# Evaluation of fracture properties by coupling digital image correlation and Finite Fracture Mechanics

Francesco Ferrian, Alberto Sapora, Rafael Estevez, Aurélien Doitrand

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## To cite this version:

Francesco Ferrian, Alberto Sapora, Rafael Estevez, Aurélien Doitrand. Evaluation of fracture properties by coupling digital image correlation and Finite Fracture Mechanics. International Journal of Fracture, 2025, 249 (4), pp.66. 10.1007/s10704-025-00885-9. hal-05244282

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# Evaluation of fracture properties by coupling Digital Image Correlation and Finite Fracture Mechanics

Francesco Ferrian¹, Alberto Sapora¹, Rafael Estevez², Aurélien Doitrand³

¹Department of Structural, Geotechnical and Building Engineering, Politecnico di Torino.

²Université Grenoble-Alpes - CNRS UMR 5266, SIMaP, F-38000 Grenoble, France

³INSA-Lyon, Université Claude Bernard Lyon 1, CNRS, MATEIS, UMR5510, 69621, Villeurbanne, France.

## Abstract

Crack initiation in flattened disks under compression containing either a central or eccentric circular hole is investigated through the Finite Fracture Mechanics approach. An implementation of the Finite Fracture Mechanics (FFM) criterion based on Digital Image Correlation (DIC) full-field measurement is proposed. The coupling between FFM and DIC is provided through boundary conditions taken from the measured displacement fields. This approach offers a more accurate representation of the actual loading conditions compared to the use of idealized prescribed force or displacement in standard Finite Fracture Mechanics implementations. Furthermore, by exploiting the value of the critical energy release rate obtained from the stable crack growth phase analysis, this method enables precise estimations of the inherent material strength and critical crack advance.

**Keywords:** FFM, DIC, size effect, energy balance, elastic properties, fracture parameters.

## Nomenclature

<table><tr><td>e</td><td>eccentricity of the hole</td><td>P<sub>c</sub></td><td>experimental initiation load</td></tr><tr><td>R</td><td>external radius of the disk</td><td>P<sub>c,avg</sub></td><td>average value of the experimental initiation loads</td></tr><tr><td>t</td><td>thickness of the sample</td><td>U<sub>c</sub></td><td>critical applied displacement</td></tr><tr><td>α</td><td>loading angle</td><td>σ<sub>UTS</sub></td><td>ultimate tensile strength</td></tr><tr><td>ρ</td><td>hole radius</td><td>σ<sub>0</sub></td><td>inherent strength of the material</td></tr><tr><td>E</td><td>Young's modulus</td><td>σ<sub>xx</sub></td><td>normal stress field</td></tr><tr><td>ν</td><td>Poisson's ratio</td><td>(x, y)</td><td>cartesian coordinate system</td></tr><tr><td>l<sub>c</sub></td><td>initiation crack length</td><td>DIC</td><td>Digital Image Correlation</td></tr><tr><td>l<sub>ch</sub></td><td>Irwin's length</td><td>FFM</td><td>Finite Fracture Mechanics</td></tr><tr><td>l<sub>exp</sub></td><td>experimental crack length</td><td>FEA</td><td>Finite Element Analyses</td></tr><tr><td>G<sub>inc</sub></td><td>incremental energy release rate</td><td>PMMA</td><td>Polymethyl Methacrylate</td></tr><tr><td>G<sub>ic</sub></td><td>critical value of the energy release rate</td><td>SIF</td><td>Stress Intensity Factor</td></tr></table>
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# 1. Introduction

The strength of structures and mechanical components can be significantly affected by the presence of stress concentrators such as notches or holes. Acting as stress raisers, they result in localized stress concentrations, thereby reducing the load span necessary for maintaining mechanical integrity. Different approaches have been proposed to address the decrease in the load-bearing capacity of structures caused by such stress raisers. Neuber [1] proposed an approach based on the calculation of an effective stress by averaging the linear elastic stress field ahead of the notch tip over a material characteristic length. Neuber argued that the effective stress should be evaluated within a structural finite volume rather than an infinitesimal volume, as assumed by classical theories. A few years later, Peterson [2] simplified Neuber's approach by observing that the reference stress could be calculated at a specified distance from the notch tip. Later, Whitney and Nuismer [3] and Tanaka [4], proposed combining ultimate tensile strength and fracture toughness to assess the material characteristic length for predicting brittle failure in notched components. Subsequently, Taylor [5] formalized these stress-based methodologies under the name of the Theory of Critical Distances (TCD). The simplest form of the TCD is the Point Method, which requires that for fracture to occur, the stress must equal the inherent strength of the material $\sigma_0$ at a finite distance $L/2$ along the critical crack path. According to the TCD approach, the characteristic material length $L$ depends on the squared ratio between the fracture toughness $K_{Ic}$ and $\sigma_0$. On the other hand, different researchers (Awerbuch and Madhukar [6], Pipes et al. [7], and Tan [8]) demonstrated a dependence of the critical distance on other parameters, such as geometry features. To overcome this issue, the Finite Fracture Mechanics (FFM) criterion was introduced by coupling a stress condition and an energy balance ([9], [10], [11]). Unlike TCD, the FFM approach enables determining an initiation crack length, which is not solely a material property but also dependent on both loading conditions and geometry under investigation. FFM was implemented to assess strength predictions for a wide range of structural configurations subjected to different loading conditions. Focusing on the failure behavior of circular holes under biaxial loading, Torabi et al. [12] investigated the strength of Polymethyl Methacrylate (PMMA) Brazilian disk samples with a central circular hole. Sapora and Cornetti [13] analyzed different biaxiality ratios considering remote loading configurations. Furthermore, Chao Correas et al. [14] conducted a
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comprehensive study on crack initiation from circular holes under biaxial loading by showing agreement among FFM, Cohesive Zone Model, and Phase Field estimations.

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In order to provide accurate strength estimations, both TCD and FFM approaches require the knowledge of K<sub>Ic</sub> and σ<sub>0</sub>. Regarding the value of the fracture toughness, it can be experimentally determined following established standards for polymers ([15], [16]). These procedures involve testing Compact Tension or Three Point Bending samples at a constant rate until failure occurs. Initially, a sharp notch has to be machined and then a natural crack, with a precise shape and length, needs to be carefully generated at the notch tip. If this is not possible, either due to the sample fracturing during tapping (as observed in some brittle materials) or the crack being not visible (as seen in some tough materials), then the pre-crack can be machined using a fresh razor blade in a single motion or with a sawing motion. These standard procedures are thus usually very complex and highly time-consuming to perform [17]. Alternatively, the value of K<sub>Ic</sub> can be derived from the Izod absorbed energy measured through Izod impact tests, which are less demanding in terms of time and costs ([18], [19]). However, the main drawback of this procedure is that it offers an indirect measure of fracture toughness based on the energy absorbed during the test, rather than a direct measurement. This absorbed energy is influenced by various factors, including sample configuration, shape of the notch, temperature, strain rate, and loading conditions. Consequently, the Izod test results may not accurately reflect the material's actual fracture behavior [20].

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An accurate estimate of the fracture toughness can be obtained by monitoring crack propagation and using DIC-based full-field measurements. DIC ([21], [22], [23]) is a practical and effective tool for full-field deformation measurement, which has been widely accepted and commonly implemented within experimental mechanics for the determination of material properties and fracture parameters. Indeed, measuring the displacements in the region surrounding the crack tip makes it possible to determine fracture parameters such as cohesive properties, J-integral and Stress Intensity Factor (SIF). In this framework, Shen and Paulino [24] proposed a hybrid inverse method based on the combined use of DIC and Finite Element Analyses (FEA) to compute the cohesive properties of a ductile adhesive and a quasi-brittle plastic. In this study, they performed tests on single edge-notched beam samples under four-point bending in order to derive the mode I traction-separation relationship. Réthoré and Estevez [25] performed a micron-scale analysis on PMMA samples to capture and investigate the crazing failure mechanism. During the tests, they used DIC to determine the mode I and II SIFs and the crack tip displacement fields. Furthermore, the onset and growth of the process zone were monitored up to crack onset by using the variation of the equivalent crack tip location. These experimental measurements were then used to accurately define the traction-separation profile of a cohesive zone model. Focusing on the determination of the stress intensity factor value, McNeill

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et al. [26] presented a DIC method to determine the SIF in pre-cracked C-specimens and three-point bending samples. Roux and Hild [27] performed SIF measurements from DIC considering Sandwiched-Beam tests. In this study, they inserted a notched SiC sample between two steel bars and determined the SIF value based on displacements measured through DIC. Hamam et al. [28] proposed a methodology to determine through DIC techniques the SIF value in Center-Cracked Tension (CTT) specimens made of steel under fatigue loading. The change in the SIF during one fatigue cycle was measured by decomposing the displacement field into a set of elastic fields. More recently, Doitrand et al. [29] conducted tensile tests on PMMA plates with a central square hole, which resulted in mixed-mode crack onset at the corners of the square hole. The mode I and II generalized SIFs at the V-notches were determined through a path-independent integral, using displacement and strain fields measured directly by DIC. Brittle fracture in sharp V-notched samples was investigated also by Yue et al. [30]. Specifically, the generalized notch SIF and fracture parameters of V-notched rock samples under dynamic impact were measured using DIC techniques. Regarding crack onset and propagation in flattened Brazilian disks, Yan et al. [31] evaluated the mode I SIF performing experimental tests on sandstone samples and measuring strains and displacements through DIC methods.

As concerns the value of the inherent strength $\sigma_0$, it can be considered equal to the ultimate tensile strength $\sigma_{UTS}$ for brittle and quasi-brittle materials such as ceramics [32] and laminated composites [3]. However, if plastic deformation occurs before the final failure, the inherent strength could differ from the value of $\sigma_{UTS}$ [33]. Specifically for polymers, due to the presence of defects or micro-cracks and crazing phenomena, the ratio $\sigma_0/\sigma_{UTS}$ typically ranges between 1 and 2 ([34], [35], [36]). When the inherent strength differs from the ultimate tensile strength, $\sigma_0$ can be determined by testing samples of the same material containing sharp and blunt notches, characterized by different stress distributions in the process zone. Indeed, according to the Point Method, the linear-elastic stress field curves at incipient failure should intersect at the same point for all the different notches. The inherent strength can thus directly be estimated by considering the stress value at this intersection point ([34], [35]).

In the framework of FFM, Duminy et al. [37] proposed a method to determine fracture parameters from DIC performing wedge splitting tests on millimeter-size PMMA specimens. Identification of fracture parameters using the FFM approach usually exploits idealized prescribed force or displacement as boundary conditions. Instead, in this study, Duminy et al. [37] determined boundary conditions performing full-field measurements during the tests, thus better representing loading conditions on a submodel surrounding the initiation crack location. The FFM approach was used for the inverse identification of fracture properties also by Girard et al. ([38], [39]). Specifically, in these
studies, the fiber-matrix debonding in a single fiber was investigated numerically and experimentally.
Then, interface fracture properties were determined to obtain crack initiation/propagation stresses and
propagation/debonding arrest angles consistent with experimental observation.

In the present study, a finite element implementation of the FFM approach is presented to investigate
crack initiation on disk samples weakened by the presence of either a central or eccentric circular
hole. Firstly, elastic material properties are determined by considering the displacement fields
measured by DIC and performing 2D linear elastic FEA. To validate the proposed procedure, the
obtained values are compared with those derived by following the ASTM D638-14 [40] standard
code. Then, the FFM criterion is implemented considering a circular submodel and boundary
conditions based on displacements measured using DIC. Following this approach, the inherent
material strength and critical crack advance are thus precisely estimated using the value of the critical
energy release rate obtained from the analysis of the stable crack growth phase.

The structure of the paper is organized as follows: Section 2 presents the experimental campaign with
the obtained initiation loads. In Section 3, the DIC technique is introduced, followed by a discussion
on the procedure implemented to determine the elastic properties in Section 3.1. Section 4 outlines
the FFM formulation for the geometries investigated in this study. Then, the finite element
implementation of the FFM criterion, using boundary conditions derived from DIC-measured
displacements, is detailed in Section 5. Finally, conclusions are drawn in Section 6.

## 2. Experimental results

Compression tests were performed at a constant displacement rate of 1 mm/min, under uniaxial
loading conditions, on disk-shaped samples made of PMMA (depicted in Fig. 1) by Ferrian *et al.* [41].
The thickness of the samples, denoted as *t*, was set at 8 mm to ensure plane strain conditions,
satisfying the condition $t \ge 2.5 \ell_{ch}$ [42]. Here, $\ell_{ch} = (K_{Ic}/\sigma_{0})^{2}$, representing Irwin's length, typically
falls within the range of 0.2 to 1 mm for PMMA ([42], [43]).
<figure><img src="image_3.png" /><figcaption>Fig. 1 Sample configuration investigated in the current study presenting a hole with radius $ρ$ and eccentricity $e$.</figcaption></figure>

Two configurations were tested, featuring a central circular hole or an eccentric one with $e = 20$ mm eccentricity. Four hole radii $ρ = 0.5, 1, 2$ and $4$ mm were considered for each configuration. For all samples, the outer radius $R$ of the disk is set to $40$ mm. Furthermore, the lower and upper regions of the disk were slightly flattened to facilitate sample positioning within the testing machine, as depicted in Fig. 1. This solution, is a common practice aimed at ensuring test effectiveness and enhancing the accuracy of tensile strength estimation. Indeed, this loading configuration ensures crack initiation at the disk center, averting localized cracking near the contact zone. The flattened Brazilian Disk (BD) test was introduced by Wang and Xing [44], and subsequent investigations on the influence of loading angles were conducted by Wang *et al.* [45] and Lin *et al.* [46]. The loading angle was maintained at $20°$, consistently with typical experimental practices ([44]-[47]).

<table><thead><tr><th>e<br>[mm]</th><th>ρ<br>[mm]</th><th colspan="5">P<sub>c</sub><br>[N]</th><th>P<sub>c, avg</sub><br>[N]</th></tr><tr><th>Sample n°</th><th></th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th></th></tr></thead><tbody><tr><td rowspan="4">0</td><td>0.5</td><td>15303</td><td>15077</td><td>14984</td><td>-</td><td>-</td><td>15122 ± 134</td></tr><tr><td>1</td><td>12642</td><td>10779</td><td>11420</td><td>-</td><td>-</td><td>11614 ± 773</td></tr><tr><td>2</td><td>12238</td><td>11888</td><td>9014</td><td>9070</td><td>-</td><td>10553 ± 1516</td></tr><tr><td>4</td><td>9919</td><td>10650</td><td>10404</td><td>11906</td><td>-</td><td>10720 ± 734</td></tr><tr><td rowspan="3">20</td><td>0.5</td><td>13244</td><td>14113</td><td>13974</td><td>13553</td><td>13081</td><td>13593 ± 400</td></tr><tr><td>1</td><td>12532</td><td>10620</td><td>10071</td><td>12556</td><td>-</td><td>11445 ± 1117</td></tr><tr><td>2</td><td>11220</td><td>12886</td><td>9270</td><td>7701</td><td>10381</td><td>10292 ± 1752</td></tr></tbody></table>
4 9873 9753 9668 9868 - 9791 ± 85

<caption>Table 1 Recorded initiation loads $P_c$ and standard deviations for each tested sample.</caption>

Five specimens were considered for each value of hole radius and eccentricity. Although some samples experienced failure within the contact zone, at least three initiation load values $P_c$ were recorded for each geometry. The initiation loads obtained for 32 samples are reported in Table 1.

Notably, the presence of an eccentric hole resulted in decreased strength compared to geometries featuring a central hole. Specifically, for radii values of 0.5 mm and 4 mm, $P_c$ values related to the geometries with a central hole were 10% higher than those registered for $e = 20$ mm. This discrepancy diminishes to approximately 2% for $\rho = 1$ mm and 2 mm.

Finally, it is important to highlight that stable crack growth was observed for all the samples before the final failure, except for $\rho = 4$ mm, $e = 0$, 20 mm and $\rho = 2$ mm, $e = 0$ mm. When stable crack propagation was detected, failure loads were approximately 2-3% higher than the initiation ones.

### 3. Digital image correlation

DIC stands out as an effective non-contact optical technique for full-field displacement and strain measurement. It works by capturing sequential images of the same object and extracting the displacement fields to attain the most accurate match in the Region Of Interest (ROI) between the reference image and the deformed ones ([48], [49]). This can be accomplished by employing several criteria and optimization algorithms.

In this study, an algorithm is utilized to minimize the sum of the squared grey level differences between the reference image and the deformed one, adjusted by the displacement field estimation $U_e$. Denoting with $f$ and $g$ be the grey level fields of reference and deformed images, respectively, and with $U$ the sought displacement field, the following expression is derived [50]:

$$U = \operatorname{Argmin}_{U_e} \sum_{\text{ROI}} [f(x) - g(x + U_e(x))]^2 \quad (1)$$

This minimization problem is solved by exploiting the software Ufreckles [51] which implements a FE-based nonlinear least-squares algorithm.

During the test campaign performed by Ferrian et al. [41], images were acquired exploiting a Prosilica GX 6600 camera, with a 2392 (H) × 2774 (V) pixel resolution and considering an acquisition rate of 1 fps. DIC measurements are inherently influenced by acquisition noise related especially to the quality of the speckles, DIC mesh size, lighting and acquisition equipment setup. In order to minimize the noise influence, samples were carefully prepared coating the surface with white and black sprays to obtain high-quality speckles. Furthermore, an unstructured mesh with 3-node triangular elements
181 is adopted setting the mesh size equal to 60 pixels, corresponding to ≈ 1.25 mm. In this way, a good balance between surface partitioning and measurement noise is obtained. A rigid body motion of the 182 disk is analyzed through DIC to estimate the displacement measurement uncertainty. By computing 183 the standard deviation of the displacements, a measurement uncertainty of ± 0.0011 mm and ± 0.0015 184 mm is determined for $U_x$ and $U_y$, respectively. 185

Figures 2a and 2b illustrate the displacements along the x-axis and y-axis, respectively, for a disk 186 sample with $e = 0$ mm and $\rho = 4$ mm. For the same sample, the x-, y-, and xy- components of the 187 strain field are depicted in Figure 3. To reduce the measurement noise, the ROI contour shape is 188 defined inside the sample, sufficiently far from the contact zones and the hole edge. 189

<figure><img src="image_4.png" /></figure>

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Fig. 2 (a) horizontal and (b) vertical displacements measured by DIC, immediately before crack initiation, 191 for a sample with $e = 0$ mm and $\rho = 4$ mm. 192
<figure><img src="image_5.png" /><figcaption>Fig. 3 Strain field components obtained by DIC, immediately before crack initiation, for a sample with $e = 0$ mm and $\rho = 4$ mm.</figcaption></figure>

4. Finite Fracture Mechanics

Considering the geometries with a central circular hole, asymmetric crack initiation from the hole edge along the y-axis is investigated. Theoretically, a symmetric crack initiation should be preferred since it is more critical from an energetic standpoint [56]. However, experimentally, an asymmetric crack initiation is observed for all the tested samples. Additionally, theoretical predictions provided by FFM for symmetric and asymmetric crack initiation are very close, with differences lower than 1% [41].

Analogously, when considering the geometries with $e = 20$ mm, crack initiation is evaluated according to the FFM approach, considering asymmetric crack initiation from the top of the hole edge. Indeed, this scenario results to be the most critical one, as presented by Ferrian *et al.* [41].
According to the FFM approach proposed by Leguillon [9], finite crack initiation occurs when stress and energy conditions are simultaneously satisfied. The stress requirement imposes that the normal stress $\sigma_{xx}$ must exceed the material inherent strength $\sigma_0$ over a finite distance $\ell$. Alternatively, the approach can be implemented considering an average stress condition [10], resulting in the following expression:

$$ \bar{\sigma}_{xx}(\ell) = \frac{1}{\ell} \int_{0}^{\ell} \sigma_{xx}(y) dy \geq \sigma_{0}. \qquad (2) $$

Conversely, the energy balance ensures that the Incremental Energy Release Rate (IERR) $G_{inc}$ is larger than the critical ERR value $G_{lc}$:

$$ G_{inc}(\ell) = - \frac{\Delta W_{el}(\ell)}{\ell} \geq G_{lc}, \qquad (3) $$

where $\Delta W_{el}$ is the elastic strain energy variation due to finite crack advance $\ell$. It is worth noting that this expression is valid for a displacement-controlled test ($\Delta W_{ext} = 0$) and considering quasi-static loading conditions ($\Delta W_k \geq 0$). Moreover, Eq. (3) does not account for the energy dissipated by non-linear mechanisms.

Following the formalism presented by Weißgraeb er et al. [57], the energy release rate function can either increase (positive geometry) or decrease (negative geometry) as the crack propagates. This behavior is influenced by various factors, including material anisotropy, specimen geometry and loading conditions ([58], [59]).

Coupling Eqs. (2) and (3), and considering a monotonically increasing IERR function (positive geometry), the following system is obtained:

$$ \left\{ \begin{array}{l} \bar{\sigma}_{xx}(\ell_c, U_c) = \sigma_0, \\ G_{inc}(\ell_c, U_c) = G_{lc}, \end{array} \right. \qquad (4) $$

where $\ell_c$ is the finite crack advance and $U_c$ the critical applied displacement.

System (4) holds also for an initially positive, globally negative geometry ([60], [57]) if $\ell_c < \ell^*$, where $\ell^*$ denotes the crack length maximizing $G_{inc}$. Instead, if the solution of the system yields $\ell_c > \ell^*$, the FFM criterion is expressed by imposing $\ell = \ell^*$ in the energy balance (4) [60].

# 5. FFM DIC-based implementation

## 5.1 Determination of submodel size

To perform the DIC-based implementation of the FFM criterion, a circular submodel concentric with the hole is considered (Fig. 6a). To ensure accurate computation of crack initiation, it is essential to
first determine the appropriate submodel size in order to correctly compute $\bar{\sigma}_{xx}(\ell)$ and $G_{\text{inc}}(\ell)$ functions. For this purpose, considering configurations with a central hole, a FE model of the full geometry is developed for each radius value, with an applied displacement $U = 1$ mm (Fig. 6a). Then, the average stress field and IERR functions are obtained exploiting the “unbuttoning” method through Eqs. (2) and (3). A mesh with 4-node bilinear plane strain elements is adopted with a minimum element size equal to 0.05 mm, following a convergence analysis. Analogously, the functions related to a submodel with a radius of 23 mm (corresponding to approximately 900 pixels) are derived by applying the displacement values determined from the FE full model as boundary conditions (Fig. 6b).

<figure><img src="image_6.png" /><figcaption>Fig. 6 (a) Schematization of FE model implemented to determine the submodel size and (b) detail of FE mesh characterizing the submodel.</figcaption></figure>

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The average stress functions obtained by FEA on the submodel are identical to those computed from the full model, for all radii values. Instead, differences are higher when considering the IERR functions, especially as the crack length $\ell$ increases. In particular, for $\rho = 2$ mm and 4 mm, the discrepancy is lower than 1% for $\ell < 0.5$ mm and $\ell < 0.4$ mm, respectively. The comparison for these radii values is depicted in Fig. 7a. Considering a higher value of the initiation length $\ell = 3$ mm, the deviation is slightly lower than 5% for $\rho = 2$ mm and around 7% for $\rho = 4$ mm, respectively.
<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 7 Comparison between IERR functions obtained by FEA on full model and submodel related to configurations with a (a) central and (b) eccentric hole.</figcaption></figure>

The same analysis is developed on the geometries with an eccentric hole. Due to the proximity of the hole to the contact zone, the submodel size for these configurations is smaller, set at 18 mm (approximately 700 pixels). The comparison between IERR functions for $ρ = 2$ mm and 4 mm is depicted in Fig. 7b. In this case discrepancies are higher due to the smaller submodel size, lower than 1% considering $ℓ < 0.3$ mm and $ℓ < 0.2$ mm for $ρ = 2$ mm and 4 mm, respectively. Taking into account a higher value of $ℓ = 3$ mm, the deviation is around 8% and 12% for $ρ = 2$ mm and $ρ = 4$ mm, respectively.

Based on these analyses, it can be concluded that crack initiation can be successfully computed by considering the $\bar{\sigma}_{xx}(ℓ)$ and $G_{inc}(ℓ)$ functions derived from the circular submodel for crack initiation lengths of up to 0.3 mm and 0.2 mm for geometries with a central and eccentric hole, respectively.

## 5.2 Results discussion

Once the appropriate submodel size is determined, for each tested sample the average stress field and IERR functions are obtained through linear elastic FEA using plane strain elements. As presented in Section 2 and summarized in Table 1, 32 tested samples are analyzed featuring a central hole or an eccentric one with $e = 20$ mm. For each configuration, four hole radii $ρ = 0.5, 1, 2$ and 4 mm are considered. Boundary conditions are set by applying to the contour of the circular subdomain the displacements $U_x$ and $U_y$ measured by DIC, just before crack initiation. These displacements are depicted in Fig. 8 as a function of the curvilinear abscissae $s$, for a sample with $ρ = 4$ mm and $e = 0$ mm.
<figure><img src="image_9.png" /><figcaption>Fig. 8 Displacements (a) $U_x$ and (b) $U_y$ measured by DIC just before crack initiation and imposed as boundary conditions to the contour of the circular FE model. Displacements are reported as a function of the curvilinear abscissae $s$.</figcaption></figure>

This approach enables the calculation of the average stress field $\overline{\sigma}_{xx}(\ell)$ and IERR $G_{\text{inc}}(\ell)$ functions corresponding to the initiation load $U_c$. Considering the FFM criterion expressed by Eq. (4), the three unknowns are: $\sigma_0$, $G_{\text{lc}}$ and $\ell_c$.

Estimating the finite crack advance $\ell_c$ experimentally is not trivial since small length values are difficult to detect and measure accurately. Additionally, in most configurations, crack initiation is followed by unstable crack propagation, which further complicates the determination of $\ell_c$. To investigate the initial crack initiation phase in more detail, Ferrian *et al.* [41] employed an ultrahigh-speed acquisition camera with a sampling rate of 240000 fps. However, even with this setup, it was not possible to accurately capture and measure the initiation length $\ell_c$.

Regarding the determination of the critical energy release rate $G_{\text{lc}}$, it can be obtained by analyzing the stable crack growth phase. Indeed, considering the LEFM condition for crack growth, $G(\ell_{\text{exp}})$ equals $G_{\text{lc}}$, where $\ell_{\text{exp}}$ represents the experimental crack length. The value of $G(\ell_{\text{exp}})$ is computed by considering a circular 2D FE model concentric with the hole, through the J-integral calculation. The displacements determined from DIC for the experimental crack length $\ell_{\text{exp}}$, are applied as boundary conditions of the FE model. Specifically, with the experimental setup presented in Section 2.1, we are able to accurately monitor by DIC the crack length during the stable crack propagation phase for three samples presenting $\rho = 2$ mm and $e = 20$ mm (refer to Fig. 9). For the other specimens showing stable crack growth, with the experimental setup, we are not able to track the crack tip due to the acquisition rate of 1 fps and the small values of the crack lengths.
<figure><img src="image_10.png" /><figcaption>(b)</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>(c)</figcaption></figure>

<figure><img src="image_12.png" /></figure>

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296 Fig. 9 Images of three different samples with $\rho = 2$ mm and $e = 20$ mm with the detail of the crack length $\ell_{exp}$

297 measured experimentally.

298 Following this procedure, a value of $G_{Ic} = 0.16 \pm 0.03$ MPa·mm is obtained.

299 Considering this value of the critical energy release rate, $\sigma_0$ and $\ell_c$ are determined, for each tested sample, as the values satisfying Eq. (4), since $\ell_c < \ell^*$ for all the investigated configurations.

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301 Consequently, following this procedure for all the tested samples, $\sigma_0 = 61 \pm 17$ MPa and $\ell_c = 0.09 \pm 0.04$ mm are determined. It is worth noting that these values are influenced by the assumptions made during the analyses, particularly the submodeling hypothesis. In this study, we considered a 2D FE model with plane strain conditions, whereas slightly different results could be obtained if a 3D FE model was considered.

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307 The value of the inherent material strength is close to the tensile strength determined by Ferrian *et al.* [41] ($\sigma_{UTS} = 67 \pm 4.8$ MPa), which was derived for the same material from the bending strength experimentally measured through TPB tests. This finding also aligns with observations regarding brittle and quasi-brittle materials ([32], [34], [35]). Regarding the critical crack advance $\ell_c$, it keeps approximately constant as the hole eccentricity and radius vary, in agreement with the observations of Ferrian *et al.* [41] and the results obtained by [13] for a circular hole subjected to biaxial loading.
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312

313 For the sake of completeness, after computing the average stress field $\bar{\sigma}_{xx}(\ell)$ and IERR $G_{inc}(\ell)$ functions corresponding to the initiation load $U_c$, the pairs of values ($\sigma_0, G_{Ic}$) that satisfy Eq. (4) are depicted in Figs. 10a and b, for the tested samples with $\rho = 4$ mm, $e = 0$ mm and $\rho = 4$ mm, $e = 20$ mm, respectively. In both figures, red circles highlight the $\sigma_0$ values obtained by considering $G_{Ic} = 0.16$ MPa·mm.
314
315
316
317
Moreover, the critical crack advance $\ell_c$ corresponding to each couple of values ($\sigma_0$, $G_{lc}$) solving Eq. (4) are reported in Figs. 10c and d. Considering $G_{lc} = 0.16 \text{ MPa}\cdot\text{mm}$, in both plots the critical crack advance assumes small values ranging between 0.04 and 0.05 mm.

<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /></figure>

<figure><img src="image_16.png" /></figure>

Fig. 10 Couples of values $\sigma_0$ and $G_{lc}$ satisfying FFM system (4) for the four tested samples with $\rho = 4$ mm, (a) $e = 0$ mm and (b) $e = 20$ mm. Additionally, critical crack advance values for each ($\sigma_0$, $G_{lc}$) pair are depicted in Fig. 10c and 10d for $e = 0$ mm and $e = 20$ mm, respectively.

Finally, in Fig. 11a and b the stress and energy conditions imposed by the FFM system (4), are reported for the samples with $\rho = 4$ mm, $e = 0$ mm and $\rho = 4$ mm, $e = 20$ mm, respectively, considering $G_{lc} = 0.16 \pm 0.03$ and $\sigma_0 = 61 \pm 17$ MPa. For these values of $G_{lc}$ and $\sigma_0$, the value of the intersection between the stress and energy conditions, is ranging between 1.1 and 1.3 for the geometries with a central hole and between 1.3 and 1.4 for $e = 20$ mm.
<figure><img src="image_17.png" /><figcaption>Fig. 11 Stress and energy conditions for the four samples with $ρ = 4$ mm, (a) $e = 0$ mm and (b) $e = 20$ mm.</figcaption></figure>

## 6. Conclusions

In this study, we investigated the failure behavior of PMMA disks under uniaxial compression loading. Samples were characterized by the presence of either a central or eccentric circular hole. Initially, elastic material properties were determined by implementing a coupled approach, considering the displacement fields measured through DIC techniques and performing 2D linear elastic FEA. The values resulting for Young's modulus and Poisson's ratio were compared with those obtained by tensile testing of plain samples, validating the effectiveness of the proposed methodology. Then, a full finite element implementation of the FFM approach is presented. This criterion is implemented by application to the contour of a circular subdomain of the displacements measured by DIC, just before crack initiation. In this way, a better representation of real loading conditions is achieved compared to standard FFM implementations where idealized prescribed force or displacement are used. The value of the critical energy release rate $G_{Ic}$ is determined from the analysis of the stable crack growth phase using displacements determined from DIC. Considering the value of $G_{Ic}$ thus estimated, this method enables precise estimation of the critical crack advance and inherent material strength.

Based on these findings, future research directions could involve the development of an experimental setup, for instance considering a different material or geometrical features, to measure accurately the critical crack advance. The experimental values obtained could then be compared to those predicted by the FFM criterion, further validating the approach.
## 353 Appenxdix: Determination of elastic properties

The elastic properties reported in Table 2 are determined by exploiting the DIC technique. The tensile and compressive moduli ($E_t$ and $E_c$ respectively) are compared for the PMMA tested by Ferrian *et al.* [41]. Regarding the tensile modulus $E_t$ and Poisson's ratio $\nu$, they were estimated following ASTM D638-14 [40] standard code. Tensile tests were performed on plain specimens with dimensions of 70 mm length, 8 mm thickness, and 20 mm width. The strain fields, necessary to compute $E_t$ and $\nu$, are obtained by implementing DIC analysis and exploiting the strain gage tool available in the software Ufreckles in a zone spanning over the specimen width.

<table><caption>Table 2 PMMA mechanical properties.</caption><thead><tr><th>E<sub>t</sub> [GPa]</th><th>E<sub>c</sub> [GPa]</th><th>ν</th></tr></thead><tbody><tr><td>2.7 ± 0.03</td><td>2.7 ± 0.12</td><td>0.35 ± 0.014</td></tr></tbody></table>

<figure><img src="image_18.png" /></figure>

<caption>Fig. 4 (a) Detail of the investigated ROI enclosed by the two concentrical dashed circles and (b) displacement field $U_y$ obtained by DIC for a sample with $e = 0$ mm and $\rho = 4$ mm. Please note that $U_{top}$ and $U_{bottom}$ denote the displacements close to the upper and lower contact zone, respectively.</caption>

The compressive modulus $E_c$ is ultimately determined through a coupled approach, considering the displacement fields measured through DIC techniques and performing 2D linear elastic FEA. While in some studies, such as Molimard *et al.* [52] and Lecompte *et al.* [53], strain fields are measured to determine elastic properties, the present work directly compares the displacement fields instead. This approach aims to avoid amplifying measurement noise associated with numerical differentiation, as highlighted by Wang *et al.* [54] and Azzouna *et al.* [55].
372 Focusing on the configurations with a central hole presented in the previous section, the displacement
373 fields $U_x$ and $U_y$ characterizing the sample surface under a specific applied load are determined using
374 DIC for each sample. As presented in Section 3, the DIC measurements are performed using a camera
375 with a resolution of 2392 (H) × 2774 (V) pixels and an acquisition rate of 1 fps. To achieve a good
376 balance between surface partitioning and measurement noise, an unstructured mesh with 3-node
377 triangular elements is applied using the software Ufreckles [51], considering a mesh size of 60 pixels,
378 equal to approximately 1.25 mm. The displacement values of the mesh nodes are evaluated within
379 the ROI enclosed by the two concentric dashed circles depicted in Fig. 4a. Furthermore, the
380 displacements close to the two contact zones, denoted as $U_{\text{top}}$ and $U_{\text{bottom}}$, are also obtained using DIC
381 (Fig. 4a).

382 To assess the values of $E_c$ and $\nu$, for each tested sample presenting a central hole, 2D linear elastic
383 FEA are performed by considering plane strain conditions and modeling the full geometry of the
384 sample. A mesh with 4-node bilinear elements is adopted setting the minimum mesh size equal to
385 0.05 mm, following a convergence analysis. Boundary conditions are set by imposing $U_{\text{top}}$ and $U_{\text{bottom}}$
386 on the two flat ends. Subsequently, the displacement fields $U_x$ and $U_y$ are extracted from the same
387 points evaluated by DIC. Finally, $E_c$ and $\nu$ are determined by satisfying the following two
388 requirements:

389 - the reaction force provided by FEA must equal the experimental one;

390 - the total deviation between the displacement fields determined by DIC and those obtained by
391 FEA must be minimized. At each point, the percentage deviation is computed as
392 $\text{dev}U_i(x, y) = |U_i^{\text{DIC}}(x, y) - U_i^{\text{FEA}}(x, y)| / (\max(U_i^{\text{DIC}}) - \min(U_i^{\text{DIC}})) \times 100$, where $(i = x, y)$,
393 $U_i^{\text{DIC}}$ and $U_i^{\text{FEA}}$ are the displacement values determined through DIC and FEA, respectively.

394 This procedure is repeated for each of the 15 tested samples presenting a central hole. For the sake of
395 clarity, the displacement field $U_y$ obtained by DIC is depicted for only one tested sample with
396 $e = 0$ mm and $\rho = 4$ mm, as shown in Fig. 4b. For this specimen, deviations between the displacement
397 values obtained by DIC and FEA are lower than 5%, as illustrated in Figs. 5 a, b.
<figure><img src="image_19.png" /><figcaption>Fig. 5 Percentual deviations between displacement fields (a) $U_x$ and (b) $U_y$ obtained by DIC and FEA, for a sample with $e = 0$ mm and $\rho = 4$ mm.</figcaption></figure>

The average values of the compressive modulus and Poisson's ratio estimated with this procedure are $E_c = 2.7 \pm 0.12$ GPa and $\nu = 0.36 \pm 0.02$, respectively. These are typical results for PPMA and closely align with those obtained from tensile tests on plain specimens [41], indicating the isotropic behavior of the tested PMMA and affirming the effectiveness of the proposed methodology.

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