GENETIC PROGRAMMING IN MODELLING OF FATIGUE LIFE OF
COMPOSITE MATERIALS
Anastasios P. Vassilopoulos1, Efstratios F. Georgopoulos2, Thomas Keller1
1. Composite Construction Laboratory (CCLab), School of Architecture Civil and Environmental
Engineering, Swiss Federal Institute of Technology, EPFL, Station 16, CH-1015 Lausanne, Switzerland
2. Technological Educational Institute of Kalamata, 241 00 Kalamata, Greece
e-mail: 1: [email protected], 2: [email protected], 3: [email protected]
Abstract
In the current paper a novel method that uses a Genetic Programming (GP) tool is proposed for modelling fatigue life of
multidirectional laminates made of GFRP composite materials and tested under constant amplitude loading patterns. It has
been shown that GP is a powerful tool for such modelling exercises. The application of this technique over a data base
containing more than 250 fatigue data points of the same material system under different loading conditions and off-axis
angles was realized with remarkable efficiency. On- and off-axis fatigue behaviour has been accurately modelled based on a
small portion, in the range of 50%-60%, of the available fatigue data. Thus, expensive tests for determination of S-N curves
could be significantly reduced without noteworthy loss of accuracy. Modelling efficiency of the genetic programming tool is
satisfactory for the current material system, irrespective of the test conditions, i.e., R-ratio, that defines the developed stress
state on the coupon. Tension-tension, compression-compression and even tension-compression loading patterns were
investigated and modelling accuracy of the proposed method was validated. Because of its simplicity and flexibility, this
technique seems to be very promising for application on other engineering problems as well.
Introduction
Although the study of fatigue behaviour of composite materials has been the subject of a significant number of research works,
it still faces a major problem; the need for extensive experimental effort to describe fatigue behaviour of examined material
systems. This effort is directly linked to the lack of generalised theoretical models. These models should be established based
on experimental data for a specific material and under specific loading conditions and then be able to extrapolate their
predictions outside the space that is determined by known variables. This problem already appears when somebody is trying
to represent constant amplitude fatigue life data. In general, representation of fatigue life data is preferably performed on the
S-N or ε-N plane and deterministic models are used to produce regression lines that could be of several different kinds, i.e.,
power curve or log base 10 curve etc. These models capture the trend of the fatigue data and it is desired that they have the
ability to extrapolate this trend outside the region of the existent experimental data sets. S-N or ε-N curves, irrespective of their
mathematical form, are useful to compare one material system to another and also in design processes. To strengthen the
modelling ability of S-N curves statistical methods were developed especially for the analysis of fatigue data.
In the last decade, novel methods are also used for modelling the behaviour of composite materials under fatigue loading
patterns. These methods, based on the artificial intelligence concept were initially developed for the optimization of design
processes, using genetic algorithms as the optimizing technique e.g. [1-3]. A genetic algorithm is presented in [1] for the
optimization of laminate’s design against buckling. The optimization of a composite laminate to maximize the strength is the
subject of another article [2]. A user friendly expert system is presented based on the combination of genetic algorithms with
other computer tools, like finite element analysis tool. In another paper [3], a genetic algorithm is used in conjunction with
stress analysis to achieve optimum design of bolted composite lap joints. The objective of the optimization was to ensure the
maximum strength for the joint. Artificial Neural Networks were later on proved efficient to accurately model fatigue behaviour
of unidirectional [4-5] and multidirectional [6-7] GFRP and CFRP materials under different loading conditions. Artificial neural
Networks were also proved helpful in constructing Constant Life Diagrams which are necessary when spectrum fatigue loads
are present [8-9]
In the present work, constant amplitude fatigue life of a Glass Fibre Reinforced Plastic (GFRP) composite laminate is modelled
by genetic programming, It is proved that this tool is very powerful since fatigue behaviour of material system could be
efficiently modelled using only 50%-60% of the available experimental data. Compared to conventional modelling techniques,
e.g., regression lines for the determination of a power S-N curve, genetic programming can potentially save considerable
amounts of time and money that are spent for the completion of extensive experimental programs.
Material data base
For the validation of the modelling ability of the genetic programming tool, a fatigue data base that has been developed by one
of the present authors was used [10].This data base contains 257 valid fatigue data points concerning coupons cut on- and offaxis from a GFRP multidirectional laminate.
For all the tests straight edge coupons were cut from a multidirectional composite laminate consisting of four layers. Two
2
2
unidirectional laminae of 100% aligned fibres, with a weight of 700 g/m and two stitched ±45, in the middle, each of 450 g/m ,
225 g/m2 in each off-axis direction. The material system was a typical glass/polyester composite, E-glass from AHLSTROM
GLASSFIBRE, while the polyester resin was CHEMPOL 80 THIX by INTERCHEM. A thixotropic unsaturated polyester was
used for the matrix material mixed with 0.4%, Cobalt naphthenate solution (6% Co), accelerator and 1.5% Methyl Ethyl Ketone
Peroxide, MEKP, (50% solution), catalyst. About 10 rectangular plates were fabricated by hand lay-up technique under
industrial conditions and cured at room temperature. Considering as 0o direction that of the fibres of the unidirectional layers,
the lay-up can be encoded as [0/(±45)2/0]T. Coupons were cut, by a diamond saw wheel, at several on- and off-axis directions,
0o (on-axis) and 15o, 30o, 45o, 60o, 75o and 90o off-axis directions.
All coupons were prepared according to ASTM 3039-76 standard, their edges were trimmed with sandpaper and aluminium
tabs were glued at both their ends. Nominal dimensions in mm (length x width x thickness) were 250 x 25 x 2.6. The length of
the 2 mm thick tabs was 45 mm leaving a gauge length of 160 mm. All fatigue tests were conducted in a servo-hydraulic MTS
testing machine of 250 kN capacity, under load control. It was decided to keep the frequency constant for all tests, irrespective
of material system and load level, at 10 Hz. As it was proved later on, no significant temperature rises were recorded in any
case. The entire test program was realized under non controlled laboratory conditions, i.e., room temperature and humidity
according to weather conditions. However, no significant fluctuations were observed during the tests irrespective of the time of
testing.
Cyclic tests, of constant amplitude, sinusoidal waveform, were carried out in the aforementioned MTS test rig. In total, 257
valid test results were obtained and comprise the constant amplitude fatigue data base. Fatigue data were used for the
determination of 17 S-N curves at various on- and off-axis loading directions, under four different stress ratios, namely, R=10
(C-C), R=-1 (T-C), R=0.1 and R=0.5 (T-T). Seven different material systems were tested as coupons were cut at seven
different angles from the aforementioned laminates.
6
Tests were continued until coupon ultimate failure or 10 cycles, whichever occurred first. In particular, for the on-axis
o
6
coupons, 0 , under reversed loading, R=-1, tests were continued for up to 5×10 cycles. Coupons that did not fail, and were
taken out at determined number of cycles were marked as “run outs”. For all the tests with compressive cycles, an antibuckling guide was used. At least three coupons were tested at each one of the four or five stress levels that were preassigned for the determination of each S-N curve, according to the material under examination. Thus, 12-18 coupons were
tested for the determination of each one of the 17 S-N curves.
Genetic Programming
Genetic Programming (GP) belongs to the big family of Evolutionary Methods. Other well known methods that belong to the
same family are: Genetic Algorithms, Evolution Strategies, Evolutionary Programming, Swarm Intelligence, Artificial Immune
System, e.t.c. According to Koza [11], GP is “a domain-independent problem solving approach in which computer programs
are evolved to solve, or approximately solve, problems. Genetic Programming is based on the Darwinian principle of
reproduction and survival of the fittest and analogs of naturally occurring genetic operations as crossover (sexual
recombination) and mutation”. GP is a special case of evolutionary method that works by emulating natural evolution, in order
to produce a system (this may be a system model or a computer program) that optimizes some fitness measure (Mean Square
Error, for example) according to a number of given data.
For the case of system modelling GP uses a population of models, which may be represented in various forms like tree, linear,
or graph structures and evolves it through many generations towards finding some solution. The GP method for the production
of a system model works very briefly as follows:
1. An initial population (generation 0) of models is generated in random; every model (structure and parameters) of
this initial population is produced randomly. The models usually are represented in tree structure like the one in
figure 1, or in linear structure (the tree representation is used for demonstration purposes because it is the most
common in GP community). If this tree representation is been turned clockwise it can be viewed as a
conventional system block diagram. Each tree is of variable length, it is constructed of nodes and represents one
candidate model. The nodes can be terminal nodes (called also leafs) placed at the end of a branch signifying an
2.
3.
input or a constant, or non-terminal nodes representing functions performing some action on their terminal nodes
(see Fig. 1).
The performance of each model in the population is evaluated by simulating the corresponding model and
calculating some fitness measure like Mean Square Error, Mean Relative Error and so on, that defines the
quality of the model with respect to the experimental data.
A new population of models is created, using certain selection schemes (like proportional selection, tournament
selection, rank based selection, e.t.c.) and evolutionary operators like crossover and mutation.
Then the algorithm proceeds with the evaluation of this new population (go to step 2) and so on. After some number of
generations the algorithm converges at a near-optimum for the problem model.
Figure 1: A tree representation of a simple model
The Genetic Programming method has been applied using the experimental data of the aforementioned data base. The
objective was to model the fatigue behaviour of this GFRP material system using a much smaller set of experimental data
compared to that needed for the determination of stress- or strain-life curves (S-N or ε-N curves) by the conventional way., i.e.,
regression analysis and curve fitting. For the application of GP method the GP software tool of RML Technologies, Inc,
TM
TM
Discipulus [12] was used. Discipulus genetic programming software is a powerful regression and classification tool. A
simple model of 4 input and one output parameter was established. Stress ratio, R, on-, off-axis angle theta, along cyclic
stress amplitude, σa, and maximum cyclic stress, σmax, were considered as input parameters. The requested output parameter
was the anticipated number of loading cycles for the corresponding material under these loading conditions. Although two of
the input parameters are connected to each other, σa and σmax, it was decided to used them both. Based on previous
experience of the authors [8], use of all four parameters as input, instead of the three independent ones, assisted an artificial
neural network to perform better without any significant increase in its complexity.
Application of the Genetic Programming to Fatigue data
All fatigue data sets were processed by the genetic programming tool DiscipulusTM. Fatigue life of the material system under
investigation was modelled with good accuracy for almost all cases studied, even when few of the experimental data were
used for as input for the necessary training of the genetic programming tool.
A parametric analysis was also realized, as fatigue life modelling was performed each time using different portions of the
experimental data set as the training set. One of the main objectives of this study was to investigate the influence of the
number of experimental data points that were necessary for the training and test set on the modelling ability of the genetic
programming tool. To this end, different results were produced using a varying percentage of the 257 experimental data points
in the training and test set. Nine different realizations of training and test sets were finally used, namely: 90-10, 80-20, 70-30,
60-40, 50-50, 40-60, 30-70, 20-80, and 10-90. In each case, the first number represents the portion of the available
experimental data that is used to form the training set, while the latter represents the portion of the data that was used as test
set. For example, 80-20 means that 80% of the 257 data, i.e., 206 data points, were used to form the training set, while the
remaining 51 data points were used to evaluate the modelling ability of the computer tool.
The performance of the Genetic programming tool for selected cases between all the nine different combinations of training
and tests sets is graphically depicted in Figs. 2-4. It is clearly depicted in these figures that the modelling ability of the
computer tool becomes poorer as the portion of experimental data points assigned to the training set is reduced. The closer
the model outputs to the straight diagonal line, which indicates a perfect match between actual and predicted loading cycles,
the better the modelling efficiency.
Predicted loading cycles (Log10)
7
Train data set
Test data set
6
5
4
80% train set-20% test set
3
3
4
5
Actual loading cycles (Log10)
6
7
Figure 2. Modelling ability of Genetic Programming. 80% of the available data points were used as the training set, 20% for the
test set
Predicted loading cycles (Log10)
7
Train data set
Test data set
6
5
4
50% train set-50% test set
3
3
4
5
Actual loading cycles (Log10)
6
7
Figure 3. Modelling ability of Genetic Programming. 50% of the available data points were used as the training set, 50% for the
test set
Predicted loading cycles (Log10)
7
Train data set
Test data set
6
5
4
20% train set-80% test set
3
3
4
5
Actual loading cycles (Log10)
6
7
Figure 4. Modelling ability of Genetic Programming. 20% of the available data points were used as the training set, 80% for the
test set
Previous comments are verified by Figs. 5-8 where S-N curves for different cases are presented. The analysis of the results
showed that in some cases even 30% of the available fatigue data were enough to form the training set. However, in the
majority of the examined cases a portion of 50%-60% was necessary to model fatigue life with acceptable accuracy. As
depicted in Fig. 5., random selection of 30% of the available fatigue data points as the train set produced a very conservative
S-N curve. On the other hand, it was clearly demonstrated that half of the available experimental data are enough for the GP
model to produce reliable and accurate predictions.
Exp. data
GP preds, 30% train set
GP preds, 50% train set
σ a (MPa)
45
40
35
30
R=0.1, 15o off-axis angle
3
4
5
Log(N)
6
7
Figure 5. Experimental data vs. GP modelling, Comparison between two different alternatives of training data set population;
30% and 50% of the available fatigue data points.
This was the rule for almost all other cases independent of the off-axis angle or the applied loading pattern. Randomly
selected cases are presented in the two following Figs 6-7. As aforementioned, theoretical predictions were corroborated very
o
well by experimental data in most of the cases studied. In Fig. 6, theoretical predictions for on-axis and 45 off-axis coupons
tested under tension-tension, R=0.5, are presented and affirm the general good comparison between experimental data and
o
o
GP modelling. Relevant comments apply also on the next figure, Fig. 7, where fatigue life of off-axis coupons cut at 15 , 45 ,
o
o
75 and 90 from the multidirectional laminate and loaded under constant amplitude tensile loading, R=0.1, is examined.
Exp. data
GP preds, 50% train set
60
0o
σ a (MPa)
50
40
30
45o
20
R=0.5
10
3
4
5
Log(N)
6
7
o
Figure 6. Experimental data vs. GP modelling. Comparison for the case of on-axis and 45 off-axis coupons under tensiontension loading, R=0.5. Train and test sets were equally populated.
Exp. data
GP preds, 50% train set
40
σ a (MPa)
15o
30
45o
20
75o
R=0.1
10
90o
3
4
5
Log(N)
6
7
Figure 7.Comparisons between experimental fatigue data and corresponding GP calculations for the case of tension-tension
o
o
o
o
loading, R=0.1. Data from coupons cut at the off-axis angles of 15 , 45 , 75 and 90 are depicted.
The outcome of the aforementioned results is that modelling of fatigue life of this kind of material under the current loading
conditions can be achieved using the genetic programming technique that has been presented herein.
Conclusions
A new tool for modelling fatigue life of composite materials is presented in the current article. It has been proven in this study
that fatigue life of multidirectional GFRP composite laminates under tension-tension, tension-compression and compressioncompression loading can be efficiently be modelled using a genetic programming method. The application of the model over
17 different loading/material configurations derived very accurate results for almost all of the cases. The usefulness of this
modelling technique is strengthened by the fact that accurate results can be obtained, even when only a small portion in the
order of 50% of the available experimental data is used to form the training set. Thus, time consuming and costly fatigue
experiments can be substantially reduced.
This technique seems very promising for modelling fatigue life of composite materials, and can be classified along with other
computer techniques, like artificial neural networks, in a new category of novel methods for the analysis of fatigue behaviour of
composite materials. The results of these new methods should thoroughly compared to corresponding results from the
conventional methods for fatigue life modelling.
References
1.
Nagendra S., Jestin D., Gürdal Z., Haftka R. T., and Watson L. T., “Improved genetic algorithm for the design of
stiffened composite panels” Comput Struct, 58, (3), 543-555, 1996
2.
Jung-Seok Kim, “Development of a user-friendly expert system for composite laminate design” Compos Struct, 79,
(1), 76-83, 2007
3.
Kradinov V., Madenci E., and Ambur D.R., “Application of genetic algorithm for optimum design of bolted composite
lap joints”, Compos Struct, 77, (2), 148-159, 2007
4.
Al-Assaf, Y., and El Kadi, H., “Fatigue life prediction of unidirectional glass fiber/epoxy composite laminae using
neural networks”, Compos Struct, 53, 65-71, 2001
5.
El Kadi, H., and, Al-Assaf, Y., “Prediction of the fatigue life of unidirectional glass fiber/epoxy composite laminae
using different neural network paradigms”, Compos Struct, 55, 239-246, 2002
6.
Lee, J. A., Almond, D. P., and Harris, B., “The use of neural networks for the prediction of fatigue lives of composite
materials”, Compos Part A-Appl S, 30, 1159-1169, 1999
7.
Vassilopoulos A. P., Georgopoulos E. F., and Dionyssopoulos V., “Modelling fatigue life of multidirectional GFRP
laminates under constant amplitude loading with artificial neural networks”, Adv. Compos Lett, 15 (2), 43-51, 2006
8.
Vassilopoulos A. P., Georgopoulos E. F., and Dionyssopoulos V., “Artificial neural networks in spectrum fatigue life
prediction of composite materials”, Int. J. Fatigue, 29 (1), 20-29, Jan 2007
9.
Raimundo Carlos Silverio Freire Junior, Adriao Duarte Doria Neto and Eve Maria Freiri de Aquino, “Building of
constant life diagrams of fatigue using artificial neural networks” Int J Fatigue 27, 746-751, 2005
10. Philippidis T. P., and Vassilopoulos A. P., “Complex stress state effect on fatigue life of GRP laminates. Part I,
experimental”, Int. J. Fatigue,24, 813-823, 2002
11. Koza J.R., Genetic programming: on the programming of computers by means of natural selection, MIT Press,
Cambridge, MA, 1992
12. AIM Learning Technology, http://www.aimlearning.com.