Cumulative damage theory for composite materials: Residual life and residual strength methods

Composites Science and Technology 23 (1985) 1-19

Cumulative Damage Theory for Composite Materials: Residual Life and Residual Strength Methods Z. Hashin Department of Solid Mechanics, Materials and Structures, Tel-Aviv University, Tei-Aviv 69 978 (Israel)

SUMMARY The cumulative damage problem is formulated in terms of a damage function which must satisfy certain conditions. It is shown that such a damage function can be used with residual life theory as well as with residual strength theory, and that these two approaches are equivalent. Various assumptions regarding the dependence of the damage function on fatigue variables lead to both known and new results of life prediction. Comparison of analytical results with available test data did not reveal a best fit of any one prediction to the data.

1 INTRODUCTION A central problem in fatigue of structures is the prediction of lifetime under a variable amplitude cyclic loading program, also called the cumulative damage (CD) problem. The problem has been considered for many years for metal fatigue. In this context the two major analytical approaches are the phenomenological approach and the crack propagation approach. The former is concerned with lifetime prediction for complex loading programs in terms of lifetime test data for sample loading programs, mostly the S-N data for constant amplitude loading, without enquiring into the microstructural nature of fatigue failure. The second approach is concerned with the prediction of the slow growth of a dominant crack due to cyclic load, and has evolved into a practical 1

Composites Science and Technology 0266-3538/85/$03.30 © Elsevier Applied Science Publishers Ltd, England, 1985. Printed in Great Britain

2

Z. Hashin

procedure based on semi-empirical crack growth laws and on-site crack inspection. The introduction of fiber composite materials for aircraft structures requires solution of the CD problem for these materials. A major difference between metals and composite materials is that fatigue failure in fiber composite laminates occurs as a result of accumulation of many cracks, rather than by the propagation of one dominant crack, and these cracks are not easily detected. Consequently, the phenomenological approach assumes revived importance for the CD problem of composite materials. Because of the progressive accumulation of cracks in composite materials during fatigue cycling the static strength and stiffness may be progressively and significantly reduced, a phenomenon which is of much less significance in metal fatigue. Consequently, much cumulative damage research in composite materials is concerned with exploitation of strength and stiffness degradation during the fatigue process. In the residual strength approach it is postulated that failure occurs when the residual static strength deteriorates to the value of the maximum stress amplitude of the last cycle before failure, this cycle thus being considered as a static test. The first attempt to predict lifetime under variable amplitude loading in terms of this phenomenon seems to be due to Broutman and Sahu,’ who were concerned with two-stage (two constant amplitudes) loading of glass/epoxy cross-ply laminates. Such an approach has also been discussed in terms of fracture mechanics concepts by Halpin et al.,’ and has been named the wear-out method. Much cumulative damage work in terms of the residual strength method has been done by Yang and coworkers3 - ’ and will be discussed further below. A major achievement of this work is the incorporation of statistical scatter into the analysis. The residual stiffness approach is similar in concept, but with some important differences. The advantages of this concept are that (a) residual stiffness is a non-destructive parameter which can in principle be determined by on-site measurement on the structure and (b) residual stiffness exhibits much less statistical scatter than residual strength. Much important work on residual stiffness has been done by Reifsnider et al. (for a recent summary of achievement see Ref. 6). Residual stiffness is much easier to determine analytically in terms of crack distribution than residual strength. However, the major disadvantage is that a failure criterion in terms of residual stiffness is not known and, indeed, the question may be raised as to whether such a failure criterion exists.

Cumulative damage theory ]or composites

3

There is an intrinsic similarity between the residual life and the residual strength methods. In both cases failure is defined by a specific value of a physical quantity. In the former case failure is defined by zero residual life, and in the latter case by the residual strength being equal to the stress amplitude of the last cycle. It will be shown in this paper that these two methods are actually equivalent, and that there is thus little reason for preferring one to the other. 2

RESIDUAL STRENGTH CONCEPTS

It is assumed that ideal specimens which exhibit negligible scatter of fatigue lifetime are available. Suppose that such a specimen is cycled for n cycles at constant maximum amplitude tr and constant minimum amplitude Ra, where R is defined as the stress ratio. The specimen is then removed from the testing machine and is subjected to a static failure test. The static strength thus obtained is defined as the residual strength %. Because of the damage produced during cycling trr < a o, where a o is the initial static strength. The residual strength tr, is at least a function of tr, R and n. Assuming constant R, this can be expressed as

o, = f ( n , a)

(1)

Equation (1) is a measure of the damage suffered by the specimen during cycling. Obviously the value of o, for no cycling is the static strength. Thus a,I, =o =f(0, a) = a o (2) Furthermore, experience shows that the initial reduction of residual strength is very small; thus, in the initial cycling range a, is practically equal to a o. This may be expressed as

% dn ,=o = 0

(3)

The fundamental assumption of residual strength theory is that failure occurs when the residual strength becomes equal to the stress amplitude a. Thus, the last stress cycle before failure is considered as a static test. As this occurs the number of cycles is N(a), which is the lifetime for constant amplitude cycling and defines the S-N curve of the specimens. Thus, eqn (1) must obey the condition a (4)

4

Z. Hashin

~'1.o

L

Ul

s3 o

S-N CURVE

.~ i

._~

I

o

Ioq N~

E

Fig. 1.

,

Io~1 Nz

g Ni

IogN log n

N o r m a l i z e d residual s t r e n g t h c u r v e family.

which must also be equivalent to the S-N representation of the specimens. Equation (1) may be regarded as a parametric family of residual strength curves with parameter a. Introducing non-dimensional variables S r = 6r/O- o

S -~ a / a o

(5)

eqn (1) can also be written in the form ~s, = s,(n, s)

(6)

A typical family of residual strength curves is shown in Fig. 1. For purposes of residual strength analysis, the form of the functions (1) or (6) must be known. Unfortunately, experimental determination of such curves is problematical because of the considerable scatter of test data. Such functions have been assumed to be linear in n in Ref. 1. A more versatile form of residual strength curves has been given in the work of Yang and co-workers; see, for example, Ref. 3, where it was assumed that d°'r

dn

h

g ( a , a o ) / ~ a r~ ~

(7)

where ~ is an empirical parameter. Integration o f eqn (7) from n o to n yields the result ~ ( n ) = a~(no) - g ( ~ , ~ro)(n - no)

(8)

Cumulative damage theoryfor composites

5

and in the special case n o = O, since a,(O) = ao, a~(n) = a~ - g ( a , ao)n

(9)

Applying condition (4) to eqn (9) gives g ( a , a o) =

O"0 - -

N(a)

( 1O)

and therefore the equation of the residual strength curves is a r~-- a o~- (a~ - a ' ) n / N ( a )

(11)

For the special case ~ = 1 the linear residual strength curves used in Ref. 1 are obtained. The residual strength curves (eqn (11)) have no intrinsic physical basis and are merely a possible mathematical representation. These results can be generalized as follows. Assume that da~ d n = - g ( a, ao)/~O( a r)

(12)

where qJ is an arbitrary function, and let tI'/(O'r) = j" ¢ ( O ' r ) do" r

(13)

Then the equations of the residual strength curves are given by qJ(o-,) = ~ ( a o ) - [~(ao) - t F ( a ) ] n / N ( a )

(14)

Note that in eqn (11) re(a)

= a

Since the abscissa of the S - N curve is usually plotted in log N, it is also reasonable to represent residual strength curves in terms of log n, where n is again the number of elapsed cycles. Defining the variable r/= log n, eqn (12) may be modified into

dar

_

dtl

Then all the results derived above are simply modified by replacing n by log n and N by log N. Thus ar~ = a~) - (a~ - a ~) log n/log N W(ar) = tP(ao) - [tP(ao) - q~(a)] logn/log N Equation (2) now holds at log n = 0, and thus at n = 1 which from the physical point of view is the same as n = 0.

6

Z. Hashin

3

C U M U L A T I V E D A M A G E : R E S I D U A L LIFE T H E O R Y

Various methods of residual life theory have been discussed over the years. For a review see, for example, Ref. 7. A theory in terms of damage curves has been given in Ref. 8, and has been generalized to incorporate statistical scatter of test data in Ref. 9. Here a simple approach based on the concept of the damage function is considered. Earlier work in terms of the damage function has been discussed in Ref. 7. For reasons of simplicity, ideal specimens with negligible scatter are considered here. Suppose a specimen has been subjected to n 1 cycles at constant maximum amplitude (from now onwards referred to as amplitude) al and constant minimum amplitude Ra~, The stress amplitude is now changed to a 2. The number of additional cycles required to fail the specimen at this amplitude is the residual lifetime n2~. Define a single-valued non-dimensional damage function D(n, N . . . . ) which is at least a function of the number of elapsed cycles n and ot' the constant amplitude lifetime N(a). Dependence on additional, dimensional, parameters must be in terms of non-dimensional combinations, e.g. a/a e, where a is stress amplitude and a e is fatigue limit. The function D has the following properties: D(0, N) = 0 (t 5) D(N, N) = 1

(16)

which implies that there is no damage for no cycling, and that failure is identified with D = 1. The fundamental property of D is that, for twostage loading, D(n~, N1) = D(n 2, N2) (17) n 2 +n2r = N 2

(18)

where N 1 and N 2 denote, from now onwards, N(a 1) and N(o'2). Equation (17) determines the number of cycles, n2, which produce at stress amplitude o-2 the 'same damage' as n~ cycles at stress amplitude o~. This determines the residual life n2r from eqn (18). The word 'damage' should not be taken literally. What is implied here is that the residual life is a measure of the damage sustained and thus eqns (17) and (18) determine residual life in two-stage loading. The damage function defined has much wider significance than twostage loading and, indeed, solves the cumulative damage problem for any loading program. Consider for example a three-stage loading program.

Cumulativedamagetheoryfor composites

7

After n I cycles at al the specimen is subjected to An2 cycles at tr2 and subsequently to n3r cycles to failure. Now n 2 cycles, as determined by eqn (17), produce the same damage at a 2 as n I cycles at al. Therefore the damage produced after an additional An2 cycles is the same as by (n 2 + An,_) cycles at o"2. Thus the three-stage loading problem has been reduced to a two-stage loading, and therefore n3r is defined by eqns (17) and (18). Thus, D(n 2 + An 2, N2) -- D(n3, N3) (19) F/3 + F/3r = N 3

The same method can be applied for any number of constant amplitude stages, and also for the case of cyclic loadings with continuously variable amplitude. It is thus seen that the information contained in the solution of the two-stage loading problem given by eqns (17) and (18) is sufficient to solve the problem of general cyclic loading. This is in accordance with the theory developed in Ref. 8, where it was shown that the damage curves for two-stage loading are sufficient information to determine lifetime for any loading program. Next, consider simple forms of the damage function D. The simplest assumption is D = D(n/N) (20) Introducing this into eqn (17) gives D ( n l / N I) = D(n2/N2)

and since D is single-valued, n l / Na = n2/ N 2

(21) (22)

It follows that the function n / N can be considered to be the damage function D. It is easily seen that this function obeys the conditions given in eqns (15) and (16). More generally, if D = D[cb(n, N)], then without loss of generality the damage function may be taken as 4~(n, N). Introducing eqn (18) into eqn (22) gives the Palmgren-Miner rule H1

n2r

N--~-+ ~

= 1

(23)

This may be generalized to multi-stage loading in the manner described above to obtain

~ i

Ani/Ni = 1

(24)

8

Z. Hashin

where An i is the n u m b e r of cycles at stress amplitude ai and the last An i is the residual lifetime. It is thus seen that the seemingly quite general assumption given in eqn (20) leads directly to the P a l m g r e n - M i n e r rule (this has been noted before; see, for example, Ref. 7) which is of very restrictive nature. The main problem with eqn (24) is that it ignores sequence of loading effects. Since fatigue damage varies more nearly logarithmically than linearly with the number of cycles, it would seem that a better assumption than eqn (20) would be D = D (log n/log N) i 25) or, more generally, D = D [log (n/Ne)/log (N/Ne) ]

(26)

where N e is a parameter. Following the same procedure as before, the damage function may be taken as the argument function in eqn (26). This function obeys eqn (15) for n = 1, which is practically the same as n = 0, and also eqn (16). Then the residual lifetime in two-stage loading is given by n I ~l,,gl~',,~'<,)logiN,,~'<,~ n2, = 1 + x>

(27)

This coincides with the result first obtained in Ref. 8 in terms of damage curves linear in (s, log n) or (log s, log n) coordinates which converge into the fatigue limit defined by stress a e and n u m b e r of cycles N e. However, note that in the present derivation no assumptions with respect to damage curves or the shape of the S-N curve are necessary. The predictions of eqn (27) have been shown to be in reasonably good agreement with metal fatigue data. 8"9 In the case Ne = 1, the results for damage curves converging into the static ultimate stress point are obtained from eqn (26). 8 Furthermore, there is no difficulty in generalizing eqn (27) to any loading program. 8

4

CUMULATIVE DAMAGE: RESIDUAL STRENGTH

Since the residual strength is a measure of the damage sustained, we define a damage function in terms of this quantity. The damage function D may be written as D = D(a, a r, n, N) (28)

Cumulative damage theory for composites

9

where a is the constant stress amplitude and n is again the number of elapsed cycles. The quantities appearing in eqn (28) are not independent since the S-N lifetime N is a function of a, and o-r is related to a and n by eqn (1). Thus, the damage function can also be written in the form D = O(a, at)

(29)

or in terms of the non-dimensional stresses (eqn (5)) D = D(s, st)

(30)

Specifying again that no damage is characterized by D = 0 and failure by D=I, D(a, % ) = 0 = D ( s , 1) (31a) D(a, a) = 1 = D ( s , s )

(31b)

The fundamental property of D for two-stage loading is now, in analogy to eqn (17),

D(al, O'rl )

~" D(a2, O'r2)

(32)

The meaning of eqn (32) is as follows. If a specimen is cycled at constant amplitude a 1 for n 1cycles at which instant the residual strength is a~l then the equivalent 'damage' for o"2 cycling is produced when the residual strength reaches O'r2. Thus, eqn (32) defines residual strengths at two different amplitudes which are equivalent in terms of damage. If the residual strength curves are known, O'rl and O'r2 uniquely define damage equivalent n 1 and n2, just as by the condition given in eqn (17). With n 2 defined, the residual life n2r is again determined by eqn (18). Again, generalization to multi-stage loading proceeds as in the case of residual life theory discussed above. Some methods of solving the cumulative damage problem in terms of the residual strength curves are now discussed. First, consider the method used by Broutman and Sahu 1 and by Yang and co-workers. 3's Let a specimen be subjected to n 1 cycles at stress o-1. Then, from eqn (11) the residual strength is given by a~,(n,) = a~ - (a; - a ~ ) n l / N j

(33)

Now apply An z cycles at stress a z. It is a s s u m e d that in order to evaluate the residual strength degradation at this stress, eqn (11) can be used with o-o replaced by ara. This yields, from eqn (11),

O'ras(Z~kn2)

=

ffarl(n 1) -- ( a~ -- a~) z~ri2/ N 2

(34)

Z. Hashm

10

and, substituting from eqn (33), (7~2(An2) = (7"o- ((7"o - a ~ ) n t / N l -- ((7"o -- a~z) a n z / N 2

(35)

Failure occurs at An 2 = n2~, when G2 = (72- It follows from eqn (35)that n r -, = N

2 1

a oat - a l

at

(36)

nl/N l

(70 - - (72

This is easily generalized to multi-stage loading. It must be assumed that the residual strength at the termination of any loading stage is considered as a new static strength in the subsequent loading stage. With these assumptions, St

nrk= Nk

1

GO--(72 t ~ 2 / N 2 ~ (70 -- (73

%--(71nl/N l (70 --

(72

at . . . .

at

(70 - - O k - I A n k at at (70 -- (Tk

7

/N 1/

k-'i

A

(37)

where k indicates the last loading stage. Furthermore, this is easily generalized to the case of the more general residual strength curves given in eqn (14). The two-stage result of eqn (36) then assumes the form nr2= N211

~P((7o)W((7°)-qJ((71)W(o2 ) n ' / N l ]

(38)

and generalization to multi-stage loading is obvious. The problematic assumption underlying these results is that the residual strength after n 1 cycles at stress amplitude a l can be considered as initial static strength for subsequent cycling at stress amplitude a2. This implies that application of An 2 cycles at stress a 2 to a virgin specimen or to one which has previously been subjected to n 1 cycles at stress a 1 results in the same degradation in residual strength. Such an assumption is difficult to justify. A different procedure has been used by Sendeckyj et al., lo and it will be seen that it leads to the same results as above. Consider again the case of two-stage loading. After application of n 1 cycles at stress a 1 the residual strength is (7rl(n t)" It is assumed that for continuation of cycling at stress a 2 the equivalent initial point on the a~2 residual strength curve has abscissa n 2 defined by G l ( n t ) = (7,2(n2)

(39)

With this assumption and the results ofeqns (8) and (10) it follows that the

Cumulative damage theoryfor composites

11

o Ul o

s~

HO0 S-N ¢)

rA,

CL IRVE

s=

g

k, N Z

Fig. 2.

~nl

k~nz

~n I

~n

Damage function assumed as residual strength reduction 1 - s,. Two-stage cumulative damage problem.

residual strength after an additional An 2 cycles at 0 2 is given by 0"r2(~2) ~---0"r~l( h i ) -- (0"~ -- 0"~) ~rl2/N 2

(40)

It is seen that this is identical to eqn (34). Therefore, this procedure predicts the same degradation of residual strength as the previous one and therefore also all of the remaining results (eqns (35)-(38)). A graphical interpretation of this procedure is given in Fig. 2. Recalling eqn (32), it is seen that the assumption given by eqn (39) implies postulation of a damage function which is best written in the form D- °o-or _l_s

r

(41)

o"o

which is a special case ofeqn (30). Thus, the measure of damage is defined here as the decrease of residual strength. It is seen that eqn (41) satisfies eqn (31a) but not eqn (31b), and it is therefore inadmissible. The significance of eqn (31b) is not the value 1, but the requirement that damage at failure must be defined by the same number for all loadings. There is no loss of generality in assigning to this number the value 1, as the damage function can always be normalized by this number. The problematic nature of the assumption given by eqn (39) can also be seen on the basis of the graphical construction in Fig. 2, for if n t is large enough the residual strength a r ~(nl) will be smaller than the minimum of

12

Z. Hashin

0-r2, which is 0-2, and in this case the equality of eqn (39) cannot be satisfied. The problem indicated is also apparent from the result given in eqn (36). It is seen that when 0-~ < 0-2 the fraction multiplying n l / N 1 is larger than unity, and it is therefore always possible to find a range of n which will make nzr negative, which is, of course, an unacceptable result. A simple damage function which fulfils the necessary requirements given by eqns (15) and (16), or by eqn (31), is D = 0"o - 0-,~= 1_~-.~ fr0--O"

(42)

I --S

This is m u c h more reasonable than eqn (41) since it defines the damage as the current reduction of residual strength divided by the m a x i m u m possible reduction of residual strength at stress amplitude 0-. If the residual strength curves are known, then the two-stage cumulative damage problem is solved by eqn (42) analogously to the procedure of eqns (17) and (18). Thus, 1 -srl(nl,sl) 1 -

-

s 1

1 -- Sr2(H2, S2) 1 s2

n 2 + n2~ = N 2

(43a) (43b)

Equation (43a) defines a point with abscissa n 2 on the st2 residual strength curve, which is 'damage equivalent' to the point with abscissa n~ on the s~ residual strength curve. Then the residual life is given by eqn (43b), This is easily generalized to multi-stage loading by a procedure entirely analogous to the one outlined for residual life theory in Section 3. Now consider specific cases of residual strength curves. First consider the seemingly general form: (44) 1 - s~(n, s) = O(s)h[n/N(s)] Since st(0, s) = 1

(45a)

s~(N,s) =s

(45b)

it follows from eqn (44) that h(0) = 0

(46a)

q,(s) = (1 - s)h(l)

(46b)

Normalizing h with respect to h(1) it follows that 1 - sr = (1 - s ) h ( n / N )

(47)

Cumulative damage theoryfor composites

13

where h must comply with the condition given by eqn (46a). C o m p a r i n g eqn (47) with eqn (42) it is seen that this is equivalent to the statement

O = h(n/N)

(48)

However, this form of damage function has already been considered before, see eqn (20), and it has also been shown that it leads inevitably to the Palmgren-Miner rule. Next, eqn (44) is modified to the form 1 -

Sr(n, s) = @(s)h [log (n/N~)/log (N/N~)]

(49)

As previously, n = 0 will be replaced by n = 1 to comply with the conditions of eqn (45a). It then follows from eqns (49) and (45) that 1 1-s --

S t

=

O = h [log (n/N~)/log (NINe) ]

(50)

This form of the damage function leads at once to the result given in eqn (27) and to all other results obtained by Hashin and R o t e m 8 and by Hashin. 9 It is thus seen that residual life and residual strength CD theory are completely equivalent since an assumed functional form of residual strength curves determines a damage function and thus the residual life. Now two-stage loading will be analyzed in terms of the residual strength curves given by eqn (11), which on dividing both sides by a m assume the form s: = 1 - (1 - s')n/N

(51)

and therefore, after n I cycles at normalized amplitude s 1,

S~rl(nx) = 1 - (1 - s~)nl/N a

(52)

To find the equivalent n 2 on the residual strength curve sr2 the equal damage condition given by eqn (43a) is used: Sr~2(n2) = 1 - (1 - s ~ 2 ) n 2 / N 2

(53)

Thus, eqn (43a) with eqns (52) and (53) defines n2, and n2r is then given by eqn (43b). In the special case 0~= 1 the residual strength curves are linear. It then follows from eqns (42) and (51) that D = (1 - Sr)/(1 -- S) = n/N

(54)

and it has been shown above in eqns (20)-(23) that this form of damage function leads directly to the P a l m g r e n - M i n e r rule. Since Broutman and

Z. Hashin

14

Sahu 1 have used such linear residual strength curves in their treatment, this implies that with the present definition of damage their approach is actually equivalent to the Palmgren-Miner rule.

5

THEORY VERSUS EXPERIMENT

Very little systematic experimental investigation of cumulative damage for composite materials has appeared in the literature. The most comprehensive work appears to be due to Broutman and Sahu, 1 who tested glass/epoxy cross-ply specimens under 24 different combinations of two-stage cyclic loadings for four different stress amplitudes. Another investigation has been reported by Yang and Johnson, 5 who tested _+45 ° angle ply laminates of graphite/epoxy, but only for one case each of low-high and high-low two-stage cyclic loading. It should be noted that the specimens tested by Broutman and Sahu were not of the usual rectangular form, but were machined in the central part to produce circular boundaries with minimum central section. This involves cutting of fibers which are in the load direction and it is possible that this may adversely affect the reliability of test data. All fatigue test data exhibit considerable scatter, and any comparison between experiment and theory must take this into account. Ideally, a cumulative damage theory should predict the probability distribution function (PDF) of the random lifetime variable, or at least the mean and standard deviation. Such theories have been constructed in Ref. 9 in terms of the clone concept, and in Ref. 3 where they have been applied to evaluate the PDF of the Broutman and Sahu lifetime data. The deterministic approaches to the problem, such as have been discussed above, are hopefully interpreted as predictions of mean lifetime in terms of means of other random variables. For example, in a prediction of the type given by eqn (27), all quantities except n t would be interpreted as means. It is intuitively clear that such an interpretation is useful only when the scatter of the quantities involved is not large. The statistical analysis given in Ref. 9, based on eqn (27), will give some idea of the validity of such an assumption for different degrees of scatter, as expressed by the magnitudes of the standard deviations of the pertinent random variables. The Broutman and Sahu data have been presented in terms of the mean and the standard deviations of the logarithm of the lifetime. These results for the S-N curve and the two-stage Ioadings are listed in Table 1. All data

Cumulative damage theory f o r composites

15

TABLE 1 Test Data a S - N Data a

56 49 42 35

(logN)

2.6930 3.3930 4-167 5 5.2360

s,

0'230 0"162 0"262 0"262

(N)

567 2649 17 641 206551

p

1.15 1.07 1'20 1'20

Two-stage Loading Data crI

~r2

56 56 56 56 56 56 49 49 49 49 42 42 35 35 35 35 35 42 42 42 49 49

35 35 42 42 49 49 35 35 42 42 35 35 42 42 49 49 56 49 49 56 56 56

nI

250 100 250 100 250 100 i 000 250 1 000 250 10000 2 000 49 940 19 975 49 940 19975 19 975 10000 2000 2 000 1 000 250

(logn2~)

s,

(n2~)

p

5.283 0 5.285 3 3.7665 4.078 2 3'097 1 3.213 5 4'935 5 5'2104 3.938 1 3.902 8 4'8934 5.045 0 3.571 8 3.977 3 2.592 4 2"9051 2'094 6 2.4463 3.1104 2.550 7 2'473 0 2'701 0

0'308 0.310 0.194 0"203 0"278 0"238 0"435 0'195 0"260 0"256 0"326 0.292 0,362 0,338 0.368 0.376 0.443 0.272 0.222 0-266 0.270 0.173

246 726 248 850 6454 13 354 1 534 1 899 142 348 179546 10 373 9 511 103694 139 047 5 280 12 847 484 1 169 209 339 1 469 428 360 543

1.29 1.29 1.11 1-12 1.23 1.16 1.66 1.10 1-20 1.19 1.07 1.25 1.42 1.35 1.24 1.45 1.69 1.16 1.14 1"21 1.08 1.08

" Stress in ksi (1 ksi = 6'89 MPa).

Z. Hashin

16

TABLE 2 Test Data and Analytical Results"

61

62

rl 1

Exp. (n2c)

l (r12~)

2 (112~)

3 (n2~)

4 (n27

56 56 56 56 56 56 49 49 49 49 42 42 35 35 35 35 35 42 42 42 49 49

35 35 42 42 49 49 35 35 42 42 35 35 42 42 49 49 56 49 49 56 56 56

250 100 250 100 250 100 1000 250 1000 250 10000 2000 49940 19975 49940 19975 19975 10000 2000 2000 1 000 250

246 726 248850 6454 13354 1534 1899 142348 179 546 10373 9 511 103694 139047 5280 12847 484 1169 209 339 1469 428 360 543

115535 170122 9863 14529 1481 2182 128578 187058 10981 15 976 89465 183134 13375 15935 2008 2392 513 1147 2349 503 353 513

179 316 195 717 14597 16 423 1992 2386 165045 196249 14089 16 753 116841 188689 12080 15416 1448 2168 384 490 2217 402 186 471

164034 199299 12651 16426 1691 2342 161121 201361 12372 16697 105102 193015 11968 14913 1587 2060 397 972 2190 428 308 482

60948 135883 6702 12648 1245 2048 95490 175711 9569 t5 510 71 886 177572 14165 16273 2 236 2489 541 1 381 2414 528 395 526

" Stress in ksi (1 ksi =6.89 MPa).

were found to follow the lognormal PDF. This is mostly a quite accurate assumption for S-N data (see also Ref. 9). It cannot in principle be expected that lognormal distribution of S-N lifetimes lead to lognormal distribution of two-stage loading lifetimes, but such an assumption appears to be reasonably accurate, 9 and this is also borne out by the data of Ref. 1. It has been shown in Ref. 9 that when r/= log N is normally distributed, then the mean ( N ) is given by (N)

2

= 10 ("> +~ln lOs,i

(55)

where (r/) is the mean of r/and s, is its standard deviation. Mean lifetimes as given by eqn (55) are listed in Table 1. Broutman and Sahu have

Cumulative damage theoryfor composites

17

presented the data as 10<">. The statistical significance of this quantity is not clear. The ratios of such lifetimes to eqn (55) are given by p = 10½1.10s,~

(56)

and are also listed in Table 1. Table 2 shows the results of four different predictions for the mean lifetime (/~2r)" 1. The Palmgren-Miner rule

- -n+l ( n 2 r ~ ) (N,)

--1

(N2)

2. The Broutman-Sahu approach, eqn (36), with ~ = 1 1 --s t

nI

1 . s 2 (mr)

(nr2)

~---1

(m2)

3. The Hashin-Rotem approach, eqn (27), with Are = 1 (this corresponds to damage curves originating at the point s = 1, log n = 0) 9 n, "~,o,/ios ( n 2 " ) = 1 (N,),] + (N2) 4. The residual strength approach contained in eqns (43), (52) and (53) at Srl

=l-(1-si)at

1 -s 2 sr2=l-(1-S,)l

s1 at

= 1 -s~z 1--_-7 S2 -

(/'/2r) "= ( m 2 )

-

- (n2)

In the fourth approach the value ofa = 7 has been chosen to fit the data. It is seen that none of these approaches can be considered to be a satisfactory prediction of all the data. In particular, none of these approaches gives a better fit than the simple Palmgren-Miner rule. All of the analyses given here can be generalized to take into account the statistical scatter of data. This can be done on the basis of the lognormal distribution of the S-N lifetimes. Such a distribution is completely

18

Z. Hashin

determined in terms of the mean and the standard deviation of the logarithm. A number of trial analyses have been performed on the basis of the method given in Ref. 9 to evaluate the mean (n2r) in terms of the means and standard deviations of log N, but it appears that this does not significantly affect the nature of the fit to the data as obtained in terms of the simple approach listed above. It would be premature to draw general conclusions on the basis of one experimental investigation. Much further experimental work is needed. 6

CONCLUSION

It has been shown that the CD problem can be resolved in terms of a damage function which must obey certain conditions. Such a damage function unifies the residual life and the residual strength methods, and therefore the two methods are equivalent. Various residual life theories given in the literature can b e simply reproduced merely by assuming a general functional dependence of the damage function on fatigue variables. Similar results, and others, can be obtained by assuming reasonable mathematical forms of residual strength curves. Comparison of prediction with available experimental data have not revealed definite trends as to which prediction should be preferred~ More testing programs must be carried out to resolve such questions. ACKNOWLEDGEMENT Support of the United States Air Force Office of Scientific Research under contract AFOSR-83-0370 with the University of Pennsylvania is gratefully acknowledged. REFERENCES 1. L. J. Broutman and S. Sahu, A new theory to predict cumulative fatigue damage in fiberglass reinforced plastics, in: Composite Materials: Testing and Design, ASTM STP 497, 1972, pp. 170-88. 2. J.C. Halpin, K. L. Jerina and T. A. Johnson, Characterizationofcomposites for the purpose of reliability evaluation, in: Analysis of the Test MethodsJbr High Modulus Fibers and Composites, ASTM STP 521, 1973, pp.5-64.

Cumulative damage theoryfor composites

19

3. J. N. Yang and D. L. Jones, Load sequence effects on the fatigue of unnotched composite materials, in: Fatigue of Fibrous Composite Materials, ASTM STP 723, 1981, pp. 213-31. 4. J. N. Yang and L. J. Johnson, Load sequence effects on graphite/epoxy [_+3512s laminates, in: Long-term Behavior of Composites, ASTM STP 813, 1983, pp. 246-62. 5. J. N. Yang and L. J. Johnson, The effect of load sequence on statistical fatigue of composites, AIAA J., 18 (1980) pp. 1525-31. 6. K. L. Reifsnider, K. Schulte and J. C. Duke, Long-term fatigue behavior of composite materials, in: Long-term Behavior of Composites, ASTM STP 813, 1983, pp. 136-59. 7. H. L. Leve, Cumulative damage theories, in: Metal Fatigue: Theory and Design (ed. A. F. Madayag), Wiley, New York, 1969. 8. Z. Hashin and A. Rotem, A cumulative damage theory of fatigue failure, Mater. Sci. Engng, 34 (1978) pp. 147-60. 9. Z. Hashin, Statistical cumulative damage theory for fatigue life prediction, J. Appl. Mech., 50 (1983) pp. 571-9. 10. G. P. Sendeckyj, H. D. Stalnaker, L. G. Bates, R. A. Kleismit and J. V. Smith, Effect of stress ratio on fatigue behavior of composite materials, Internal Rep., AFFDL, Wright-Patterson Air Force Base, Ohio, 1983.