An Influence of Geometrical Parameters of the Lock of the Blade-Disc Joint on Stress in the FEA and DIC Methods

In this paper, an assessment of strength properties of the existing blade-disc dovetail joint was presented, taking the parametric optimization of its geometry. A sensitivity of the analysis was carried out to determine the relationship between the selected geometric parameters and strength properties of the structure questioned. The results from experiments and computer simulations were compared. In addition, the fatigue life of the structure was analyzed for different materials and using the change of speed ranges and parameterization of geometry.


INTRODUCTION
A rotor assembly is a basic component of a turbine jet engine. Usually, this element, including a compressor, is subjected to centrifugal force [16]. The value of this loading results from the operating conditions, i.e. it strongly depends on the rotational speed of the rotor. The transferred loading depends on a temperature during exploitation, kind of material as well as geometry of the structural system. The numerical approach to the lock region containing partial parameterization of geometry by means the finite element method (FEM) can be found in a lot of papers [1,17,22]. In the case of lock connections used in compressor rotors, the correctly selected geometrical features of the zone allow reducing the stress [12]. From the operation point of view, the resistance on highcycle fatigue plays an important role. Therefore, determination of the fatigue durability is considered in many research groups [4,7,10,11,12,15,19,23,29]. These authors analyze the fatigue durability of rotor elements at various operational stages containing additional working conditions: (Fig. 1), corrosion, defects caused by collisions with sucked-in foreign bodies (scratches, cracks), abrasive effects (so-called fretting) up to failure. Therefore, the aim of the paper focuses on the assessment of fatigue compressor rotor using experimental and numerical approaches.
The assessment of strength properties of the blade-disc dovetail joint was carried out, taking into account the parametric optimization of its geometry. The analysis was performed to examine the relationship between the selected geometric features and the strength properties of the structure. It was done by comparing the results from experiments and simulations. In addition, the fatigue durability of the structure was analyzed selecting different materials and speed ranges, as well as parameterization of the geometry.

FE model
In the finite element method, or analysis, (FEM or FEA) the discretization of region is an essential step to help the finite element method solve the governing differential equations by approximating the solution within each finite element [24]. Mesh generation, as a preprocessing step, involves discretization of a solid model into a set of points called nodes on which the numerical solution is An Influence of Geometrical Parameters of the Lock of the Blade-Disc Joint on Stress in the FEA and DIC Methods to be based [24]. Finite elements are then formed by combining the nodes in a predetermined topology (linear, triangular, quadrilateral, tetrahedral or hexahedral) [24]. A more detailed discussion of the theoretical basis of the FEM method can be found in works [2,20,21,24,28,30].
In FEM for three-dimensional problems, the displacement vector consists of three components: (1) For example, for four-node tetrahedral element, the matrix of nodal displacement values can be presented as follows [3,18]: (2) where: u i = (u i , v i , w i ), i = 1, 2, 3, 4.
The relationship between the strain and displacement, on the other hand, is represented by: (3) [T] is commonly referred to as the linear-differential operator [18]. The stress field in the three-dimensional problem is related to the strains as follows: (4) where: [v] is the elasticity matrix for isotropic materials [3,18].
The basic FEM equation for the element is [3]: (5) where: [K e ] is the stiffness matrix [18,21]: (6) Matrix [B] is called the displacement differentiation matrix. It can be obtained by differentiation of displacements expressed through shape functions and nodal displacements; V is the volume of the element [18,20].
In turn, {Q e } is a load matrix, in which [N] represents shape functions [3,18]: (7) For the analyzed model of the dovetail joint, the mesh generated in ANSYS consists of threedimensional ten-node tetrahedral elements, shown in Figure 2.
The created element has ten nodes, each node having three translational degrees of freedom in the nodal x, y, and z-directions. The displacement field is represented by the following equations [31]: Expressions of the type N = (2L 1 -1) L 1 are shape functions of corner nodes, while N = 4L 3 L 4 are the shape functions of intermediate nodes.

Parameterization of geometry and sensitivity analysis
The computational model was developed based on the designing requirements to the high pressure compressor of the turbofan engine. The geometry of the elements was captured using the reverse engineering methods. For this purpose, a real blade of the engine and generally available technical documentation were used. The analysis, the geometry of the lock and the rim section of the disc were parameterized. Parameterization of the geometric model and sensitivity analysis made it possible to effectively follow the influence of the variability of the selected parameters on the operation and behavior of a given structure. It enabled developing many variants of the structure, with different geometrical dimensions and features in a relatively short time.
The initial geometry of the lock was characterized by several geometric features (Fig. 3). Taking into account the previous analysis [14] and the condition of dimensional invariance, the following two variables were selected ( Fig. 3): − An angle specifying the width of the lock (the P1 parameter), − An angle specifying the thickness of the lock (the P2 parameter). The analysis allowed determining the influence of the selected features on stress in the lock. It was done using the variability of the operating ranges of the adopted engine, i.e. 60%, 85% and 100% of the maximum speed of the high pressure rotor (N2 max = 15.183 rpm). The equivalent Huber-von Mises-Hencky stress was taken for the analysis. The Von Mises or equivalent stress σ e is presented by the following equation [3,5]: The analysis provided the information on the influence of the P1 and P2 geometrical parameters on the changes of the equivalent stresses (P4) at different ranges of rotational speed. There is a significant peak of the stress at P1 values at the range of 90-92° for engine operation in the idle speed range 60% N2 max (Fig. 4a, red). The lowest value of the equivalent stresses occurs at the angle P1 87-88°. As a result of the increase in the rotational speed (Fig. 4a, green) to the cruising speed range (85% N2 max ), no increase was evidenced. The lowest values of the equivalent stress occur at the range of angle P1 from 86° to 88°. For the values above 89°, the values of the stress remain at a similar level; the values are within the range of 750-765 [MPa]. A comparison of these data with results at 85%/N2 and 100%/N2 indicated on maximum values of the stress at greater values of P1 features.
For this parameter, it can be determined that the P1 parameter has very low influence on the equivalent stress values in the joint. In the case of the P2 parameter (Fig. 4b), the region of small values of the stress was obtained for the P2 values reaching 1.1-1.35°. A similar course was obtained for all considered ranges of rotational speed. An increase of value of the P2 parameter leads to a gradual increase of the stress in the joint. For this parameter, it can be concluded that the P4 parameter of the system has a high sensitivity to P2.

PARAMETRIC OPTIMIZATION
Parametric optimization enables to use the finite element method (FEM) and geometry parameterization to perform optimization of shape, transverse dimensions, material, etc., stress and strain criteria. These methods include non-gradient and gradient algorithms that enable to determine the minimization of single variable functions, as well as the algorithms of minimization in independent or conjugate directions of functions of many variables, etc. [9,25]. The following analysis used the ANSYS program and the optimization procedures implemented in it. The set of solutions was processed using the Response Surface Optimization tool. As a result of the optimization, two values of the width angle parameter (P1) and the thickness angle of the lock (P2), which for the assumed rotational speed N2 best met the following criteria were distinguished (Table 2): − The criterion of maximum stress (W1), − The criterion of minimum strain (W2). (11) The ranges of variability of the considered parameters are presented in Table 1, and the obtained results are shown in Figure 4.  The obtained distributions of the equivalent stress and relative strain of the selected variants are presented in Table 3. The optimization out gave the intended result for both variants. In the case of the initial model, the highest value of the equivalent stress in the joint is close to 490 MPa. The maximum value of the stress appearing in the W1 model is 30MPa greater than it was calculated for W1. In the case of the W2, 485 MPa represented the maximum value of the stress. In search of a solution for the variant of the minimum strain criterion (W2), the minimum relative strain achieved was at a level of 3.65e-04, while in the case of the base model it was 3.32e-04, and 3.6e-04 for the W1 model. The increase of the value of parameters P1 and P2 by only 5-6% at variant W2, enabled to significantly reduce the minimum relative strain in the joint considered.

EXPERIMENTAL ANALYSIS
In order to verify of the results obtained from the numerical analysis, an experiment was carried  out. It included the production of samples, having geometry of the basic model of the lock and the W1 and W2 variants obtained in the optimization process. The samples were made of low-carbon steel with a density of 7850 kg/m 3 . A view of an example of a sample of the lock in the form of a connected and disconnected system is shown in Figure 5. The laboratory tests consisted of a static tensile test. The aim of this experiment was to determine the full fields of strain. The Dantec Q400 digital image correlation system was used to capture distribution of strain. The principal stage of the DIC test was to prepare a special pattern on a measurement zone of a sample, adjust lightening conditions, capture a reference photo of the pattern and follow its variations under loading [6]. In the experiment 60 measurements by means of DIC were done. The loading was used at the range of 0.5 -10.0 kN. Table 4 shows the selected strain maps. They illustrate the full field strain distribution of total and relative strains, at loading of 5 and 10 kN respectively. The value of 10 kN can be considered as corresponding to the loading of the blade caused by a centrifugal force at the cruising speed for this engine. The results obtained by means of the DIC system enabled to obtain strain distribution at the following values of tensile force: 5 kN and 10 kN. The obtained experimental results are comparable with the results of numerical analysis. The value of stress in the joint of the lock was determined on the basis of the values of relative strains. For example, in the case of the two geometrical variants, at a force of 10 kN, the maximum value the stress were between 460-500 MPa. It was very close to the stress value determined by means of the numerical approach.

NUMERICAL ANALYSIS
A characteristic feature of loading in rotor assemblies (including compressor sections) is related to the variations of centrifugal forces caused by changes in the rotational speed. This chapter presents the influence of P1 and P2 geometric parameters on the fatigue life of the compressor lock and disc rim, using the geometry parameterization introduced in Chapter 3. Three different speed ranges were considered: 60%N2, 85%N2 and 100%N2 (N2 -rotational speed of the high   pressure compressor adopted in accordance with the rotational speed range of turbofan engines [26]). Fatigue life of the lock was analyzed for three different engineering materials: typical steel with density ρ = 7850 kg/m 3 , the Ti-6Al-4V titanium alloy (ρ = 4508 kg/m 3 ) and Inconel 718 nickel-chromium alloy (ρ = 8220 kg/m 3 ). The mechanical properties of these materials are taken from the study [33]. Because the analysis below only considers the basic load from the tensile force, the simulations were carried out taking zero value of the minimum major stress σ min and the stress coefficient R. In the case of such a load, the mean value of stress σ m is equal to the stress amplitude σ a , so the maximum stress σ max is equal to the doubled value of the stress amplitude σ a , i.e. σ max = 2σ a [8,13]. Fatigue strength correction was carried out according to the Goodman's theory. The results of the analysis are included in the charts presented in Figure 6.
Within the considered speed ranges, there were some differences in the influence of the parameters of the increase of fatigue life. In the case of an engine at the RPM (60%N2) and connecting elements made of steel or Inc718 alloy, the reduction in the thickness angle (P2) played a significant role, which gave a clear increase in the durability of the joint.
The influence of parameter P2 on titanium elements occurs at the range of 100%N2, and for values P2 > 1.35. It is expressed by a significant decrease of the number of cycles from 10e6 to 10e5. For higher rotational speeds (85%N2 and 100%N2), it can be concluded that the increase in fatigue life is affected by decreasing P2 at each range of P1. These results enabled to indicate the range of P2 value when the super alloy fatigue life decreased significantly, i.e. 1.35 mm. In the range of low and medium engine speeds, the joint made of nickel-chromium alloy are characterized by the longest life. In the range of 60%N2, this reaches 10e9 cycles, while at 85%N2 it is around 10e7 cycles. It is only at the maximum speed range that the Inconel-based joint is less durable than titanium alloy. However, it should be noticed that the operating time of the engine at the minimum ranges of is smaller than its nominal value because of the thermal conditions.

CONCLUSIONS
The problem of strength analysis (including the fatigue strength problem) of the lock joint in compressor sections of aircraft turbine engines is a significant problem from the point of safety of flight, which is also present in new constructions. The issue comes down to the attempts to determine the number of elements cycles, subassemblies, assemblies and the complete engine allowing the performance of an aerial task. A detailed description of the significance for the optimization process of individual parameters in the issues related to fatigue life can be obtained by means of the "sensitivity" analysis. A series of analyses were carried out to check the weight of the geometrical parameters of the blade-disc joint defined in point 3. The studies confirmed that irrespective of the rotational speed adopted, the most important impact (weight) on the fatigue life of the joint was shown by the thickness angle as well as, to a slight degree, the width of the lock. The unequivocal influence of the significance of the lock thickness angle P2 is observed for higher rotational speeds (the maximum range of engine operation).
A very important stage of the research was the validation of the numerical results obtained with the help of an experiment. The physical model and the digital image correlation system were used for this purpose. The results of the conducted experiment confirmed the results of numerical analyses related to optimization. The obtained results of fatigue life should be considered satisfactory; the useful life of the complete CFM56-5 engine, according to the manufacturer's data, is from 0.9e5 to 3.0e5 cycles depending on the version [32]. The expected next stages of the work regarding this issue will involve dynamic loads and fatigue failures caused by corrosion or defects resulting from the collisions with sucked-in foreign bodies (FOD).