The global FE model considered is representative of a statoric segment (made of 6 airfoils) of low pressure turbine stage, modelled considering a typical turbine blade superalloy, with mechanical and thermal isotropic material properties whose variations vs. temperature are shown in Fig. (1) and Table (1).

Fig. (1). Material properties vs. temperature for the considered superalloy: (a) Young’s modulus, (b) Expansion coefficient; (c) Poisson’s ratio.

Fluid pressure was modeled as a mechanical load applied on both sides of the airfoils, together with a temperature scenario previously calculated by thermo-fluid-dynamic analyses. In addition, cyclic symmetry boundary conditions Fig. (2a) were enforced on a casing that couples with a statoric segment, in order to simulate the periodicity of the whole stage. Surface to surface contacts were applied on the interfaces between statoric hooks and casing. Thermal and mechanical loads applied on the global model are representative of the most severe conditions, reached at take-off, for an aircraft engine during its mission. As a matter of fact, at take-off there is the need for the maximum boost of engines and consequently vane temperatures Fig. (2b) get the highest magnitudes and the highest gradients, with consequent enhancement of thermal stresses.

Fig. (2). (a) Cyclic symmetry FEM boundary conditions; (b) FEM thermal scenario; (c) FEM max principal stresses.

Due to a combination of mechanical and thermal cyclic stresses, fatigue cracks can nucleate, most likely from those locations with highest stresses localized in-between airfoils and casing (Fig. 2c).

A simplified thermo-mechanical fatigue load cycle was extracted from the mission profile Fig. (3), with its maximum value corresponding to the take-off phase and its minimum value to the engine shut off (zero load).

Fig. (3). Engine mission profile.

Mechanical, thermal and fatigue properties for the considered superalloy were evaluated at the average temperature of sub-model. In particular the calibration of the Paris’ law (Eq. 1), adopted to calculate crack growth rates, was performed at such average temperature.

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The DBEM submodel cannot allow for the spatial variability of the material properties, caused by temperature gradients and, consequently, uniform material properties were used, as evaluated in correspondence of an average temperature. Nevertheless, for the analyzed problem the impact of such approximation on the final results (e.g. Stress Intensity Factors (SIFs) along the crack front) turns out to be negligible, due to the limited temperature variation in a very small DBEM submodel.

SIFs are calculated using the J-integral approach [25-27] and the crack path assessment is based on the Minimum Strain Energy Density (MSED) criterion [28] (actually, the comparison between outcomes of different crack path criteria was not provided because the crack propagation evolves under nearly pure mode I).

DBEM Modelling Considering “LC” and “DC” Approaches

Two different methodologies were considered in order to calculate a SIF distribution along crack front:

• Crack surface loading problem (hereafter named “LC”): tractions, evaluated along the virtual surface traced by the advancing crack by means of a FEM analysis of the uncracked component, were applied on the crack faces in the DBEM submodel. In particular, the DBEM evolving crack geometry “filters”, at each step of crack propagation, the stresses to be extracted from the solution of the FE uncracked model; then such stresses are converted in tractions to be applied on the crack faces (Fig. 4a).
• Displacement Control (DC) approach: displacement and temperature scenarios, calculated by an FEM analysis of the global uncracked component, were imported into DBEM environment and applied on cut surfaces; in addition, FE temperatures were applied on all the remaining DBEM elements Fig. (4b). Fluid pressure on airfoil was not modelled because causing negligible effects, but no difficulties would arise in adding such pressure too. Moreover, no volume forces are present in the submodel domain (it is part of a vane so with no centrifugal loads).

Both DC and LC methodologies eventually provide SIF distributions along the crack front but following two different approaches: the former solves a thermal-stress analysis; the latter solely need a pure stress analysis. LC approach thus avoids the need for a combined thermal-stress problem and can leverage on a coarser convergent mesh with consequent run time reduction. On the other hand, LC approach can provide a realistic stress scenario just in an infinitesimal area surrounding the crack tip, so cannot be adopted when an overall knowledge of stress distribution is needed.

Fig. (4). Considered loading strategies for DBEM analyses: (a) LC and (b) DC.

Fig. (5). Superposition principle applied to a thermal-stress crack problem.

In particular, a superposition principle is implemented through the following steps (Fig 5):

• starting from an original uncracked model (a), a crack can be opened (b) and then closed again by the distribution of tractions calculated over the dashed line of the virtual crack in (a);
• the new configuration (b), perfectly equivalent to the previous one (a), can be solved using a superposition principle by splitting the boundary conditions as illustrated in (c) and (d): (c) represents the original problem that we need to solve whereas, the problem (d) that is identical to (e) but for the sign of crack tractions, represents an alternative equivalent problem that will be solved, gaining substantial improvements in terms of accuracy and computational burden. As a matter of fact, using boundary conditions coming from the considered thermal-stress uncracked problem, a pure stress crack problem (e) can be solved, in which crack faces are subjected to tractions equal in magnitude but opposite in sign to those calculated over the dashed line in (a).

In conclusion, using the superposition principle it is possible to state that SIFs for case (c) are equal to those calculated for case (e):

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Using the LC approach, an uncracked FEM model is solved to accurately calculate the stress field in the surroundings the cracked zone, whereas, the fracture problem is entirely demanded to the DBEM analysis. All the loads applied to the real component (e.g. thermal, electro-magnetic, dead weight, bolt preloadings) are included in the FEM analysis, whereas, the SIFs evaluation and subsequently the whole crack-growth is worked out into the DBEM environment by means of a step-by-step pure stress analyses.

Fig. (6). DBEM submodel sizes: mesh before crack introduction and highlight of crack site.

A further benefit of such LC approach is given by the possibility to simulate crack propagations with a considerably smaller model than needed by DC approach, as enabled by the self-equilibrated nature of the load applied with LC and its rapidly vanishing effects when getting far apart from the crack. Such reduction of the DBEM domain will strongly speed up the analyses.

As previously mentioned, two submodels with different sizes were extracted by a sphere cutting and then solved. The aim was to assess the minimum required submodel size when using DC approach, “compatible” with boundary conditions that were not updated during crack growth.

In more details, submodel boundary conditions for both DC and LC approaches come from the FEM analysis of the uncracked global model but, for the DC approach this represents an element of approximation because the displacement boundary conditions applied on the cutting surfaces should, in principle, be updated at each step of crack advance, so as to allow for the submodel stiffness variation when solving the global model. On the contrary, in LC approach the self-equilibrated tractions applied on crack faces are correctly extracted from an uncracked body solution, as rigorously dictated by the superposition principle implementation and, during crack propagation, are updated step by step because new crack surface is continuously created and loaded. Consequently, for LC approach there is no need to guarantee a submodel much larger than crack extension, being sufficient to enclose that part of the volume surrounding the crack where stresses are non negligible; moreover it is worth to point out that the stress “fading distance” is very short when a self-equilibrated load is applied on the crack.

Fig. (7). (a) Von Mises stresses [psi] for the initial cracked DC and (b) LC models, with close-up of the cracked area (cutting sphere radius R= 1 inch).

A crack was modeled and inserted in both LC and DC models; crack sizes are representative of the maximum tolerable defect as dictated by GE-AVIO regulations. Introducing such a crack into a DBEM submodel, with consequent automatic remeshing in the cracked area, two different stress scenarios came out from DC and LC approaches Fig. (7): the former produces a realistic stress scenario throughout the whole DBEM submodel whereas the latter provides a realistic stress scenario just in the surrounding of the infinitesimal crack tip area; hence, a comparison between the two methodologies can just concern SIFs along the crack front.

A convergence study Fig. (8) was provided for each of the two methodologies (DC and LC), varying the submodel size and the mesh on and nearby the crack, in order to benchmark the respective convergence rates and run time efficiencies.

Fig. (8). Different meshes on the crack face, as used for the convergence study (the setting parameter “cfes” is the crack front element size” [24]).

Analyzing the SIFs trend, it is possible to observe a high convergence rate for LC methodology, rapidly providing a convergent SIF distribution. As a matter of fact, from Fig. (9) it is possible to see that the distribution of K_I along the crack front is insensitive of mesh refining and consequently results corresponding to the mesh with crack front element size equal to 0.003 in are already convergent; moreover same results are obtained with different submodel sizes (R=1 in and R=0.3 in) proving that the smaller submodel size R=0.3 is sufficient to provide accurate results. On the contrary the DC approach need a stronger mesh refinement in order to reach convergence (cfes=0.0006 in); again the smaller submodel size is sufficient to provide accurate results as proven by the overlap of K_I profiles for both submodel sizes (R=1 in and R=0.3 in).

Fig. (9). KI values along crack front of the initial crack: LC (a) and DC (b) methodologies, considering different mesh (cfes=…) and submodel (R=…) sizes.

Even if K_II and K_III values are one order of magnitude lower than K_I values, from Fig. (10) it is possible to appreciate the substantial coincidence of respective convergent results when using DC and LC approach, but it is worth to emphasize that the DC methodology can only reach convergence on K_II and K_III values after a stronger mesh refinement.

Fig. (10). Comparison between LC and DC convergent KII and KIII values for the larger submodel (R=1).

The nearly perfect coincidence of the results provided by the two LC and DC methodologies in correspondence of the initial cracked configuration Figs. (9b-10) guarantees the correct implementation of the proposed LC alternative approach.

Afterwards, a crack propagation simulation has been performed with both LC and DC methodologies but convergent crack propagation results related to DC approach could not be calculated due to the excessive mesh refinement needed and consequent prohibitive run times.

Crack shape vs. cycles, calculated with LC and DC methodologies, are compared in Fig. (11): again a strong similarity of the DC vs. LC crack configurations can be noted, showing the equivalence for the two proposed methods. Crack propagation has been stopped when the maximum SIF effective (Eq. 4) [24], which in this case is nearly coincident with K_I due to the reduced level of mode-mixity, reaches 96% of fracture toughness K_c Fig. (12). As previously said, it was difficult to obtain convergence along crack propagation with DC approach and that is why in (Fig. 12) only the LC results are displayed.

Fig. (11). Crack shapes as provided by LC approach, with highlight of normal tractions (psi) introduced on crack faces (up), and DC approach (down).

Fig. (12). KI values along the crack front during propagation with LC approach (R=1).

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LC SIFs for both smaller and larger submodels are nearly overlapped along the whole propagation Fig. (13a) and consequently the same holds true for the crack size vs. cycles Fig. (13b): this shows that the smaller submodel (R=0.3) can be used for the whole crack propagation analysis when using the LC approach. On the contrary, SIFs from DC approach are overlapped for smaller and larger submodels just for the initial crack propagation steps Fig. (14a), when cutting surfaces are still sufficiently far from advancing crack, but with a progressive divarication between the curves of crack size vs. cycles as the crack grows Fig. (14b). This show that the smaller submodel is not suitable to provide accurate results along the whole crack propagation when using DC approach. In conclusion, the performed analyses clearly shows that the LC approach allows to consider a smaller submodel than DC approach, without affecting the accuracy, with a consequent benefit in terms of run times for the problem solution.

Fig. (13). Comparison of Keff along crack front (a) and equivalent crack size vs. cycles (b) for smaller and larger submodels with LC approach.

Fig. (14). Comparison of Keff along crack front and crack size vs. cycles for smaller and larger submodels with DC approach.

From Fig (13), it is interesting to observe how the K_eff values becomes substantially constant along the crack front when the crack propagation proceeds, as a consequence of the crack front profile changes dictated by implementation of Eq. 1, with ΔK= ΔK_eff. Such behavior is not replicated in Fig. (14), where a certain degree of fluctuations on K_eff along the crack fronts is obtained, but this is expected considering the lack of a strict convergence when simulating crack propagation with DC approach.