A comparative study of 2D and 3D Digital Image Correlation approaches for the characterization and numerical analysis of composite materials.

This article makes a comparison between different Digital Image Correlation methods to determinate the main mechanical characteristics of composite materials. More specifically Carbon Fiber Reinforced Polymers. For this purpose, several tensile tests were carried out using the same camera and lens model. Different statistical methods as well as probabilistic numerical simulations were performed with the aim of evaluating the discrepancies between methods, and between different mechanical parameters. We want to highlight the consistency of the results, enabling the possibility of using 3D methods with non-planar specimen for determining the mechanical properties of Carbon Fiber Reinforced Polymers. In this case, the novelty is focused on the use of different configurations (2D and 3D) to study the differences in terms of results. the objective is not the specific characterization of CFRP, but to analyze the way in which the use of a dataset from DIC3D or, on the contrary, from DIC2D affects the final results. According to this, it is possible to concluded that significative differences arise in the evaluation of the elastic properties that could be assigned to the uncertainties of the methods. However, this significance does not appear in the results of the probabilistic simulation.


Iº INTRODUCTION
Composite materials are a real alternative to the traditional steel-based solutions in engineering due to their mechanical properties, environmental resistance (especially in corrosive atmospheres) [1,2], fatigue performance [3][4][5] and a high weight-resistance ratio [6]. This type of material could be applied to different products such as pressure pipes and vessels [7,8], most of them made of Carbon Fiber Reinforced Polymer (CFRP) [9,10] and Glass Fiber Reinforced Polymer (GFRP). From a numerical point of view, composite materials are more complex than metallic materials. This is due to the wide range of uncertainties during manufacturing and the own complexity of materials [11,12]. Fiber orientation is one of the main factors, generating an intrinsic heterogeneity to the composite [13][14][15]. Due to this variability of performance, the product design is more complicated than other orthotropic materials, requiring the use of advanced numerical simulations able to represent this heterogeneity. In this context the use of probabilistic approaches based on the use of the Finite Element Method (FEM) could be considered as a feasible solution [16][17][18]. These approaches require the proper definition of the mechanical properties of the composite solution through the application of large experimental campaigns that allow to reproduce their probabilistic density functions (PDFs). Different devices can be used for the direct measurement of the material´s properties like Linear Displacement Transducer Design (LVDT) [19], sensors as laser transducers, extensometers or strains gauges, with the same precision as the DIC [20]. But the presence of different factors or alterations in these sensors might entail inaccuracies and erroneous data measurements that could vary the inputs of the PDFs. For more details about the different factors the reader is refereed to García-Martín et al. [21]. In order to avoid these limitations, the Scientific Community has developed different non-contact techniques which can evaluate all the strains over the specimen. Among the main techniques available to determinate the global mechanical characteristics are Moiré Interferometry [22]; Particle Image Velocimetry (PIV) [23]; photoelasticity [24]; multi-camera tracking and the Digital Image Correlation (DIC) [25][26][27]. In this context, DIC supposes a good technique for the characterisation of heterogeneous materials such as wood [28], concrete [29] or composites [25,[30][31][32]. There are two types of DIC strategies: the first one consists of a two-dimensional approach (2D DIC), while the second consists of a threedimensional approach (3D DIC) [6,31]. While the 2D DIC approach allows the measurement of displacements and strains in a single study plane, the 3D DIC approach allows the measurement of displacements that take place in different directions out of the main plane [33]. The main advantage of the 2D DIC is the possibility to use a simple setup using a single camera and without the need for external calibration, so the processing time is faster. However, the camera must be positioned completely orthogonally to the study plane and the uncertainties that can occur out of plane can be a disadvantage when this technique is applied [31]. For its part, 3D DIC requires a more complex setup using several cameras, which must be synchronised, in order to solve their relative orientation, requiring longer processing times due to its inner calibration and the high number of images acquired. However, in applications where the geometry is non-planar is the unique approach valid for assessing the mechanical characteristics of the material. This issue is especially relevant in composite designs made of CFRP and using filament winding processes, in which the manufacture of flat specimens for mechanical analysis is not common [34][35][36]. In these specimens the stresses are mainly produced in the plane of the shell (thin shell theory) [37], making use of 3D approaches necessary the strains produced. In this context, it is necessary to make a comparison between the 2D and 3D DIC approaches in order to verify the suitability of the 3D method for determining the mechanical properties of non-planar CFRP specimens for numerical simulations. This comparison has to be exhaustive, not only analyzing the results obtained by the simulations or the results of the test. It is necessary to carry on a statistical comparison. In this case, the novelty is focused on the use of different configurations (2D and 3D) to study the differences in terms of results. The objective is not the specific characterization of the CFRP, but to analyze the way in which the use of a dataset from DIC3D or, on the contrary, from DIC2D affects the results. After this introduction, Section 2 describes the materials and methods used for this study, namely 2D DIC, 3D DIC and probabilistic numerical simulations. In Section 3 the experimental results are outlined. Finally, Section 4 shows the conclusions of the research and future projects.

Aº . Composite solution and specimen preparation
For this work, the composite solution used was a CFRP manufactured with fiber CC 200 T-120, epoxy resin CR82 and complemented by a hardener Biresin CH80-10. The mechanical properties of the different constituents can be consulted in [38]. The manufacturing process of the specimen has been based on ISO 527:1997 [39,40]. A total of 25 specimens were done, each one with a total of 9 layers with orthogonal orientation among them. The process of manufacturing was the following one: (i) fiber placement with the corresponding orientation; (ii) resin impregnation; and (iii) pressure application to avoid possible resin defects. After placing all the layers, the entire block was introduced in a drying oven at a temperature of 50 °C during 50 minutes. As a result, a CFRP composite with an average thickness of 2 mm apt to the manufacturing of pressure vessels was obtained. When the composite block was completed, it was cut with a Computer Numerical Control (CNC) machine.

Bº . Test Set Up and Specimen Preparation
A Servosis ME-405/50/5 test machine was used for performing the tensile test. This machine provides a maximum load of 500 kN, with a TC50Kn REP transducer as the load cell and MTS Model XSA304A Grips as the used in past works [38]. Added to the test machine, it is necessary to use cameras for performing the DIC tests. Specifically, a total of three DSLR Canon 700D cameras, equipped with a 60 mm macro-lens and a CMOS APS-C sensor of 22.3x14.9 mm, were used for performing the DIC tests. The image size was of 5184 x 3456 px with a pixel size of 4.3 μm. Two cameras were used for 3D DIC and one camera was used for 2D DIC. The 2D DIC camera was placed orthogonal to the plane of the specimen with the aim of minimizing possible uncertainties. The verticality of the specimen has been guaranteed with an inclinometer and with a micrometric ball joint. So, the orthogonality between optic axes of specimen and camera is guaranteed. 3D cameras are placed in both sides and the 2D DIC camera is the one in the middle, orthogonal to the specimen. [41]. Meanwhile, the cameras used for the 3D-DIC were placed forming a stereo angle of 15° ( Figure 1) in order to avoid possible depth of field problems [42]. For automating the acquisition process, the cameras were connected to a Programmable Logic Controller (PLC), Siemens Logo. The PLC was programmed to simultaneously send a signal to the cameras and thus capture the images. Also, the machine and the PLC were connected to a data acquisition system, QUANTUM, which collects all data. Additionally, two led spotlights were used with the aim of providing a good illumination during the tensile tests ( Figure 1). Javier Pisonero: A comparative Study of 2D and 3D Digital Image Correlation approaches for the characterization and numerical analysis of composites materials (February 2017) VOLUME XX, 2017 3 FIGURE 1º. General view of the a) image acquisition system and b) data acquisition system. 3D cameras are placed in both sides and the 2D DIC camera is the one in the middle, orthogonal to the specimen.
Finally, the load increment of the test was calculated according with ISO 527-4 [39,40]. Result to this, the cameras were programmed for capturing one image per second. For the specimen preparation, first of all is necessary to apply the Speckle pattern. Speckle pattern is one of the most important factors for the final results of the DIC [43]. For this work, a computer speckle pattern was used, since could be considered the most robust and reliable way to create it. To this end, the following considerations were taken into account [44][45][46][47][48]: i) pattern randomness; ii) circular spots; and iii) covering among 40-70% of the specimen surface, in order to reduce homogeneity. The Mean Intensity Gradient (MIG) was used for assessing the quality of the pattern [49,50]. Taking into consideration the previous specification, a speckle pattern was designed following the approach proposed by [38]. First of all, a regular mesh with circular specks was created, being the principal parameters for this mesh the diameter, d, of the spots, which is comprised between 3-5 pixels, and the separation (step) among centres. The randomness of the pattern was generated applying a Gaussian random factor. The pattern was developed with a Matlab© script obtaining a MIG of 56, considering it as acceptable [51]. The defined diameter, d, was 0.324 mm and the step was 0.432 mm. Figure 2 shows the designed speckle impregnate into the specimen.

Cº . Digital Image Correlation approaches
This section describes the different DIC strategies used for estimating the mechanical properties of the composite solution, namely Young Modulus (E), Poisson ratio (v) and Tensile strength (T). To this end, an ad-hoc set-up was designed to compare the two techniques: on the one hand, the 2D DIC method was applied with the images captured by the central camera. Meanwhile, the 3D DIC method was fed with the images captured by the stereo system ( Figure 1). After completion of the test and the data acquisition, the mechanical properties have been defined following the workflow showed in Figure 3.

1) 2D digital image correlation
The Bundle Adjustment (BA) approach was applied in order to obtain the inner calibration of the camera. This method needs a set of images -about 20 -of a flat checkboard calibration target, with variations of position and orientation. The BA algorithm allows to minimise the overall re-projection error of the corners of squares (control points) extracted from the checkerboard calibration target. Then, a Gaussian radial distortion model is applied in order to compensate the distortion of the lens (Eq. 1).
After calibration, a set of images was processed using a 2D DIC approach ( Figure 3) [52]. The DIC method divides the image into square regions, called sub-sets, which are compared between two consecutive images (i and i+1), i being the reference image and i+1 the deformed image. The calculation of displacements and strains of each subset started with the centroid determination in the deformed image (i+1) with the use of [53]. The difference between the subset centroid positions provided the displacement vector, A, between the reference image (i) and the deformed image (i+1) (Figure 4).  To reach sub-pixel precision in DIC, the Quintic B-spline function and a non-linear optimization has been carried out as it was described in past works [21,54]. It is worth mentioning that the Inverse-Compositional Gauss-Newton (IC-GN) algorithm was used for obtaining the solution to the optimisation problem. [55]. The strategy previously shown was applied over the entire Region of Interest (ROI). With the aim of minimizing the error propagation as well as optimizing the computation time, the reliability-guided digital image correlation algorithm (RG-DIC) was used [56].

2) 3D digital image correlation
The main difference between 3D DIC and 2D DIC relies on the use of two or more additional cameras to estimate the displacements out of the main plane of the camera. To this end, the cameras need to be positioned in a non-orthogonal position [57]. In line with this, the 3D DIC method requires to solve the inner as well as the relative orientation of the cameras. On the one hand, the inner calibration of the cameras was carried out using the 2D DIC calibration approach. Then, the relative orientation of the cameras was performed using the Direct Linear Transformation (DLT) [57,58]. The use of the DLT allows to relate the image coordinates (xp, yp) with the object coordinates (X´, Y´, Z´) (Eq. 2). With the aim of minimising the uncertainty along the depth-axis, a cylindrical calibration target with control points was used. (2) = 5 + 6 + 7 + 9 9 + 10 11 + 1 Where x p and y p are the image point coordinates and L 1 -L 11 are the mathematical DLT parameters.
The relative orientation of the cameras allows them to obtain the 3D coordinates of homologous points which were determined in the following way ( Figure 5): i) matching between the pair of images (Left-Right) ; ii) matching between the images captured at the time i and i+1 (Reference-Deformed). This matching was carried out in the same way that the 2D DIC. Then, the DLT transformation was applied in order to obtain the displacements of the subsets using the following least square approach (Eq. 3).
Where A is a design matrix which connects the DLT parameters and the parameters presented in equation (2). M is the displacement matrix for every point. Finally, U is the difference matrix between DLT coordinates and camera parameters. Thanks to this, it was possible to assess the in-plane displacements (x,y) and the out-of-plane movements (z) depth direction thanks to the use of the DLT. As a result, it was possible to obtain a 3D point cloud of the specimen evaluated which allowed to obtain the full-field displacements and strains using the 2D DIC approach previously defined.

Dº . Statistical analysis
Once the process was finished following the 2D DIC and 3D DIC methods, parametric and non-parametric statistical analysis were performed to detect if there were significant differences among the mechanical properties. First, the data obtained for virtual extensometer in each experiment (DIC 2D and DIC 3D) were compared using the Kruskal-Wallis Test for independent samples. This non-parametric test allowed us to demonstrate whether a group of data comes from the same population.
If the existence of a statistically significant difference is not demonstrated, the parameters obtained with all the virtual extensometer for each experiment are compared using different parametric statistics: Levene Test and T-test. In addition, to strengthen the statistical comparison, non-parametric tests were applied: Median test, Kolmogorov-Smirnov, Moose's Test, Mann-Whitney U Test and Kruskal Wallis Test. These tests allowed us to evaluate the existence of statistically significant differences in each test.

Eº . Numerical analysis
In order to evaluate how the material uncertainties (input) Javier Pisonero: A comparative Study of 2D and 3D Digital Image Correlation approaches for the characterization and numerical analysis of composites materials (February 2017) VOLUME XX, 2017 5 influence the numerical simulations (output), an advanced numerical strategy was followed. This strategy, which is based on a probabilistic approach, followed two stages: i) the definition of the numerical model and; ii) the reliability analysis.

3) Definition of the numerical model
A FEM model was implemented following the indications established in García-Martín et al. [38]. This numerical model consisted in a real case of repaired corroded steel pipe by a composite wrap made of different composite plies. This model required the definition of the following parts ( Figure 6): i) the steel pipe; ii) the corroded part and iii) the wrap. The wrap part was meshed as a continuum shell with reduced integration and eight nodes SC8R elements, obtaining a mesh with 5,779 elements. This model will be the start of the construction of the metamodel which will be explained later. And the final results are obtained because of the execution of this model. Also, this is a practical application of the material which is currently used in several simulations [59,60]. The failure criterion of the models was the Tsai-Wu criterion [61]. This criterion assumes that failure occurs when the failure index (FI) is bigger than 1. Fi was calculated by the expression (Eq. 4): Where: ) √ 33 11 σ n ≡ Uniaxial tension in the different directions. X T ≡ Tensile load X C ≡ Compression load Y T ≡ Transversal tensile load, in the y direction Z T ≡ Transversal tensile load, in the z direction Y C ≡ Transversal compression load, in the y direction Z C ≡ Transversal compression load, in the z direction S xy ; S yz ; S zx ≡Shear transversal load

4) Reliability analysis
As it was stated in the introduction, the proper analysis of composite materials requires the use of probabilistic approaches. These approaches allowed us to determine the probability of failure of a system (Eq. 5): Where: • P fk is the failure probability. • G k are the restrictions. • f X (X) is the probability density function (PDF) of the random vector X ∈ R m and the design vector d ∈ R n .
However, the solution to this equation is not trivial and requires the use of methods able to approximate this value. For the present case study it was decided to use the Monte Carlo method for approximating this value. This method requires the use of thousands of simulations in order to obtain reliable results, requiring a high computational cost. In order to solve this drawback, a metamodeling strategy was used, in our case the Polynomial Chaos Expansion (PCE) method [62]. This model shows a good performance on those systems with probabilistic inputs. The PCE approximates the behaviour of the numerical simulation using multivariate polynomials, assuming that the real model can be represented as a finite model whose inputs are considered independent variables [63]. In this way, the polynomials were constructed with respect to the PDFs of the independent input parameters by estimating their coefficients through least squares minimisation of the input variables and the real responses. In this case, the adaptive sparse PCE based on the least angle regression [64] was used along with the least angle regression algorithm [65], in order to avoid an over-fitting. In order to evaluate the quality of the metamodel obtained, the modified version of the Leave-one-out error (LOO) [66] was used (Eq. 6-7). This parameter allowed us to perform an estimation of the error with an acceptable computational cost.
Where Y(X (i) ) is the computational model; f~P CE (X (i) ) is the surrogate model obtained from a specific DoE with N samples; Javier Pisonero: A comparative Study of 2D and 3D Digital Image Correlation approaches for the characterization and numerical analysis of composites materials (February 2017) VOLUME XX, 2017 6 h i is the i-th diagonal term of the matrix A(A T A) -1 A T ; A is the experimental matrix; cardA is the number of terms in the truncate series and; φ={φ ij =φ j (x (i) ), i=1,…,N; j=1,…,cardA}. All the process and simulations were executed using a Intel ® XEON E3-1240 v3 processor running at 3,4GHz with 8GB RAM DDRII.

IIIº EXPERIMENTAL RESULTS
A total of 25 composite specimens were manufactured and tested specifically for this research. During each test the 2D and 3D DIC methods were simultaneously applied using the set-up described in the previous section. Thanks to this, it was possible to obtain different mechanical variables such as the Young Modulus (E), the Poisson ratio (v), and the maximum strength (T).
Previously to perform each test, the specimen was properly prepared according with the suggestion done in Section 2.1. In this case, we started with the specimen preparation applying the speckle pattern. It is worth mentioning that all the images were acquired in RAW format with the aim of enhancing the contrast during the post-processing stage, allowing a better correlation between subsets of different images until the fracture of the specimen. Please note that the cameras were synchronized to always shoot at the same time. The average Ground Sample Distance (GSD) was 0.07 mm/px in all the cases. The test speed was of 2 mm/min and the time among shoots was 2 seconds.

1) 2D DIC approach
The images acquired by the 2D DIC camera were firstly corrected from lens distortion using the pattern and calibration model defined in Section 2.3.2 (Table I). To this end, a total of 74 images per specimen were used. For the 2D DIC post-processing the open-source software Ncorr [67], developed in Matlab ©, was used. This software implements the procedures showed in Section 2, allowing to obtain full-field displacements and strains. During this stage a subset size of 20x20 pixels and a step of 7 pixels were used for calculating the full-field displacements. In order to obtain the mechanical properties of each specimen, a total of six virtual extensometers were placed on the surface. Three along the vertical direction and three along the horizontal direction. The first one was placed at the center of the Region of Interest (ROI), the other two were located at 3 mm of distance ( Figure  7). Thanks to this, it was possible to obtain for each specimen a total of three Young Modulus (E) and Poisson´s ratio (ν) and a maximum strain (T) (Figure 7). Furthermore, the full field of displacements was calculated in order to compare the 2D and 3D approaches ( Figure 7B and 7C). Although the angular deviations in 2D-DIC can cause errors greater than 103 μstrains [68], the procedure used to ensure the orthogonality avoided these errors and differences were not found with respect to the results obtained in 3D. The equality of outcome of both approaches also makes it possible to demonstrate that out-of-plane motions do not occur, something already known in plane tensile tests. Furthermore, a macro lens was used in order to minimize out-of-plane motion and that this does not contribute significantly to in-plane strain measurement error [31].

FIGURE 7º. Fields of DIC into the specimen; a) Position of the different virtual extensometers used for each specimen; b) Field in the 2D DIC and c) Field in the 3D DIC reconstruction.
A non-parametric comparison was performed to analyze the existence of statistical significant differences between the calculated parameters (Young's Modulus (E) and Poisson's ratio (ν)) from each one of the three virtual extensometers within the 2D DIC experiment. For this purpose, a Kruskal Wallis test was applied between the E parameter of each virtual extensometer. This test did not demonstrate the existence of significant differences between extensometers in the variable E (p-value 0.938), nor for the variable v (p-value 0.694). Also, a parametric T-test was applied for E (p-value 0.950) and v (lower obtained p-value 0.259). The existence of significate statistical differences among the data provided by each virtual extensometer could not be demonstrated. Therefore, it was possible to work under the hypothesis that the data was homogeneous.  Finally, the PDF of each mechanical variable was computed, allowing to obtain the best PDF model. Then, the different statistics were calculated. Table III shows the results of using the Chi-square, Kolmogorov-Smirnov and Anderson-Darling tests.
Based on Table III, Young Modulus (E) was adapted to a Log-Normal function (LN), Poisson ratio (ν) to a Log-Normal function (LN) and tensile strength (T) to a Weibull function (W). In order to reinforce this Figure 8 shows the graphic representation of the PDFs of each parameter. The principal values of the PDF distributions are shown in Figure 8. Complementary to the estimation of the different PDF functions, a correlation analysis was carried out. For the present study case it was used the Spearman correlation index [69,70]. This index varies from 1 (complete positive correlation), 0 (no correlation) and -1 (a complete inverse correlation) according with the following equation (Eq. 8):

FIGURE 8º. Graphical representation of the PDFs obtained by 2D DIC approach. a) Log-Normal distribution of the Young's Modulus (E); b) Log-Normal Distribution of the Poisson's Ratio (ν) and c) Weibull distribution of the Maximum principal tensile (T)
Where x and y are the two vectors of the different parameters and m x and m y are the corresponding means. The correlation between parameters was shown in Table V. According to this, it was possible to conclude that the Young Modulus (E) and the tensile strength (T) of the specimen were highly correlated (R2  0.8527). Meanwhile the Poisson ratio (v) did not have correlation with the other two variables.

2) Results from the 3D DIC
For the 3D DIC post-processing the open-source software MultiDIC was used [71]. This software is based on Ncorr [65], using the same algorithms for processing the displacements and strains but including the relative orientation of the cameras. Firstly, the inner calibration of the cameras was obtained by applying the method used in the previous section, obtaining the following results (Table VI).  Then the relative orientation of the cameras was obtained using the inner calibration parameters as well as the cylindrical calibration target showed in Section 2.3.2 to solve the DLT transformation. These parameters allowed us to reconstruct the position of each subset´s centroid in 3D and thus their displacement vector as the difference of the centroid in the pair of images i (reference) and the pair of images i+1 (deformed).
Similarly to the 2D DIC, during the 3D DIC reconstruction a total of six extensometers placed in the same position were used. Thanks to this it was possible to obtain a set of values of the Young Modulus (E), Poisson ratio (v) and maximum strain (T) ( Table VII). These values were used to calculate the PDF functions following the method used in the previous section (Table VIII, Table IX) (Figure 9). A non-parametric comparison was performed to analyse the existence of statistically significant differences between the calculated parameters (E and T) from each one of the three virtual extensometers within the 3D DIC experiment. For this purpose, a Kruskal Wallis Test was applied first. This test did not demonstrate the existence of significant differences among the extensometers for the variable E (p-value 0.820) nor for the variable v (p-value 0.693). Also, a parametric T-test was applied for E (p-value 0.805) and v (p-value 0.277). The existence of statistically significate differences between the data of each virtual extensometer could not be demonstrated. Therefore, it was possible to work under the hypothesis that the data was homogeneous.

FIGURE 9º. Graphical representation of the PDFs obtained by 3D DIC approach. a) Log-Normal distribution of the Young's Modulus (E); b) Log-Normal Distribution of the Poisson's Ratio (ν) and c) Weibull distribution of the Maximum principal tensile (T)
As it was done in the previous section, the Spearman correlation factor was calculated (Table X). The R 2 between E and T was 0.6989 and the RSME was 26.41. Table XI shows a comparison between the PDFs obtained by the 2D DIC and the 3D DIC.

Bº Statistical comparison
As it was noted in the previous sections, significant differences were not detected between the data provided by each of the three virtual extensometers (within the same experiment). Therefore, using the results of all virtual extensometer, a general statistical comparison between 2D DIC and 3D DIC, can be made using parametric and non-parametric methods. The parametric statistics applied were Levene Test to study equality of variances, and the T-test to study equality of means. The nonparametric statistics applied were the Median Equality Test and the Moses Test. To analyze the equality of the distribution, the Mann-Whitney U Test, the Kolmogorov-Smirnov Test (K-S) and the Kruskal Wallis Test were used. The results are shown in the Table XII. For the variable E, both parametric and non-parametric tests indicate that there are statistically significant differences (p<0.05), except in the case of equality of variances. For the variable v, the existence of significant differences between the two methods cannot be demonstrated. The histogram of both variables can be visualised in Figure 10, observing a significate difference in the central tendency for the variable E. These significant differences can be explained from the point of view of the different uncertainties of each of the methods used. While in the 2D approach the main source of uncertainty are out-ofplane strains, in the 3D approach they are mostly related to the calibration of the cameras. This means that although the test procedures are similar, the uncertainties are independent for each technique. In this case, the difference between the strains is approximately 10-4 for values of an order of 20 times greater, so it is acceptable.

Cº Global Reliability analysis
From the results previously obtained, a global reliability analysis was performed over the deterministic mesh considering the Young Modulus, E, and the Poisson's coefficient, v, obtained in 2D DIC and 3D DIC as the input variables. As it was described in section 2.5, it was necessary to have a tool to determinate the Tsai Wu maximum in every simulation. So, for this reason a FEM analysis was executed. After that, a metamodel was built. In this way, the computer cost was greatly reduced, and it was possible to carry out a huge number of simulations in less time. As it was commented in section 2.5.2, the LOO error results obtained by the different metamodels were 0.032 in the 2D DIC methodology and 0.0034 in the 3D DIC methodology. A total of 200 executions to build the metamodel were carried out, and then a total of 100,000 executions of the Reliability analysis were carried out obtaining the probability of failure and the concentration of that as it is shown in Figure 11, where it was compared the probability with the parameters obtained by 2D DIC and 3D DIC. After the simulations, in the 2D and 3D only one case of 100,000 provided a Tsai Wu higher than 1. Next table (Table  XIII) outlines the Tsai Wu values at different quartiles. The inter-quartile (IQR) and inter-percentile (IPR) ranges (Table XIII) were extracted from the data generated in the Tsai-Wu analysis. The data show that the distribution for 2D DIC and 3D DIC are similar. This can be corroborated through the visualization of the histogram and failure plot ( Figure 11).

IVº CONCLUSIONS
This work aims to compare two DIC methods for determining the mechanical properties of CFRP composite solutions. To this end, a total of 25 composite specimens were tested using 3 different cameras at the same time: one for the 2D DIC method and two for the 3D DIC. The mechanical properties of the composite were determined using virtual extensometers in each one of the samples. Then, the Young Modulus, E, and Poisson ratio, v, were extracted. The Kruskal Wallis test performed on each of these variables reveals that the data captured is homogenous and could be considered as a unique population: one for the Young Modulus, E, and one for the Poisson ratio, v. According with different parametric and non-parametric tests performed, significant differences for the Young Modulus, E, were detected between the two tests. Specifically, the mean and median of the Young Module obtained by 3D-DIC was slightly higher than the one obtained by 2D-DIC. Despite this, the differences obtained were acceptable and may be due to the uncertainties of each test setup. Finally, a set of probabilistic numerical analysis were carried out with the aim of analysing the discrepancies in the numerical simulations for a real study case, fed by a FEM model, simulating a repaired corroded pipe. To this end it was used the PDF functions of each variable, as well as a metamodeling strategy for reducing the computational costs since it was required to perform a Monte Carlo analysis. The results obtained in both numerical analyses revealed a great similarity between both approaches in terms of interquartile, accumulative histograms and probability of failure. Also, a statistical comparison between the principal mechanical properties shows the same similarity. These results show that the small differences found between 2D and 3D are no longer significant when probabilistic numerical analysis methods are applied. In this way, the uncertainties are blurred and similar valid results are obtained for both methods, since the validity of the 2D-DIC method has been verified in previous works. In accordance with the results obtained during this work it was possible to conclude that the 3D DIC method was able to provide similar results than the 2D DIC with the advantage of being possible its use in non-planar surfaces. These surfaces were used for the determination of the mechanical properties in CFRP solutions. Future works will be focused on integrating the 3D approach in all the design chain, starting from the mechanical characterization and ending in the verification of the numerical model using physical specimens.