Comparison of genetic programming with conventional methods for fatigue life modeling of FRP composite materials
Introduction
The realization of comprehensive experimental programs is very important when trying to understand and describe the fatigue behavior of composite materials. Over the last decades a significant number of databases have been developed for that reason. Some of them are relatively limited and refer to a specific material system, aiming primarily at the verification of new theoretical models, e.g. [1], [2], [3]. On the other hand, other databases are more extensive and were developed for the characterization of entire categories of materials primarily used in specific applications, such as databases DOE/MSU [4] and Optidat [5] for materials used in the wind turbine rotor blade industry. When fatigue data are available, the fatigue model is established in terms of a mathematical expression. The parameters of the fatigue model are then estimated by fitting the experimental data.
One of the most explicit and straightforward ways to represent experimental fatigue data is the S–N diagram. It is preferred to other types of representations, e.g. stiffness degradation or crack propagation, since only very simple recording devices are required. All that needs to be done is to measure load fluctuations during fatigue testing and the number of cycles after which the material fails. A fatigue model is then established, based on experimental data. The selection of the fatigue model is of paramount importance for any fatigue analysis. The fatigue model converts experimental data into theoretical equations that are subsequently used during design calculations. A number of different types of fatigue models (or types of S–N curves) have been presented in the literature, the most “famous” being the semi-logarithmic (also called Lin–Log) and the logarithmic (Log–Log) relations. Based on these it is assumed that the logarithm of the loading cycles is linearly dependent on the cyclic stress parameter, or its logarithm. Fatigue models defined in this way do not take different stress ratios or frequencies into account, i.e. different model parameters should be determined for different loading conditions. A drawback of these methods is that they are case-sensitive, since they may provide very accurate modeling results for one material system, but very poor for another. Other types of fatigue formulations that take the influence of stress ratio and/or frequency into account were also reported [6], [7]. A unified fatigue function that permits the representation of fatigue data under different loading conditions (different R-ratios) in a single two-parameter fatigue curve is proposed by Adam et al. [6]. In another work by Epaarachchi and Clausen [7], an empirical model that takes the influence of stress ratio and loading frequency into account is presented and validated against experimental data for different glass fiber–reinforced plastic composites. Although these models seem promising, their empirical nature is a disadvantage as their predictive ability is strongly affected by the selection of a number of parameters that should be estimated or even, in some cases, assumed.
Recently, methods of artificial intelligence have been adopted for the interpretation of the fatigue data of composite materials. Such methods have been previously used and validated in a number of different fields. They appear to offer a means of dealing with many multivariate problems for which an accurate analytical model does not exist or would be very difficult to develop. Artificial neural networks (ANN) have proved to be very powerful tools for pattern recognition, data clustering, signal processing, etc. During the last 10 years, ANN have been used to model the fatigue life of composite materials [8], [9], [10], [11], [12], [13], but they have also been used in other engineering problems. For example, Lee et al. [14] have used ANN for the prediction of the multiaxial strength of composite materials. Jia and Davalos [15] have used the same tool for modeling the fatigue crack growth rate of bonded FRP–wood interfaces. A hybrid Neuro-Fuzzy method designated ANFIS (Adaptive Neuro-Fuzzy Inference System) has been used to model the fatigue life of unidirectional and multidirectional composite laminates. Results of its application to two material systems have been presented in two references [16], [17]. Finally, genetic programming (GP) has been successfully used as a tool for modeling the fatigue behavior of composite materials, as presented by Vassilopoulos and Georgopoulos [18].
This innovative fatigue life modeling tool, genetic programming, is also presented in the current paper and its effectiveness is evaluated against other conventional methods, such as linear regression and Weibull statistics that are commonly used for this type of material analysis. Selected experimental data from the literature have been used for the application of the different models.
Section snippets
The S–N formulation
Traditionally, the S–N data are fitted by a semi-logarithmic or logarithmic equation. In the first case, it is assumed that the stress parameter is analogous to the logarithm of the number of cycles, while in the second, the stress parameter is linearly dependent on the logarithm of the number of cycles. The stress parameter S could refer to any cyclic stress definition, σmax (maximum stress), σa (stress amplitude), or even Δσ (stress range). The mathematical expression of the aforementioned
Experimental data
The fatigue behavior of four different composite material systems tested under several fatigue loading conditions of constant amplitude have been modeled using the aforementioned methods and results were compared. Fatigue data from tests at tension–tension, tension–compression, and compression–compression loading were retrieved from the literature. All of the materials are fiberglass–polyester or fiberglass–epoxy laminates, typical materials used in the wind turbine rotor blade construction
Results and discussion
In the present study, the idea of modeling fatigue life with genetic programming was applied on the four data sets that were described in the previous paragraphs. Fatigue data were considered as pairs of maximum cyclic stress and the corresponding cycles to failure. All data were handled as follows:
- 1.
The available data set was divided into training and validation data using a randomization technique; approximately 50% of the data were used for training and the remainder for the validation and
Conclusions
Genetic programming has been proved to be a very powerful tool for modeling the non-linear behavior of composite laminates subjected to cyclic constant amplitude loading. It can be used to model the fatigue life of several composite material systems, and compares favorably with other modeling techniques.
In its present form, genetic programming has been used as a stochastic non-linear regression analysis tool. As the training data set was structured with a single input for a single output, the
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