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Mixed-mode fracture assessment of wooden structures with cracks initiated along and across the fibers considering non-singular T-stress term


A new criterion for investigating the failure of wooden structures using maximum shear stress (MSS) under mixed mode I/II was presented for two conditions in which the crack is oriented along and across the fibers. The impact of non-singular stress terms in Williams series expansion (T-stress) is also included. This criterion is presented by considering the reinforcing effects of fibers in which the orthotropic material is assumed as an isotropic material that is reinforced with the fibers. The proposed failure criterion includes a damage coefficient that establishes an effective relationship between the fracture toughness of isotropic and orthotropic materials. The assumption is that at the onset of fracture, from the microscopic point of view, the crack makes a small kink in a direction where the shear stress has the critical value at a critical distance based on the MSS criterion. Then, the crack will propagate along the fibers after colliding with them. The results are verified by using available experimental data of different wood species. It was shown that with the proposed criteria both the crack growth path and the moment of crack growth are well predicted.

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\(C_{ij}\) :

Components of the compliance matrix

\(C^{\prime}_{ij}\) :

Components of compliance matrix in-plane strain condition

\(E_{ij}\) :

Young’s modulus

\(E_{L} ,E_{R} ,E_{T}\) :

Longitudinal, radial, and tangential Young’s modulus for wood specimens

\(f_{ij} (\theta )\,,\,\,g_{ij} (\theta )\) :

Angular functions of orthotropic stress state around the crack tip

\(G_{ij}\) :

Shear modulus of wood material

\(K_{I} ,K_{II}\) :

Mode I and mode II stress intensity factors (SIF)

\(K_{Ic} ,K_{IIc}\) :

Mode I and mode II fracture toughness

\(K_{I}^{kink} ,K_{II}^{kink}\) :

Mode I and mode II stress intensity factors at the tip of the crack kink

\(L,R,T\) :

Longitudinal, radial, and tangential orthotropic axes in the wood specimen

\(n_{i}\) :

Reinforcement factor in the theoretical criteria

\(p_{ij} (\theta )\,,\,\,q_{ij} (\theta )\) :

Angular functions of isotropic stress state around the crack tip

\(V_{f} ,V_{m}\) :

The volume fraction of fibers and the matrix in a composite

\(\beta_{i}\) :

Damage coefficient

\(\delta_{i} \,(i = 1 - 3)\) :

Reinforcement factor in RIS concept

\(\theta\) :

Arbitrary angle to show stress state in the crack tip

\(\theta_{c}\) :

Crack kink between fibers in microscopic point of view

\(\nu_{ij}\) :

Poisson’s ratio

\(\nu_{LR}\) :

Poisson’s ratio in RL direction

\(\nu_{LT}\) :

Poisson’s ratio in TL direction

\(\nu_{TR}\) :

Poisson’s ratio in TR direction

\(\sigma_{ij}^{iso}\) :

Isotropic stress state at the crack tip

\(\sigma_{ij}^{ortho}\) :

Orthotropic stress state at the crack tip

\(\tau\) :

Shear stress

\(\tau_{cr}\) :

Critical shear stress

\(\tau_{\max }\) :

Maximum shear stress

\(\varphi\) :

Crack-fiber angle


Fracture limited curve


Double cantilever beam


Fracture process zone


Maximum shear stress


Generalized reinforcement isotropic solid maximum shear stress


Single-edge-notched tension


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Correspondence to Mahdi Fakoor.

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Fakoor, M., Shahsavar, S. & Berto, F. Mixed-mode fracture assessment of wooden structures with cracks initiated along and across the fibers considering non-singular T-stress term. Wood Sci Technol 56, 1261–1291 (2022).

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