Thermo-mechanical fatigue
crack propagation in a
single-crystal turbine blade
Andreas Koernig Nicke Andersson
Link¨oping University Division of Solid Mechanics
Link¨oping University Department of Management and Engineering Division of Solid Mechanics Master Thesis 2016|LIU-IEI-TEK-A–16/02524—SE
Thermo-mechanical fatigue
crack propagation in a
single-crystal turbine blade
Andreas Koernig Nicke Andersson
Academic supervisor: Kjell Simonsson
Industrial supervisors: H˚akan Lindgren & Magnus Hasselqvist Examiner: Daniel Leidermark
Link¨oping University SE-581 83 Link¨oping, Sweden
Abstract
Simulation of crack growth in the internal cooling system of a blade in a Siemens gas turbine has been studied by inserting and propagating cracks at appropriate locations. The softwares used are ABAQUS and FRANC3D, where the latter sup-ports finite element meshing of a crack and calculation of the stress intensities along the crack front based on the results from an external finite element program. The blade is subjected to thermo-mechanical fatigue and the cracks are grown subjected to in-phase loading conditions.
The material of the blade is STAL15SX, a nickel-base single-crystal superalloy. The <001> crystalline direction is aligned with the loading direction of the blade, while the secondary crystalline directions are varied to examine how it affects the thermo-mechanical crack propagation fatigue life of the blade.
The finite element model is set up using a submodeling technique to reduce the computational time for the simulations. Investigations to validate the submodeling technique are conducted.
From the work it can be concluded that a crack located at a critical location in the cooling lattice reach above the crack propagation target life. Cracks located at noncritical locations have crack propagation lives of a factor 5.2 times the life of the critical crack.
Sammanfattning
Sprickpropageringssimulering i en skovels inre kylsystem tillh¨orande en gasturbin fr˚an Siemens har studerats genom att s¨atta in och propagera sprickor vid l¨ampliga platser. Mjukvarorna ABAQUS och FRANC3D har anv¨ants, d¨ar den senare st¨odjer meshning av en spricka samt ber¨akning av sprickintensitetsfaktorer l¨angst med sprickfronten, baserat p˚a resultat fr˚an en extern finita element-mjukvara. Skoveln ¨
ar utsatt f¨or termo-mekanisk utmattning och sprickorna propageras under in-phase f¨orh˚allanden.
Skovelns material ¨ar STAL15SX, en nickel-baserad enkristallin superlegering. Kristall-riktningen <001> ¨ar parallell med skovelns lastriktning, medan konfigurationen av de sekund¨ara kristallriktningarna varieras f¨or att unders¨oka hur detta p˚averkar det termo-mekaniska sprickpropageringslivet.
Finita element-modellen anv¨ander en submodelleringsteknik f¨or att reducera den kr¨avda CPU-tiden f¨or simuleringarna. Utredningar f¨or att validera submodellering-stekniken utf¨ors ¨aven.
En slutsats fr˚an arbetet ¨ar att en spricka som ¨ar placerad vid ett kritiskt omr˚ade i det inre kylsystemet har en tillr¨ackligt l˚ang livsl¨angd. Sprickor placerade vid mindre kritiska omr˚aden har sprickpropageringsliv 5,2 g˚anger l¨angre ¨an livsl¨angden hos den kritiska sprickan.
Acknowledgments
We would like to thank our supervisors at Siemens, H˚akan Lindgren and Magnus Hasselqvist, for granting us the opportunity of conducting this project. Special thanks goes out to David Gustafsson, Erik Lundholm, Fredrik Lundblad, Henrik Petersson, Omid Lorestani and Patrik Rasmusson for their interest in our project, always being available to provide help and discussions when needed.
Furthermore, we would also like to thank our supervisor at Link¨oping University, Kjell Simonsson, for his great enthusiasm and support throughout the project.
Andreas Koernig & Nicke Andersson
Nomenclature
Abbreviations and acronyms
Abbreviation Meaning
Al Aluminum
Ce Cerium
Co Cobalt
CPU Central processing unit
Cr Chromium
FCC Face-centered cubic FE Finite element
FEM Finite element method
Hf Hafnium
IP In-phase
LEFM Linear elastic fracture mechanics Mo Molybdenum
Ni Nickel
OP Out-of-phase RT Room temperature
Si Silicon
SIF Stress intensity factor
SIT Siemens Industrial Turbomachinery AB
Ta Tantalum
Tmax Service temperature
TMF Thermo-mechanical fatigue
W Tungsten
Latin symbols
Symbol Description Units A Time dependent Paris law coefficient [−]
a Crack length [m]
C Paris law coefficient [−] E Young’s modulus [P a] f Geometry function [−] G Energy release rate J/m2 KI Stress intensity factor mode I [P a
√ m] KII Stress intensity factor mode II [P a
√ m] KIII Stress intensity factor mode III [P a
√ m] ∆KI Stress intensity range mode I [P a
√ m] Kmax Maximum stress intensity [P a
√ m] Kmin Minimum stress intensity [P a
√ m]
Symbol Description Units Kopen Stress intensity at crack opening [P a
√ m] m Paris law exponent [−] n Time dependent Paris law exponent [−]
N Number of cycles [−]
s Number of crack steps [−]
tdwell Dwell time [h]
U Displacement [m]
ui Displacement vector [m]
W Strain energy [J ]
Ws Surface energy [J ]
ws Fracture surface energy [J/m]
xj Coordinate vector [−]
< hkl > Crystal direction [−]
Greek symbols
Symbol Description Units δij Kronecker delta [−] ε Strain [−] γ Microstructure phase [−] γ0 Microstructure phase [−] µ Shear modulus [P a] ν Poisson’s ratio [−] Ω Integration domain [−] Π Potential energy [J ] σ Stress [P a] σf Fracture stress [P a]
Contents
1 Introduction 1
1.1 Background . . . 1
1.2 Single-crystal superalloys in gas turbines . . . 2
1.3 Using FEM for crack propagation . . . 2
1.4 Aim of the work . . . 3
1.5 Delimitations . . . 4
1.6 Other considerations . . . 4
2 Fundamentals 5 2.1 Thermo-mechanical fatigue . . . 5
2.2 Single-crystal materials . . . 6
2.3 Linear elastic fracture mechanics . . . 6
2.4 Stress intensity factors . . . 7
2.5 Calculation of stress intensity factors . . . 8
2.6 Fatigue life analysis . . . 9
2.7 Shakedown . . . 10
3 Method 11 3.1 Presentation of model . . . 11
3.2 Pre-processing for crack insertion . . . 12
3.3 Crack insertion . . . 13
3.4 Propagating the cracks . . . 14
3.5 Computational limitations . . . 15
3.5.1 Tied model validation . . . 15
3.5.2 Submodel validation . . . 16
3.5.3 CPU time reduction . . . 18
3.6 Crack increment size study . . . 19
3.7 Evaluating the TMF life . . . 20
3.8 Varying the crystalline directions . . . 21
4 Results 23 4.1 Crack 1 . . . 23
4.2 Crack 2 . . . 25
5 Discussion 29 5.1 Pre-processing and FE-model . . . 29
5.2 Crack propagation method . . . 29
5.3 Computational limitations . . . 30
5.4 Submodeling technique . . . 31
5.5 Evaluating the TMF life . . . 31
6 Conclusion and future work 33 6.1 Conclusion . . . 33
Appendices 37
A Appendix 37
A.1 Mesh of the FE-model . . . 37 A.2 Temperature field . . . 38
1
Introduction
Gas turbines are used for various applications, such as power generation and propul-sion of ships and airplanes. One growing application of gas turbines as a power source is to complement e.g. wind and sun power when needed. When developing gas tur-bines, it is important to investigate the life of different components. Turbine blades are subjected to large stresses and high temperatures. At critical locations in such a blade, crack initiation or defects may lead to crack propagation.
This work investigates methods for crack propagation life estimation in the internal cooling system of a hollow nickel-base single-crystal turbine blade.
1.1
Background
Siemens Industrial Turbomachinery AB (SIT) develops and manufactures gas and steam turbines. A gas turbine consists of three main parts: a compressor, a com-bustor and a turbine (cf. Figure 1). Air is compressed in the compressor and is mixed with fuel when entering the combustor. The high pressure is then used in the turbine to create mechanical torque to drive a generator or create driving thrust (as in the case of a ship engine).
Compressor
Combustor
Turbine
Figure 1: The interior of an industrial gas turbine.
The high temperature, up to 1 500 ◦C [1], and pressure, set high demands on the material in the turbine. Regular steels are not applicable, instead nickel-base su-peralloys are used. These alloys have excellent qualities with respect to mechanical strength, creep resistance and resistance to corrosion and oxidation [2].
1.2
Single-crystal superalloys in gas turbines
Since the efficiency of gas turbines increase with increasing inlet temperatures, a substantial amount of research has been carried out with focus on developing better materials applicable to higher temperatures. As an example, single-crystal superal-loys allow for higher temperatures compared to polycrystal superalsuperal-loys since they have better resistance to creep. Creep and oxidation usually develops at the bound-aries of the grains in the material [3] and since single-crystal materials consist of only one grain (i.e. no grain boundaries exist) it is more resistant to these effects. Therefore, single-crystal materials have grown more popular for critical turbine com-ponents, such as turbine blades.
The material investigated in this work is a single-crystal nickel-base superalloy named STAL15SX. Nickel is used mainly for its face-centered cubic (FCC) crys-tal structure, which is both ductile and tough and has good resistance to thermally activated effects (such as creep) [4].
The microstructure of STAL15SX consists of γ0 particles in a γ matrix. γ0 is es-sentially Ni3Al with elements of e.g. Ta in solution, whereas γ is Ni with elements
of e.g. Cr, Co, Si, Mo, W in solution, cf. Table 1 for the nominal composition of STAL15SX.
Table 1: Nominal composition of STAL15SX in vol. %.
Ni Cr Ta Co Al W Mo Si Hf Ce Base 15 8 5 4.6 4 1 0.25 0.1 0.005
STAL15SX contains about 45 vol. % of γ0. It is the γ/γ0 structure which provides the comparatively extreme creep strength of this class of superalloys.
It should be noted that the γ matrix and the γ0 particles have the same crystal ori-entation. Hence, STAL15SX consists of a two-phase single-crystal structure.
1.3
Using FEM for crack propagation
Investigation of crack propagation can be challenging, since crack growth usually changes the stress field. When using the finite element method (FEM) for crack propagation calculations, it therefore requires remeshing of the grid to take care of both the insertion of a crack and the propagation of it1. A program that manages this is FRANC3D (v. 7.0.1) [6–8]. It uses an existing mesh and lets the user insert a crack into it. It then remeshes the grid with respect to the inserted crack and an external finite element (FE) program is used to conduct the FE-analysis. After the FE-analysis, FRANC3D uses the derived stress field to calculate the stress intensity
1It is to be noted that FE-formulations for propagating a crack through elements exist, cf. XFEM
factors (SIF) to propagate the crack, cf. Figure 2. The external FE-program used in this work is ABAQUS (v. 6.14) [9].
EXTERNAL FE-PROGRAM
Analyze global model
Define submodel
FRANC3D
Insert crack into submodel
EXTERNAL FE-PROGRAM
Stress analysis
Compute SIFs
Extend crack
Uncracked model Model with initial crack
Model with extended crack
Figure 2: Schematic view of how FRANC3D works.
1.4
Aim of the work
It is essential for SIT to keep track of how cracks propagate in critical parts of the gas turbine. Therefore, an investigation on how to perform crack propagation sim-ulations and calcsim-ulations of the thermo-mechanical fatigue (TMF) life of a turbine blade is to be conducted. It should be noted that in this work, only the crack prop-agation life is investigated, i.e. crack initiation life is not taken into account.
The aims of this work can be summarized as follows:
• Develop a method for propagating cracks in a single-crystal material turbine blade subjected to in-phase (IP) TMF conditions.
• Investigate how a crack at a certain location in a turbine blade with a very complex geometry will propagate.
• Investigate how different secondary crystalline directions influence the crack propagation.
• If time allows for it, investigate how a crack at a certain location in the turbine blade will propagate under out-of-phase (OP) TMF conditions.
For a further discussion about fundamental concepts, such as TMF under IP- and OP-conditions, the reader is referred to Section 2.
1.5
Delimitations
The following delimitations are set for the work:
• Crystallographic crack propagation will not be considered. • No more than two cracks will be propagated.
• The locations of the cracks are limited to the internal cooling lattice of the blade.
• Cracks will only be propagated through the ribs and not through the thickness of the blade.
1.6
Other considerations
No ethical or gender related questions are raised by this work. Nor does it include any direct connections to environmental questions. However, efficient gas turbines constitute an important part in the strive towards a sustainable society.
2
Fundamentals
This section introduces the reader to some relevant theory concerning TMF, nickel-base single-crystals, SIFs and fatigue life calculations.
2.1
Thermo-mechanical fatigue
A first stage gas turbine blade is subjected to TMF, i.e. large variations in temper-ature and mechanical strain due to startups and shutdowns [10]. TMF is a primary cause of failure of hot turbine components [11].
Due to the non-homogeneous temperature field of the blade, local spots will be cooler or hotter than the surrounding material in service. Therefore, those spots will experience tensile and compressive stresses respectively, due to the difference in thermal expansion. A situation in which the temperature and mechanical loading varies in phase is referred to as IP-loading, while the contrary is referred to as OP-loading. Thus, regarding IP TMF situations, the maximum principal stress is tensile in service, and the crack will be open during operation [12]. For OP TMF situations on the other hand, crack opening will take place after unloading, cf. Figure 3.
Figure 3: OP load cycling (left) and IP load cycling (right) where the temperature varies from room temperature (RT) to service tem-perature (Tmax).
If the service temperature is sufficiently high, the IP crack propagation life will be shorter than the OP life, because the combination of high temperature and tensile stress accelerates the creep and oxidation effects [13].
2.2
Single-crystal materials
As mentioned previously, nickel-base single-crystal materials are widely used in gas turbines, due to their excellent high temperature capability. The need for single-crystal materials has increased, due to increasing inlet temperatures [14].
Single-crystal turbine blades are cast using a method called investment casting or ’lost-wax’ process [15], in which the <001> crystal direction constitutes the growth direction. However, when casting the specimens, misalignments from the nominal primary direction <001> will occur, leading to a deviation of the elastic stiffness with respect to the loading direction, which is an important factor to consider when determining the TMF life of a single-crystal component [16].
Initiated cracks in nickel-base single-crystal superalloys generally propagate per-pendicular to the axial stress with the aid of oxidation. During propagation, the crack path may sometimes change to a crystallographic one. Different physical ex-planations for this behavior exist, such as twinning phenomena [17]. The risk for crystallographic crack propagation is in the component investigated believed to be limited [18]. Hence, in this work crystallographic crack propagation will not be modeled. Instead, the cracks will be assumed to propagate according to Mode I (see Section 2.4).
2.3
Linear elastic fracture mechanics
In linear elastic fracture mechanics (LEFM), originally developed by A.A. Griffith [19], the stress field near the crack tip is calculated through linear elasticity theory. For monotonic loadings, a crack will propagate when the stress intensity at the crack tip exceeds the so called fracture toughness.
Griffith proposed that for a crack to grow, it requires two new free surfaces to be created and thus an increase in surface energy. The reduction of potential energy due to the formation of a crack must be equal or greater than the increase in surface energy. This can be mathematically expressed as
−dΠ da ≥
dWs
da (1)
where Π is the potential energy, Ws is the surface energy and a is the crack length.
Griffith showed that the potential energy (per thickness unit) released when intro-ducing a crack (of length a) in an infinite plate subjected to a uniformly applied stress σ can be written as
Π = −σ
2a2π
where E is the Young’s modulus. Furthermore, the total surface energy (per thick-ness unit) for two surfaces is
Ws = 2wsa (3)
where ws is the fracture surface energy. With equations (1) to (3), the following
expression for the critical crack length is obtained, cf. [20]
d (Π + Ws)
da = − σ2πa
E + 2ws= 0 (4)
If this condition is satisfied, the fracture is imminent and the stress can therefore be written as σf, which gives
σf =
r 2Ews
πa (5)
Beyond this point, for this specific case of load control, more energy becomes avail-able than is required to create new crack surfaces, which in turn will lead to unstavail-able propagation of the crack and fracture of the specimen [21].
In a turbine blade, significant plasticity and creep will occur, but eventually an almost elastic state will be obtained (see Section 2.7). The crack will therefore prop-agate in an essentially elastic body, just as LEFM assumes. LEFM can therefore be applied provided that the residual stress field generated is taken into account.
2.4
Stress intensity factors
A local increase in the stress field will occur at notches and similar geometric fea-tures in loaded components. These stress concentrations will generally be the origin of fatigue cracks.
The SIFs give a measure of the intensity of the stress singularity at the crack tip. The SIFs are expressed for three modes, corresponding to the different types of load-ing a crack can be exposed to: KI, KII and KIII, cf. Figure 4. These modes of
loading corresponds to opening (KI), in-plane shear (KII) and out-of-plane shear
(a) Mode I (b) Mode II (c) Mode III
Figure 4: Crack modes.
2.5
Calculation of stress intensity factors
To calculate the energy release rate G in an FE-context, the J -integral can be used J = Z Ω σij ∂ui ∂x1 − W δ1j ∂q ∂xj dS (6)
where Ω is the integration domain, σij represents the stress tensor, ui represents
the displacement vector, W is the strain energy (in an elastic setting), δ1j is the
Kronecker delta and q is a function that is 1 at the crack tip and 0 on the boundary of the integration domain. Finally, xj represents the coordinate vector.
Clearly, the ’problem’ with the J -integral is that it only gives a scalar for the energy release rate. For an isotropic material we have
J = G = K 2 I E0 + KII2 E0 + KIII2 2µ (7)
where µ is the shear modulus, and where
E0 = (
E plane stress
E/ 1 − ν2 plane strain (8)
For an anisotropic material, a more complicated expression is found, see e.g. [22] and further references therein.
To handle this, the M -integral (or interaction integral) can be used to calculate the SIFs [22]. To obtain the M -integral, two solutions are assumed and superposed, which is possible if the material is assumed to be linear elastic (note that the M -integral is valid for a general nonlinear elastic behavior). For more information regarding the theory behind the M -integral, the reader is referred to [22]. Summing it up, the J -integral is calculated, and the M -integral is then used to extract KI,
KII and KIII from J .
When unloading a cracked structure, the crack may close before the remote stress has reached zero. The crack will then not open again until the magnitude of the stress reaches the value at which the crack was closed [20]. The stress intensity range can be written as
∆KIef f ective= KImax− KIopen (9)
However, in this work crack closure will not be considered, and KIminis set to zero,
which leads to the following
∆KI= KImax (10)
Note that excluding crack closure generally leads to a shorter TMF life. However, under IP TMF loading when time effects (creep and oxidation) dominate, it is KImax
that governs propagation, not ∆KI [18]. Thus, this assumption is supported.
2.6
Fatigue life analysis
In the intermediate region of a diagram representing the logarithm of fatigue crack growth rate da/dN as a function of the logarithm of ∆KI, there is a more or less
linear relationship between log da/dN and log ∆KI, the so called Paris law [23],
given by
da
dN = C (∆KI)
n (11)
where C, giving the magnitude of the curve and n giving its slope, represent material parameters. N is the number of cycles.
In order to calculate the life as a function of crack length, Paris law can in this simple case be integrated in closed form. One has with the general expression for the stress intensity factor K = σ√πaf (a), that
da
dN = C (∆KI)
n= C ∆σ√πaf (a)n
(12)
where f is a geometry dependent function. Integration then gives
N = 1 C (∆σ√π)n Z af inal ainitial da (√af (a))n (13)
2.7
Shakedown
If the loading in an IP point in the blade is moderate, the following will occur: • The stress will increase until ’yielding’, and after subsequent stress relaxation,
an elastic residual stress is obtained (after unloading), cf. Figure 5.
• The next startup will again be elastic and the stress relaxation will continue as if the elastic shutdown/startup ramps had not occurred.
• Eventually, the difference in relaxed stress from one cycle to another is essen-tially zero, i.e. a stabilized response has been reached. Similar reasoning holds for an OP point.
• If almost all points are moderately loaded, the result is shakedown to an almost elastic stabilized state.
• Unless the stabilized shutdown ramp is essentially elastic, the crack initiation life is very low, hence a well designed blade must end up in an almost elastic stabilized state. This is a consequence of the strong but brittle nature of the cast material.
• Those points where reversed plasticity are obtained are assumed to not affect the overall stress redistribution process.
Mech. strain Stress Stabilized loop First cycle Creep relaxation Residual stress after first cycle
Figure 5: General idea of the shakedown analysis.
Based on this idea, a converged hysteresis and thus a shakedown state can be ob-tained through simulating only one loading cycle with a long enough hold time (in this case 5 000 h).
3
Method
This section presents the method used in this work for insertion and propagation of cracks as well as calculations of the TMF life in a turbine blade.
3.1
Presentation of model
The full FE-model of the turbine blade and the disc can be seen in Figure 27 in Appendix A.1. The disc is fixed at the bottom and the blade is attached to the disc using contact conditions. The program used for FE-simulations is, as previously mentioned, ABAQUS. The ABAQUS FE-simulation model consists of three steps. Note that this model is used only to acquire the residual stress field.
The first step, which is a static step, contains a ramp-up of the load, represent-ing a centrifugal load associated to a variation from 0 to 6 600 rpm. Temperature loads are introduced, cf. Figure 28 in Appendix A.2. Furthermore, the blade is subjected to pressure and suction loads from the hot gas stream and the cooling air. An anisotropic elasto-plastic material model is used.
The second step contains the dynamic visco-plastic creep analysis of the model. Note that in order to allow for LEFM crack propagation calculations, a closed hys-teresis loop must be obtained. Instead of running shorter cycles until the loop has closed (which requires a high computational cost), a shakedown analysis is con-ducted. The shakedown analysis is conducted using a hold time of 5 000 h.
The third step (also static) contains the unloading of the blade. The blade is here considered to be fully elastic (’pseudo elastic’), which is modeled by increasing the yield limit and removing all plastic data in the FE-model. A residual stress field is then obtained through elastically unloading the blade from the point where creep effects have relaxed the blade. The steps are represented graphically in Figure 6.
Time [h]
0 1000 2000 3000 4000 5000
Normalized load amplitude
0 1
3.2
Pre-processing for crack insertion
When deciding if one should analyze a point corresponding to the maximum or the minimum principal stresses (tension or compression), one must first decide whether IP or OP crack growth should be analyzed. In this work, IP TMF life has been investigated since, as mentioned previously, it is believed to be shorter than OP because of creep and oxidation effects. IP is also a higher risk because the load carrying parts of a blade are in tension during service, leading to propagation into load carrying structures.
Appropriate crack locations in the cooling lattice of the blade for IP TMF life cal-culations can be seen in Figure 7.
Crack 1
Crack 2
Figure 7: Crack locations for Crack 1 and Crack 2 as well as a maxi-mum principal stress contour plot of the ribs (cooling lattice).
Before insertion of the crack, FRANC3D needs a different set up than the FE-model used to calculate the residual stress field. Since the hysteresis loop is considered closed after the shakedown analysis, LEFM is applicable for crack propagation anal-ysis. Therefore, the FE-model for FRANC3D was set up by removing all plastic data and increasing the yield limit to ensure that the loading operates within elastic boundaries. The residual stresses were mapped from the mesh without a crack into the FRANC3D mesh (after crack insertion) using a Python script provided by SIT.
The FRANC3D input file consists of two static steps. During Step 1, the resid-ual stresses from the unloaded state’s stress field are mapped from the uncracked to the cracked model. During Step 2, the blade is subjected to the centrifugal load, temperature and pressure load corresponding to IP loading. The cyclic loading behavior can be seen in Figure 8.
Crack growth steps
1 2 3
Normalized load amplitude
0 1
Figure 8: Normalized load amplitude for the FRANC3D crack simu-lations.
3.3
Crack insertion
Before the crack propagation analysis, appropriate locations to insert cracks were identified. To find appropriate locations, the maximum principal stress was investi-gated. Since cracks grow according to Mode I, the cracks were inserted perpendicular to the vectors of the largest maximum principal stress, cf. Figure 9.
X Y
Z
(a) Location for Crack 1 (b) Location for Crack 2
Figure 9: Top view of a cut-plane of the cooling lattice showing max-imum principal stress represented by vectors at the crack insertion locations for Crack 1 and Crack 2. As can be seen, the general behav-ior is that the vectors of the maximum principal stress are oriented along the ribs, implying that the cracks should be inserted perpen-dicular to the surfaces.
The crack propagation calculations were conducted with FRANC3D and ABAQUS, where the former is used for calculating the SIFs and kink angles, propagating the crack and remeshing the region of the crack after every growth increment. Quarter point wedge elements are introduced at the crack tip [24]. ABAQUS was used to perform the FE-analysis during each iteration.
where the local subdomain is specified by the user. The nodes which are shared by the local and global model are merged so that the boundary of the local submodel coincides with the global model, cf. Figure 10.
Figure 10: An example of how FRANC3D divides the domain into a local domain (red) and global domain (white).
Though ABAQUS resolves the stress and strain fields for the whole domain, it is important that the submodel is large enough so that the transition of the mesh between the global and local model is smooth enough. Furthermore, if the submodel is too small, FRANC3D encounters difficulties when meshing the domain. The shapes and dimensions of the inserted cracks were for simplicity circular ’penny’ cracks with a radius of 2e-4 m.
3.4
Propagating the cracks
Cracks are propagated through an incremental procedure handled by FRANC3D2. When growing the cracks, it is important that the mesh is adapted to fit the new crack front which is calculated using the SIFs and kink angles. This is accomplished through fitting a polynomial and ignoring nonphysical crack-front endpoints (the M -integral assumes plane strain which will not be the case for the endpoints located near the surface).
The cracks grow in the direction according to the kink angle, which is defined as the amount the cracks will deviate from the self-similar direction measured in a plane perpendicular to the crack front [6].
Finally, the density of the mesh at the crack front is specified, and was in this work set to 20 % of the median crack increment size (a value between 10 to 30 % is recommended by FRANC3D [6]).
2It is to be noted that there is not ’one iteration per cycle’. Each iteration with a predefined
3.5
Computational limitations
It was noted that crack propagation simulations for the full model were unreason-ably expensive concerning CPU time. Therefore, different approaches to reduce the CPU time were investigated.
The first approach was to simply tie the contact nodes between the disc and the blade. Through this approach extra stiffness might be introduced into the model which is why the SIFs of the tied model were compared to the full model, see Section 3.5.1.
The second approach was to use a submodeling technique. The submodel is con-nected to a results file of the full uncracked model, the nodes located at the bound-ary of the submodel were set to have the same displacements as their corresponding nodes in the full model (for each step of the analysis). The subdomain has to be large enough so that the crack does not affect the boundary conditions for the sub-domain as it is inserted and propagated. Investigations to validate the submodeling technique is found in Section 3.5.2.
3.5.1
Tied model validation
To validate the tie modeling technique, the SIFs of the tied model were compared to the full model.
Normalized distance along front
0 0.2 0.4 0.6 0.8 1 K I [MPa m ] 0.5 1 1.5 2 2.5 3 3.5 4 Tied model Full model
The tied model has a relatively good agreement with the full model, cf. Figure 11. Some but small changes regarding the characteristics of the SIF along the crack front can be seen. However, though the CPU time is reduced, it is still considered too long.
3.5.2
Submodel validation
In order to validate the submodeling technique before introducing it to FRANC3D, the stress field of a number of uncracked submodels with different domain sizes were compared to the uncracked global model (which drives the submodel). Note that the pressure load was modified into only affecting the elements in the submodel.
Comparisons of stresses and displacements were conducted. The investigated quan-tities were extracted from the nodes in the internal cooling lattice (which is the region of interest) and were compared with the same nodes of the full model. The comparison between the full model and the largest submodel (Submodel 4) can be seen in Table 2. As can be seen the submodeling technique has a relatively good agreement with the full model before crack insertion.
Table 2: Comparison of Submodel 4 and full model.
Physical quantity Average disparity (%) Standard deviation U (magnitude) 0.0943 0.1459e-5 [m]
U1 0.2149 0.1520e-5 [m]
U2 31.3009 0.3560e-5 [m]
U3 0.0672 0.1389e-5 [m]
von Mises 0.6499 6.0966e+5 [Pa] Max. Principal 3.8051 8.2186e+5 [Pa] S11 10.4566 9.1171e+5 [Pa] S22 5.1179 7.7093e+5 [Pa] S33 11.3863 6.8467e+5 [Pa] S12 13.8618 5.2556e+5 [Pa] S13 13.4612 3.9289e+5 [Pa] S23 11.0895 3.9020e+5 [Pa]
To make sure that the submodel is large enough to handle the impact the crack has on the stress field in an appropriate way, a comparison of KI between different
submodel sizes and the full model was conducted.
Unfortunately, the submodel keyword that ABAQUS uses was, at the time this work was conducted, unsupported by FRANC3D. Therefore, the nodal displacements at the boundary which drive the submodel were extracted manually from the full model and specified in the FRANC3D input file. The displacements corresponding to the residual stress field were prescribed in Step 1 and the displacements corresponding to the loaded state where creep is taken into account were prescribed in Step 2.
In Figure 12 it can be seen that a too small submodel domain leads to inaccu-rate results, which is why the largest possible submodel allowed without including the contact regions (which are computationally expensive) is required, i.e. Submodel 4.
Normalized distance along front
0 0.2 0.4 0.6 0.8 1 K I [MPa m ] 0 2 4 6 8 10 12 14 Submodel 1 Submodel 2 Submodel 3 Submodel 4 Full model
Figure 12: KI for different submodels compared to the full model.
The total number of elements in each model can be seen in Table 3.
Table 3: Number of elements in each model.
Model Number of elements Submodel 1 71206
Submodel 2 95351 Submodel 3 165924 Submodel 4 185883 Full model 456021
Quantification of the comparison for Submodel 4 versus the full model can be seen in Table 4.
Table 4: Comparison of KI for the full model and Submodel 4.
Physical quantity Average disparity (%) Standard deviation KI 5.6252 1.8332e+4 [P a
√ m]
If the submodel is too small, the stress intensities of a crack might relax as it propa-gates due to the specified displacements in the submodel. Therefore, it is convenient to investigate whether the error of the SIFs between the full model and submodel grows during crack propagation. Since it is expensive with respect to CPU time to propagate a crack using the full model, crack front geometries obtained from crack growth with the submodel were imported into the full model from three different crack growth increment steps.
Figure 13 shows a comparison of SIFs between Submodel 4 and the full model using a crack geometry for three different crack growth steps. As can be seen, the SIFs of the submodel and the full model have good agreement for each respective step. This implies that the crack is not dramatically affected by the boundary conditions introduced from the submodeling technique.
Normalized distance along front
0 0.2 0.4 0.6 0.8 1 K I [MPa m ] 4 6 8 10 12 14 16
Full model step 10 Submodel step 10 Full model step 20 Submodel step 20 Full model step 30 Submodel step 30
Figure 13: Comparison of SIFs for different crack growth steps be-tween Submodel 4 and the full model.
3.5.3
CPU time reduction
In Figure 14 the advantage of using a submodeling technique is visualized through presenting the required CPU time for one crack growth analysis using the different modeling techniques. The simulation time is reduced with a factor 40 compared to using the full model.
Full model Tied model Submodel CPU time [h] 0 10 20 30 40 50 60 70 80 90 100
Figure 14: CPU time for different modeling techniques.
Though the results lose accuracy compared to the full model, the submodeling tech-nique is considered an acceptable payoff due to the dramatic reduction in CPU time, and therefore this method was henceforth used.
3.6
Crack increment size study
In order to investigate how the size of the crack growth increments affects the results, crack propagation simulations where different increment sizes are compared through evaluating KI were conducted. Note that a mesh size equal to 20 % of the increment
size was used for all increments, which implies that the number of elements varies.
The crack increment size study was conducted using five different median crack increment sizes. The crack was grown a total median extension of 0.15 mm.
Crack length [mm] 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 K I [MPa m ] 4.6 4.8 5 5.2 5.4 5.6 5.8 6 6.2 6.4 6.6 Increment 0.75e-5 Increment 1e-5 Increment 1.5e-5 Increment 2.5e-5 Increment 5e-5
Figure 15: Crack increment size study for different step sizes.
In Figure 15 it can be noted that the responses of the different increment sizes are similar. However, from a conservative point of view either the increment size of 0.75e-5 m or 1e-5 m should be used. The 0.75e-5 m increment size oscillates more and additionally would require more crack growth increments, which is why the increment size of 1e-5 m was henceforth used as the initial increment size for crack propagation. As the crack was grown a number of steps, the median increment size was increased gradually from 1e-5 m to 3e-5 m.
3.7
Evaluating the TMF life
When a crack has been fully propagated, the SIFs were evaluated through defining a path along the crack fronts. Note that the paths should not be located too close to a free surface (i.e. along the end points of the crack front). This is because the accuracy of the SIFs are questionable there [6]. Three paths per crack were defined, cf. Figures 16-17. The TMF life can be calculated with use of Paris law as described in Section 2.6. The equation used to calculate the life, which accounts for hold time, has the general form as follows
da dN = ∂a ∂Ncyclic + Z tdwell 0 ∂a ∂tdt = C∆K m+ Z tdwell 0 AKndt (14)
where C and A are material parameters. Note that Kmin, as previously mentioned,
∆K is then equal to Kmax.
The SIFs were extracted for each path, and through using a simple integration scheme the number of cycles were calculated according to the following equation
N = s−1 X i=1 1 2 1 CKim+ AtdwellKin + 1
CKi+1m + AtdwellKi+1n
(ai+1− ai) (15)
where s is the total number of crack steps. It can be noted that ∆K is calculated as a mean value for each interval of the integration domain.
(a) Path 1 (b) Path 2 (c) Path 3
Figure 16: Illustration of Path 1-3 for Crack 1.
(a) Path 1 (b) Path 2 (c) Path 3
Figure 17: Illustration of Path 1-3 for Crack 2.
The paths were defined by using the nearest subsequent node for each crack front.
3.8
Varying the crystalline directions
The <001> direction of the blade is aligned with the centrifugal load i.e. along the axial direction of the blade, which is the best configuration of the crystal orienta-tions concerning the TMF life [16].
Since the configuration of the secondary crystalline direction can not be controlled as the blade is cast, it is relevant to examine how the TMF life is affected by differ-ent crystal rotations around the <001> (axial) direction. Therefore, the secondary
directions were varied in the model and cracks were inserted and grown using identi-cal crack dimensions and locations. The different crystalline direction configurations can be seen in Figure 18.
X Y Z Rotation 0˚ Rotation 22.5˚ Rotation 45˚
Figure 18: Top view of the blade showing the three different sec-ondary crystalline direction configurations investigated.
4
Results
This section presents results of the life calculations for the propagated cracks. The number of cycles are normalized to the longest life.
4.1
Crack 1
Crack length [mm] 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 KI [MPa m ] 3.5 4 4.5 5 5.5 6 6.5 Path 1 Rotation 0° Rotation 22.5° Rotation 45°Normalized number of cycles
0 0.2 0.4 0.6 0.8 1 Crack length [mm] 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Path 1 Rotation 0° Rotation 22.5° Rotation 45°
Figure 19: SIF as a function of crack length (left) and crack length as a function of cycles (right) for Path 1.
In Figure 19 it can be seen that the differences when varying the secondary crys-talline directions are relatively small for Path 1. It can also be noted that the number of cycles are accelerating with increasing crack length, but fortunately not severely.
The stress intensity factor KI is rather oscillatory, but not showing dramatic slope
changes. Crack length [mm] 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 KI [MPa m ] 3 4 5 6 7 8 9 Path 2 Rotation 0° Rotation 22.5° Rotation 45°
Normalized number of cycles
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Crack length [mm] 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Path 2 Rotation 0° Rotation 22.5° Rotation 45°
Figure 20: SIF as a function of crack length (left) and crack length as a function of cycles (right) for Path 2.
respect to crack length. It can also be noted that the 0◦ secondary crystalline direction rotation provides the shortest life for Path 2.
Crack length [mm] 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 KI [MPa m ] 3 4 5 6 7 8 9 10 11 Path 3 Rotation 0° Rotation 22.5° Rotation 45°
Normalized number of cycles
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Crack length [mm] 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Path 3 Rotation 0° Rotation 22.5° Rotation 45°
Figure 21: SIF as a function of crack length (left) and crack length as a function of cycles (right) for Path 3.
Path 3 shows more distinct differences in number of cycles for the different secondary direction rotations compared to Path 1 and Path 2. It can also be noted that Path 3 shows the shortest life for all secondary directions. The crack propagates along the wall perpendicular to the crack insertion surface which might explain why the crack propagation life of Path 3 is the shortest. Rotation 45◦ shows the shortest life with respect to crack length.
It can be noted that Crack 1 generally provides a long life, reaching well above the target life when a hold time of 12 h is used.
Normalized distance along front
0 0.2 0.4 0.6 0.8 1 K I [MPa m ] 4 6 8 10 12 14 16 Rotation 0° Rotation 22.5° Rotation 45°
Figure 22: SIF along the front corresponding to the last crack growth increment for the different secondary direction rotations.
In Figure 22 it can be seen that the SIF for the last crack growth increment is rather similar for all secondary direction rotations.
4.2
Crack 2
Crack length [mm] 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 KI [MPa m ] 4 6 8 10 12 14 16 18 Path 1 Rotation 0° Rotation 22.5° Rotation 45°Normalized number of cycles
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Crack length [mm] 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 Path 1 Rotation 0° Rotation 22.5° Rotation 45°
Figure 23: SIF as a function of crack length (left) and crack length as a function of cycles (right) for Path 1.
Crack 2 has significantly shorter life than Crack 1. The number of cycles versus crack length plot indicates an asymptote as af inal is reached, cf. Figure 23. Note
also the increased slope of the SIF at crack length 0.8 mm, which could be caused by when the crack reaches the surface at the side of the rib. Note that the 22.5◦ rotated secondary direction rotation shows the shortest life, but is still above the target life of the component.
Crack length [mm] 0.2 0.3 0.4 0.5 0.6 0.7 0.8 KI [MPa m ] 4 5 6 7 8 9 10 11 12 Path 2 Rotation 0° Rotation 22.5° Rotation 45°
Normalized number of cycles
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Crack length [mm] 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Path 2 Rotation 0° Rotation 22.5° Rotation 45°
Figure 24: SIF as a function of crack length (left) and crack length as a function of cycles (right) for Path 2.
Path 2 exhibits a longer life than Path 1, cf. Figure 24. The differences between the different secondary direction rotations are relatively small.
Crack length [mm] 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 KI [MPa m ] 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 Path 3 Rotation 0° Rotation 22.5° Rotation 45°
Normalized number of cycles
0 0.05 0.1 0.15 0.2 Crack length [mm] 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Path 3 Rotation 0° Rotation 22.5° Rotation 45°
Figure 25: SIF as a function of crack length (left) and crack length as a function of cycles (right) for Path 3.
The gradient of the SIF KI, similarly to Path 1, shows a change around a crack
length of 0.8 mm. It can be noted that Path 2 and Path 3 have by estimate equally long lives, whilst Path 1 shows the shortest life.
To conclude, Path 1 is the deciding factor regarding the life of the component. In analogy to Crack 1, Crack 2 also propagates along the wall perpendicular to the crack insertion surface which might explain why the crack propagation life of Path 1 is the shortest. The secondary direction rotation which shows the shortest life is 45◦.
Crack 2 generally has small but influencing differences regarding the impact of ro-tating the secondary crystalline directions.
Normalized distance along front
0 0.2 0.4 0.6 0.8 1 K I [MPa m ] 8 10 12 14 16 18 20 22 Rotation 0° Rotation 22.5° Rotation 45°
Figure 26: SIF along the front corresponding to the last crack growth increment for the different secondary direction rotations.
In Figure 26 it can be seen that the SIF for the last crack growth increment is rather similar for all secondary direction rotations. Note that the cracks have not been grown equally far due to variations of the crack growth characteristics, where e.g. the 22.5◦rotation is grown the shortest distance and therefore the SIFs generally have a smaller magnitude.
5
Discussion
In this chapter, discussions regarding the results and the choices made when devel-oping the method are presented.
5.1
Pre-processing and FE-model
When evaluating where it is appropriate to insert and propagate cracks, the un-cracked FE-model was used. Since the cooling lattice is of interest, the maximum principal stresses were examined for that region (cf. Figure 7). Obviously, it is ap-propriate to conduct crack growth for a location with the highest maximum principal stress, which was the motivation behind Crack 2. However, a majority of ribs in the lattice have a relatively even stress distribution, though lower than the maximum. Crack 1 was therefore motivated since e.g. voids or other damaged locations due to manufacturing defects have a high probability to be located in that region. Crack 1 can be interpreted as a benchmark for a crack somewhere in the middle to upper part of the lattice.
The initial orientation of the crack was decided from the vector plot of the maxi-mum principal stress. Here one can see that the vectors are oriented along the ribs, implying that the crack should be inserted perpendicular to the face of a rib. A verification that the crack is inserted with an appropriate orientation is to examine KII, which should be close to zero. It can be concluded that finding the correct
initial orientation of the crack is hard due to the rough mesh at the ribs, but after a few increments KII oscillates around zero which validates the initial orientation of
the crack.
The mesh density at the lattice is quite low, and the stress field might be mesh dependent. However, the characteristics of the stress field along the ribs are rel-atively homogeneous. Additionally, FRANC3D refines the grid locally during the remeshing, which, though not removing the mesh dependency, leads to a refined stress distribution especially near the crack.
The ribs examined are relatively small, with a cross section of ∼ 0.9x2.2 mm. The inserted cracks have a radius of 0.2 mm, which is relatively large compared to the dimensions of the ribs. It would be interesting to investigate how the TMF life is affected when varying the initial crack size.
5.2
Crack propagation method
The crack propagation method rests upon LEFM conditions, which is obtained through the shakedown analysis. When the hysteresis loop has closed, elastic
behav-ior was modeled. This simplification might however not be fully correct. The blade might still experience visco-plastic behavior (creep) due to e.g. the temperature field, but the impact of this is assumed to be negligible.
The cracks were grown according to Mode I, i.e. no crystallographic crack growth behavior was modeled. This probably leads to a longer TMF life as well as different crack propagation paths. The life calculations should therefore be interpreted as to the order of magnitude of the life.
5.3
Computational limitations
The CPU time for running crack growth simulations using the full FE-model was unreasonably long. The reason for this is both the contact region between the blade and the disc (approximately 50 % of the simulation time was reduced when the contact nodes were tied), but also the number of elements in the analysis. Through using the submodeling technique, the CPU time was reduced to a more reasonable level. Because of this, the loss of accuracy when using the submodeling technique is deemed acceptable.
A large portion of the required CPU time when analyzing the full model is the interpolation of the residual stresses. This is partly due to the variation of the element sizes, where the interpolation script is adapted to the largest element in the model. When removing the disc (and therefore reducing the size of the largest element in the domain), the script did not need as much CPU time to perform the interpolation. It is probably possible to reduce the CPU time for the full model if the mapping script is modified.
It can be noted from the crack increment size study that the step size holds no larger impact on the results. The differences between using a median increment size of 5e-5 m and 0.75e-5 m are relatively small. Using a step size of 0.75e-5 m leads to quite large oscillations of the stress intensity factor with respect to crack length, whilst larger increments yield a more stable curve. A too large median increment size seem to yield non-conservative results, especially when the crack length is small (i.e. the early crack propagation increments). Therefore, the cracks were initially grown using smaller median increment sizes and were gradually increased.
Another thing to note is that the mesh for the step analysis was set to 20 % of the median increment size, implying that the mesh density changed as well. In or-der to isolate the impact of the step size, the mesh for the crack front would have to be set to a fixed size. This however leads to difficulties with respect to FRANC3D meshing.
5.4
Submodeling technique
Figure 12 illustrates the importance of using a sufficiently large submodel domain. The average disparity of KI for Submodel 4 is 5.6 % which is acceptable. Since
the submodel is driven by the displacements extracted from the full model, it is important to investigate if the SIFs relax as the crack propagates. Since it is not realistic to compare the SIFs through propagating a crack using the full model, the crack geometries from propagated cracks when using Submodel 4 were extracted and inserted in the full model. Figure 13 shows that the crack intensities are very similar, which could imply that the stress intensities were not relaxed as the crack propagates. This might indicate that the submodel boundary has a sufficiently large distance between the crack. However, this might become problematic if the crack is located too close to the boundary and should be examined for such a case.
5.5
Evaluating the TMF life
Since the secondary crystalline direction varies when the specimens are cast, it is important for SIT to investigate how this influences TMF life in the airfoil. As can be noted in the figures in Section 4, varying the secondary directions affects the life of the component. However, the differences are small which is good from an indus-trial point of view. Again, it is important to underline that crystallographic crack propagation mode is not used and hence, the differences might be larger if such a model was used.
As can be seen in the results section, Crack 1 shows a long life when a hold time of 12 h is used. The conclusion from this is that a crack or void in the middle to upper part of the lattice would probably not be a risk for failure.
In Figures 22 and 26 it can be seen that the crack grows faster on one side of the crack front. This could be explained by the fact that the crack front reaches another surface and therefore a dramatic redistribution of the stress carrying vol-umes occurs, which in this case increases the SIFs for that part of the crack front. This can also explain why the lives are shorter for the paths that experience this transition. It can be noted that the cracks have not been grown entirely through the ribs (i.e. the ribs are not fully cut through), due to difficulties which occur as the cracks extend to the ends of the surface where the cracks were inserted. Therefore, these difficulties defined the stopping criteria for the crack propagations in this work. Note that a crack inserted to propagate through both the rib and thickness of the blade will most likely show a longer crack propagation life and would be interesting to investigate further.
Figure 26 shows that SIFs along the front corresponding to the last increment for each respective secondary direction rotation generally have the same characteristics for both cracks. However, for Crack 2 it can be seen that the 22.5◦ rotation has its minimum closer to the center of the front. Therefore it can be concluded that though
the life of the cracks are not dramatically affected, the crack growth characteristics and paths are affected when rotating the secondary crystalline direction.
For both cracks, KI increases with increasing crack length. This indicates that they
propagate in a relatively homogeneous stress field. In contrast, a crack initiated in a notch may well see a decreasing KI as it grows, since stresses tend to decrease fast.
It should be noted that for a small part of the front of Crack 2, the SIFs are above zero when unloaded. This implies that the assumption Kmin = 0 might be
unacceptable. To investigate this further, the crack can be grown using Step 1 (i.e. the residual stress field). Comparisons between using the stress field in a loaded state and unloaded state with Paris data for high temperature and room tempera-ture respectively, would then show which case is the most conservative. However, since KI and not ∆KI governs the TMF life calculations, this assumption is
6
Conclusion and future work
In this section some conclusions from the work can be found, as well as some com-ments on possible future work.
6.1
Conclusion
A method for propagating and analyzing cracks in the cooling lattice of a turbine blade has been developed. Life estimations have been carried out with the help of Paris law, where input parameters have been provided by SIT. The main findings and conclusions drawn from the work can be summarized as follows:
(i) The submodeling technique is an effective way in terms of computational effi-ciency, reducing the computational time with a factor 40. The loss of accuracy compared to using the full model is deemed to be acceptable. The choice of submodel domain size is important, and the required size when studying crack propagation in a component should be investigated thoroughly.
(ii) The initial crack increment size should be relatively small in order to achieve conservative SIFs. Since the ribs in the cooling lattice are thin, large crack increments are hard to conduct, resulting in relatively small increments for the entire crack growth sequence.
(iii) The mesh of the airfoil is relatively rough (some locations in the cooling lattice only consist of 2 elements in thickness) which may result in poor accuracy. However, after crack insertion, FRANC3D remeshes the cooling lattice which provides a better stress distribution near the crack.
(iv) Rotations of the secondary crystalline direction have some, but not major impacts on TMF life with respect to crack propagation in the cooling lattice. (v) The path evaluations show that a crack at a critical location in the lattice has
a crack propagation life reaching above the target life, where the life is defined as when the crack has cut through the rib. However, since the results might be mesh dependent, the results should be interpreted as to the order of magnitude of the life.
(vi) Cracks located in the middle to upper part of the lattice (where the stress field is distributed relatively even) have crack propagation lives reaching well above the target life, showing around 5.2 times longer lives than the critical crack for the cracks to cut through the ribs.
Furthermore, it can be said that the automatic crack growth analysis provided in FRANC3D not always works as intended. The most efficient way to obtain a good crack growth is to manually attend each crack increment to make sure that the curve fitting and mesh are well adjusted.
6.2
Future work
In the extension of this work there are further issues that can be investigated. It would be interesting to investigate how a refinement of the mesh at the cooling lat-tice would affect the outcome of the stress field and thus the locations of critical cracks.
Furthermore, growing and evaluating an OP crack would be interesting. The SIFs for Crack 2 show that Kmin for a part of the front is above zero when the blade is
unloaded, while Kmin is set to 0 when calculating the TMF life. Therefore, a study
investigating the most conservative contribution to the TMF life through comparing the life when the crack is grown in loaded respectively unloaded state should be conducted.
It would also be interesting to make more thorough attempts on growing the cracks further to investigate how the crack propagates when reaching the end of the inser-tion surface of the rib. Growing a crack through both the rib and thickness of the blade should also be investigated.
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A
Appendix
A.1
Mesh of the FE-model
X Y Z X Y Z X Y Z
Figure 27: Mesh of full model (left), Submodel 4 (upper right) and the interior (lattice) of the airfoil (bottom right).
A.2
Temperature field
Figure 28: Temperature field of the airfoil in service where blue rep-resents cooler areas and red reprep-resents hotter areas.
