Fatigue life of individual composite specimens based on intrinsic fatigue behavior

Int. J. Faligue Vol

19. Nc~ 5. pp. 369 377, 1996 ': 19~17 Elsevier Science l,td All righls reserved. Printed m Great Britain 0142 I 123/97/$17.00+.1)0

EI.SEVIER

PIh S0142-1123(97)00004-2

Fatigue life of individual composite specimens based on intrinsic fatigue behavior Howard G. Halverson*, William A. Curtin and Kenneth L. Reifsnider

Materials Response Group, Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061, USA (Received 25 September 1996; revised 13 January 1997; accepted 21 January 1997) A method is presented to predict fatigue life which explicitly accounts for specimen specific damage histories and the statistical distribution of quasi-static tensile strengths. First a 'critical element' is identified as that portion of the composite which controls failure, and the stress on that elemenl is monitored. Then a cumulative damage model based on the remaining strength of the critical element, is used to extract an "intrinsic' S-N curve for the critical element fatigue response from the fatigue data and the stress history. Using unidirectional specimens, such an intrinsic S-N curve was derived, and subsequent fatigue predictions were vastly improved over the traditional S-N curve. Fatigue tests on (90°/0°)~ crossply materials were then run and lives predicted based on the intrinsic S-N curve with no adjustable parameters. The agreement between experiment and prediction validates the method. This method may have application in real-time durability evaluation of individual composite components using real-time stiffness to continuously evaluate remaining life during service. © 1997 Elsevier Science Ltd.

(Keywords: composite; fatigue; life prediction)

INTRODUCTION

studied for many years. Many fatigue mechanisms have been identified and their effects characterized ~ 4. Fatigue has typically been treated as a deterministic process: however complicated the mechanistic modeling, most models yield a single value of predicted life for an applied stress. Of course, real materials display a great deal of experimental scatter in fatigue. There are two major components to this scatter, variations in the static strength and variations in the propagation rates associated with the fatigue mechanisms 5. One method of accounting for the static strength variation is the strength-life equal-rank assumption (SLERA) 6. In essence, this assumption states that if a set of specimens could simultaneously be tested tk)r static strength and fatigue life expectancy, each individual specimen would occupy the same rank in both data sets. Hence, by knowing the quasi-static strength distribution, a range of fatigue lives for a given applied stress can be obtained. However, the SLERA does not consider the variation in the effects of the fatigue mechanisms among individual specimens. As a result, it gives only probabilities involving multiple specimens. The question we must ask is whether there is a methodology of observing and/or predicting the progression of fatigue mechanisms in any particular specimen, based on experimentally determined parameters. If this is possible, we can use this information, in conjunction with the SLERA, to improve the fatigue life prediction for that specimen. Such a methodology is proposed

Engineers who use composite materials for design are often confronted with a trade-off between reliability and efficiency. Both the strength and fatigue response of composite materials may vary widely, even among nominally identical specimens. As a result, designs are intentionally kept to low strains, well below the fatigue limit. While structurally sound, this overdesign of structures is undesirable, especially for those design applications where weight is critical. One example of this situation is the problem with determining the fatigue life of composite laminates. Testing is expensive and time-consuming, and must be repeated if the arrangement or number of plies is changed. Even when testing is done, the variability from one sample to another still cannot preclude failure at (what are considered to be) moderate strains. As a result, there has been much research on the fatigue properties of composite materials, for unidirectional, crossply, and other arrangements. We now have a tar better understanding of fatigue mechanisms than we did ten years ago, but one major problem remains--that of predicting the fatigue life of a particular specimen, or a particular piece of the airframe, or any specific composite structure placed under load. The mechanisms of composite fatigue have been *Corresponding author. Ph: 540-231-7493, Fax: 540-231-9187

369

H.G. Halverson et al.

370

below. It is demonstrated and validated through testing and analysis of fatigue failure on model composite systems. The method we propose is based on a concept that has been used successfully to examine composite fatigue for over a decade. The reader is referred to other references 7'8 for details, but there are two main ideas in this methodology: the separation of the structure into critical and sub-critical elements and a damage evolution scheme for evaluating the critical element remaining strength. The critical element is that portion of the structure which controls and/or defines failure-when it fails, the structure as a whole fails. Sub-critical element failure does not result in structural failure, but it does affect the stress state on the critical element. For example, in the uniaxial fatigue of composites containing plies aligned with the load direction, the failure of these 0 ° plies often leads directly to failure of the composite as a whole. These plies as a group are then the critical element. The sub-critical elements are the off-axis plies which typically sustain a great deal of damage during the fatigue process. This damage transfers stress to the axial plies, which then accelerates fatigue degradation of the critical element, and hence the laminate as a whole. We characterize the critical element by its normalized remaining strength Fr (normalized by an undamaged static strength). The damage evolution scheme then accounts for decreasing Fr during the load history of the critical element. Simply put, the current normalized strength of the critical element, Fr, is reduced by the effect of some normalized applied stress function, Fa (also normalized by the undamaged static strength), acting over some generalized time r according to

Fr=l-f[(1-Fa)j(r'y

1dr'

(1)

For fatigue with no thermal or other effects, r is the current cycle count normalized by the expected number of cycles to produce failure at the current Fa, i.e. n

r = "N(Fa)

problem to relate sub-critical element damage to the increase in applied load on the critical element. For unidirectionally reinforced composites, fatigue loading often results in the fracture and subsequent splitting of strips of fibers. These strips often completely debond from the specimen. If we assume that this strip carries no load, then the intact area of the composite is decreased, leading to a decrease in structural stiffness and an increase in cross-section stress (in a loadcontrolled test). This may occur several times during a fatigue test. We treat the broken strips of composite as sub-critical elements, while the remaining intact region contains the critical element and other, as yel unbroken, sub-critical elements. Even though we do not know beforehand exactly which portion of the composite is the critical element, by monitoring the stiffness of the specimen during the test we can determine the stress, and thus the normalized applied load Fa, on it at each point during the test. Second, we postulate that the critical element has an intrinsic stress-life curve, one that is obscured at the macroscopic level by the damage progression during a test. The form of this S - N curve is assumed for simplicity to be log-linear: O"

x,

= Fa = A + B log(N)

(3)

where ~r is the applied load on the critical element, X, is the initial quasi-static strength of the critical element. N is the number of cycles to failure, and A and B are material constants. Implicit in this equation is the SLERA: as X, increases, N must also increase (with B negative). We describe the cumulative distribution of quasi-static strength, X,, by the well-known Weibull distribution,

= , -e

(L)"

where Pj(o-) is the probability of specimen failure at or below applied stress o-, with a scale factor X,o and a shape factor p. This distribution is obtained through a series of quasi-static tests. Once X,o and p have been determined, the strength Xt associated with a probability of failure Pr follows from Equation (4) as I

(2)

although in other cases it may be a normalized time. Here j is a material constant, typically taken from experience 7"8 to be 1.2. Failure occurs when Fr = Fa, i.e., the remaining strength of the critical element has been reduced to such a degree that it is no longer able to withstand the applied loads. Fa includes the applied loads plus the loads transferred to the critical element due to damage in sub-critical elements, and is therefore a function of the specific sample history. The remainder of this paper is as follows. In Section 2 we discuss the methodology in detail. Section 3 provides experimental verification. Discussion and conclusions are contained in Section 4. M E T H O D O L O G Y FOR F A T I G U E LIFE PREDICTION First, we conceptually divide the structure into critical and sub-critical elements and then solve a mechanics

Xt = X~[ - ln(1 - Pf)] , .

(5)

If we assume that the critical element in quasi-static tension is identical to the critical element in tensiontension fatigue, the distribution of quasi-static strength can then be taken identical to that of the initial fatigue strength. Thus, we can combine Equations (3) and (5) to calculate the probability distribution of fatigue lives for the critical element under some constant local applied stress. Third, we relate sub-critical damage, critical element fatigue, and remaining strength through the aforementioned damage evolution scheme. Consider first a constant local stress Fa in the critical element as if the critical element alone was being tested in fatigue. Then we can directly integrate Equation (1) to yield

f r = 1 - (1 - F a ) ~ = l - (1 - f a ) ~

(6)

where n is the elapsed number of cycles and N is the life at Fa as given by Equation (3). For different

Fatigue life of individual composite specimens based on intrinsic fatigue behavior

371

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Failure occurs when Fr, = f'a, and lhis locus of

strength Fr~. If the local load on the critical element then changes to Faz, Ft" does not change, and the initial location on the new damage curve corresponding to stress Fa2 must be that cycle n0 which has exactly the remaining strength Fr~, as if the material had been subjected to stress Fa2 for /10 cycles. Mathematically, this can be expressed as the equivalence of two damage states, one defined by (Fa~, nJN~) and the other by (Fa2, n°/N2):

applied loads Fa this yields a family of remaining strength curves due to evolving fatigue of the critical element and relates Fr, Fa, and n as shown in Figure 1. Now consider the composite containing sub-critical elements. If, after some time, damage occurs in the sub-critical elements then the specimen stiffness decreases and the stress Fa acting on the critical element changes. Further reductions in remaining strength are then governed by the remaining strength curve corresponding to the new applied stress, Fa2 and starting at the current value of the remaining strength Fr. The transition to the new remaining strength curve thus uses F r as a measure of damage, i.e. Fr completely describes the state of damage of the critical element and the exact load history leading to the current Fr value is inconsequential. For example, as indicated in Figure 2, when we apply a load Faj for nl cycles the critical element has some remaining

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The principle of equivalent damage states. The sample is subjected to a load Fa~ tbr n~ cycles followed by a load Fa~ to failure. The total fatigue life is n~ + (N~ - n2°)

H.G. Halverson et al.

372

In this case, n o does not represent an actual cycle, but some 'pseudo-cycle' which is a starting point for further material degradation. This is in contrast to the Palmgren-Miner Rule, which uses life fraction as a damage metric. If the new load F'a2 is applied for a duration of An2, there is then an additional change in the remaining strength given by the integral tl 0 + aX n 2

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(8)

As applied cycles progress, the applied loads may continue to change, and the incremental process of shifting among different remaining strength curves is repeated. The remaining strength is the summation of the increments AFri of material degradation occuring at each load level Fai: after i increments of loading change, i

Fri = 1 - '~. AFri

(9)

j-I Failure occurs when Fri = Fag, and the life is fig, ^ where N is given by

i (10) j=l

A schematic of the evolution of remaining strength with a simple load history is shown in Figure 2. In this formulation, the specimen damage history, as reflected by the stiffness reduction, is thus intimately related to the measured life of the sample. Fourth, we derive the intrinsic critical element S-N curve, i.e. the S-N curve in the absence of sub-critical element damage, by applying the methodology to a sequence of fatigue tests on unidirectional materials. By continuously monitoring the specimen stiffness, we obtain the local applied stress o- vs cycles throughout the test, and measure the number of cycles to failure. From separate quasi-static tensile tests, we know what the distribution {X,} of tensile strengths should be for the fatigue-tested samples. What remains unknown are the values of A and B in the intrinsic S-N curve, and the exact value of X, for each specimen tested in fatigue. The curve fitting procedure used to determine the best values for A and B is straightforward. We first choose specific values for A and for B. Then for each individual sample we find that value for X, which, when combined with the methodology of Equations (7)-(9) and our experimentally determined local applied stress for that sample, yields failure at the measured number of cycles. After determining the X, for all the fatigue specimens in this manner at the chosen A and B values, we then examine the distribution {X,} to determine if it is an identical distribution to the quasistatic tests previously done. The values of A and B which describe the intrinsic S-N curve are those which

yield a distribution for X, that is identical to the true distribution, as characterized by the Weibull parameters X,o and p. Finally, we apply the entire methodology to the S-N behaviour of specific 00/90 ° crossply specimens, using the now-established intrinsic S-N curve for the critical element. For crossply specimens, the 90 ° plies usually crack very early during the test, transferring stress to the 0 ° plies. In many cases, as the test progresses the 0 ° plies exhibit the splintering behavior previously described m. In this case, we can still use stiffness as a measure of the stress increase on the critical elements. This does, however, ignore many details about the stress state of the 0 ° plies, such as the 90 ° ply constraint and stress concentrations due to matrix cracks, but appears sufficient to capture the major effects of damage on fatigue failure. EXPERIMENTAL VERIFICATION Two plates of a graphite-bismaleimide material were manufactured from BASF prepreg (G40-700/5245C). One plate consisted of two plies arranged in the same direction. The second was in a (90°/0°k arrangement. The first plate provided tensile specimens used for unidirectional fatigue tests and quasi-static strength, while the second was used to demonstrate the applicability of the method to a slightly more complex laminate geometry.

Unidirectional laminates The tensile tests yielded the quasi-static tensile strength distribution required for the analysis. A fit to the Weibull distribution leads to a scale factor of X~) = 2.28 GPa (330.6ksi) and a shape factor of p = 16.55 ~. The raw results of the fatigue tests are shown in Figure 3 as a standard S-N curve. The measured cycles-to-failure are shown against the applied stress ~rapp normalized to the mean strength (50% failure probability) of 2.23 GPa (323.5 ksi). A least squares fit of the the raw data to the general log-linear form yields O'~pp - 0.861 - 0.010log(N) 2.23 GPa

(11)

Also shown in Figure 3 are the predicted results for the 10th percentile (1.99 GPa) and 90th percentile (2.40 GPa) strengths using the SLERA. As can be seen, the data points all fall within the bounds, but these bounds are much too wide to be of any practical use. For all the specimens but one, the 10th percentile strength prediction is less than one cycle. Clearly, this method does not allow one to predict fatigue lives with any accuracy. The very broad 10-90 bounds found here stem from the very shallow slope of the raw S N curve for this particular material. We now turn to the stiffness history of the specimens. The solid line in Figure 4 shows a typical stiffness vs cycles curve for a unidirectional material which demonstrates three distinct stiffness drops. From observations during the test, each stiffness drop was associated with the longitudinal splitting and subsequent transverse cracking of a small region of the composite.

Fatigue life of individual composite specimens based on intrinsic fatigue behavior

373

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Cycle Figure 4 Stiffness history for one particular unidirectional specimen ( specimen failure

Using the fatigue data, the stiffness history for each sample, and the quasi-static tensile strength distribution, we use the method described earlier to derive the intrinsic S-N curve parameters. This intrinsic S-N curve is o-

2.23 GPa = 1.0395 - 0.02078log(N)

(12)

and is shown in Figure 3. As required, the derived

) and one particular crossply specimen

( .......... ~. × r e p r e s e n t s

Weibull parameters X~ = 2.28 GPa and p = 16.5 are very close to those obtained from quasi-static testing. As with Equation (11), this curve describes the response of a laminate with the mean strength of 2.23GPa. This intrinsic S-N curve represents the fatigue behaviour of the critical element alone, without the severe effects of sub-critical element damage which serve to increase the local loads and accelerate the fatigue failure. The intrinsic S-N curve is rather different than that suggested by the raw data: the slope B

H.G. Halverson et al.

374

in Figure 6 along with a raw S - N curve and the 10th and 90th percentile strength bands. The raw S - N curve is described by

is twice as steep, and the A value greater by 21%. Again, it is the damage history in these composites that leads to an observed S-N curve given by Equation (11) which is rather different than the intrinsic fatigue behavior of the critical 0 ° element. From the derived intrinsic S - N curve we reverse the process to predict life vs initial sample strength for each sample given the actual fitted X, values. The difference between the experimentally measured life and the prediction for the actual fitted value of X,o is generally too small to be shown graphically, even for the specimen which lies outside the 10-90 prediction bands, which has a very low fitted strength. We also take the stress history of each specimen and predict the fatigue lives for hypothetical laminates of 10th and 90th percentile quasistatic strength which follow that same stress history. These predictions are shown in Figure 5. By comparing the bounds in Figures 3 and 5, we see that incorporation of the damage history of a specific material does, for most cases, greatly decrease the statistical spread in predicted fatigue lives. Since the intrinsic S - N curve is obtained from the fitting of the A and B parameters, we certainly expect good predictions to the fitted data, of course. The power of this approach is in application to other laminates once the intrinsic 0 ° critical element S - N curve has been determined.

O-~pp(O,.~ = 0.966 - 0.019log(N) 1.07 GPa

(13)

As with the unidirectional raw S - N curve in Figure 3 these bands are extremely wide. Now suppose we try to relate the raw S - N curves for the 0 ° and 90°/0 ° specimen. First, we calculate the stress carried by the 0 ° plies of the crossply materials as found by a standard CLT analysis, with all plies completely undamaged. After doing so we may apply the raw S - N curve for the unidirectional materials, Equation (l 1), directly to the cross-ply data, as shown in Figure 7. Although many of the points do lie within the 10-90 bounds of the S - N curve, the data and trends are not predicted well at all; this straightforward attempt to relate the two fatigue sets does not yield useful results. Even completely discounting the 90 ° plies in the CLT does not improve the comparison. The unidirectional S - N data predict much lower failure times over the stress range considered, but also have a much shallower slope and so would not be conservative at lower applied loads. The nearly twofold difference in slope between the crossply and unidirectional S - N curves would also suggest that the failure and damage mechanisms in these two materials progress rather differently. As we show below, when stress history is taken into account, the fatigue behavior of the 0 ° and 90°/0 ° are closely related and the failure mechanisms are similar. We now apply the proposed methodology to predict the crossply data. A typical crossply stiffness degradation curve is shown as the dashed line in Figure 4 and we see that this material also demonstrates distinct

Crossply laminates We now perform similar tests on crossply materials. We assume that the strength of the crossply materials is one-half that of the unidirectional materials (i.e. use a ply-discount method), with an additional 5% decrease due to the larger volume of the crossply specimens with Weibull modulus 16.55. Normalizing by this strength value (1.07 GPa), the fatigue test data is shown

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fatigue lives

Fatigue life of individual composite specimens based on intrinsic fatigue behavior

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stiffness drops, although the magnitudes of these drops are lower. We begin the analysis with completely undamaged plies, then increase the stress in the 0 ° plies inversely with the stiffness decrease. The predictions of specimen life resulting from the remaining strength methodology using the inputs of (i) the specimen stiffness history, (ii) the intrinsic S-N curve for the critical element as obtained from the unidirectional data, and (iii) initial strength distribution as for the unidirectional

materials are shown in Figure 8. The proposed analysis predicts the fatigue of crossply materials in rather good agreement with the experimental data. It is important to emphasize that these predictions are based solely on the information derived only from the unidirectional results and the individual specimen specific stiffness degradations experimentally obtained during the crossply fatigue tests. The uncertainty in predicted life is now due solely to statistical strength variability, which cannot be minimized or determined a priori.

H.G. Halverson et al.

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DISCUSSION AND C O N C L U S I O N We have postulated that unidirectional continuous fiber reinforced composite materials possess an intrinsic S N curve, which is typically obscured by 'splintering' damage during a fatigue test. This splintering reduces the effective cross-sectional area, increasing the local applied stress and thus has a significant effect on the subsequent behavior of the composite. Ignoring this stress redistribution leads to broad bounds on fatigue life when the statistical nature of initial strength is used in the SLERA, as shown in Figure 3. These bounds are broad due to the very shallow slope of the raw S - N curve, which was - 0.01 for these materials, and imply that samples at the 90th percentile strength live 10 ~5 times longer than samples at the 10th percentile strength. By using a cumulative damage scheme, along with specimen specific stiffness reductions, we have determined the parameters for the intrinsic S - N curve. Combining this intrinsic S - N curve with the specimen specific stiffness reduction and the quasi-static strength distribution greatly reduces the bounds on fatigue life, and produces predictions for individual specimens, as shown in Figure 5. These improvements are due mainly to two factors. First, the analysis yields a higher slope for the intrinsic S - N curve (0.0208). The second factor which reduces the spread of fatigue lives is that the method accounts for the increases in stress on the critical element over the duration of the test. For unnotched specimens of these two geometries, the applied stress on the critical element is always increased by damage. Increasing the applied stress increases the rate of critical element damage, and reduces the remaining number of cycles to failure of the specimen. The difference in the parameters which define the intrinsic and raw S - N curves for the 0 ° laminates is rather large, and is related to the fatigue mechanisms

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methodology for each specific sample, and actual measured

which 'translate' the specimen from the intrinsic curve to the raw one. The value of A is increased by 21% for the intrinsic curve over the raw one, and the B value is doubled. Both curves are shown for comparison in Figure 3, along with the experimental data. The change in the value of A is related to the large stress increase due to sub-critical element failure (typically 15%-25%) which occurs over the duration of the fatigue test. The B value is related to the progression of sub-critical element failure with cycles. There are two competing processes which affect this difference in slope. Firstly, for specimens cycled at lower stresses fatigue lives are longer. This allows more time for sub-critical element damage to occur and tends to make the raw S - N curve steeper than the intrinsic S-N curve. Secondly, at lower stresses, the rate of subcritical element damage is reduced, which would lead to a raw S - N curve that is shallower than the intrinsic S - N curve. In this case, the second factor appears to control the raw S - N curve. In the second phase of this work we related the crossply fatigue behavior to the unidirectional fatigue behavior. Using the raw S - N curve for the unidirectional materials to explain crossply fatigue behavior proved unsatisfactory, and suggested that the crossply materials show a different fatigue mechanism and response. We then used the intrinsic S - N curve determined for the 0 ° laminate and the crossply stiffness history to predict the fatigue response of crossply materials (see Figure 8), i.e. failure occurred between our 10th and 90th percentile strength endpoints for individual specimens. The good predictions arise even though crossply materials typically exhibit far more complex damage modes than unidirectional materials. The fact that the intrinsic S - N curve for the unidirectional materials does satisfactorily explain crossply fatigue indicates that these additional damage modes may not play an important role in explaining crossply

Fatigue life of individual composite specimens based on intrinsic fatigue behavior fatigue behavior. Rather, the crossply fatigue is controlled by the fatigue of an underlying 0 ° critical element and damage serves mainly to increase the stress on that critical element. This cumulative damage scheme uses an exponent, j, to define the shape of the remaining strength curve with time and/or cycles. For this work, j was taken to be 1.2, although other values may be used. For high values of,j, the shape of the curve indicates a 'sudden death' behavior, while for j less than 1, the remaining strength initially drops very quickly, then levels off until failure. However, for values of j ranging from 1.2 to 2, there was almost no change in the constant A of the intrinsic S-N curve, and B was changed only slightly. For j = 1 there was a greater difference in the values ot: A and B ( - 3% each). In all cases, however, the subsequent life predictions are not much changed by these slight changes in the intrinsic S-N curve parameters, and hence the methodology is not sensitive to the detailed shape of the remaining strength curves. The uncertainty in fatigue lives stems from two sources, the initial strength of the specimen in question, and the propagation rates of the fatigue damage modes during the test. As long as we have no way of determining the quasi-static strength non-destructively, we will be unable to totally eliminate uncertainty in fatigue life predictions. However, this method uses the stiffness changes during the test to monitor the damage and corresponding stress redistribution, and so acts to eliminate one of the two sources of uncertainty. By using techniques which yield additional information about the global specimen response, such as frequency response ~, it may be possible to improve even further our understanding of the state of damage in a material during a fatigue test, and to use this to improve fatigue life predictions. Given the success of the present methodology in predicting life under well-controlled fatigue conditions, this method may prove useful for in-service life prediction of individual components under more complex stress histories.

REFERENCES 1

2 3 4

5

6 7

8

9

I0

II

Research sponsored by Office of Fossil Energy, Advanced Research Technology Development Materials Program, [DOE/FE AA 15 10 100, Work Breakdown Structure Element ORNL-VPI-1], W.S. Department of Energy under contract number DEAC05-96OR22464 with Lockheed Martin Energy Research Corp and the NSF Center for High Performance Polymeric Adhesives and Composites under grant number DMR9120004.

Petitpas, E., Renault, M. and Valenlin. 1).. Fatigue behavior of crossply CFRP laminates made of T~O0 or T400 fibres. I111..7. kklti~ue, 1990. 12, 245-251. Curtis. P. T., Tensile fatigue mechanisms in unidirectiomd polymer matrix composites. Int. J. Fatigue, 1991. 13, 377-382. Reifsnider, K.L.. ed., Fatigue of Compogih' Maleriolx. Elsevier Publications Inc., New York, 1991. Razvan, A.. Fiber fracture in continuous-liber reinforced corn posite materials during cyclic loading. Ph.D dissertation. College of Engineering, Virginia Polytechnic ]nslilule and State (!nivcr sity. Blacksburg, Virginia, 1992. Barnard, P. M., Butler, R. ,I. and Curtis, P. T., The strength lil;e equal rank assumption and its application to lhc fatigue life prediction of composite materials, htt..I, kati~,uc, 1988. 10, 171 177. Hahn, H. T. and Kim, R. Y., Proof rusting ol composilc materials. J. Comp. Maler., 1975, 9. 297 311. Rcifsnider, K.L. and Stinchcomb, W.W.. A critical-clemen! model of the residual strength and life of fatigue loaded corn posite coupons. Composite Materialw F, ti~ue aml fractltrc, ASTM STP 907, Philadelphia. Pennsylvania, 1986. Rcifnsider, K.L., Xu, Y.L., lyengar, N. and Case, S., Durability find Damage Tolerance: Re~'enl Adt'tlltce.~, Center liar Composite Materials and Structures Report 95-07, Blacksburg. Virginia, 1995. Halverson, H.G., Improving Fatigue Life Predictions: Theory and Experiment on Unidirectional and Crossply Polymer Matrix Composites. Master's Thesis, College of Engineering, Virginia Polytechnic Institute and State Universily, Blacksburg, Virginia. 1996. Jameson, R.D., Advanced Fatigue Damage [)e,~elopmcn! in Graphite Epoxy Lanfinates. Ph.D. dissertation, College of Engineering, Virginia Polytechnic lnstittne and State University, Blacksburg, Virginia, 1982. Elahi. M.. Razvau, A. and Reifsuider, K.L, Characterization of Composite Materials Dynamic Response [!sing a I,oad/Sm)ke Frequency Response Measurement. ('ompo.~ile Material,w f'atiguc and t:racture. ASTM STP 1156, Philadelphia, Pennsyl vania, 1992.

NOTATION A,B Fa Ft" j

ACKNOWLEDGEMENTS

377

n n° N fi/ Pf X, X,o p (Tap p

r

constants for strength-life curve applied stress function normalized by the initial strength of the critical element strength of the critical element normalized by its initial strength constant exponent in damage evolution equation current cycle current 'pseudo-cycle' predicted number of cycles required for fatigue failure experimentally determined fatigue life probability of failure initial strength of the critical element scale factor of Weibull Distribution shape factor of Weibull Distribution applied stress generalized time