# Analysis of mode I fracture propagation in wood using high-resolution full-field measurements and imaging

Arthur Bontemps, Raphaël Langlois, Benoît Blaysat, Gaël Godi, Rostand

Moutou Pitti, Emmy Voyer, Joseph Gril

► To cite this version:

Arthur Bontemps, Raphaël Langlois, Benoît Blaysat, Gaël Godi, Rostand Moutou Pitti, et al. Analy-sis of mode I fracture propagation in wood using high-resolution full-field measurements and imaging. 2025. hal-05248863

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Analysis of mode I fracture propagation in wood
using high-resolution full-field measurements and
imaging

Arthur Bontemps<sup>1,2*†</sup>, Raphaël Langlois<sup>1,3*†</sup>, Benoît Blaysat<sup>1,4</sup>, Gaël Godi<sup>1</sup>, Rostand Moutou Pitti<sup>1,5</sup>, Emmy Voyer<sup>6</sup>, Joseph Gril<sup>1,7</sup>

<sup>1</sup>Clermont Auvergne Université, CNRS, Clermont Auvergne INP,
Institut Pascal, F-63000, Clermont-Ferrand.

<sup>2</sup>now at INSA Lyon, CNRS, GEOMAS, EA 7495, F-69621 Villeurbanne,
France.

<sup>3</sup>now at INSA Lyon, CNRS, LaMCoS, UMR5259, F-69621 Villeurbanne,
France.

<sup>4</sup>Institut Universitaire de France (IUF), Paris, France.

<sup>5</sup>CENAREST, IRT, BP 14070, Libreville, Gabon.

<sup>6</sup>Clermont Auvergne Université, CNRS, Laboratoire Magma et Volcans,
F-63000, Clermont-Ferrand, France.

<sup>7</sup>Université Clermont Auvergne, INRAE, PIAF, F-63000,
Clermont-Ferrand, France.

*Corresponding author(s). E-mail(s): arthur.bontemps@insa-lyon.fr; raphael.langlois@insa-lyon.fr;

Contributing authors: benoit.blaysat@uca.fr; gael.godi@uca.fr; rostand.moutou_pitti@uca.fr; emmy.voyer@uca.fr; joseph.gril@cnrs.fr; †These authors contributed equally to this work.

License: CC-BY @TheAuthors
Abstract

Experimental characterization of wood fracture properties raises a number of challenges: crack tip location or the application of equivalent linear elastic fracture mechanics theory to a heterogeneous and anisotropic material. This article presents the application of the Localized Spectrum Analysis, a full-field measurement method, to obtain high-resolution strain maps for the analysis of three mode I crack propagation tests on TDCB silver fir wood specimens. The strain maps presented enabled us to identify annual rings through stiffness variations, to observe internal cracks through abnormal compression zones, and to determine the material's resistance-curves, therefore its critical energy release rate. This work opens up interesting prospects for the use of new measurement methods in the characterization of wood fracture properties.

**Keywords:** Fracture mechanics, Full-field measurement method, Internal cracks, R-curve, Wood
# Nomenclature

## Abbreviations

41 DCB

42 DIC

43 eqLEFM

44 FEM

45 FPZ

46 LEFM

47 LSA

48 LTENF

49 MC

50 MMCG

51 RL and TL

53 TDCB

54 WST

## Notations

56 $\Delta$

57 $S_i$

## Variable names

59 $S$

60 $u$

61 $X$

62 $\varphi$

63 $\rho$

Double Cantilever Beam

Digital Image Correlation

Equivalent Linear Elastic Fracture Mechanics

Finite Element Model

Fracture Process Zone

Linear Elastic Fracture Mechanics

Localized Spectrum Analysis

Linear Tapered End Notched Flexure

Moisture Content

Mixed-Mode Crack Growth

Respectively Radial-Longitudinal and Tangential-Longitudinal orientation

Tapered Double Cantilever Beam

Wedge Splitting Test

Variation

Beam number $i$

Gray-level distribution on a given image

Displacement vector

Position vector

Phase vector

Density
<table>
  <tr>
    <td>64</td>
    <td>σ</td>
    <td>Standard deviation</td>
  </tr>
  <tr>
    <td>65</td>
    <td>a, a<sub>c</sub></td>
    <td>Respectively crack length and equivalent critical crack length when the FPZ is developed</td>
  </tr>
  <tr>
    <td>66</td>
    <td>d</td>
    <td>Thickness of a sample</td>
  </tr>
  <tr>
    <td>67</td>
    <td>E<sub>L</sub></td>
    <td>Longitudinal elastic modulus</td>
  </tr>
  <tr>
    <td>68</td>
    <td>F</td>
    <td>Force measured by the load cell</td>
  </tr>
  <tr>
    <td>69</td>
    <td>G<sub>Ic</sub></td>
    <td>Critical energy release rate in mode I</td>
  </tr>
  <tr>
    <td>70</td>
    <td>G<sub>ic</sub>, G<sub>mc</sub></td>
    <td>Minimum and maximum energy release rate estimated in mode I</td>
  </tr>
  <tr>
    <td>71</td>
    <td>h<sub>1</sub>, h<sub>2</sub></td>
    <td>Sample's height 1 and 2</td>
  </tr>
  <tr>
    <td>72</td>
    <td>J</td>
    <td>Compliance</td>
  </tr>
  <tr>
    <td>73</td>
    <td>k</td>
    <td>Stiffness of a sample</td>
  </tr>
  <tr>
    <td>74</td>
    <td>L</td>
    <td>Length of a sample</td>
  </tr>
  <tr>
    <td>75</td>
    <td>L<sub>n</sub></td>
    <td>Notch length</td>
  </tr>
  <tr>
    <td>76</td>
    <td>p</td>
    <td>Checkerboard period</td>
  </tr>
  <tr>
    <td>77</td>
    <td>r(X)</td>
    <td>Gray level residual</td>
  </tr>
  <tr>
    <td>78</td>
    <td>u</td>
    <td>Displacement measured by Zwick machine</td>
  </tr>
</table>
# 1 Introduction

The structural integrity and reliability of wooden constructions are of paramount importance in civil engineering. This is a crucial topic for predicting and preventing structural failures. Wood, being a heterogeneous, anisotropic and hygroscopic material with inter and intra species variability, presents unique challenges in understanding its fracture mechanics. Studying diverse wood species, including hardwood, is essential given the difficulty in predicting which species will survive to climate change and what the resource will be in the future (Menezes-Silva et al. 2019). We must gain a deeper understanding of the properties of wood used in structures, encompassing both solid timber and processed solid wood products (e.g. glued laminated beams or cross-laminated panels). Furthermore, the current uncertainty associated with the diversity of mechanical performances of timber impedes the use of species that hold future potential in their forest availability or classification quality. Presently, the load-bearing capacity of a wooden notched-beam design with Eurocode 5 is determined using the Gustafsson method (AFNOR 2010; Gustafsson 1988). This approach uses Linear Elastic Fracture Mechanics (LEFM) that fits to brittle material only, with Spruce (*Picea abies*) properties. In addition to being more than just Spruce, wood is not a brittle material either (Morel et al. 2005).

Recent publications focused on the measurement of mixed-mode fracture prop- erties of temperate softwoods and tropical hardwoods (Biyo'o et al. 2024; Sorin et al. 2022; Odounga et al. 2019). The main difficulties encountered in measuring such properties are: (i) wood is a quasi-brittle material, hence LEFM theory should not be directly applied otherwise the energy release rate is overestimated (Kunecký et al. 2024); (ii) experimental measurement of the crack tip position is delicate because of the Fracture Process Zone (FPZ) that is developing in a quasi-brittle material (Sorin et al. 2022). The quasi-brittle behaviour is currently analysed using equivalent LEFM (eqLEFM), that considers the FPZ development as a fictitious crack propagation
in a brittle material, and then uses the tool of LEFM theory such as compliance method with an equivalent crack length (Morel et al. 2005). In the same time, two main approaches are currently considered to estimate this equivalent crack length: (i) using a finite element model (FEM) that correlates the compliance loss due to FPZ development with an equivalent crack propagation (Sorin et al. 2022); (ii) directly estimating the crack tip position using strain field obtained with full-field measurement. The problems with using a FEM are the hypothesis related to the material behaviour: homogeneity, elasticity, etc. On the other hand, locating the crack tip in wood is a sensitive step since micro-cracking or crack-bridging occur in its vicinity. Based on a strain field measurement, different criteria are proposed to locate the crack tip (Kunecký et al. 2024; Ostapska et al. 2020). In any case, the outcome strongly depends on the strain field resolution. Recent studies in experimental mechanics have shown that a checkerboard with periodic pattern provides the optimal resolution for full-field measurements (Bomarito et al. 2017). Such checkerboard is not easily processed through Digital Image Correlation (DIC) but rather through a Fourier-based method like Localized Spectrum Analysis (LSA) to minimize the optical residual and extract the displacement field (Grédiac et al. 2019).

The present article presents an analysis of mode I crack tests on Tapered Double Cantilever Beam (TDCB) samples of Silver Fir (*Abies alba*) wood using checkerboard processed by LSA. There are 18 failure modes in wood (6 orientations with 3 modes), but the topical mode in civil engineering is mode I in the longitudinal direction and is therefore the one studied here. The article starts with material description, tensile tests conditions and methods used, reminders on the LSA and the crack tip estimation processes. In the results section the force-displacement curves and strain fields obtained are presented, highlighting the precision of the observations that can be achieved, allowing a fine analysis of a bridging event. The analysis includes energy release rate estimation and comparison with the literature, and internal crack identification thanks to strain field observation.
## 2 Material and methods

### 2.1 Wood TDCB specimens

Three samples of silver fir, denoted S1, S2, and S3, were machined into Tapered Double Cantilever Beam (TDCB) specimen format. The wood originates from a beam used in a previous study aimed at characterizing the silver fir of the Auvergne Rhône-Alpes region (Bontemps et al. 2024). Samples properties are given in Table 1, with geometrical variables shown in Fig. 1, $\rho$ the density, $MC_e$ the estimated moisture content (MC) and $E_L$ the longitudinal modulus of elasticity. This latter was measured based on the eigen frequencies measured in bending vibration (i.e. BING© method) (Brancheriau 2024). The geometry of the TDCB specimens (Fig. 1) is based on a slender increase of the cross section along the extent of the crack, which leads to an increasing stiffness that ensures stable crack propagation. The consequence is a much clearer horizontal asymptote in the Resistance-curves, thus providing a better estimation of fracture energy release rate. Pictures of all samples are presented in Section A. As all three specimens were stabilized in a constant environment with relative humidity of 40% and temperature of 20°C, corresponding to an estimated 8% MC using hygroscopic equilibrium abacus of Eurocode 5 (AFNOR 2010). As much as possible, the samples were quarter-sawn (Section A), so that the crack only propagates in the Radial-Longitudinal (RL) mode (the first letter indicating the normal direction of the crack plane and the second the direction of crack propagation). An initial notch of 30 mm was made using a bandsaw. Such notch went through earlywood for S2, while for S1 and S3 they were cut through both earlywood and latewood due to annual ring angles (Section A). The initial notch tips are shown in Section A. The bandsaw did not result in a smooth rounded notch tip due to the saw pattern. Therefore, the crack did not propagate in the middle of the notch but rather along its edges (Section A).
<figure><img src="image_2.png" /><figcaption>**Fig. 1:** Drawing and dimensionning of the TDCB specimens. Initial notch is 30 mm long and annual rings are quarter-sawn oriented. Crack propagates longitudinally.</figcaption></figure>

<table><caption>**Table 1:** Sample characterization</caption><thead><tr><th>Sample</th><th>d (mm)</th><th>h<sub>1</sub> (mm)</th><th>h<sub>2</sub> (mm)</th><th>L (mm)</th><th>ρ (kg.m<sup>-3</sup>)</th><th>E<sub>L</sub> (MPa)</th><th>MC<sub>e</sub> (%)</th><th>L<sub>n</sub></th></tr></thead><tbody><tr><td>S1</td><td>12.75</td><td>58.4</td><td>86.9</td><td>107.7</td><td>526</td><td>9590</td><td>8</td><td>32</td></tr><tr><td>S2</td><td>12.73</td><td>57.7</td><td>86.2</td><td>106.6</td><td>454</td><td>9590</td><td>8</td><td>31.5</td></tr><tr><td>S3</td><td>12.93</td><td>57.7</td><td>86.7</td><td>106.5</td><td>472</td><td>9590</td><td>8</td><td>32.5</td></tr></tbody></table>

Before testing, a face of the samples was painted white and then a checkerboard was “printed” using a laser engraving machine (Bouyra et al. 2022). The obtained surface texture will be used as an information carrier for full-field kinematic measurements, Fig. 2.
<figure><img src="image_3.png" /><figcaption>Fig. 2: S2 sample: (a) the reference image, which defines the unloaded state, and (b) close-up of the checkerboard pattern. This figure corresponds to the red square of (a)</figcaption></figure>

## 2.2 Tensile tests

The tensile tests were performed with a Zwick/Roell machine, using parameters given in Table 2. An Allied Vision Prosilica GT6600 B/C GigE camera was used to film the checkerboard deformation, with parameters given in Table 3. The displacement imposed speed of 0.005 mm.s⁻¹ ensured a stable crack propagation (Odounga et al. 2018). The tensile test of specimen Si is illustrated in Fig. 3.

<table><caption>Table 2: Tensile tests machine characteristics</caption><thead><tr><td>Load cell</td><td>Accuracy class</td><td>Range</td><td>Error (% of the nominal force)</td><td>Displacement imposed speed</td></tr></thead><tbody><tr><td>Xforce P Zwick Roell</td><td>0.5</td><td>from 5 N to 10 kN</td><td>0.2</td><td>0.005 mm.s<sup>-1</sup></td></tr></tbody></table>

Synchronisation between camera and force measurements with the load cell are explained in Section D. The illumination of the observed surface is achieved using two light spots, in order to maximize homogeneity.
<table><caption>**Table 3:** Images and acquisition characteristics</caption><thead><tr><th>Experiment</th><th>S1</th><th>S2</th><th>S3</th></tr></thead><tbody><tr><td colspan="4">Camera characteristics</td></tr><tr><td>Model</td><td colspan="3">AVT Prosilica GT6600</td></tr><tr><td>Sensor size</td><td colspan="3">6576 × 4384 pixel<sup>2</sup></td></tr><tr><td>Dynamic range</td><td colspan="3">8 bit</td></tr><tr><td>Lens</td><td colspan="3">Samyang T3.1/100 mm</td></tr><tr><td>Aperture</td><td colspan="3">f/11</td></tr><tr><td>Magnification</td><td colspan="3">1.2</td></tr><tr><td>Exposure</td><td colspan="3">425 ms</td></tr><tr><td>Acquisition frequency</td><td colspan="3">1 Hz</td></tr><tr><td colspan="4">Method characteristics</td></tr><tr><td>Pitch <sup>p</sup>&frasl;<sub>&radic;2</sub> (μm)</td><td colspan="2">45</td><td>70</td></tr><tr><td>Pixel size (μm/Pix)</td><td>9.95</td><td>9.86</td><td>20.60</td></tr><tr><td>Extraction parameter σ (Pix)</td><td>6.34</td><td>6.45</td><td>4.80</td></tr></tbody></table>

<figure><img src="image_4.png" /><figcaption>**Fig. 3:** Tensile test conditions illustration.</figcaption></figure>

## 2.3 Kinematic measurement and crack length estimation

### 2.3.1 Localized Spectrum Analysis

2D displacement fields are retrieved from pairs of checkerboard images using the so-called Localized Spectrum Analysis (LSA) (Grédiac et al. 2019).
Let be $\mathbf{S}_{\tau}$ and $\mathbf{S}_{0}$ respectively the gray-level distribution of the deformed image and of the reference image. Assuming a periodic pattern of pitch $p$ in directions $\mathbf{e}_{\alpha}$ and $\mathbf{e}_{\beta}$ (with $\beta = \alpha \pm \frac{\pi}{2}$), $\mathbf{S}_{\tau}$ writes for $\tau \in [0, t]$:

$$ \mathbf{S}_{\tau}(\mathbf{X}) = \sum_{\ell \in [\alpha, \beta]} \text{frng}(2\pi\mathbf{X} \cdot \mathbf{e}_{\ell}/p + \varphi_{\tau, \ell}(\mathbf{X})) \quad (1) $$

where frng denotes a $2\pi$ periodic function and $\varphi_{k,\ell}$ represents the phase modulation of the periodic carrier wave along the direction $\mathbf{e}_{\ell}$ at time $k$.

The gray level conservation, i.e. $\mathbf{S}_{\tau}(\mathbf{X} + \mathbf{u}(\mathbf{X})) = \mathbf{S}_{0}(\mathbf{X})$, with $\mathbf{u}(\mathbf{X})$ the displacement vector at position $\mathbf{X}$, thus leads to the following link between phase modulations and displacement:

$$ \mathbf{u}_{\tau}(\mathbf{X}) = \sum_{\ell \in [\alpha, \beta]} -\frac{p}{2\pi}(\varphi_{\tau, \ell}(\mathbf{X} + \mathbf{u}(\mathbf{X})) - \varphi_{0, \ell}(\mathbf{X}))\mathbf{e}_{\ell}, \quad (2) $$

In practice, considering the checkerboard pattern, the carrier waves are defined by the square repetition, following their diagonals. $p$ is thus the length of the square diagonal, and $\mathbf{e}_{\alpha}$ the corresponding direction. $\mathbf{e}_{\beta}$ is equal to $\mathbf{e}_{\alpha}$, rotated by $\pi/2$. Phase modulations are extracted by convolution. An interested reader can refer to (OpenLSA). Fig. 2 presents an example of a pattern used to compute the displacement/phases.

### 2.3.2 Crack features estimation

Different techniques are used to estimate crack tip position, such as an optimization of a kinematically admissible displacement field (Ostapska et al. 2020) or a principal strain criterion (Kunecký et al. 2024).
As LSA minimizes optical flow, we can compute a gray level residual $r(\mathbf{X})$ as:

$$r(\mathbf{X}) = \sqrt{(S_0(\mathbf{X}) - S_t(\mathbf{X} + \mathbf{u}(\mathbf{X})))^2}, \quad (3)$$

which showcases the local mismatch between the reference image $S_0(\mathbf{X})$ and the deformed image mapped onto the reference configuration $S_t(\mathbf{X} + \mathbf{u}(\mathbf{X}))$. An underlying assumption used in LSA is that the displacement is continuous within kernel window introduced to compute the phase vectors (similar to the assumption of continuity within subsets for most DIC libraries). Therefore, it is expected to rise sharply around the crack as it naturally induces a displacement discontinuity.

The residual is therefore a good indicator to detect the presence of cracks within a sample (Langlois 2023) and allows one to evaluate the full crack path in the reference configuration as presented in Fig. 4. This map is then used to evaluate a spline that defines the crack path presented in Fig. 5. Using this spline in conjunction with the residual maps for all images, the tip of the crack can be robustly tracked and the length of the crack can be estimated as proposed in (Langlois 2023).

<figure><img src="image_5.png" /><figcaption>Fig. 4: Residual maps for sample S1</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Fig. 5: Crack path (in blue) defined using a single spline</figcaption></figure>
# 3 Results and Discussion

## 3.1 Kinematic results

Fig. 6 presents typical results of displacement, after the removal of rigid-body dis-placements for sample S3 and Fig. 7 presents in-plane strain components for sample S1. Similar strain fields maps for the two other samples are provided in Section C with Figs. 16 and 17. All strain maps show two interesting results:

* Stiffness variations allow to clearly identify earlywood and latewood: high tensile strain is reached in earlywood (red in the strain map), while negative strain is reached in latewood (compressive strain). These annual rings are comparable to the real one (Section A), with greater spacing between annual rings under the crack than above. The crack seems to propagate at the boundary between latewood (under the crack) and earlywood (above the crack), i.e. in the less dense region. A FEM was performed to verify the consistency of these strain maps, and is presented in Section E.

* The crack tip area involves a cross-shape shear stress, suggesting that mode II is also involved in crack propagation. This process was already observed for wood by different authors (Toussaint et al. 2016; Moura et al. 2018), and is observed in the FEM (Section E).

As the field of view used for sample S3 is larger than the two other samples, we can also see a heterogeneous compressive strain localization at the outer right boundary, which is also due to the stiffness discrepancies between early and latewood. Finally, it should be noted that despite the heterogeneities and the complex loading conditions (i.e. crack propagation and sample geometry) the intensity of the $ε_{xx}$ strain is 10 times lower compared to the other strain components which leads to noisier maps. Also some defects on the sample surface are present and lead to artifacts on the strain maps.
<figure><img src="image_7.png" /><figcaption>Fig. 6: Displacement fields provided by LSA for sample S3 at time $t = 178$ s, rigid body motion are removed following the procedure detailed in Section B.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 7: Strain fields provided by LSA for sample S1 for time-step 430</figcaption></figure>

## 3.2 Analysis of crack bridging

Let us now focus on a crack bridging event occurring in sample S1 between the timesteps 437 and 438 whose corresponding strain maps (*yy* components) are given in Fig. 8(a). On these maps, three dotted lines are present and are used to extract the strain profiles
around the crack(s) at three distinct *x* positions chosen close to the crack tip. At timestep 437: (1) the green leftmost line in the damage material, (2) the black middle line is approximately at the crack tip and (3) the purple rightmost line in “uncracked” material.

Fig. 8(b) present the evolution between the timesteps 424 and 444 of the strain profiles (i.e. $\varepsilon_{yy}$ vs $y$) along the aforementioned dotted lines, each plot corresponding to a single line, the leftmost plot to the green line, the middle plot corresponding to the black line, and the rightmost plot to the purple line. The time evolution is given by the color bar, and each color corresponds to an individual timestep going from 424 (dark blue) to 444 (bright yellow).

In these profiles, three features can be observed:

1. the presence of a fully formed crack leads to a “singular” strain (dark gray area on Fig. 8(b),

2. the crack propagation leads to a local unloading which can be observed by the decreasing positive strains for all $y < 2200$,

3. a compressive strain localization is observed around $y = 2275$ (light gray area on Fig. 8(b)) before timestep 438 on all three plots. This suggests that an internal crack can be identified by a compression zone in a strain field.

After the secondary crack apparition, the strain in the leftmost plot remains compressive around $y = 2275$ (light gray area on Fig. 8(b)) while in the middle and rightmost plots it switches to tension. For the rightmost plot it even becomes “singular” as the displacement discontinuity appears and slightly propagates.

Those observations are consistent with a qualitative simulation carried out and briefly presented in Section E, where the presence of a non-through crack introduces a compressive load locally as depicted on Fig. 9, which presents the strain field $yy$-component generated by the introduction of a secondary crack (positive strain means local decrease of the strain intensity), more details are provided in Section E.
<figure><img src="image_9.png" /><figcaption>(a) Strain field $\varepsilon_{yy}$ before and after fiber bridging occurs. The colored vertical lines define the position used in (b) to extract the strain. The dark and ligh gray rectangles correspond to the highlighted position in Fig. 8(b).</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>(b) Evolution of the strain $\varepsilon_{yy}$ for several time-steps along three lines “perpendicular” to the crack propagation direction. The black dotted line corresponds to zero strains. From left to right, each plot corresponds to a vertical line of a given color in (a). The dark (resp. light) gray area corresponds to the vertical position of the initial (resp. secondary) crack showed within the dark (resp. light) gray rectangle of figure 8(a) before (resp. after) bridging.</figcaption></figure>

<caption>**Fig. 8:** Evolution of the strain during a fiber bridging event.</caption>

## 3.3 Evaluated crack length and stiffness results

Fig. 10 shows the force-displacement curve (based on the testing machine measurements) and the crack length-displacement curve (based on the camera measurements). For each sample a linear part during the beginning of the test is identified and allows to calculate a stiffness $k = F/u$, where F and u are respectively the force and the displacement imposed measured by the Zwick machine. The stiffness results are reported in Table 4. All samples also exhibits a softening stage, represented in grey in Fig. 10, a distinctive feature of quasi-brittle materials that highlight a Fracture Process Zone
<figure><img src="image_11.png" /><figcaption>**Fig. 9:** Stress introduced by the addition of a secondary non-through crack, here positive strain means local decrease of the strain intensity.</figcaption></figure>

(FPZ) (Yu et al. 2019). Crack measurements represented by dots in Fig. 10 show that when the load-deflection curve starts decreasing, the crack finally propagates. A sudden load drop is also identified for S3, before any crack propagation was observed. This can be explained by crack bridges as observed in Section A.

<table><caption>**Table 4:** Tensile tests raw results</caption><thead><tr><td>Sample</td><td>Stiffness k (N.mm<sup>-1</sup>)</td></tr></thead><tbody><tr><td>S1</td><td>532</td></tr><tr><td>S2</td><td>493</td></tr><tr><td>S3</td><td>728</td></tr></tbody></table>

## 3.4 Compliance method

This type of test enables the energy release rate to be estimated on the basis of an energetic approach. Here, a simple LEFM approach is used. For a brittle material, one can assume that all work given by the external load is converted into an elastic energy
<figure><img src="image_12.png" /><figcaption>Fig. 10: Load-displacement and crack length-displacement curve for all three specimens. The stiffness and the FPZ development are illustrated.</figcaption></figure>

and a possible crack propagation energy. This latter is assumed to be $G_c\Delta S$ where $G_c$ is the critical energy release rate and $\Delta S = d.\Delta a$ the new external surfaces created by crack propagation. Based on these assumptions, the compliance method provides an expression for $G_c$ (Bouvet 2020):

$$G_c = \frac{F^2 \partial J}{2.d \partial a} \qquad (4)$$

where $J = 1/k = u/F$ is the compliance of the sample at a given crack length $a$. In Fig. 11 the compliance and the estimated $G_c$ are plotted as a function of the measured crack length $a$ (so-called Resistance-curve). A cubic polynomial function is
fitted to correlate *J* and *a*, so that compliance and crack length experimental noise
does not affect the derivative in Eq. (4). χ² is returned in Fig. 11 as it represents the
root mean square (RMS) of the error between polynomial fitting and experimental
results. The maximum and minimum *G*<sub>c</sub> are respectively noted *G*<sub>mc</sub> and *G*<sub>ic</sub>. As
expected for a Resistance-curve evolution, *G*<sub>c</sub> starts to increase, normally while the
FPZ is developping until it reaches its maximum size, then *G*<sub>c</sub> shows a plateau before
it starts decreasing for S2 and S3 samples. For S1, *G*<sub>c</sub> quickly decreases after reaching
a maximum value. Possibly, *G*<sub>c</sub> decreases when the FPZ boundaries reaches the edges
of the specimens, leading to a precocious failure. Therefore, S1 sample may be a
little too short to determine *G*<sub>c</sub>. *G*<sub>mc</sub> is reached at the critical crack length noted *a*<sub>c</sub>,
between 22 and 28 mm for all samples, when the FPZ has fully developed. In Table 5
the results are compared with results from literature concerning softwoods. Different
types of test are considered, such as Linear Tapered End Notched Flexure (LTENF),
Wedge Splitting Test (WST), Double Cantilever Beam (DCB). Wood density and
fracture properties are not correlated whence note indicated in the table (Sorin et
al. 2022). The results from this study should be compare to those of (Angellier et
al. 2017) with DCB specimens of silver fir (570 J.m<sup>-2</sup>). Only S2 sample is close to
this results, emphasizing the variability of *G*<sub>mc</sub> measurements. Sample S3 stands out
due to its specific characteristics. It exhibits the highest stiffness (728 N.mm<sup>-1</sup>) while
simultaneously having the lowest *G*<sub>ic</sub> (215 J.m<sup>-2</sup>). Possibly, an internal crack can
explain the sudden load drop not visible in the strain map (Section 3.3), decreasing
the apparent energy release rate. Moreover, a crack bridging event is clearly observed
in the post-mortem sample image (Fig. 15). Some of the anatomical features of sample
S3 are observed in section Section 3.5
<table><caption>**Table 5:** Comparison of the energy release rate of softwoods from literature. $G_{mc}$ column give the average of all tests and Std corresponds to their standard deviation. TL orientation refers to Tangential-Longitudinal crack propagation</caption><thead><tr><th>References</th><th>Wood species</th><th>Geometry</th><th>Orientation</th><th>MC [%]</th><th>G<sub>mc</sub> [J.m<sup>-2</sup>]</th><th>Std</th><th>a<sub>c</sub> (mm)</th></tr></thead><tbody><tr><td>Current study S1</td><td></td><td></td><td></td><td></td><td>688</td><td></td><td>27.4</td></tr><tr><td>Current study S2</td><td>Silver Fir</td><td>TDCB</td><td>RL</td><td>8</td><td>588</td><td></td><td>22.2</td></tr><tr><td>Current study S3</td><td></td><td></td><td></td><td></td><td>215</td><td></td><td>26.7</td></tr><tr><td>(Sorin et al. 2022)</td><td>Spruce</td><td>LTENF</td><td>RL &amp; TL</td><td>12</td><td>190</td><td>69</td><td>12</td></tr><tr><td>(Sorin et al. 2022)</td><td>Maritime Pine</td><td>LTENF</td><td>RL &amp; TL</td><td>12</td><td>500</td><td>210</td><td>24</td></tr><tr><td>(Ostapska et al. 2020)</td><td>Norway spruce</td><td>WST</td><td>RL</td><td>12</td><td>500</td><td>210</td><td>24</td></tr><tr><td>(Angellier et al. 2017)</td><td>Silver fir</td><td>DCB</td><td>RL</td><td>9</td><td>570</td><td></td><td>40</td></tr><tr><td>(Angellier et al. 2017)</td><td>Douglas fir</td><td>DCB</td><td>RL</td><td>9</td><td>750</td><td></td><td>45</td></tr><tr><td>(Morel et al. 2005)</td><td>Norway spruce</td><td>CIB</td><td>RL</td><td>12</td><td>149</td><td></td><td>47</td></tr></tbody></table>

## 3.5 Identification of 3D effects and fibrous bridging

Crack propagation in wood is inherently a 3D phenomenon due to several factors: sample asymmetry caused by the orientation of annual rings, wood anatomy leading to crack bridging, and experimental setup imperfections, among others. This is precisely illustrated in section 3.2. Consequently, calculating a critical energy release rate based solely on crack propagation measurements from a single face, assuming a 2D behavior, inevitably introduces an error. The magnitude of this error depends on the significance of these 3D effects.

Some observations made using a Scanning Electron Microscope (SEM) have allowed us to visualize these effects. The Figs. 12 and 13 presents SEM images of sample 3. It
<figure><img src="image_13.png" /><figcaption>(a) Compliance *J* as a function of the crack length *a* with the cubic polynomial fitting. *G*<sub>mc</sub>, *G*<sub>ic</sub> and *a*<sub>c</sub> are indicated in red.</figcaption></figure>

<caption>Fig. 11: Experimental results analyzed with the compliance method.</caption>

<footnote n="323">323 can be observed that wood rays promote 3D crack propagation, as illustrated in the
Fig. 12, where they facilitate the crack’s transition from one plane to another.
324</footnote>
<figure><img src="image_14.png" /><figcaption>**Fig. 12:** Observation of a crack propagation plane shift in sample 3. Two propagation planes at different heights are observed</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>**Fig. 13:** Illustration of crack deviation caused by wood rays in sample 3</figcaption></figure>
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# 4 Conclusion and perspectives

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The present study analyzes mode I fracture propagation in silver fir wood using full-field measurements with a checkerboard processed via LSA, providing high-resolution strain fields. Three samples were tested. The experimental method reveals annual rings in the strain field due to stiffness variations, even for low strains such as $\epsilon_{xy}$. The crack tip position is identified without assumptions on the material's constitutive model, allowing precise crack path tracking. These data enhance fracture test analysis, for instance, by feeding a FEM that correlates compliance loss with equivalent crack propagation. Internal cracks were also observed before they bridge with the main crack because of an abnormal compression zone in the strain fields. Moreover the energy release rate in mode I ($G_{Ic}$) and the critical crack length ($a_c$) are estimated using LEFM, yielding results consistent with literature for silver fir and similar geometries. The key contribution of this work is the high-resolution strain field and crack tip location, improving mechanical characterization in fracture testing. This is the first study applying the latest full-field measurement techniques to wood.

The main limitation is that 3D effects are not taken into account, requiring symmetry assumptions on crack propagation. Wood fracture behavior is strongly influenced by its anatomical structure, as evidenced by the significant discrepancy between sample S3—exhibiting pronounced anatomical singularities—and the other two. Large standard deviations are commonly reported in fracture tests (Sorin et al. 2022; Ostapska et al. 2020). A multi-scale approach linking wood anatomy to fracture properties would benefit the field. Additionally, real structures are force-loaded rather than displacement-controlled, leading to rapid crack propagation beyond the critical limit. Determining $G_{Ic}$ in a force-imposed test remains challenging due to camera acquisition frequency limitations but would benefit the civil engineering research
community.

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Given the complexity of the tests analyzed through LSA-processed checkerboards, it seems wiser to use this method to better understand the fracture mechanics in wood, rather than to carry out batteries of property characterization tests. For instance, it could be applied on fracture tests with the aim of : (i) comparing temperate and tropical woods in order to better understand the mixed-mode crack propagation dependency with annual rings; (ii) evaluating the impact of water content on crack propagation or (iii) understanding the coupling between wood anatomy and crack path.

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Acknowledgements

The authors would like to take this opportunity to thank the laboratory "Magma et Volcans" of Clermont Ferrand for allowing us to use their Scanning Electron Microscope. Raphaël Langlois and Benoît Blaysat acknowledge support from the ANR Grant ANR-18-CE08-0028-01. This work was also sponsored by the French government research program "Investissements d'Avenir" through the IDEX-ISITE initiative 16-IDEX-0001 (CAP 20-25), and from the AURA regional council.

5 Compliance with ethical standards

On behalf of all authors, the corresponding author states that there is no conflict of
interest.

371 6 Declaration

A CC-BY public copyright license has been applied by the authors to the present
document and will be applied to all subsequent versions up to the Author Accepted
Manuscript arising from this submission, in accordance with the grant's open access
conditions.

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448
449 A Samples pictures

<figure><img src="image_16.png" /><figcaption>**Fig. 14:** Post-mortem pictures of all samples, allowing to observe annual rings and crack propagation.</figcaption></figure>
<figure><img src="image_17.png" /><figcaption>(a) S1</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>(b) S2</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>(c) S3</figcaption></figure>

<caption>**Fig. 15:** Initial notch tip observation using binocular loupe</caption>
## B Rigid body motion estimation

Figure 6 presents displacement fields where the rigid body motion are removed:

$$ \mathbf{u}(\mathbf{X}) = \mathbf{u}_{\text{LSA}}(\mathbf{X}) - \mathbf{u}_{\text{RBM}}(\mathbf{X}) \quad (5) $$

All 2D rigid body displacement $\mathbf{u}_{\text{RBM}}$ reads as:

$$ \mathbf{u}_{\text{RBM}}(\mathbf{X}) = \underbrace{(\mathbf{R} - \mathbf{I}) \times \mathbf{X}}_{\text{Rigid rotation}} + \underbrace{\mathbf{c}}_{\text{Rigid translation}}, \quad (6) $$

$$ = \begin{bmatrix} \cos \theta - 1 & -\sin \theta \\ \sin \theta & \cos \theta - 1 \end{bmatrix} \times \begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} c_x \\ c_y \end{pmatrix}, \quad (7) $$

under the small-angle assumption $\theta \approx 0$, we get $\cos \theta \approx 1$ and $\sin \theta \approx \theta$, allowing the following simplification:

$$ \mathbf{u}_{\text{RBM}}(\mathbf{X}) = \begin{bmatrix} 0 & -\theta \\ \theta & 0 \end{bmatrix} \times \begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} c_x \\ c_y \end{pmatrix}, \quad (8) $$

$$ = \underbrace{\begin{bmatrix} -y & 1 & 0 \\ x & 0 & 1 \end{bmatrix}}_{\mathbf{L}} \times \underbrace{\begin{pmatrix} \theta \\ c_x \\ c_y \end{pmatrix}}_{\mathbf{P}}, \quad (9) $$

projecting the raw displacement results onto the rigid body transformation basis leads to a least-square minimization problem:

$$ \arg \min_{\mathbf{p}} \| \mathbf{u}_{\text{LSA}}(\mathbf{X}) - \mathbf{L} \times \mathbf{p} \|_{2}^{2}, \quad (10) $$
from which the following normal equation rises:

$$L^T L p = L^T u_{LSA}(X). \qquad (11)$$

## C Other kinematic results

<figure><img src="image_20.png" /><figcaption>Fig. 16: Strain fields provided by LSA for sample S2 for time-step 325.</figcaption></figure>

## D Time synchronisation

During each test a deloading-reloading phase was applied from 70 N to 50 N in order to synchronize camera and Zwick datas. This creates a displacement reduction that can be observed in both the displacement measured by the camera and measured by the Zwick machine. The phenomenon is showed in Fig. 18 for sample S2, presenting the
<figure><img src="image_21.png" /><figcaption>Fig. 17: Strain fields provided by LSA for sample S3 for time-step 178.</figcaption></figure>

displacement as a function of time for both camera and Zwick measurements. For each measurement device, 2 points can be identified: the intersection between the loading and deloading phase; the intersection between the deloading and reloading phase. For each point a time interval $\Delta t$ between each measurement is computed. Finally, camera times are translated by $\Delta t_{moy} = \frac{\Delta t_1 + \Delta t_2}{2}$ and the devices are considered synchronized.
<figure><img src="image_22.png" /><figcaption>**Fig. 18:** Determination of the time shifting between camera and Zwick loading machine based on unload-reload step. Sample S2.</figcaption></figure>

## E Finite-Element Modelling

In order to obtain a qualitative comparison for the strain measurements, some simulations were carried out.

### Geometry and mesh

The geometry considered is the one presented in Fig. 1 omitting the loading holes for ease of meshing. A finite element structured mesh of first order hexahedra (and wedges for the top and bottom edges) was generated using `gmsh` (Remacle et al. 2012).

### Constitutive models and crack representation

Perfectly horizontal layers are introduced to represent annual rings with the constitutive parameters for each phase taken from (Phan et al. 2024) (average values) and given in Table 6. It is considered that the anisotropy directions are aligned with the
<table><caption>**Table 6:** Properties of Earlywood (EW) and Latewood (LW)</caption><thead><tr><th>Elastic constants</th><th>EW</th><th>LW</th></tr></thead><tbody><tr><td><i>E</i><sub>L</sub> (GPa)</td><td>8.69</td><td>29.93</td></tr><tr><td><i>E</i><sub>R</sub> (GPa)</td><td>0.2601</td><td>1.862</td></tr><tr><td><i>E</i><sub>T</sub> (GPa)</td><td>0.0589</td><td>2.030</td></tr><tr><td><i>G</i><sub>RT</sub> (GPa)</td><td>0.0103</td><td>0.1951</td></tr><tr><td><i>G</i><sub>LT</sub> (GPa)</td><td>0.386</td><td>1.834</td></tr><tr><td><i>G</i><sub>LR</sub> (GPa)</td><td>0.491</td><td>0.971</td></tr><tr><td><i>ν</i><sub>RT</sub></td><td>0.398</td><td>0.154</td></tr><tr><td><i>ν</i><sub>LT</sub></td><td>0.36</td><td>0.504</td></tr><tr><td><i>ν</i><sub>LR</sub></td><td>0.42</td><td>0.37</td></tr></tbody></table>

480 Cartesian reference frame such that:

$$ \vec{e}_x = \vec{L}, \quad \vec{e}_y = \vec{R}, \quad \vec{e}_z = \vec{T}. $$

481 Finally, cracks are represented by "damaging" some elements, i.e. drastically reducing their stiffness. Two cases are considered: (1) a sample with a unique crack along the middle of the sample and (2) the same with an additional circular non-through crack with -1.5 mm vertical and 3.75 mm horizontal offsets compared to the initial crack tip. The cracks are introduced in the latewood phase as in experiment of sample S1.

486 Loading conditions

487 A simplified loading scenario is used which is not fully representative of the experimental loads, a unitary vertical displacement is imposed on the left boundary: positive above and negative under the prescribed main crack position.

490 Results

491 Figure 19 presents the resulting in-plane components of the computed strain field on the surface of the domain. The simulation fields qualitatively agree with the experimental observation as the following features are captured:

494 1. strain localized in earlywood,
2. “star”-shaped pattern in the shear component around the crack tip,

3. compressive $\epsilon_{xx}$ above and under the crack tip.

The introduction of a secondary mode 1 crack in the heterogeneous anisotropic medium is equivalent, at first order, to the introduction of a local stress field $\tilde{\sigma}$ at the second crack location such that:

$$ (\sigma(x, y, z) + \tilde{\sigma}(x, y, z)) \cdot n(x, y, z) = 0, \qquad (12) $$

which corresponds to the introduction of a free-boundary. The field $\tilde{\sigma}$ introduces a perturbation strain $\tilde{\epsilon}$ which can be approximated:

$$ \tilde{\epsilon} = \epsilon_{SC} - \epsilon_{TC}, \qquad (13) $$

with $\bullet_{SC}$ corresponding to the Single Crack simulation and $\bullet_{TC}$ to the Two Cracks simulation.

Figure 20 presents the perturbation strain field $yy$-component.

This figure showcases that $\tilde{\sigma}$ introduces a local strain unloading (which is expected) which is consistent with the observed compressive strain observed before the crack bridging observed in sample S1.

### Some remarks

The simulation result presented here correspond to vastly simplified model of the experiment on sample S1. There are a numerous improvements which could be made, for instance:

1. annual rings (angles and size variations),

2. better material model (correct orthopric direction + damage model),

3. experimental load applied.
<figure><img src="image_23.png" /><figcaption>**Fig. 19:** Strain fields provided by the simulation for a single crack case</figcaption></figure>
<figure><img src="image_24.png" /><figcaption>Fig. 20: Difference between the second and first simulation</figcaption></figure>