A micromechanistic-based approach to fatigue life modeling of titanium-matrix composites

PII: S1359-8368(96)00083-2

ELSEVIER

Composites Part B 28B (1997) 507-521 © 1997 Published by Elsevier Science Limited Printed in Great Britain 1359-8368/97/$17.00

A micromechanistic-based approach to fatigue life modeling of titanium-matrix composites M.A. Foringer, D.D. Robertson and S. Mall* Department of Aeronautics and Astronautics, Air Force Institute of Technology, Wright-Patterson AFB, OH 45433, USA (Received 5 December 1995; accepted 20 September 1996) Fatigue life modeling of titanium-based metal-matrix composites (MMCs) was accomplished by combining a unified viscoplastic theory, a non-linear micromechanics analysis and a damage accumulation model. The micromechanics analysis employed the Bodner-Partom unified viscoplastic theory with directional hardening. This analysis was then combined with a life-fraction fatigue model to account for the time-dependent component of fatigue damage. The life-fraction fatigue model involved the linear summation of damage from the fiber and matrix constituents of the composite. A single set of empirical constants for the life-fraction fatigue model were established for each of two titanium MMCs reinforced with silicon carbide fibers: SCS-6/Ti-15-3 and SCS-6/ TIMETAL®21s. The predicted fatigue lives were within one order of magnitude of the experimental data for different loading conditions: isothermal fatigue, and both in-phase and out-of-phase thermomechanical fatigue. MMCs modeled included cross-ply, quasi-isotropic and unidirectional SCS-6/TIMETAL®21s, and cross-ply and quasi-isotropic SCS-6/Ti-15-3 laminates. © 1997 Elsevier Science Limited. (Keywords: A. metal-matrix composites (MMCs); B. fatigue; C. thermomechanical; D. damage mechanics)

INTRODUCTION Titanium-based metal-matrix composites (MMCs) have been the focus of study in recent years and show good promise for high-temperature use. Two such systems, SCS6/TIMETAL®21s ( T i - 1 5 M o - 3 N b - 3 A 1 - 0 . 2 S i w t % ) and SCS-6/Ti- 15-3 (Ti- 15V-3Cr-3A1-3 Sn wt%), have been tested extensively under both isothermal and thermomechanical fatigue (TMF) loading 1-1°. The present study seeks to develop a fatigue life methodology for these composites by employing the damage accumulation model of Nicholas et a l ) '11-14 coupled with a three-dimensional non-linear micromechanics analysis 15-18 along with a unified viscoplastic t h e o r y 19-21. Titanium alloys behave as a viscoplastic material. They can be accurately modeled with the Bodner-Partom unified viscoplastic theory with directional hardening a9-21. A set of viscoplastic material parameters for TIMETAL®21s are already available 22, but this is not the case for T i - 1 5 - 3 . A previous study listed some of these material constants 15, but they were obtained from various experimental sources which employed a mixed set of data from both as-received and the heat-treated condition of this material. Rogacki and Tuttle accomplished constant-strain-rate testing of T i - 1 5 - 3 at temperatures above 480°C 23, but there were no data * To whom correspondence should be addressed

available for the lower temperatures. Therefore, a few experiments were conducted at the lower temperatures in the present effort to characterize the viscoplastic material parameters of the heat-treated T i - 1 5 - 3 material over a full spectrum of temperatures. The viscoplastic material parameters, obtained in this study for the T i - 15 - 3 material and obtained from the previous study for TIMETAL®21 s, were then integrated into a modified method-of-cells micromechanics model 15-18'24 to predict fatigue lives of these titanium-matrix composites under a variety of mechanical and thermal loading cycles. The constituent microstress calculations from the micromechanics solutions were used as input for a fatigue life model 1A1-14. This model is a mechanistic-based lifefraction model involving the summation of damage from cycle-dependent and time-dependent effects. This modeling approach has been used by Nicholas and colleagues where these two effects are modeled by empirical-based cycle- and time-dependent terms in several linear and non-linear lifefraction models 1Al-14. Because the majority of the timedependent fatigue damage in titanium-matrix composites is due to viscoplastic deformation of the matrix, the timedependent damage term can be replaced by information from the viscoplastic micromechanics analysis. Replacing the time-dependent terms reduces the number of empirical parameters required to predict the composite's behavior. Additionally, allowing the micromechanics model to

507

Fatigue life modeling of MMCs: M. A. Foringer et al.

m V - ~

Cross-Section of Composite

/ RW J Analysis Cell

A

J

int 2

int 1 Figure

1 Micromechanicsunit cell

address the time dependence through a viscoplastic theory provides a more physically based model. Results from this model are compared with experimental data for TIMETAL®21s and T i - 1 5 - 3 matrix composites with some different ply lay-ups for different frequencies in both isothermal fatigue and TMF loading.

of the deviatoric stress tensor defined as

J2=~oq

t

00

r

and Do and n are material constants. Z I is given by (.I_~r,

ml Wp'Z1- Zi"] - A1ZI oz, + L\Z1 -Z2,I ~-T+ \Z1

MICROMECHANICS WITH VISCOPLASTICITY To predict the behavior of a metal-matrix composite, it is necessary to understand bow the constituents of the composite behave micromechanically. The approach used in the present study is a modified method-of-cells approach which has successfully been applied to predict composite material behavior 15. It is based on Aboudi's unit cell approach z4 as shown in Figure 1. The stresses in each of the four regions are assumed to be constant• The model uses the Bodner-Partom theory of viscoplasticity to characterize the matrix material. It treats the fiber material as linear elastic and allows for a statistical distribution of damage at the interface between the fiber and matrix 16'18. The Bodner-Partom theory for viscoplastic materials with directional hardening is classified as a unified theory and, as such, possesses a single inelastic strain term in its constitutive relationship• The rate of inelastic strain deformation is given by

~I=Doexp[-~((ZI+zD)2"~n] 3J2

uiJ'

(1)

J Jv~

where Z I and Z D represent the isotropic and directional hardening, o/j' is the deviatoric stress, J2 is the second invariant

508

(2)

Z1 .] - Z 2 J OT-J

(3)

where the plastic work rate is • .i Wp = aijeij

(4)

Z D = ~ijuij

(5)

Z D is defined as

where

30 and uij are further defined as

~ij=m2Wp(Z3uij_fJij)_A2Z 1 +

3o oz3

~ ~-~

~ij ( ~kl~I~ r2 (6)

and Uij =

%

(7)

In the above equations, T is temperature and A1, A2, rl, r2, Z1, Z2, Z3, ml and mE, as well as the previously mentioned Do and n, are material constants. Viscoplastic characterization of the T i - 1 5 - 3 alloy involved solving the above equations for the stress-strain

Fatigue life modeling of MMCs: M. A. Foringer et al. response at constant strain rate for comparison with experiment. The sequence of steps used in this study to compute the stress for a single uniaxial strain-time increment is given below. This is the same basic algorithm as is used in the micromechanics program except it has been adapted to handle uniaxial strain-control mode in the present study 25. (1) Initialize the stress and elastic strain rate at interval tp = tp_l + At by assuming that the response is purely elastic: ap = Op_ 1 "~ EkAt

(8)

~I= 0

(9)

(2) Calculate the inelastic strain rate arising from the directional and isotropic hardening resulting from the current values of a v and kI by using the Bodner-Partom theory: Wp = tTpbI

-

m, Wp

-

alZ}l

ZI

( il)

ZI mlZ3Wp-Q p-I

(12)

1~At- Q

Zip= Z p1 - i + dZI

(13)

(z4

-' )

-m=%-a2tz)

(14) (15)

dZ D = m2Z3Wp + Q2ZD- 1 1~At - Q2

= Zp_ 1 -[-dZO

ca,=

(16)

exp - - -

% /

(17)

(3) Determine the change in the effective inelastic strain rate from the calculated value in step (2) and the value used for the previous iteration: •I __ ~ I = ~cal

(18)

~I

(6) Use the new inelastic strain rate to calculate the stress for the current iteration: O-p= Orp_1 + E ( k - kl)At

(22)

(7) Check convergence. If: - < 0.001 6~:max

(23)

then use the value of ap calculated in step (6) and proceed to step (1) with the next time step. If not, return to step (2), using the new values of stress and inelastic strain rate.

- 6~I

6~ I -- kt~kmax

6kI

6~I + k6kmax

Many researchers have developed models to predict the fatigue life of titanium-matrix composites under isothermal or TMF conditions. A few have proposed models based on statistical considerations of the behavior exhibited by one or more constituents of the composite 26-2s. Other researchers have developed empirical functions to model fatigue based on the behavior of the components ling. Still other researchers have examined the stress redistribution between layers of the composite as the various layers reach their maximum strength 3°. In this latter approach, the fatigue of a composite laminate containing 0 ° plies is assumed to be controlled primarily by the behavior of the 0 ° plies 3°. The present effort simplifies one of the empirical models by replacing the time-dependent term with viscoplastic micromechanics solutions for the matrix. This model was developed by Nicholas et al. in a series of publicationsl'l 1-14. The basic hypothesis of this approach (life-fraction model) under isothermal fatigue is that the life of the composite (N) can be determined by a combination of a cyclic term (Nc) and a time-dependent term (Nt) such that 1

1

1

1

U - Nc +- Nt + Ni

(24)

where Ni is an interaction term between the cyclic and timedependent effects

(4) Calculate the scaling ratio, R, by:

R=

(21)

FATIGUE MODELING

Z2) r'

i rl ~

.~___

dZi_

k I ----k I q- RtS~max

(10) /

a

(5) Update the inelastic strain rate with the new scaling ratio:

for

6k I <

0

N i = CV/NcNt

(19) for 6~ I ~ 0

where k is a user-supplied constant (usually between 2 and 30) to control convergelace stability and ~bmax is the maximum change in strain rate between iterations and is given by: ~i~:max= 7 ~ [~2 --k2(~'~)2]

(20)

(25)

and C is an empirical constant. In this model, the empirical parameters are functions of temperature. For the present effort, the N t and Ni terms are removed, and the timedependent behavior of the composite is accounted for by employing the viscoplasticity-based micromechanics analysis. A model to predict fatigue life under TMF has been developed which was also based on the isothermal nonlinear model described by eqn (24) 1. In this model, the cyclic term is further broken down into terms governed by

509

Fatigue life modeling of MMCs: M. A. Foringer et al. Table 1 Bodner-Partom temperature dependent viscoplastic constants for Ti-15-3 Temp (°C)

E (GPa)

Z2 (MPA)

Z3 (MPA)

n

a] = a2 (sec -i)

m2 (MPa-l)

25 315 427 482 566 650

86.3 80.4 77.5 72.2 64.4 53.0

1200 1070 1020 850 750 650

250 454 550 1100 2400 3000

4.5 2.9 2.7 1.6 1.05 0.9

10 -8 4.4 × 10 -6 10 -5 1 2.5 3

0.005 0.04 0.05 5 15 20

the fiber and matrix stresses as follows: 1

1

+

1

Uc Uf Um

(26)

where Nf and Nr~ represent the number of cycles to failure when fibers and matrix stress contribute to the fatigue damage, respectively. The terms Nf and Nm are represented as follows: NO(1 - 0rf'max~

Nf = l0

a0 /

Nm = n m (Atrm) - n o

(27) (28)

where fff,maxis maximum stress in the fiber, Airm is the stress range in the matrix, and No, a0, Bm, nm are empirical constants, a,, and (re must be determined by a micromechanics model of the composite. Unlike in earlier versions of this model 13, o0 does not have a defined physical significance in this study. It is used here as an additional empirical constant. A characteristic of the above approach is the inability to predict the fatigue life of the composite under different temperatures and loading conditions by using just one form of model with one set of constants. The present effort attempts to alleviate this by using a single set of constants for a material system. With the removal of the timedependent terms by basing the time-dependent fatigue characteristics on the viscoplastic behavior, eqns (26)-(28) are used with the micromechanics analysis in the present study to predict both isothermal and thermomechanical fatigue lives.

VISCOPLASTIC CHARACTERIZATION As mentioned, one part of this study was viscoplastic characterization of the T i - 1 5 - 3 matrix which required two separate tasks. The first was to select and perform experiments at appropriate temperatures to complement the data already available in the literature 23. The second task was to determine which constants in the Bodner-Partom model will best predict the experimental stress-strain curves. Prior to testing, the panel was heat-treated at 700°C for 24 h in an argon environment to stabilize the/3 phase of the material 1°. The panel was then cut into individual rectangular samples, 152.4 mm × 12.7 mm × 1.84 mm in size. A total of five tests was accomplished: one at room temperature with a strain rate of 0.001 s -1, and two

510

each at 315°C and 427°C with strain rates of 0.01 s -1 and 0.0001 S -1. Although a systematic approach to find the BodnerPartom constants has been developed 31, it was not used because it requires more experimental data than were available for the present study. Instead, the parameters were developed empirically by comparing the stress-strain curves generated by the theory with experimental results. Some initial assumptions were made to facilitate this effort: (1) the viscoplastic behavior of T i - 1 5 - 3 is similar to that of TIMETAL®21 s which has been characterized previously22; (2) the viscoplastic constants are functions of temperature, but not of strain rate; and (3) the elastic constants (E and v) are as listed in a previous work 32. Since there are more constants than data curves, there are probably multiple sets of constants that can satisfactorily model the available experimental data. The emphasis in this effort was to determine constants that would accurately predict the elastic behavior, the yield point, and the plastic hardening behavior at the higher strain rates. In this effort, emphasis was placed on the lower temperatures owing to the fact that there are more fatigue data available for SCS-6/ T i - 1 5 - 3 below 450°C. The initial guess for the Bodner-Partom constants was taken from 15. These previous constants were determined from an incomplete set of experimental data and were found to predict the elastic behavior and the yield point reasonably well, but did a poor job of predicting plastic hardening. With the addition of the experimental data from the present study, and employing the approach outlined above, the BodnerPartom constants for the titanium matrix material, T i - 1 5 - 3 , are as shown in Table 1. Figures 2-7 show the stress-strain response of T i - 1 5 - 3 at different temperatures and strain rates along with the predicted response based on the previous constants 15, and from the new constants at the six temperatures listed in Table 1. As Figure 2 shows, both the set of constants from the previous work 15 and those determined in the present effort correlate well with the experimental results at room temperature. The present method, however, is better in predicting the actual behavior. As temperature increases, the difference between the stress-strain curves resulting from the old and new constants is more pronounced (Figures 3-7). Experimental data from the previous study 23 were used to generate the constants for temperatures above 427°C. At these higher temperatures, strain-rate effects increased. Figures 5-7 show the stress-strain curves for 482°C, 566°C and 649°C obtained from experiments and from viscoplastic analysis using the previously developed

Fatigue life modeling of MMCs: M. A. Foringer et al. IOOC

"~" r/l

I

J

500

r/l

t12

j

¢/3

I

I

-'"2---___2___2__22222

Experiment .... Present Constants - - Previous Constants [15]

"

I

I

I

0

0.005

0.01

0.015

0.02

Strain Figure 2

T i - 15-3 stress-strain response, 25°C

100(

'

.01 Strain/see [] .0001 Strain/see

~

G"

50C ¢/'3

B

/

J

0

f

i 0.005

'

'

,, ,_.- ~:~ . . . . . . .

/

- ~2 - -

__._._.

.,.~...: . . . . . . . . ~:1. . . . . . . . . . . . . .

- - Experiment .... Present Constants i-- Previ°us ~ °nstants [15]

0.01

0.015

0.02

Strain Figure 3

T i - 15-3 stress-strain response, 315°C

constants and the new constants for both fast and slow strain rates. As can be seen in these figures, T i - 1 5 - 3 behaves substantially differently at temperatures above 450°C than at temperatures below 450°C. Strain rate has much more effect on the stress-strain response at the higher temperatures. Additionally, the behavior after yield at the higher temperatures possesses little strain hardening. The new constants tend to overpredict plastic hardening at the slower strain rates, but do a good job of predicting the plastic behavior at the high strain rates. Since the frequencies in the fatigue studies used in the present effort tend to be greater than 1 Hz, more emphasis was put on establishing a good curve fit at the higher strain rates. The viscoplastic constants affect the predicted stressstrain curve in various interactive ways. The constants Z1 and n primarily govern the yield point and the magnitude of the stress in the plastic region while m l and a l jointly determine the level of plastic hardening. The constant a l

also controls the strain-rate sensitivity when Z3 is greater than Z2. Figure 8 shows Z2 and Z3 plotted against temperature. A transition occurs at about 450°C. Below this temperature, Z2 is greater than Z3, and the stress-strain behavior is relatively independent of strain rate. Above this temperature, Z3 is greater, and the stress-strain behavior is highly dependent on strain rate. The values of r2 and r 3 were not changed from those determined in the previous study 15. In addition, previous studies 21-23 assume that Do is equal to 10 4 S -1, and that assumption is also made for the present study. Once viscoplastic characterization was accomplished, these new constants were incorporated into the micromechanics analysis which was then used to determine if the new constants improved on the predictive accuracy for a laminate when compared with experimental data 1°. The angle-ply lay-up ([ --- 45]s) was used for this purpose since it is assumed that the matrix dominates the behavior of this particular laminate under load. These results are shown in

511

Fatigue life modeling of MMCs: M. A. Foringer et al.

1000

I

I

I

.01 Strain/sec e~.0001 Strain/sec s

S~'~-

IB1 .

.

.

.

.

.

.

FI-

.

_

"v'

........ -_

500 j~'

- - Experiment ' Present Constants ~ - Previous Constants[ 15]

~" /

,

0

0.005

0.01

0.015

0.02

Strain Figure 4

Ti-15-3 stress-strain response, 427°C

1000

I

I

I

.01 Strain/sec 17 .0001 Strain/sec ......

_

-- -- _" ~'--'--"

:,. . . . . . . . . . . .

500 --

Experiment

[23]

• .. P r e s e n t C o n s t a n t s - - Previous

I

0

0.005

I

0.01

Constants

[15]

I

0.015

0.02

Strain Figure 5 Ti-15-3 stress-strain response, 482°C

Figure 9. As this figure shows, the micromechanics model with the new viscoplastic constants for the matrix do an excellent job in predicting the behavior of the angle-ply titanium-matrix composite.

FATIGUE LIFE PREDICTIONS The fatigue life prediction model discussed earlier requires constituent microstress values. Therefore, the micromechanics program LISOL 17 was used with BodnerPartom constants 21 for TIMETAL®21s obtained from a previous study and the Bodner-Partom constants developed for T i - 1 5 - 3 in this effort to determine the microstresses in the composite. A variety of laminate orientations and load cases were examined in the analysis. For instance, laminates with unidirectional, cross-ply and quasi-isotropic lay-ups were examined under isothermal fatigue and TMF loadings.

512

With the exception of one case involving tensioncompression loading (R = - 1 ) , all the cases involved tension-tension (R = 0.1) loading. In each case, the maximum fiber stress (Of,max), usually in the 0 ° ply, and the maximum matrix stress (AOm) , usually in matrix region 2 of the 0 ° ply (Figure 1), were used. The microstresses computed from LISOL were related to the applied composite stress by using a second- or third-degree polynomial fit. These relationships for o'f,max and A0" m w e r e subsequently inserted into eqns (26)-(28), and the computed fatigue lives were compared with isothermal fatigue life data at elevated temperature and TMF life data (R=0.1). A single set of constants was developed for each matrix material (Table 2). If the mechanistic effects during cycling are correctly modeled through the micromechanics and fatigue life-fraction models, then a single set of fatigue life constants should apply under many different types of fatigue loading.

Fatigue life modeling of MMCs: M. A. Foringer et 1000

I

I

I

_Experiment [23] •.. Present Constants - - Previous Constants [15]

-

al.

-

.01 Strain/sec El .0001 Strain/sec

• . . . . " .-'"" .~.-.=-=-c _'L_-'L'2L': "L'-'L'-:. r~

500

o,) .::'~

..... q.3"--

.............°i . . . . .

"~

-(3"-

'

-

".."..'. 7..'..".

tJ

0 0

0.005

0.015

0.01

0.02

Strain Figure 6 T i - ] 5-3 stress-strain response, 566°C

1000

I

I

I

- - Experiment [23] .01 Strain/sec •.. Present Constants D .0001 Strain/sec - . Previous Constants [ 15] 500

0J

z•:..........._.

1:ft. G , ,: :', r,.,. . . . . . . . . .

i

0

0

I

0.005

I

0.01

.

0.015

0.02

Strain Figure 7 Ti-15-3 stress-strain response, 649°C

t~

4000

I

I

I

- 0 - Z2 •~I- Z3 ¢.q

°

2000

°"

© r.~

o

• .... . 0

,

o

I

I

200

400

~

.I 600

800

Temperature (C) Figure 8 Z2 and Z3 for T i - 1 5 - 3

versus

temperature

513

Fatigue life modeling of MMCs: M. A. Foringer et al. 400

I

I ~

~

.°°

.....

...°

300

200

O') O')

j t

100

0

- - Experiment [ 10] ... P r e s e n t C o n s t a n t s -'- P r e v i o u s C o n s t a n t s [15]

0

I

I

0.005

0.01

0.015

Strain Figure 9

SCS-6/Ti-15-3 angle-ply stress-strain response, 427°C

1000

~t~,

I

I

I

J

800 600 o, ..... °.°°.°..°,

r~

.....

•

°

400

.01 H z .... . 1 H z

.i-.~

200 0 0

-I 5

I 10

1Hz I 15

-

20

Cycle Figure 10

SCS-6/TIMETAL21s cross-ply microstress v e r s u s cycle, 650°C isothermal fatigue, R = 0.1, O'max = 200 MPa

Initially, tension-tension fatigue loading of cross-ply SCS-6/TIMETAL®21s under both isothermal fatigue at 650°C and TMF with 150°C/650°C cycles was analyzed 1. The frequencies were 0.01, 0.1 and 1 Hz for the isothermal case and 0.00556 Hz for TMF. The maximum applied loads for each frequency ranged from 200 to 600 MPa. These values were chosen to match the experimental conditions 1. In each case, the microstresses were stabilized by the 20th cycle (i.e. viscoplastic effects were saturated). The differences between fiber stresses at the 19th and 20th cycle for 0.01, 0.1 and 1 Hz were 0.5%, 0.8% and 0.5%, respectively, at a maximum applied stress of 200 MPa. Figure 10 shows the fiber stress as calculated by the micromechanics model plotted against the cycle count. Similar behavior was exhibited at higher stress levels with the maximum difference being 0.6%. At 400 and 500 MPa and 0.1 Hz, the fiber stresses achieved a constant value before the 20th cycle. The matrix microstresses also stabilized along with the fiber stresses at the 20th cycle, with a worst-case

514

difference between the 19th and 20th cycles of 1.7%. Generally, this difference was less than 1%. The microstresses for TMF showed similar behavior. Figure 11 shows the fiber stresses as determined by LISOL during the first 20 cycles. As Figure 11 demonstrates, the viscoplastic effects as shown by the change in the 0 ° fiber stress are fairly well stabilized by the 20th cycle. The differences between the fiber stresses in the 19th and 20th cycles for in-phase and out-of-phase fatigue were 0.3% and 0.2%, respectively. Once fiber and matrix microstresses were computed from the micromechanics analysis, LISOL, they were combined with the damage accumulation model [eqns (26)-(28)]. The S - N curves generated by this model for the various frequencies were then compared with experimental data. The fatigue model constants (No, a0, Bm, nm) were then varied to produce the best match between the predicted and experimental fatigue life of the cross-ply laminate for isothermal fatigue at elevated temperature, in-phase TMF and out-of-phase TMF. This provided the values of these

Fatigue life modeling of MMCs: M. A. Foringer et al.

2000

I

I

I

t~

1500 1000 x

In Phase .... Out of Phase

500

t~

I

I

I

5

10

15

20

Cycle Figure 11

SCS-6/TIMETAL21s cross-ply fiber stress, in-phase and out-of-phase thermomechanical fatigue, 3 min 150/650°C cycles

700

I

I

I

I

I

500 r/)

- - Prediction (1 Hz; "~?'..xN,x 0 Experiment (l Hz) [1]" - ,'O..N,N " - ~...'?':~ 300 .... Prediction (0.1 Hz) 1:3 Experiment (0.1 H'z) [1] ~-~',~'-.'~. PredmUon (0.01 Hz) ~".'N -t- Experiment (0.01 Hz) [1] I I I I I 100 4 5 6 10 100 1000 10 10 10 •

.

%

Oo

10

C y c l e s to Failure Figure 12

SCS-6/TIMETAL21 s cross-ply, 650°C isothermal fatigue S - N curves, various frequencies

700

I

I

I

0 IV exp [1] Prediction (IP) 0 0 P exp [1] n (OP)

500 "".O

I

~

O'1

ID r/3

300

-

~

-

D

100 10

I

I

100

1000

'o

"O

'1

104

I

105

106

Cycles to Failure Figure 13

SCS-6/TIMETAL21s cross-ply thermomechanical fatigue S - N curves, in-phase and out-of-phase

515

Fatigue life modeling of MMCs: M. A. Foringer et al. Table 2

106

i

i

I

i©

/i

105

E Z

Fatigue model constants

Matrix material

No

a0

Bm

nm

TIMETAL®21s Ti-15-3

7.5 6.5

3800 2800

1.4 × 1018 1.2 × 10 TM

5.50 5.35

10 4 of the cross-ply laminate under TMF loading. Figure 15 compares the predicted cycles to failure with the actual cycles to failure for both in-phase and out-of-phase TMF. The accuracy of the model is significantly greater for out-ofphase TMF than for in-phase. This is probably due to the greater scatter in the in-phase TMF data for this laminate 1. For both high-cycle and low-cycle fatigue, the predicted results deviate from the actual results by less than one order of magnitude. The present model was then used to predict the fatigue lives of quasi-isotropic ([0/+45/90]s) and unidirectional laminates under TMF loading using the same constants that were derived for the cross-ply laminate as shown in Table 2. The TMF cycle time was again 3 min, and the load ratio was again equal to 0.1. In each case, as with the cross-ply laminate, the fiber and matrix microstresses were taken from the 0 ° ply in the 20th cycle as generated by LISOL. Figures 16 and 17 show the S - N curves predicted by the model along with experimental data 2. The present model is fairly accurate at predicting thermomechanical fatigue lives for the quasi-isotropic laminate by using the same empirical constants as developed for the cross-ply laminate. However, owing to the presence of different failure mechanisms between unidirectional laminates and laminates containing off-axis lamina, the model underpredicts out-of-phase TMF life in the unidirectional case by roughly one order of magnitude. Unlike in

1000 [] / / / ( ~ ~ 1 Hz // -I[ ] .1 Hz t .01 Hz /

100 10

/

// 10

i 100

I 1000

I 105

104

-

106

NExp [11 Figure 14 SCS-6/TIMETAL21s cross-ply, 650°(2 isothermal fatigue life prediction comparison, various frequencies

constants as listed in Table 2. The resulting S - N predictions and experimental points 1 are shown in Figures 12 and 13. A comparison of predicted cycles to failure and the actual cycles to failure for the isothermal fatigue is shown in Figure 14. It can be seen in this figure that the model predicts fatigue life with a good level of accuracy except in the low-cycle range. Overall, use of the strain-rate effects on the matrix as determined by the viscoplastic micromechanics model to account for the time-dependent fatigue behavior seemed to produce fairly accurate results over the range of frequencies examined. The present model also correlated well with the behavior

!

105

_

'

(~)Q InPhase

'/ ]

104

"o 1000

Z 100

10

- OI - I© I 10

100

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Cycles to Failure Figure 20 SCS-6/Ti-15-3 quasi-isotropic thermomechanical fatigue S - N curves, in-phase and out-of-phase cross-ply and quasi-isotropic lay-ups, the fiber/matrix interface is not subjected to normal stresses in a unidirectional composite under axial load, leading to different damage mechanisms between these two cases. The present fatigue life methodology was applied similarly to the isothermal and thermomechanical fatigue of laminates with Ti-15-3 as the matrix. The constants developed for the fatigue life model with SCS-6/Ti-15-3 are shown in Table 2. The resulting S - N curve for isothermal fatigue of the cross-ply laminate at 427°C and 10 Hz is shown in Figure 18. The fatigue life model with the constants as listed in Table 2 were also applied to cross-ply and quasi-isotropic SCS-6/Ti-15-3 laminates under TMF loading. The temperature was cycled between 149°C and 427°C both in-phase and out-of-phase with the mechanical loading. The cycle period was 48 s and the load ratio was 0.1. Figures 19 and 20 show the resulting S - N curves for in-phase (IP) and

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out-of-phase (OP) TMF for the two laminates as compared with experiment. Figures 18-20 provide similar conclusions to those drawn from the SCS-6fFIMETAL®21s laminate fatigue results. A single fatigue life model is sufficient to predict the fatigue behavior for both cross-ply and quasi-isotropic laminates. In addition to isothermal fatigue at elevated temperature (427°C), a room temperature (25°C) comparison with the SCS-6/Ti-15-3 cross-ply laminate is made in Figure 21. Although the constants for the fatigue life model were developed for isothermal fatigue at elevated temperature (427°C) and TMF (149-427°C), the mechanistic-based lifefraction model also provides a good correlation to the roomtemperature data. A constituent's damage accumulation is primarily controlled by the microstresses in the constituent. Therefore, if a realistic micromechanics model is employed to evaluate these constituent microstresses for use in a damage accumulation model such as the life-fraction model

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discussed in this study [eqns (26)-(28)], then an estimate of the fatigue life is readily obtained. To evaluate the present model's ability to predict the fatigue life in a different fatigue loading condition, it was applied to isothermal tension-compression (R = - 1 ) fatigue of cross-ply SCS-6/Ti-15-3 at 427°C as shown in Figure 22. As in the tension-tension cases, the microstresses were obtained from LISOL and then employed in the fatigue model to predict life. The predicted fatigue lives were then compared with experimental data 33. The predicted lives are within half an order of magnitude of the experimental life. Thus, once again, the model demonstrates its versatility since it did reasonably well in predicting tension-compression fatigue life despite being developed solely from tension-tension fatigue data.

In all cases, both the fiber and matrix microstresses used in this model were determined in the 0 ° ply. The fiber stresses in the 90 ° lamina were a small fraction of those in the 0 ° ply. The matrix stresses in the off-axis ply were lower than in the on-axis layer by only about 15% in the worst case. The microstresses were also determined during the 20th cycle to allow the viscoplastic effects to stabilize. By the 20th cycle, either a constant value of maximum fiber stress was reached, or the difference in the fiber stress between consecutive cycles was less than 1%. This behavior was seen for both isothermal fatigue and TMF. The matrix microstresses behaved in a similar manner, except that the matrix stress range decreased with successive cycles as opposed to increasing. A single set of constants in the present fatigue model can

519

Fatigue life modeling of MMCs: M. A. Foringer et al. be used to predict fatigue life within one order of magnitude of experimental results for both isothermal fatigue and TMF (in-phase and out-of-phase) loading conditions. The present model is capable of providing at least, and usually better than, an order-of-magnitude estimate for the fatigue life of cross-ply, quasi-isotropic and unidirectional titanium-based metal-matrix composites under different fatigue loading conditions. The higher levels of accuracy are achievable on the laminates containing off-axis plies.

2. 3.

4.

CONCLUSIONS Fatigue life modeling of titanium-based MMCs was accomplished by combining a unified viscoplastic theory, a non-linear micromechanics analysis and a damage accumulation model. The viscoplastic theory is the Bodner-Partom theory with directional hardening and was used to model the matrix material in the titanium-matrix composites SCS-6/TIMETAL®21s and SCS-6/Ti-15-3. The appropriate viscoplastic material parameters were obtained from the literature or from tests of the pure matrix, as required 19-23. The micromechanics analysis involved a unit cell or representative volume element (RVE) approach, where the RVE was partitioned into regions of c o n s t a n t stress 15-18"24'25. This analysis was combined with a life-fraction fatigue model 1'2'12-14 to predict the fatigue life of the titanium-based MMC laminates under a variety of fatigue loading conditions. The life-fraction model involved the linear summation of damage from the fiber and matrix constituents of the composite. In the present study, the time-dependent damage term was removed from the life-fraction fatigue model of the previous studies 1'2'~2-~4, and instead, time-dependent effects were accounted for through the viscoplasticity-based micromechanics. A single set of empirical constants for the linear lifefraction fatigue model was established for each of two titanium metal-matrix composites reinforced with silicon carbide fibers; SCS-6/Ti- 15-3 and SCS-6/TIMETAL®21 s. Fatigue life predictions for the various loading conditions of isothermal fatigue (tension-tension and tensioncompression) and thermomechanical fatigue (in-phase and out-of-phase) were accomplished for unidirectional, crossply and quasi-isotropic laminates. The predicted fatigue lives correlated well with the experimental data for laminates with off-axis plies, but the model needs some refinement to accurately model unidirectional lay-ups. In addition, comparison with experimental data of different frequencies demonstrated the capability of the present approach to account for viscoplastic time-dependent fatigue effects in the loadcontrolled cases examined. At this point, the model does not address other types of time-dependent effects.

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