A novel methodology for estimating tensile properties in a small punch test employing in-situ DIC based deflection mapping

Abstract Determining tensile properties from small punch test is being pursued actively in the nuclear industry due to the limited volume of material such tests use compared with standard tests which can be critical when considering active or development samples. One of the crucial challenges in harnessing the full potential of this technique is formulating methodologies which correlate the small punch specimen’s deflection to equivalent uniaxial tensile properties. Existing approaches for correlation rely on deflection obtained from a single point on the small punch test specimen, used with empirical equations to make the correlation. However, the deflection and strain accumulation in a small punch specimen is highly heterogeneous and data from a single point does not represent the gross deformation evolving in the specimen. This data when used in conjunction with the empirical formulations for deriving equivalent uniaxial tensile properties, would not result in accurate identification of material properties. In this work we offer an alternative approach which uses the full field deflection of the specimen mapped through in-situ digital image correlation. The use of digital image correlation combined with inverse finite element analysis augments the existing method of material properties identification from single point deflection data thereby significantly improving the reliability of the measurements.


Introduction
Over the last few decades there has been a considerable impetus for developing small scale specimen testing techniques for evaluating mechanical properties of materials. It is the multiple advantages these techniques offer that has propelled its sustained advancement and is being actively pursued as a reliable method in situations where estimation of mechanical properties using conventional testing techniques is not viable. The paramount application of the small specimen techniques is in the nuclear industry where determining mechanical properties of irradiated material using small specimen volume is crucial [1e5]. Small specimen testing has also found application in remnant life assessment of in-service components [6e10].
Among several small specimen testing techniques, the small punch test is one of the most versatile as it has been used to deduce tensile [2,11e15], creep [8,9,16e18], fracture [2,19e21], ductile to brittle transition temperature [11,22e24] and fatigue properties [25]. In a small punch test, a thin specimen is clamped between two nondeformable dies and a rigid spherical punch is forced into the specimen. Elevated temperature tests carried out at constant load in force-controlled mode are used to obtain the time-displacement characteristics. Creep properties can hence be estimated from the time-displacement characteristics. Load-displacement curves can be obtained if the tests are carried out in a displacement-controlled mode when the penetration rate of the punch is held constant. The load-displacement characteristics can be used to obtain equivalent tensile and fracture properties. The wide acceptance of the small punch test has led to the development of a pre-normative code of practice by CEN [26]. This code lists the guidelines for designing a small punch test rig and gives details about the testing procedure. The code also describes various approaches for deriving creep, tensile and fracture properties.
The CEN code of practice outlines two approaches for obtaining equivalent tensile properties from displacement-controlled small punch tests. The first approach which has been widely pursued by several researches uses empirical correlations to convert certain parameters characterising the load-displacement curve to equivalent 0.2% proof strength and ultimate tensile strength [3,4,10,11,27e31]. Substantial research has also been pursued employing the second approach which envisages the use of Finite Element Modelling (FEM) as a tool to back-calculate the elastic and plastic properties to match the profile of the experimentally obtained load-displacement curve [32e39].
The inherent limitation in both these approaches stems from the fact that the load-displacement/deflection characteristics of the material which is pivotal in both these approaches is acquired from a single point on the specimen. In case of the load-deflection curve it is obtained from the central point of the deforming specimen using a linear variable differential transformer (LVDT). The loaddisplacement curve on the other hand is obtained by recording the displacement of the punch. Both these methods do not consider the specimen's gross deflection contour which resembles a hemispherical shell owing to the geometry of the punch. Mapping the entire deflection contour of the specimen would offer new perspectives to existing approaches for estimating equivalent uniaxial properties. Though DIC based studies have been attempted in studying deformation behaviour during small punch test [40], a comprehensive account on the limitation and applicability of this method is not well documented.
In this work a new methodology has been developed for estimating the elastic and plastic properties of 316L stainless steel utilising Digital Image Correlation (DIC) based in-situ deflection mapping and inverse finite element modelling (iFEM). Using DIC a wide deflection field could be mapped on the specimen. The mechanical properties were then back-calculated using FEM by matching the load-deflection characteristics obtained from multiple locations in the specimen.

Material and characterisation
The chemical composition of 316L stainless steel used in this study is given in Table 1. Rods of 8 mm diameter were machined from 12 mm thick plates which were solution annealed at 1323 ± 5 K for 30 min and water quenched. The initial grain size of the material which was determined by Heyn's intercept method was around 50 ± 15 mm. Specimens of 1 mm were sliced from the heat-treated rods by electrical discharge machining. The final thickness of 0.5 ± 0.005 mm of the small punch specimens was achieved by carefully grinding them manually with P800 and subsequently P1200 silicon carbide sandpapers. For DIC imaging a white primer was first applied on the specimen before spraying black speckle patterns on to its surface using an airbrush. Tensile specimen of gauge length 12 mm and diameter 6 mm was also machined from the solution treated rods. Uniaxial tensile testing was carried out on this specimen at a nominal strain rate of 1 Â 10 À3 s À1 . Both small punch and uniaxial tensile loading were carried out along the longitudinal direction of the rods from which the respective specimens were machined. Optical micrographs were taken using Zeiss Axio Imager M2 optical microscope. Electron Back Scatter Diffraction (EBSD) images were taken using Zeiss Sigma HD VP field emission Scanning Electron Microscope operated at an accelerating voltage of 20 kV. TSL OIM software was used for plotting Kernel Average Misorientation (KAM) Maps. An upper threshold of 1.5 was used to map the misorientation in these maps. The threshold misorientation used for grain boundary detection was 10 .

Small punch test rig
The small punch test rig was attached to a servo-hydraulic benchtop system and the load cell equipped with it was operated in the range of ±5 kN. A crosshead displacement rate of 0.005 mms À1 was used in this study. It should be noted that unlike a uniaxial tensile test, the strain rate during a small punch test is not uniform across the specimen. However, the peak strain rate in a small punch test can be compared to the uniaxial strain rate using an empirical correlation [26]. In this investigation, the crosshead velocity of 0.005 mms À1 used in the small punch test would generate a maximum strain rate of 5 Â 10 À3 s À1 . Since the variation in the maximum small punch strain rate with respect to that of the tensile test (1 Â 10 À3 s À1 ) is within the same order, its influence on the deformation behaviour is negligible. Further, it has been shown that the room temperature strain rate sensitivity of 316 stainless steel does not vary significantly within the strain rate range of 10 À2 to 10 À4 s À1 [41]. The schematic of the small punch test rig used in this study is given in Fig. 1. The dimensions of the specimen, dies, and punch (ceramic ball) were compliant with the CEN standard [26]. The principal dimensions of the test rig are given in Table 2. A first surface mirror inclined at an angle of 45 was placed in between the lower dies. The inclined mirror facilitated a less cumbersome access for acquiring DIC images from the speckled specimen surface. The test rig was aligned on the servo-hydraulic machine in such a way that the optical axis of the lens system, the plane normal of the mirror and the loading axis were all coplanar. furnished in Table 3. DaVis software Version 10.0.5 was used to process the acquired image and obtain the out of plane deflections. A subset size of 55 pixels and step size of 25 pixels were used for the analysis. Procedures outlined by the Good Practices Guide for Digital Image Correlation were followed for patterning, calibration and processing stages of the analysis [42]. As the specimen could undergo rigid body movement between the scans, corrections were made using DaVis software Version 10.0.5 which considers both the translational and rotational displacements. Stereo DIC systems maps have been widely used to estimate changes in shape and out of plane displacements [43e47]. These systems estimate the out of plane displacement (Z D ) in addition to in-plane displacements (X D and Y D ) using a triangulation algorithm [48]. The triangulation algorithm establishes an analytic linear relationship between each of the camera's pixel coordinates and the actual coordinates. The actual coordinates of the point of interest are then estimated by solving the two linear systems corresponding to the two cameras by least square technique. The solution for the analytic relationship requires estimation of the cameras' extrinsic and intrinsic parameters which can be obtained by a calibration procedure. The extrinsic parameters are related to the positioning and orientation of the cameras. The intrinsic parameters are dependent upon the camera's optic centre and focal length. The calibration gives the scaling for in-plane displacements (X D and Y D ), these parameters are essential for estimating Z D . Though the calibration establishes the relationship between the actual and the camera coordinates, there are several uncertainties in Z D values which are inherent due to the use of stereo system [48,49]. Further uncertainties were generated in this study due to the use of an inclined mirror. These uncertainties, classified as systematic errors, can be quantified as opposed to random errors which arise due to quality of speckle patterning and image noise level [50].

Quantifying uncertainties attributed to stereo DIC system
Both the stereo angle [49] and the base line distance [48] influence the precision of the out of plane displacement (Z D ). The  precision of Z D deteriorates with decrease in stereo angle as well as the base line distance. As the mirror used in this study was inclined at an angle of 45 , the plane of the lens and the plane of the mirror are not parallel. This implies that while estimating the Z D all the longitudinal points (points along the vertical direction) in the image are not in focus at the same instance. Further, the resolution in the out of plane direction is also dictated by the depth of field (DOF) of the imaging system [42]. The DOF gives the estimate of how much the object can be moved perpendicular to the line of sight without appreciable change is focus. The DOF is related to the numerical aperture (NA) of the lens and the wavelength of light (l) used for illumination through the following equation [51].
For a wavelength of 550 nm (the average wavelength of light in the visible spectrum) the DOF of the system is around 0.6 mm. This implies that within the Z D range of 0.6 mm, uncertainties attributed to the limitation in the lens system's DOF is minimum.
In order to quantify the uncertainties in Z D arising due to the use of the mirror, stereo DIC measurements were carried out on a zerostrain field test in the calibrated setup. For this test a speckled surface was displaced towards the mirror (in the Z D direction) at a crosshead displacement rate of 0.005 mms À1 . The speckle pattern on this surface was consistent with the one used for the small punch specimen. The crosshead displacement was then compared to Z D estimated by the stereo DIC setup on the speckled surface. The direction of Z D in this test was congruent to the direction of the out of plane deflection of the deforming small punch specimen. This test was used to establish the 1) uncertainties of the Z D in the central point and 2) the variation in the displacement values in the contour map. A circular region of interest (ROI) with a radius of approximately 1.6 mm was chosen for contour mapping using the DIC data during the zero-strain field test. It was ensured that subsequent DIC analysis on the actual small punch specimen was carried out on a ROI almost equal and concentric with respect to the one used in the zero-strain field test.
The variations between the crosshead and the DIC estimated displacements values taken from the central point (CP) in the ROI for the crosshead-based displacement range between 0 and 3 mm is shown in Fig. 2. The region of minimum deviation (±0.01 mm) between the two displacements is indicated in the figure. This region (which spanned around 0.65 mm) was between the crosshead-based displacement range of 0.6e1.25 mm. The extent of this region was approximately equal to the DOF calculated for this system. Despite the uncertainties arising due to the stereo set up and the usage of mirror, the deviation between crosshead and DIC estimated (Z D ) displacements was in the range of ±0.07 mm through the crosshead displacement range of 0e2.7 mm. The variations start to enhance for crosshead displacement values greater than 2.7 mm. The small punch tests in the current work were carried out in such a way that during the test the crosshead displacement values lie within the range of 0.6e2.7 mm, which is indicated as -total displacement range used in Fig. 2. This was done in order to obtain minimum deviation during the initial stages of the test. The displacement values during the initial stages of the test were crucial as they were used subsequently for estimating the elastic and plastic properties using iFEM.
The displacement contour map taken at a crosshead displacement of 1.15 mm is shown in Fig. 3. The scatter in the displacement values estimated by DIC in the transverse (X direction) and longitudinal (Y direction) directions for this crosshead displacement is compared in Fig. 4. The scatter in the transverse direction was comparatively lower than what was observed in the longitudinal direction. The gradient in Z D along the longitudinal direction is due to the use of the inclined mirror. In the current analysis the Z D along one half of the transverse direction was used for further analysis. This segment of interest (SOI) is indicated in both Figs. 3 and 4. The variation in Z D along the SOI was in the range of ±0.005 mm in the crosshead-based displacement range of 0.6e2.7 mm. Further, the DOF is same for all the points in the SOI as they lie along the transverse direction. Hence, there was no DOF attributed uncertainties in the selected SOI.
Using the zero-strain field test, the SOI which had the least spread of uncertainties could be determined and subsequent iFEM based investigations were confined to this segment.

Load-deflection characteristics
The load-deflection curve of 316 L stainless steel obtained from the central point of the region of interest (ROI) is shown in Fig. 5a. The stages corresponding to various deformation mechanisms is also indicated in the figure. The enlarged portion of the first two stages of the load-deflection curve is given in Fig. 5b. The loaddeflection data in these two regimes were subsequently used for estimating the mechanical properties incorporating iFEM. The positional coordinates of the ROI were equivalent to the one used in the zero-strain field test. The DIC images estimated for central point deflections of 0 mm, 0.54 mm, 1.16 mm, 1.75 mm and the image after failure are given in Fig. 6. The DIC estimated 3D-contour maps at these deflections (except for the failed instance) are shown in Fig. 7. The DIC estimated deflection contours were almost concentric to the central point. The contour maps start to degenerate along the circumferential regions during the later stages of specimen deformation. This is because progressive deformation of the specimen generates sharp curvature in the circumferential regions which obstructs precise resolution of the speckle pattern by the DIC system. However, the deflections along the regions adjoining the central point could be mapped satisfactorily until the peak load. Beyond the central point deflection of 2.05, the DIC system could not resolve any data from the speckled specimen surface.
The five stages which correspond to various deformation modes [5,11,12,14,29,30] during the small punch are depicted in Fig. 5. Macrographs corresponding to these stages taken along the longitudinal cross-sections of the small punch specimens are given Fig. 8(aed). These macrographs were taken on specimens interrupted after deflections which roughly correspond (±0.01 mm) to the deflections depicted in Figs. 6 and 7. Stages I and II in the loaddeflection curve comprises of the elastic and plastic bending regimes respectively. During the first two stages the contact area between the punch and the specimen increases rapidly and there is no appreciable reduction in specimen thickness [2]. Stage III is the membrane stretching regime when the contact area does not increase significantly but the specimen thickness starts to reduce. Initiation of localised thinning was evident even at this stage (Fig. 8b). Further thinning and localised necking constitute the plastic instability regime (Stage IV). Thereafter, in Stage V due to material softening there is a drop in load and fracture propagates rapidly resulting in failure. In Stages I and II, the deformation of the specimen is governed only by elastic and plastic properties of the material. With the onset of Stage III, voids start to nucleate at regions which are highly strained. These voids coalescence during Stage IV which subsequently result in further localised thinning and failure [52]. The deformation behaviour of the specimen after the onset of Stage III can be completely simulated only by incorporating damage models in addition to using suitable elasto-plastic models [11,14,33,52].
Though there is no apparent specimen thickness reduction in the plastic-elastic bending stage, strain gradients could still accumulate across the specimen which had a central point deflection of 0.54 mm. To probe localised strains evolved in this specimen, EBSD based KAM maps (Fig. 9) were obtained from the central point (location A) and from a location at a radial distance of 0.6 mm from the central point (location B) (Fig. 8). The KAM maps are used to get  an estimate of the geometrically necessary dislocations which can be used to map localised strain gradients in the material [53e55]. It could be seen that the misorientation spread in location B was higher than in location A. This indicates that the strain accumulation in Location B was higher than what was observed in Location A, despite the deflection in the later location being higher. This observation ascertains that the deflection and the strain accumulation are not directly correlated.
During a constant displacement rate controlled uniaxial tension test, the strain accumulation within the gauge length of the specimen is relatively uniform up to the point of instability. This enables a straightforward conversion of the load-displacement data obtained in a uniaxial tensile test to engineering data. However, in a small punch test considerable strain gradient evolve in the specimen even during the initial stages of deformation. Therefore, load deflection characteristics obtained only from the central point or the singular punch displacement is not an ideal representation of the gross deformation behaviour occurring within the material during testing. Estimating several location specific deflections would enhance the understanding about the deformation characteristics of the material paving way for formulating better correlation with respect to uniaxial tensile properties.

Inverse finite element method (iFEM)
An effort was made to determine the elastic and plastic      properties of the material based on the deflection values up to the onset of stage III obtained from multiple locations on the specimen incorporating an inverse finite element method (iFEM). The scheme of the iFEM used in this work is depicted in Fig. 10. There three main constituents of the of the iFEM framework which was used in the current analysis are -a) geometrically representative finite element model b) constitutive model for characterizing elastic and plastic flow behaviour of the material and c) optimisation procedure. These three aspects of the iFEM framework are described in the following sections.

Finite element model
The finite element model was generated using Abaqus CAE 6.14-1, the geometry of the model was identical to the dimensions used in the experiment. The punch, lower and upper dies were considered as rigid bodies. The deformable element type used for the modelling the specimen was a 4-node bilinear axisymmetric quadrilateral (CAX4R). A uniform element size of 0.03 mm was used for meshing the specimen geometry. The total number of elements in the specimen model was 2261. The specimen was deformed until a punch displacement of 0.65 mm. This punch displacement would generate specimen deflection within the Stage II deformation regime. The value of the friction coefficient used for modelling interaction between the punch and specimen was 0.2. A frictionless interaction was assumed between the die and specimen interface. Though values of the friction coefficient are material specific, its influence on the load deflection characteristics is negligible during the initial stages of specimen deformation [56].

Constitutive flow equation
The Young's Modulus (E) and the Poisson's ratio(n) were used to characterize the deformation behaviour in the elastic regime. The values of both these constants were optimised using iFEM. Since DIC is a non-contact strain measurement method which directly acquires the deflection from the specimen, the influence of the testrigs compliance can be disregarded [40,57]. DIC based methods have been previously used to estimate elastic properties of materials with reasonable accuracy [58]. The Ludwigson constitutive relationship was used to model the plastic flow behaviour as it gives a closer description of flow behaviour of austenitic stainless steels [59e63]. The Ludwigson equation has an additional exponential term which considers the deviation from the Ramberg-Osgood model at lower strains which is observed for lowstacking fault energy materials like austenitic stainless steel [64].
The Ludwigson equation takes the following form Where s p and ε p are the true stress and true plastic strain respectively, K 1 ; n 1 ; K 2 and n 2 are empirical constants. The upper and lower bounds of the parameter values used in the analysis are given in Table 4.

Optimisation procedure
A genetic algorithm (GA) based procedure was implemented using MATLAB R2018a programming language for optimising the elastic and plastic properties. GA is an optimisation tool which searches global minima of objective functions [38,65]. Initially GA chooses a random population of parameter values within the given upper and lower bounds. It then obtains the values of the objective function based on various combination of the parameter values for each generation. The rank of each of these combinations is evaluated based on a fitness function which gauges the proximity to the minima. GA then refines the population in subsequent generations to obtain a better combination of parameters with improved fitness function-based ranking. The options set in the GA module are given in Table 5.
Two optimisation techniques were formulated in this study, a single-point optimisation (SPO) using the deflection obtained from the central point and another multi-point optimisation (MPO) using the deflections obtained from four points with equal weighting factors. Since the data extracted for the SPO technique was from the central point it is equivalent to acquiring specimen deflection from an LVDT. It can be therefore be argued that the SPO technique is equivalent of performing iFEM using LVDT based deflection data. The four points which were chosen for the MPO technique lie along the SOI indicated in Figs. 3 and 4. Apart from the one central point the other three points were placed at a distance of 0.6, 0.9 and 1.2 mm from the central point in the specimen's pre-deformed condition (Fig. 11). The rationale behind choosing these points was that they had distinct load-deflection characteristics, which would make the MPO technique more effective. The load displacement curve from the point placed at a distance of 0.3 mm from the central point was not considered in this analysis as its load-deflection characteristics was similar to the one obtained from the central point.
During the optimisation process, the deflections obtained from each simulation were compared to the deflection obtained from DIC for each iteration. In both SPO and MPO techniques, the values of the parameters were refined to get the least difference with respect to the DIC obtained deflection values. The MPO technique generated a set of non-dominated solutions as there were four objective functions (corresponding to the four points-central point, 0.6,0.9, and 1.2 mm from the central point) with equal weighting factors. The parameter combination which gave the least difference between the DIC and iFEM estimated deflection values at the central point was chosen as the optimised solution for the MPO technique. This is due to negligible curvature-dependent errors which might be associated with the central point. The limitation of the DIC system in reconstructing curved surfaces of the deforming small punch specimen has already been indicated. The termination criteria for the GA was set to 20 generations for both SPO and MPO techniques as there was no significant improvement in the results for subsequent generations.

Optimisation results
The true stress-true plastic strain curve estimated by SPO and MPO techniques are compared with the values obtained from the experimental true stress-true plastic strain in Fig. 12. The values of Ludwigson parameters fitted with the actual true-stress true strain data along with the parameters obtained from the SPO and MPO techniques are given in Table 6. The values of 0.2% proof stress (PS) for both the MPO and SPO techniques were estimated from the engineering stress strain data, which was calculated from the true stress-true plastic strain data obtained using the Ludwigson parameters. The ultimate tensile strength (UTS) reported for SPO and MPO techniques were determined from the values of true stress which were equal to the corresponding work hardening rates (slope of the true stress-true strain curve) [66]. The experimental Youngs modulus and Poisson's ratio used for comparison were typical values mentioned in ASME [67]. Though the value of the 0.2% PS could be obtained with reasonable precision by both SPO and MPO techniques, the values of the elastic constants, UTS and the plastic flow curve derived from MPO technique were in greater agreement with the experimental values as against the values obtained by SPO technique.
The maximum in-plane plastic strain distribution in the specimen after punch displacement of 0.6 mm using the optimised elastic and plastic parameters by MPO technique is given in Fig. 13a. The variation of maximum in-plane plastic strain distribution on the outer surface of the modelled specimen for punch displacements ranging from 0.1 to 0.6 mm is given in Fig. 13b. The figure also shows co-ordinate lines drawn at approximate locations were the DIC deflections were obtained from the corresponding small punch specimen. The strain evolution around the pre-deformed distance of 0.6 mm was higher than what was observed at the central point. This finding substantiated the results of strain gradients estimated using KAM maps. The load-deflection curves obtained from each of the four locations (on the modelled specimen) by FEM using the parameters optimised by MPO technique shows a good agreement to the and ones obtained using DIC on the actual small punch specimen (Fig. 14).
A retrospective insight on the precision of both the optimisation techniques showed that the effectiveness of the optimisation process depended on the spread of strain values at the point(s) which were chosen in each of the techniques. The SPO technique used deflections from the central point which experienced relatively lower strain during deformation. The MPO technique on the other hand included deflections from the point located at 0.6 mm from the central point which had a relatively larger strain accumulation in spite of the specimen deflection in this region being relatively lower. Further, the accuracy of the MPO technique could have been possibly enhanced as it utilised deflections from regions which showed a larger spread of maximum in-plane strain value (Fig. 13b) which ranged from À0.02 (at location of 1.2 mm) to 0.15 (at location of 0.6 mm). An elaborate study on optimising the number of locations and the position of these locations from the central point where DIC and FEM based deflections can be obtained would further enhance the accuracy of the MPO based results.
In this study, an iFEM based approach employing multiple DIC based deflections obtained from Stages I and II could estimate uniaxial tensile properties to a reasonable precision. However, several improvements in the acquisition system could enhance the applicability this methodology. Designing a more versatile DIC system taking into consideration the uncertainties which arise due to the stereo system and DIC image corrections can help obtain gross deflection contours throughout the course of the test up to the specimen failure. This knowledge would be crucial for comprehensively estimating the damage characteristics of the material.

Conclusion
A new system integrating small punch test and DIC was developed for in-situ deflection mapping of the specimen. The systematic errors arising due to the use of the stereo-DIC setup and inclined mirror was estimated using a zero-strain field test. An inverse finite element-based methodology was developed to obtain the elastic and plastic properties of the material using the DIC data. The use of DIC data from multiple points in the small punch specimen gave a closer estimate of the uniaxial tensile properties as it used deflections from locations which were subjected to higher and larger spread of strain during deformation.   Table 6 Comparison of the parameters obtained from experimental, multi-point (MPO) and single-point optimisation (SPO) iFEM techniques. The deviation between the actual and estimated properties is also indicated.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.