# Modeling the influence of crack-lip pressure on the energy release rate in orthotropic materials: Effects of moisture content and mixed mode decoupling

Loic Chrislin Nguedjio, Jérôme Afoutou, Rostand Moutou Pitti, Benoit

Blaysat, Frédéric Dubois, Naman Recho, Pierre Kisito Talla

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Loic Chrislin Nguedjio, Jérôme Afoutou, Rostand Moutou Pitti, Benoit Blaysat, Frédéric Dubois, et al. Modeling the influence of crack-lip pressure on the energy release rate in orthotropic materials: Effects of moisture content and mixed mode decoupling. 26th French Mechanics Congress, AFM, Aug 2025, Metz, France. hal-05229964

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# Modeling the influence of crack-lip pressure on the energy release rate in orthotropic materials : Effects of moisture content and mixed mode decoupling

Loïc Chrislin NGUEDJIO¹,², Jérôme AFOUTOU³, Rostand MOUTOU PITTI¹,⁴, Benoit BLAYSAT¹,⁵, Frédéric DUBOIS⁶, Naman RECHO¹, Pierre Kisito TALLA²

1. Université Clermont Auvergne, Clermont Auvergne INP, CNRS, Institut Pascal, F-63000, Clermont-Ferrand, France : loic.nguedjio@doctorant.uca.fr, rostand.moutou_pitti@uca.fr, benoit.blaysat@uca.fr, naman.recho@uca.fr

2. Unité de Recherche de Mécanique et de Modélisation des Systèmes des Physiques (UR2MSP) ; Département de Physiques, Faculté des Sciences, Université de Dschang, BP 67, Dschang, Cameroun : loic.nguedjio@doctorant.uca.fr, tpierrekisito@yahoo.com

3. Laboratoire des Technologies Innovantes, EA 3899 – Université de Picardie Jules Verne, IUT Amiens, Avenue des Facultés – Le Bailly, 80025 Amiens Cedex 1, France : sonagnon.jerome.afoutou@u-picardie.fr

4. CENAREST, IRT, BP14070, Libreville, Gabon : rostand.moutou_pitti@uca.fr

5. Institut Universitaire de France, Paris, France ; benoit.blaysat@uca.fr

6. Université de Limoges, GC2D, UR 14477, 19300, Egletons, France : frederic.dubois@unilim.fr

## Abstract :

The aim of this work is to incorporate the influence of pressure at the crack lips into a two-dimensional calculation of the energy release rate. Cracks in materials like wood often result not only from mechanical stresses but also from residual internal stresses induced by thermo-hydric environmental gradients. These factors significantly affect crack opening and propagation, and accounting for them enhances the predictive accuracy of models for cracked structures in complex environments. In this study, the energy release rate is determined by introducing a pressure condition at the crack lips, utilizing the generalized T invariant integral. This approach is adapted to orthotropic behavior in a two-dimensional framework and leveraging real and virtual fields to enable the decoupling of mixed fracture modes. The proposed new analytical path-independent integral is implemented in a finite element model using Cast3M software, allowing the calculation of the energy release rate for opening mode, shear mode, and mixed mode fractures. Simulations performed on an MMCG wood specimen geometry validate the path integration domain independency by analyzing the energy release rate as a function of integration crowns. Moreover, this study evaluates the influence of crack-lip pressure on the energy release rate, providing more realistic insights into fracture behavior.

**Key words :** Crack lips, orthotropic material, MMCG specimen, path-independent integral, energy release rate.
# 1 Introduction

The issue of crack formation in wooden structural elements remains one of the most critical challenges today, particularly when external thermal or moisture-induced stresses are involved. These additional loads complicate the analysis of wood behavior due to its heterogeneous and orthotropic nature. Crack initiation in wood often results from mechanical stresses, typically sustained over long durations in defect-free specimens. Under variable environmental conditions, crack propagation becomes even more complex, as thermo-hydro stresses induce dimensional changes in the structural elements. Linear Elastic Fracture Mechanics (LEFM) provides an energy-based framework, originating from the work of Griffith [1], to evaluate the energy release rate required for the formation of new crack surfaces. In this study, the energy release rate is computed using invariant integrals [2], which due to their path-independence allow evaluation far from the crack tip [3]. Among these, the J-integral, introduced by Rice [4], is widely employed to calculate the energy release rate in elastic media under Mode I and Mode II loading, in both two- and three-dimensional configurations [5].

However, for orthotropic materials like wood, there is a growing need for an adapted version of the J-integral that ensures mode decoupling under mixed-mode loading. Moutou Pitti et al. [6] addressed this issue by introducing the invariant M-integral. Nevertheless, this approach remains inadequate for analyzing wood behavior under varying moisture and temperature conditions. To this end, the T-integral [7] was proposed to evaluate the energy release rate in the presence of thermal and moisture gradients. While its formulation is promising, its numerical implementation remains complex, despite significant contributions from Riahi et al. [8] and Seif et al. [9]. Additionally, crack edges in wood are often subjected to pressures from various sources, which alter the distribution of the energy release rate along the crack front and influence the crack propagation kinetics. For instance, in winter, water may infiltrate an open crack and freeze, expanding the crack edges due to ice formation. Similarly, in timber assemblies exposed to wind loading, the resulting tensile and bending stresses can further open the crack. Other phenomena, such as differential moisture expansion, residual compressive stresses due to shrinkage and swelling in the radial-tangential plane, and delamination, also contribute to pressure on the crack lips either promoting closure or further opening.

This study aims to model these different pressure-generating scenarios at the crack edges and to evaluate their influence on the global energy release rate, particularly in the presence of moisture gradients. A new two-dimensional formulation of the T-integral is proposed and implemented numerically in the Cast3M software using the MMCG geometry of the wood specimen [10].

# 2 Methods

## 2.1 T and $A\theta$ integrals formulations

The initial concept of an invariant integral for evaluating the energy release rate under variable environmental conditions was introduced by Bui [7]. A more comprehensive formulation was later developed by Moutou Pitti et al. [11], based on a generalization of Noether's theorem [12] to account for external thermo-hydro stresses. In the specific case of a two-dimensional crack propagating within an $\Omega$ shaped
domain subjected to a humidity variation $\delta w$, the T-integral takes the following form :

$$ T = \frac{1}{2} \int_{\Gamma} \left[ \sigma_{ij,k}^{(v)} u_i - \sigma_{ij}^{(u)} v_{i,k} + \delta w_{,k}^{(u)} v_i + \delta w^{(u)} v_{i,k} \right] n_j \, dS - \frac{1}{2} \int_{L} \left[ \sigma_{ij}^{(v)} u_{i,k} + \left( \sigma_{ij}^{(u)} - \delta w_{ij}^{(u)} \right) v_{i,k} \right] n_j \, dx. \quad (1) $$

In this equation, $\sigma_{ij}^{(u)}$ is the stress field tensor taken in the real configuration, $\sigma_{ij}^{(v)}$ is the equivalent of this tensor in the virtual configuration and $\sigma_{ij,k}^{(v)}$ its divergence. $u_i$ and $v_i$ are respectively the real and virtual displacement fields, $u_{i,k}$ and $v_{i,k}$ their respective gradients. The integral is evaluated on a curvilinear contour $\Gamma$ directed by the vector $\vec{n}$ having components $\vec{n}_j$ ($j = 1, 2$) and $L$ represents the lines of the upper and lower crack lips, while $x$ is the preferential direction of crack propagation (Fig. 1). Finally, $\delta w$ is the humidity variation of the medium which is calculated from a reference humidity $w_{ref}$ and $\delta w_{,k}^{(u)}$ is the humidity gradient in a given direction. All terms related to humidity are taken only in the real configuration. The T-integral studied in this paper therefore mainly comprises two terms : The first term represents the energy required for stationary cracking with consideration of hygroscopic effects and is used for mixed mode decoupling. The second term introduces a pressure at the crack lips influenced by the presence of a humidity variation. Numerical integration over curvilinear domains has been shown to exhibit convergence issues [13], prompting a transition to a surface contour-based approach. This led to the development of a new formulation, referred to as the A$\theta$-integral :

$$ A\theta = \frac{1}{2} \int_{\Gamma} \left[ \sigma_{ij,k}^{(v)} u_i - \sigma_{ij}^{(u)} v_{i,k} + \gamma_{ij} \left( \delta w_{,k}^{(u)} v_i + \delta w^{(u)} v_{i,k} \right) \right] \theta_{k,j} \, dS - \frac{1}{2} \int_{L} \left[ (F_i - \gamma_{ij} \delta w_{ij}^{(u)}) v_{i,k} \right] \theta_k \, dx. \quad (2) $$

<figure><img src="image_3.png" /><figcaption>FIGURE 1 – Parameters around the crack tip and $\vec{\theta}^{2D}$ field in plan configuration.</figcaption></figure>

The stability of the numerical computations is ensured by the smoothness and differentiability of the field $\vec{\theta}$ over the integration domain, where it takes a unit value within the domain and vanishes outside. The terms $F_i$ ($i = 1, 2$) correspond to the equivalent nodal forces resulting from pressures acting on the crack edges. These pressures may arise from water confinement or freezing within the crack, or from stress development due to the shrinkage and swelling behavior of wood. $\gamma_{ij}$ is the coupling tensor between the mechanical behavior of wood and its hygroscopic behavior and its components are defined as follow :

$$ \gamma_{ij} = \sum_{k,l} C_{ijkl} \alpha_{kl}, \quad (3) $$

where $C_{ijkl}$ is the compliance matrix and $\alpha_{kl}$ characterizes the hygroscopic shrinkage-swelling behavior.
## 2.2 Hygro-mechanical fields at the crack tip

For an anisotropic material in the Radial-Longitudinal plane, the constitutive law including the hygroscopic effects represented by the tensor $\alpha_{kl}$ and the elastic compliance tensor $C_{ij}$ is given by the following relation :

$$ \begin{bmatrix} \varepsilon_L \\ \varepsilon_R \\ 2\varepsilon_{RL} \end{bmatrix} = \begin{bmatrix} \frac{1}{E_L} & -\frac{\nu_{LR}}{E_R} & 0 \\ -\frac{\nu_{RL}}{E_L} & \frac{1}{E_R} & 0 \\ 0 & 0 & \frac{1}{G_{LR}} \end{bmatrix} \begin{bmatrix} \sigma_L \\ \sigma_R \\ \sigma_{RL} \end{bmatrix} - \begin{bmatrix} \alpha_L \\ \alpha_R \\ 0 \end{bmatrix} \delta w. \quad (4) $$

Since the hygroscopic variations lead to changes in the components of the compliance tensor, we used the modified components according to Guitard's predictions [14].

The stress fields at the crack tip are obtained for an orthotropic material using the singular Williams series [15] :

$$ \sigma_{ij} = \frac{K_I}{\sqrt{2\pi r}} f_{ij}(\theta) + \frac{K_{II}}{\sqrt{2\pi r}} g_{ij}(\theta) + \mathcal{O}(r^0) + \mathcal{O}(r^{1/2}) + \dots \quad (5) $$

$K_I$ and $K_{II}$ are the stress intensity factors in mode I and mode II, respectively. The functions $f_{ij}(\theta)$ and $g_{ij}(\theta)$ are, in turn, given by the following relations :

$$ f_{11}(\theta) = \Re \left( \frac{a_1 a_2 (b_2 a_2 - a_1 b_1)}{a_1 - a_2} \right), \qquad g_{11}(\theta) = \Re \left( \frac{b_2 a_2^2 - a_1^2 b_1}{a_1 - a_2} \right), \quad (6) $$

$$ f_{22}(\theta) = \Re \left( \frac{b_2 a_1 - a_1 b_2}{a_1 - a_2} \right), \qquad g_{22}(\theta) = \Re \left( \frac{b_2 - b_1}{a_1 - a_2} \right), \quad (7) $$

$$ f_{12}(\theta) = \Re \left( \frac{a_1 a_2 (b_2 - b_1)}{a_1 - a_2} \right), \qquad g_{12}(\theta) = \Re \left( \frac{b_1 a_1 - a_2 b_2}{a_1 - a_2} \right), \quad (8) $$

where $b_1 = \frac{1}{\sqrt{\cos(\theta)+a_1 \sin(\theta)}}$, $b_2 = \frac{1}{\sqrt{\cos(\theta)+a_2 \sin(\theta)}}$ and $\Re$ the real part of a complex number.

Furthermore, the parameters $a_1$ and $a_2$ represent the roots of the characteristic equation of the material, defined as follows :

$$ \xi_{11}a^4 - 2\xi_{16}a^3 + (2\xi_{12} + \xi_{66})a^2 - 2\xi_{26}a + \xi_{22} = 0, \quad (9) $$

with $\xi_{ij} = C_{ij}^w = C_{ij}^{w_{ref}}(1 - \chi \delta w)$, $\delta w = w - w_{ref}$.

To account for the hygroscopic effects in the distribution of stresses at the crack tip, a non-singular component called the T-stress [16] has been added to the expression of the Williams series :

$$ \sigma_{ij} = \frac{K_I}{\sqrt{2\pi r}} f_{ij}(\theta) + \frac{K_{II}}{\sqrt{2\pi r}} g_{ij}(\theta) + \mathcal{O}(r^0) + \mathcal{O}(r^{1/2}) + T_i \delta_{ij} + \dots \quad (10) $$

, where $T_i = E_i \alpha_i \delta w$.

## 2.3 Specimen and numerical hypothesis

For the numerical simulations, the wood specimen used features a Mixed Mode Crack Growth (MMCG) geometry, with its dimensions and finite element mesh shown in Fig. 2. Mechanical loading is applied via an Arcan device, which enables force transmission through appropriately inclined loading points. In this study, a mechanical force of 400 N was applied in each configuration to generate mode I, mode II, and mixed mode I+II loading conditions. An initial crack of 83.5 mm in length was introduced, and the
finite element analysis was carried out under plane stress assumptions using the Cast3M software. The wood material was modeled with hygro-mechanical properties similar to those of Scots pine [14] and the crack lips were subjected to a tensile pressure of 20 KPa.

<figure><img src="image_4.png" /><figcaption>FIGURE 2 – MMCG specimen : dimensions and finite elements mesh.</figcaption></figure>

## 3 Results and Discussion

The numerical calculations involve evaluating the energy release rate at various positions within the integration domain, which is here subdivided into eight concentric rings. Domain independence is achieved when the energy release rate *G* remains nearly constant across all rings. Fig. 3 illustrates the variation of *G* across the different integration rings, showing the respective effects of pressure.

<figure><img src="image_5.png" /><figcaption>FIGURE 3 – Verification of domain independence in the presence of tensile crack lips pressure : (a) Part of mode I, (b) Part of mode II.</figcaption></figure>

It is observed that domain independence is maintained for rings 3 to 7, regardless of pressure applied to the crack lips. This indicates the stability of the numerical calculations and, consequently, the reliability of the energy release rate obtained through the implementation of the $A\theta$ integral. However, deviations are noted in rings 1 and 2 due to proximity to the crack tip, where singularities affect the results, and in ring 8, which is influenced by the specimen's boundary conditions. Overall, the domain independence of the invariant integral is preserved in the presence of crack lips pressure, with average fluctuations remaining below 5% across the numerical results.
# 4 Conclusion and outlooks

This paper presents a two-dimensional numerical validation of a new mathematical formulation of the invariant $A\theta$ integral, developed to compute the energy release rate during the cracking process of wood elements subjected to a moisture gradient. The model incorporates the effect of pressure applied to the crack lips, which was numerically implemented to evaluate its contribution to crack process through its influence on the energy release rate. The numerical results demonstrate that, in the presence of such pressure, the proposed integral retains its domain independence provided that it is neither too close to the crack tip nor too close to the edge of the integration domain. As a perspective, future work will focus on modeling the influence of crack edges pressure on the onset of crack propagation and extending the current approach to a three-dimensional framework.

# Acknowledgments

The authors gratefully acknowledge financial support from the French government through the Eiffel Excellence Scholarship Program, and from the French National Research Agency (ANR) under the DIAM-WOOD project (ANR PRCE No. 23-CE22-0006-03), which made this work possible.

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