Ana Vrgoč

Viktor Kosin

Benjamin Smaniotto

Clément Jailin

Zvonimir Tomičević

François Hild

4D full-field measurements over the entire loading history: Evaluation of different temporal interpolations

Keywords: Fiber reinforced polymer, In-situ tests, Projection-based Digital Volume Correlation, Tomography

Introduction

Due to their advantageous material properties and unique ability to tailor them to meet specific design requirements, fiber reinforced polymers (FRPs) are increasingly being regarded as efficient substitutes for conventional engineering materials in critical-safety and lightweight structural applications [START_REF] Prashanth | Fiber reinforced composites -A review[END_REF]. The failure of such materials is induced by various damage mechanisms, which initiate at different length scales and often interact [START_REF] Brunner | Identification of damage mechanisms in fiber-reinforced polymer-matrix composites with acoustic emission and the challenge of assessing structural integrity and service-life[END_REF][START_REF] Vrgoč | Damage characterization in fiber reinforced polymer via Digital Volume Correlation[END_REF]. These phenomena call for advanced techniques for damage detection and quantification. X-Ray Computed Tomography (XCT) has proven to be a powerful tool in experimental solid mechanics to assess bulk kinematics and microstructural changes (Buljac et al. 2018a, Buljac et al. 2018b[START_REF] Tomičević | Damage observation in glass fiber reinforced composites[END_REF]. When coupled with Digital Volume Correlation (DVC), quantitative analyses of specimen deformation can be performed in-situ between two discrete states. However, the material behavior between two consecutive scans is not accessible (as acquisitions may last for few minutes up to few hours). Thus, time-dependent phenomena (such as creep or relaxation) cannot be captured. As FRPs are likely to undergo stress relaxation during experiments [START_REF] Maurer | Stress relaxation behaviour of glass fibre reinforced thermoplastic composites and its application to the design of interrupted in situ tensile tests for investigations by X-ray computed tomography[END_REF][START_REF] Vrgoč | Characterization of glass fiber reinforced polymer via Digital Volume Correlation: Investigation of notch sensitivity[END_REF], the reconstructed volumes may be impacted by motion artifacts. These restrictions can be circumvented by performing 4D (i.e., space and time) measurements employing the recently developed Projection-based Digital Volume Correlation (P-DVC). Instead of working with fully reconstructed volumes, P-DVC measures displacement fields from series of 2D radiographs acquired at different angles and loadings, thereby resulting in richer temporal resolutions without degrading too much the spatial resolution (Buljac et al. 2018, Jailin et al. 2018a). In general, P-DVC relies on space-time discretizations of displacement fields (Jailin et al. 2018a). The following study utilizes an alternative route in which spatial modes are constructed via scan-wise DVC, and thus only the corresponding temporal modes are sought via P-DVC.

The present work aims for 4D (i.e., space and time) characterizations of a glass fiber mat reinforced polyester resin subjected to in-situ cyclic tension. The dogbone specimen (containing a machined rectangular notch to induce high strain gradients) was subjected to cyclic tensile loading and simultaneously imaged in an XCT scanner. First, the properties and fabrication procedure of the investigated composite are presented, followed by a brief description of P-DVC in which the spatial modes are described by DVC fields. The investigated specimen geometry is presented together with the employed mechanical and imaging setup. Finally, the results of 4D measurements are discussed.

Material and experimental setup

The investigated material is a polyester resin reinforced with a continuous glass fiber mat. It was produced by manual lay-up followed by compression molding. The laminate consisted of 12 plies of R-glass fiber mat. The composition (in volume fraction) was 40% of fibers, 55% of matrix and, as vacuum was not applied during molding, the volume fraction of air voids was equal to 5%. The enhanced contrast between the constituents (in this specific case, between matrix, fibers and air voids, as shown in Fig. 1) makes the investigated material suitable for DVC analyses as it relies on natural patterns (Buljac et al. 2018[START_REF] Bartulović | Assessment of DVC measurement uncertainty on GFRPs with various fiber architectures[END_REF]. The 5.2 mm thick dogbone specimen contained a machined rectangular notch to induce high strain gradients (Fig. 2(a)). The notch depth was 1.6 mm and its width was equal to 0.6 mm. The radiographs were acquired continuously over the entire loading history in the X50+ scanner (North Star Imaging) of LMPS. First, the reference volume was acquired in the unloaded state (marked with black dots in Fig. 3). This scan consisted of 800 radiographs captured at equally spaced angles ranging over a full 360° revolution, and was acquired with so-called high quality (HQ) scanning parameters (Table 1). This acquisition took approximately 2 h to be completed. To conduct the experiment in a reasonable time and to mitigate e.g., stress-relaxation, the number of averaging frames for all subsequent acquisitions was lowered to 1 (i.e., continuous scanning parameters). The latter settings led to 4 min acquisitions per revolution. After acquiring the reference scan, the specimen was subjected to cyclic tension at a constant stroke velocity of 4 μm/s, and a continuous rotation of the specimen was set with 768 acquisitions per turn (see the so-called low quality (LQ) in Table 1). The radiographs were continuously acquired during the prescribed loading history with angular increments of 0.46º.

The deformed scans were reconstructed from the stages where a constant stroke was applied (marked with red dots in Fig. 3). These scans were used to calculate displacement fields via finite element (FE) based DVC implemented within the Correli 3.0 framework [START_REF] Leclerc | Correli 3.0[END_REF]. A total of 17,442 projections was acquired during the prescribed loading history. The radiographs had an initial definition of 1944 × 1536 px. Coarse graining of 4 × 4 elementary pixels into one superpixel was performed. This down-sampling led to 486 × 384 px projections (called images at scale 4), while the physical length of one voxel was equal to 58 μm. given by tomograph calibration and a filter back-projection approach. The inspected Region of Interest (ROI) was discretized with first-order tetrahedral elements tailored to the actual specimen geometry (Fig. 2(b)). The average element length was 20 vx.

Projection-based Digital Volume Correlation

This section outlines the basic principles of DVC and P-DVC. The notations used herein are introduced as well. The projections of the volume f(x) (i.e., the sum of absorption coefficient along each material point x of the beam ray hitting the detector at position r) are denoted as p(r,θ(t)), with θ the rotation angles at different time t. This results in a linear relationship between f and p  

() ( ) ( , ( )), t f p t    xr (1)
with () t   being the projection operator at angle θ(t). The inversion of this linear system for a large sequence of angles is called reconstruction. The 3D kinematics of a medium can be followed by reconstructing a series of such tomographic images. The full-field displacement field is estimated via DVC, which relies on the conservation of gray levels

  0 ˆˆˆ( ( , )), T T f f T  x x u x (2)
where 𝐱 ̂ denotes the position of any material point in the reference configuration, and û is the sought displacement field that describes the change of the reference volume 0 T f (usually captured in the undeformed state) to the deformed configuration (here marked as T f ). It is important to note the different time scales, i.e., the 3D images are reconstructed over the duration needed for one turn T.

To reconstruct such 3D images, hundreds up to thousands instantaneous radiographs (i.e., projections) captured at θ(t) are needed over a full revolution (or turn) of the specimen. The turn parameter T marks the different states at which a 3D scan was captured, while T0 simply denotes the reference scan. The sought displacement field is obtained from the minimization of the quadratic differences

      0 2 ˆˆˆˆˆ( , ) Argmin + , ( ) . T T T f T f   v x u x x v x x (3)
As a global (i.e., FE-based) approach was employed in the present study, the displacement field is written as follows ˆ( , ) ( ),

jj j Tv   v x x  (4)
where vj are nodal displacements, and ˆj  the corresponding FE shape functions. The present DVC framework is Lagrangian since the sought displacement field is determined with respect to the reference configuration. Instead of utilizing fully reconstructed volumes, P-DVC aims to measure 4D (i.e., space and time) fields from a series of 2D projections acquired at different angles θ(t) and loading steps (Jailin et al. 2018b). The minimization problem is thus rewritten as follows

        0 2 , ( ) Argmin , ( , ( )) . T t t t f t p t          v r u x, x v x r (5)
Thus, the problem is no longer stated in the Lagrangian framework, but in the Eulerian system (contrary to Eq. ( 3)). To complete the notations needed to describe the motion in Lagrangian and Eulerian settings, it is worth noting that the displacement at any spatial location x reads ˆˆ( , ) = = ( , ). tt  u x x x u x (6)

The P-DVC method employed herein requires the acquisition of one reference volume 0 T f . As the microstructure of the sample is known from 0 T f , the remaining unknowns are the displacements sought in the spatiotemporal framework. In the present work, the spatial component is constructed with scan-wise DVC, while only the temporal modes are sought via P-

DVC DVC ˆ( , ) ( , ) ( ), t T t     u x U x  (7)
where σ τ (t) gathers temporal basis function(s), ατ is the temporal amplitude(s), while the DVC field reads

  DVC ˆ( , ) ( ) . jj j T v T   U x x  (8)
The displacement fields are thus sought in a vector space generated by a reduced kinematic basis.

The interpolation functions σ naturally introduce temporal regularization (Jailin et al. 2018b). In such a way, only one projection per loading level is needed. In the present work, the temporal basis σ(t) consisted in the stroke history or the measured uniaxial force, and only one DVC field was utilized as spatial mode. Thus, the nodal displacements vj(T) were measured via FE-DVC, and the temporal amplitude is sought via P-DVC and iteratively updated

In Eq. ( 10), the Hessian l M is updated for each iteration

            , , , l l l t M S t t S t t    rr (11)
where S is the projected sensitivity

          0 DVC ˆ, , , .
T ll t S t T f t        r U x x u x (12)
Furthermore, the second member m l is based on the projected residual fields

      , , , ( ) ,
l l l C t m S t t t      r rr (13)
where the projection residual field per angle at each iteration is expressed as

          0 , ( ) , ( ) , . T ll C t t p t f t           r r x u x (14)
Thus, the reference volume, corrected by the displacement field u l (x,t), and projected for each angle θ(t) should match the acquired projection p(r, θ(t)). The projection residual field for all angles is to be minimized during the iterative procedure. As in DVC, these residuals reveal what was not captured by the 4D corrections, herein projected in the 2D detector space. A general overview of 4D analyses in which spatial modes are controlled by DVC fields is shown in Algorithm 1.

Algorithm 1 P-DVC enhanced DVC Perform DVC analysis DVC ˆ( , ) T  Ux Choose an initial guess α 0 while |δα l |>1•10 -3 do Corrections       0 , T ll f t f t  x x -u x, Update l M and l m Solve l l l m M   Update temporal amplitudes 1 , l l l      Update displacement fields 1 1 DVC ˆ( , ) ( , ) ( ) ll t T t    u x U x
end while

Application to in-situ test on single-notched specimen

In this section, the kinematics of a single-notched specimen subjected to in-situ cyclic tension is analyzed. The P-DVC measurements were performed up to specimen failure. Prior to performing 4D measurements, the reconstruction errors were evaluated. Then the results of 4D full-field measurements over the entire loading history are presented. They all converged, i.e., the change of displacement correction amplitude δα (Eq. ( 10)) between two iterations became less than 10 -3 .

Evaluation of reconstruction error

As shown in Eq. ( 14), the P-DVC code minimizes the projection residual fields, i.e., differences between the acquired radiograph p(r,θ(t)) and the reference volume corrected by the sought displacement field, and projected according to the angle θ(t). It is interesting to extend this procedure to the initial state (corresponding to the reference scan 0), where u is equal to 0 by definition. Thus, the reconstruction error was quantified by evaluating the difference between the corresponding projections and re-projections of the reconstructed reference volume 0 (acquired in the unloaded stage, as marked in Fig. 3). Due to acquisition noise, uncertainties in the geometric parameters used for the reconstruction and assumptions in the projection operator (linearity, pixel/voxel integration, …), the initial residual field was not equal to 0. Figure 4 shows the root mean square (rms) residuals (in arbitrary units) evaluated for the first 360° revolution (scan 0). The residuals were low with respect to the mean dynamic range of the original radiographs (0.7 a.u.). The highest rms residual levels (θ = 90º and θ = 270º) stem from the specimen orientation with respect to the X-ray source (i.e., the thickness of the specimen with respect to the X-ray source was highest at those angles, which influenced X-ray attenuation). The projection residual fields for two different angles are presented in Fig. 5. These initial residual maps are shown with a divergent colormap to highlight positive and negative values. The initial residual field for θ = 0º revealed elevated levels close to the edges of the notch. The residual could be reduced by further adjusting the geometric parameters of the tomography setup. Another source for such error could be small motions during scanning or beam hardening. The latter was not corrected in the present work. 4.2 Full-field measurements over the entire loading history First, the 4D kinematics was measured using the normalized force signal as temporal basis. As shown in Fig. 3, the measured uniaxial force history highlighted rapid but limited relaxation during the acquisition of scans 001, 003 and 005 (the drop of the peak force was approximately 160 N, see Fig. 3). At the peak of the fourth loading cycle, the fast relaxation stage was followed by a more gradual decrease. During the acquisition of scan 007, the force decrease was approximately 50 N. Next, the normalized stroke was used as temporal basis. The results obtained with the two temporal interpolations are compared in the sequel. In both cases, only one DVC field was utilized as spatial mode, i.e., the displacement field of scan 007 (corresponding to the last loading cycle prior to specimen failure). This field was obtained by performing volumetric correlation between scans 0 (reference) and 007. It is important to emphasize that the P-DVC measurements were performed until specimen failure (i.e., after acquiring scan 007). The projections to be analyzed with P-DVC were selected so that the angle between two consecutive projections was equal to 5º. A total of 1,338 projections was analyzed (out of 17,442 radiographs acquired during the whole loading history), which led to 70 projections per turn (360º revolution). The chosen sampling was also governed by the computation time. In P-DVC procedures, the heaviest operations are the computation of Hessian (i.e., sensitivity) matrices and the correction of the reference volume (Eqs. ( 11)-( 12)). Thus, to perform measurements in reasonable time, a compromise had to be made between temporal resolution and computation time. Last, the initial value of α was set to 1 (Eq. ( 8)).

The convergence of the algorithm was first evaluated from the change of displacements corrections during the iterative procedure (Fig. 6(a)). Both analyses took 4 iterations to converge. The measured displacement fields yielded a global decrease of the residuals within the inspected ROI, as shown in Fig. 6(b). Using the stroke history resulted in a significantly lower global residual. The same conclusions are drawn when observing the rms residual for each analyzed angle/time-step (to be compared with the mean dynamic range of original radiographs (0.7. a.u.)) shown in Fig. 6(c). Their differences were especially pronounced in the unloaded stages of the conducted experiment (Fig. 3). The residuals obtained using the stroke history remained lower during the analyzed loading history, while those obtained using the measured force signal were higher in unloaded stages. In both analyzed cases, the peaks of the rms residuals were observed for the radiographs acquired at 90º and 270º degrees with respect to the detector (as the thickness of the specimen with respect to the X-ray source was the highest). The comparison of the two temporal interpolations was performed with the converged value of α (Eq. ( 10)). The amplitude  was very close in both cases (Table 2), and greater than 1, since the DVC displacement field of the last loading cycle prior to specimen failure was used as spatial mode (Fig. 3), while the 4D characterization was performed beyond this stage (i.e., up to the specimen failure). Thus, despite this extrapolation, the displacement fields for all analyzed radiographs converged and were consistent (Fig. 6). With the chosen temporal sampling, the kinematics was very well captured even though with the increase in load levels the displacement fields became more heterogeneous. Furthermore, the rms differences between the 3D nodal displacements measured from the registration of two 3D images (with FE-based DVC) and projection-based measurements were calculated only for displacements in the z (i.e., loading) direction. The DVC displacement fields of the scans acquired during the prescribed loading history were obtained by performing volumetric correlations with scan 0 being the reference. It is important to highlight that large rigid body rotations (RBRs) in x and y directions were observed during the experiment. Thus, the rms differences were only computed for z direction displacements as the mechanical component was dominant in that direction. The rms differences were calculated between DVC displacement fields and converged P-DVC displacement fields for the time-steps corresponding to the beginning of each constant stroke stage (Table 3). As the series of projections acquired at the maximum load levels were affected by timedependent motions (i.e., stress-relaxation phenomena), which influenced the reconstruction of the full volume, differences were expected. From the values reported in Table 3, it is concluded that for both temporal regularization procedures the rms differences were higher for the unloaded stages of the conducted experiment (which was especially pronounced when using the measured uniaxial force as the temporal interpolation basis). This was due to the fact that part of the deformation mechanisms was deactivated [START_REF] Vrgoč | Characterization of glass fiber reinforced polymer via Digital Volume Correlation: Investigation of notch sensitivity[END_REF]. For the loaded stages, the differences were very small. This observation confirms that, despite using only one spatial mode (i.e., the displacement field of scan 007) when performing 4D measurements, the kinematics was very well captured. In addition, employing the stroke history as the temporal interpolation better captured the temporal fluctuations (as the rms differences were smaller compared to those obtained when using the measured uniaxial force).

Table 3. Comparison of rms differences in z direction between the 3D nodal displacements measured from the registration of two 3D images (with FE-DVC, see Fig. 3) and projection-based measurements. The comparison was performed for converged P-DVC displacement fields for the angles at the beginning of each constant stroke stage rms(Uz DVC -uz The displacement fields at the beginning of the last loading cycle obtained with P-DVC employing both temporal interpolations are shown in Fig. 7. They correspond to time-step 865 and are compared to the FE-DVC field of scan 007. From the presented displacement fields, it is seen that RBRs occurred during the conducted cyclic tensile test. The comparison of displacement fields further confirmed that the kinematics was properly captured at this stage with the P-DVC framework employed herein. However, higher differences in the z direction displacement fields were observed when performing P-DVC employing the measured force as temporal basis (Table 4).

To further analyze the latter statement, Table 4. reports the rms differences in z direction between the projection-based measurements at the beginning (time-step 865) and the end (time-step 1115) of the last loading cycle prior to specimen failure and displacement field of scan 007 obtained with DVC. At the beginning of the last loading cycle prior to specimen failure, using the measured uniaxial force as the temporal basis resulted in higher rms displacement difference. At the end of the loading cycle, this rms difference significantly decreased and reached the level obtained by using the stroke history as temporal basis (which was constant through the loading plateau). This effect stems from the fact that the measured uniaxial force revealed relaxation phenomena, and thus, the projection-based measured displacements differed. Conversely, the stroke history was constant (Fig. 3) and thus the projection-based displacements were constant during the analyzed loading. Figure 8 shows the displacement fields at the end of the last loading cycle obtained with P-DVC employing both temporal interpolations. These fields correspond to time-step 1115 and are compared to FE-DVC fields of scan 007. At the end of the loading cycle the differences between the projection-based and DVC fields of scan 007 are negligible (which is further confirmed by the values reported in Table 4). Figure 9 shows the projection residual map and major eigen strain field when θ = 105º captured at very end of the analyzed loading history, i.e., just before specimen failure. This map was obtained via P-DVC employing the stroke history as temporal basis. The measured displacement field correctly reduced the residual and thus was deemed trustworthy. Some residuals are still visible within the inspected ROI. The elevated residuals in the notched region were expected to occur as high strain gradients appeared in this region (Fig. 9(b)). In a previous work [START_REF] Vrgoč | Characterization of glass fiber reinforced polymer via Digital Volume Correlation: Investigation of notch sensitivity[END_REF], when only DVC measurements were used, it was shown that multiple cracks initiated and propagated within the notched region.

Conclusions

In the present work, 4D mechanical characterization of a glass fiber mat reinforced polyester resin was performed by employing P-DVC enhanced DVC that relied on space-time discretization of measured displacement fields. An alternative route in which spatial modes are constructed with DVC, and only temporal modes were sought via P-DVC was exploited. The main conclusions are as follows:

 By continuously acquiring the radiographs over the entire loading history and using only one radiograph per loading step, it was possible to enrich the temporal sampling compared to scanwise DVC. Within such a framework, it may be possible to capture time-dependent phenomena such as e.g., stress-relaxation and crack propagation.  The 4D kinematics was measured using two different temporal bases (i.e., the normalized stroke history or normalized measured uniaxial force). In both cases, only one spatial mode was used, namely, the displacement field of scan 007 (corresponding to the peak of the last loading cycle prior to specimen failure) measured via FE-DVC. It is important to note that 4D displacement fields were measured beyond the last loading cycle (up to specimen failure). Despite this extrapolation, the kinematics was very well captured (even though with the increase in load levels the displacement fields became more heterogeneous). These results prove that using only one spatial mode was sufficient to measure the entire 4D kinematics.  Both temporal interpolations led to a global decrease of the residuals over the entire ROI. Thus, the measured displacement fields correctly reduced the residual and were deemed trustworthy. However, using the stroke history resulted in a significantly lower global residual.

One of the principal aspects of future investigations will be the enrichment of the temporal basis and thus the spatial modes to further probe the solution faithfulness. Further, the presented results were obtained using coarse grained images (i.e., the resulting definition was down-sampled over a 4 × 4 regular square grid). This may lead to a loss of information beneficial for mechanical characterization. Another aspect will thus be to perform 4D measurements at the original scale. In addition, the measurements may be improved by further adjusting the geometrical parameters of the tomography setup, as well as performing beam hardening corrections (which was not carried out in the current work). The final objective will be to characterize time-dependent phenomena, especially the inception and propagation of damage as a previous work [START_REF] Vrgoč | Characterization of glass fiber reinforced polymer via Digital Volume Correlation: Investigation of notch sensitivity[END_REF]) revealed damaged zones within the investigated specimen via DVC residual fields.

Fig. 1

 1 Fig. 1 Optical micrograph of the glass fiber mat reinforced polyester resin composite studied herein

  Fig. 2 (a) Geometry of the investigated single notched dogbone specimen together with the Region of Interest (depicted with red contour), the size of which is 384 × 384 × 486 vx. The dimensions are expressed in mm. (b) Reconstructed reference volume together with first-order tetrahedral mesh employed herein

Fig. 3

 3 Fig.3Measured uniaxial force (blue) and stroke history (green) of the studied in-situ tensile test. The black dots mark the acquisition of the high-quality reference scan (0). The red dots depict the load levels at which full volumes were reconstructed. These volumes were employed in the scan-wise DVC analyses

Fig. 4

 4 Fig. 4 Root mean square (rms) residuals evaluated for the first 360° revolution (scan 0)

  projection residual fields for (a) θ = 0º and (b) θ = 90º

  a) Change of displacement corrections δα during the iterative procedure. (b) Change of rms projection residuals over the entire Region of Interest during the iterative procedure. (c) Change of rms projection residuals at convergence for each analyzed radiograph over the entire loading history

Fig. 7

 7 Fig. 7 Comparison of displacement fields at the beginning of the last loading cycle (projection 865) prior to specimen failure obtained with DVC (top) and P-DVC using the stroke history (middle) or measured uniaxial force (bottom) as temporal basis. The red dots on rms(uz P.DVC ) using the measured uniaxial force (blue) and prescribed stroke history (green) marks the time-step at which the results were compared

Fig. 8

 8 Fig.8Comparison of displacement fields at the end of the last loading cycle (time-step 1115) prior to specimen failure obtained with DVC (top) and P-DVC using the stroke history (middle) and measured uniaxial force (bottom) as temporal basis. The red dots on the rms(uz P.DVC ) using the measured uniaxial force (blue) and prescribed stroke history (green) marks the time-step at which the results were compared

  Fig. 9 (a) Projection residual map and (b) major eigen strain field obtained with P-DVC employing the stroke history as temporal basis calculated at the end of the analyzed loading history (time-step 1338), i.e., just before specimen failure. (c) Measured uniaxial force history. The red dot denotes the time step 1338 at which the residual map and major eigen strain field are shown

Table 1 .

 1 Scanning parameters

	Tomograph	North Star Imaging X50+
	X-ray source	XRayWorX XWT-240-CT
	Target / Anode	W (reflection mode)
	Filter	None
	Voltage	145 kV
	Current	78 µA
	Focal spot size	5 µm
	Tube to detector	910 mm
	Detector	Dexela 2922
	Definition	1507 x 1849 px (2 × 2 binning)
	Scanning parameters	High quality (HQ)	Low quality (LQ)
	Number of projections	800	768
	Angular amplitude	360°	360°
	Frame average	20 per projection	Continuous (1 per projection)
	Frame rate	3 fps	3 fps
	Acquisition duration	1 h 46 min 26 s	4 min 26 s
	Reconstruction algorithm	Filtered back-projection	Filtered back-projection
	Gray levels amplitude	8 bits	8 bits
	Field of view

Table 2 .

 2 Converged temporal amplitudes α obtained with the two temporal interpolations

	Signal	F	U
	Initial α	1	1
	Converged α	1.206	1.197

Acknowledgments

This work was performed within the FULLINSPECT project supported by the Croatian Science Foundation (UIP-2019-04-5460 Grant). This work was also partially supported by the French "Agence nationale de la Recherche" through the "Investissements d'avenir" program (ANR-10-EQPX-37 MATMECA Grant).


Andrija Zaplatić
email: andrija.zaplatic@fsb.hr

Zvonimir Tomičević
email: zvonimir.tomicevic@fsb.hr

Xuyang Chang
email: xuyang.chang@ens-paris-saclay.fr

Ivica Skozrit
email: ivica.skozrit@fsb.hr

Stephane Roux
email: stephane.roux@ens-paris-saclay.fr

François Hild
email: francois.hild@ens-paris-saclay.fr

Hybrid infrared-visible multiview correlation to study damage in a woven composite complex-shaped specimen

Keywords: Arcan fixture, full-field measurements, stereocorrelation, woven composite, damage

In this study, a cyclic tensile test on a notched butterfly specimen made of woven glass fiber composite was performed on a modified Arcan fixture. During the mechanical test, the sample was monitored with a hybrid stereoscopic system comprised of two visible lights and one infrared camera. The visible light cameras were employed for kinematic measurements using a finite-element-based multiview correlation technique. A semi-hybrid correlation approach was followed, providing Lagrangian temperature fields of the Region of Interest. Due to the complex composite architecture and specimen shape, localized shearing was observed during the tensile loading. Furthermore, asymmetrical damage developed around the notches as revealed by localized strains and thermal hot spots.

Introduction

The growing industry standards impose strict requirements for modern materials. Nowadays, conventional engineering materials are substituted with fiber-reinforced polymer matrix composites, especially in high-tech engineering. Composites are lightweight materials, and their architecture can be finely tailored to specific loading conditions. By predicting the loading regimes of in-service conditions, their mass and manufacturing cost are then reduced. However, since composite architectures are complex [START_REF] Vrgoč | Damage characterization in fiber reinforced polymer via Digital Volume Correlation. Coupled systems mechanics[END_REF], extensive mechanical investigations are required for a comprehensive characterization of their degradation under different loading conditions.

Since composite materials are subjected to complex loading regimes during their operational time, it is necessary to carry out mechanical tests that simulate realistic working conditions. Tensile tests are the most common due to their simplicity and availability. To simulate different loading types, composite samples are cut at different angles [START_REF] Vrgoč | Damage characterization in fiber reinforced polymer via Digital Volume Correlation. Coupled systems mechanics[END_REF]. However, using special loading devices, different loading types can be prescribed to the sample. The modified Arcan fixture (MAF) [START_REF] Zaplatić | Identifiability of Ludwik's law parameters depending on the sample geometry via inverse identification procedure[END_REF]) is a well-established loading apparatus, which provides a wide range of loading types by simply rotating the device. The MAF was used for a variety of materials such as metals [START_REF] Zaplatić | Boundary condition effect on the evaluation of stress triaxiality fields[END_REF][START_REF] Zaplatić | Identifiability of Ludwik's law parameters depending on the sample geometry via inverse identification procedure[END_REF], composites [START_REF] Holmes | Digital image and volume correlation for deformation and damage characterisation of fibre-reinforced composites: A review[END_REF][START_REF] De Sousa | Failure prediction of hybrid composite using Arcan's device and Drucker-Prager model[END_REF][START_REF] Hao | Development of Modified Arcan Fixture for biaxial loading response of fiber-reinforced composites[END_REF][START_REF] Ud Din | Design of a New Arcan Fixture for In-plane Pure Shear and Combined Normal/Shear Stress Characterization of Fiber Reinforced Polymer Composites[END_REF], adhesive joints [START_REF] Benelli | Experimental and numerical characterization of adhesive joints with composite substrates by means of the Arcan test[END_REF], or wood [START_REF] Xavier | Measurement of the shear properties of clear wood by the Arcan test[END_REF]. However, advanced measurement protocols are necessary for the full-field characterization of Arcan samples.

The development of visible light cameras prompted the emergence of contactless full-field measurement methods where Digital Image Correlation (DIC) stands out as the most used tool [START_REF] Sutton | Image correlation for shape, motion and deformation measurements: Basic concepts, theory and applications[END_REF]. In its fundamental form, DIC discretizes the observed region of interest (ROI) into a number of subsets (or facets) and measures the displacements of the center of each facet independently. The drawback of such approaches is the independently measured kinematics, which results in discontinuities, thus requiring additional post-processing interpolation/smoothing. This step can be avoided by using finite element (FE) based DIC, which assumes the continuity of the displacement field over the entire ROI [START_REF] Hild | Comparison of Local and Global Approaches to Digital Image Correlation[END_REF]. For complex sample geometries, Multiview Correlation (MC) (aka stereocorrelation) may be used. MC may also exploit FE meshes [START_REF] Berny | Displacement Uncertainty Quantifications in T3-Stereocorrelation[END_REF]. Furthermore, the optical setup can be enriched by adding an infrared camera in so-called hybrid environments [START_REF] Charbal | Hybrid Stereocorrelation Using Infrared and Visible Light Cameras[END_REF][START_REF] Wang | Hybrid Multiview Correlation for Measuring and Monitoring Thermomechanical Fatigue Test[END_REF]. With hybrid setups, displacements and temperatures are measured simultaneously using hybrid multiview correlation (HMC) algorithms [START_REF] Dufour | Shape, displacement and mechanical properties from isogeometric multiview stereocorrelation[END_REF][START_REF] Charbal | Hybrid Stereocorrelation Using Infrared and Visible Light Cameras[END_REF][START_REF] Wang | Hybrid Multiview Correlation for Measuring and Monitoring Thermomechanical Fatigue Test[END_REF].

The aim of this paper is to characterize the response of a woven glass fiber sample under tensile loading. The MAF is used to load the butterfly specimen. A hybrid stereoscopic setup consisting of two visible light and one infrared cameras is employed. FE-based MC was used for displacement and strain measurements. Moreover, a semi-hybrid algorithm was implemented in order to obtain Lagrangian temperature fields using the measured displacement fields.

Experimental investigation

In this research, a cyclic tensile test was carried out with the MAF. The 3-mm thick butterfly specimen was made of woven glass fibers infused with a vinylester resin. The yarns were oriented at 0°/90° along the loading direction. Two notches were cut with a band saw in the central gauge area of the sample. The length of the notches was approximately 2 mm long and 1 mm wide. The speckle pattern on the sample was created by spraying a thin layer of white paint on the surface, and then fine black speckles were applied using a paintbrush. The speckle pattern was not visible in the IR spectrum, hence uniform emissivity ( 0.96  = [START_REF] Leistner | Model validation of polymer curing processes using thermography[END_REF]) was assumed over the observed Region of Interest (ROI). The sample was loaded with a 0.24 mm/min rate in displacement control mode. The experimental and optical setups are shown in Fig. 1. The sample surface was monitored by a stereo-optical camera pair (VL) and one infrared camera (IC) with two light sources (L). In this work, only visible light cameras were used for displacement measurements. The hardware parameters of the optical setup are listed in Table 1.

Multiview calibration and correlation

In this section, the FE-based MC algorithm is presented. It is based on the pinhole camera model, which allows the 3D FE nodes to be projected onto 2D image planes of each camera c

1 1 c c c c X x Y sy Z       =             Π , (1)
where [Π c ] is the projection matrix, s c the scale factor, x c and y c the projected image coordinates of the FE nodal coordinates X, Y and Z.

               ==                   Π K T (2)
where fx and fy are the horizontal and vertical scale factors expressed in pixels, cx and cy the positions of the optical center, and ξ is the skew angle which is often disregarded since the horizontal and vertical scale factors are similar. The parameters rij in the extrinsic matrix [T c ] refer to three Euler angles ψ, φ and θ and their combinations. Finally, the last three parameters of the extrinsic matrix [T c ] are three translations tx, ty and tz in x, y and z directions, respectively. During the calibration procedure, the extrinsic and intrinsic parameters are determined for each camera to construct the projection matrices [Π c ].

Camera calibration procedure

The calibration procedure was initialized by Point and Perspective (PnP) [START_REF] Lepetit | EPnP: An accurate O(n) solution to the PnP problem[END_REF] solutions to obtain initial guesses of the extrinsic and intrinsic parameters from which the initial projection matrices [Π c ] are constructed for each camera c. At least 6 points are chosen on the FE mesh (X), and their respective positions in the images are selected (x c ). For calibration, grey level conservation is assumed such that the pictures captured by the left

  ( ) ( ) ( ) ( ) ( ) 2 2 1 1 1 2 2 2 ,, ff      =-    PX Π X Π xx (4)
is minimized over the chosen ROI where ρ denotes the calibration residuals and {P} collects the intrinsic and extrinsic parameters of the two cameras. For this integrated approach, the intrinsic and extrinsic parameters are first calibrated, and from which the projection matrices [Π 1,2 ] are then obtained. The sensitivity fields with respect to each parameter are computed via finite differences. The small parameter variations lead to pseudo-displacement fields in the 2D camera planes for each 3D point from c

x to cc  + xx from which the sensitivity field for each camera is expressed as

c c c   = P S P x (5)
which is a 2D vector field defined at each evaluation point X. In the calibration procedure, the intrinsic and extrinsic parameters are progressively updated to minimize the cost function (4) via Gauss-Newton iterations. After computing the sensitivity fields, the Hessian [H c ] for each camera is computed as

( ) ( ) ( ) ( ) T ,, c c c c c c c c c ROI ff           =                    PP HX Π S X Π S xx (6)
All camera-wise Hessian matrices are gathered in the global Hessian matrix

[C]   1 2 0 0     =     H C H (7)
Next, the camera parameter-based second member {b c } is expressed as

  ( ) ( ) ( ) ( ) ( ) ( )   ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 1 1 1 ,, ,, T ROI T ROI f f f f f f      =    -           =    -        P P b S X Π X Π b S X Π X Π x x x x x x (8)
and are gathered in the global second member

      1 2   =    b B b (9)
At each iteration step the corrections to the sought parameters for each camera are calculated

      1  - = P C B (10
) until the convergence criterion over the stationarity of the residuals, expressed in gray levels, is satisfied

15 10 nn    --  = -  ( 11
)
where n is the number of the current iteration.

Kinematic measurements

After calibrating the cameras, the nodal-wise kinematic field U(X) is measured at any given time t.

The new SSD cost function considers the reference images c f and corrected deformed image () c gt at time t for each respective camera

( ) ( ) ( ) ( ) ( ) ( ) 2 , 1 ( , ) min , , , c Nc c c c c c c t ROI t g t f =     =-     UX U X X Π X Π xx (12)
where

( ) ( ) ( ) , ,, c c c t gt   UX X Π x
represents the deformed image acquired by camera c at time t corrected by the displacement field ( )

,t UX
and Nc the number of cameras. The regularity of the kinematic field ( )

,t UX
in space, is ensured using a set of spatial shape functions Φ(X)

( ) ( ) ( ) 1 , DOF iN ii t u t = =  UX ΦX ( 13
)
where NDOF is the number of degrees of freedom, ui the nodal displacements determined at each step t. In the present case, three-noded (T3) elements were considered [START_REF] Berny | Displacement Uncertainty Quantifications in T3-Stereocorrelation[END_REF]). An iterative Gauss-Newton scheme was employed for the FE-based MC procedure. Therefore, the Hessian matrix [HU] and second member {BU} need to be computed

  ( ) ( ) ( ) ( ) ( ) ( ) T 1 ,, c cN cc c c c c c c ROI ff =          =                     U H Φ X X Π Φ X X Π XX xx xx (14) and   ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) T , 1 , , , , c cN c c c c c c c c c c t ROI f g t f =         =     -           U UX B Φ X X Π X Π X Π X x x x x (15)
where c  X x is the displacement sensitivity. The nodal displacement amplitude corrections are calculated

      1  - = UU HB u
(16) and the nodal displacement vector is updated

( )   ( )   ( )   . t t t  =+ u u u ( 17
)
The camera-wise gray-level residuals c  are computed for each evaluation point

( ) ( ) ( ) ( ) ( ) ( ) , , , , c c c c c c c c t g t f      =-    UX X Π X Π x x x ( 18
)

IR Camera

It should be noted that, since the speckle pattern was not visible in IR images, the IR camera could not be calibrated with the approach presented in Section 3.1. Hence only the PnP solution of the projection matrix [Π 3 ] was used. Furthermore, no prior IR camera calibration for the tested sample surface was performed and a uniform emissivity 0.96

 =
was assumed over the entire ROI.

The measured displacements ( , ) t UX need to be transformed from the FE mesh perspective to the camera plane. The nodal displacements in the IR camera plane IR u are defined ( ) ( )

IR IR IR fg tt =- u x x (19)
where IR f

x is the position of any projected FE node onto the IR camera plane in the reference configuration, and ()

IR g t
x that in the deformed configuration. The next step was to determine the projector matrix [p], which is used to interpolate nodal values to pixel positions. With the known nodal displacements, it is possible to compute the displacement fields for any pixel position

( )     ( )   ( )     ( )   px IR xx px IR yy tt tt = = u p u u p u (20)
where px u are the pixel-wise displacements of the ROI. Finally, the IR images were then corrected using the pixel-wise displacements

( ) ( ) ( ) , , , IR px T t g t t =+ x x u x (21)
where ( ) , Tt x is the pixel-wise Lagrangian temperature field of the ROI. Since the displacement fields were obtained via FE-based MC performed with images acquired with visible light cameras, it is a semi-hybrid procedure (in contrast with a full hybrid that would include the IR images in the determination of the displacement field [START_REF] Wang | Crack initiation and propagation under thermal fatigue of austenitic stainless steel[END_REF].

Experimental results

In this section, the multimodal results are discussed. Before the experiment, a set of 10 images was captured in the unloaded state for each camera, which were used for the calculation of measurements uncertainties (Table 2). A single pair of visible light images was used for the full calibration of the projection matrices. The geometry of the sample and its relation to the MAF was known a priori. Hence, characteristic points related to the sample and MAF were used to initialize the calibration procedure. The final projection matrices [Π c ] provided good results for the 3D mesh, once projected onto each image plane of the three cameras (Fig. 2). Moreover, a calibration object may be desirable given the planar shape of the sample. The tensile test consisted of eight loading / unloading cycles. In Fig. 3, the mean net section stress is displayed, which was calculated by dividing the measured force with the initial ligament area. The cycles were defined a priori by conducting a monotonic tensile test with the same experimental setup and same material. The sample was removed five times from the MAF during the test in order to access any microstructural changes via X-ray computed tomography (CT) [START_REF] Bartulović | Assessment of DVC measurement uncertainty on GFRPs with various fiber architectures[END_REF]. The effects of unmounting and remounting of the sample are discussed hereafter. The presented IR images are corrected by subtracting the current temperature field T with the first image of the corresponding loading cycle (Fig. 3). This operation is possible thanks to the fact that Lagrangian fields were obtained. The first results are for the images captured for the peak of the elastic regime of the material. The displacements for a stress level of 145 MPa are displayed in Fig. 4. The tensile loading axis corresponds to the y-direction (vertical direction in Fig. 2). The largest displacements are reported for the top left and right parts of the ROI. This is due to the geometry of the sample where the central part exhibited higher stiffness than the peripheral zones of the ROI. Displacement gradients are detected in a narrow zone around the notches as well as along vertical lines emanating from the notches. These phenomena are also observed in Fig. 5 and Fig. 6. The highest strain levels (Fig. 5) are concentrated in small zones around the notches. On both longitudinal (Fig. 5(a)) and shear (Fig. 5(b)) strain fields, two bands oriented along the loading direction are observed. The strain levels on both fields are of the same order of magnitude. Furthermore, the temperature field (Fig. 5(c)) reveals slight decreases in temperature in the center of the sample between the notches, which correspond to thermoelastic effects [START_REF] Crupi | Experimental analyses of SFRP material under static and fatigue loading by means of thermographic and DIC techniques[END_REF]. The correlation residuals (Fig. 6) for this stage between the reference and corrected deformed images reveal very small differences, which may be safely attributed to acquisition noise thereby indicating full convergence of the FE-based MC algorithm. The correlation residual levels (Fig. 9) increased overall. The main cause is attributed to changes in brightness and contrast (BC) compared to the reference states can be seen in the residual levels between the left and right images. It is especially apparent for the second camera. Such a uniform change may result from a rotation of the sample surface with regards to the lighting sources, due to non-Lambertian surface reflectivity/diffusivity. The highest residuals occurred on the left peripheral part of the sample where BC changed the most for both cameras. Increasing the applied stress by 50 MPa, the displacement fields in the y-direction (Fig. 10(b)) experienced the largest change. The displacements in the y-direction display much higher gradients due to shear than could be seen in Fig. 7(b). Moreover, the displacement fields in x and z-directions (Fig. 10(a,c)) experienced very small changes. The high gradients observed in the x-direction for Uy displacement field are not observed in the x and z displacement fields. strain fields. They are more pronounced than for the previous load level. Both longitudinal and shear strains have similar levels. Increased temperatures are noted in the lower right corner of the ROI (see Fig. 11(c)) and the same localization is present in the strain fields, but not in the residual fields (Fig. 12). Since the increase temperature is minimal, it could be attributed to artefactual heating originated from the testing machine and lighting. The gray level residual fields (Fig. 12) were similar to the previous cycle (Fig. 9), thereby indicating that the BC conditions did not change much during one further unmounting / remounting step. However, increased residuals are noted perpendicular to the left notch, oriented along the loading direction. This location matches that of the shear strain bands. Moreover, below the right hand side notch, the increased residuals coincide with the highest strains (Fig. 11). The next step corresponds to the ultimate tensile load level tested to investigate whether any damage initiated on the monitored surface. Before this cycle, the sample was removed and mounted again. Rigid body motions are observed from the displacement fields in x and z-directions (Fig. 13(a,c)). High gradients are detected in the displacement field in the y-direction (Fig. 13(b)) between the center and peripheral parts of the sample. Several shear bands are observed in the strain fields shown in Fig. 14. As seen from the high gradients in Fig. 13, increased strains occurred around the notches. The highest strain levels were reached above the left notch where also a faint variation in temperature can be observed indicating that the strain levels were reached earlier, leading to lower temperature and more diffuse signature. Above the right notch, increased strain levels are also present. This is also observable in the IR image (Fig. 14(c)) where increased changes in temperature are present due to energy dissipation during damage initiation. The correlation residuals (Fig. 15) reveal changes in BC conditions as for the previously discussed results. For the second camera, the overall residuals were higher than for the first one. Increased residuals were located around the notches. On the notch roots, small areas with higher levels indicate the presence of initiated cracks as was observed in the strain and displacement fields. Furthermore, below the right notch, higher residuals correspond to the shear band that propagated on its edge. A spot with high residuals is also present in the lower central part of the ROI. This was caused by a damaged speckle pattern, which was not related to the loading of the sample but due to its mounting/unmounting. For the final loading step, the maximum load was lower than the previous one, thus indicating that damage further developed in the sample. Severe gradients were detected in the displacement fields (Fig. 16), especially around the right notch root from which another strain band emanated (Fig. 17). The sample was removed from the testing machine before this last cycle, which caused rigid body motions seen in the y and z-directions (Fig. 17 The strain fields reveal several shear bands, with two in the vicinity of the right notch where severe damage occurred. In all strain fields (Fig. 17 (a,b)), the bands around the right notch are visible. Moreover, higher changes in temperature are observed in that area, highlighting energy dissipation due to damage. A change in temperature also occurred around the left notch in the same location as high strain levels are reported. The correlation residuals (Fig. 18) also reveal high levels in the same locations as the strain bands evidenced in the displacement and strain fields. The highest residuals are located under the right notch. Damage also occurred above the left notch, which is clearer for the second camera. The gray level residuals are compared to the raw images (Fig. 19). The increased residuals correspond to damage observed on the images. Due to differences in BC between images, the residual fields are also slightly different. Moreover, the peripheral areas of the ROI exhibit increased residuals for both cameras. The underlying damage caused the speckled surface to break, hence the FE-based MC algorithm could not converge between the reference and the deformed image since the gray level conservation was violated. From Fig. 19, it is confirmed that damage was more extensive in the vicinity of the right notch (Fig. 19

Conclusions

In this study, a cyclic tensile test was performed on a notched glass fiber woven composite via an MAF. A hybrid optical setup with two visible light and one infrared was used to monitor the surface motion of the sample. An FE-based multiview correlation algorithm was employed to measure displacement, strain and temperature fields. The main results of this work are:

• A semi-hybrid multiview procedure was implemented in which Lagrangian temperature fields could be reported over the investigated ROI in addition to displacement and strain fields using the exact same spatial coordinates. • Although unidirectional tensile load was applied on the sample, shear strain bands emanated from the two notches because of the sample geometry and the underlying woven architecture. • Damage initiated and developed asymmetrically around both notches, which indicates a significant influence of the woven fabric. Improvements to the experimental setup could enable for a complete multiview and hybrid analysis to be performed. Last, brightness and contrast corrections could be implemented.

  Fig. 1 (a) Test setup with the Modified Arcan fixture and cameras; see text for description of the labels. (b) Modified Arcan fixture.

  is only valid if the calibration parameters are properly determined without any acquisition noise. Therefore, a global formulation is performed where the Sum of Squared Differences (SSD)

  3D mesh with the calibrated projection matrices [Π c ] on the image plane of the first (a) and second (b) visible light cameras as well as the IR camera (c).

Fig. 3

 3 Fig. 3 Loading history of the tensile test. Five characteristic points are outlined by red circles for which kinematic and temperature measurements are reported hereafter. The chosen net section stress levels are 145, 196, 245, 289 and 262 MPa. The blue point denotes 99% of the ultimate tensile strength (289 MPa). The results for this level are not displayed since they were very close to the next one. The green rectangles denote the first image of each loading cycle between CT scans.

  displacement fields expressed in mm for the first cycle peak (145 MPa). Components Ux (a), Uy (b) and Uz (c).

  strain and temperature fields for the first cycle peak (145 MPa). (a) yy  , (b) xy  and (c) ΔT = T -T1 (in °C).

  level residuals for the first cycle peak (145 MPa) for the first (a) and second (b) cameras. The second set of results (Figs. 7 & 8) correspond to early onset of mechanical nonlinearity. Increased out-of-plane z displacements should be noted (Fig. 7(c)). The rise of these displacements was caused by the first unmounting and remounting of the sample. These perturbations caused rigid body motions, which do not influence the strain fields. The displacements in the y-direction indicate high gradients due to the obvious shear bands. displacement fields expressed in mm for the second cycle peak (196 MPa). Components Ux (a), Uy (b) and Uz (c) Strained bands are observed on both longitudinal (Fig. 8(a)) and shear (Fig. 8(b)) strain fields. Higher strain levels are reported for the left notch. The temperature field (Fig. 8(c)) still reveals thermoelastic effects in in the entire area between the notches. strain and temperature fields for the first cycle peak (196 MPa). (a) yy  , (b) xy  and (c) ΔT = T -T2 (in °C).

  level residuals for the second cycle peak (196 MPa) for the first (a) and second (b) cameras.

  displacement fields expressed in mm for the third cycle peak (245 MPa). Components Ux (a), Uy (b) and Uz (c) The gradients observed in the previous figure have a direct signature in the strain fields shown in Fig. 11. Strain (shear) bands are visible in both longitudinal (Fig. 11(a)) and shear (Fig. 11(b))

  Measured strain and temperature fields for the first cycle peak (245 MPa). (a) yy  , (b) xy  and (c) ΔT = T -T2 (in °C).

  Gray level residuals for the third cycle peak (245 MPa) for the first (a) and second (b) cameras.

  displacement fields expressed in mm for the fourth cycle peak (289 MPa). Components Ux (a), Uy (b) and Uz (c).

  Measured strain and temperature fields for the first cycle peak (289 MPa). (a) yy  , (b) xy  and (c) ΔT = T -T3 (in °C).

  level residuals for the fourth cycle peak (289 MPa) for the first (a) and second (b) cameras.

  displacement fields expressed in mm for the fifth cycle peak (262 MPa). Components Ux (a), Uy (b) and Uz (c).

  strain and temperature fields for the first cycle peak (262 MPa). (a) yy  , (b) xy  and (c) ΔT = T -T4 (in °C).

  level residuals for the fifth cycle peak (262 MPa) for the first (a) and second (b) cameras.

  Images of the damaged sample acquired by the first (a) and second (b) visible light cameras at the fifth cycle peak. The red circle indicates surface damage. The green circle indicates the surface deformity which originates from the manufacturing process.

Table 1 .

 1 Hardware parameters of the hybrid system.

	Cameras	2 × Dalsa Falcon 4M60	Flir A6751
	Definition	2358 × 1728 px	640 × 512 px
	Digitization	8-bits	16-bits
	Lens	Titanar 50 mm	Flir 50 mm
	Distance ring	7 mm	6.35 mm
	Aperture	f/5.6	-
	Stereo-angle	25°	0°
	Stand-off distance	260 mm	265 mm
	Camera distance	88 mm	-
	Image acquisition rate	1 fps	1 fps
	Patterning technique	B/W paint	-

Table 2

 2 Hybrid multiview analysis parameters

	DIC software		StereoCorreli 3.0		Correli 3.0
	Image filtering			None
	Average element length			0.7 mm
	Shape functions		Linear (T3 elements)
	Matching criterion	Sum of squared differences	-
	Temperature noise floor		-		0.05°C
	Displacement noise floor Strain noise floor	Ux 0.3 μm εx 5×10 -4	Uy 0.4 μm εy 6×10 -4	Uz 2 μm γxy 4×10 -4	-

Acknowledgments

This work was performed within the FULLINSPECT project supported by the Croatian Science Foundation (UIP-2019-04-5460 Grant). AZ was also supported through the Incoming International Scholarship Program of ENS Paris-Saclay. This work was also partially supported by the European Union through the European Regional Development Fund, Operational Programme "Competitiveness and Cohesion" 2014-2020 of the Republic of Croatia, project "Improvement of high-efficiency welding technology" (ImproWE, KK.01.1.1.07.0076).