Combining Kriging meta models with U-function and K-Means clustering for prediction of fracture energy of concrete
Introduction
The growth and expansion of micro-cracks and their transformation into large and continuous cracks will lead to the fracture of concrete [1]. Investigating the solutions to prevent and control the growth of micro-cracks in concrete elements increases safety and reduces the economic losses and casualties caused by the fracture and collapse [2]. As micro-cracks cover a large area and prevent the rapid expansion of cracks, structural engineers consider the size effect parameter (fracture energy (Gf) (initial fracture energy)) to examine the impact of the member's dimensions on its ultimate strength [3]. Therefore, when the purpose is to examine the performance of the real elements using small-scale samples in a laboratory, the significance of the size effect concept becomes more apparent. In addition, the classification of materials (brittle, quasi-brittle, and elastic-plastic) and the determination of their linear or nonlinear performance in structures are based on the size and recognition of the fracture process zone (FPZ) at the end of the crack [4,5]. To study the fracture parameters of different materials, use can be made of the size effect method [4,6], work fracture method [4,7], crack band method [8], effective crack method [9], two parameter model [10], cohesive crack method or fictitious crack method [11,12], double-G fracture method [13], KR-curve method [14,15], double-K fracture method [[16], [17], [18], [19]], boundary effect method and simplified boundary effect method [20,21].
In calculating the size effect fracture energy (Gf), the test specimens have the same geometry (shape), but vary in size. For this reason, the fracture energy calculated in size effect method (SEM) does not depend on the size, shape and geometry of the samples. As the maximum applied load and the area under the load-displacement graph before the maximum load is used to calculate the Gf, it is called the initial fracture energy (Fig. 1) [5]. Another method used to study the fracture energy is called work fracture method (WFM). In this method, the specific fracture energy (GF) is the same energy used to create the unit area of crack. Conventional-sized samples are used in WFM to calculate the GF, and the amount of the fracture energy depends entirely on the size and shape of the samples. In this method the fracture energy will be calculated by calculating the total area under the load-displacement diagram (Fig. 1), [5].
The fracture energy in WFM is always greater than the fracture energy obtained from SEM [22,23]. There are different points of view to the existence or non-existence of a clear relationship between these two parameters [24,25]. This difference is caused by various reasons such as the size and shape of the samples, how the experiments were performed, the distribution and the uncertainty of the fracture energy values obtained from the WFM after the maximum load and in the final part of the load-displacement diagram compared to the SEM fracture energy [4,23]. Bazant and Becq-giradun [24] reported a value of about 2.5 for the GF/Gf in typical concrete. While Beygi et al. [10] claimed that for self-compacting concrete (SCC), this ratio is 3.11. Ghasemi et al. [27] reported GF/Gf ratio for SCC with steel fibers to be 9.66. Researches by Kazemi et al. [28] showed that in high-strength concrete with steel fibers, the GF/Gf value is approximately equal to 10.5. The WFM method yields better fracture energy results than SEM because it considers the total area under the load-displacement diagram and the post cracking behavior of concrete [4]. Furthermore, the physical sample size is not adequate to describe the size effect on the fracture properties observed on laboratory-size samples because depending on the relationship between the sample width and the ligament transition length, a boundary effect and the size effect can exist [21].
Water-to-cement ratio (W/C), aggregate properties, amount and characteristics of admixtures, etc. are among the parameters affecting crack propagation and accelerating the fracture phenomenon in concrete [29]. According to research by Wittmann et al. [30], increasing the W/C in conventional concrete reduces the fracture energy. Carpinteri and Brighenti [31] reported that using the optimal amount of W/C generates the maximum fracture energy due to increased quality and strength of cement paste, especially in the area of contact with aggregates. They believed that this happens because the high strength of the cement paste almost prevents the spread of micro-cracks in this area. According to the studies by Nallathambi et al. [32], when the amount of W/C is high, the expansion and growth of cracks occurs more rapidly. Previous studies reported a decrease in fracture energy when W/C is too low or too high [26]. However, some researchers have presented different results on the relationship between W/C ratio and fracture parameters. For example, Mindess et al. [33] and Phillips and Binsheng [34] reported that the fracture energy does not change with increasing or decreasing the W/C ratio. According to Zhao et al. [35], there is no clear and tangible relationship between these two parameters. However, Ince and Alyamac [36] used Abrams’ law to confirm the existence of a specific relationship between W/C ratio and fracture characteristics.
Aggregates make up more than 70% of the concrete volume. For this reason, the performance and behavior of concrete at different ages are strongly influenced by the properties of these materials [37]. Although there is no specific relationship between the size of the aggregate and fracture toughness [38], increasing the amount and maximum size of aggregates (dmax) will have a positive effect on the failure properties [39,40]. Many researchers have reported that increasing the maximum size of the aggregates increases the fracture energy of concrete [[41], [42], [43]]. The studies by Petersson [44] showed that improving the quality and strength of the aggregates will have a positive effect on the fracture characteristics. Ince and Alyamac [36] reported that when the compressive strength of typical concrete (normal vibrating concrete) increases, the characteristics of the failure phenomenon such as energy and toughness increase. Compressive strength plays an important role in increasing fracture energy and toughness of high strength concretes. Although increasing the maximum size of the aggregates in this type of concrete will increase the compressive strength, unlike ordinary concretes, the failure energy of high-strength concretes is more affected by the type of aggregates [[32], [33], [34]].
Providing prediction models that have the desired strength, accuracy, and low cost to estimate the various parameters of materials, based on existing experimental input and output data sets, has always been interesting for researchers. In addition to multiple linear regressions, artificial intelligence methods such as artificial neural networks (ANN), adaptive neuro-fuzzy inference systems (ANFIS), and fuzzy and neuro-fuzzy systems have been used to estimate the complex performance of the structural materials [23,45,46]. Some Meta models such as Kriging surrogate, response surface methodology, dimension reduction prediction, etc. have also been used for this purpose [[47], [48], [49]]. Because concrete falls into the category of quasi-brittle materials and has a complex nonlinear behavior in fracture, assessment and obtaining an accurate fracture behavior model for this material play a key role in for an ideal, optimal, and safe design. Due to the cost and time reduction, it is very useful to provide an accurate model of failure behavior using existing experimental data sets and based on prediction methods and formulas [23].
Section snippets
Significance of the study
Investigating the concrete characteristics requires time, money, designs of experiments (DoE), preparation of materials and testing with special equipment at the desired ages. Therefore, the use of solutions that prevents or reduces the loss of cost, time and other disadvantages is needed in the concrete industry. In addition, human error due to the lack of skills and adequate knowledge of basic theories, the poor performance of the researcher during the experiment and equipment errors such as
Existing relationships to predict concrete fracture energy
Various equations and relationships have been proposed in recent decades to estimate the fracture energy of typical concrete using regression and the experimental samples. Some of these equations that have been used by researchers and codes are shown in Table 1. These relationships, which are often based on statistics and regression, are not accurate enough. Therefore surrogate methods are needed for more accurate assessments.
Kriging method
The original idea for the Kriging surrogate was developed by Daniel Krige and very fast became a common method as a low-cost simulation approach [60]. Kriging method is widely used in reliability assessment and collapse probability problems. Compared to the other surrogate methods, this method is known as an accurate interpolation method and the design points predicted by this method are exactly the same as the actual output values. The user can also increase the prediction accuracy of the
U learning function
In the past, the proposed Kriging models were presented inactive or fixed with the introduction of many points based on DoE concepts [69]. But Kriging's ability to determine the variance of the estimated data makes it possible to improve the DoE points with the help of an adaptive structure. To increase the accuracy of result prediction, when there is data with high variance, that point will be added to the DoE data set [48]. It is difficult to achieve the desired level of accuracy and
K-means clustering
The unsupervised classification process (clustering) is used to classify and place the data with similar characteristics in the same groups [73,74]. K-means clustering is widely used for data sets classification in K cluster. Despite the simplicity and power of discovering the hidden patterns of the data set, as well as the very good convergence rate of K-means clustering in achieving the optimal local point, the use of this method is limited. These limitations are due to introducing the number
Test data and input and output parameters
A data set collected by Nikbin et al. [23] where used in this study to predict the fracture energy of the concrete as an output parameter. A total of 246 data series where used. In order to summarize the information, the values of the input and output parameters are shown as domains in Table 2. Table 3 shows the details of each parameter. Many researchers have investigated the effects of various parameters such as aggregate type, aggregate content, aggregate size distribution, curing
Methodology
In prediction methods such as neural networks and ANFIS, determining a certain percentage of data for better network learning and providing an accurate model for estimating the engineering properties and materials is a big problem and in most cases, a lot of basic data sets are needed. In this study, the Kriging method, U-learning function and K-means clustering where combined to assess models that show the best performance in concrete fracture energy estimation with the lowest percentage of
Base kriging models
Fig. 3 shows the Gf and GF results of the base Kriging models versus experimental results. The results of the statistical parameters are given in Table 6 and Table 7 to examine the accuracy of the basic Kriging models. According to Table 6, Table 7, considering the aggregate shape, the WFM and SEM fracture energy prediction for all basic Kriging models have less error compared to the previous available relationships. However, predicting the fracture energy in the WFM using the basic Kriging
Conclusions
In this study, a database of 246 data sets was collected from past studies in which compressive strength, water-to-cement ratio and maximum aggregate size are considered as input parameters and the fracture energy as the output. In order to predict the fracture energy parameter of concrete, Kriging surrogate method was used by selecting 10%, 30%, and 50% of the data for training and combining this method with U-learning function and K-Means clustering. After analyzing the performance of the
Author statement
Iman Afshoon: Resources, Writing - Original Draft, Data curation, Writing- Original draft preparation, Software.
Mahmoud Miri: Conceptualization, Methodology, Investigation, Reviewing and Editing, Supervision, Project administration, funding acquisition.
Seyed Roohollah Mousavi: Resources, Writing - Original Draft, Data curation, Writing- Original draft preparation, Software.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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