A new approach to fatigue analysis in composites based on residual strength degradation

Composite Structures 48 (2000) 183±186

A new approach to fatigue analysis in composites based on residual strength degradation D. Revuelta, J. Cuartero, A. Miravete *, R. Clemente Department of Mechanical Engineering, University of Zaragoza, C/Marõa de Luna 3, 50015 Zaragoza, Spain

Abstract Composite materials applied to transportation are usually subjected to cyclic loads that extend for periods of time longer than the ones traditionally studied. Therefore, it seems necessary to de®ne a prediction model which would extend its range beyond the traditional 106 cycles of the S±N curves. This paper develops a prediction model based on the residual strength degradation of composites when subjected to cyclic loads. Ó 1999 Elsevier Science Ltd. All rights reserved. Keywords: Fatigue; Residual strength; Composites; Degradation; Cyclic loads; Prediction model

1. Fatigue theories based on residual strength degradation

1.2. First approach

1.1. Hypothesis

Like A and B are independent from r, Eq. (2) can be integrated to yield:

Composite fatigue theories based on residual strength degradation are based on three assumptions: 1. The static strength follows a two-parameter Weibull distribution   a  re ; …1† P …re † ˆ exp ÿ b where a and b are the shape and scale parameters of the Weibull distribution. A maximum-likelihood estimation method is used in order to determine both parameters from the static data. 2. Residual strength, rr , after n cycles of constant amplitude alternating load is related to the initial static strength, re , through a deterministic equation. dr 1ÿ1=A ; …2† ˆ ÿABr1=A a r dn where r, ra , A and B are the instantaneous material strength, the maximum applied load, and two dimensionless functions that do not depend on r. 3. Fatigue failure takes place when the residual strength decreases to the maximum cyclic applied load, i.e., rr ˆ ra .

*

Corresponding author. Fax: +349-76-761861. E-mail address: [email protected] (A. Miravete)

r1=A ˆ r1=A ÿ Br1=A r e a …n ÿ 1†;

…3†

where integration limits are chosen to provide that, if failure takes place over the ®rst cycle, the residual strength, rr , equals the initial static strength, re . Eq. (3) may be rewritten as: "  #A 1=A rr …4† ‡ B…n ÿ 1† : re ˆ ra ra Assumption 3 stated that fatigue failure takes place when rr ˆ ra . Therefore: re ˆ ra ‰1 ‡ …N ÿ 1†BŠA ;

…5†

where N are the cycles to failure when the applied cyclic stress is ra . Eq. (5) describes the S±N curve of the material as a function of the two earlier mentioned dimensionless functions, A and B. These two parameters adopt di€erent forms which may even include the dependence upon the stress ratio, R. However, test data for di€erent stress ratio was not available, so only two fatigue models were assumed depending on the shape of A and B. The two models considered in this paper are: Model

A

B

M1 M2

A0 A0

1 B0

0263-8223/00/$ ± see front matter Ó 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 2 3 ( 9 9 ) 0 0 0 9 3 - 8

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D. Revuelta et al. / Composite Structures 48 (2000) 183±186

Model 1 is the most commonly used power law, where B ˆ 1, that results in a straight line when plotting the S±N curve on a log±log plot. This model is the simplest one, being broadly used by Ran Kim (Ref. [1]). Model 2 is the wearout model, where A and B are constants, and has been used by Sendeckyj and coworkers (Ref. [2]). Fatigue theory parameters may be obtained by di€erent estimation procedures, as regression analysis. If the possibility of failure during the ®rst cycle is disregarded, then the life distribution can be obtained by substituting Eq. (5) into Eq. (1), to get: a    N ÿD f ; …6† P …N † ˆ exp ÿ bf

( P …rr † ˆ exp ‡

"

rr b

ÿ

a=af #af )

…n ÿ 1† ………re =ra †1=A ÿ 1†=B† ‡ 1 "

P …N † ˆ exp ÿ

N ÿD …re =ra †

" P …N † ˆ exp ÿ

‡

1=A

…Model M2†; …10†

!af # …Model M1†; …N ÿ D†

…Model M2†;

where af ˆ Aa; bf ˆ …b=ra † =B; D ˆ ÿ…1 ÿ B†=B: Finally, the residual strength distribution can be obtained by substituting Eq. (3) into Eq. (1), to yield: ( "  #aA )  1=A 1=A rr ra B…n ÿ 1† ‡ : P …rr † ˆ exp ÿ b b …7† Eq. (5) resembles a three-parameter Weibull distribution. Ran Kim [1] uses this resemblance to formulate his residual strength degradation model. However, he does not take into account the in¯uence of more than oneparameter S±N curves. It is observed through experimental tests that the shape parameter, af ˆ aA, does not depend upon the applied stress level in fatigue tests. Therefore, it may be inferred that a combined shape parameter exists for the whole fatigue distribution in the material. This parameter is ®gured out by normalizing the cycles for each stress level over the corresponding pooled Weibull scale parameter, bi , for that stress level. Maximum-likelihood method may be used again to calculate the combined shape parameter once the data have been normalized. 1=A On the other hand, Eq. (6) de®nes bf as …b=aa † =B. The bf parameter is the scale parameter of a Weibull distribution. Then, due to the own nature of the distribution, this parameter means the value of the characteristic life of the distribution. In the case of fatigue, the characteristic life is the fatigue life, N. …8†

The life N is found out from the S±N curve equation, Eq. (5). Introducing these changes into Eqs. (6) and (7), the new residual strength distribution and fatigue life distribution equations for each fatigue model are obtained: ( "  #af ) a=af rr …n ÿ 1† ‡ P …rr † ˆ exp ÿ 1=A b …re =ra † …Model M1†;

…9†

!af #

………re =ra †1=A ÿ 1†=B† ‡ 1

1=A

bf  N :

…11†

Fig. 1. Residual strength distributions.

…12†

D. Revuelta et al. / Composite Structures 48 (2000) 183±186

Theoretical curves described by Eqs. (9) and (10) are presented together with the experimental residual strength data for di€erent fatigue lives and applied stresses, for three di€erent material systems: a glass ®bre reinforced polyester, a glass ®bre reinforced epoxy, and a carbon ®bre reinforced epoxy (Fig. 1). Probabilities of survival for the residual strength data are obtained through the median rank formula, f …x† ˆ 1 ÿ …i ÿ 0:3†= …n ‡ 0:4†, where i is the ith test specimen for a sample size of n specimens. It is noticed that, for glass ®bre reinforced composites, experimental data appear to be located to the left of the theoretical prediction. Therefore, Eqs. (9) and (10) are not a good prediction method. In the case of the carbon ®bre reinforced composite, the model seems to ®t more satisfactorily to the exper-

185

imental data. A possible explanation about this divergence between glass and carbon ®bre may be the worse glass composite fatigue behaviour than carbon composites, which translates into a greater data dispersion. Thus, it was necessary to revise the statistical parameter determination procedure by introducing con®dence bounds which may be able to issue a safe design value. 2. Second approach (probabilistic) The great existing divergence between the theoretical prediction and the experimental data made necessary the revision of the probabilistic procedures for estimating the shape and scale parameters, a and b, for both Weibull distributions, fatigue and static distributions.

Fig. 2. Residual strength curves (probabilistic approach) and fatigue life distributions.

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D. Revuelta et al. / Composite Structures 48 (2000) 183±186

Fortunately, maximum-likelihood method issues the advantage that con®dence bounds may be calculated for the a and b estimations. If b means the Weibull scale parameter estimation issued from the static data through the maximum-likelihood method ± Eq. (11), the 95% con®dence bound for the scale parameter, b, may be obtained from: ~ P …~ a ln …b=b† < l0:95 † ˆ 0:95;

…13†

where a is the shape Weibull parameter estimation from the static strength distribution, and l0:95 the statistic critical value that veri®es Eq. (13). If b names the lower limit of the 95% con®dence bound, then: b ˆ b~ exp‰ÿ…l0:95 =~ a†Š:

…14†

Admissible value B, XB , is a fatigue design value for a material. It is de®ned by the probabilistic assessment that is 95% reliable the statement that the probability of survival of the admissible value B is 90%. Thus, admissible value B, XB , may be obtained from: ~ a~Š ˆ 0:90 P …XB † ˆ exp‰ÿ…XB =b† 1=~ a

XB ˆ b‰ÿ ln …0:90†Š

:

) …15†

The admissible value XB is introduced in the formulation instead of b to get: In fatigue, the lower limit for the con®dence bound is obtained at 95% certainty for each stress level; afterwards, the admissible value XBi is calculated for each stress level. Fatigue data is then normalised by XBi to get the shape and scale parameter of the fatigue life distribution. The admissible value XB is introduced in the formulation instead of b to get: ( "  #af ) a=af rr …n ÿ 1† P …rr † ˆ exp ÿ ‡ 1=A XB …re =ra † …Model M1†;

…16†

( P …rr † ˆ exp ÿ …Model M2†:

"

rr XB

a=af

‡

#af )

…n ÿ 1† ………re =ra †

1=A

ÿ 1†=B† ‡ 1 …17†

Eqs. (16) and (17) are plotted with the residual strength experimental data, their probabilities of survival obtained through the median rank formula. At last, fatigue life distributions are plotted according to Eqs. (11) and (12) (Fig. 2).

3. Conclusions · It has been shown that the Weibull distribution also models the fatigue response of polymeric matrix composites. · The introduction of more complex fatigue models than the traditionally used linear model to characterise the S±N curve does not yield a signi®cant better ®t of the curves to the experimental data. · The introduction of con®dence bounds and admissible design values in the fatigue parameters estimation procedures is necessary in glass ®bre reinforced composites. · The use of con®dence bounds in carbon ®bre reinforced composites does not appear to be needed, maybe due to the better fatigue response of this kind of composites.

References [1] Kim RY, Pipes SL. Life prediction of glass/vinylester and glass/ polyester composites under fatigue loading. ICCM 1995;10. [2] Sendeckyj GP, Reifsnider KL, editors. Life prediction for resin± matrix composite materials. In: Composite materials series, vols. 4, 10. Amsterdam: Elsevier, 1991:431.