Microsoft Word - 5-Bendouba.doc
Engineering, Technology & Applied Science Research Vol. 4, No. 1, 2014, 587-590 587
www.etasr.com Bendouba et al.: Fatigue Life Prediction of Composite Under Two Block Loading
Fatigue Life Prediction of Composite Under Two
Block Loading
Abstract— The damage evolution mechanism is one of the
important focuses of fatigue behaviour investigation of composite
materials and also the foundation to predict fatigue life of
composite structures for engineering applications. This paper is
dedicated to damage investigation of composite materials under
two block loading cycle fatigue conditions. The loading sequence
effect and the influence of the cycle ratio of the first stage on the
cumulative fatigue life are studied. Two loading sequences, i.e.,
high-to-low and low-to-high cases are considered. The proposed
damage indicator is connected cycle by cycle to the S-N curve and
the experimental results are in agreement with model
expectations. Previous experimental research is employed for
validation.
Keywords- fatigue; damage acumulation; composite
I. INTRODUCTION
Composite materials were first used in aircraft engine rotor
blades in the 1960s [1] and their use became more and more
important in the construction of several framework in various
domain. Fatigue behavior of these materials was a subject of
thorough and extensive studies, due to the large utilization of
these materials in different applications. Fatigue life assessment
has been described with more than 70 cumulative damage
hypotheses [2], the best known is the Miner rule [3]. Several
researchers have investigated the fatigue phenomenon in
composite materials [4-9]
Howe and Owen [10] studied the accumulation of damage
during cyclic loading with the objective of obtaining useful
working relationships of the Miner-rule that might be used in
design. With the aid of optical microscopy they studied the
development of debonding sites and resin cracks in chopped-
strand-mat/polyester composites and they suggested that,
although debonding did not itself cause reductions in strength,
it served to initiate resin cracks which did weaken the material.
Mandell, et al [11] demonstrated that the data from various
fiberglass composite materials in the data base may be
characterized by a power law curve fit when they are
normalized to the ultimate tensile or compressive strength of
the composite. Wohler curve for different loading ratios (R),
require a correction using the Goodman diagram
There are many studies of the behavior of composite
materials under cyclic loading, and reviews are given in [12-
14]. Approaches in the fatigue problems of composites can be
divided into two classes: the Wöhler curve method and the
damage accumulation theory. The Wöhler curve method [15–
19] has been widely employed in engineering to deal with the
fatigue issue of composites. However, only under the
conditions of low stress and simple stress state, is the method
suitable. The damage accumulation theory, which can be
applied under complex loading conditions, is a hotspot in the
research of the fatigue of composites. Several approaches have
been proposed, such as the residual strength model presented in
[20-24].
II. SOME DAMAGE MODELS FOR COMPOSITE MATERIALS
The mechanism of damage in composites is one of the
important topics in the study of the fatigue behavior of
composite materials and also the basis for predicting the fatigue
life of composite structures for engineering applications [25].
The fatigue damage of composites is more complex than
those of metals. Failure of composite materials under cyclic
loading can occur following four scenarios:
Cracking of the matrix
Interfacial debonding
Delamination
Breaking Fibers
A. The Dzenis model
The Dzenis model [26] treats the process of fatigue damage
in composite materials as a process related to the load. That is
to say, the accumulation of damage in the load cycles is time
dependent. For this study the effects of variable amplitude,
frequency and shape of the cycle on the fatigue behavior of
composite materials are considered. The Dzenis model is given
by the following formula:
iji sjjji
DtKsD )(2,
2
,
Mostefa Bendouba
Mascara University
Algeria
bendoubamos@yahoo.fr
Abdelkrim Aid
Mascara University
Algeria
aid_abdelkrim@yahoo.com
Mohamed Benguediab
Djillali Liabes University
of Sidi Bel Abbes, Algeria
benguediab_m@yahoo.fr
Engineering, Technology & Applied Science Research Vol. 4, No. 1, 2014, 587-590 588
www.etasr.com Bendouba et al.: Fatigue Life Prediction of Composite Under Two Block Loading
where ,i js is the laminate compliances, , jK is the
correlation functions, ( )j t is the applied stresses and
ij
sD is
the dispersion.
B. The Kang-Kim model
Kang and Kim [27] presented the fatigue behaviour of
laminated carbon/epoxy with an impact-induced damage under
two blocks tensile loading. To describe this behaviour, the
concept of reduction in the strength of the material is
introduced.
The model is given by the equation:
,1 ,20 Re e 1 Re 2
0 2 0 2 ,1 0 2 ,2
. .
imp imp RR
R
imp R imp R
n n
D
N N
where
0
is the ultimate tensile strength,
1
,
2
are the
applied stresses,
Re
is the residual tensile strength,
,1imp
n is
the number of cycles at
1
,
, 2imp R
n is the number of cycles at
2
,
,1imp R
N is the residual life in the first loading and
, 2imp R
N is
the residual life in the second loading.
C. The Rognin et al model
The authors [28] used experimental data and numerical
methods to characterized the composite material. The effects of
fatigue are often evaluated by conducting experiments with two
blocks loading ( high-low/low-high ). The purpose of the study
was to predict, using probability methods, the fatigue resistance
of the coupon and show the relationship between fractions
accumulation of damage during the experiment. The authors
propose a formula for the accumulation of damage as follows:
1
1
m
i m
R
i i m
n n
D
N N
where DR is the fatigue damage variable, ni and Ni are
respectively the actual applied number of cycles and the
number of cycles to failure
D. The Jen-Yang model
Jen and Yang [29] studied experimentally the cumulative
damage of carbon nanotubes in composite material under two
blocks loading. The content of the chemically modified carbon
multiwall nanotubes used for the sample is 0.5% by weight.
The effect of loading and the influence of the cycle rate of the
first block on the cumulative damage were studied. The
authors make their proposal as following:
n f
s
i f
S S
D
S S
where ,
i
S ,
n
S and
f
S are the magnitudes of stiffness
corresponding to the initial cycle, the nth cycle, and the final
stable cycle, respectively.
E. The proposed model
Under cyclic stress, structural loading will occur in the field
of micro cracks in composite materials and these loads lead to
fatigue damage. With an increase in the number of charging
cycles, the amount of the loading increases and the damage to
the material will accumulate in phase that leads to a change in
the microscopic and macroscopic mechanical properties of
materials. Based on experimental studies [4, 11, 18, 26, 29] we
can conclude that the damage evolution of composite material
is not linear. During the initial period of loading cracks appear
in the matrix and the matrix cracks when it reaches saturation,
fiber breakage occurs, and the damage is growing rapidly in
this material as we can as shown in Figure 1.
Fig. 1. The evolution of fatigue damage in a unidirectional composite
material.
According to the mechanisms of fatigue damage of
composite materials, findings and observations from previous
models, a new comprehensive model of fatigue damage is
presented to describe the rule and stiffness degradation of
composite materials for two blocks loading, as follows:
1
2
2
1
1
1
1
ff N
n
N
n
u
where
1
n is the cycle number corresponding to
1
,
2
n is the
cycle number corresponding to
2
,
1f
N is the number of
cycles to failure corresponding to
1
,
2f
N is the number of
cycles to failure corresponding to
2
and
u
is the ultimate
tensile strength.
Engineering, Technology & Applied Science Research Vol. 4, No. 1, 2014, 587-590 589
www.etasr.com Bendouba et al.: Fatigue Life Prediction of Composite Under Two Block Loading
III. APPLICATION AND VALIDATION OF THE PROPOSED
MODEL
The proposed model is verified using experimental results
from the literature. These results are consisted of two-block
loading sequences with transitions from low to high (L–H) and
high to low (H–L) load levels. Plumtree et al [30] conducted
tests for fatigue in cyclic tests on [±45]2S angle ply carbon–
epoxy specimens using stress ratios with an R (minimum/
maximum stress) of 0.1 and -1.0. After a given number of
cycles under known loading conditions, the cyclic stresses were
changed and the test continued to failure under the new
conditions. The loading conditions, the test results reported in
[30], the theoretical predictions of the proposed model and
Miner’s rule are given in Table I for increasing and decreasing
block types of loading respectively.
TABLE I. EXPERIMENTAL RESULTS AND THE PREDICTIONS OF THE
PROPOSED MODEL.
1
(MPa
2
(MPa)
1
n
1f
N
1f
N
2
n (Exp) 2n
(predicted)
2
n
(Miner)
105 64 675 1406 196807 108244 158733 102340
96 58 511 4258 410417 578688 408543 361167
101 61 511 2222 285291 407966 276482 219674
106 64 511 1246 199721 131816 171820 117836
110 66 511 763 155130 141168 87982 51193
64 110 85960 199907 764 1153 744 435
64 108 85960 199907 972 846 947 554
64 109 85960 199907 866 745 844 494
According to the experimental data [30], the results show
that the difference between the predicted residual fatigue life
and the experimental data are acceptable because of the big
scatter of fatigue life and most points are within 1.5 times range
as shown in the Figure 2, on the other side predicted life
calculated by the Miner’s rule are 2.5 greater than the
experimental results in two cases. The predicted residual
fatigue life by the proposed algorithm is in good agreement
with the experiment, considering the bigger scatter of
composites.
Fig. 2. Experimental and predicted residual fatigue lives of laminates
Table II compares the experimental results reported in [31]
and the predictions of model proposed in this paper. As shown
in Figure 3, the majority of the results calculated by the
proposed prediction model is conservative, as they are in the
neighborhood of the experimental results.
TABLE II. EXPERIMENTAL RESULTS AND THE PREDICTIONS OF THE
PROPOSED MODEL.
1
(MPa
2
(MPa)
1
n
1f
N
1f
N
2
n (Exp) 2n
(predicted)
2
n
(Miner)
315 340 87200 115150 8800 520 343 2136
315 340 87000 115150 8800 150 345 2151
315 340 86300 115150 8800 1408 355 2205
315 340 57700 115150 8800 1750 827 4390
315 340 57550 115150 8800 2280 830 4402
315 340 40300 115150 8800 2027 1226 5720
315 340 28700 115150 8800 3320 1584 6607
315 340 26500 115150 8800 2640 1666 6775
315 340 25300 115150 8800 2464 1713 6867
315 340 17650 115150 8800 6170 2068 7451
315 340 17000 115150 8800 38140 2104 7501
315 340 13000 115150 8800 14300 2356 7807
315 340 12500 115150 8800 24030 2392 7845
340 315 8500 8800 115150 15250 235 3926
340 315 7480 8800 115150 17060 1096 17273
Fig. 3. Experimental and predicted residual fatigue lives of laminates
In this investigation, the relative error of prediction
represents the relative difference between the experimental and
calculated lines using the proposed model and the Miner’s rule.
The REP is defined by :
exprimental calculated
exprimental
N N
REP(%) 100
N
x
(6)
The corresponding predictions of the proposed model and
those calculate by Miner’s rule are gathered and presented in
Figure 4. It is clear in this figure that the predictions are very
good. All the relative errors in the proposed model are less than
10% except for one load condition, which leads to an error of
Engineering, Technology & Applied Science Research Vol. 4, No. 1, 2014, 587-590 590
www.etasr.com Bendouba et al.: Fatigue Life Prediction of Composite Under Two Block Loading
28.42% (Decreasing blocks). It should also be noticed that the
REP in the absolute value calculated by the proposed model are
lower than the REP calculated by Miner’s rule.
Fig. 4. Relative errors of prediction for calculate lives using the proposed
model and Miner’s rule
IV. CONCUSION
The paper presents a non-linear damage accumulation
model to predict the remaining fatigue life of the second stage.
The use of this model is simple, it has no parameters to be
determined, and requires only the knowledge of the S-N curve.
A comparison between our proposition and the Miner’s rule
was made and some deviation is evident. The two-level loading
examples show that the model can predict residual fatigue life
of composite materials quite well. The theoretical analyses are
well in conformance and are in good agreement with the
experimental data for all materials tested in this investigation
for the residual life as well as for the cumulative damage. From
this viewpoint, we hope that our model may eventually find
broad use. The proposed model may be extended to complex
random loading.
V. REFERENCE
[1] D. H. Middleton, Composite materials in aircraft structures, Longman
Scientific & Technical, Harlow, 1990 New York, J. Wiley, 1990
[2] A. Fatemi, L. Yang, “Cumulative fatigue damage and life prediction
theories: a survey of the state of the art for homogeneous materials”,
International Journal of Fatigue, Vol. 20, No 1, pp. 9–34, 1998
[3] M. A. Miner, “Cumulative damage in fatigue”, Journal of Applied
Mechanics, Vol. 12, pp.159-164, 1945
[4] A. Varschavsky, “The matrix fatigue behaviour of fibre composites
subjected to repeated tensile loads”, Journal of Materials Science, Vol. 7,
No. 22, pp. 159-167, 1972
[5] K. Reifsnider, “Fatigue behavior of composite materials”, International
Journal of Fracture, Vol. 16, No. 6, pp. 563-583, 1980
[6] Z. Hashin, C. Laird, “Cumulative damage under two level cycling: some
theoretical predictions and test data”, Fatigue & Fracture of Engineering
Materials & Structures, Vol. 2, No. 4, pp. 345-350, 1979
[7] M. Aslam, S. Jeelani, “Prediction of cumulative fatigue damage”,
Journal of Materials Science, Vol. 20, No. 9, pp. 3239–3244, 1985.
[8] A. Poursartip, P.W.R. Beaumont, “The fatigue damage mechanics of a
carbon fibre composite laminate: II-life prediction”, Composites Science
and Technology, Vol. 25, No 4, pp. 283-299, 1986
[9] T. Adam, G. Fernando, R. F. Dickson, H. Reiter, B. Harris, “Fatigue life
prediction for hybrid composites’, International Journal of Fatigue, Vol.
11, No 4, pp. 233-237, 1989
[10] M. J. Owen, R. J. Howe, “The accumulation of damage in a glass-
reinforced plastic under tensile and fatigue loading”, Journal of Physics
D: Applied Physics, Vol. 5, No. 9, 1972
[11] J. F. Mandell, H. J. Sutherland, R. J. Creed, A. J. Belinky, G. Wei, “High
cycle tensile and compressive fatigue of glass fiber-dominated
composites”, ASTM, Sixth Symposium on Composites: Fatigue and
Fracture, Denver, Colorado, 1995
[12] K. L. Reifsnider, “Life prediction analysis: directions and divagations”,
6th International Conference on Composite Materials (ICCM-VI),
London, England, 1987
[13] P. P. Oldyrev, V. P. Tamuzh, "Multicycle fatigue of composite
materials", Zhurnal Vsesoyuznogo Khimicheskogo Obshchestva imeni
D.I. Mendeleeva, 24, No. 5, 545- 552, 1989 (in Russian).
[14] G. P. Sendeckyj, "Life prediction for resin-matrix composite materials",
in: Composite Materials Series, Vol. 4, Fatigue of Composite Materials,
Elsevier, 1991
[15] R. Talerja, Fatigue of composite materials, Technomic Publishing
Company Inc., Lancaster, 1987
[16] K. L. Reifsnider, Fatigue of Composite Materials, Elsevier, New York
1991
[17] Z. Hashin, A. Rotem, “A fatigue failure criterion for fiber reinforced
materials”, Journal of Composite Materials, Vol. 7, No. 4, pp. 448-464.
1973
[18] F. Wang, X. Zeng, J. Zhang, “Predictive approach to failure of
composite laminates with equivalent constraint model”, Acta Mechanica
Solida Sinica, Vol. 23. No. 3, pp. 240-247, 2010
[19] E. K. Gamstedt, R. Talreja, “Fatigue damage mechanisms in
unidirectional carbon-fiber-reinforced plastics”, Journal of Materials
Science, Vol. 34, No. 11, pp. 2535-2546, 1999
[20] M. Quaresimin, L. Susmel, R. Talreja, “Fatigue behaviour and life
assessment of composite laminates under multiaxial loadings”,
International Journal of Fatigue, Vol. 32, No 1, pp. 2-16, 2010
[21] C. Tang, J. Fan, C. Tsui, T. Lee, L. Chan, B. Rao, “Quantification of
shear damage evolution in aluminum alloy 2024T3”, Acta Mechanica
Solida Sinica, Vol. 20, No. 1, pp. 57-64, 2007
[22] W. X. Yao, N. Himmel, “A new cumulative fatigue damage model for
fibre-reinforced plastics”, Composites Science and Technology, Vol. 60,
No. 1, pp. 59-64. 2000
[23] R. Talerja, “Stiffness properties of composite laminates with matrix
cracking and interior delamination”, Engineering Fracture Mechanics,
Vol. 25, No. 5-6, pp. 751-762. 1986
[24] J. Zhang, K. Hermann, “Stiffness degradation Induced by multilayer
intralaminar cracking in composite laminates”, Composites Part A:
Applied Science and Manufacturing, Vol. 30, No. 5, pp. 683-706, 1999
[25] M. Kawai, S. Yajima, A. Hachinohe, Y. Takano, “Off-axis fatigue
behavior of unidirectional carbon fiber-reinforced composites at room
and high temperatures”, Journal of Composite Materials, Vol. 35, No 7,
pp. 545–576, 2001
[26] Y. A. Dzenis, “Cycle-based analysis of damage and failure in advanced
composites under fatigue, 2. Stochastic mesomechanics modeling”,
International Journal of Fatigue, Vol. 25., No. 6, pp. 511–520, 2003
[27] K. Kang, J. Kim, “Fatigue life prediction for impacted carbon/epoxy
laminates under 2-stage loading”, Composites Part A: Applied Science
and Manufacturing, Vol. 37, No. 9, pp. 1451-1457, 2006
[28] F. Rognin, F. Abdi, V. Kunc, M. Lee, K. Nikbin, “Probabilistic methods
in predicting damage under multi-stage fatigue of composites using load
block sequences”, Procedia Engineering, Vol. 1, No. 1, pp. 55-58, 2009
[29] Y. Jen, Y. Yang, “A study of two-stage cumulative fatigue behavior for
CNT/epoxy”, Procedia Engineering, Vol. 2, No. 1, pp. 2111–2120, 2010
[30] A. Plumtree, M. Melo, J. Dahl, “Damage evolution in a [±45]2S CFRP
laminate under block loading conditions”, International Journal of
Fatigue, Vol. 32, No. 1, pp 139–145, 2010
[31] F. Wu, W. Yao, “A fatigue damage model of composite materials”,
International Journal of Fatigue, Vol. 32, No. 1, pp. 134-138, 2010