## Mechanisms Governing The Distortion of Aluminaforming Alloys Upon Cyclic Oxidation

1. Synopsis

The problem is addressed by adapting an approach developed for understanding TGO displacement instabilities. The following material properties are incorporated. The FeCrAlY is assigned a yield strength that varies with temperature. It is constant, °r , between temperature T1 and the maximum, Tmax. Below T1 the strength increases rapidly with decreasing temperature. The TGO is allowed tgo E

to yield at the peak temperature, with yield strength, rY , but otherwise is elastic with modulus, tgo. It has a thermal expansion coefficient lower than the FeCrAlY, with misfit, D a .

The strains generated by the TGO are basic. One component is caused by thermal expansion misfit and the other by TGO growth. The growth strains require some explanation. An element of the FeCrAlY changes composition with a net diminution of Al. This Al reacts with ingressing O to form A2O

3 with an associated increase in volume. The new TGO forms primarily on the interface. Since this interface is incoherent, on a planar segment, the volume increase (thickening) is accommodated by an upward, rigid body displacement of the TGO, obviating the development of stress. However, some of the new TGO resides on the internal grain boundaries causing it to elongate when the constraint from the bond coat is relaxed. This strain induces in-plane compression and causes the observed curvature changes.

Based on prior understanding of TGO induced displacements, acquired through an analytic model with spherical symmetry, the following assessment provides a framework for performing simulations and analysis. To obtain a permanent curvature, the convex TGO layer must exhibit plastic elongation and therefore, must yield at the temperature maximum, during the growth step. Correspondingly, the FeCrAlY must exhibit plastic bending and elongation, requiring that through-thickness yielding occur upon cooling, motivated by growth and thermal expansion misfit. Accordingly, the fundamental requirement for cyclic curvature is that both the bond coat and the TGO must experience yielding during at least one of the thermal stages within every cycle. A consequence is that curvature change occurs only within a defined window of properties and thickness. For the TGO to yield at the temperature maximum, the substrate thickness, H, must be thick enough to satisfy:

where H is the thickness of the FeCrAlY, h1 the thickness of the convex TGO and h2 the thickness of the concave TGO. For the bond coat to yield on cooling, its strength must satisfy,

where DT = Tmax - T1. To realize large-scale curvature changes, both inequalities must be satisfied simultaneously. When only one is satisfied (or neither), the system can elongate, but the curvature change will be small.

### 2. Numerical Results

Finite element simulations have been performed using the ABAQUS code. The approach has been described elsewhere. The first variable to explore has been the elongation strain. Initially, to differentiate the roles of this strain and the thickening, any relationship between them has been neglected. Instead, the TGO thickness is held constant and the elongation strain, Ae , is introduced in each cycle. It is a parameter in the simulations. The calculations are performed for different TGO thickness. Once the trends have been identified, a full simulation is performed in accordance with parabolic thickening and corresponding elongation. Coincidence with the measurements will establish the magnitude of Ae relative to the thickening.

Preliminary calculations probe the property ranges that result in curvature changes, with insights from (1), starting with the final TGO thickness found experimentally (hl = 2.8jm, h2 = 1.8 jm, H = 700 jm ). The yield strength range for the TGO is assessed from growth stress measurements: 500 MPa < dg < 2 GPa . Then, (1) is used to establish the range of bond coat yield strengths that might be expected to produce curvature change. This range is: 5 MPa < <dyc < 20 MPa . Within these property ranges, preliminary results for the stresses affirm that the response is fully-elastic at temperatures below Ti (T < 750C), enabling further calculations to be confined to higher temperatures, between T and Tmax. In this range, the bond coat responds as follows. Upon initial cooling, the displacements are elastic. It then reaches yield and the stress remains essentially constant down to T1. Reheating beyond T1 elastically unloads the alloy and induces compression. In some cases, reverse yielding occurs. The stress in the TGO layers is always compressive. At the start of a cycle, typically it is below the yield strength. Imposing the elongation strain during growth increases the compression until the yield strength is reached. Thereafter, the stress remains constant and there is no further elongation, only thickening. The initial stress is much closer to yield on the concave than the convex TGO layer, allowing a larger component of elongation strain. This difference is the basis for the permanent change in curvature.

Calculations of the curvature made for a wide property range affirm that, for the chosen thickness, large scale curvature changes occur only when the TGO yield strength is in the range, 1. 5 GPa > df > 0.7 GPa with an alloy having strength in the range, 5 to 10MPa.

### 3. Analytical Approximation

An analytical solution can be obtained by using the insights gained from the numerical results, particularly the responses that apply when (1) is satisfied. After the first few cycles, the TGO on the concave side acquires a stress at the peak temperature about equal to its yield strength. There are two main consequences. (a) The stress in this layer cycles in a linear elastic manner. (b) At the peak temperature, the layer yields immediately upon application of the growth strain, such that it thickens without elongation. Conversely, on the convex side, the TGO is below yield upon reheating to Tmax, allowing some elongation strain to be added during the growth step. On cooling, the bond coat yields throughout and attains a state of uniform tension equal to its high temperature yield strength at T1. On reheating, there is no reverse yielding. The system is elastic below T1.

Since the concave TGO is at its yield strength at Tmax and otherwise, behaves in an elastic manner, then at instant A in the reheat cycle (coincident with T1), the stress in this layer is:

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