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Modelling thermomechanical fatigue crack growth rates in Ti24Al11Nb
Modelling thermomechanical fatigue crack growth rates in Ti-24AI- 11Nb J.J. Pernot*, T. Nicholas* and S, Mall* *Metals and Ceramics Division, Materials Directorate, Wright Laboratory, WrightPatterson AFB, OH 45433, USA tDepartment of Aernonautics and Astronautics, Air Force Institute of Technology, Wright-Patterson AFB, OH 45433, USA (Received 29 October 1992; revised 7 July 1993) A model is developed for predicting the crack growth rate in a titanium aluminide alloy under thermomechanical cycling. The model utilizes a linear summation of cycle-dependent and timedependent terms, where the cycle-dependent behaviour is dependent on temperture as well as AK and R. The time-dependent term is obtained through an integration of da/dt over any cycle, where de/dr depends on K and temperature. A unique feature of this model is the use of a retardation coefficient with the crack growth terms to account for creep or stress relaxation under load at high temperatures. Evolution equations are derived for the retardation term, which can vary during an individual load cycle. The model is calibrated using solely isothermal data and applied to several thermomechanical fatigue conditions, where good correlation is obtained between experimental data and model predictions. (Keywords:TI-24AI-IINb; thermomeehanialcycling;crack growth)
To meet damage tolerance design requirements for aerospace structures, crack growth rate models are required to predict crack propagation from initial or service-induced material defects in order to assure safe life. Therefore, the capability must exist to predict crack growth rates under the environmental conditions encountered in service for each particular material and structural application. For materials used in hightemperature applications, the ability to predict crack growth rates at temperatures up to and including their maximum-use temperature is an important requirement. Further, for some applications, crack growth under conditions of simultaneous cycling of load and temperature, referred to as thermomechanical fatigue (TMF), is another important design consideration. One class of alloys that has received significant attention in recent years is the titanium aluminides. These materials have shown promise for replacing heavier nickel-base superalloys and extending the temperature range of current titanium alloys in turbine engine components. For such applications, TMF is of particular importance in this class of materials, and prediction of crack growth rates presents a formidable challenge. The challenge is even more difficult because of the wide variation in properties such as yield strength and ductility that is observed with temperature 1. BACKGROUND Linear-summation models have been used for a number of years to predict crack growth due to a combination of cycle-dependent and time-dependent mechanisms. 0142-1123/94/020111-12 © 1994Butterworth-HeinemannLtd
In that situation, total crack growth rate, ( d a / d N ) t o t , can be expressed as the sum of two crack growth rate components: a mechanical fatigue contribution, (da/ dN)m, and an environmental contribution, (da/dN)e. This linear-superposition concept, which has been used effectively for corrosive fatigue2-4, has also been used to predict crack growth rates at elevated temperatures5-1°. This form of modelling works effectively for frequency effects in fatigue5,6, as well as for hold-time conditions7-1°. In the application of this type of modelling to elevated temperature conditions, the total crack growth rate is expressed as the sum of a fatigue (or cyclic) contribution and a time-dependent term based on sustained-load crack growth rate, (de/ dt),t data. The fatigue term is generally obtained from high-frequency and/or low-temperature data, and is considered to be independent of temperature for many materials. The time-dependent contribution for a fatigue cycle is computed from an integration of (de/ dt)s~ over the cycle. When a hold time is added to the cycle, (da/dt)s] is multiplied by the hold time to yield the crack growth rate contribution. When these linear-summation concepts were applied to the titanium aluminide alloy Ti-24Al-11Nb, they were found to be inadequate in predicting crack propagation under purely isothermal conditions1Lt2, and isothermal fatigue with superimposed load hold times ~3'~4. Mall et a111 attempted to model the crack growth behaviour in Ti-24AI-11Nb under hold-time conditions at 750 °C using the linear-summation model developed by Nicholas and Weerasooriya9, which predicted crack growth rates in Inconel 718 under
Fatigue, 1994, Vol 16, February
111
Modelling thermomechanical fatigue crack growth rates in Ti-24AI- 11Nb: J.J. Pernot et al. similar conditions. They found that, unlike the results for nickel-based superalloys, hold times did not increase the growth rate per cycle, even though a reduction in frequency resulted in a higher growth rate. They attributed these observations to crack-tip blunting or stress relaxation due to creep at the crack tip, which retards crack growth in Ti-24AI-11Nb, and suggested that this phenomenon has to be accounted for in the model. Parida and Nicholas 13 observed a net retardation due to hold times in this same alloy, whereas lowering the frequency caused an increase in growth rate. Further, hold times at minimum load were found to increase the net growth rate, thereby confirming the existence of an environmental effect that leads to the observed frequency variation in growth rate, as well as a blunting effect that causes retardation when hold times are applied at maximum load. To account for these observed effects in Ti-24AI-11Nb, models have been developed that account for the retardation while still maintaining an environmental contribution 11-~5. In general, these models consider the total crack growth rate as the sum of an effective cycle-dependent crack growth rate and an effective time-dependent crack growth rate, where the effective values account for the creep (retardation) effect. These effective values are obtained by multiplying an unretarded growth rate by a blunting coefficient, which simply shifts the d a / d N versus AK curve. This effective crack growth rate form is very similar in concept to that used by Wheeler ~6 to account for overload crack growth rate retardation. The unretarded growth rates for the cyclic term are those that occur at high frequencies in the absence of hold times. For sustained loading, the growth rates are those that occur immediately after high-frequency loading, when the crack tip is sharp. The linear-superposition concept, which involves adding cycle-dependent and time-dependent contributions, has been adapted to predict crack growth rates under non-isothermal conditions. For the simple case involving no mechanical fatigue component, Haritos et al. 17 successfully modelled sustained-load crack growth during superimposed thermal cycling using a linear-summation technique. The total growth rate of the cycle was calculated from the time integral of sustained-load crack growth rates over the entire thermal cycle. This technique then was extended to TMF conditions ~8-22. Heil et al. 2° modelled crack growth rates during TMF cycling by combining the Wei and Landes 2 approach of adding the mechanical fatigue and environmental crack growth contributions with the Haritos et al. 17 integration procedure. The model of Heil et al. 2° (referred to as HNH) uses the linear sum of the cycle-dependent crack growth rate, (da/dN)cd, and the time-dependent crack growth rate, (da)dN)t d, to predict the total crack growth rate, (da/ d N ) t o t , during a TMF cycle. The (da/dN)ca is obtained from high-frequency isothermal fatigue tests at a temperature where cycle-dependent behaviour dominates, while ( d a / d N ) t d is determined by integrating elevated-temperature sustained-load crack growth rate data, (da/dt)sl, over the loading portion of the fatigue cycle, while (da/dt)sl is non-decreasing. The computation of time-dependent contributions had previously been found to be most accurate when the integration
112 Fatigue, 1994, Vol 16, February
is taken over only the loading portion of the cycle by Nicholas et al. 8 for isothermal fatigue and Jordan and Meyers TM for TMF. Recently, Pernot et al. 23 evaluated the crack growth rate behaviour of Ti-24AI-11Nb over a range of temperatures under cyclic fatigue, fatigue with superimposed hold times, and TMF conditions. They found that the linear-summation models including the H N H TMF model are not sufficient to account for the observed behaviour of the aluminides. This investigation was conducted in order to develop a model capable of predicting the complex isothermal and TMF crack growth rate behaviour of Ti-24AI-11Nb. EQUIPMENT AND EXPERIMENTAL PROCEDURES The material used in this investigation is the alpha2 based titanium aluminide alloy whose nominal composition is T i - 2 4 A l - l l N b (at.%). Details of this alloy are provided in Ref. 24. Compact tension (C(T)) specimens were used throughout the study as the material was in short supply. The specimens were machined to ASTM specifications E 647. The nominal width and thickness are 40.0 mm and 2.54 mm, respectively, and H / W = 0.6. The equipment used in this study comprised components that load, heat and cool the specimen. The loading was performed with a computer-controlled MTS servohydraulic load frame, and the heating and cooling were accomplished with four quartz lamp heaters and compressed air jets, respectively. Direct current electric potential was used to monitor the crack length. Details of the testing system are provided in Ref. 24. Constant Pmaxand constant Kmaxexperiments were used to generate the crack growth rate data. Triangular waveforms were used for both load and temperature, and all experiments were performed with a load ratio, R, of 0.1. Thermomechanical fatigue experiments involved simultaneously cycling load and temperature. All TMF experiments were performed at a frequency of approximately 0.01 (96 s cycles) between 315 °C as the minimum temperature and 650 °C as the maximum temperature. The phase angle between load and temperature, where load leads temperature, included 0 ° (in phase), 180° (out of phase), 90° and 270°. These wave profiles are depicted in Figure 1. Additional TMF tests were conducted using upper-triangularphase and lower-triangular-phase wave profiles as shown in Figure 1. In these profiles, the loading portion of the cycle is identical to that of the in-phase cycle. Additional time is spent at either maximum load or maximum temperature, while the other variable is decreased. The entire cycle time is increased from 96 s for the in-phase cycle to 144 s for the uppertriangular or lower-triangular wave profiles. DEVELOPMENT OF THE TMF MODEL The model proposed for predicting crack growth in Ti-24Al-11Nb under elevated temperature isothermal as well as TMF conditions is based on linear-summation and integration concepts embodied in the TMF model described above (HNH model). It also uses retardation coefficients, described earlier, to account for crack-tip
Modelling thermomechanical fatigue crack yrowth rates i n Tir2VlAI- I 1Nb: J.J. Pernot e t In-Phase
I LT
90° Out-of-Phase
-I
/
\.
I',T
time
time
180° Out-of-Phase
270 °
Out-of-Phase
LT
P, T
time
Ti-24AI-11Nb become frequency-independent, the cycle-dependent crack growth rate still changes as a function of temperature. The model developed here requires integration of both cycle-dependent and time-dependent contributions over an entire TMF cycle. As the cycledependent term is now a function of temperature, its contribution to the total crack growth rate must be determined from an integration, as temperature varies during a TMF cycle. For this reason, the purely cyclic growth rate contribution must be distributed over an entire cycle. Further, the retardation coefficient can vary during the TMF cycle as a function of cycle time. Therefore this retardation coefficient has to appear inside the integral terms of cycle-dependent and time-dependent crack growth rates. Following these procedures, the new model is written in a form similar to the H N H model described earlier as da
Upper-Triangular-Phase
Lower-Triangular-Phase
time
time
Figure I Schematic of thermomechanical wave profiles blunting. While various empirical constants can be chosen to model different materials, temperatures, frequencies and hold times, the major limitation of these retardation models is that they cannot be used when temperature changes during a cycle (that is, during a TMF cycle). Thus, to use the retardation coefficient concept for TMF cycles, a model must be developed that allows parameters to change during the cycle. The present model uses the linear-summation concept, whereby the two crack growth rate components are retarded by a continuously varying coefficient to account for the observed crack growth behaviour, resulting in effective cycle-dependent and time-dependent contributions.
al.
da
where (da/dN)to, the total crack growth rate, is shown as a linear combination of a retarded cycle-dependent crack growth rate and a retarded time-dependent crack growth rate.
Cycle-dependent term The cycle-dependent retarded crack growth rate shown in Equation (1) is expressed as
(d/%)ret cd = f~ul 13 (t) where: 13(0
(!) ( ~da ) u r eddt
(2)
the retardation coefficient, which varies continuously with time; (da/dN)u, oa = the unretarded cycle-dependent crack growth rate, which is a function of temperature; the cycle period, defined as the T inverse of the frequency of the pure fatigue cycle; and tul -~- the uploading time of the cycle, defined as the time required to reach maximum load. =
Basic formulation of the model This modified TMF crack growth rate model differs from the H N H linear-cumulative damage model, described earlier, in two significant ways: first, it utilizes a coefficient to account for the retardation of crack growth due to crack-tip blunting and, second, it allows the cycle-dependent contribution to crack growth to be a function of temperature, as observed in the isothermal crack growth rate results presented in Ref. 23. The retardation coefficient is introduced because the observed hold-time and frequency effects cannot be accounted for in the H N H model under isothermal conditions. Further, it has been demonstrated that the introduction of a retardation coefficient can account for decreases in the crack growth rate contribution during a hold time in Ti-24AI-11Nb, while maintaining the observed frequency effect of the crack growth rates. The temperature dependence of the cycle-dependent crack growth rate contribution is necessary because, although the crack growth rates in
The unretarded crack growth rate is considered as the growth rate of a crack that is not blunted by creep deformation near the crack tip. The unretarded crack growth rates are obtained from very high-frequency fatigue cycles, which are described later. The integration is limited to the loading portion of the fatigue cycle to be consistent with previous modelling efforts 9,18,2°, where it was shown that damage occurs only during loading. In Equation (2), the contribution of the unretarded fatigue crack growth is distributed uniformly with time over the loading half of the cycle. In general, 13 changes as load (P) and temperature (T) change; thus 13 must remain inside the integral. The evolution equation, which provides the rate of change of 13with time, is discussed later. When using this model for alloys that do not exhibit crack growth retardation, such as Inconel 718, the coefficient 13 is equal to unity, and the retarded and unretarded crack growth rates are identical.
Fatigue, 1994, Vol 16, February 113
Modelfing thermomechanical fatigue crack growth rates in Ti-24AI-11Nb: J.J. Pernot et al. In general, the unretarded growth rate, (da/dN)u rcd, is a function of AK, R and T. This unretarded crack growth rate is expressed in terms of the modified sigmoidal equation (MSE), which has been used successfully to represent crack growth rate data over a range of temperatures 25. In the MSE, (da/dN)ur cd is expressed as a function of AK and six independent parameters:
\AKth/J
\~]J
obtained from a fit to the data at v = 5 Hz, 649 °C, and is assumed not to be a function of R or T. With Q taken as a constant, there are only four remaining parameters to define the MSE completely. The cycle-dependent crack growth rate is expressed only as a function of AK, R and T, as the basic premise of cycle dependence is that the growth rates are frequency-independent. The expression used to define the four MSE parameters for the unretarded cycle-dependent crack growth rate in terms of R and T is
(3)
where Agth and AKc are, respectively, the values of AK at the lower and upper asymptotes of the da/dN vs AK curve, AK~ is the value of AK at the curve's inflection point, and Q and D are the lower and upper shaping coefficients, respectively. The parameter Bm is defined as Bm= In ( d~)i
(4)
where (da/dN)i is the value of daMN at the curve's inflection point, and the exponent C is defined as
Q In ( A K i / A K t h )
log (da/dN)i
(daMN);
(da/d 0
In (AKdAK,)
(5)
where (daMN)[ is the slope of the curve at the inflection point when plotted as log (da/dN) vs log (AK). The six independent parameters are AKc, AKth, AKi, (da/dN)i, (da/dN)'i and Q. The MSE is simplified by making the da/dN vs AK curve symmetric about the inflection point; this is accomplished by setting D = - Q . This assumption allows AKc to be expressed as a function of AKi and AKth: A K ¢ - Agth
(6)
With this simplification, the independent parameters that define the symmetric curve are reduced to five: AKth, AKi, (da/dN)i, (da/dN)'i and Q. Subsequently, when the lower and middle regions of the da/dN vs AK curve are defined, the upper region of the curve also is defined. This was used in a previous modelling effort involving the MSE 2°, and it did not restrict the modelling as the region near the upper asymptote of the curve was not used. Similarly, in the current effort, the symmetric assumption is employed, and the upper asymptotic region of the crack growth curve is not modelled. Previous studies involving the modelling of Inconel 718 data 19,26 have shown that the lower shaping coefficient Q was not a function of frequency v, load ratio R or temperature T; therefore assuming that Q is equal to a constant did not impair the modelling process. In the current study, there are insufficient crack growth rate data in the threshold stress intensity region to determine Q adequately for the range of frequencies and temperatures studied; therefore, this region of crack growth is not modelled. Instead, Q is
114 Fatigue, 1994, Vol 16, February
log
ref
(7)
[ (T- T~a) L
where the subscript 'ref' refers to an arbitrary reference load ratio and temperature and the Aq coefficients determine the change in position of the crack growth curve with respect to the reference curve.
Time-dependent term
D
+
log(aKi)
A21 A22
A41 A42
[ \AKi]J
C = \dNJi -
log (da/dN)i
A31 A32
- Dln In
(da I'
log (aKth)
AliA12 [ +
-Qln[ ln(AKil]\AKth/j
log (AKth) log(AKi)
The retarded time-dependent contribution to crack growth, the second term of the TMF crack growth rate model, Equation (1), is defined as da = f£,a 13(t) (da) ~ dt (~1) ret td ur td
(8)
where 13 is the same retardation coefficient as in Equation (2); tnd, the upper limit of integration, is the time when the load begins to decrease; and (da/ dour td is the unretarded time-dependent crack growth rate contribution. The time when the load begins to decrease, tna, is defined as the sum of the uploading time tul and any hold time at maximum load, th. For pure fatigue cycles th = 0; therefore tna = tul. Recall that the upper limit of the cycle-dependent integration is tub as the cycle-dependent crack growth contribution is assumed to occur only during the fatigue portion of the cycle. Heil et al.2° found that a slight modification to the limits of integration for the time-dependent term was necessary for TMF modelling using the H N H model. Under isothermal conditions, the integrand, (da/dt)st, for the time-dependent term in the H N H model begins to decrease when the load begins to decrease, but when temperature and load are both changing during a cycle, this is not necessarily the case. Therefore, in that study, the time-dependent integration was carried out while the integrand was a non-decreasing function. This additional restriction of the integration is not imposed in the current modelling effort. Here, the retardation coefficient 13is multiplied by the unretarded time-dependent crack growth rate, and the integration is carried out until t = tnd regardless of how (da/ dour td is changing. In this case, 13 has the capability to reduce the effect of (da/dt)~t ta if necessary as in
Modelfing thermomechanical fatigue crack growth rates in Ti-24AI- 1lNb: J.J. Pernot et al. the case for the load holds at Pm~ during hold-time tests. The unretarded time-dependent crack growth rate in Equation (8) represents the hypothetical growth rate of a crack if it experienced no retardation effects (that is, 13 -- 1) under a sustained load. This is similar to the unretarded cycle-dependent crack growth rate, where it also represents a condition of no retardation. These crack growth rates cannot be measured directly as in the HNH model, as the sustained-load crack growth rates for the titanium aluminide alloy are, for all practical purposes, equal to zero. Under sustained load, crack growth is totally retarded (the crack tip is assumed to be blunt); therefore the unretarded crack growth rates have to be determined indirectly using the results from isothermal-fatigue and hold-time tests. The procedure for determining (da/dt)urta will be discussed later. The unretarded time-dependent crack growth rate, (da/dt)u~ to, which is a function of the stress intensity K and temperature T, can be expressed in terms of K and T using the MSE. The form of the equation is identical to that shown in Equation (3) except that AK is replaced by K. The unretarded da/dt vs K curves are modelled as symmetric about the inflection point, as were the cycle-dependent unretarded crack growth curves, such that D -- - Q . This simplification does not restrict the capability of the equations in this case, since both asymptotic regions need not be defined. With this simplification, K¢ is defined as shown in Equation (6) if AK is replaced with K. In similar fashion to the unretarded cycle-dependent crack growth rate contribution, the independent MSE parameters are reduced to five when using the assumption of symmetry. As in the cycle-dependent term, Q is not a function of temperature, leaving four parameters to be defined as functions of T: log (Kth)
log (Kth)
log(Ki)
log(Ki)
log (da/dt)i
log (da/dt)i
=
(da/dt)[
(da/dt)~
The coefficient 13 represents the degree of crack blunting at the crack tip. A value of 13 equal to unity represents a totally unretarded crack growth rate condition resulting from high-frequency fatigue. When 13is less than unity, some crack growth rate retardation has occurred, and the amount of retardation is a function of temperature, load and loading frequency. Experimental data suggest that as the temperature increases, 13 decreases, and as frequency decreases, 13 decreases. Also, 13 decreases more quickly for a hold at Pmax than for a hold at Pmin; therefore 13 decreases with increasing load level. A lower limit of retardation, expressed as 13o, defines the level of retardation during steady-state sustained-load crack growth. The retardation coefficient 13is expressed in integral form of a continually changing function of time as 13(t) = 13(to) +
dt- dt
(10)
The associated evolution equation that satisfies the above requirements can be expressed as the following sum of two components: d13 d13 + ~-~ (11) d-t-= ~ - i n c dec The evolution equation, Equation (11), has an increasing (crack-tip sharpening) term, (d13/dt)inc, and a decreasing (crack-tip blunting) term, (d13/dt)dec. These components represent the change in the amount (level) of retardation that is thought to be caused by the sharpening and blunting of the crack tip. The rate of change of 13 is expressed as a function of 13, v, T, and P/Pm~x, where v is the frequency of the fatigue portion of the loading cycle, T is the instantaneous temperature during a cycle, P is the instantaneous load during a cycle, and Pmax is the maximum load encountered during a cycle. The increasing (crack-tip sharpening) contribution to Equation (11) is assumed to have the following form:
ref
( 12) 81
+
B2 B3
0 ( T - Tref)
(9)
B4
Retardation coefficient The retardation coefficient 13 in the integrand of both the cycle-dependent and time-dependent crack growth expressions, Equations (2) and (8), is required to account for the retardation in crack growth rates that has been observed experimentally at elevated temperatures, particularly during the hold-time experiments. This retardation coefficient varies continuously with time during a TMF cycle, as it is a function of both temperature and load. While it is sufficient to use an average or effective value of retardation (cracktip blunting) over an entire cycle when considering isothermal fatigue with and without hold times, for TMF, 13 is required to change during a cycle as the temperature and loading conditions dictate.
fordt = 0
where C1 is a function of v, which is defined later. By definition, this increasing component must be active only during fatigue cycling; therefore (d13/dt)i~ must be equal to zero when the load is held constant (that is, dP/dt = 0). It is postulated that a fatigue cycle sharpens a crack, which is reflected by an increase in d13/dt. The increasing term is constantly driving the retardation coefficient towards unity, which is the steady-state high-frequency fatigue value. If the frequency of the fatigue cycle is increased, 13 approaches unity faster (that is, high-frequency fatigue cycles sharpen the crack faster than lower-frequency cycles). Therefore C1 is a function of frequency that decreases as loading frequency increases and is determined by curve-fitting isothermal crack growth data obtained at different frequencies, as discussed later. The decreasing contribution to Equation (11) is expressed as
Fatigue, 1994, Vol 16, February
115
Modelling thermomechanical 'fatigue crack growth" rates, in Ti;- 2 4 A I . 11Nb : ~J.d. Pernot letal.
where 13ois the value of 13during steady-state sustainedload crack growth, and (72 is a function of temperature. Equation (13) tends to drive the retardation coefficient towards 130 when P is at its maximum value (Pm~) for long periods of time: that is, under steady-state sustained-load crack growth. Crack-tip blunting, and the associated retardation of growth rate, are postulated to occur when a hold time at Pmax is superimposed on a fatigue cycle. This produces a decrease in 13, where the rate of decrease is determined by the current state, 13, and C2, which increases as temperature increases. The function C2 is determined from crack growth rate data from isothermal fatigue at different temperatures and isothermal fatigue with superimposed hold times. P/Pmax is incorporated into Equation (13) to account for the effect of load on the rate of retardation, because hold times at Pmax tend to retard growth much more than those at lower load values. DETERMINATION OF MODEL CONSTANTS
Cycle-dependent term Calibration of the constants for the cycle-dependent term for the Ti-24AI-11Nb alloy involves fitting the MSE to data obtained at the lowest obtainable growth rate at any given temperature. The unretarded growth rate, (da/dN)urcd, is assumed to be equal to the growth rate at the highest frequency tested, since the crack growth rates vary continuously with frequency for all temperatures between 315 and 649 °C and never become purely cycle-dependent. For the purposes of this study, the pure cycle-dependent crack growth rate condition is defined as 100 Hz, as this is the highest frequency for which data are obtained either during this study or found in available literature 27,28. The model is based on the assumption that further increases in frequency will not reduce the growth rate further; therefore there is no basis for using this model above 100 Hz. At this freuency, it is assumed that no retardation occurs; thus 13 is assumed to be equal to unity when defining the parameters for the pure cycledependent crack growth condition. Calibration requires fitting the MSE to data at the reference condition to determine the reference parameters shown in Equation (7). Second, crack growth data at different temperatures and load ratios are performed to establish the A# in Equation (7). In this investigation, the reference condition for the MSE parameters that define (da/dN)ur ¢d is taken as T = 250 °C and R = 0.1. Although this temperature is less than the TMF tempeature range, constant K data at R = 0.1 show a distinct discontinuity in the cycle-dependent crack growth rates at that temperature 23. Complete da/dN vs AK data are not available for this reference condition; however, it is possible to use other fatigue data 13,23 to define the curve completely at T = 250 °C, R = 0.1. For this study, limitations in the database made it desirable not to model behaviour near threshold; therefore a value of AKth is chosen such that the value is less than all the data being modelled. The 5 Hz, 649 °C test produced crack growth rates for the lowest values of AK studied; therefore, the reference AKth is selected
116 Fatigue, 1994, Vol 16, February
based on this test, and is equal to 4.0 MPa m 1/2. Also, as the regions near the threshold and critical values of AK are not modelled, AKth and AKc are taken to be independent of temperature. With these constraints, and the requirement that the da/dN vs AK curve is symmetric (that is, D = - Q ) , the reference value of AK1 is 30.0 MPa m 1/2. The isothermal test data obtained at 649 and 482 °C show that the da/dN vs AK curves shift as temperature varies but remain parallel 24. This makes the slope of the linear region of the curve temperature-independent, further simplifying the modelling. This makes (da/ dN)'i, the slope at the inflection point, a constant, which is easily determined by fitting crack growth data at different temperatures, and then selecting an average value of 2.5, which best describes the data. The parameter (da/dN)i in Equation (7) is taken as 4.5 x 10-6m cycle -1 to match the 5.0 Hz, 649°C isothermal fatigue data, and Q is chosen as 0.4 to provide the best shaping for the curves that are matched. The MSE curve fit to the 5 Hz data is shown in Figure 2. The remaining coefficient A32, which accounts for the temperature dependence of the inflection point, is determined by using 100 Hz data available at 25 and 650 °C. Because there is a considerable change in (da/dN)i for temperatures between 25 and 250 °C, but (da/dN)i remains relatively constant for temperatures between 250 and 650 °C, A32 is determined separately for temperatures below and above 250 °C. The R dependence of the MSE parameters appears in Equation (7) to maintain generality, but is not treated here as all tests considered in this study are performed with R = 0.1. The MSE parameters that represent the unretarded cycle-dependent crack growth rate contribution at R = 0.1 are
10-5 D Experimental Data --MSE Curve Fit /
~D "-~ 10-6
Z
10 -7 T = 649°C v=5Hz R=0.1
10-8
2
1
3
4 5 6789
2
3
10
4 5 6789
100
AK (MPaqm) Figure 2
MSE fit to 5 Hz, 649 °C isothermal crack growth data
Modelling thermomechanicat fatigue crack growth rates in Ti-24A1-11Nb: J.J. Pernot et aL 6.02 x 10-1
J log (AKth)
log(aKi)
1.48
log [(da/dN)i ]
-6.10
(daMN)"
2.50
10 -2
+
u
|
i
|
| m |
I
w
|
i
|
| 1 |
/
10 -3
0.0 0.0
|
10 .4 [ T - 250 °C] (14)
A32 0.0 where A32 is equal to -4.44 x 10-3 (°C) -1 for temperatures less than or equal to 250 °C and equal to 4.44 x 10-4 (°C)-1 for temperatures greater than 250 °C. The MSE parameters that define the unretarded cycle-dependent crack growth rate are determined based on the assumptions that the crack growth rates at and below 250 °C are purely cycle-dependent and the retardation effects can be ignored at 100 Hz.
O O cD
10-5
r~
/ /
(D cD
10 -6
/
/
E 10 -7
/
'a "O
/
10-8
Time-dependent term As for the cycle-dependent term, the threshold and fracture toughness regions of K are not critical to this modelling effort; therefore the values of Kth and Kc are made constant. Equation (6) requires that Ki be constant also. Following a procedure similar to that for the cycle-dependent term, an assumption is made that the unretarded time-dependent crack growth curves remain parallel as temperature changes; thus (da/dt)[ is constant. Only the parameter (da/dt)i varies with temperature, which means that the crack growth curves retain their shape but translate vertically on the da/dt vs K plot as (da/dt)i is varied with temperature. After accounting for the contributions of crack growth rate retardation (the retardation parameter is discussed later), the MSE parameters are defined at a reference condition of 350 °C. To account for the increase in (da/dt)i with temperature, B3 is chosen as 5.0 x 10-3 (°C)-1, and the MSE parameters for the time-dependent unretarded crack growth rate are I log ( K t h ) log(Ki)
0.6021 =
1.6990
log (da/dt)i
-4.5458
(da/dt)[
2.30
0.00 +
(T - 350 °C)
I
I
I
2
3
4 5 6789
I
I
,,,I
1
I
,
*
2
3
4 5 6789
,
,.,
10
100
K (MPaqm) Fr,~-e 3 Unretarded time-dependent crack growth curves
imposed hold times. In matching data, retardation effects (13 < 1) contribute to retardation of crack growth; thus the unretarded (da/dt)i are increased to compensate for the reductions imposed by 13. The values shown in Equation (15) have been adjusted to compensate for such retardation effects.
Retardation coefficient Determination of the expressions for C 1 and (72 and the value of 13o in Equations (12) and (13) involves matching the crack growth rate data of various isothermal and hold-time tests. Following an iterative procedure, crack growth rate data for specific values of AK, taken from constant Km~x and constant Pm~ tests, are used to determine constants at several temperatures. The values of C1, (72 and 130 are then evaluated over the full range of AK for the required range of temperatures and frequencies. This results in expressions for C1 as a function of frequency and (72 as a function of temperature. The details of the iterative procedure are presented in Ref. 24. The expression for C~ is C~ = 1.418 x 10 -3 + 5.816 x 10-2v
500 X 10 -3
0.00
10 -9
(15)
with Q = 0.4. The unretarded crack growth rate curves for T = 350, 500, and 650 °C are presented in Figure 3. These curves represent hypothetical sustained-load crack growth in T i - 2 4 A l - l l N b if the crack were propagating unretarded, where 13 = 1. Since the crack blunts and fails to propagate under a sustained load, these curves are obtained from an iteration process to fit crack growth data from tests under elevatedtemperature fatigue and isothermal fatigue with super-
+ 4.182 x 10-% 2
(16)
where v is frequency, and the expression for (?2 is C2 = 0.3344 - 9.889 x 10-4T + 8.463 x 10-6T 2 - 2.452 × 10-aT 3 + 2.504 x 10-11T 4
(17)
where T is temperature. The value of retardation representing steady-state sustained-load crack growth, 13o, is equal to 1.0 x 10-4.
Fatigue, 1994, Vol 16, February
117
Modelling thermomechanical fatigue crack growth rates in Ti-24AI- 11Nb: J.J. Pernot e t Predictions of crack growth rates for different frequencies, temperatures and hold times were made, to check the validity of the functions and coefficients. The fit of the model to the experimental data used in its calibration is illustrated in Figures 4-6. The predicted crack growth rates for different frequencies at 649 °C for values of AK = 14.85 MPa m 1/2 a r e presented in Figure 4. The crack growth rate predictions adequately represent the experimental data, a result that is expected since the 0.01, 1.0, and 100.0 Hz data are used primarily to calibrate the frequency dependence of the function C1 (Equation (16)). The crack growth rate predictions for temperatures between 315 and 649 °C for frequencies of 0.01 and 1.0 Hz are shown in Figure 5, along with the experimental crack growth rate data. The 0.01 Hz data are used to determine the constants affecting the effect of temperature; therefore it is not surprising that this fit to the data is almost perfect. Conversely, the 1.0 Hz crack growth rate predictions are based on both the fit to the 0.01 Hz data at different temperatures shown in Figure 5 and the fit to the 649 °C data at different frequencies shown in Figure 4. The only data point at 1.0 Hz used during model calibration is the 649 °C point, and the other 1.0 Hz crack growth rate predictions match the experimental data very well. The crack growth rate predictions for hold-time tests are based on both (da/dN)i and (?2, any combination of which can accurately model isothermal fatigue crack growth rates. Hold-time experiments are used to identify the best combination of (da/dN)i and C2. The crack growth rates for 1.0 s cycles with hold times of 10, 100 and 1000 s are shown in Figure 6 with the model predictions for each case. The predicted crack
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Fatigue,
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Modelling thermomechanical fatigue crack growth rates in Ti-24AI- 1lNb: J.J. Pernot et al. growth rates are somewhat high for the 10 and 100 s hold-times, and this may suggest either that the combination of (cla/dN)i and C2 is not the optimum, or that this is a limitation of the model. The 1000 s predicted crack growth rate matches rather well with experimental data, but this is expected as this prediction is a result of fitting [3o to the experimental data.
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MODEL PREDICTIONS AND DISCUSSION OF RESULTS The capability of the model to represent data over a wide range of isothermal conditions is illustrated in Figures 7 and 8, which show crack growth rate predictions for the full range of AK tested. Crack growth rates for frequencies of 0.01, 0.1 and 5.0 Hz are presented in Figure 7. The 5.0 Hz data are used to generate the MSE coefficients that define the shape of the crack growth curve, and the crack growth rates for AK = 14.85 MPa m 1/2 from all three frequencies are used to model the frequency behaviour as shown in Figure 4. The predictions for crack growth rates over the range of AK for 0.01 and 0.1 Hz compare very well with the experimental results. Similarly, the temperature effect is determined from the growth rates obtained at AK = 14.85 MPa m 1/2, and the predictions for the other values of AK at 482 and 649 °C compare well with the experimental data, as shown in Figure 8. Only one data point was obtained at 315 °C, so no further comparisons can be made at this temperature for behaviour at 0.01 Hz. Predictions for TMF crack growth rates, where the temperature is cycled between 315 and 649 °C using a 96 s triangular waveform for load and temperature, are presented in Figures 9 and 10. These comparisons
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Fatigue, 1994, Vol 16, February 119
•Modelling thermomechanical fatigtm crack growth rates in: Ti-24AI~. 11Nb: J.J. Pernot et
out-of-phase results are presented in Figure 10. The crack growth rate predictions for the baseline TMF tests are presented in Figures 9 and 10. Figure 9 shows that the in-phase TMF crack growth rate predictions match the two experimental data sets quite well. The 180° out-of-phase TMF crack growth rate predictions match the data extremely well. In Figure 10, the 90° out-of-phase TMF crack growth rate predictions are observed to be slightly lower than the experimental data for the entire range of the AK values studied. However, the predicted crack growth rates differ from the experimental data by no more than a factor of two, which is considered to be typical specimen-tospecimen variation in data of this type, as illustrated by the in-phase data of Figure 9. The 270° out-ofphase TMF crack growth rate predictions match the data quite well at higher values of AK, while the predictions are slightly greater than the experimental data at the lower values of AK. None the less, all these crack growth rates are within a factor of two of the data, so the predictions are considered to be quite acceptable. Crack growth rate predictions under TMF were also conducted on several proof tests, which are termed the upper-triangular-phase and lower-triangular-phase TMF tests (see Figure 1). Both of the upper- and lower-triangular-phase TMF tests are performed with 48 s load and temperature ramps and 48 s load and temperature holds. These cycles are used to identify the portions of the TMF cycles where damage occurs, and also to verify the limits of integration on the retarded time-dependent calculation (Equation (8)). As discussed previously in the model development, it is postulated that the integration is continued until the
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load begins to decrease. The results presented here support this assumption. The crack growth rate predictions for the proof TMF tests are presented in Figure 11. The model prediction for the two wave profiles is nearly identical and is shown as a single line. Figure 11 shows that the upper-triangular-phase TMF crack growth rate data match the predictions extremely well for the entire range of the AK values studied. However, the lower-triangular-phase TMF crack growth rate data are slightly above the predicted curve, but the two differ by no more than a factor of two. Similar observations in Inconel 718 are discussed by Mall et al. n The results for the two proof tests can be compared with those for the in-phase TMF condition (Figure 9). The predictions for all three cases are nearly identical, a result that is based on the assumption that damage occurs only during the rising-load portion of the cycle. In the proof tests, the upper and lower triangular wave shapes involve longer cycle times than the inphase TMF because of time at maximum load while temperature decreases or at maximum temperature while load decreases, respectively. In either case, if damage were simply an integration of effects due to load at temperture, considerably more damage would occur in the proof tests than in the in-phase TMF condition. Clearly, this is not the case, since the experimental data match the predictions, which are nearly identical for all three cases. Additional evidence for the assumption of damage occurring only during the loading portion of the cycle can be shown by comparing the results for the and 270° TMF conditions shown in Figure 10. Predictions and data for the 270° condition are dearly
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Fatigue, 1994, Vol 16, February
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Modelling thermomechanical fatigue crack growth rates in Ti-24AI- 11Nb: J.d. Pernot et al. higher than those for the 90° case. Examination of the phase diagrams, however, shows that each cycle covers exactly the same conditions, but the loading history is different. For the 270° condition, the load rises when the temperature is towards the upper end of the cycle, while for the 90° condition, the temperature is near the lower end while the load rises. The higher growth rate at elevated temperatures compared with that at lower temperatures results in more crack growth contribution for the 270° condition than for the 90° condition while the load is increasing. As crack growth contributions are not present during the decreasing portion of the cycle, according to our hypothesis, the 270° TMF cycle should have a higher growth rate than the 90° cycle. Both the analysis and experimental results presented in Figure 10 clearly demonstrate this to be the case. SUMMARY AND CONCLUSIONS A model has been developed that has the capability to predict the crack growth rates for isothermal fatigue at various frequencies and temperatures, isothermal fatigue with superimposed hold times at Pmax, and thermomechanical fatigue (TMF) of the titaniumaluminide alloy,Ti-24Al-llNb. The complete model is summarized as ( ~ ) t o t = f~ul ~ (t) ~ 2 ( ~da) u r cd dt + f~nd
da 13(t) ( ~ ) u r td dt
(18)
where the two terms on the right-hand side of the equation are the cycle-dependent and time-dependent crack growth rate contributions, respectively. The two significant features of this model, which no previous efforts have incorporated, are the ability to account for first, the cycle-dependent crack growth rate as a function of temperature and second, the retardation of crack growth rates that is attributed to creep blunting of the crack tip. The cycle-dependent contribution to the total crack growth rate is accounted for in integral form as the reference growth rate is a function of temperature. This temperature-dependent form is a modification of the HNH model 2°, which expressed the cycle-dependent crack growth rate as a function of only R and K. This contribution to crack growth of Inconel 718 at elevated temperatures is equal to that observed at room temperature, which is the case for most materials. The current modelling effort requires the dependence on temperature to model the observed cycle-dependent crack growth behaviour of Ti-24AI-11Nb. The expressions that define ( d a / d N ) u r c d are shown in Equations (3)-(7). The second significant feature of the TMF model developed in this study is that it accounts for crack growth rate retardation. This is accomplished by introducing a coefficient 13 into the expressions for the cycle-dependent and time-dependent contributions to the total crack growth rate, as shown in Equation (18). The coefficient 13 numercially accounts for the experimentally observed retardation effect that is attributed to blunting of the crack tip. The retardation
coefficient 13 appears inside the integral for both crack growth rate contributions as 13 changes continuously during the cycle. The expression for the evolution of 13 (d13/dt) is a function of v, P/Pmax, T and 13. 13 is defined by Equations (10)-(13). The use of this retardation coefficient provides the model's ability to predict crack growth rates for a wider range of materials than any previous modelling effort. Since the coefficients in the expression for the evolution of [3 can vary with materials, 13 can represent the case where creep blunting dominates the crack growth rates (for example, hold-time tests with T i - 2 4 A l - l l N b ) , as well as the case where creep is not present (for example, hold-time tests with the nickel-base superalloy, Inconel 718). As an example, the model presented in this work predicts the crack growth rates for hold-time tests for Ti-24AI-11Nb with 100 s fatigue cycles, where very minor increases in crack growth rates are observed with additions of hold times, which represent one of the more severe cases of crack growth rate retardation. However, for 1 s fatigue cycles with superimposed holds, T i - 2 4 A l - l l N b shows substantial increases in growth rates, and the model predicts increased crack growth rates, although the predictions are somewhat higher than what is observed experimentally. For the hold-time tests described above, the model demonstrates the ability to predict no measurable increases in crack growth rates, while it is also able to predict substantial increases in crack growth rates, depending on both the time of the fatigue portion of the cycle and the time of the hold at Pmx. The model also has the ability to predict purely timedependent crack growth rate behaviour as seen in hold-time tests with Inconel 718. This represents the extreme case where there is no crack growth rate retardation, and 13 is equal to unity. In this latter case, the model reduces to the HNH model, which has been shown to provide excellent predictive capability. ACKNOWLEDGEMENTS This research was part of a Ph.D. dissertation by the first author at the US Air Force Institute of Technology, Wright-Patterson AFB, OH, USA. This work was supported by the Air Force Materials Directorate and the Air Force Office of Scientific Research. The technical support of Mr George Hartman of the University of Dayton Research Institute is gratefully acknowledged. © US Government. REFERENCES 1 Larsen,JM., Williams, K.A., Balsone, S.J. and Stucke, M.A. in 'High Temperature Aluminides and Intermetallics', TMS/ASM International, Metals Park, OH, 1990, pp. 521-530 2 Wei, R. P. and Landes, J. D. Mater. Res. Stand. 1969, 25, 25 3 Kim, Y.H. and Manning, S.D. 'Fracture Mechanics 14th Symposium, Vol 1, Theory and Analysis', ASTM STP 791, American Society for Testing and Materials, PA, 1983, pp. 1-446-I-462. 4 Gabetta,G., Rinaldi, C. and Pozzi, D. in 'Environmentally Assisted Cracking: Science and Engineering', ASTM STP 1049, AmericanSocietyfor Testingand Materials,PA, 1990, pp. 266-282 5 Winstone,M.R., Nikbin, K.M. and Webster, G.A.J. Mater. Sci. 1985, 20, 2471. 6 Dimopulos, V., Nikbin, K.M. and Webster, G.A. Metall. Trans. A 1988, 19A, 873
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Modelling thermomechanical fatigue crack growth rates in Ti-24AI- 11Nb: d.d. Pernot et al. 7 8
9 10 11 12 13 14 15 16 17 18
Saxena, A., Williams, R.S. and Shih, T.T. in 'Fracture Mechanics: Thirteenth Conference', ASTM STP 743, American Society for Testing and Materials, PA,1981, pp. 86-99 Nicholas,T., Weerasooriya, T. and Ashbaugh, N.E. in 'Fracture Mechanics: Sixteenth Symposium', ASTM STP 868, Amerian Society for Testing and Materials, PA, 1985, pp. 167-180 Nicholas, T. and Weerasooriya, T. in 'Fracture Mechanics: Seventeenth Volume', ASTM STP 905, American Society for Testing and Materials, PA, 1986, pp. 155-168 Plumtree, A. and Nal-yong Tang Fatigue Fract. Eng. Mater. Struct. 1989, 12, 377 Mall, S., Nicholas, T., Pernot, J.J. and Burgess, D.G. Fatigue Fract. Eng. Mater. Struct. 1991, 14, 79 Nicholas, T. and Mall, S. in 'Advances in Fatigue Liftime Predictive Techniques', ASTM STP 1122, American Society for Testing and Materials, PA, 1992, pp. 143-157 Parida, B.K. and Nicholas, T. Mater. Sci. Eng. 1992, A153, 493 Mall, S., Staubs, E. A. and Nicholas, T. J. Eng. Mater. Tech. 1990, 112, 435 Nicholas, T. in 'Elevated Temperature Crack Growth',(Eds. S. Mall and T. Nicholas), MD-Vol. 18, American Society of Mechanical Engineers, New York, 1990, pp. 107-112 Wheeler, O.E.J. Basic Eng. 1972, D94, 181 Haritos, G.K., Miller, D.L. and Nicholas, T. J. Eng. Mater. Technol. 1985, 107, 172 Jordan, E.H. and Meyers, G.J. Eng. Fract. Mech. 1986, 23, 345
122 Fatigue, 1994, Vol 16, February
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21 22 23 24 25 26 27
28
Heil, M.L. Ph.D. dissertation, Air Force Institute of Technology, Wright-Patterson AFB OH, 1986 Heil, M.L., Nicholas,T. and Haritos,G.K. in 'Thermal Stress, Material Deformation, and Thermo-Mechanical Fatigue' (Eds. H. Sehitoglu and S.Y. Zamrik), ASME PVP-Vol. 123, American Society of Mechanical Engineers, New York, 1987, pp. 23-29 Nicholas, T., Heil, M.L. and Haritos,G.K. Int. J. Fract. 1989, 41, 157 Harmon, D.M., Saff, C.R. and Burns, J.G. in 'Elevated Temperature Crack Growth', ASME MD-Vol 18, 1990, pp. 37-51 Pernot, J.J., Nicholas, T. and Mall, S. ASME Winter Meeting, Anaheim, CA, USA, 1992, pp. 109-117 Pernot, J.J., Ph.D. dissertation, Air Force Institute of Technology, Wright-Patterson AFB OH, 1991 Nicholas, T., Haritos, G.K. and Christoff, J.R.J. Propulsion and Power, 1985, 1, 131 Painter, G.O.M.S. thesis, Air Force Institute of Technology, Wright-Patterson AFB OH, December 1984 Balsone, S.J., Khobaib, M., Maxwell, D.C. and Nicholas, T. in'Elevated Temperature Crack Growth', (Eds. S. Mall and T. Nicholas), MD-Vol. 18,American Society of Mechanical Engineers, New York, 1990, pp. 87-91 Parida, B.K. and Nicholas, T. Joint FEFG/ICF International Conference on Fracture of Engineering Materials and Structures, Singapore, 6-8 August 1991, pp. 685-690
To meet damage tolerance design requirements for aerospace structures, crack growth rate models are required to predict crack propagation from initial or service-induced material defects in order to assure safe life. Therefore, the capability must exist to predict crack growth rates under the environmental conditions encountered in service for each particular material and structural application. For materials used in hightemperature applications, the ability to predict crack growth rates at temperatures up to and including their maximum-use temperature is an important requirement. Further, for some applications, crack growth under conditions of simultaneous cycling of load and temperature, referred to as thermomechanical fatigue (TMF), is another important design consideration. One class of alloys that has received significant attention in recent years is the titanium aluminides. These materials have shown promise for replacing heavier nickel-base superalloys and extending the temperature range of current titanium alloys in turbine engine components. For such applications, TMF is of particular importance in this class of materials, and prediction of crack growth rates presents a formidable challenge. The challenge is even more difficult because of the wide variation in properties such as yield strength and ductility that is observed with temperature 1. BACKGROUND Linear-summation models have been used for a number of years to predict crack growth due to a combination of cycle-dependent and time-dependent mechanisms. 0142-1123/94/020111-12 © 1994Butterworth-HeinemannLtd
In that situation, total crack growth rate, ( d a / d N ) t o t , can be expressed as the sum of two crack growth rate components: a mechanical fatigue contribution, (da/ dN)m, and an environmental contribution, (da/dN)e. This linear-superposition concept, which has been used effectively for corrosive fatigue2-4, has also been used to predict crack growth rates at elevated temperatures5-1°. This form of modelling works effectively for frequency effects in fatigue5,6, as well as for hold-time conditions7-1°. In the application of this type of modelling to elevated temperature conditions, the total crack growth rate is expressed as the sum of a fatigue (or cyclic) contribution and a time-dependent term based on sustained-load crack growth rate, (de/ dt),t data. The fatigue term is generally obtained from high-frequency and/or low-temperature data, and is considered to be independent of temperature for many materials. The time-dependent contribution for a fatigue cycle is computed from an integration of (de/ dt)s~ over the cycle. When a hold time is added to the cycle, (da/dt)s] is multiplied by the hold time to yield the crack growth rate contribution. When these linear-summation concepts were applied to the titanium aluminide alloy Ti-24Al-11Nb, they were found to be inadequate in predicting crack propagation under purely isothermal conditions1Lt2, and isothermal fatigue with superimposed load hold times ~3'~4. Mall et a111 attempted to model the crack growth behaviour in Ti-24AI-11Nb under hold-time conditions at 750 °C using the linear-summation model developed by Nicholas and Weerasooriya9, which predicted crack growth rates in Inconel 718 under
Fatigue, 1994, Vol 16, February
111
Modelling thermomechanical fatigue crack growth rates in Ti-24AI- 11Nb: J.J. Pernot et al. similar conditions. They found that, unlike the results for nickel-based superalloys, hold times did not increase the growth rate per cycle, even though a reduction in frequency resulted in a higher growth rate. They attributed these observations to crack-tip blunting or stress relaxation due to creep at the crack tip, which retards crack growth in Ti-24AI-11Nb, and suggested that this phenomenon has to be accounted for in the model. Parida and Nicholas 13 observed a net retardation due to hold times in this same alloy, whereas lowering the frequency caused an increase in growth rate. Further, hold times at minimum load were found to increase the net growth rate, thereby confirming the existence of an environmental effect that leads to the observed frequency variation in growth rate, as well as a blunting effect that causes retardation when hold times are applied at maximum load. To account for these observed effects in Ti-24AI-11Nb, models have been developed that account for the retardation while still maintaining an environmental contribution 11-~5. In general, these models consider the total crack growth rate as the sum of an effective cycle-dependent crack growth rate and an effective time-dependent crack growth rate, where the effective values account for the creep (retardation) effect. These effective values are obtained by multiplying an unretarded growth rate by a blunting coefficient, which simply shifts the d a / d N versus AK curve. This effective crack growth rate form is very similar in concept to that used by Wheeler ~6 to account for overload crack growth rate retardation. The unretarded growth rates for the cyclic term are those that occur at high frequencies in the absence of hold times. For sustained loading, the growth rates are those that occur immediately after high-frequency loading, when the crack tip is sharp. The linear-superposition concept, which involves adding cycle-dependent and time-dependent contributions, has been adapted to predict crack growth rates under non-isothermal conditions. For the simple case involving no mechanical fatigue component, Haritos et al. 17 successfully modelled sustained-load crack growth during superimposed thermal cycling using a linear-summation technique. The total growth rate of the cycle was calculated from the time integral of sustained-load crack growth rates over the entire thermal cycle. This technique then was extended to TMF conditions ~8-22. Heil et al. 2° modelled crack growth rates during TMF cycling by combining the Wei and Landes 2 approach of adding the mechanical fatigue and environmental crack growth contributions with the Haritos et al. 17 integration procedure. The model of Heil et al. 2° (referred to as HNH) uses the linear sum of the cycle-dependent crack growth rate, (da/dN)cd, and the time-dependent crack growth rate, (da)dN)t d, to predict the total crack growth rate, (da/ d N ) t o t , during a TMF cycle. The (da/dN)ca is obtained from high-frequency isothermal fatigue tests at a temperature where cycle-dependent behaviour dominates, while ( d a / d N ) t d is determined by integrating elevated-temperature sustained-load crack growth rate data, (da/dt)sl, over the loading portion of the fatigue cycle, while (da/dt)sl is non-decreasing. The computation of time-dependent contributions had previously been found to be most accurate when the integration
112 Fatigue, 1994, Vol 16, February
is taken over only the loading portion of the cycle by Nicholas et al. 8 for isothermal fatigue and Jordan and Meyers TM for TMF. Recently, Pernot et al. 23 evaluated the crack growth rate behaviour of Ti-24AI-11Nb over a range of temperatures under cyclic fatigue, fatigue with superimposed hold times, and TMF conditions. They found that the linear-summation models including the H N H TMF model are not sufficient to account for the observed behaviour of the aluminides. This investigation was conducted in order to develop a model capable of predicting the complex isothermal and TMF crack growth rate behaviour of Ti-24AI-11Nb. EQUIPMENT AND EXPERIMENTAL PROCEDURES The material used in this investigation is the alpha2 based titanium aluminide alloy whose nominal composition is T i - 2 4 A l - l l N b (at.%). Details of this alloy are provided in Ref. 24. Compact tension (C(T)) specimens were used throughout the study as the material was in short supply. The specimens were machined to ASTM specifications E 647. The nominal width and thickness are 40.0 mm and 2.54 mm, respectively, and H / W = 0.6. The equipment used in this study comprised components that load, heat and cool the specimen. The loading was performed with a computer-controlled MTS servohydraulic load frame, and the heating and cooling were accomplished with four quartz lamp heaters and compressed air jets, respectively. Direct current electric potential was used to monitor the crack length. Details of the testing system are provided in Ref. 24. Constant Pmaxand constant Kmaxexperiments were used to generate the crack growth rate data. Triangular waveforms were used for both load and temperature, and all experiments were performed with a load ratio, R, of 0.1. Thermomechanical fatigue experiments involved simultaneously cycling load and temperature. All TMF experiments were performed at a frequency of approximately 0.01 (96 s cycles) between 315 °C as the minimum temperature and 650 °C as the maximum temperature. The phase angle between load and temperature, where load leads temperature, included 0 ° (in phase), 180° (out of phase), 90° and 270°. These wave profiles are depicted in Figure 1. Additional TMF tests were conducted using upper-triangularphase and lower-triangular-phase wave profiles as shown in Figure 1. In these profiles, the loading portion of the cycle is identical to that of the in-phase cycle. Additional time is spent at either maximum load or maximum temperature, while the other variable is decreased. The entire cycle time is increased from 96 s for the in-phase cycle to 144 s for the uppertriangular or lower-triangular wave profiles. DEVELOPMENT OF THE TMF MODEL The model proposed for predicting crack growth in Ti-24Al-11Nb under elevated temperature isothermal as well as TMF conditions is based on linear-summation and integration concepts embodied in the TMF model described above (HNH model). It also uses retardation coefficients, described earlier, to account for crack-tip
Modelling thermomechanical fatigue crack yrowth rates i n Tir2VlAI- I 1Nb: J.J. Pernot e t In-Phase
I LT
90° Out-of-Phase
-I
/
\.
I',T
time
time
180° Out-of-Phase
270 °
Out-of-Phase
LT
P, T
time
Ti-24AI-11Nb become frequency-independent, the cycle-dependent crack growth rate still changes as a function of temperature. The model developed here requires integration of both cycle-dependent and time-dependent contributions over an entire TMF cycle. As the cycledependent term is now a function of temperature, its contribution to the total crack growth rate must be determined from an integration, as temperature varies during a TMF cycle. For this reason, the purely cyclic growth rate contribution must be distributed over an entire cycle. Further, the retardation coefficient can vary during the TMF cycle as a function of cycle time. Therefore this retardation coefficient has to appear inside the integral terms of cycle-dependent and time-dependent crack growth rates. Following these procedures, the new model is written in a form similar to the H N H model described earlier as da
Upper-Triangular-Phase
Lower-Triangular-Phase
time
time
Figure I Schematic of thermomechanical wave profiles blunting. While various empirical constants can be chosen to model different materials, temperatures, frequencies and hold times, the major limitation of these retardation models is that they cannot be used when temperature changes during a cycle (that is, during a TMF cycle). Thus, to use the retardation coefficient concept for TMF cycles, a model must be developed that allows parameters to change during the cycle. The present model uses the linear-summation concept, whereby the two crack growth rate components are retarded by a continuously varying coefficient to account for the observed crack growth behaviour, resulting in effective cycle-dependent and time-dependent contributions.
al.
da
where (da/dN)to, the total crack growth rate, is shown as a linear combination of a retarded cycle-dependent crack growth rate and a retarded time-dependent crack growth rate.
Cycle-dependent term The cycle-dependent retarded crack growth rate shown in Equation (1) is expressed as
(d/%)ret cd = f~ul 13 (t) where: 13(0
(!) ( ~da ) u r eddt
(2)
the retardation coefficient, which varies continuously with time; (da/dN)u, oa = the unretarded cycle-dependent crack growth rate, which is a function of temperature; the cycle period, defined as the T inverse of the frequency of the pure fatigue cycle; and tul -~- the uploading time of the cycle, defined as the time required to reach maximum load. =
Basic formulation of the model This modified TMF crack growth rate model differs from the H N H linear-cumulative damage model, described earlier, in two significant ways: first, it utilizes a coefficient to account for the retardation of crack growth due to crack-tip blunting and, second, it allows the cycle-dependent contribution to crack growth to be a function of temperature, as observed in the isothermal crack growth rate results presented in Ref. 23. The retardation coefficient is introduced because the observed hold-time and frequency effects cannot be accounted for in the H N H model under isothermal conditions. Further, it has been demonstrated that the introduction of a retardation coefficient can account for decreases in the crack growth rate contribution during a hold time in Ti-24AI-11Nb, while maintaining the observed frequency effect of the crack growth rates. The temperature dependence of the cycle-dependent crack growth rate contribution is necessary because, although the crack growth rates in
The unretarded crack growth rate is considered as the growth rate of a crack that is not blunted by creep deformation near the crack tip. The unretarded crack growth rates are obtained from very high-frequency fatigue cycles, which are described later. The integration is limited to the loading portion of the fatigue cycle to be consistent with previous modelling efforts 9,18,2°, where it was shown that damage occurs only during loading. In Equation (2), the contribution of the unretarded fatigue crack growth is distributed uniformly with time over the loading half of the cycle. In general, 13 changes as load (P) and temperature (T) change; thus 13 must remain inside the integral. The evolution equation, which provides the rate of change of 13with time, is discussed later. When using this model for alloys that do not exhibit crack growth retardation, such as Inconel 718, the coefficient 13 is equal to unity, and the retarded and unretarded crack growth rates are identical.
Fatigue, 1994, Vol 16, February 113
Modelfing thermomechanical fatigue crack growth rates in Ti-24AI-11Nb: J.J. Pernot et al. In general, the unretarded growth rate, (da/dN)u rcd, is a function of AK, R and T. This unretarded crack growth rate is expressed in terms of the modified sigmoidal equation (MSE), which has been used successfully to represent crack growth rate data over a range of temperatures 25. In the MSE, (da/dN)ur cd is expressed as a function of AK and six independent parameters:
\AKth/J
\~]J
obtained from a fit to the data at v = 5 Hz, 649 °C, and is assumed not to be a function of R or T. With Q taken as a constant, there are only four remaining parameters to define the MSE completely. The cycle-dependent crack growth rate is expressed only as a function of AK, R and T, as the basic premise of cycle dependence is that the growth rates are frequency-independent. The expression used to define the four MSE parameters for the unretarded cycle-dependent crack growth rate in terms of R and T is
(3)
where Agth and AKc are, respectively, the values of AK at the lower and upper asymptotes of the da/dN vs AK curve, AK~ is the value of AK at the curve's inflection point, and Q and D are the lower and upper shaping coefficients, respectively. The parameter Bm is defined as Bm= In ( d~)i
(4)
where (da/dN)i is the value of daMN at the curve's inflection point, and the exponent C is defined as
Q In ( A K i / A K t h )
log (da/dN)i
(daMN);
(da/d 0
In (AKdAK,)
(5)
where (daMN)[ is the slope of the curve at the inflection point when plotted as log (da/dN) vs log (AK). The six independent parameters are AKc, AKth, AKi, (da/dN)i, (da/dN)'i and Q. The MSE is simplified by making the da/dN vs AK curve symmetric about the inflection point; this is accomplished by setting D = - Q . This assumption allows AKc to be expressed as a function of AKi and AKth: A K ¢ - Agth
(6)
With this simplification, the independent parameters that define the symmetric curve are reduced to five: AKth, AKi, (da/dN)i, (da/dN)'i and Q. Subsequently, when the lower and middle regions of the da/dN vs AK curve are defined, the upper region of the curve also is defined. This was used in a previous modelling effort involving the MSE 2°, and it did not restrict the modelling as the region near the upper asymptote of the curve was not used. Similarly, in the current effort, the symmetric assumption is employed, and the upper asymptotic region of the crack growth curve is not modelled. Previous studies involving the modelling of Inconel 718 data 19,26 have shown that the lower shaping coefficient Q was not a function of frequency v, load ratio R or temperature T; therefore assuming that Q is equal to a constant did not impair the modelling process. In the current study, there are insufficient crack growth rate data in the threshold stress intensity region to determine Q adequately for the range of frequencies and temperatures studied; therefore, this region of crack growth is not modelled. Instead, Q is
114 Fatigue, 1994, Vol 16, February
log
ref
(7)
[ (T- T~a) L
where the subscript 'ref' refers to an arbitrary reference load ratio and temperature and the Aq coefficients determine the change in position of the crack growth curve with respect to the reference curve.
Time-dependent term
D
+
log(aKi)
A21 A22
A41 A42
[ \AKi]J
C = \dNJi -
log (da/dN)i
A31 A32
- Dln In
(da I'
log (aKth)
AliA12 [ +
-Qln[ ln(AKil]\AKth/j
log (AKth) log(AKi)
The retarded time-dependent contribution to crack growth, the second term of the TMF crack growth rate model, Equation (1), is defined as da = f£,a 13(t) (da) ~ dt (~1) ret td ur td
(8)
where 13 is the same retardation coefficient as in Equation (2); tnd, the upper limit of integration, is the time when the load begins to decrease; and (da/ dour td is the unretarded time-dependent crack growth rate contribution. The time when the load begins to decrease, tna, is defined as the sum of the uploading time tul and any hold time at maximum load, th. For pure fatigue cycles th = 0; therefore tna = tul. Recall that the upper limit of the cycle-dependent integration is tub as the cycle-dependent crack growth contribution is assumed to occur only during the fatigue portion of the cycle. Heil et al.2° found that a slight modification to the limits of integration for the time-dependent term was necessary for TMF modelling using the H N H model. Under isothermal conditions, the integrand, (da/dt)st, for the time-dependent term in the H N H model begins to decrease when the load begins to decrease, but when temperature and load are both changing during a cycle, this is not necessarily the case. Therefore, in that study, the time-dependent integration was carried out while the integrand was a non-decreasing function. This additional restriction of the integration is not imposed in the current modelling effort. Here, the retardation coefficient 13is multiplied by the unretarded time-dependent crack growth rate, and the integration is carried out until t = tnd regardless of how (da/ dour td is changing. In this case, 13 has the capability to reduce the effect of (da/dt)~t ta if necessary as in
Modelfing thermomechanical fatigue crack growth rates in Ti-24AI- 1lNb: J.J. Pernot et al. the case for the load holds at Pm~ during hold-time tests. The unretarded time-dependent crack growth rate in Equation (8) represents the hypothetical growth rate of a crack if it experienced no retardation effects (that is, 13 -- 1) under a sustained load. This is similar to the unretarded cycle-dependent crack growth rate, where it also represents a condition of no retardation. These crack growth rates cannot be measured directly as in the HNH model, as the sustained-load crack growth rates for the titanium aluminide alloy are, for all practical purposes, equal to zero. Under sustained load, crack growth is totally retarded (the crack tip is assumed to be blunt); therefore the unretarded crack growth rates have to be determined indirectly using the results from isothermal-fatigue and hold-time tests. The procedure for determining (da/dt)urta will be discussed later. The unretarded time-dependent crack growth rate, (da/dt)u~ to, which is a function of the stress intensity K and temperature T, can be expressed in terms of K and T using the MSE. The form of the equation is identical to that shown in Equation (3) except that AK is replaced by K. The unretarded da/dt vs K curves are modelled as symmetric about the inflection point, as were the cycle-dependent unretarded crack growth curves, such that D -- - Q . This simplification does not restrict the capability of the equations in this case, since both asymptotic regions need not be defined. With this simplification, K¢ is defined as shown in Equation (6) if AK is replaced with K. In similar fashion to the unretarded cycle-dependent crack growth rate contribution, the independent MSE parameters are reduced to five when using the assumption of symmetry. As in the cycle-dependent term, Q is not a function of temperature, leaving four parameters to be defined as functions of T: log (Kth)
log (Kth)
log(Ki)
log(Ki)
log (da/dt)i
log (da/dt)i
=
(da/dt)[
(da/dt)~
The coefficient 13 represents the degree of crack blunting at the crack tip. A value of 13 equal to unity represents a totally unretarded crack growth rate condition resulting from high-frequency fatigue. When 13is less than unity, some crack growth rate retardation has occurred, and the amount of retardation is a function of temperature, load and loading frequency. Experimental data suggest that as the temperature increases, 13 decreases, and as frequency decreases, 13 decreases. Also, 13 decreases more quickly for a hold at Pmax than for a hold at Pmin; therefore 13 decreases with increasing load level. A lower limit of retardation, expressed as 13o, defines the level of retardation during steady-state sustained-load crack growth. The retardation coefficient 13is expressed in integral form of a continually changing function of time as 13(t) = 13(to) +
dt- dt
(10)
The associated evolution equation that satisfies the above requirements can be expressed as the following sum of two components: d13 d13 + ~-~ (11) d-t-= ~ - i n c dec The evolution equation, Equation (11), has an increasing (crack-tip sharpening) term, (d13/dt)inc, and a decreasing (crack-tip blunting) term, (d13/dt)dec. These components represent the change in the amount (level) of retardation that is thought to be caused by the sharpening and blunting of the crack tip. The rate of change of 13 is expressed as a function of 13, v, T, and P/Pm~x, where v is the frequency of the fatigue portion of the loading cycle, T is the instantaneous temperature during a cycle, P is the instantaneous load during a cycle, and Pmax is the maximum load encountered during a cycle. The increasing (crack-tip sharpening) contribution to Equation (11) is assumed to have the following form:
ref
( 12) 81
+
B2 B3
0 ( T - Tref)
(9)
B4
Retardation coefficient The retardation coefficient 13 in the integrand of both the cycle-dependent and time-dependent crack growth expressions, Equations (2) and (8), is required to account for the retardation in crack growth rates that has been observed experimentally at elevated temperatures, particularly during the hold-time experiments. This retardation coefficient varies continuously with time during a TMF cycle, as it is a function of both temperature and load. While it is sufficient to use an average or effective value of retardation (cracktip blunting) over an entire cycle when considering isothermal fatigue with and without hold times, for TMF, 13 is required to change during a cycle as the temperature and loading conditions dictate.
fordt = 0
where C1 is a function of v, which is defined later. By definition, this increasing component must be active only during fatigue cycling; therefore (d13/dt)i~ must be equal to zero when the load is held constant (that is, dP/dt = 0). It is postulated that a fatigue cycle sharpens a crack, which is reflected by an increase in d13/dt. The increasing term is constantly driving the retardation coefficient towards unity, which is the steady-state high-frequency fatigue value. If the frequency of the fatigue cycle is increased, 13 approaches unity faster (that is, high-frequency fatigue cycles sharpen the crack faster than lower-frequency cycles). Therefore C1 is a function of frequency that decreases as loading frequency increases and is determined by curve-fitting isothermal crack growth data obtained at different frequencies, as discussed later. The decreasing contribution to Equation (11) is expressed as
Fatigue, 1994, Vol 16, February
115
Modelling thermomechanical 'fatigue crack growth" rates, in Ti;- 2 4 A I . 11Nb : ~J.d. Pernot letal.
where 13ois the value of 13during steady-state sustainedload crack growth, and (72 is a function of temperature. Equation (13) tends to drive the retardation coefficient towards 130 when P is at its maximum value (Pm~) for long periods of time: that is, under steady-state sustained-load crack growth. Crack-tip blunting, and the associated retardation of growth rate, are postulated to occur when a hold time at Pmax is superimposed on a fatigue cycle. This produces a decrease in 13, where the rate of decrease is determined by the current state, 13, and C2, which increases as temperature increases. The function C2 is determined from crack growth rate data from isothermal fatigue at different temperatures and isothermal fatigue with superimposed hold times. P/Pmax is incorporated into Equation (13) to account for the effect of load on the rate of retardation, because hold times at Pmax tend to retard growth much more than those at lower load values. DETERMINATION OF MODEL CONSTANTS
Cycle-dependent term Calibration of the constants for the cycle-dependent term for the Ti-24AI-11Nb alloy involves fitting the MSE to data obtained at the lowest obtainable growth rate at any given temperature. The unretarded growth rate, (da/dN)urcd, is assumed to be equal to the growth rate at the highest frequency tested, since the crack growth rates vary continuously with frequency for all temperatures between 315 and 649 °C and never become purely cycle-dependent. For the purposes of this study, the pure cycle-dependent crack growth rate condition is defined as 100 Hz, as this is the highest frequency for which data are obtained either during this study or found in available literature 27,28. The model is based on the assumption that further increases in frequency will not reduce the growth rate further; therefore there is no basis for using this model above 100 Hz. At this freuency, it is assumed that no retardation occurs; thus 13 is assumed to be equal to unity when defining the parameters for the pure cycledependent crack growth condition. Calibration requires fitting the MSE to data at the reference condition to determine the reference parameters shown in Equation (7). Second, crack growth data at different temperatures and load ratios are performed to establish the A# in Equation (7). In this investigation, the reference condition for the MSE parameters that define (da/dN)ur ¢d is taken as T = 250 °C and R = 0.1. Although this temperature is less than the TMF tempeature range, constant K data at R = 0.1 show a distinct discontinuity in the cycle-dependent crack growth rates at that temperature 23. Complete da/dN vs AK data are not available for this reference condition; however, it is possible to use other fatigue data 13,23 to define the curve completely at T = 250 °C, R = 0.1. For this study, limitations in the database made it desirable not to model behaviour near threshold; therefore a value of AKth is chosen such that the value is less than all the data being modelled. The 5 Hz, 649 °C test produced crack growth rates for the lowest values of AK studied; therefore, the reference AKth is selected
116 Fatigue, 1994, Vol 16, February
based on this test, and is equal to 4.0 MPa m 1/2. Also, as the regions near the threshold and critical values of AK are not modelled, AKth and AKc are taken to be independent of temperature. With these constraints, and the requirement that the da/dN vs AK curve is symmetric (that is, D = - Q ) , the reference value of AK1 is 30.0 MPa m 1/2. The isothermal test data obtained at 649 and 482 °C show that the da/dN vs AK curves shift as temperature varies but remain parallel 24. This makes the slope of the linear region of the curve temperature-independent, further simplifying the modelling. This makes (da/ dN)'i, the slope at the inflection point, a constant, which is easily determined by fitting crack growth data at different temperatures, and then selecting an average value of 2.5, which best describes the data. The parameter (da/dN)i in Equation (7) is taken as 4.5 x 10-6m cycle -1 to match the 5.0 Hz, 649°C isothermal fatigue data, and Q is chosen as 0.4 to provide the best shaping for the curves that are matched. The MSE curve fit to the 5 Hz data is shown in Figure 2. The remaining coefficient A32, which accounts for the temperature dependence of the inflection point, is determined by using 100 Hz data available at 25 and 650 °C. Because there is a considerable change in (da/dN)i for temperatures between 25 and 250 °C, but (da/dN)i remains relatively constant for temperatures between 250 and 650 °C, A32 is determined separately for temperatures below and above 250 °C. The R dependence of the MSE parameters appears in Equation (7) to maintain generality, but is not treated here as all tests considered in this study are performed with R = 0.1. The MSE parameters that represent the unretarded cycle-dependent crack growth rate contribution at R = 0.1 are
10-5 D Experimental Data --MSE Curve Fit /
~D "-~ 10-6
Z
10 -7 T = 649°C v=5Hz R=0.1
10-8
2
1
3
4 5 6789
2
3
10
4 5 6789
100
AK (MPaqm) Figure 2
MSE fit to 5 Hz, 649 °C isothermal crack growth data
Modelling thermomechanicat fatigue crack growth rates in Ti-24A1-11Nb: J.J. Pernot et aL 6.02 x 10-1
J log (AKth)
log(aKi)
1.48
log [(da/dN)i ]
-6.10
(daMN)"
2.50
10 -2
+
u
|
i
|
| m |
I
w
|
i
|
| 1 |
/
10 -3
0.0 0.0
|
10 .4 [ T - 250 °C] (14)
A32 0.0 where A32 is equal to -4.44 x 10-3 (°C) -1 for temperatures less than or equal to 250 °C and equal to 4.44 x 10-4 (°C)-1 for temperatures greater than 250 °C. The MSE parameters that define the unretarded cycle-dependent crack growth rate are determined based on the assumptions that the crack growth rates at and below 250 °C are purely cycle-dependent and the retardation effects can be ignored at 100 Hz.
O O cD
10-5
r~
/ /
(D cD
10 -6
/
/
E 10 -7
/
'a "O
/
10-8
Time-dependent term As for the cycle-dependent term, the threshold and fracture toughness regions of K are not critical to this modelling effort; therefore the values of Kth and Kc are made constant. Equation (6) requires that Ki be constant also. Following a procedure similar to that for the cycle-dependent term, an assumption is made that the unretarded time-dependent crack growth curves remain parallel as temperature changes; thus (da/dt)[ is constant. Only the parameter (da/dt)i varies with temperature, which means that the crack growth curves retain their shape but translate vertically on the da/dt vs K plot as (da/dt)i is varied with temperature. After accounting for the contributions of crack growth rate retardation (the retardation parameter is discussed later), the MSE parameters are defined at a reference condition of 350 °C. To account for the increase in (da/dt)i with temperature, B3 is chosen as 5.0 x 10-3 (°C)-1, and the MSE parameters for the time-dependent unretarded crack growth rate are I log ( K t h ) log(Ki)
0.6021 =
1.6990
log (da/dt)i
-4.5458
(da/dt)[
2.30
0.00 +
(T - 350 °C)
I
I
I
2
3
4 5 6789
I
I
,,,I
1
I
,
*
2
3
4 5 6789
,
,.,
10
100
K (MPaqm) Fr,~-e 3 Unretarded time-dependent crack growth curves
imposed hold times. In matching data, retardation effects (13 < 1) contribute to retardation of crack growth; thus the unretarded (da/dt)i are increased to compensate for the reductions imposed by 13. The values shown in Equation (15) have been adjusted to compensate for such retardation effects.
Retardation coefficient Determination of the expressions for C 1 and (72 and the value of 13o in Equations (12) and (13) involves matching the crack growth rate data of various isothermal and hold-time tests. Following an iterative procedure, crack growth rate data for specific values of AK, taken from constant Km~x and constant Pm~ tests, are used to determine constants at several temperatures. The values of C1, (72 and 130 are then evaluated over the full range of AK for the required range of temperatures and frequencies. This results in expressions for C1 as a function of frequency and (72 as a function of temperature. The details of the iterative procedure are presented in Ref. 24. The expression for C~ is C~ = 1.418 x 10 -3 + 5.816 x 10-2v
500 X 10 -3
0.00
10 -9
(15)
with Q = 0.4. The unretarded crack growth rate curves for T = 350, 500, and 650 °C are presented in Figure 3. These curves represent hypothetical sustained-load crack growth in T i - 2 4 A l - l l N b if the crack were propagating unretarded, where 13 = 1. Since the crack blunts and fails to propagate under a sustained load, these curves are obtained from an iteration process to fit crack growth data from tests under elevatedtemperature fatigue and isothermal fatigue with super-
+ 4.182 x 10-% 2
(16)
where v is frequency, and the expression for (?2 is C2 = 0.3344 - 9.889 x 10-4T + 8.463 x 10-6T 2 - 2.452 × 10-aT 3 + 2.504 x 10-11T 4
(17)
where T is temperature. The value of retardation representing steady-state sustained-load crack growth, 13o, is equal to 1.0 x 10-4.
Fatigue, 1994, Vol 16, February
117
Modelling thermomechanical fatigue crack growth rates in Ti-24AI- 11Nb: J.J. Pernot e t Predictions of crack growth rates for different frequencies, temperatures and hold times were made, to check the validity of the functions and coefficients. The fit of the model to the experimental data used in its calibration is illustrated in Figures 4-6. The predicted crack growth rates for different frequencies at 649 °C for values of AK = 14.85 MPa m 1/2 a r e presented in Figure 4. The crack growth rate predictions adequately represent the experimental data, a result that is expected since the 0.01, 1.0, and 100.0 Hz data are used primarily to calibrate the frequency dependence of the function C1 (Equation (16)). The crack growth rate predictions for temperatures between 315 and 649 °C for frequencies of 0.01 and 1.0 Hz are shown in Figure 5, along with the experimental crack growth rate data. The 0.01 Hz data are used to determine the constants affecting the effect of temperature; therefore it is not surprising that this fit to the data is almost perfect. Conversely, the 1.0 Hz crack growth rate predictions are based on both the fit to the 0.01 Hz data at different temperatures shown in Figure 5 and the fit to the 649 °C data at different frequencies shown in Figure 4. The only data point at 1.0 Hz used during model calibration is the 649 °C point, and the other 1.0 Hz crack growth rate predictions match the experimental data very well. The crack growth rate predictions for hold-time tests are based on both (da/dN)i and (?2, any combination of which can accurately model isothermal fatigue crack growth rates. Hold-time experiments are used to identify the best combination of (da/dN)i and C2. The crack growth rates for 1.0 s cycles with hold times of 10, 100 and 1000 s are shown in Figure 6 with the model predictions for each case. The predicted crack
10
-, ........
........ I
........ I
........ I
,
. . . . . . . . .
I . . . . . . . . .
I . . . . . . . .
Ti-24AI-I I N b AK = 14.85 MPa m 1/2 R= 0 . I ~
~ O O
10 -6
I--4
Z "~
/O
/ 10 -7
Model Prediction~
/o
- --
0.01 Hz
--
- l . O H z
Experimental Data [] 0.01 Hz o 1.0 Hz Used to Determine C.
• 1('l-8 ~
.......
i .........
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Temperature (°C) Figure $ Crack growth rate predictions at different temperatures
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Fatigue,
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Modelling thermomechanical fatigue crack growth rates in Ti-24AI- 1lNb: J.J. Pernot et al. growth rates are somewhat high for the 10 and 100 s hold-times, and this may suggest either that the combination of (cla/dN)i and C2 is not the optimum, or that this is a limitation of the model. The 1000 s predicted crack growth rate matches rather well with experimental data, but this is expected as this prediction is a result of fitting [3o to the experimental data.
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MODEL PREDICTIONS AND DISCUSSION OF RESULTS The capability of the model to represent data over a wide range of isothermal conditions is illustrated in Figures 7 and 8, which show crack growth rate predictions for the full range of AK tested. Crack growth rates for frequencies of 0.01, 0.1 and 5.0 Hz are presented in Figure 7. The 5.0 Hz data are used to generate the MSE coefficients that define the shape of the crack growth curve, and the crack growth rates for AK = 14.85 MPa m 1/2 from all three frequencies are used to model the frequency behaviour as shown in Figure 4. The predictions for crack growth rates over the range of AK for 0.01 and 0.1 Hz compare very well with the experimental results. Similarly, the temperature effect is determined from the growth rates obtained at AK = 14.85 MPa m 1/2, and the predictions for the other values of AK at 482 and 649 °C compare well with the experimental data, as shown in Figure 8. Only one data point was obtained at 315 °C, so no further comparisons can be made at this temperature for behaviour at 0.01 Hz. Predictions for TMF crack growth rates, where the temperature is cycled between 315 and 649 °C using a 96 s triangular waveform for load and temperature, are presented in Figures 9 and 10. These comparisons
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of experimental data with the model predictions are for different phase angles between load and temperature. In-phase and 180° out-of-phase results are shown in Figure 9, while 90° out-of-phase and 270°
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Fatigue, 1994, Vol 16, February 119
•Modelling thermomechanical fatigtm crack growth rates in: Ti-24AI~. 11Nb: J.J. Pernot et
out-of-phase results are presented in Figure 10. The crack growth rate predictions for the baseline TMF tests are presented in Figures 9 and 10. Figure 9 shows that the in-phase TMF crack growth rate predictions match the two experimental data sets quite well. The 180° out-of-phase TMF crack growth rate predictions match the data extremely well. In Figure 10, the 90° out-of-phase TMF crack growth rate predictions are observed to be slightly lower than the experimental data for the entire range of the AK values studied. However, the predicted crack growth rates differ from the experimental data by no more than a factor of two, which is considered to be typical specimen-tospecimen variation in data of this type, as illustrated by the in-phase data of Figure 9. The 270° out-ofphase TMF crack growth rate predictions match the data quite well at higher values of AK, while the predictions are slightly greater than the experimental data at the lower values of AK. None the less, all these crack growth rates are within a factor of two of the data, so the predictions are considered to be quite acceptable. Crack growth rate predictions under TMF were also conducted on several proof tests, which are termed the upper-triangular-phase and lower-triangular-phase TMF tests (see Figure 1). Both of the upper- and lower-triangular-phase TMF tests are performed with 48 s load and temperature ramps and 48 s load and temperature holds. These cycles are used to identify the portions of the TMF cycles where damage occurs, and also to verify the limits of integration on the retarded time-dependent calculation (Equation (8)). As discussed previously in the model development, it is postulated that the integration is continued until the
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load begins to decrease. The results presented here support this assumption. The crack growth rate predictions for the proof TMF tests are presented in Figure 11. The model prediction for the two wave profiles is nearly identical and is shown as a single line. Figure 11 shows that the upper-triangular-phase TMF crack growth rate data match the predictions extremely well for the entire range of the AK values studied. However, the lower-triangular-phase TMF crack growth rate data are slightly above the predicted curve, but the two differ by no more than a factor of two. Similar observations in Inconel 718 are discussed by Mall et al. n The results for the two proof tests can be compared with those for the in-phase TMF condition (Figure 9). The predictions for all three cases are nearly identical, a result that is based on the assumption that damage occurs only during the rising-load portion of the cycle. In the proof tests, the upper and lower triangular wave shapes involve longer cycle times than the inphase TMF because of time at maximum load while temperature decreases or at maximum temperature while load decreases, respectively. In either case, if damage were simply an integration of effects due to load at temperture, considerably more damage would occur in the proof tests than in the in-phase TMF condition. Clearly, this is not the case, since the experimental data match the predictions, which are nearly identical for all three cases. Additional evidence for the assumption of damage occurring only during the loading portion of the cycle can be shown by comparing the results for the and 270° TMF conditions shown in Figure 10. Predictions and data for the 270° condition are dearly
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Fatigue, 1994, Vol 16, February
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Modelling thermomechanical fatigue crack growth rates in Ti-24AI- 11Nb: J.d. Pernot et al. higher than those for the 90° case. Examination of the phase diagrams, however, shows that each cycle covers exactly the same conditions, but the loading history is different. For the 270° condition, the load rises when the temperature is towards the upper end of the cycle, while for the 90° condition, the temperature is near the lower end while the load rises. The higher growth rate at elevated temperatures compared with that at lower temperatures results in more crack growth contribution for the 270° condition than for the 90° condition while the load is increasing. As crack growth contributions are not present during the decreasing portion of the cycle, according to our hypothesis, the 270° TMF cycle should have a higher growth rate than the 90° cycle. Both the analysis and experimental results presented in Figure 10 clearly demonstrate this to be the case. SUMMARY AND CONCLUSIONS A model has been developed that has the capability to predict the crack growth rates for isothermal fatigue at various frequencies and temperatures, isothermal fatigue with superimposed hold times at Pmax, and thermomechanical fatigue (TMF) of the titaniumaluminide alloy,Ti-24Al-llNb. The complete model is summarized as ( ~ ) t o t = f~ul ~ (t) ~ 2 ( ~da) u r cd dt + f~nd
da 13(t) ( ~ ) u r td dt
(18)
where the two terms on the right-hand side of the equation are the cycle-dependent and time-dependent crack growth rate contributions, respectively. The two significant features of this model, which no previous efforts have incorporated, are the ability to account for first, the cycle-dependent crack growth rate as a function of temperature and second, the retardation of crack growth rates that is attributed to creep blunting of the crack tip. The cycle-dependent contribution to the total crack growth rate is accounted for in integral form as the reference growth rate is a function of temperature. This temperature-dependent form is a modification of the HNH model 2°, which expressed the cycle-dependent crack growth rate as a function of only R and K. This contribution to crack growth of Inconel 718 at elevated temperatures is equal to that observed at room temperature, which is the case for most materials. The current modelling effort requires the dependence on temperature to model the observed cycle-dependent crack growth behaviour of Ti-24AI-11Nb. The expressions that define ( d a / d N ) u r c d are shown in Equations (3)-(7). The second significant feature of the TMF model developed in this study is that it accounts for crack growth rate retardation. This is accomplished by introducing a coefficient 13 into the expressions for the cycle-dependent and time-dependent contributions to the total crack growth rate, as shown in Equation (18). The coefficient 13 numercially accounts for the experimentally observed retardation effect that is attributed to blunting of the crack tip. The retardation
coefficient 13 appears inside the integral for both crack growth rate contributions as 13 changes continuously during the cycle. The expression for the evolution of 13 (d13/dt) is a function of v, P/Pmax, T and 13. 13 is defined by Equations (10)-(13). The use of this retardation coefficient provides the model's ability to predict crack growth rates for a wider range of materials than any previous modelling effort. Since the coefficients in the expression for the evolution of [3 can vary with materials, 13 can represent the case where creep blunting dominates the crack growth rates (for example, hold-time tests with T i - 2 4 A l - l l N b ) , as well as the case where creep is not present (for example, hold-time tests with the nickel-base superalloy, Inconel 718). As an example, the model presented in this work predicts the crack growth rates for hold-time tests for Ti-24AI-11Nb with 100 s fatigue cycles, where very minor increases in crack growth rates are observed with additions of hold times, which represent one of the more severe cases of crack growth rate retardation. However, for 1 s fatigue cycles with superimposed holds, T i - 2 4 A l - l l N b shows substantial increases in growth rates, and the model predicts increased crack growth rates, although the predictions are somewhat higher than what is observed experimentally. For the hold-time tests described above, the model demonstrates the ability to predict no measurable increases in crack growth rates, while it is also able to predict substantial increases in crack growth rates, depending on both the time of the fatigue portion of the cycle and the time of the hold at Pmx. The model also has the ability to predict purely timedependent crack growth rate behaviour as seen in hold-time tests with Inconel 718. This represents the extreme case where there is no crack growth rate retardation, and 13 is equal to unity. In this latter case, the model reduces to the HNH model, which has been shown to provide excellent predictive capability. ACKNOWLEDGEMENTS This research was part of a Ph.D. dissertation by the first author at the US Air Force Institute of Technology, Wright-Patterson AFB, OH, USA. This work was supported by the Air Force Materials Directorate and the Air Force Office of Scientific Research. The technical support of Mr George Hartman of the University of Dayton Research Institute is gratefully acknowledged. © US Government. REFERENCES 1 Larsen,JM., Williams, K.A., Balsone, S.J. and Stucke, M.A. in 'High Temperature Aluminides and Intermetallics', TMS/ASM International, Metals Park, OH, 1990, pp. 521-530 2 Wei, R. P. and Landes, J. D. Mater. Res. Stand. 1969, 25, 25 3 Kim, Y.H. and Manning, S.D. 'Fracture Mechanics 14th Symposium, Vol 1, Theory and Analysis', ASTM STP 791, American Society for Testing and Materials, PA, 1983, pp. 1-446-I-462. 4 Gabetta,G., Rinaldi, C. and Pozzi, D. in 'Environmentally Assisted Cracking: Science and Engineering', ASTM STP 1049, AmericanSocietyfor Testingand Materials,PA, 1990, pp. 266-282 5 Winstone,M.R., Nikbin, K.M. and Webster, G.A.J. Mater. Sci. 1985, 20, 2471. 6 Dimopulos, V., Nikbin, K.M. and Webster, G.A. Metall. Trans. A 1988, 19A, 873
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Modelling thermomechanical fatigue crack growth rates in Ti-24AI- 11Nb: d.d. Pernot et al. 7 8
9 10 11 12 13 14 15 16 17 18
Saxena, A., Williams, R.S. and Shih, T.T. in 'Fracture Mechanics: Thirteenth Conference', ASTM STP 743, American Society for Testing and Materials, PA,1981, pp. 86-99 Nicholas,T., Weerasooriya, T. and Ashbaugh, N.E. in 'Fracture Mechanics: Sixteenth Symposium', ASTM STP 868, Amerian Society for Testing and Materials, PA, 1985, pp. 167-180 Nicholas, T. and Weerasooriya, T. in 'Fracture Mechanics: Seventeenth Volume', ASTM STP 905, American Society for Testing and Materials, PA, 1986, pp. 155-168 Plumtree, A. and Nal-yong Tang Fatigue Fract. Eng. Mater. Struct. 1989, 12, 377 Mall, S., Nicholas, T., Pernot, J.J. and Burgess, D.G. Fatigue Fract. Eng. Mater. Struct. 1991, 14, 79 Nicholas, T. and Mall, S. in 'Advances in Fatigue Liftime Predictive Techniques', ASTM STP 1122, American Society for Testing and Materials, PA, 1992, pp. 143-157 Parida, B.K. and Nicholas, T. Mater. Sci. Eng. 1992, A153, 493 Mall, S., Staubs, E. A. and Nicholas, T. J. Eng. Mater. Tech. 1990, 112, 435 Nicholas, T. in 'Elevated Temperature Crack Growth',(Eds. S. Mall and T. Nicholas), MD-Vol. 18, American Society of Mechanical Engineers, New York, 1990, pp. 107-112 Wheeler, O.E.J. Basic Eng. 1972, D94, 181 Haritos, G.K., Miller, D.L. and Nicholas, T. J. Eng. Mater. Technol. 1985, 107, 172 Jordan, E.H. and Meyers, G.J. Eng. Fract. Mech. 1986, 23, 345
122 Fatigue, 1994, Vol 16, February
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Heil, M.L. Ph.D. dissertation, Air Force Institute of Technology, Wright-Patterson AFB OH, 1986 Heil, M.L., Nicholas,T. and Haritos,G.K. in 'Thermal Stress, Material Deformation, and Thermo-Mechanical Fatigue' (Eds. H. Sehitoglu and S.Y. Zamrik), ASME PVP-Vol. 123, American Society of Mechanical Engineers, New York, 1987, pp. 23-29 Nicholas, T., Heil, M.L. and Haritos,G.K. Int. J. Fract. 1989, 41, 157 Harmon, D.M., Saff, C.R. and Burns, J.G. in 'Elevated Temperature Crack Growth', ASME MD-Vol 18, 1990, pp. 37-51 Pernot, J.J., Nicholas, T. and Mall, S. ASME Winter Meeting, Anaheim, CA, USA, 1992, pp. 109-117 Pernot, J.J., Ph.D. dissertation, Air Force Institute of Technology, Wright-Patterson AFB OH, 1991 Nicholas, T., Haritos, G.K. and Christoff, J.R.J. Propulsion and Power, 1985, 1, 131 Painter, G.O.M.S. thesis, Air Force Institute of Technology, Wright-Patterson AFB OH, December 1984 Balsone, S.J., Khobaib, M., Maxwell, D.C. and Nicholas, T. in'Elevated Temperature Crack Growth', (Eds. S. Mall and T. Nicholas), MD-Vol. 18,American Society of Mechanical Engineers, New York, 1990, pp. 87-91 Parida, B.K. and Nicholas, T. Joint FEFG/ICF International Conference on Fracture of Engineering Materials and Structures, Singapore, 6-8 August 1991, pp. 685-690