A high-fidelity strain-mapping framework using digital image correlation

Materials Science & Engineering A 594 (2014) 394–403

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Materials Science & Engineering A journal homepage: www.elsevier.com/locate/msea

A high-fidelity strain-mapping framework using digital image correlation Shahram Amini, Rajesh S. Kumar n United Technologies Research Center (UTRC), 411 Silver Lane, East Hartford, CT 06108, USA

art ic l e i nf o

a b s t r a c t

Article history: Received 17 September 2013 Received in revised form 2 November 2013 Accepted 6 November 2013 Available online 15 November 2013

A practical framework is developed in this work for extracting high-fidelity quantitative strain information from three-dimensional digital image correlation (3D-DIC) experiments. The framework is applicable for continuum-scale deformation in elastic and plastic regimes in the presence of macroscopic strain gradients. The framework is developed, demonstrated, and validated by conducting 3D-DIC experiments and corresponding finite element analysis (FEA) on polycrystalline aluminum tensile specimens with and without macroscopic strain gradient subjected to uniaxial tensile deformation in both elastic and plastic regimes. The developed framework is expected to be applicable for continuumscale deformation in other classes of materials. & 2013 United Technologies Corporation. Published by Elsevier B.V. All rights reserved.

Keywords: Digital Image Correlation (DIC) Strain mapping Strain gradient Length-scales Elastic deformation Plastic deformation

1. Introduction For decades, improvements in materials have played a significant role in advancing aerospace mechanical and structural designs. Super-alloys, ceramic matrix composites, polymer matrix composites, and nano-enhanced composites have been the focus of aerospace industry for low- and high-temperature structural applications such as: gas turbine engine components (blades, vanes, combustor liners, turbine disks, rotor and fan blades), wing leading edges of hypersonic flight vehicles, nozzle exit ramps for advanced rocket engines, helicopter rotor blades and gearbox housings, among many others [1–13]. While some of these advanced materials concepts are at the emerging stage of the applications delineated above, their success would proliferate throughout the aerospace industry as a common benefit to result in more economical, lighter, stronger and safer aircrafts. However, this success has been partly inhibited by a lack of understanding of the fundamental behavior of these materials at multiple length scales (e.g. nano-, submicron-, and macro-scales) under complex loading and severe environmental conditions. This lack of understanding is further exacerbated by the highly heterogeneous and locally anisotropic character of some of these materials, e.g. composites, on distribution of stresses and strains, as well as their

n

Corresponding author. Tel.: þ 1 860 610 7045; fax: þ1 860 353 2928. E-mail address: [email protected] (R.S. Kumar).

failure mechanisms. Fundamental understandings of material behavior and failure can enable efficient design of lower cost, safer and more robust aerospace structures. On the other hand, inadequate understanding of failure mechanisms in these materials – usually obscured by the difficulty of visualizing damage evolution and presence of multiple interacting failure mechanisms – is yet another stumbling block for their use in critical aerospace applications. Many studies [14–21] have been performed to identify and understand the complex failure mechanisms in heterogeneous materials and to propose solutions to improve their resistance against failure. However, further understanding of the micromechanics of damage and failure processes in these materials under applied load will not only make these materials more useful but also will contribute to an improvement in safety and design of engineering components made from them. More importantly, as materials complexity evolves to meet the extreme challenges of their environments, it is critical to characterize and understand their fundamental behavior. In recent years, optical full-field strain-mapping techniques such as Digital Image Correlation (DIC) have increasingly been used in research and industry for in-situ displacement and strain measurements of a specimen subjected to external stimulus (mechanical, thermal, thermo-mechanical, etc.), enabling improved characterization of materials and components at various length scales [16,22–31]. DIC is a noncontacting technique for measurement of surface displacements of an object by tracking deformation of a speckle pattern through a series of digital images acquired during a test [32–34]. Displacements are

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S. Amini, R.S. Kumar / Materials Science & Engineering A 594 (2014) 394–403

obtained by imaging a speckle pattern on the material surface during a test and subsequently correlating each image of the deformed pattern to that in the un-deformed state. Strains are obtained by differentiating the displacement fields. The technique combines pixel level displacement accuracy with high spatial resolution, where strain gauges and extensometers lack the requisite resolution and fidelity. Due to the miniscule motions that are often of interest in engineering applications, the resolution requirements are much higher than those for most other applications. DIC has been performed with many types of object-based patterns, including lines, grids, dots and random speckle patterns [35], however, it is most common to employ stochastic patterns. In the past several years, two-dimensional DIC (2D-DIC) has undergone rapid growth worldwide [35]. However, as 2D-DIC uses predominantly in-plane displacements and strains, relatively small out-of-plane motions will change the magnification and may introduce errors in measured in-plane displacements [35]. As a result, three-dimensional DIC (3D-DIC) has been used for a wide range of applications and, hence, the present work is focused on 3D-DIC. Novak and Zok [23] demonstrated that DIC could be used when high thermal loads are present in a specimen. DIC is also suitable for studies spanning a large range of strain rates that typically span several orders of magnitude from creep [36–38] to shock and impact [39–41]. In principle, these attributes make DIC eminently suitable for probing strain distributions in various materials for a variety of testing conditions. However, to date, for the most part, DIC measurements have been focused on qualitative understanding of deformation and fracture rather than quantitative correlation with analyses. Despite numerous advantages DIC measurements offer, obtaining high-fidelity quantitative information from such measurements is still challenging, specifically when strain gradients and materials heterogeneities are present. Design of high-fidelity DIC experiments is complex, because numerous pre- and post-testing parameters at various length scales must be selected simultaneously to achieve requisite fidelity. Some of these parameters must be chosen before an experiment commences and some can be selected after the experiment concludes [22]. Furthermore, some of these parameters are intimately related to the length-scales associated with the material and the structure under consideration. In addition, if the results from DIC are to be used for model calibration and/or validation, the parameters should be consistent with the length-scale resolved in the models. Material behavior at various length- and time-scales could potentially span multiple orders of magnitude thereby adding further challenges and complexity for a common framework for DIC measurements. Therefore, validated DIC methodologies are essential to apply this useful experimental method to various deformation, time and materials length scales, and for quantitative extraction of strains specifically in the presence of strain gradients and inhomogeneities. A robust DIC methodology can enable efficient design and validation of mechanics and manufacturing models. In essence, various DIC parameters could be categorized as follows: a) Optics and correlation: stochastic speckle pattern, hardware setup, camera noise and illumination conditions such as lighting, magnification, brightness, contrast, blurring, imaging system calibration, and correlation criteria b) Post-processing: facet (or subset) size, facet step and filter size c) Length scale: microstructure and continuum length-scales d) Deformation gradients: gradients due to inherent heterogeneities e.g. in woven composites or those due to specific geometric and loading configurations in the structure The reliability of DIC measurements depend on the knowledge of uncertainty and the sources of errors. There have been numerous

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studies in the literature where effect of one or more of these parameters on the fidelity of DIC results was examined: Siebert et al. [42] demonstrated the influence of different camera parameters on the DIC accuracy, uncertainty and errors. Triconnet et al. [26] proposed guidelines to choose correlation parameters as a function of various stochastic speckle patterns and studied their effect on the uncertainty of measured displacement and strain quantities. Tong [43] examined performance of various DIC criteria using several digital images with various characteristics and assessed relative robustness, computational cost, and reliability of each criterion for strain mapping applications. Bornert et al. [24] determined the performances of several image processing algorithms for displacement error assessment using various speckle patterns, where they used several DIC packages based on different formulations. Berfield et al. [25] demonstrated the feasibility of using DIC at micrometer and nanometer length-scales and showed that successful application of DIC required the ability to generate a speckle pattern at those scales. Yaofeng and Pang [27] investigated the effect of facet size and image quality on the accuracy of deformation measurements by DIC. They provided guidelines for sample preparation, estimation of displacement errors, and facet size normalization for DIC. Recently, in a comprehensive study, Rajan et al. [22] provided guidelines for selecting DIC test parameters to maximize the extent of correlation and to minimize errors in displacements and elastic strains using specimens with various geometric configurations. However, their work is restricted to purely elastic deformation and their guidelines do not take into account the inherent microstructural length-scales such as grain size and its relation with the DIC parameters. Furthermore, their guidelines do not explicitly account for the length-scale associated with the strain gradient. The present paper overcomes these limitations by accounting for both the microstructural and strain gradient length-scales. In addition, the methodology is applied and demonstrated for both the elastic and plastic deformation regimes. The objectives of this work are to develop a methodology for high-fidelity full-field strain-mapping of elastic and plastic deformations of polycrystalline materials at macroscopic continuum length-scales subjected to quasi-static loading conditions and to validate the methodology by conducting DIC experiments and comparing the results against Finite Element Analysis (FEA). In Section 2 we outline the experimental procedure, while Section 3 is concerned with brief description of the FEA procedure used. In Section 4, we develop and outline a step-by-step methodology for obtaining high-fidelity quantitative strains from DIC measurements, which is demonstrated and validated in Section 5.

2. Experiments Uniaxial tension tests were performed on 3 mm thick aluminum 6061-T6 specimens with and without an open-hole. The specimens were of ‘dog-bone’ geometry with a gauge length of 60 mm and width of 12.5 mm. The open-hole specimens had a central hole of 3 mm diameter. Two speckling techniques were used for the DIC experiments. In the first technique, the samples were initially sprayed with a flat white spray paint and subsequently sprayed using a spray canister with flat black spray paint (referred to as “spray” method hereafter). In the second technique an airbrush with black water-soluble paint was used (referred to as “airbrush” method hereafter). As will be discussed later, the two techniques result in very different speckle sizes. The speckle size distribution was determined using ImageJ software (US National Institutes of Health, Bethesda, MD, USA). The mechanical tensile tests were performed using an MTS servo-hydraulic test frame (MTS 810, Minneapolis, MN) at a displacement rate of 0.05 mm/s. Threedimensional (3D) DIC was employed to monitor speckled face of

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Fig. 1. FE model and corresponding mesh of the specimen with a circular hole.

the specimens to obtain full-field displacement and strain maps using a commercially available ARAMIS system (GOM mbH, Braunschweig, Germany). In this system, a set of images is acquired with a calibrated pair of stereo cameras using appropriate lenses and illumination. The ARAMIS software is then used to correlate the acquired images and post-process the data. Visual representation of the crystallographic microstructure was performed using Orientation Imaging Microscopy, OIM, with an attached electron backscatter diffraction (EBSD) system (EDAX-TSL, USA) in a scanning electron microscope (SEM) (Leo 1455-VP, Germany) with a thermionic LaB6 filament. Five OIM maps were taken starting from the edge of the tensile bars to the edge of the hole prior to mechanical testing. Each scan area was approximately 1.7 mm2. The maps were overlapped for an approximate reconstruction of the region and the average grain size was estimated from image analysis.

4. Methodology for high-fidelity quantitative DIC DIC can be readily used to provide a qualitative understanding of strain distributions and strain localizations due to various mechanisms. However, extracting high-fidelity quantitative strain measurements from DIC data is not trivial, and requires a careful selection of various pre- and post-test parameters with respect to microstructure, acceptable strain error, and strain gradients. In this section, we aim to develop a methodology for such high-fidelity quantitative determination of macroscopic strains. The methodology is developed based on experiments (where uniform macroscopic elastic strains are applied), heuristic estimations and/or prior works. The methodology is then validated using experiments involving elastic and plastic deformations in the presence of strain gradients in a model polycrystalline material. 4.1. Relevant DIC length-scales

3. Finite element analysis The FEA of the open-hole specimen subjected to quasi-static tension was performed using Abaqus finite element code (version 6.10, Dassault Systemes Simulia Corp., Providence, RI, USA). Full 3D analysis was conducted to accurately capture the local strain in the vicinity of the hole and the specimen edge. Furthermore, due to geometrical symmetry only 1/8th of the specimen was modeled as shown in Fig. 1. This figure also shows the mesh for the analysis. The mesh was chosen based on a mesh convergence study and that further refinement of the mesh did not significantly change the distribution of the strain in front of the hole. Fully integrated brick elements were used. For elastic analysis, material properties of interest are elastic modulus and Poisson's ratio. These properties were taken from a handbook [44], E ¼69 GPa and ν¼ 0.3. On the other hand, for elastoplastic analysis, complete stress–strain response of the material is needed. The material was modeled using classical J2 plasticity model available in Abaqus. The stress versus plastic strain data required for the model was derived from uniaxial stress–strain response of an un-notched specimen (see Fig. 5) and specified in a tabular form in the FE model. For comparison with the experimental DIC results, strains on the free surface in front of the hole were extracted at a specified value of applied far-field vertical strain.

There are three critical parameters and length scales associated with DIC: speckle size, facet size, and facet step. There are numerous studies in the literature that have discussed various methods of characterizing speckle size and its distribution. Speckle size, as discussed by Rajan et al. [22], is defined as the 90th percentile value obtained from particle analysis. Their study demonstrated that the median speckle size is not a suitable parameter for characterizing speckle size distribution as it is relatively insensitive to the presence of very large speckles. Speckle pattern and inherently its size should be chosen and applied prior to an experiment based upon deformation, features of interest, field-of-view, resolution, and structure length scale. Facet (or subset) size is essentially defined as the correlation window, i.e. a relatively small aperture that comprises multiple speckles used for intensity pattern matching or correlation. Lastly, facet step (or step size) is defined as the facet overlap length. The latter two parameters – facet size and facet step – are chosen once the experiment is completed, or in other words, during the postprocessing of DIC data. Fig. 2 shows a schematic diagram of speckle size, facet size, and facet step superimposed on a typical stochastic speckle pattern. An additional parameter called filter length may also be used to smooth out the results during DIC data post-processing. Although use of filtering in many DIC experiments  generally in the absence of strain gradients – is

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Fig. 3. Schematic diagram of the relationship between microstructure and DIC facet size.

Fig. 2. Schematics of speckle size, facet size, facet step, and filter length overlaid on a sample speckle pattern.

beneficial, it is not used in the present work and hence filter length is not used in the guidelines. 4.2. Microstructural length-scales and facet size All materials are heterogeneous when observed at some appropriate length-scale. In the Al alloy considered in this work, the microstructure of relevance is the grain size. At the microstructural length-scale deformation depends on the details of heterogeneities: for example, in polycrystalline materials, individual grains, their size, orientation and misorientation with respect to their neighboring grains can affect the deformation, and in composite materials, tow geometry, weave architecture, and defects at the tow scale are relevant microstructural features. Such microstructure-scale deformation  while important for understanding the mechanisms of failure and fracture or for validating microstructure-scale models  is not the focus of the present work. Herein, we focus on engineering problems where macroscopic strains are of interest and it is not essential to resolve strains at the length-scale of microstructural heterogeneities. In such cases, the material is treated as a homogeneous continuum. However, information regarding the material microstructural length-scale is still important as the homogeneous assumption implicitly assumes existence of a suitable Representative Volume Element (RVE). In the case of surface strain measurements associated with DIC experiments, we are dealing with a representative area rather than volume; however, we still continue to refer to it as a RVE. In DIC experiments where strains at macroscopic length-scales need to be determined, the lower bound on the facet size should correspond to the RVE size, otherwise the strain would capture the local fluctuations associated with the micro-scale heterogeneities. The RVE size for the polycrystalline material of interest in this work is a function of the average grain size (dg ), which can be determined using standard microstructural characterization tools. If facet size (df ) is smaller than dg , which means several facets span each grain, then DIC is resolving deformation at the scale of individual grains. On the other hand, if each facet contains several grains of a polycrystalline material, then deformation regime may

Fig. 4. (a) Schematic of the test specimen; the physical strain gauge was attached to the opposite face of the specimen located at the center of the dog-bone and (b) the speckle pattern and representative grain distribution superimposed on it.

be classified as continuum regime and governed by continuum mechanics models and predictions. Fig. 3 illustrates schematic diagrams of deformation regimes derived from comparison of grain size (dg ) and facet size (df ). Based on the notion of RVE, it is intuitive that the facet should encompass multiple grains in order to capture the strains associated with a material point in the continuum mechanics sense. In order to determine an appropriate lower bound to the facet LB size, df , we use data derived from experiments where a macroscopically uniform strain is applied on the specimen without a

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1 Applied far field Strain ()

0.2%

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Strain (%)

Stress (MPa)

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=3%

250 200 150

=0.2%

DIC, df =60 (pixels)

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DIC, df =30 (pixels)

DIC, d =30 (pixels) f

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50

f

-1

Strain Gauge

0 1

2

3 Strain (%)

4

5

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Fig. 5. Comparison of strain gauge data and DIC data obtained by choosing various facet sizes from the specimen without a hole as shown in Fig. 4.

hole. Fig. 4a shows a schematic of the test specimen. The speckle pattern and representative grain distribution superimposed on the speckle pattern applied by the airbrush method are illustrated in Fig. 4b. The overall macroscopic stress–strain response of the specimen was determined by averaging the DIC data over a small area (shown as the green window in Fig. 4a). The physical strain gauge was attached to the opposite face of the dog-bone specimen at its center. Fig. 5 compares the overall response from the DIC and strain gauge for various facet sizes, keeping the step size to be half of the facet size and no filtering was applied. It is obvious that the overall response from the DIC matches well with the strain gauge data in both elastic and plastic regimes. Furthermore, it is clearly observed that the overall response is not sensitive to the facet size. This result is not surprising as the area over which averaging is performed for computing the macroscopic strains is much larger than the facet size. Next, we consider the local strain variations along path A–B as shown in Fig. 4 at fixed values of applied macroscopic strain. Fig. 6 shows the variation of εyy from DIC data at the applied macroscopic strain of  0.2%, which is well within the linear elastic regime, for various facet sizes. Because deformation is homogeneous, the macroscopic applied strain should be constant along the y-coordinate. On the other hand, the strain derived from DIC data shows fluctuations when the facet size is small. These fluctuations become small when facet size is increased, and beyond a facet size of 60 pixels, the strain from DIC data becomes uniform and equal to the applied far-field strain. When the applied macroscopic strain is  3%, i.e., the deformation is well within the plastic regime, the variation of the local εyy from DIC as a function of facet size is shown in Fig. 7. It is observed that the strain is not constant along the length, possibly attributed to localization and/ or inhomogeneity associated with plastic deformation. However, similar to the elastic case, fluctuation in the strain reduces with increasing facet size, and again, a facet size of 60 pixels generates reasonable strains compared to the far-field applied strain. It should be noted that when facet size is extremely small, the noise in the results could be attributed to presence of insufficient speckles in the facets for accurate correlation. Furthermore, the average grain size for the material used in this study was determined to be dg  10 7 5 pixels. Combining this grain size

5 Length [A-B] (mm)

10

Fig. 6. Effect of facet size on strain variation for a macroscopically uniform applied strain (  0.2%) in the elastic deformation regime obtained from the specimen without a hole as shown in Fig. 4.

5 Applied far field Strain ( ) ~ 3% 4

Strain (%)

0

0

3

2 DIC, df =60 (pixels) DIC, df =30 (pixels)

1

DIC, df =15 (pixels) 0

0

5 Length [A-B] (mm)

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Fig. 7. Effect of facet size on strain variation for a macroscopically uniform applied strain (  3%) in the plastic deformation regime obtained from the specimen without a hole as shown in Fig. 4.

information with the observation noted earlier that a facet size of at least 60 pixels is needed to yield uniform strain from DIC under applied uniform strain, a lower bound to the facet size is calculated LB as df  4dg . Thus, at least 4 grains are required along the facet edge in order to capture continuum macro-scale deformation, and is recommended as a guideline. As discussed by Rajan et al. [22], another lower bound to the facet size is set by the requirement that an individual facet contains unique features that distinguish it from neighboring facets. The larger of the two lower bounds can then be used as the lower bound. However, in our proposed approach, the requirement that each facet contains unique features is ensured by choosing an appropriate speckle size once facet size is determined within the bounds established by microstructural considerations as delineated earlier and strain gradient considerations to be discussed next.

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4.3. Macroscopic strain gradient and facet size If deformation is uniform, any facet size larger than the lower bound determined based on RVE consideration in Section 4.2 should provide enough fidelity in strains measured by DIC. However, in most engineering applications of interest, deformation is not uniform. This is especially the case when stress-risers such as holes and notches and/or complex loading conditions are introduced for attachments and for other design considerations. Furthermore, strain localization in the vicinity of such stress-rising features leads to damage and fracture initiation in such areas. Thus, it is essential to capture such strain gradients as accurately as possible. The desired spatial resolution of strain in the presence of strain gradient in a structure essentially sets an upper-bound to facet size. If facet size is larger than this upper-bound, strain gradient would be averaged out and strain distributions will not be accurately captured, therefore DIC strain measurement loses its fidelity. On the other hand, based on the discussion in Section 4.2, facet size cannot be reduced indefinitely to capture the steep strain gradients as the minimum size is set by microstructural considerations. Thus, the facet size should be between upper and lower bound set by strain gradient and microstructural considerations, respectively. It should be noted that if the objective is to resolve strains at microstructural length-scales, facet size can be smaller than the lower-bound determined above based on continuum considerations. In order to estimate an upper bound of facet size, we introduce two quantities: (1) an acceptable error in the maximum strain, and, (2) an average strain gradient. Acceptable error in strain, εa , is problem-dependent and is the level of error that is acceptable in the region of highest strain gradient. Acceptable strain error will typically be dependent on the maximum strain. The value of this parameter should be as small as possible; however, as discussed in the previous paragraph, this parameter cannot be arbitrarily small due to the lower bound on facet size set by microstructural considerations. We recommend that the acceptable strain error should be at least one order of magnitude smaller than the maximum strain that is being resolved, i.e., the acceptable error should be at most 10%. A lower value of error is certainly desirable for applications demanding higher quantitative fidelity. It is also suggested that the value of the acceptable error should be parametrically varied in order to understand and quantify its effect on the desired strain output. The average strain gradient (εS:G: ) is defined as a ratio of the strain difference (Δε ¼ εmax  εmin ) to the length (l) over which the gradient spans (Fig. 8a), i.e., εS:G:  ðΔε=lÞ, where εmax and εmin are maximum and minimum strains present over this length. The maximum and minimum strain values are not known prior to the experiments, but may be estimated based on analytical formulae, FEA predictions, or previous experiments. For a specific test configuration, e.g., open-hole specimen subjected to uniaxial tension (Fig. 8b), εmax and εmin are strains at the edge of the hole and edge of the specimen, respectively. Based on the two quantities defined in the foregoing, we UB estimate the upper-bound to the facet size as: df ¼ εa =εS:G: . UB LB Typically, df Z df , and the facet size should be chosen to lie UB LB between the two bounds. However, if df o df , then the facet size LB should be taken to be equal to df in order to satisfy continuum requirements, and, in this case, the resulting strain in the high gradient regions will display higher levels of error. 4.4. Speckle pattern and size For a chosen facet size that satisfies the bounds established based on the procedure outlined in Sections 4.2 and 4.3, the next consideration is to determine desired speckle size and identify a

Fig. 8. Schematic diagram representing maximum strain (εmax ), minimum strain (εmin ), and average strain gradient (εS:G: ) for: (a) a generic problem and (b) an openhole test configuration.

method of application that will result in the desired speckle size and pattern. There are two governing considerations in determining the appropriate speckle pattern. First, speckle pattern should be such that each facet is different from its neighbor, and, second, multiple pixels should span each speckle in order to avoid aliasing [35]. Based on the work of Rajan et al. [22], the rule-of-thumb for establishing the speckle size is: dsp Z 3 pixels and dsp r df =4, LB UB where the facet size satisfies the bounds, df r df r df . Thus, in the proposed methodology, the lower bound to speckle size is determined based on the magnification of the instrument and the upper bound is determined from an appropriately chosen facet size, established as per the bounding criteria discussed in Sections 4.2 and 4.3. The speckle pattern can be applied on the specimen using various techniques and their size distribution can be characterized using established approaches such as those discussed in [22,35,45]. Characterization of speckle pattern helps confirm that the speckling technique has yielded the desired speckle size distribution per abovementioned requirements. As discussed in Section 2, two methods (“spray” and “airbrush”) were used for speckling in this work and ImageJ software was used for speckle characterization. 4.5. Step size and filtering In order to uniquely identify each facet during the course of deformation and subsequently obtain correlation, some overlap is necessary between the adjacent facets. This overlap distance is inversely proportional to the distance between centroids of adjacent facets, which is called facet step (Fig. 2). Displacement and strain associated with each facet are assigned to the centroidal point of the facet. Error in strain is associated with facet step: large facet steps lead to a loss in spatial resolution of displacement and strain measurements. On the other hand, small facet steps lead to noise and larger strain errors. As strain is calculated by numerical differentiation of displacement field, noise in the displacement measurement is further amplified in strain calculations. Filtering or averaging is typically applied over a certain filtering length to reduce such noise. However, filtering can reduce the accuracy with which strain can be captured, especially in the regions of high strain gradients. Filter length (df il ) and facet step (dst ) can together influence the accuracy and fidelity of calculated strains. Rajan et al. [22] showed that if facet step is smaller than facet size, filtering is necessary to reduce the strain error. Furthermore, in this case, there is an intricate interplay between facet size, facet step, and filter length requiring careful determination of these parameters for high fidelity strain measurements. In the present work, we will not use any filtering but instead establish our framework based upon facet size

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Step 4: Establish facet step as, dst ¼ df =2, and choose no filtering. Effect of filtering was not studied in this work and hence this recommendation. Filtering may or may not affect the other parameters. Step 5: Determine acceptable average speckle size to follow the bounding relation: ð3 pixelsÞ r dsp r df =4, and determine an appropriate speckling technique that is likely to provide the required speckle size. Once the speckle pattern is applied on the specimen, the speckle pattern can be characterized to confirm whether acceptable speckle pattern and size have been obtained or not. It should be noted that the proposed methodology is valid for macroscopic strains and not for strain determination at microstructural length-scales.

5. Demonstration and validation of the methodology Fig. 9. Effect of facet step on strain variation for a macroscopically uniform applied strain (  0.1%) in the elastic deformation regime obtained from the specimen without hole as shown in Fig. 4 (Note: dst and df are specified in pixels).

and facet step. Fig. 9 shows variation of εyy from DIC data along path A–B (see Fig. 4) at an applied macroscopic strain of 0.1% within the linear elastic regime at a constant facet size of 60 pixels for various facet steps (5, 30 and 55 pixels). As discussed earlier, because deformation is homogeneous, the macroscopic applied strain should be relatively constant along the y-coordinate on path A–B at 0.1% axial elastic strain. When the facet step is small (5 pixels, which translates to very large overlaps between facets), the strain from DIC data shows large fluctuations. Standard deviation of strain data is reduced by at least one order of magnitude when facet step is increased to at least half of the facet size (df =2), beyond which there is an insignificant reduction in strain fluctuations; however, strain resolution decreases as facet step increases beyond df =2, because less number of facets are available for correlation, therefore resolution is reduced. Thus, in order to maintain requisite fidelity in strain measurements on one hand, and resolution (most important where strain gradients are present) on the other hand, facet step is recommended to be half of facet size, i.e., dst ¼ df =2. However, if facet step is chosen to be greater than half of facet size, i.e., dst 4 df =2, strain fluctuations will be minimized but resolution is decreased; this might be beneficial where no strain gradients are present because it can potentially decrease computation/correlation time involved during post-processing.

4.6. Summary of the proposed methodology The following steps summarize our proposed methodology for obtaining high-fidelity quantitative strains from DIC experiments: Step 1: Estimate the size of the microstructural features, average grain size (dg ) in the present case, and determine LB lower bound to facet size as, df  4dg . Step 2: Estimate average strain gradient (εS:G: ) in the problem of interest as well as acceptable error in strain (εa ). Average strain gradient may be estimated based on prior experiments or analysis. Acceptable error in strain is dependent on application and a rough guideline is that it should be at least one order of magnitude smaller than expected maximum strain. Based on this information, determine upper bound to facet size as, UB df ¼ εa =εS:G: . Step 3: Establish the facet size between the lower and the upper LB UB UB LB LB bounds: df r df rdf . If df o df , then df ¼ df .

In order to demonstrate and validate the proposed methodology established above, we consider strain concentration ahead of a circular hole in a flat aluminum specimen subjected to uniaxial tension. Furthermore, we consider both linear elastic and plastic regimes of deformation to validate our methodology. The strain in the y-direction along a horizontal line ahead of the hole is determined from DIC data and compared against linear elastic and elastoplastic FEA at the same level of applied macroscopic strain. 5.1. Elastic deformation regime Following the methodology outlined in Section 4, we determined average grain size of the aluminum sample used for making the specimens. Using OIM data, the average grain size was found to be 10 75 pixels, which translates to 57 728.5 μm for the magnification used. Thus, the lower bound for facet size is: LB df  4dg ¼ 40  60 pixels ¼ 228 342 μm. This lower bound estimate is only a function of grain size and hence remains same for both elastic and plastic deformation regimes. In order to determine the upper bound to the facet size we need to establish average strain gradient and acceptable error in maximum strain. In the experiment involving uniaxial deformation of the specimen with a hole, the maximum strain in the vertical direction, εyy , occurs at the edge of the hole. We first consider elastic deformation regime where a vertical strain, εapp ¼ 0:07%, was applied. In this linear elastic regime, the stress concentration factor, and correspondingly, the strain concentration factor in front of a circular hole in an infinite plate is 3.0. The far field strain will be equal to the applied uniform strain, εapp . Thus, the average strain gradient over half width of the specimen is approximately, εS:G:  2εapp = ð0:5W RÞ, where W is the specimen width and R is the radius of the hole. Note that the strain gradient is approximate as the strain concentration for the real specimen will be different from the infinite plate solution due to finite width and 3D effects. Also note that the true strain gradient (slope of the strain plot) in the vicinity of the hole will be higher, but as discussed in Section 4.3, we consider an average strain gradient, which still provides useful information for selecting the facet size. The upper bound to the facet size is given by UB df  εa =εS:G: ¼ ð0:5W  RÞεa =ð2εapp Þ. Now for the specimen under consideration, the dimensions are W ¼ 12:5 mm and R ¼ 1:5 mm. Assuming an acceptable level of strain error in the elastic deformation regime to be εa ¼ 0:01% (this is approximately one order of magnitude smaller than the maximum strain of 3  0.07%¼0.21%), UB the upper bound to facet size is estimated as df  0:4 mm. Furthermore, in the experiments, the magnification was such that 1 pixel corresponded to 5.7  10  3 mm. Thus, based on these

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Fig. 10. Length-scales associated with microstructure, speckling, and DIC parameters, and schematic of the specimen with open-hole geometry for the specimen speckled using the spray method; also shown is the microstructure overlaid on the speckle pattern wherein white squares represent facets chosen for this experiment.

Fig. 12. Length-scales associated with microstructure, speckling, and DIC parameters, and schematic of the specimen with open-hole geometry for the specimen speckled using the airbrush method; also shown is the microstructure overlaid on the speckle pattern wherein white squares represent facets chosen for this experiment.

0.15

0.3

Applied far field Strain () ~ 0.07%

Applied far field Strain () ~ 0.1% 0.25

FEA 0.1

DIC

FEA

Strain (%)

Strain (%)

0.2

0.05

DIC

0.15 0.1

C

0

401

D

0.05 0

0

1

2

3

4

5

Length [C-D] (mm)

E

0

1

2

3

4

F

5

Length [E-F] (mm)

Fig. 11. Comparison of vertical strain (εyy ) from DIC and FEA along a path C–D in front of the hole as shown in the inset.

Fig. 13. Comparison of vertical strain (εyy ) from DIC and FEA along a path E–F in front of the hole as shown in the inset.

bounding considerations, the facet size should follow 0:34 mm rdf r 0:4 mm or, equivalently, 60 pixels r df r 70 pixels. The facet step is then determined as dst ¼ 0:5df ¼ 30  35 pixels ¼ 170  200 μm. Furthermore, following the guidelines, the average speckle size should satisfy: 3 pixels r dsp r ð15  18Þ pixels or, equivalently, 17 μm r dsp r ð86  103Þμm. Experiments on the specimen with circular hole were conducted using two different speckling techniques. The speckle size distributions were characterized for both techniques. The first technique (spray method) resulted in the 90th percentile speckle size of dsp ¼ d90th  50 pixels ¼ 285 μm. This is clearly out of the range derived using the guidelines discussed in the previous paragraph. In actual workflow, such speckle pattern should be rejected and an alternate method must be sought that yields a speckle size within the established bound before proceeding to testing. However, to illustrate how DIC results would compare against FE prediction, we tested the specimen with this coarse speckle pattern. As the speckle size is large, facet size will have to be larger than the recommended guidelines as well so that enough speckles are present in each facet for correlation. Assuming that the facet size should span at least 3–4 speckles, a facet size of 200 pixels (1.14 mm) is chosen. The facet step should then be 100 pixels (0.57 mm). Fig. 10 shows the microstructure overlaid on the speckle pattern, and length-scales associated with this speckling technique. The vertical strain (εyy ) in front of the hole as obtained

from DIC is compared with FE results in Fig. 11. It is clear that strain in the vicinity of the hole is not captured accurately by DIC experiments, and DIC strain is approximately one order of magnitude smaller than the reference linear elastic FEA solution. The second specimen was speckled with a different speckling approach (airbrush method) that resulted in the 90th percentile speckle size of dsp ¼ d90th  15 pixels ¼ 85:5 μm. This speckle size satisfies the bounds discussed previously. Now following the earlier guidelines we choose the facet size df ¼ 60 pixels (0.342 mm) and the facet step dst ¼ 0:5df ¼ 30 pixels (0.171 mm). Thus, in this case, all the control parameters associated with the DIC measurements are within the recommended bounds and a better correlation between the experimental data obtained from DIC and FEA is expected. The microstructure, speckle pattern, and length-scales associated with this specimen are shown in Fig. 12. Fig. 13 compares vertical strain as measured by DIC and linear elastic FEA. It is obvious that the entire strain distribution is accurately captured by DIC with judicious and systematic selection of speckle pattern and corresponding DIC post-processing parameters.

5.2. Plastic deformation regime The lower bound to the facet size is a function of grain size and hence should be independent of the deformation regimes.

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12

Applied far field Strain () ~ 1.8% 10

Strain (%)

8

FEA DIC

6 4 2 0

E

0

1

2 3 Length [E-F] (mm)

4

F

5

Fig. 14. Comparison of vertical strain (εyy ) from DIC and FEA along a path E–F in front of the hole as shown in the inset.

LB

Thus, similar to the elastic regime, df  4dg ¼ 40 60 pixels¼ 228–342 μm. In order to compute the upper bound to the facet size, once again we consider average strain gradient in front of the hole. Comparison between FEA and experiments will be made at the applied macroscopic strain of  1.8% (corresponding stress of 330 MPa), which is well within the plastic regime of the material behavior. Local strain concentration in front of the hole is expected to be much higher. From FEA, the maximum strain in front of the hole is 10%. Thus the average strain gradient over half width of the specimen is εS:G:  ð0:1  0:018Þ=ð0:5W  RÞ ¼ 0:017 mm  1 . Now assuming the acceptable error to be one order of magnitude smaller than the maximum strain, i.e., εa ¼ 1%, the upper bound to facet size UB is calculated as df  εa =εS:G: ¼ 570μm¼ 100pixels. Thus, the bounds on facet size are described by: 60 pixels r df r100 pixels (342 μm r df r 570 μm). Step size is then determined as dst ¼ 0:5df ¼ 30  50 pixels¼171–285 μm. Following the guidelines, average speckle size should satisfy: 3 pixels r dsp r ð15  25Þ pixels or, equivalently, 17:1 μm rdsp r(85.5–142.5)μm. The airbrush speckling method that provided a fine speckle pattern with 90th percentile speckle size of 15 pixels (85.5 μm) satisfies the bounding requirement stated above for deformation in the plastic regime. Corresponding facet size of 60 pixels (342 μm) and step size of 30 pixels (171 μm) also satisfy the recommendation. Fig. 14 compares vertical strain in front of the hole along path E–F as shown in Fig. 12 from DIC using recommended parameters against reference FEA solution. A reasonable agreement is observed between the two over the entire half-width of the specimen. Peak strain and strain gradient in the vicinity of the hole are well captured. However, some discrepancies are noted away from the hole. The reasons for this discrepancy could be attributed to the fact that in plastic deformation regime, FEA solution itself may not be an accurate representation of material behavior. For example, the J2 plasticity model may not accurately represent material behavior due to intense plastic localization and/ or damage in the highly deformed regions. In any case, the overall strain response from DIC is captured reasonably well for engineering analysis purposes.

high-fidelity quantitative measurements to better understand deformation and fracture is still challenging, specifically when strain gradients and material heterogeneities are present. In this study, we developed a practical methodology for highfidelity full-field strain-mapping of elastic and plastic deformations of materials at macroscopic continuum length-scales under quasi-static loading conditions. The methodology was then validated by comparing DIC results with Finite Element Analysis. It was emphasized that three critical parameters and length scales associated with DIC, namely speckle size, facet size, and facet step were linked to microstructural length-scales inherent to the material under study. A lower bound to facet size was defined that takes into account microstructural features of the material. Furthermore, an upper bound to the facet size was defined in order to accurately capture strains in the vicinity of high deformation gradients and to maintain requisite fidelity. This is essential because most engineering problems of interest deal with nonuniform deformations due to the presence of stress-risers and complex loading conditions in structural components. Subsequently, in order to maintain optimal balance between fidelity and resolution of strain measurements, a bound on facet step was also determined. Finally, an acceptable size of speckle pattern was determined as a natural outcome of the bounding relations determined for facet size and step size, and it was recommended that the speckle pattern be characterized via image analysis methods to ensure compliance with the requirements. It was demonstrated that when DIC parameters are not selected carefully, strain in the vicinity of high deformation gradients were not captured accurately, and in some cases DIC strains were approximately one order of magnitude smaller than a reference linear elastic FEA solution. On the other hand, when all the associated DIC parameters were within the recommended bounds defined in our methodology, a reasonable agreement between the experimental data obtained from DIC and FEA was achieved for both elastic and plastic deformation regimes. Therefore, it is essential to judiciously and systematically select all the parameters and length-scales associated with DIC in an integrated manner, taking into account the length-scales associated with microstructure and deformation gradient, in order to ensure that the measured strain is of requisite fidelity. The methodology presented in this paper was demonstrated on a specific problem of a plate with a circular hole. Future work could focus on application and refinement of the methodology to other complicated stress/strain concentrators such as a plate with a sharp notch. As a final remark, it should be noted that the proposed methodology should be applicable to other material systems such as fiber-reinforced composites with and without strain concentrators, provided the interest is in resolving strains at the macroscopic length-scales, i.e., length-scale greater than or equal to the size of associated RVE.

Acknowledgments The authors gratefully acknowledge the assistance of Mr. Patrick Clavette and Mr. Daniel Collins with mechanical testing, Ms. Julie Wittenzellner and Mr. Frederick Espinosa for microstructural characterization, and Dr. Mark Vogel for image analysis. The authors would also like to thank Drs. Eric Amis, Bill Tredway, Ellen Sun, GV Srinivasan and Tania Kashyap for their support, discussions and feedback throughout the course of this work.

6. Concluding remarks References Digital Image Correlation (DIC) is a powerful technique for mapping strain and displacement distributions in various materials and structures subjected to complex loading conditions. However,

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