Extracting fracture properties from digital image and volume correlation displacement data: A review

The advent of digital image and volume correlation has attracted wide use in fracture mechanics. The full‐field nature of digital image and volume correlation allows for the integration of computational fracture mechanics to analyse cracked samples quantitatively. This review provides a comprehensive overview of current methods used to extract fracture properties from full‐field displacement data. The term full‐field fracture mechanics is introduced to highlight the uniqueness of using inherently noisy experiential data to extract fracture properties. The review focuses on post‐processing‐based approaches rather than integrated approaches, as these have less limitations and are more commonly employed. There are four approaches that are discussed in extracting fracture properties from experimentally computed displacement data: field‐fitting, integral, crack‐opening and cohesive zone modelling approaches. This is further developed to discuss problems associated with using digital image and volume correlation to extract properties, including application examples.


| INTRODUCTION
The ability to predict fracture is fundamentally important for the reliable performance of components and structures.Design or analysis of such components and structures usually involves considering stress or strain fields in conjunction with a postulate that predicts the event of failure.Criteria are developed to predict failure that relies on describing the material as a continuum and can be roughly classified as ductile or brittle at either extreme.In the former, failure is associated with yielding and plastic flow, providing some sense of warning before failure, whereas in the latter, failure is sudden and catastrophic.Historically, the phenomenon of brittle fracture has been encountered more frequently, even in materials that exhibit ductile behaviour.Efforts to better predict brittle failure have led to the discipline of fracture mechanics.In contrast to conventional failure criteria, fracture mechanics assumes that materials inevitably contain defects.Such defects may form during material processing and manufacturing methods or during mechanical, thermal or corrosive service loading, to name a few.
It is essential to recognise that nondestructive testing methods aim to measure defects to a physical size.For this reason, fracture mechanics is based on the existence of a defect modelled as a crack size, a.This crack is surrounded by defect-free material, which is described using continuum mechanics material laws.However, the discontinuous state across the crack flanks, and the deformation developed at the crack tip which theoretically results in a stress singularity (in an elastic solid), cannot be described using classical continuum mechanics concepts.Fracture mechanics describes this singularity by, for example, the stress intensity factor (SIF), which allows crack fields to be uniquely described.
Fracture mechanics has been established in the last 60 years as an independent scientific discipline and combines knowledge and modelling approaches from mechanical, materials and solid-state physics disciplines.It has become widely accepted and has been incorporated into many design codes.2][3][4] The method enables the processing of digital images captured before and after the application of load to obtain deformations across the field of view.Within areas of continuity, accurate displacement measurements can be obtained from arbitrary selections of subsets that are correlated between image pairs.The resulting displacement vectors occupy the entire field of view, resulting in DIC, often being referred to as a full-field measurement technique.Like fracture mechanics, DIC has become ubiquitous in experiential mechanics and many other applications. 5IC differentiates itself from other optical flow-based optical methods because it focuses on the high accuracy required to study strains exhibited in stiff structures.This typically leverages the continuity assumptions while assuming an underlying deformation field of, typically first-or second-order polynomial deformations.The extensions to two or more cameras, called stereo-DIC, allow for the measurement of out-of-plane deformations (versus purely in-plane or planar).The technique can also be applied within the volume when volumetric images are available (e.g., via X-ray or synchronised imaging), called digital volume correlation (DVC). 6IC and DVC are classified as either local (subset-based) or global.The terms local and global refer to whether the displacements are solved in a modular subset-by-subset type approach or within a framework which enforces continuity between neighbouring displacements, usually within the finite element (FE) framework. 7Global approaches based on the FE method can apply h-adaptivity to increase spatial resolution and p-adaptivity when higher order basis functions are required, [8][9][10] and X-FEM-based approaches in fracture cases. 115][16] The accuracy and efficiency of the two methods must be similar because Hild and Roux 7 reported that global approaches are the more accurate, whereas Wang and Pan 17 reported the opposite as true.A possible advantage of the global approach is that it allows for direct integration with finite-element capabilities, which in local DIC is a postprocessing step. 7Most readily available DIC and DVC approaches are based on local approaches. 5he full-field nature of DIC and DVC allows for complex constitutive equations to be used to determine more than one material property at a time, using methods such as the virtual field method 18 and the FE model updating method. 191][22] These attribute also transfer well to fracture mechanics applications.The fact that displacement uncertainties of the order of $0.01 pixels * are achievable allows DIC and DVC to reveal cracks that cannot be seen if their opening is less than one pixel wide. 23umerical methods can, moreover, compute SIFs from displacement data around a crack tip.For example, crack tip opening angles or displacements are measured with excellent accuracy by employing DIC [24][25][26][27] and DVC. 28,29The computation of SIFs has been reported to use field-fitting, 30,31 integral [32][33][34] or more generalised crack-opening displacement 35 approaches.Alternatively, cohesive zone models (CZMs) have also been computed from displacement data. 36,37In a fieldfitting approach, the computation of SIFs from image data can be undertaken using two different ways, as a two-step process using the computed displacement data and subsequently extracting SIFs, or as an integrated approach (often referred to as IDIC) such as presented by Roux and Hild. 26IDIC uses prior information, such as the constituent laws and the applied boundary conditions, to allow for the direct identification of the relevant material parameters of the model.IDIC typically outperforms a two-step process, as it provided high general accuracy and robustness in terms of measurement uncertainty. 38owever, integral approaches can only be undertaken as a post-pressing step like its application in the FE method.
These techniques, i.e., field-fitting, integral and crack-opening and CZM approaches, can be applied to microscopic 39 and large-scale 40,41 images using the same algorithms.
Although the use of fracture mechanics established techniques to extract SIFs from displacement data is not new with most of their development attributed to the advent of the FEM, intricate nuances arise when applying them to experimentally obtained data.For example, the accuracy achievable with image correlation in continuous regions is reduced in the presence of high-gradient material nonlinearities such as plasticity 42 or fracture. 435][46] Several strategies have been proposed that maximise the remaining data through selective masking strategies. 47,48his review aims to provide an extensive overview of the three postprocessing techniques reported in the literature to extract fracture properties from DIC and DVC computed displacement data.The article provides insight into the assumptions, advantages and limitations of the various methods.It also aims to provide common application examples in extracting fracture toughness and mixed-mode fracture.

| BASIC PRINCIPLES OF DIC AND DVC
Several techniques exist to extract full-field surface deformations; early methods used incoherent light and separate gratings for each deformation state, e.g., moiré. 49Later, coherent light-based methods, such as holographic interferometry, 50 electronic speckle interferometry 51 and moiré interferometry, 52 were developed.These offer high spatial resolution but require specialised equipment and complex sample preparations.White-light methods, such as the grid method and DIC, 53 have gained popularity due to their simpler setup.The grid method requires regular predetermined patterns, while DIC processes random patterns, making it less prescriptive in terms of sample preparation.DIC stands out from other methods, as it focusses on high accuracy for studying strains in stiff structures, leveraging assumptions of the materials mechanics.As such, DIC has become synonymous in experimental mechanics, as reflected in the huge body of literature available on this technique. 5For this reason, we focus on DIC in the following section.However, it is worth noting that the subsequently presented methods for extracting compute SIFs from displacement data are applicable to any of the above-mentioned methods, provided sufficient accuracy is achievable.
DIC consists of four steps, namely, (i) calibration, which involves determining the parameters that relate the realworld coordinates to that of the image sensor coordinates; (ii) correlation, which performs the mapping of subsets between image pairs to obtain the displacement field data; (iii) displacement transformation, which uses the calibration parameters to transform sensor displacements determined by the correlation step to the real world; and (iv) strain computation from the derivatives of the displacement fields.DVC differs in that reconstruction of the volume includes calibration, so calibration and displacement transformation steps are unnecessary.
Several excellent review articles describe the steps mentioned above in detail for DIC 5,54,55 and DVC. 6We highlight our previous work, which provides a detailed outline of both planar- 12 and stereo-DIC 13 and its implementation in code using the inverse compositional Gauss-Newton method, which is considered to be the current state-of-the-art optimisation method that offers the best accuracy, noise resistance and computational efficiency in practice. 56,57riefly, image correlation ‡ considers two images, a reference and a deformed image.The reference image is broken up into subsets, which are groups of neighbouring pixels.Conceptually, correlation attempts to map a reference subset to a corresponding subset in the deformed image using a predefined shape function.Since images are discrete (i.e., measurements are only taken at pixel or voxel locations), interpolation is used to match light intensities at noninteger locations, making correlation a complex problem.
DIC and DVC require a choice of subset size and the shape function order to set up the analysis (in global approaches, these are termed mesh size).Correlation then determines the displacement at a query point (typically the subset centre), which dictates the displacement as the average of that experienced by pixels within the subset.Accuracy and computational efficiency are high if the subset regions exhibit low deformation gradients.This allows for the selection of large subsets while maintaining a low order of deformation complexity.However, a limit is reached when increasing the subset size further yields a reduction in accuracy.This occurs when the subset contains deformations that cannot be fully accounted for by the allowable deformations prescribed by the shape function.Complex deformations are mainly the result of geometric boundaries, damage, complex loading, stress concentrations and various unwanted artefacts.asking implies removing data from the analysis altogether.
‡ We recognise that DIC and DVC are fundamentally similar.In DIC, measurements are taken on the surface using digital images that are described in pixels.DVC is the volumetric equivalent; however, images are based on volumetric reconstructions and are described in voxels.We use the term DIC to both encompass planar and stereo deformation measurements and DVC to encompass volumetric deformation measurements.
It is challenging to ascertain the 'best' subset size and shape function order to minimise displacement errors.An example of this is shown in Figure 1, where a smaller subset size leads to lower systematic errors due to subset smearing across the crack flanks, and a larger subset size leads to better random noise suppression.9][60] For example, the International DIC Society (iDICs) provides a 'good practice guide', 62,63  This dilemma is a pertinent issue when employing displacement measurements in fracture mechanics, in which high accuracy is needed for the various approaches to extract fracture parameters.Crack tip displacement fields are discontinuous along the crack faces and asymptotic as they approach the crack tip, leading to significant inaccuracies and failed correlations over the crack region.Reducing the subset size leads to a reduction in accuracy in this region, which is worsened if the nonlinear effects of damage have spread ahead of the crack tip. 64Some examples that address these issues are the subset-splitting-based strategy to enable crack face discontinuities first introduced by Poissant and Barthelat, 14 regularisation to reduce errors in regions of high deformation gradients and continuity 65 or the inclusion of analytical crack tip fields in the basis set of the correlation. 11,66,67Baldi 68 proposed the use of approach for both displacement (and strain) computations across discontinuities using principle of RANSAC (Random Sample and Consensus, an iterative algorithm used to estimate parameters from a set of observed data that contains outliers) to isolate the dominant statistical population within the subset.Despite these advances, such methodologies still assume some form of material continuity or have yet to be established in readily available DIC and DVC software packages.
The resultant uncertainty in displacement measurement is divided into two types of errors, namely, variance and bias. 62Variance refers to a random error, typically normally distributed with the mean representing the true value, and bias refers to a systematic error that shifts the mean value from the true value.Although the variance is representative of the measurement noise in the system, for example, from image sensor noise, bias occurs from factors that include smoothing, calibration, or out-of-plane motion in the case of planar-DIC to name a few.Ideally, bias is removed from the measurement in the analysis to achieve better accuracy.Variance, on the other hand, is an indicator of the precision of the experimental measurement system.

| FULL-FIELD FRACTURE MECHANICS
In fracture mechanics, the behaviour of the crack is described from a macroscopic point of view in the context of continuum mechanics.Under the assumption of infinitesimal strain theory, a distinction in its deformation is drawn between three independent movements of the two crack faces relative to each other; crack opening modes, i.e., Modes I-III, are defined as crack opening, in-plane sliding and out-of-plane sliding, respectively.Complex crack deformation can be regarded as a superposition of the three modes. 69This important feature allows for mode-specific SIF extraction, as we will see later.Only Modes I and II are observable in planar-DIC.In 3D (i.e., stereo-DIC and DVC), all modes are discernible.Furthermore, in DVC, SIFs are considered at discrete points along the crack front, as shown in Figure 2c.
In linear elastic fracture mechanics (LEFM), the deformation behaviour is assumed linear elastic according to the generalised Hooke's law.However, significant nonlinearities occur around the crack tip (which, in many cases, are limited by an assumption of small-scale yielding, SSY § ) and may require elastic plastic fracture mechanics (EPFM) formulations. 69Historically, LEFM used the Williams ¶ series that uniquely describes the stress field around a crack tip with a single parameter, K, giving rise to the well-known relation , where K I is the mode I SIF, σ is the applied stress, a is the crack length and Y is a geometry function to allow for different loading configurations.Under large-scale nonlinearities, two other parameters are used: the crack tip opening displacement, δ, and the J-integral, J.
Traditionally employed standardised testing aims to yield a single SIF value or fracture resistance curve. 69Test configurations have been extensively developed to allow for test conditions to measure the applied load and, in some instances, crack-opening displacements.Crack length measurements and extensions are either measured postmortem, inferred through compliance relations or measured by techniques such as potential drop methods.Estimates of SIFs are then based on the weight functions that relate a, the applied load and sample geometry to a SIF.
The use of DIC and DVC in fracture investigations provides an additional tool to describe the fracture process and thus gain deeper insight into the behaviour of the materials.We use the term full-field fracture mechanics to emphasise that full-field deformation data are used.In full-field fracture mechanics, only the local displacement field and material constants are needed (e.g., Young's modulus, E, and poisons ratio, v) and the approximate crack location and orientation to compute SIFs.No load and exact crack length measurements are necessarily needed.This unique ability makes full-field fracture mechanics highly desirable.
As with any measurement technique, nuances must be discussed before introducing the respective methods to extract SIFs.
We consider the crack front depicted in Figure 2. Three steps are typically required for the computation of SIFs from full-field displacement data: (i) defining a local crack tip coordinate system, (ii) masking of nonphysical data along the crack flanks and (iii) SIF extraction.In DVC, crack-tip segmentation is required as a precursor to these steps to define the crack front (s) location for SIF extraction, as illustrated in Figure 2c.
A local coordinate system is defined such that the crack tip is located at the origin, x 1 is aligned with the crack plane and is assumed to be the same as the direction of crack propagation, x 2 is normal to the crack plane and x 3 is aligned to the crack front.In planar-and stereo-DIC, x 3 is assumed to be normal to the sample surface.This, however, assumes the crack front is perpendicular to the surface, which may not always be the case.Locating the crack tip precisely and hence the coordinate system's origin from experimental data is challenging.Thus, SIF extraction involves either optimisation of the crack tip location, as in field-fitting techniques, or crack tip location independent formulations, such as energy integral methods.
Masking of the crack flanks is common practice to omit regions with high strain gradients or poor correlation.A straightforward approach to remove unreliable data is visually by selecting and masking data around the crack.Alternatively, thresholding data from the correlation coefficient can be used to eliminate crack-biassed data. 70The study by Molteno and Becker 33 highlighted that in Mode I loading, SIFs are relatively unaffected by masking; however, under mixed-mode loading, data along the crack flanks contain high strain energy densities that can significantly affect the SIF computation, 33 mainly if a J-integral approach is used.Interpolation of the masked data is recommended using either polynomial fitting Garcia 71 or FE-based filters, 44,72 or alternatively using RANSAC as proposed by Bardi. 68IF extraction uses displacement data around the crack tip.Since the exact location of the crack tip is seldom precisely known and the computation of displacement data is erroneous close to the crack tip and the crack flanks, fullfield approaches aim to use displacement data that are some distance from the location of the crack tip.This contrasts with FEM, where great effort is spent to best resolve crack tip deformations using special quadratic elements with quarter nodes and collapsed elements. 735][76] Roux and Hild 26,75 discussed strategies for locating the crack tip using the amplitudes associated with the Eigen functions for Modes I and II.In so doing, they reported an uncertainty of crack tip position of $20 μm for a crack of 15 mm.It is worth noting that Roux and Hild assume a linear elastic behaviour, and therefore, the measured crack tip is an Local crack tip coordinate system, x 1 , x 2 and x 3 for (a and b) DIC and (c) DVC, highlighting the line (a), area (b) and volumetric (c) formulations for the J-integral. 32,34Uppercase X i represented the global coordinate system.Reproduced with permission equivalent elastic crack tip.Feld-Payet et al. 77 proposed empirical criterion uses the evolution of the standard deviation of the displacement gradient along a potential crack path to locate the tip of the crack.The method is reported to be robust enough to deal with materials with not perfectly straight crack paths (due to the loading conditions and/or the microstructure) and blunted crack fronts.Alternatively, Sciuti et al. 78 use an adaptive meshing technique to better capture displacements near discontinuities, such as cracks, and thereby the ability to more accurately locate the crack tip.
Full-field fracture mechanics employs the Williams series using a field-fitting approach, the J-integral by integrating stress-strain data around the tip of the crack or δ approach by measuring displacements over the crack flanks.Recent attempts to derive CZM laws from experimental measurements have also been proposed by mapping stress softening ahead of the crack tip.
The field-fitting and J-integral approaches are reported most.This is because measuring the displacement accurately across the crack flanks and knowing the exact position of the crack tip is difficult.Field-fitting approaches are limited to LEFM under SSY conditions, although Yoneyama et al. 79 have adopted the field-fitting for EPFM conditions using Hutchinson, Rice, Rosengren (HRR) fields.Under LEFM conditions, the field-fitting approach relies on fitting the Williams series to the measured displacement field.J-integral approaches can be employed for both LEFM and EPFM; a linearelastic material model can still be assumed as long as contours are taken sufficiently far away from the crack tip.Largescale plasticity requires complex material models that allow stress-strain relationships under plastic deformations. 80

| EXTRACTING SIFS
In analytical form, the SIFs are the 1st-order coefficients of a series expansion describing the stress and displacement fields surrounding the crack tip. 81The equations can be fitted to experimental data to obtain SIFs, as was first shown by McNeil et al. 82 and later by Yoneyama et al. 30,31 for mixed-mode SIFs and higher-order terms.The effects of anisotropy on SIFs were studied 83,84 in single-crystal materials, where an anisotropic least squares regression algorithm was used to find SIFs.The inclusion of T-stress and higher order terms was shown to improve the accuracy of the regression. 85he J-integral is based on the formulation of the elastic energy-momentum tensor, introduced by Eshelby. 86The use of the J-integral on cracks was first suggested by Rice, 87 with the key result that J values may be reliably calculated on arbitrary contours encircling the crack tip.34]79 In these approaches, the fields required for integration (stresses, strains and displacement gradient tensors) have been computed directly from the experimental displacement data 33,34,79 or calculated within a numerical framework by applying the displacements as boundary conditions (e.g., using FEM 32 ).
Wells 88 first introduced the concept of δ, which describes the distance between crack faces before fracture.Plastic deformation blunts an initially sharp crack prior to fracture; the degree of crack blunting, quantified using δ, is seen as the toughness of the material.In this approach, the displacement across the crack flanks is computed and related to δ.

| Field-fitting approach
The stress distribution around the crack tip relative to the applied stress was described by Williams 81 and is usually referred to as the Williams series.The Williams series expresses the stress, strain or displacement fields in terms of elastic constants (the shear modulus, μ, and the Kolosov constant, κ) and the coefficients for the number of terms considered in the series expansion of modes, M, one to three (A n ð Þ IÀIII ).Its derivation assumes an infinite centre cracked plate under linear elastic and isotropic conditions with traction free crack flanks.The displacement fields in u 1 , u 2 and u 3 are given with respect to r, θ (the polar coordinates of x 1 and x 2 ) measured along the crack front s by where, M are polar functions given by , and μ and κ are typically written in terms of E and v, so that μ ¼ E=2 1þ ν ð Þ and, for surface measurements, i.e., DIC, a state of plane stress is assumed with , whereas for volumetric measurements, i.e., DVC, a state of plane strain is assumed with κ ¼ 3 À 4v.Commonly, Equations ( 2) is written as a function θ only.Including r, as well as the elastic constants, allows for the simple linearisation required for field-fitting.Equation ( 1) can be rewritten as u ¼ C r,θ, s ð Þa, where u is an appended displacement vector; C corresponds to M , and h n ð Þ M ; and a provides the respective coefficients of the Williams expansion-for further details, see Ayatollahi et al. 89 Although seldom reported in the literature, fitting Mode III data are also possible. 90he beauty of the field-fitting approach lies in the ability to naturally include mixed-mode and higher order terms in the solution.SIFs are extracted by fitting the Williams expansion to the experimentally measured displacement data.The unknown Williams coefficients can be calculated in the least-square sense using a ¼ Irwin 91 showed that A III relate to the SIFs as The T-stress is simply 4A 2 ð Þ I .Higher order terms for each mode M (i.e., A M ) dominate at increasing distances from the crack tip with increasing n.Notably, A 2 ð Þ II describes a rigid body rotation (about x 3 ), and similarly, A 2 ð Þ III describes a constant shear stress, τ 13 .# By setting n ¼ 0, the terms corresponding to A 0 ð Þ IÀIII are independent of r and θ and can be taken as the rigid body translations.
In practicality, when using experimentally measured DIC or DVC displacement data, the field-fitting approach requires further consideration: First, using higher order terms makes it possible to use displacement data far from the crack tip, thereby removing the need to determine displacement data at the crack tip accurately.Moreover, to allow for a more accurate solution that is independent of the selected displacement data domain, the number of measurement points (i.e., DIC subsets) chosen should be greater than the number required to solve the set of equations (i.e., 2k > 2N þ 2, where k is the number of data points in the case of planar DIC and N is the total number of terms considered).This leads to an overdetermined set of equations and improved noise robustness.
Second, the precise location of the crack tip is not known.Accounting for the exact location of the crack tip is important in extracting accurate SIFs.For example, Yates et al. 92 evaluated the effects of crack tip position on the computed SIFs and found that underestimation of crack length increased SIFs.Accounting for the location of the crack tip requires modifying Equation (2) to allow for unknown coordinates of the crack tip, given in the polar coordinates as III would appear as a rigid body rotation about x 1 .However, this is not the case as with x ip being the coordinate of the crack tip, given by Here, x ic is a correction applied in the assumed location of the crack tip.Note that in DVC, x 3 depends on the location of the crack front along s.This results in the fitting procedure becoming nonlinear and thus making linear least squares fitting not possible.Various solution methods have been reported; Yoneyama et al. 30,31 proposed a Newton-Raphson nonlinear least squares solver to solve simultaneously for A n ð Þ IÀII and x ic .Other solvers have been investigated by Harilal et al. 93 using an exhaustive search, thereby maintaining the more efficient linear least-squares solver, and by Zanganeh et al. 94 including Nelder-Mead simplex, 95 pattern search 96 and genetic algorithms. 97In all cases, a cost function is defined that is typically related to the sum of the absolute differences, SAD (although Yoneyama et al. 31 reported both a SAD and a normalised correlation criterion, NCC).
In-house studies have shown that a nonlinear least-squares solver is generally more stable, although otherwise reported by Harilal et al. 93 and Zanganeh et al. 94 Harilal et al.'s approach seems highly dependent on the search window and its step size.For the nonlinear least-squares solver, when including several higher order terms, results in a multivariable optimisation problem.Considering five terms, for example, equates to 12 or 17 optimisation variables, depending on whether Modes I to II or I to III are considered, respectively.This may result in unstable solutions; in some cases, the optimisation does not converge to a reasonable result.Subsequently, Yoneyama et al. 30 sequentially increase the number of terms; the initial crack tip location is solved for using a series with N ¼ 1.For N greater than two, the SIFs and crack-tip location computed in previous iterations are used as the initial guess for the subsequent addition of terms, allowing for a more stable optimisation problem.Moreover, the optimisation of higher order terms is often poorly scaled; the first-order terms, directly related to K IÀIII , are typically several orders of magnitude larger than the higher order terms.This may necessitate more complex nonlinear optimisers, such as the Levenberg-Marquardt-Fletcher method, 98 which allows for scaling of the Jacobian matrix.
Third, DIC and DVC displacement data inherently include rigid body translations and rotations.These need to be considered in the fitting procedure, including translation and rotation terms in Equation (2).We need to differentiate between planar-DIC and stereo-DIC or DVC.In planar-DIC, rigid body translations are in x 1 and x 2 , and rotations are about x 3 .These rigid body translations and rotations are inherently included in the Williams series, as II , respectively, and adding additional terms to account for rigid body motion in Equation ( 2), as previously done, 30 is folly.In stereo-DIC or DVC, rigid body translations are included in the expansion terms, i.e., by A 0 ð Þ IÀIII ; however, rotations about x 1 and x 2 are not present; a series cannot establish rotations in three dimensions using a series of linear terms.Thus, rigid body rotations should be included in the optimisation problem in stereo-DIC or DVC.A more pragmatic approach is removing rigid body rotations before fitting using approaches suggested by Mostafavi et al. 99 or singular value decomposition. 100 The general procedure for the field-fitting approach is to start with a best estimate of the crack tip location and to define the domain (i.e., the measurement points to be considered) for the fitting procedure.SIF extraction is generally reported to consider several terms in the Williams formula to illustrate convergence.Typically, five terms are reported, although some argue that the use of higher than three terms does not improve the accuracy of the fitted values of SIF. 101Some authors have reported that measurement points should be located 2/3rd ahead and 1/3rd behind the crack in x 1 . 92,102An example of fitted displacement data is shown in Figure 3 using seven terms.
It should be noted that because the Williams series relies on the assumptions of a linear elastic material, the obtained location and orientation of the crack tip (using Equations ( 4) and ( 5)) are equivalent to that of an elastic crack tip, that is, the position of the crack tip in an elastic medium leading to the measured displacements.
An advantage of the field-fitting approach is that analytical fields may be fitted directly to experimental displacement fields without computing mesh parameters or stresses, and it naturally extends to mixed-mode applications.This makes the field-fitting approach more noise-robust, especially if the fitting procedure uses a large domain.The limitation of the field-fitting approach is the requirement for reasonably accurate crack tip location estimates.

| Energy integral approaches
The J-integral was first introduced by Cherepanov 103 and Rice 87 as a two-dimensional line integral criterion to describe the strain energy release rate for crack growth in linear elastic or elastic-plastic materials.An essential property of the J-integral is its contour independence; all contours surrounding a notch or crack tip provide the same value.Initially, its application was limited to loaded cracks with no internal stress/strains or edge traction 104 and no significant plasticity. 105The classical form of the J-integral is restricted to linear and non-linear (monotonic loading) deformation theory.Nevertheless, with modifications, the J-integral can be applied to other conditions, including EPFM conditions including incremental plasticity theory and large deformations. 106The formulation of the J-integral easily facilitates its evaluation with FE methods, typically by application of the divergence theorem to form an equivalent domain integral, as outlined later. 107he classical form of the J-integral is given in Equation ( 7) in indicial notation with respect to the crack coordinate system shown in Figure 2a along the crack front s.
where Γ is an arbitrary contour path that typically proceeds in the clockwise direction, W is the strain energy density, n k is the outward unit normal along the contour, T i ¼ σ ij n j and is the traction on Γ with σ ij as the Cauchy stress tensor, u i are the displacement vector components, and J k is J-integral evaluated along the The vectorial presentation of J k leads to numerical difficulties in the analysis in mixed mode loading.Budiansky and Rice 108 and Hussain et al. 109 argued that J k represents specific energy release rates related to the movements of the crack surfaces along x k .The J-integral thus cannot distinguish between the portions contributed by the opening and shearing modes, which is a disadvantage when studying mixed mode crack propagation.
To simplify the J-integral in a way that makes J equivalent to the SIFs, while avoiding the difficulty of the physical interpretations of J k , [110][111][112][113][114][115] the analysis can be resolved to a coordinate system consistent with the crack geometry and F I G U R E 3 Experimental (red markers) and reconstructed displacement fields obtained for seven terms under mixed mode loading.(a) u 1 and (b) u 2 in mm. 93Reproduced with permission a predefined direction of crack propagation, i.e., k ¼ 1, which assumes the crack advances only in x 1 .The total energy released per unit crack extension is then equal to the sum of the independent J values obtained from Modes I to III, by Equation ( 8) is an underdetermined system.Unless loading is in a pure mode-for example, in Mode I, the computation of J 1 gives directly J I , so that K I ¼ ffiffiffiffiffiffiffi ffi J 1 E p (plane stress, i.e., DIC) or (plane strain, i.e., DVC)mode-specific SIFs cannot be directly obtained.Two approaches have been proposed to facilitate mixed-mode analysis; the interaction integral, introduced by Stern, 116 and the decomposition method, introduced by Ishikawa et al. 117 The interaction integral uses the Maxwell-Betti reciprocal theorem to decouple the J values into separate SIFs.In this approach, direct use is usually made of analytical crack tip fields to provide physically permissible auxiliary fields, 118 for which the placement and definition on the surface or in the volume are essentially prone to the same challenges as the field-fitting approach.Importantly, the errors due to crack tip position uncertainty will strongly affect the accuracy of mixed mode extraction and have not been proposed for use in DIC or DVC displacement data but only as an integrated DIC approach. 90

| Mode decomposition
The decomposition method is derived from the observation that the singular stress terms of the Williams series expansion can be directly separated based on their unique symmetry around the crack plane under the assumption of infinitesimal strain theory.0][121][122][123] Molteno and Becker 33 first reported the use of the decomposition method to extract mixed SIFs using the J-integral on DIC displacement data by separating the displacement field in the same manner that any function may be separated into symmetric (sym) and antisymmetric (as) parts as where the notation u 0 i represents a displacement field that has been reflected about the crack plane.The symmetric part equates to Mode I loading.The antisymmetric modes represent combined Modes II and III and can be divided.Extraction of the three modes is thus possible by considering the mode-dependent displacement data, i.e., u ¼ Assuming that stresses are linear-elastic functions of strain, Equation (10), with k = 1, is used to obtain the modespecific J-integral, i.e., with elastic stresses, strains and strain energy derived from decomposed displacement data u M i with and The components of Mode I have full entries (for i = 1, 2, 3), whereas the antisymmetric entries are shared between Modes II and III.When symmetric and antisymmetric displacement gradients are desired, summation of the antisymmetric terms yields a compact form using traditional gradients to differentiate Equation (10).Decomposition of the antisymmetric term, i.e., the decomposition of the out-of-plane components, has been debated by Nikishkov and Atluri, 119 Shivakumar and Raju, 122 Huber et al., 121 and Rigby and Aliabadi. 123n DIC, displacement measurements are on the surface, and Mode III anti-plane shear loading requires that the tractions are zero, i.e., ∂u III i,3 ¼ 0 for i ¼ 1, 2. 119 Neglecting these components allows for the separation of Modes II and III. 33The decomposition of the J-integral was applied to mixed-mode loading using an Arcan geometry to extract mode-specific SIFs.The example is shown in Figure 4.
In DVC, where complex strain fields exist, decomposition leads to a loss of stress equilibrium.Rigby and Aliabadi 123 provide this reasoning directly by manipulating the stress terms in the Eshelby tensor.In their work, it is shown that the usual commutative property of partial derivatives used to derive strain does not apply between Modes II and III; that is, Thus, the full decomposition of displacement data is not possible for volumetric data; separation is only possible into J 1I and J 1IIÀIII .
The decomposition method has the drawback in that fields emanating from other nearby stress sources, such as edges or neighbouring cracks, can become involved in the symmetry operation, resulting in loss of stress equilibrium as discussed by Shivakumar and Raju. 122Furthermore, accurate strain measurements are required at the crack flanks, where DIC and DVC perform poorly due to subset smearing.This requires careful consideration of how strains are extracted from displacement data along the crack flanks.

| Equivalent domain integral
Equation ( 7) can be used to extract J from planar stereo volumetric problems.However, its formulation as a line integral is not numerically robust, such that equivalent domain integral forms have been introduced. 115This approach transforms the 2D path integral into the area integral and the 3D path-area integral into a volume integral.For displacement data computed by DIC and DVC, the equivalent domain offers a significant advantage by allowing more data points to be considered, thereby reducing the noise in the computation of J. 32 For an equivalent domain integral, the crack front is assumed to be approximately straight, ** over the distance Δs, as shown in Figure 2c. 125A crack front function, q s ð Þ, is introduced that is zero outside the arc of interest and defines a virtual crack extension area along Δs as A c ¼ R q s ð Þds.Furthermore, a virtual crack extension field, Q, is defined over an integration volume, V , that is equal to q(s) on the crack front, zero on the outer boundary, and arbitrary and differentiable within.Q essentially implies a virtual crack extension aligned with the x 1 axis.Examples of Q functions are shown in Figure 5.Using the divergence theorem allows for the formulation of the classical volume integral for quasistatic planar straight cracks in isotropic materials, 115 which is given by **Alternate curves crack front formulations have been presented, for example, by Benhamena et al. 124 where, δ is the Kronecker delta and Q has the constraints, as out outlined in Shih et al. 125 and de Lorenzi, 126 on the outer boundary on the crack front For DIC applications, noting that the gradients of Q and the virtual crack extension in the x 3 direction are zero in plane strain or plane stress conditions, Equation ( 16) can be rewritten as the classical area integral, as Arcan test setup in 45 mixed-mode loading and (b) SIF estimates computed using Equation ( 18) of the decompose displacement fields to show the Modes I-III specific loading (A-C).r i indicates the mask size. 33Reproduced with permission where A is the integration area, as shown in Figure 2b.Q is a planar cross-section of the volume case in the plane of constant x 3 .
The decomposition of Equations ( 17) or ( 18) is equally possible by computing stresses, strains and strain energy derived from decomposed displacement data.However, the same limitations apply to the volume, i.e., DVC data, allowing only for symmetric, i.e., Mode I, and antisymmetric decomposition, i.e., Modes II and III combined.
The choice in Q is an important consideration when using experimentally computed displacement data 34 ; for the evaluation of the J-integral, Equation ( 16) or (18) requires the derivates of Q with respect to x i .Thus, Q can either allow for simple numerical implementation (i.e., constant gradients) or inherently remove poorly defined displacement data near the crack tip by setting a near-zero gradient at the crack tip.In DVC, J encapsulates the segment along a crack front, i.e., q s ð Þ; therefore, the crack-tip data will be included.On the surface, however, the crack-tip data can be entirely excluded by choosing Q with a zero gradient near and at the crack-tip.This makes the J-integral especially attractive in DIC applications.
In a similar manner to the field-fitting approach, the application of the J-integral to experimentally measured DIC or DVC displacement data require further consideration: First, the J-integral approaches are more susceptible to variance noise in displacement data, as they require information on strain and stress obtained from the numerical differentiation of displacement data and the constitutive relationship, respectively.We need to differentiate between gradients and strain; the former is the derivative of the measured displacement field, also often described using the deformation tensor, whereas the latter relies on assumptions such as infinitesimal strain theory.The constitutive relations that relate strain and stress rely on the symmetry of the strain tensor, necessitating its symmetrisation.Computation of displacement gradients, strains and stresses requires numerical differentiation of DIC or DVC displacement data.Due to the relatively noisy nature of displacement data, various approaches have been proposed to best perform this, including the use of FE shape functions, 32 polynomial fitting 127 or FE extrapolation of the entire crack region. 48In-house experience shows that simple differentiation using, for example, finite differencing suffices if a larger domain is used over which J is computed.Global DIC offers the advantage that displacement gradients and strains are inherently computed through its formulation.
Second, the J-integral is well suited for DIC and DVC data in that careful selection of Q allows the crack-tip data to be excluded and contours to be taken far from erroneous data.However, in the volume, i.e., DVC data, data in x 3 cannot be excluded.This makes the J-integral less attractive in the volume, especially if mixed mode loading is present.Mixed mode loading results in high strain energy densities along the crack flanks that contribute to mode II and III contributions in J. Ignoring contributions in x 3 will result in contour dependence.In DIC applications, the J-integral does not require the exact location of the crack tip, and contours can be arbitrarily placed.This is a desirable feature in experimentally computed displacement data, where the location of the crack tip is not known precisely.If mode-specific F I G U R E 5 Q function gradients for linear, polynomial (fourth order) and contour (piecewise linear) functions. 34Reproduced with permission J is required, the location of the crack flanks in x 2 is needed.In volume, J computation requires three aspects: crack-tip segmentation, integration volume mapping masking and the summation of the integrand.If the data are in a gridded and uniform format, which is often the case for DIC and DVC data, the summation of the integrand is straightforward.The procedure is described in Becker et al. 34 Importantly, segmenting the crack tip requires knowledge of the crack path through the volume.
Third, the J-integral is immune to rigid body translations and rotations, assuming the displacement gradients are computed in such a way that they are unaffected by rigid body rotations.However, unlike the field-fitting approach, data along the crack flanks cannot be simply excluded.Molteno and Becker 33 tested the relative sensitivities of Modes I to III, in which it was shown that the relative sensitivity of the computed SIF to masking was related to the strain energy field of the masked region-a consistently low value for Mode I in which the crack faces and flanking regions contain minimal strain energy relative to Modes II and III.This highlights the importance of masking.Suitable methodologies have been proposed to replace crack face data accurately. 39,59The use of strain energy as an indicator of J accuracy is only qualitative since J estimates are susceptible to various parameters (e.g., selection of large contours on the surface or modification of the Q function in the volume).

| Crack tip opening displacement approaches (less common)
One of the earliest and simplest approaches in extracting fracture properties using full-field fracture mechanics is to measure δ. 69 Measuring δ (defined by two that secants that are placed at an angle of ± 45 from the crack tip) is difficult, as this requires the exact location of the crack tip and accurate displacement measurements at the crack tip, exactly where DIC and DVC perform poorly.Because of this, the crack-opening displacements are usually measured across the crack flanks and extrapolated to the crack tip by means of geometrical assumptions (e.g., intercept theorem for rotation around a plastic hinge 69 ).The δ can then be related to a mode I SIF, using where m ≈ 1 for plane stress and m ≈ 2 for plane strain and σ y is the Yield stress.
A more pragmatic approach is measuring crack-opening displacements along the crack flanks.Under LEFM assumptions, the discrete displacement can be related to the SIF using where r is the distance from the crack tip.This approach, in a way, is identical to the field-fitting approach.Using Equation ( 20) and only considering u 2 and Mode I loading, with θ = 180 , provides estimates for K I .An example of measuring crack-opening displacements in nuclear-grade graphite is shown in Figure 6.
Although computationally simpler than using the William series, Equation ( 20) seems questionable since only crack-opening displacements are considered as opposed to full displacement fields around the crack tip.However, δ measurements may prove useful in materials that exhibit a large fracture process zone (such as ductility or microcracking) and thus significant stretching and blunting at the crack tip even before the crack initiates, as shown, for example, in Figure 6.The use of crack-opening displacements has also been useful in investigating fracture process zone and CZM. 128,129

| CZM approaches
CZMs were first introduced by Dugdale 130 and Barenblatt 131 and have since been shown to be suitable for simulating fracture in a wide range of materials and to account for heterogeneities at various size scales 132,133 ; they have been successfully used to simulate and predict the entire fracture process from the onset of the crack propagation to the final rupture, including crack growth, propagation, potential bifurcation and multiple fracturing.Broadly speaking, CZMs are a way of bridging the gap between fracture mechanics and damage mechanics: CZMs assume that the material failure process occurs in a strip-shaped zone ahead of the crack.Damage to the material until its final separation is described using the boundary tractions as a function of the separation distance.The energy required for total separation equates to the fracture energy, as used in classical fracture mechanics.
Typically, the CZM traction-separation law can comprise two regions, comprising of a linear-elastic state (δ ≤ δ e , where δ e denotes the elastic limit of the separation at the interface) and a damage evolution state (δ e < δ ≤ δ c , where δ c is the maximum relative displacement at the interface in the normal direction prior to failure, i.e., the crack tip).When presenting an opened displacement (δ ≥ δ c ), indicating formation of a crack, the material stiffness degenerates to zero.The traction-separation law can be expressed in mathematical form as 134 t ¼ kδ where k denotes the initial linear-elastic stiffness and t is the traction at the interface.f δ ð Þ represents a function that describes the traction-separation law.Within the framework of LEFM, the correlation with the critical energy release rate, G c , by For EPFM conditions, the separation energy correlates with the physical crack initiation value G c ¼ J 1c , the critical value of J, as for example defined in ASTM 1820.For a widely used bi-linear traction-separation law, δ c ¼ G c =t max , where t max is the maximum traction in f δ ð Þ.Other forms of commonly employed traction-separation laws include trapezoid, polynomial or exponential functions.In Mode I, δ ¼ δ n , and t ¼ σ 22 , where δ n is the separation normal to the assumed crack plane.Ιn mixed mode loading, the tangential shifts (i.e., Modes II and III) are also considered, requiring unique traction-separation laws, i.e., δ ¼ δ n , δ t , δ s ½ , providing the separation in the normal and tangential shifts with respect to the traction directions, i.e., t ¼ σ 22 , σ 12 , σ 23 ½ . 73Some recent attempts have been proposed to derive the cohesive laws from DIC data.The approach differs from the field-fitting, J or δ approaches in that the cohesive laws cannot directly be obtained from measured displacement data as estimates of the tractions are required (and possibly an assumed shape to describe the traction-separation law, e.g., bi-linear).
Shen and Paulino 36 combined DIC and FE method (to compute tractions) to measure the mode I traction-separation relation of a ductile adhesive (Devcon Plastic Welder II) using an inverse analysis approach through an optimization procedure (Nelder-Mead solver).The unknown CZM was constructed using a flexible b-spline without a priori assumption of its shape.Their method assumed linear elasticity in a state of plane stress, where only material in a predefined path contributed the cohesive zone.
Conversely, Rajan et al. 37 characterised the law of traction-separation for modified bitumen polymers in Mode I by differentiating Equation ( 22) with respect to dδ to solve for the traction, i.e., assuming only mode I loading.J 1c was computed using sample specific relations, requiring knowledge of the exact crack length and applied load (and sample dimensions) at the onset of crack propagation to determine the tractionseparation law assuming a third-order polynomial.Gorman and Thouless 135 used a similar approach to Rajan et al. and compared their results with the inverse analysis approach, as, for example, presented by Shen and Paulino.They observed that agreement was 'close to the range that one might attribute to experimental error'.Recent work by Hou et al. 128 focused on rather computing k, t max and δ c of the CZM (in Mode I) without inferring a relationship between the traction-separation law a priori.Hou et al. achieved this by assuming the balance of the traction forces along the boundary in a double-cantilever sample.In doing so, the authors were able to establish a CZM by relating k, t max and δ c to G Ic using either a bi-linear or polynomial traction-separation law posteriori.Blaysat et al. 136 and Ruybalid et al. 137 used IDIC for the parametric identification of CZM in metal joints.The proposed approaches account for the traction-separation profile (a higher order polynomial) along the interface using few degrees of freedom, i.e., crack tip position, maximum stress and size of the process zone.The work of Ruybalid et al. is noteworthy because it considered mixed-mode loading on a microstructural scale in microelectronic devices.
There are a few considerations that need to be made when computed CZM parameters from DIC computed displacement data.The crack path generally needs to be defined, and the location of the crack tip needs to be known exactly.The cohesive zone needs to be large enough so not to be limited by the inherent smearing of DIC in highly strained regions.High displacement gradient resolution is required, especially if a shorter cohesive zone and lower cohesive tractions are expected. 128

| APPLICATIONS
The use of full-field fracture mechanics to determine fracture properties from the specimens and the loading conditions studied offers significant progress in the studies of the crack driving force and other mechanical parameters for structural integrity evaluations of engineering structures and components.It is reassuring that, for most cases reported, the fitted values of the parameters from the DIC and DVC measurements appear close to the analytical solutions when SSY conditions are satisfied.Field-fitting approaches are particularly limited in that the solution assumes linear-elastic conditions.On the other hand, the J-integral offers the advantage that it allows for elastic-plastic fracture mechanics even if linear elastic conditions are assumed-as long as contours are taken sufficiently far away from the crack where linear elastic conditions still apply.For fully plastic behaviour or other nonlinear crack mechanisms, a δ approach is likely to yield better approximations.
The field-fitting approach lends itself well to establishing mixed-mode parameters.The formulation establishes the location of the crack tip and thereby offers the additional capability of tracking crack propagation, as demonstrated by Huchzermeyer and Becker 102 and Zanganeh et al. 94 The field-fitting approach has been predominantly applied in applications investigating fatigue crack growth behaviour, particularly crack closure phenomena.For example, Carroll et al. 85 used the field-fitting approach to examine the fatigue crack closure behaviour at macroscales and microscales in Grade 2 titanium.They showed that the crack-opening load increases with the reduction of the distance to the crack tip.Lopez-Crespo et al. 138 identified crack closure under mixed Modes I and II loading conditions; closure effects were attributed to combined effects of plasticity, roughness and frictional forces.O'Connor et al. 139 analysed the near-tip displacements and strains using a 6082-T6 aluminium alloy.They also showed that the opening load is higher for locations closer to the tip of the crack.Tong et al. 140 measured the crack closure phenomenon via the crack-opening displacement approach, a field-fitting approach and the classical analytical approach based on the applied load, crack length and sample geometry.Interestingly, they concluded, similar to Carroll et al. 85 and others, 3,139 that near-tip normal strains are found to increase linearly with increasing applied load and do not correlate with the trend presented in the crack-opening displacement versus load curves.
Although less often reported, the field-fitting approach has been used to measure the fracture toughness of polymethyl methacrylate (PMMA), 102,141 Polyvinyl Chloride polymer, 142 functionally graded materials 143 and composites, 144 to name a few.Applications have mainly been limited to near-linear elastic materials.On the other hand, the J-integral approach has been adopted more widely in fracture toughness investigations.Notably, it has been applied to quasi-brittle materials, such as nuclear-grade graphite, 29,145-147 steels, 148 rock, 149 plastics, 150 timber 151 and functionally graded materials, 143 to name a few.
An important consideration when using DIC is that SIFs obtained on the surface assume plane stress conditions, whereas in the volume, i.e., using DVC, plane strain conditions are assumed.This will likely result in lower estimates of SIFs on the surface. 69Moreover, the crack front may not necessarily be constant throughout the volume.Figure 7 shows the comparitive crack-tip positions obtained using a field-fitting approach, potential drop system and physical F I G U R E 7 Comparison of determined crack-tip positions using a full-field approach (using a pattern search [PS] optimisation), potential drop (PD) system and physical measurements taken using an optical microscope (red). 94Reproduced with permission measurements during fatigue testing of a Compact Tension specimen made of grade 2024 aluminium alloy. 94Surface measurements underestimate the average crack length through the volume used in analytical solutions.
Although the success of computing SIFs from the full-field displacement data can be readily compared to the analytical solutions, the significance of these exercises beyond gaining confidence in applying the presented approaches is unclear.We see three factors limiting the more widespread application of full-field fracture mechanics: First, limited error analysis has been conducted on the extraction of SIFs from inherently noisy data; second, the incorporation of DIC and DVC algorithms that better deal with discontinuous displacement data across crack flanks into readily available software packages; and third, the incorporation of the approaches into readily available software packages.We aim to elaborate further on these points below.
Several authors have tried to incorporate noise sensitivity in the extraction of SIFs.For example, Zanganeh et al. 94 considered various optimisation routines when fitting the William series.They concluded that a Newton-type optimisation routine is less accurate.However, their results are somewhat questionable as their solutions were highly sensitive to the initial guess; this was not reported in other studies. 30,31Molteno and Becker 33 considered the effect of masking nonphysical crack flank data in the computation of J from DIC data and noted its influence under mixed mode loading.Becker et al. 34 investigated the influence of random Gaussian noise on DVC computed displacement data and noted the influence in the choice of Q highlighting an error of $0.7 % in the computation of J when for a displacement noise of 1% in present.What makes an error analysis complex is that errors in DIC and DVC computed data are complex, consisting of both variance and bias types of noise.While the variance is easily simulated by, for example, creating artificial speckle patterns with known deformations and noise (see iDICs 62,63 ), incorporating bias proves challenging.Steps are underway to standardise DIC measurements (which probably should be extended to DVC), and these may aid in investigating the influence of the error of full-field fracture mechanics.
Several complex DIC algorithms have been developed that allow for discontinuous displacements across crack flanks, such as subset splitting 152 or XFEM. 11,67To our knowledge, these have not been incorporated into readily available software packages; thus, their use is minimal.When applying full-field fracture mechanics, the errors associated with poor correlation across discontinuous fields are probably one of the most significant sources of concern.Integrated approaches, such as those presented by Roux et al., 75 provide a framework incorporating enriched displacement solutions to better map displacements near cracked regions from DIC and DVC.Incorporating such techniques into readily available software packages would be highly desirable.
Similarly, full-field fracture mechanics techniques live in the realm of researchers.While incorporating published techniques into post-processing scripts is trivial, it requires a certain level of competence in solid and fracture mechanics.Incorporating these into readily available software packages would also be highly desirable.
Several new and exciting developments in full-field fracture mechanics have recently been presented, which entail in situ experiments on a microstructural scale to, for example, monitor local deformation under cyclic loading and the development of fatigue cracks.Carroll et al. 85 investigated the interactions between strain accumulation, microstructure and fatigue crack behaviour and mapped plastic strain accumulation at the subgrain level.The accumulated plastic strain fields associated with fatigue crack growth were highly inhomogeneous, varying from grain to grain and within individual grains, and strain localisation was evident in slip bands within grains and on twin and grain boundaries shown in Figure 8. Strains have been monitored at fixed observation points 76,153,154 of a growing fatigue crack, and a critical or onset strain was identified from instantaneous strains at the crack tip as the crack approached the observation points.Such information is significant in discovering a local crack driving force, albeit within a continuum mechanics domain.Microstructural information and local strain measurements can be used to identify individual slip system activities and relate them to the initiation and growth of fatigue cracks. 155Recent work by Koko et al. 156,157 used high-(angular) resolution electron backscatter diffraction (HR-EBSD), as opposed to DIC, to extract local deformation measurements to associate mixed mode SIFs quantitatively in age-hardened duplex stainless-steel.

| SUMMARY AND OUTLOOK
Full-field fracture mechanics has allowed for new ways to investigate deformations ahead and behind the crack tip, allowing for extracting SIFs directly from DIC and DVC computed displacement data.Four approaches are commonly employed, namely, the field-fitting approach, whereby the Williams series is fitted to the measured disablement data; the J-integral approach, whereby the displacement associated strains and stresses are used to compute the energy release rate by integrating around the crack tip; δ approach, whereby crack-opening displacements are used to compute SIFs; and CZMs, whereby the traction-separation law ahead of the crack front are computed.
All approaches require knowledge of material properties, i.e., linear elastic constants E and v in their simplest form, and do not necessarily require knowledge of the applied load or crack length (although CZMs generally do require the exact position of the crack tip).Field-fitting is limited to SSY conditions and thus has seen most of its application in fatigue crack growth rate type investigations.On the other hand, the J-integral allows for elastic-plastic conditions even when linear elastic constants are used if integral contours are taken sufficiently far away from the crack.δ approaches lend themselves well to complex fracture mechanisms allowing for further investigations into, for example, CZMs.
Although extracting SIFs from experimentally measured displacement data is well established, wide adoption seems hindered by limited studies into their sensitivity to the various DIC-and DVC-associated errors, thus limiting confidence in their application.Moreover, limitations of commonly employed DIC and DVC algorithms near and across cracked specimens, i.e., subset smearing and poor correlation, add complexity due to the nuances involved in dealing with erroneous displacement data.Incorporating modern DIC and DVC algorithms, as well as full-field fracture mechanics, into readily available software packages would significantly increase the adoption of extracting SIFs from experimentally measured displacement data.
Several new and exciting developments have been presented in full-field fracture mechanics, which entail in situ experiments on a microstructural scale to monitor local deformations at a microstructural scale.Such investigations could open new insights into the local crack driving force at the microscopic level, which may impact the development of characterisation techniques and future materials design.
F I G U R E 8 Accumulated plastic strain fields in x 2 , recorded at five crack lengths.Plastic strain in the wake of the crack tip is heterogeneous with high and low strain lobes at roughly ±40 with the crack plane. 85White dots show the location of the crack tip.Reproduced with permission part of which aims to limit errors in DIC measurements.F I G U R E 1 Full-field displacementerrors of the area correlated under K ¼ 20 MPa ffiffiffiffi m p : variance (random) shown in the left column (a, b and c) and bias (systematic) shown in the right column (d, e and f) for the selected subset sizes. 61Reproduced with permission

F I G U R E 6
Comparison between normalised crack-opening displacements measured by DVC and finite element simulation prediction.(a) First loading stage: average crack length is 5.7 mm, and stress intensity factor at the peak load is K = 1.60 MPa√m and (b) second loading stage: average crack length is 6.2 mm and stress intensity factor at the peak load K = 1.71MPa√m.LEFM solution is shown by the straight line. 29CMOD, crack mouth opening displacement.Reproduced with permission