A Zaplatić

Z Tomičević

D Čakmak

F Hild

Improvement of the Arcan setup for the investigation of thin sheet behavior under shear loading

Keywords: Arcan fixture, Buckling, Simple shear, Stereocorrelation, Thin sheet 1 Introduction

Background: Accurate predictions of thin sheet material springback during forming processes are of great interest in the forming industry. However, thin sheets are susceptible to buckling under shear loading. Objective: The present research aims at improving the so-called Arcan setup for testing thin (1-5 mm) sheet samples with large gauge areas (i.e., width about 21 mm) by introducing anti-buckling devices to mitigate sample buckling. Method: Three monotonic and one cyclic shear tests were carried out on 1 mm thick C60 high carbon steel. Results: The use of the proposed anti-buckling device resulted in the suppression of sample buckling. Numerical analyses of the experiment where buckling was eliminated revealed predominant shear stress states in the gauge area (i.e., stress triaxiality =

INTRODUCTION

Miyauchi [START_REF] Miyauchi | A proposal of a planar simple shear test in sheet metals[END_REF] proposed a complex geometry containing three clamped bars linked by the two sheared zones (i.e., gauge areas). By pulling the central bar, shear stress states were induced in the gauge areas. The sample geometry according to ASTM B831 standard [START_REF]ASTM B831-05, Standard Test Method for Shear Testing of Thin Aluminum Alloy Products[END_REF] contains two slant slits, thereby establishing a single shear zone in the middle of the sample. Merklein and Biasutti [START_REF] Merklein | Forward and reverse simple shear test experiments for material modeling in forming simulations[END_REF] modified the ASTM geometry to enable for compressive loadings. A comprehensive study carried out by Yin et al. [START_REF] Yin | An experimental and numerical investigation of different shear test configurations for sheet metal characterization[END_REF], where the aforementioned geometries were compared, yielded good agreement for the shear stress vs. strain responses for each investigated test.

Peirs et al. [START_REF] Peirs | Novel Technique for Static and Dynamic Shear Testing of Ti6Al4V Sheet[END_REF] developed a flat shear sample, which was based upon the geometry proposed by Bao et al. [START_REF] Bao | A comparative study on various ductile crack formation criteria[END_REF], with asymmetric rounded off notches with eccentricity that promoted shear stress states in the gauge area (i.e., stress triaxiality close to zero). Reyes et al. [START_REF] Reyes | Assessment of yield and fracture criteria using shear and bending tests[END_REF] optimized the geometry containing two slant slits with respect to stress homogeneity and triaxiality for static and dynamic shear experiments. The geometry of the slits was changed to a semi-circular shape with V notches to promote uniform strain distributions.

Roth and Mohr [START_REF] Roth | Determining the strain to fracture for simple shear for a wide range of sheet metals[END_REF] carried out a comprehensive study of Peirs' shear sample to obtain reliable shear strain measurements at fracture. As a result, three modified geometries were proposed for various levels of material ductility. Furthermore, Roth and Mohr [START_REF] Roth | Experimental investigation on shear fracture at high strain rates[END_REF][START_REF] Roth | Ductile fracture experiments with locally proportional loading histories[END_REF] optimized the so-called smiley geometry with two parallel gauge sections for in plane shear testing by modifying the notch geometry to avoid premature fracture initiation near the free gauge boundaries. To prevent buckling of the sample, the width and height of the gauge section were of the same order of magnitude as the thickness.

Iosipescu [START_REF] Iosipescu | New Accurate Procedure for Single Shear Testing of Metals[END_REF] proposed a flat sample geometry with two symmetric V notches (i.e., butterfly sample) in the gauge region to induce quasi uniform shear states between V notches.

Mohr and Henn [START_REF] Mohr | Calibration of stress-triaxiality dependent crack formation criteria: A new hybrid experimental-numerical method[END_REF] designed butterfly specimens with thickness reduction in the gauge area for investigating the fracture of metallic materials in a wide range of triaxialities. Moreover, Mohr and Oswald [START_REF] Mohr | A new experimental technique for the multi-axial testing of advanced high strength steel sheets[END_REF] improved the biaxial testing machine developed by Mohr and Doyoyo [START_REF] Mohr | A new method for the biaxial testing of cellular solids[END_REF], which was based on the Arcan fixture [START_REF] Arcan | A method to produce uniform plane-stress states with applications to fiber-reinforced materials[END_REF]. It enabled the authors to investigate the material behavior with the aforementioned butterfly sample under combined tensile and shear loadings. Further improvements to the sample geometry [START_REF] Dunand | Optimized butterfly specimen for the fracture testing of sheet materials under combined normal and shear loading[END_REF] were proposed for investigating fracture under combined normal and shear stresses. Abedini et al. [START_REF] Abedini | Fracture Characterization of Automotive Alloys in Shear Loading[END_REF][START_REF] Abedini | Fracture Characterization of Rolled Sheet Alloys in Shear Loading: Studies of Specimen Geometry, Anisotropy, and Rate Sensitivity[END_REF] analyzed the influence of two sample geometries (i.e., Peirs' mini-shear sample [START_REF] Peirs | Novel Technique for Static and Dynamic Shear Testing of Ti6Al4V Sheet[END_REF] and the butterfly geometry [START_REF] Dunand | Optimized butterfly specimen for the fracture testing of sheet materials under combined normal and shear loading[END_REF]) on the fracture of thin sheets. The butterfly sample achieved lower fracture strains than its counterpart. However, it was deemed useful for quasi-static fracture characterization in shear and plane strain/stress states. Furthermore, Peshekhodov et al. [START_REF] Peshekhodov | Experimentalnumerical evaluation of a new butterfly specimen for fracture characterisation of AHSS in a wide range of stress states[END_REF] developed a butterfly sample for fracture characterization of advanced high strength steel sheets under shear, tensile and combined stress states.

The aim of this research is to discuss a modified Arcan setup to investigate the behavior of butterfly specimens made of 1 mm C60 high carbon steel. The thin sheet samples were designed with large gauge areas (i.e., height equal to 21 mm), whereas the previously discussed shear geometries contained small gauge zones (i.e., < 3 mm). The latter ones were designed to promote uniform shear stress states and to reduce buckling [START_REF] Boni | Microscopic in situ observation of the plastic deformation of polybutene-1 films under simple shear[END_REF][START_REF] Bouvier | Simple shear tests: Experimental techniques and characterization of the plastic anisotropy of rolled sheets at large strains[END_REF][START_REF] Pucillo | On the mechanical characterization of materials by Arcan-type specimens[END_REF].

However, samples with large gauge areas are prone to buckling due to increased width to thickness ratios. Buckling restricts the use of standard optical measurement methods for determining the material response under simple shear loading. Therefore, to prevent buckling, transparent anti-buckling devices made of acrylic glass were added. Three monotonic and one cyclic shear tests were carried out and monitored with a stereovision system. Furthermore, the influence of the anti-buckling device was evaluated numerically by observing the contribution of friction to the calibrated plasticity parameters via Finite Element Model Updating (FEMU) [START_REF]Full-Field Measurements and Identification in Solid Mechanics[END_REF] and to stress triaxiality fields. Last, friction was calibrated in addition to the kinematic hardening parameters of the investigated steel.

In this section, the modifications to the Arcan fixture are presented. They include increased thickness of the fixture, upgraded gripping system of the tested samples, and modular design of the fixture. Additionally, changes to the butterfly geometry are also presented.

Modified Arcan device

The Arcan fixture [START_REF] Arcan | A method to produce uniform plane-stress states with applications to fiber-reinforced materials[END_REF] was designed as two separate semi-circular symmetrical plates made of aluminum alloy. The butterfly samples were glued to the supporting plates, and subjected to in-plane shear loadings. For the present study, the Modified Arcan Fixture (MAF, see Figure 1) was designed to study the mechanical response of thin (i.e., ≤ 5 mm in thickness) metal sheets subjected to in-plane tensile, simple shear and combined loadings with expected triaxiality values of 0.33, 0 and < 0.66 in the gauge areas, respectively. The main advantage of the MAF is an increased stiffness of the fixture to avoid out-of-plane bending, and a modular design to allow for varying sample thicknesses. Figure 1 shows the MAF that consists of two separate yet identical semi-circular 35-mm thick supporting plates (i.e., rigs), two sets of gripping jaws, and two identical testing machine adapters.

Modified Arcan device

See text for the description of the labels

The Arcan rig 1 (Figure 1) is made of M200 steel plates with a radius of 123.5 mm. The use of high strength steel (i.e., σ y = 1000 MPa) ensures that the fixture does not enter the plastic regime during tests. Near the peripheral area of the rigs, a series of twelve 13mm in diameter holes was drilled 15°apart. The prescribed load onto the butterfly sample is selected by the loading holes. For example, pure tension (α = 0°), simple shear (α = 90°) and combined loadings (0°< α < 90°) can be applied. The front central part of the rigs contains milled rectangular openings designed to hold the gripping jaws and the butterfly sample. The size of the rectangular openings is 75 × 35 mm with a depth of 25 mm from the front face. The rectangular openings result in asymmetric distributions of rig stiffness (i.e., the rear side of the fixture is stiffer than the front side). This shortcoming manifested itself in the form of out-of-plane bending of the sample since the load distribution induced spurious bending. The aforementioned imperfection was compensated with the appropriate design of the gripping jaws 2 and 3 (Figure 1).

The gripping jaw 2 serves as a placeholder to prevent any motion of the sample. The jaw 3 was designed with dual purpose. The primary purpose is to connect the sample 4 to the fixture via two sets of four M8 bolts 5 . The secondary role is to increase the rigidity of the front side of the rigs, which is established by directly connecting jaw 3 to the front side via three additional M8 bolts 6 . As a result, the rigs and griping jaws behave as a single solid part. Two additional pins 7 were added to prevent bolt shearing. Moreover, the gripping system was designed to ensure the coaxiality between the loading axis and that of the sample. To prevent slip of the sample, the contact surfaces on the gripping jaws were knurled.

The connection between the fixture and the testing machine is established via two adapters 8 . They consist of a central shaft, which is fastened to the testing machine via a single 13-mm in diameter pin 9 , the lock nut 10 , and two tabs 11 on the sides of the shaft.

Each tab consists of three loading holes equally spaced according to the layout on the rigs.

The central shaft of the Arcan adapter was designed with a radius less than the machine connector slot. As a result, a clearance is present between the Arcan adapter and the machine slot. Since the Arcan fixture is connected to the machine with a single pin, a single rotational degree of freedom is allowed. During tests, the fixture has the ability for selfalignment. However, to compensate for the aforementioned clearance, the Arcan adapter was also threaded at the connection point with the machine. This modification allows for the use of lock nut 10 . By tightening the nut, the adapter is firmly pushed into the connector slot of the machine, thereby increasing the preload on the sample.

Butterfly sample

Butterfly sample

The butterfly sample (Figure 2) was designed with two symmetric V notches for localizing the plastic strains in the narrow gauge area (i.e., between the two V notches). The surface of the sample is divided into three regions, namely, the gripped area, the observable area during tests and the Region of Interest (ROI) for stereocorrelation analyses. The dimensions of the gripped area, shown as red hatched in Figure 2(a), correspond to those of the contact surfaces of gripping jaws 2 and 3 . Since the gripped surfaces cannot be monitored during tests, only a 30 × 40-mm 2 surface area is visible. However, the chosen ROI, depicted by green hatches, was smaller than the observable area of the sample. This choice is justified by the fact that shear strains were concentrated in the gauge area close to the ligament. On each end of the sample, four 8-mm holes were drilled to allow the threaded M8 bolts to pass through to tighten the sample and the fixture together. Tightening the sample with four bolts makes the contact pressure more uniform. Two additional holes for 6-mm pins were added to prevent bolt shearing.

Experimental investigation

The experimental investigation presented herein was performed with the MAF. The latter offers three main loading modes depending on the angle α of the fixture with respect to the loading axis of the testing machine (depicted by the red arrow in Figure 1). The first loading configuration (Figure 1(a)) induces simple shear stress states in the gauge area of the sample, whereas the second loading configuration (Figure 1(b)) corresponds to tensile loading where the expected triaxiality levels in the gauge area are around 0 and 0.33, respectively.

Furthermore, the combination of both loading configurations (Figure 1(c)) enables for the application of combined shear/tensile loadings with expected triaxialities in the gauge area ranging from 0 to 0.66.

Buckling under shear loading

The accurate characterization of thin sheets subjected to shear loading is challenging [START_REF] Boni | Microscopic in situ observation of the plastic deformation of polybutene-1 films under simple shear[END_REF][START_REF] Bouvier | Simple shear tests: Experimental techniques and characterization of the plastic anisotropy of rolled sheets at large strains[END_REF][START_REF] Pucillo | On the mechanical characterization of materials by Arcan-type specimens[END_REF]. The samples are generally thin with a large gauge width to thickness ratio inducing folding of the sample (Figure 2(b)). Saurupt et al. [START_REF] Boni | Microscopic in situ observation of the plastic deformation of polybutene-1 films under simple shear[END_REF] concluded that the folding mechanism could be interpreted as buckling due to compressive stresses. Due to sample clamping, the material close to two diagonally opposed corners of the unclamped region is subjected to tension while the other corners are subjected to compression. Where compression occurs, buckling initiation was reported [START_REF] Bouvier | Simple shear tests: Experimental techniques and characterization of the plastic anisotropy of rolled sheets at large strains[END_REF]. Such mechanism may be prevented using additional supports placed on the observed region of interest [START_REF] Pham | Anti-Buckling Device for Ultra-Thin Metallic Sheets Under Large and Reversed Shear Strain Paths[END_REF], or by changing the sample gauge width [START_REF] Bouvier | Simple shear tests: Experimental techniques and characterization of the plastic anisotropy of rolled sheets at large strains[END_REF].

Experimental setup

The butterfly sample subjected to shear loading on the MAF exhibited buckling. This phenomenon was observed for sample thicknesses less than 2 mm (Figure 2(b)). In the experimental investigations presented herein, the behavior of 1-mm thick butterfly samples, made of C60 high carbon steel, was sought. Therefore, in order to suppress the initiation of buckling, additional supports in the form of two transparent plates made of Poly methyl methacrylate (PMMA) were selected. Such configuration was already used to successfully mitigate wrinkling in 1D and 2D tests on sub-millimeter sheets [START_REF] Bertin | On the identifiability of the Hill-1948 model with one uniaxial tensile test[END_REF][START_REF] Bertin | On the identifiability of Hill-1948 plasticity model with a single biaxial test on very thin sheet[END_REF]. The dimensions of the PMMA plates (i.e., 70 × 25 mm) were chosen with respect to the observable area (Figure 2(a)). Moreover, the transparency of PMMA allows optical measurement methods to be employed. Stereocorrelation was applied to measure displacement and strain fields of the ROI. The application of the supports is straightforward. Each plate was placed on the front and back surfaces of the sample and was connected via six M5 bolts.

Experimental setup

In order to determine the influence of the additional support on the mitigation of buckling, an appropriate experimental protocol had to be defined. The three configurations employed in this investigation are presented in Table 1. Furthermore, the influence of the stiffness of the connection between the MAF and the testing machine had to be studied. Pham et al [START_REF] Pham | Anti-Buckling Device for Ultra-Thin Metallic Sheets Under Large and Reversed Shear Strain Paths[END_REF] determined that clamping of the fixture (i.e., increasing the stiffness of the fixture) resulted in stress perturbations in the sample, thereby leading to buckling initiation. Higher stiffnesses of the MAF are achieved by tightening the lock nut on the testing machine adapter. Consequently, additional tensile/compressive preloads could be applied to the tested samples. The first configuration was characterized by a loosened connection between the MAF and the testing machine. Furthermore, no additional support was applied. However, even with the allowed self-alignment of the MAF, buckling of the sample occurred (Figure 2(b)).

Therefore, in the second configuration, an additional support with 5-mm thick PMMA plates was employed. Higher stiffness of the MAF was achieved and the measured preload was equal to -1000 N (Table 1). Although the PMMA support was applied, buckling of the sample was observed, which fractured the plates. For that reason, the third configuration employed 10-mm thick plates, and a loosened connection between the MAF and the testing machine. The third configuration suppressed buckling initiation. An additional cyclic test was then performed with the third configuration for further validation purposes.

The shear tests were carried out on a uniaxial testing machine Messphysik Beta 50-5 in controlled displacement with a stroke rate of 1 mm/min. The material response in the observed ROI (Figure 2) was monitored by a stereovision system. The image acquisition rate for the monotonic (i.e., E1, E2 and E3) experiments was 1 fps, whereas for the cyclic (i.e., E4) test it was 0.3 fps. The hardware parameters of the optical setup are listed in Table 2. The experimental investigation employing the MAF 1 (Figure 3) was monitored by the optical setup (Figure 3) consisting of two vertically positioned Dalsa cameras 2 and 3 (Figure 3) and two light sources 4 . The vertical layout of the cameras was chosen in order to achieve maximum spatial resolution for the ROI. The stereovision system captured the entire gauge area of the sample 5 and the additional PMMA supports 6 . For this study, FE-based stereocorrelation [START_REF] Dufour | Displacement measurement using CADbased stereo-correlation with meshes[END_REF][START_REF] Dubreuil | Mesh-based shape measurements with stereocorrelation[END_REF][START_REF] Pierré | Finite Element Stereo Digital Image Correlation: Framework and Mechanical Regularization[END_REF] was utilized for measuring displacement and strain fields in the region of interest (ROI). The aforementioned algorithm uses a continuous description of the observed surface discretized with triangular finite elements. The commercial code EikoTwin DIC [START_REF]EikoTwin DIC[END_REF] was used. The calibration was carried out in two steps. For the pre-calibration step, a first estimation of the projection matrix was sought for each camera. During this process, the scale factor (i.e., conversion between pixel and mm) was set. An FE mesh was created, which corresponded to the nominal geometry of the sample and the fixture (Figure 4(a)) that were observed by both cameras. Since each camera was independent at this stage, different points were chosen for the left and right camera pictures. These points must be distributed in the entire 3D volume that is covered by each camera. A total of 8 points was chosen in two different planes and the first estimates of the projections matrices were obtained for this self-calibration route [START_REF] Beaubier | CAD-based calibration of a 3D-DIC system: Principle and application on test and industrial parts[END_REF]. The pre-calibration step allows in particular for the verification of the reprojection of the measurement mesh (Figure 4(b)). ) by running a second series of spatial registrations in which out-of-plane corrections were allowed [START_REF] Dubreuil | Mesh-based shape measurements with stereocorrelation[END_REF].

Experimental setup

Last, displacement fields were measured by performing temporal registrations in which the nodal displacements were determined [START_REF] Dufour | Displacement measurement using CADbased stereo-correlation with meshes[END_REF][START_REF] Dufour | CAD-based displacement measurements. Principle and first validations[END_REF][START_REF] Pierré | Finite Element Stereo Digital Image Correlation: Framework and Mechanical Regularization[END_REF]. The stereocorrelation parameters are listed in Table 3. The noise floor estimates were obtained by calculating the (temporal) standard deviations of the nodal displacements and strains from the registration of ten image pairs acquired in the unloaded states. The first peak load was equal to 7.2 kN for a shear strain of 17.5%. The kinematic fields corresponding to the characteristic points shown in Figure 5(a) are reported in Appendix A.

EXPERIMENTAL INVESTIGATION

Test E2

The loading history of test E2 shown in Figure 5(b) exhibited a more stable behavior compared to the E1 loading history (Figure 5(a)). This trend was due to the application of the 5-mm thick plates, which delayed buckling inception. However, since the stiffness of the connection between the MAF and the testing machine was high, buckling was not totally suppressed. This mechanism led to the failure of the PMMA supports and the test was interrupted. The kinematic fields corresponding to the characteristic points shown in Figure 5(b) are reported in Appendix A. was essentially uniform. Furthermore, the achieved strain levels were at least one order of magnitude higher than the noise floor levels (Table 3). No damage was detected in the correlation residual map (Figure 9 . Very high gradients were observed in the strain fields (see Figures 8(e-f)), which were also present in axial displacement fields (Figure 7(c)) and correlation residual maps (Figure 9(b-c)). These "discontinuities" correspond to initiated and growing cracks. Since the sample had a tendency toward buckling, slight folding occurred in the center of the ROI, which caused changes in brightness (i.e., shadows). These changes in brightness were 20

EXPERIMENTAL INVESTIGATION

observed as an increase in the correlation residuals in the ROI center (Figure 9(b-c)). The correlation residuals are sensitive to damage initiation and growth [START_REF] Hild | Damage Measurements via DIC[END_REF]. Observing the correlation residual maps allowed mesocracks to be detected and quantified (i.e., symmetric initiation and propagation of two cracks from each V-notch, see Figure 9(c)). Last, comparing the correlation residual maps with the final deformed shape of the sample (Figure 5(c))

shows a good agreement for the location and size of the cracks. cycles. The 10-mm thick supports were utilized in addition to a relaxed connection between the MAF and the testing machine since it allowed buckling of the sample to be avoided.

Observing the final deformed shape (Figure 5(c)), it was concluded that the sample remained

Test E4

flat throughout the whole experiment. This observation was an additional validation of the usefulness of the anti-buckling device used herein.

Figure 10 displays measured out-of-plane displacement fields for test E4. Since the same configuration (i.e., PMMA supports and relaxed connection between the MAF and the testing machine) was employed in tests E3 and E4, it was expected (and confirmed) that no notable out-of-plane displacements occurred. Furthermore, the displacement levels in elasticity (Figure 10(a)) and early plasticity (Figure 10(b)) were very small as was the case in test E3. Moreover, negligible out-of-plane displacement amplitudes were observed in the whole experiment. However, as was also the case for test E3, the sample had a tendency toward buckling, which was prevented thanks to the additional supports. This phenomenon led to non-uniform distributions of out-of-plane displacements (Figures 10(c-f)). The measured axial displacement fields revealed the rotation of the ROI (Figure 11).

Such rotation was also observed for test E3. Furthermore, the levels of axial displacements achieved in this test (Figure 11(a-b)) were comparable to those of test E3. Additionally, "discontinuities" arose at the roots of both V notches (Figure 11(c)). However, compared to test E3, the present discontinuities were of different sizes, which will be discussed in the sequel. A good agreement in terms of global response is observed until instability inception due to buckling. Test E1 was stable up to 17% shear strain, which was proven by the measured out-of-plane displacement fields reported in Figure 19. Until this critical state, the magnitude of out-of-plane displacements was low. Since buckling occurred in test E1, additional supports were applied in test E2. A good reproducibility is observed for the global response between tests E2, E3 and E4 up to 34% shear strain. The instability in test E2 was due to the large level of preload (i.e., the connection between the MAF and the test machine was very stiff). No instability was observed for tests E3 and E4 (i.e., no buckling was noted when the connection between the MAF and the test machine was loosened and thicker supports were utilized). Furthermore, the agreement between tests E3 and E4 further validated the proposed setup for cyclic loadings.

Test E4

Analysis of friction between PMMA plates and the sample

In this section, the influence of friction between the PMMA plates and the sample surfaces is reviewed and discussed. It was assumed that the effect of friction was small since all experiments displayed identical responses before buckling occurred (Figure 5). Further, the speckle pattern did not degrade too much since sterecorrelation could be run until the end of each test. During the installation of the PMMA supports, the six M6 bolts were not tightened in order to minimize friction and contact pressure between the sample surface and the plates.

In the following, a numerical study is presented for more quantitative evaluations of the effect of friction.

ANALYSIS OF FRICTION BETWEEN PMMA PLATES AND THE SAMPLE

Parameter calibration

In order to numerically describe the material behavior during mechanical tests, reliable parameters are needed. FEMU was employed to calibrate the sought parameters for test E3. The material model used in the identification procedure was implemented in the commercial software Abaqus. Nonlinear kinematic hardening follows Armstrong-Frederick's model [START_REF] Armstrong | A mathematical representation of the multiaxial bauschinger effect[END_REF]. Thus, the sought parameters were the yield stress σ y , hardening modulus C, and nonlinear coefficient c.

Within FEMU frameworks, differences between measured and computed quantities are minimized by iteratively updating the material parameters based on sensitivity fields [START_REF] Tarantola | Inverse Problems Theory. Methods for Data Fitting and Model Parameter Estimation[END_REF].

If only the measured displacement fields are considered (i.e., FEMU-U), the cost function is expressed as the squared error between measured {u m } and computed {u c } nodal displacements

χ 2 u ({p}) = 1 γ 2 u N u {u m } -{u c } 2 , (1)
where {p} is the column vector gathering all unknown material parameters, γ u the standard displacement uncertainty, and N u the number of kinematic degrees of freedom [START_REF] Mathieu | Estimation of elastoplastic parameters via weighted FEMU and integrated-DIC[END_REF]. With such setting, χ u tends to 1 when the only source of error is due to measurement uncertainty.

Any deviation from 1 is an indication of model error.

The identification procedure can also be conducted with respect to measured loads {F m } and computed reaction forces {F c } (FEMU-F) extracted from edges where Dirichlet boundary conditions are prescribed

χ 2 F ({p}) = 1 γ 2 F N F {F m } -{F c } 2 , (2)
where γ F is the standard uncertainty of the load cell, and N F the number of load data.

Parameter calibration

The combination of the aforementioned cost functions (i.e., FEMU-UF) is expressed as

χ 2 tot = (1 -ω)χ 2 u + ωχ 2 F , (3)
where the parameter ω determines the contribution of each individual cost function. For the present study, the latter was set to ω = 0.5, which endowed each cost function with equal weight.

The FE mesh used in the identification procedure was constructed from the measurement mesh (Figure 4). It was converted into four-noded quadrilateral (Q4) elements, and then extruded to obtain C3D8 elements with reduced integration (C3D8R). The measured displacements were prescribed along the stressed edges of the mesh as Dirichlet boundary conditions (Figure 16(a)). The elastic parameters (i.e., Young's modulus E and Poisson's ratio ν) were kept constant and equal to their nominal values (Table 5). Only the aforementioned plastic parameters were calibrated. Once calibrated, the parameters yielded significant decreases in both displacement and force residuals (Table 5). Furthermore, the identification resulted in a very good agreement between the global reaction forces extracted from the FE model and the measured forces (Figure 15), which led to residuals that were on average only 11 times the uncertainty level.

However, in elasticity, the computed response was slightly offset from the measured curve.

This trend is believed to be due to the fact that the Dirichlet boundary conditions had to 4 ANALYSIS OF FRICTION BETWEEN PMMA PLATES AND THE SAMPLE be extruded. The identified material parameters reported in Table 5 were used in further numerical simulations. The calibrated value of the yield stress was higher than the expected value estimated from the macroscopic stress-strain curve (Figure 14). This difference was due to the fact that even in the elastic regime, the stress state was not uniform in the ligament. It was expected that the resultant forces extracted from the stressed edges of the model were sensitive to friction. In the available literature, the reported values of friction coefficient between PMMA and steel vary [START_REF] Rabbe | Fretting deterioration of orthopaedic implant materials: Search for solutions[END_REF][START_REF] Nuño | Static coefficient of friction between Ti-6A1-4V and PMMA for cemented hip and knee implants[END_REF][START_REF] Lim | Effect of Friction Coefficient on Finite Element Modeling of the Deep -Cold Rolling Process[END_REF][START_REF] Nuño | Static coefficient of friction between stainless steel and PMMA used in cemented hip and knee implants[END_REF]. However, the estimated levels generally range from 0.3 to 0.4. First, FEMU-UF was run to calibrate plastic parameters for a very low friction coefficient (i.e., µ = 0.01). The calibrated parameters are reported in Table 6;

they were similar to those obtained previously even though the boundary conditions were different (Table 5). The calibrated material parameters were then used to initialize the second FEMU-UF analysis where, additionally, the friction coefficient was sought. The initial value was set to µ = 0.3, which corresponded to the lower bound found in the surveyed To further illustrate the effect of friction on the calibrated parameters, a uniaxial tensile loading was applied to a single C3D8R element. Dirichlet boundary conditions were prescribed on four surface nodes of the cube (denoted by arrows in Figure 17(a)), whereas on the opposite surface, both rigid body displacements and rotations were disabled. The prescribed displacements ranged from 0 to 0.47 times the element height with a step size of 0.001 (i.e., corresponding to the strain levels obtained via the DIC virtual gauge before damage occurred). The differences between stresses for the frictionless (reference) case (i.e., µ = 0.01) and for the calibrated value (µ = 0.33) were not monotonic with respect to the strain level. The mean stress difference δσ was equal to 38 MPa (Figure 17 6) and the corresponding boundary conditions. (b) Stress-strain curves calculated for µ = 0.01 and µ = 0.33 with the respective material parameters (black curves) and corresponding stress differences (red curve)

From the numerical simulation with the calibrated material parameters and friction coefficient, the stress triaxiality fields were evaluated. In Figure 18, the triaxiality field is displayed for the last analyzed image when µ = 0.33. From the triaxiality field, it was concluded that the gauge area was predominantly subjected to shear stresses (i.e., stress triaxialities close to zero). This result was observed for the central points through the thickness (i.e., points 1,2 and 3), which displayed slight deviations in stress triaxialities yet close to zero. This trend was observed for all analyzed images. For the chosen points in the root of the V notch, the stress triaxiality changed significantly with the applied shear strain. For low shear strains, the stress triaxiality was closer to zero (i.e., shear stress state). At the root of the V notch, complex stress states were expected, which were observed from the deviation of the middle point (i.e., point 5), whereas the surface points had almost identical levels. Moreover, the areas around the V notch where the material was subjected to tensile stresses (i.e., stress triaxiality close to 0.33) corresponded to the locations of initiated cracks (Figure 5(c)).

Conclusion

The experimental campaign performed herein was aimed at preventing buckling to occur in thin (i.e., 1 mm) butterfly samples under simple shear loading on a modified Arcan fixture (MAF). Three monotonic shear tests and one additional cyclic test were carried out.

They were monitored with a stereovision system, and the kinematic fields were measured by means of FE-based stereocorrelation. Three different configurations of the experimental setup were studied considering anti-buckling devices and connection stiffness between the

COMPLIANCE WITH ETHICAL STANDARDS

MAF and the testing machine. Furthermore, friction between the anti-buckling device and the sample was evaluated and subsequently calibrated via FEMU-UF. The main results of this work are:

• shear buckling was successfully suppressed for Arcan butterfly samples with a large gauge area using the thickest anti-buckling device,

• FEMU using measured displacements prescribed as Dirichlet boundary conditions on the FE model was carried out to calibrate kinematic hardening parameters,

• the influence of the friction coefficient on the calibrated material parameters was studied using a numerical model of the sample and the anti-buckling device,

• the friction coefficient between the PMMA anti-buckling device and the sample surface could also be calibrated (µ = 0.33).

Last, a detailed analysis of the prescribed boundary conditions on the FE model of the butterfly sample may provide optimal routes to avoid complex numerical simulations (i.e., modeling of the sample and the anti-buckling device). V notches where the strain levels were one order of magnitude higher than on the edges of Figure 22 displays measured shear strain fields. Since buckling of the sample was delayed, the material response (i.e., strain fields) was captured more reliably. Since only elastic strains were present in Figure 22(a-b), they were rather evenly distributed over the ROI, and were at least one order of magnitude higher than the noise floor levels (Table 3). Furthermore, a single strained band developed between the two V notches (Figure 22(c)) and was observed until the end of the test (Figure 22(d-e)). Comparing strains between tests E2 and

Fig. 1 :

 1 Fig. 1: Loading configurations of the Modified Arcan Fixture. (a) Simple shear configuration, (b) tensile loading configuration, and (c) combined shear/tensile loading configuration.

Fig. 2 :

 2 Fig. 2: (a) Butterfly geometry with characteristic regions (see text). The various stress areas are depicted for simple shear loading. Dimensions are in mm. (b) Buckled E1 sample under simple shear loading in the modified Arcan fixture

Fig. 3 :

 3 Fig. 3: Experimental and optical setups used for stereocorrelation measurements. See text for the description of the labels

Fig. 4 :

 4 Fig. 4: Chosen reference points on the FE mesh and one of the reference images for the pre-calibration step. (a) User defined points (i.e., fiducials) on the FE mesh, and (b) on one of the reference pictures used in the pre-calibration step. The (green) mesh of the ROI is also shown in the 3D model and its reprojection in one of the reference pictures. The blue rectangle depicts the area over which the average strains were assessed

Figure 5

 5 Figure 5 displays the load histories of the four tests reported herein. For each test, six characteristic points were chosen for which the stereocorrelation results are discussed in the following subsections.

Fig. 5 :

 5 Fig. 5: Load histories for tests E1, E2, E3 and E4. Six characteristic points are outlined by red circles for each experiment for which the stereocorrelation results are discussed hereafter

3. 5 Figure 5

 55 Figure 5(c) shows the loading history of test E3. Compared to the loading histories of tests E1 (Figure 5(a)) and E2 (Figure 5(b)), the highest load level was reached in test E3. The 10-mm thick support did not fail. The connection stiffness between the MAF and the testing machine was loosened again. As a result, buckling was prevented, thus the material response could be observed more clearly.

Figure 6

 6 Figure 6 displays the measured out-of-plane displacement fields for test E3. Since the 10-mm thick plates were employed, negligible amplitudes were measured throughout the experiment. This result proves that the plate thickness was sufficient to significantly lower the amplitudes of out-of-plane motions. In elasticity (Figure 6(a)) and early plasticity (Figure 6(b)), the displacements were of the same order of magnitude. Increases in magnitude observed in Figure 6(c) were due to the tendency toward buckling (as seen in test E1). However, the additional supports prevented further increases. The displacement fluctuations were attributed to buckling. Even for the highest achieved load (Figure 6(d)), the out-of-plane dis-

Fig. 6 :Fig. 7 :Figure 8

 678 Fig. 6: Measured out-of-plane displacement fields for test E3. The displacements are expressed in mm

  (a)). Furthermore, a single strained band developed between the V notches (Figure8(c)). Comparing the strain levels achieved for approximately equal loads, tests E3 (Figure8(c)) and E2 (Figure22(d)) led similar values, which further confirmed that no buckling occurred in test E2 at that point. Overall, larger strain levels were reached in test E3 in comparison to tests E1 and E2.

3. 5 Fig. 8 :

 58 Fig. 8: Measured xz strain fields for test E3. The strained band is outlined by dashed blue ellipses, whereas the location of the cracks are indicated by arrows

Fig. 9 :Figure 5 (

 95 Fig. 9: Correlation residual maps for test E3. The residuals are expressed in % of the dynamic range of the reference pictures. The arrows indicate the location of propagating cracks, whereas the white circle outlines the change in brightness as a result of limited buckling

Fig. 10 :

 10 Fig. 10: Measured out-of-plane displacement fields for test E4. The displacements are expressed in mm

Fig. 11 :

 11 Fig. 11: Measured axial displacements for test E4. The arrows indicate the locations of the cracks. The displacements are expressed in mm

Fig. 12 :Fig. 13 :Fig. 14 :

 121314 Fig. 12: Measured xz strain fields for test E4. The strained band is outlined by blue ellipses, whereas the location of strain discontinuities (i.e., cracks) are indicated by arrows

Fig. 15 :

 15 Fig. 15: Comparison between measured and resultant forces extracted from the FE model of test E3

Fig. 16 :

 16 Fig. 16: FE model of test E3. (a) E3 sample with prescribed measured displacements as boundary conditions. (b) Assembly of sample E3 and PMMA plates

4. 2

 2 Effect of friction literature. The parameters calibrated along with µ all decreased (the hardening modulus C experienced the largest relative reduction). The global residual χ tot (Equation (3)) only slightly decreased.

Fig. 17 :

 17 Fig. 17: (a) Cube element used for depicting the tensile response with the calibrated parameters (Table 6) and the corresponding boundary conditions. (b) Stress-strain curves calculated

Fig. 18 :

 18 Fig. 18: Stress triaxiality history for six different points with respect to the calculated shear strain in the virtual gauge and the triaxiality field for the last computed image (i.e., frame 330) when µ = 0.33

Fig. 19 :

 19 Fig. 19: Measured out-of-plane displacement fields for test E1. The displacements are expressed in mm. F denotes the applied load, and xz the average shear strain calculated with the optical gauge

Fig. 21 :

 21 Fig. 21: Measured out-of-plane displacement fields for test E2. The displacements are expressed in mm

Table 1 :

 1 Experimental configurations (M: monotonic, C: cyclic)

	Test	E1	E2	E3	E4
	Configuration	#1	#2	#3	#3
	Loading regime	M	M	M	C
	Thickness of PMMA plates, mm	-	5	10	10
	Preload, N	-39	-1000	-40	-35

Table 2 :

 2 Hardware parameters of the stereovision system

	Cameras	Dalsa Falcon 4M60
	Definition	2358 × 1728 pixels (B/W images)
	Color filter	none
	Gray Levels rendering	8 bits
	Lens	Titanar 50 mm
	Aperture	f/2.8
	Field of view	3,996 mm 2
	Image scale	≈ 32 pixel/mm
	Stereo-angle	25°S
	tand-off distance	31.6 cm
	Image acquisition rate	1 fps (0.3 fps for test E3)
	Patterning technique	B/W paints
	Pattern feature size	15 px

Table 3 :

 3 Stereocorrelation analysis parameters

	DIC software	EikoTwin DIC
	Image filtering	none
	Element length	1 mm
	Shape functions	linear (T3 elements)
	Mesh	see Figure 4
	Matching criterion	penalized sum of squared differences
	Regularization length	5 mm
	Displacement noise floor for test E1 (x z y)	0.4 µm	0.2 µm	0.7 µm
	Displacement noise floor for test E2 (x z y)	0.1 µm	0.2 µm	0.5 µm
	Displacement noise floor for test E3 (x z y)	0.2 µm	0.1 µm	0.6 µm
	Displacement noise floor for test E4 (x z y)	0.2 µm	0.2 µm	0.6 µm
	Strain xz noise floor for test E1	6 × 10 -5
	Strain xz noise floor for test E2	6 × 10 -5
	Strain xz noise floor for test E3	7 × 10 -5
	Strain xz noise floor for test E4	8 × 10 -5

Table 4

 4 gathers characteristic values for each experiment. Similar yield stresses were observed regardless of the employed configuration. It was concluded that instability initiated in the plastic regime. Furthermore, the highest ultimate load was achieved in tests E3 and E4 for a corresponding shear strain of 48%, and an ultimate shear strength of 417 MPa. A good reproducibility between the reported quantities for tests E3 and E4 was noted, which fully validated the third configuration. With the application of the PMMA plates onto sample E2, buckling was delayed, thereby leading to a 43 MPa higher ultimate shear strength compared to test E1. With thicker supports and loosened connection, the ultimate strength further increased by 25 MPa compared to test E2. The resulting shear strains at the ultimate load are 10-30% higher for tests E3 and E4 compared to tests E2 and E1, respectively.

Table 4 :

 4 Characteristic values for the four shear tests on the Modified Arcan Fixture

	Test	Yield stress, MPa	Ultimate shear strength, MPa	Ultimate load, kN	Shear strain at ultimate load, %
	E1	259	349	7.3	17.5
	E2	243	392	8.2	35.6
	E3	258	417	8.71	48.7
	E4	261	417	8.66	47.4

Table 5 :

 5 Identification results via FEMU-UF

		E, GPa	ν, -	σ Y , MPa C, GPa c, -χ U	χ F
	Initial	210	0.25	250	6000	15	35	116
	Identified	210	0.25	365	4635	13	23	11

Table 6 :

 6 Identification results for low friction value and calibrated material parameters and

	friction coefficient µ
		E, GPa	ν, -	σ Y , MPa C, MPa	c, -	χtot	δσ, MPa
	µ = 0.01	210	0.25	365	4616	13.1	43	-
	µ = 0.33	210	0.25	357	3860	11.8	42.6	38

Acknowledgements

This work was performed within the FULLINSPECT project supported by the Croatian Science Foundation (UIP-2019-04-5460 Grant). The authors also acknowledge and thank EikoSim for providing the EikoTwin DIC software to conduct the stereocorrelation analyses reported herein.

Compliance with Ethical Standards

The authors have no conflict of interest to declare.

Appendix A

This appendix gathers stereocorrelation results of tests E1 and E2 that enabled for the improvements utilized in tests E3 and E4.

Test E1

Figure 19 shows out-of-plane displacement fields for different load levels (Figure 5(a)) and strain values. Positive displacements are oriented away from the reader. Low out-of-plane displacements were measured in the elastic regime of the sample (Figure 19(a)). Furthermore, the inception of plasticity (Figure 19(b)) was also characterized by small out-of-plane displacements. However, for the 7.2-kN load level (i.e., the first peak on the load-time plot of Figure 5), the out-of-plane displacement amplitudes increased. This growing trend continued while further loading the sample. Furthermore, the out-of-plane displacement distribution was symmetric, which was also observed on the deformed sample in Figure 5(a). The aforementioned symmetry was observed in the displacement levels (i.e., displacement amplitudes were similar in Figure 19(d-f)). Even though a stereovision system was employed, these high levels of measured displacement were not considered reliable since the out-ofplane rotation of the ROI caused some parts to be out of focus.

Test E1

the ROI (Figure 20(c)). The rotation of the ROI due to buckling is observed in Figure 20(d).

1 Further measurements (Figure 20(e-f)) were deemed unreliable. . However, clear signs of buckling initiation were observed (Figure 21(e-f)). The displacements in the center of the sample were negative, whereas on the opposite corners they were positive. This result indicated that wrinkling had started, and was also observed from the shape of the deformed sample in (Figure 5(b)). The PMMA support fractured due to increased buckling. Thus, stereocorrelation results were no longer deemed reliable beyond this point.

E1, higher levels were achieved due to the more stable material response thanks to the addi-1 tional supports.


Ana Vrgoč

Zvonimir Tomičević

Andrija Zaplatić

François Hild

DAMAGE ANALYSIS IN GLASS FIBER REINFORCED EPOXY RESIN VIA DIGITAL IMAGE CORRELATION Summary

Keywords: Damage, Digital Image Correlation, registration residuals, glass fiber composite A. Vrgoč, Z. Tomičević Damage analysis in glass fiber reinforced A. Zaplatić, F. Hild epoxy resin via Digital Image Correlation Damage analysis in fiber reinforced

The objective of the presented experimental work was to perform monotonic tensile tests and observe damage occurring on the surface of a glass fiber mat reinforced epoxy resin. The experiments were performed on dogbone specimens machined from a 5.8 mm thick composite plate. The macroscopic response was revealed from load data and by using 2D digital image correlation. FE-based DIC was employed to measure displacement fields and calculate strain fields across the entire Region of interest. In addition, the gray level residuals extracted from the correlation analyses were utilized to detect discontinuities (i.e. damage initiation and growth) on the surface of the investigated specimens.

Introduction

Due to their high stiffness-to-weight ratio and their unique ability to tailor material properties to meet specific design requirements, the use of fiber reinforced polymer (FRP) composites is in constant increase in critical-safety and lightweight structures. In addition to enhanced environmental and fatigue resistance, FRPs exhibit good formability during manufacturing and are easy to maintain [START_REF] Brighenti | Mechanics of interface debonding in fiber-reinforced materials[END_REF]- [START_REF] Schwab | A multiscale approach for modelling impact on woven composites under consideration of the fabric topology[END_REF]. Various industry branches (especially automotive, aeronautics and aerospace) aim to exploit more and more the potential of FRPs. However, the full understanding of their behavior for the whole service cycle remains a challenge and calls for advanced material characterization methods. In particular, the identification of material properties, and the observation of damage mechanisms via nondestructive methods have become crucial to accurately predict structural responses up to and beyond the inception of damage and fracture.

In recent years, Digital Image Correlation (DIC) has proven to be an accurate and versatile tool to bridge the gap between experimental testing and mechanical models [START_REF] Sutton | Image Correlation for Shape, Motion and Deformation Measurements: Basic Concepts, Theory and Applications[END_REF], [START_REF] Grédiac | Full-Field Measurements and Identification in Solid Mechanics[END_REF]. This contactless, full-field measurement technique has been widely employed in experimental solid mechanics to extract meaningful information from tests [START_REF] Sutton | Computer Vision-Based, Noncontacting Deformation Measurements in Mechanics: A Generational Transformation[END_REF]. Due to the increased use of composite materials, DIC has been utilized to identify material parameters and investigate damage mechanisms occurring within such complex and heterogeneous materials [START_REF] Gras | Identification of the Out-of-Plane Shear Modulus of a 3D Woven Composite[END_REF]- [START_REF] Montesano | Modeling fatigue damage evolution in polymer matrix composite structures and validation using in-situ digital image correlation[END_REF].

The research conducted herein focuses on the observation of damage initiation and growth in a glass fiber reinforced epoxy resin composite during monotonic tensile tests. The specimens were cut from a composite plate at different angles to estimate their global response. In the present case, regularized FE-based DIC was utilized to measure displacement fields and calculate strain fields on the surface of tested specimens. Furthermore, gray level residual maps extracted from correlation analyses were studied to detect damage initiation and monitor subsequent crack growth.

Experimental investigation

In this section, the experimental protocol employed in the present study is discussed. First, the properties of the investigated composite material are presented. The dogbone specimens were cut from a composite plate along three different directions to estimate the macroscopic response and its dependence with the angular sampling. The experimental data were the applied force measured by the load cell of the testing machine, and image series of specimen surfaces acquired during each of the three experiments. A brief overview of the global DIC algorithm is given followed by a summary of hardware parameters employed in the correlation procedure.

FRP composite

The experimental investigation was performed on a continuous glass fiber mat reinforced epoxy resin composite. The manufacturing process of the material was a manual lay-up followed by compression molding. The laminate consisting of 12 layers of R-glass fiber mat placed in 438 g of the epoxy resin mixture (LG 285 laminating resin, GRM Systems, Germany) was cured under pressure for 24 h at room temperature. Subsequently, post-curing was performed at 100 ºC for 3 h. Micrographs of the investigated material are presented in Fig. 1. Since vacuum was not applied during molding after lay-up, the micrographs reveal the presence of air bubbles (i.e. voids).

Digital Image Correlation

Digital Image Correlation is one of the most commonly used technique to perform fullfield measurements based on registering gray level images of specimen surfaces [START_REF] Sutton | Image Correlation for Shape, Motion and Deformation Measurements: Basic Concepts, Theory and Applications[END_REF]. The images acquired during the experiment are correlated to measure displacement fields over the entire Region of Interest (ROI) assuming the conservation of gray levels at any pixel location x between the reference image f(x) usually taken in the undeformed state and a deformed one g(x):

( ) ( ( )), fg  x x u x (1)
where u(x) is the sought displacement field. The DIC technique consists in minimizing the sum of square differences 2 c  over the ROI with respect to the sought displacement field [START_REF] Hild | Comparison of Local and Global Approaches to Digital Image Correlation[END_REF]:

  22 ROI d, cc    xx (2)
where () c  x defines the correlation residuals field:

( ) ( ) ( ( )). C fg     x x x u x (3)
Compared to local approaches, which analyze series of subsets independently, the global DIC approach applied herein assumes the continuity of displacement fields over the whole ROI.

Since the presented minimization is a nonlinear and ill-posed problem, the displacement field is written as:

( ) ( ), nn n u   ux ψx (4)
where un are the nodal displacements associated with the selected shape functions ψn. If small increments are assumed, a Taylor expansion is applied:

( ( ) δ ( )) ( ( )) δ ( )) ( ). g g f       x u x u x x u x u x x (5)
A Gauss-Newton iterative scheme is utilized to solve linear systems:

     δ, i  M u b (6)
where [M] is the DIC matrix:

   ROI ( ) ( ) ( ) ( ) , mn m n M f f       x ψ x x ψ x (7)
{b i } the residual vector at iteration i:

   ROI ( ) ( ) ( ) ( ) ii mm b f g f      x x x ψx (8)
and g ̃i(x) is equal to g(x+u i (x)) and referred to as the corrected deformed image by the current estimate u i of the displacement field. In the present setting, the Hessian matrix [M] is calculated once for all, whereas the residual vector {b i } has to be updated for each new iteration step. In addition, mechanical regularization based on the equilibrium gap method [START_REF] Claire | A finite element formulation to identify damage fields: the equilibrium gap method[END_REF] is employed herein to further circumvent the ill-posedness of the registration [START_REF] Tomičević | Mechanics-aided digital image correlation[END_REF]. Such approach results in a reduction of uncertainty levels [START_REF] Tomičević | Mechanics-aided digital image correlation[END_REF], [START_REF] Naylor | Mechanically regularized FE DIC for heterogeneous materials[END_REF]. If linear elasticity applies, and if the bulk nodes are free from external forces, regularized FE-DIC consists in adding a penalty term:

       T T 2 , m u K K u (9)
where [K] is the stiffness matrix, while 2 m  is the sum of the squared norm of all equilibrium gaps at inner nodes only. Further, an additional regularization is applied for the loaded (Dirichlet) boundary nodes:

       T T 2 , b u L L u (10)
where [L] is an operator acting on the ROI boundary [START_REF] Mendoza | Complete mechanical regularization applied to digital image and volume correlation[END_REF]. Thus, regularized DIC consists in minimizing the total cost function

  2 2 2 2
1,

m b t c m m b b w w w w          (11)
                   2 2 2 2 2 2 T T T TT , , , c m b c m b          v M v v K K v v L L v (12)
where {v} gathers all nodal degrees of freedom associated with the displacement field v. The

wavelength dependence of        T T v K K v and        T T v L L v is of fourth order, while      T v M
v is independent of the wavelength. Consequently, the weights wm and wb are chosen as:

    44 2π , 2π , m m b b w l w l  kk ( 13
)
where lm and lb denote regularization lengths for 2 m  and 2 b  , respectively.

As previously mentioned, the DIC technique measures displacement fields, and thus enables the calculation of strain fields on the surface of tested specimens. Additionally, the correlation residual maps are available for each analyzed image. These maps correspond to the gray level difference at the convergence of the correlation procedure (Equation ( 3)), thereby indicating the quality of the registration. High values indicate local violations to the gray level conservation or incompleteness of kinematic bases, which make correlation residuals the ideal field to reveal the presence of cracks when displacement continuities are assumed to occur [START_REF] Hild | Damage measurements via DIC[END_REF].

Experimental protocol

In order to analyze the global behavior of the investigated composite, 5.8 mm thick dogbone specimens were subjected to monotonic tensile tests. The specimens were cut from a composite plate at 0º, 45º and 90º, as shown in Fig. 2(a). To ensure that the specimen breaks in the ligament area, the central part was thinned with a radius of 49 mm, while the width at the narrowest section was equal to 5.8 mm.

Three uniaxial tensile tests were conducted in a displacement-controlled mode with a prescribed stroke rate of 2 mm/min. The tests were monitored with a single CCD camera, thus resulting in 2D-DIC measurements. The image acquisition rate was equal to 3 fps. The three experiments were carried out consecutively without any changes to the camera and lighting settings. Furthermore, in order to measure displacement fields, a fine-grained random black and Damage analysis in fiber reinforced A.Vrgoč, Z.Tomičević epoxy resin via Digital Image Correlation A.Zaplatić, F.Hild white speckle pattern was applied onto the surface of the samples. The ROI considered herein is shown in Fig. 2(b), while the hardware parameters of the optical setup for DIC measurements are summarized in Table 1. The images acquired during each test were analyzed via regularized FE-based DIC within the Correli 3.0 framework [16]. Since the correlation code estimates the nodal displacements associated with the finite element discretization, a 2D mesh corresponding to the sample geometry and consisting of three-noded triangular (T3) elements was considered herein (Fig. 2(b)). T3 elements were selected to properly describe the boundaries of each sample. The mesh topology was identical for each specimen. However, the meshes slightly differed in shape owing to variations in specimen alignment. The average element length, calculated as the square root of the mean element area, was equal to 20 px for each sample. The other parameters employed in the correlation analyses are summarized in Table 2. Before performing DIC measurements, the displacement and strain uncertainties were quantified by acquiring 50 images in the reference configuration. Their levels obtained for each test are reported in Table 3. For all three cases, the uncertainties are identical for the maximum eigen strain, and the displacement uncertainties are in the cpx range. Even though finer FE meshes better capture kinematic fields (especially in the presence of localized phenomena), they are more corrupted by acquisition noise (i.e. the lower the element size, the higher measurement uncertainties). This observation leads to a compromise between measurement uncertainty and spatial resolution [START_REF] Hild | Comparison of Local and Global Approaches to Digital Image Correlation[END_REF]. Hence, small scale displacement resolutions are hardly accessible without introducing more assumptions (e.g. mechanical regularization) in the correlation procedure. When the regularization length is greater than the element size, the Damage analysis in glass fiber reinforced A. Zaplatić, F. Hild epoxy resin via Digital Image Correlation former then controls the spatial resolution. Otherwise, the element size is proportional to the spatial resolution.

Results

The main objective of the research reported herein was to monitor damage initiation and growth on the surface of glass fiber/epoxy resin composite via DIC. FE-based DIC was carried out to measure displacement fields and calculate strain fields. In addition, the gray level residual maps were investigated to detect discontinuities and monitor damage growth on the monitored surfaces.

Global material behavior

The global material response in the uniaxial tensile tests was obtained via DIC gauges positioned over the ROI as shown in Fig. 3(a). The uniaxial stress was determined from the load data and the initial area of the ligament. From the reported stress-strain curves, Young's modulus values of 6.8 GPa, 5.4 GPa and 5.5 GPa were identified for the 0º, 45º, 90º samples, respectively. Fig. 3(a) shows a good agreement between stress-strain curves in the linear elastic regime for the 45º and 90º specimens contrary to the 0º sample. This difference arises from the sample machining conditions as well as the presence of voids (Fig. 1) within the samples. A detailed microstructural analysis was performed by acquiring X-Ray Computed Tomography (CT) scans to clarify the differences in the obtained results (Fig. 3(b)). The reconstructed mesostructure reveals a random glass fiber architecture and provides some information about the distribution of voids within the sample.

Correlation results for the 0º sample

The total amount of 313 images was analyzed for the 0º specimen. The first 50 images were acquired in the unloaded configuration for uncertainty quantification purposes (Table 3). Table 4 reports both maximum eigen strain and stress data calculated for different images acquired during the prescribed load history. In the present case, the residual maps and maximum eigen strain fields for images listed in Table 4 are reported to detect damage initiation and growth. Fig. 4 shows the corresponding stress-strain curve, in which the studied pictures are shown, and the change of RMS residual with respect to the image number. Up to Image 50, the global residual is constant since no loading was prescribed. As the loading increases The gray level residual maps and the corresponding maximum eigen strain fields for different load levels in the linear elastic regime of the material response are shown in Fig. 5. The dynamic range of the gray level residuals is selected so that an increase of fluctuations is observed. Fig. 5(a)-(b) show the initial residual map and maximum eigen strain field calculated in the unloaded regime. Very low residuals are homogeneously distributed over the ROI, thus indicating that the registration was successful. These low initial residuals were induced by camera sensor noise during image acquisition. Further, the strain fluctuations of small magnitude are due to acquisition noise. As the load levels become higher, the correlation residuals gradually increase over the entire ROI (Fig. 5(c)). The first detected discontinuities correspond to Image 144, i.e. four distinct regions of elevated residuals are observed at the upper and lower edge of the sample ligament (Fig. 5(c)). Such localized phenomena are also visible in the maximum eigen strain field (Fig. 5(d)). Thus, these strain concentrations on the edges of the sample ligament show the impact of sample machining induced damage.

With the gradual increase of load levels, localized phenomena become dominant and the presence of the macrocrack within the narrowest gauge section is visible on both fields (Fig. 6(a)-(b)). Consequently, damage growth induces nonlinear material response (Fig. 4(a)). The additional regions of elevated gray level residuals and maximum eigen strains are observed on the edges of the specimen ligament (marked with red circles in Fig. 6(c)-(d)).

Further, as shown in Fig. 6(e)-(f) corresponding to the last image acquired prior to specimen failure, strain localization phenomena become more apparent, while the correlation residual map reports discontinuities within the same regions. It is worth emphasizing that the damage process concentrates around the dominant macrocrack, even though multiple areas with Damage analysis in glass fiber reinforced A. Zaplatić, F. Hild epoxy resin via Digital Image Correlation elevated correlation residuals and maximum eigen strains are detected as well. Failure occurs within the narrowest gauge section of the dog bone specimen (which was expected from design).

Fig. 5 Gray level residual maps (left) and maximum eigen strain fields (right) for three different loading steps in the linear elastic regime of the 0º tensile test. The black circles depict the first detected discontinuities (i.e. zones in which the eigen strains increase with the prescribed load).

Fig. 6 Gray level residual maps (left) and maximum eigen strain fields (right) for different loading steps in the nonlinear regime of the 0° tensile test. The black circles depict the first detected discontinuities, while the red circles correspond to additional areas of higher correlation residuals.

Fig. 7(a) shows the displacement field measured in the longitudinal direction prior to failure. The discontinuity in the displacement field is large and corresponds to the previously observed dominant macrocrack revealed through both gray level residuals and maximum eigen strain field (Fig. 7(b)). The observed macrocrack path (marked with the black dashed line) is not straight, thereby indicating the influence of the random and heterogeneous mesotructure.

Correlation results for the 45º and 90º sample

The total amount of images analyzed to detect damage on the 45º specimen surface was equal to 314, while 306 images were analyzed for the 90º specimen. In both cases, the first 50 images were acquired in the unloaded stage to assess the measurement uncertainties (Table 3). Fig. 8(a)-(b) show the initial residual map and maximum eigen strain field of the 45º specimen corresponding to Image 50. The correlation residuals are homogeneously distributed over the ROI, while the values remain low, again indicating the influence of the optical setup. As the load levels become higher, the correlation residuals gradually increase over the entire ROI (Fig. 8(c)). The first detected discontinuities correspond to Image 154 captured in the linear elastic regime (Fig. 8(c)), i.e. three distinct regions of elevated residuals are observed at the upper and lower edge of the sample ligament. Such localized phenomena are also visible in the maximum eigen strain field (Fig. 8(d)). These strain concentrations on the edges of the sample ligament show the impact of sample machining induced damage. Further, as shown in Fig. 8(e)-(f) acquired prior to specimen failure, strain localization phenomena become even clearer, and correspond to the regions of elevated correlation residuals. The discontinuities in the measured displacement fields are large and correspond to the previously observed discontinuities revealed through the gray level residuals and maximum eigen strain fields. In both cases, the dominant macrocrack path is not straight, which indicates the influence of underlying heterogeneities.

Conclusion

Monotonic tensile tests were carried out on a glass fiber mat/epoxy resin composite. 2D-DIC was employed to measure displacements and calculate strain fields on the surface of the investigated material. First, the global material response in the uniaxial tensile tests was obtained via DIC gauges positioned over the ROI, and Young's modulus values for all three samples were identified. Good agreement between stress-strain curves in the linear elastic regime for 45º and 90º samples was observed (Young's modulus equal to 5.5 GPa and 5.4 GPa, respectively). However, the Young's modulus for the 0º specimen was 6.8 GPa. It is worth noting that the observed fluctuations in mechanical response originated from the random fiber architecture, as well as the sample machining and the presence of mesodefects (i.e. air bubbles) arising from the manufacturing process. A microstructural analysis was conducted by acquiring a 3D scan of the chosen ROI. The 3D reconstructed volume revealed the underlying fiber mat architecture, as well as the presence of voids.

The analysis of DIC residuals and maximum eigen strain fields revealed damage initiation and growth on the surface of the investigated specimens. For all three samples, the initial correlation residuals corresponding to the unloaded state were homogeneously distributed over the ROI and their levels remained low, thereby indicating the influence of the optical experimental setup (i.e. lighting, acquisition parameters, camera sensor noise). As the load levels became higher, these residuals gradually increased over the entire ROI. The first detected discontinuities for all three specimens were observed at the upper and lower edge of the sample ligament in the linear elastic regime, thus indicating the impact of sample machining induced damage (since the observed localization phenomena did not impact the global material response, i.e. the nonlinearity was still very low). Localization observed on the correlation residuals were also visible on maximum eigen strain fields. With the increase of load levels, localized phenomena became dominant and the presence of the macrocrack, as well as the macrocrack path was revealed. The observed macrocrack path for all three specimens was not straight due to the underlying heterogeneities. Because of the random mesostructure and the presence of manufacturing and machining induced defects, the specimens broke at different stress and elongation levels.

The results obtained in this study indicate that the methodology employed herein (regularized FE-DIC) can efficiently detect damage and macrocracks on the surface of composite materials. However, DIC measurements do not provide insight into the full FRP behavior and underlying damage mechanisms in the material bulk. This type of analysis can be achieved via in-situ X-ray computed tomography (CT) [START_REF] Maire | Quantitative X-ray tomography[END_REF]. The reconstructed volumes can be registered via Digital Volume Correlation (DVC) to measure 3D displacement fields [START_REF] Bay | Methods and applications of digital volume correlation[END_REF], [START_REF] Buljac | Digital Volume Correlation: Review of Progress and Challenges[END_REF]. Such approach would enable for the identification of damage mechanisms at the microscale and in the bulk, as well as constitutive parameters to be calibrated.

Fig. 1

 1 Fig. 1 Micrographs of composite cross section at two different magnifications.

t 2 c  , that of equilibrium gap   2 m

 22 comprising of the correlation cost function   and boundary fluctuations   2 b  :

Fig. 2

 2 Fig. 2 Orientation of the specimens cut from the composite plate (a) and the reference image of the uniaxial tensile experiment used for DIC analysis with 20-pixel triangular elements (b). The red contour depicts the ROI.

Fig. 3

 3 Fig. 3 Stress-strain curves of all three tensile tests (a) and mesostructure of the chosen ROI (b). The reconstructed volume reveals the random fiber orientation and the presence of voids (i.e. air bubbles).

Fig. 4

 4 Fig. 4 Stress-strain curve obtained from the monotonic uniaxial tensile test (a), and change of normalized RMS residuals as a function of image number (b) extracted from the correlation analysis of 0º sample. The solid black circles depict the images for which the correlation residual maps are shown to detect discontinuities.

Fig. 7

 7 Fig. 7 Displacement field of the 0º sample in the loading direction (i.e. x axis) expressed in pixels (a) and maximum eigen strain field (b) for Image 299. The black dashed line depicts the macrocrack path and the black arrow shows the initiation location.

Fig. 8

 8 Fig. 8 Gray level residual maps (left) and the maximum eigen strain fields (right) for different loading steps of the 45° tensile test. The black circles depict the first detected discontinuities. Fig. 9(a)-(b) show the initial correlation residual map of the 90º sample for Image 50, as well as the maximum eigen strain field. The correlation residuals are homogeneously distributed over the ROI, while the small fluctuations in the maximum eigen strain are due to measurement uncertainties. The increase of load levels results in a gradual increase of the correlation residuals over the entire ROI (Fig. 9(c)). The first detected discontinuities

Fig. 9

 9 Fig. 9 Gray level residual maps (left) and the maximum eigen strain fields (right) for different loading steps of the 90° tensile test. The black circles depict the first detected discontinuities.

Fig. 10

 10 Fig.10shows the displacement fields measured in the longitudinal direction prior to failure of the 45º and 90º sample, respectively.

Fig. 10

 10 Fig. 10 Displacement field in the loading direction (left) and maximum eigen strain field (right) of the 45º and 90º samples corresponding to the last images acquired prior to specimen failure. The displacements are expressed in pixels, blue dashed lines depict the macrocrack path and the blue arrows show the initiation locations.

  Damage analysis in glass fiber reinforced A. Zaplatić, F. Hild epoxy resin via Digital Image Correlation where wm and wb are weights defining length scales associated with 2

	denotes cost functions normalized with respect to those associated with a plane wave
	displacement field	0  exp( i v x v ( ) k x where 0 ),

m  and 2 , b  while  v is the amplitude and k the wave vector. The normalized residuals read:

Table 1

 1 DIC hardware parameters.

	Camera	CCD Dalsa
	Definition	2358×1728 pixels
	Gray levels amplitude	8 bits
	Lens	Titanar 50 mm
	Aperture	f / 2.8
	Field of view	38 × 28 mm 2
	Image scale	16 µm / pixel
	Stand-off distance	120 mm
	Image acquisition rate	3 fps
	Exposure time	26 ms
	Patterning technique	Sprayed black paint

Table 2

 2 DIC analysis parameters.

	DIC software	Correli 3.0 [16]
	Image filtering	None
	Element length	20 pixels
	Shape functions	Linear (T3)
	Mesh	See Fig. 2(b)
	Matching criterion	Regularized sum of squared differences
	Regularization length	10 pixels

Table 3

 3 Standard displacement and strain uncertainties.

	Displacement noise floor 0º (x y)	0.012 px 0.012 px
	Displacement noise floor 45º (x y)	0.016 px 0.013 px
	Displacement noise floor 90º (x y)	0.017 px 0.011 px
	Strain noise ε1 floor 0º, 45º and 90º	8×10 -5

Table 4

 4 Stress and maximum eigen strain values of the 0º sample calculated for different load levels.

	Image number Stress (MPa)	ε1
	50	-	-
	90	30	0.004
	144	76	0.012
	180	118	0.016
	245	160	0.024
	299	184	0.037

ACKNOWLEDGEMENTS

This work was performed within the FULLINSPECT project supported by the Croatian Science Foundation (UIP-2019-04-5460 Grant).