Analytic Creep Durability of Rotating Uniform Disks

Turbine disks of aircraft engines in operation are subjected to alternating thermocyclic deformation under high temperatures. Operation gives rise to sufficiently high stresses and subsequent creep damaging effects.


INTRODUCTION
Turbine disks are important elements of aircraft engines.The strength and lifetime of disks deter- mine an engine's ability to reach its required service operating parameters including durability and reliability.
In operation, disks of turbojet engines are subjected to alternating thermocyclic deformation under high temperatures.The basic components of disk loading are: centrifugal forces caused by rotation; nonstationary temperature field (essentially inhomogeneous along the radial coordinate); pressure differential on the disk web; clamping forces caused by the rotor assemblage; radial forces from the disk blading.
Thejoint action ofthe aforementioned factors can lead to high stresses and subsequent creep effects.
Experimental data obtained by Sosnin et al. (1986) allow to estimate the time for creep deformation development up to the value of 0.5% as 101-102h within the stress concentration zones (disk fillets, blade roots and shaft splines).Therefore we can state Corresponding author.Tel." (3422) 39-17-02.E-mail: nyashin@tmex.icmm.perm.su.250 Y. NYASHIN AND V. SHISHLYAEV that creep deformation is a dominant factor of material damage in comparison to elastic one.

PHENOMENOLOGICAL DESCRIPTION OF DAMAGE AND RUPTURE
The prevailing methods of engineering design are based on the various theories of plastic flow and analogous creep models.At the same time accu- mulation of the static and cyclic defects leads to untimely fracture of the disks.Therefore the durability ought to be calculated on the basis of exact estimation of stress and strain fields, varia- tion of material properties due to the cyclic loading and account of material damage.
The phenomenological approach to the descrip- tion of creep rupture was suggested by Kachanov   (1958).He introduced the scalar function b: _> b > 0, characterizing the damage (accumula- tion of microscopic cracks and material defects) within the investigated domain.For initial (undamaged) state -I; in operation, the function b is decreasing down to the zero value.The value --0 corresponds to material fracture and can be considered as the fracture criterion.To obtain the durability it is necessary to supplement the boundary-value problem by equation describing damage development with time (the so-called damage kinetics equation).
Later different models were suggested by Rabotnov (1969), Ilyushin and Pobedria (1970), Kolmogorov (1970), Murakami (1983), Lemaitre  and Chaboche (1985), Sosnin et al. (1986), Chaboche (1988) and Baltov (1993) in order to describe material fracture in the process of un- reversible deformation of material.Generally speaking, such models ought to correlate with applied model of deformation and reflect kinetics of damage development at the points of the investigated body.Usually the fracture criterion is formulated on the basis of deformation-type or energy-type criteria.As a result of aforementioned model representations, investigation of construc- tion behavior consist in step-by-step integration of the constitutive relations for different levels of external loading.
In this paper we consider the problem of description of the rupture of thin isotropic disks in the framework of constitutive relations of the creep theory.We ought to mention that similar problem was solved by Kachanov (1974) for uniformly distributed tensile load acting along external contour of the disk.In this paper we consider the centrifugal forces caused by rotation as the load factor in an attempt to more closely simulate the operation of aircraft turbine engines.

STATEMENT OF THE PROBLEM
Let us consider a rotating disk with an internal axial hole.Initial radius of the internal hole is denoted by a0, an external disk radius by b0 (see Fig. 1).A rotation rate of co was held constant.
We assume the initial thickness of the disk h to be small and independent of radial coordinate r.This assumption seems to be correct for thin turbine disks and deflectors.It means that plane stress model can be applied and consequently o-z =0.
(1) a(t) bo zone of fracture

FIGURE
Initial configuration of the investigated domain and fracture development from the surface of internal hole.
Radial and tangential strain rates can be written in the following form 01 1; r=r r, {=7, (2) where v (r, t) is radial velocity of the disk particles.
To simplify the analysis we apply the scheme of maximal tangential stress with associated flow law.
Let us assume, that for every internal point of the disk.Then in accordance with the suggested flow law r--0, and consequently, Generally speaking, Kachanov (1960) shows that the interior of a disk can experience zones with different flow rules.In this paper we consider the variant described by the inequality (6) only.
Further we show that definite restriction on the initial sizes follows from this assumption.Figure 2 presents the Treska prism cross-section by the plane Crz 0 and possible flow directions.
From Eq. ( 7) it follows that Ov/Or-O.Conse- quently, the radial velocity of the disk particles depends upon the time only: v-vo(t), where vo-v(ao, t), ( 9) and tangential strain rate is determined as { v -o -.

STRESS FIELD CALCULATION
Resolving Eq. ( 14) with respect to 0-and inte- grating the equilibrium equation (4) over the radial coordinate r taking account of the boundary conditions (5), we obtain the stress tensor com- ponents: where # 1/m.Further we need to verify the hypothesis (6).This assumption means that at every internal point of the disk the following inequalities ought to be valid 0"-0"r > O, Vr E (a,b), (16.1) 0"r > 0, Vr E (a, b).(16.2) Taking account of K < 0, b > a and r > a, Vr (a,b), we can show that inequality (16.1) is equivalent to a 1-> /zr 1-. ( 17) Relation ( 17) can be forced because rmax-b: a -" > #b -". (18) Finally, assumption (16.1) is constant at the moment of fracture for a wide class of materials (e.g., for c and c +/3 titanium alloys).In Eq. ( 20) 0"ij designates the stress tensor components, eij designates the creep strain tensor components (i, j= 1,3).This assumption was verified experimentally for various temperature regimes and deformation rates.Therefore we can consider the value of dissipated energy W as the measure of accumulated damage.
Let us introduce the scalar function asociated with the dissipated energy: where W, is the critical value of the dissipated energy corresponding to material fracture.For initial (undamaged) state -1.The value -0 corresponds to critical accumulation of micro- scopic defects and material fracture.Thus, the function is normalized (1_>_>0) and non- decreasing: d/dt <_ O.
where O'ma is the maximal principal stress, D > 0, n _> 0 material constants; rn the power index from Eq. ( 11).
In terms of the measure we can formulate the damage kinetics equations as follows where A D/WT, +l.The constants A and n ought to be found from the experiments on the creep rupture strength.
To find the durability of the disk we need to integrate Eqs. ( 23) with respect to the found stress fields.
According to assumption (6) the tangential stress is maximal for all internal points of the disk at initial moment: O'max Crop > 0, Vr E (ao, bo).Because of maxr [cr(r)] r(a0), Vr E (a0, b0), we assume that fracture starts from the surface of internal hole of the disk.Consequently, in the process of fracture external radius of the disk is constant, while the internal radius value depends upon the time: b-b0-Const, a a(t), r (a(t), bo).( 24) Let us introduce the dimensionless variables: Then Eq. ( 15.2) can be written taking account of ( 24) where s(a) By separation of variables in Eq. ( 23.1) we can describe the damage accumulation process as follows fl f0 bndb -A CWmax dt. (27) At the crack initiation stage (while < q) the stress tensor components are constant: because a ao Const and a ao p.
Moreover, the value of principal stress Crma is time independent.Consequently, if suppose b= 0 in Eq. ( 27) we can obtain the following simple relation for the value of ti oz 0 q A(n + 1)O'a x A(n + 1)sm(ozo) (29) The time ti corresponds to the critical accumula- tion of damage on the contour of the internal hole.From a physical point of view it corresponds to appearance of macroscopic cracks and material defects.Further (when > q) the disk fracture develops in a radial direction towards the external contour.This process is accompanied with varia- tion of disk geometry.
The fracture criterion for any point of the disk can be formulated as A(n / 1) Crnax(-) dT--1. (30) Let us note that in the process of fracture progression, the internal radius of the disk is increasing (a(t) > ao), and, consequently, restriction ( 19) and assumption (6) are valid.It means that Eqs. ( 15) are correct at any time and tangential stress corresponds to the maximal principal stress always: amx a S(a)p-", A(n + 1) sm(c)pd--1. (31)  In Eq. ( 31) s(c0-s[(7-)] and the value p is independent of the load history: p-c.(t).Therefore A(n + 1) lotS m (32) Differentiating Eq. ( 32) with time we obtain the following equation of fracture edge movement do A( + ),m(,).
(33) dt Further, taking account of (29) and integrating the equation of fracture edge movement (33) with initial condition a-s0 for t-tI, we obtain da.(34 The time for complete fracture of the disk t, can be found from Eq. (34 with the following parameters: rotation rate o 11 600 rpm; initial radius of internal hole a0-75 mm; external radius of the disk b0-237.9mm; material density d-4490 kg/m3.This alloy is widely used to manufacture disks, deflectors, disk blades, fasteners and body details of aircraft gas turbine engines.The ultimate stress cry, yield limit cry and creep limit or100 for titanium alloy VT9 are presented in the Table I. Aforementioned sizes and rotation rate corre- spond to the real parameters of a turbine disk of an aircraft gas turbine engine during take-off.According to experimental data the temperature of the disk hub is equal to 452C, while the rim temperature is equal to 608C.The values of material constants applied for damage description are shown in Table II.The values are calculated on the basis of the experimental data obtained by Sosnin et al..(1986).It is easy to verify that restriction (19) is valid for both of the investigated regimes.
Figure 3 shows the fracture edge position versus dimensionless time for the temperature of 600C (m= 3, Co=0.315).The fracture time was calcu- lated by numerical integrating of Eq. (34) by the Gauss method.The figure shows that the fracture develops with relatively slow speed at initial stage.
Then the fracture rate increases significantly.

NUMERICAL ANALYSIS AND DISCUSSION
As an illustration we try to find the durability of the disk made of c + fl titanium alloy VT9 (5.8-7.0%A1, 2.8-3.8%Mo, 0.8-2.5% Zr, 0.20-0.36%Si) Complete fracture of the disk (i.e. the state when the fracture edge reaches external radius) corre- sponds to the moment of t. 1o017ti.
The calculation of crack initiation time according with Eq. ( 30) shows the following results: ti 82 845 h at 550C; q 48 600 h at 600C.
The obtained solution can be applied to estimate the lifetime of machinery components when creep effects are dominant.Use of an overly simpli- fied disk geometry makes the problem tract- able from an analytic point of view, but the presented model of damage development can be realized numerically as an element of stress field calculation.
Turbine disks in aeronautical gas turbines are life limited not by creep only, but by low-cycle fatigue too due to each start-stop cycle of loading.This would cause a reversal in stress direction within a local region (the inner bore in the present problem) and re-initiation of primary creep upon the start of the second flight, etc. Therefore the model can be applied for some long-range flights (e.g.supersonic cruise of transcontinental flights) when the contribution of the creep portion to the overall damage is dominant.

FIGURE 2
FIGURE 2 Creep flow surface cross-section by the plane c: 0 and possible flow directions.
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