et al [118] realised that the failure is associated with interface irregularities that are not always sinusoidal, therefore, the finite element models were built with actual interface geometries from the image processing of metallographic sections, which were found in the experiments by them. Generally the actual interface geometries have higher local curvature in place of the more commonly used sinusoidal geometries, to use of actual interface geometries results in the calculation of higher local stresses. All the analyses were carried out using the finite element code Abaqus version 5.6.

In the same year, a FE model was used by A.M. Freborg et al. [119] to simulate the oxidation induced stresses in thermal barrier coatings. The finite element model was developed to evaluate stresses induced by thermal cycling of a typical plasma sprayed TBC system. The failure mechanisms of thermal barrier coatings have been examined through a finite element model of residual stress generation due to oxidation, topcoat creep and bond coat creep. The results indicate that topcoat and bond coat creep generate tensile stresses at bond coat peak and off-peak locations, while generating compressive stresses in the valley regions. In this FE model, the element “birth” and “death” techniques are utilised to replace the bond coat element with the same size of oxide bond coat elements at defined time intervals. The first oxide birth was in the elements located at the ceramic metal interface. Subsequent growth occurred through oxide elements birthed at the scale/metal interface at the start of the high temperature of the cycle. The oxide elements were allowed to ‘‘grow’’ during steady state to the full volume. In this way, the stress due to oxide growth, relaxation of growth stresses and

83 thermal cycling was incorporated in the model. Since the oxide elements were birthed at the oxide/bond coat interface, the model simulates the inward growth of an oxide scale.

The next is to introduce a 2D axisymmetric FE Model, which was built using ABAQUS to simulate an oxidation process in 2001 [120]. In this FE model, an oxidation process wherein the new TGO forms at the interface, an anisotropic growth law is used. The growth of the TGO is simulated by imposing stress-free strains in accordance with a user subroutine. It consists of two components,  and _g _t. The magnitude of the strain per cycle normal to the interface, _t (which governs the thickening), is taken to be much larger than the strain per cycle,  parallel to the interface (which causes lengthening of the TGO and induces the _g growth stress). Thickening of the TGO is modelled by adding a strain, _t in the row of TGO elements next to the bond coat. Moreover, in order to limit the growth stress to levels found experimentally, the TGO is allowed to undergo high temperature stress relaxation.

The next important contribution to the FE simulation of TBC is the work done by U.

Hermosilla et al. in 2009 [121], a coupled FE model integrating the mechanical response and microstructure evolution with considering the TGO growth, volume changes and the impact to TBC damage. A coupled microstructural–mechanical analysis was used to study the high temperature behaviour of coatings and the accumulation and concentration of stresses that may responsible for spallation upon cooling. A microstructure evolution model, a 1D diffusion model is developed to simulate phase changes under different temperature conditions using thermodynamic phase equilibrium calculation.

One of the major purposes of the work [121] is to understand the development of stresses due to the growth of the oxide layer. Initial models assumed a simple parabolic growth law for the oxide layer; the models were then developed to consider the evolving properties of the substrate and bond coat, and a more rigorous model of the oxidation process was implemented. The formation of the thermally grown oxide (TGO) is modelled by considering the volume change due to oxidation. In turn, the model predicts the evolution of stresses at the positions within the TGO layer. The TGO growth was modelled by applying swelling strain rates to the material that composes the initial oxide layer.

Similar FE model is built by U. Hermosilla et al. in 2013 [122] to simulate microstructure phase transformation and material degradation of TBC material system. A one-dimensional finite element diffusion model is developed to simulate diffusion of elements between the

84 substrate material and the bond coat in TBC. The TGO growth is simulated within the model and coupled with diffusion of the oxide-forming element. Microstructure changes and material degradation under different temperatures with varied bond coat compositions are simulated using this FE model. The results show that the accumulation of high out-of-plane tensile stress within the alumina layer as an additional phenomenon that could drive high temperature crack nucleation.

Based on the studies presented above, it is seen that the finite element method is capable to predict some aspects of TBC failures. It is the work in the future to extend the analytical model developed in this chapter to a complete TBC life prediction model with the supporting of numerical and experimental method.