Fracture study of wood considering the effect of T-stress term based on matrix reinforcement coefficients model

Abstract

A new comprehensive criterion has been presented to assess the fracture response in cracked wood as natural orthotropic material based on the maximum strain energy release rate criterion (SER) under mixed-mode I/II loading. A general off-axis case is assumed for desired crack-fiber angle. The most important purpose of the proposed paper is to accurately predict the fracture behavior of orthotropic materials. In this article, two influencing factors on fracture assessment are discussed comprehensively. The first one is employing the accurate model of the orthotropic material named as reinforcement isotropic solid (RIS) model, and the second factor is considering the effects of the T-stress term. The SER concept is extended in combination with the reinforcement isotropic solid (RIS) model for orthotropic materials. Using RIS, a mapping between the stress fields in the isotropic and orthotropic materials is established. In this article, the reinforcement coefficients are extracted using a new method called "corresponding stresses" for the desired crack-fiber angles. These coefficients depend on the mechanical properties of the orthotropic material and crack-fiber angle. The results obtained from the curves of reinforcement coefficients indicate that as the crack-fiber angle increases, the reinforcement coefficients decrease drastically. The SER around the crack tip is extracted considering the normal and shear stress fields and the non-singular T-stress term. Using extracted results, it can be shown that the T-stress term has a serious role in predicting the crack behavior of orthotropic materials. It is practically impossible to calculate the fracture toughness of pure modes I and II when the crack is not along the fibers. Based on the proposed extended maximum strain energy release rate (EMSER) criterion, a new concept called equivalent fracture toughness (EFT) is introduced as a fracture property of orthotropic materials. To validate and evaluate the accuracy of the EMSER criterion, the fracture limit curves (FLCs) for different angles between crack and fibers are compared with available experimental data from the literature. Compared to other criteria, a prominent correlation is obtained between the theoretical results evaluated by the EMSER criterion and available test data.

Appendix A

$F_{ij} (\theta )$ and $g_{ij} (\theta )$ coefficients in Eqs. (11)-(13) are as follows:

$$f_{11} (\theta ) = \cos \frac{\theta }{2}\left( {1 - \sin \frac{\theta }{2}\sin \frac{3\theta }{2}} \right)$$

(A1)

$$f_{22} (\theta ) = \cos \frac{\theta }{2}\left( {1 + \sin \frac{\theta }{2}\sin \frac{3\theta }{2}} \right)$$

(A2)

$$f_{12} (\theta ) = \sin \frac{\theta }{2}\cos \frac{\theta }{2}\cos \frac{3\theta }{2}$$

(A3)

$$g_{11} (\theta ) = - \sin \frac{\theta }{2}\left( {2 + \cos \frac{\theta }{2}\cos \frac{3\theta }{2}} \right)$$

(A4)

$$g_{22} (\theta ) = \sin \frac{\theta }{2}\cos \frac{\theta }{2}\cos \frac{3\theta }{2}$$

(A5)

$$g_{12} (\theta ) = \cos \frac{\theta }{2}\left( {1 - \sin \frac{\theta }{2}\sin \frac{3\theta }{2}} \right)$$

(A6)

In addition, $\hat{f}_{ij} (\theta )$ and $\hat{g}_{ij} (\theta )$ coefficients in Eqs. (8)-(10) are expressed as follows:

$$\hat{f}_{11} (\theta ) = {\text{Re}} \left( {\frac{{(\mu_{1} \mu_{2} (\mu_{2} \nu_{2} - \mu_{1} \nu_{1} )}}{{\mu_{1} - \mu_{2} }}} \right)$$

(A7)

$$\hat{g}_{11} (\theta ) = {\text{Re}} \left( {\frac{{(\mu_{2}^{2} \nu_{2} - \mu_{1}^{2} \nu_{1} )}}{{\mu_{1} - \mu_{2} }}} \right)$$

(A8)

$$\hat{f}_{12} (\theta ) = {\text{Re}} \left( {\frac{{\mu_{1} \mu_{2} (\nu_{1} - \nu_{2} )}}{{\mu_{1} - \mu_{2} }}} \right)$$

(A9)

$$\hat{g}_{12} (\theta ) = {\text{Re}} \left( {\frac{{(\mu_{1} \nu_{1} - \mu_{2} \nu_{2} )}}{{\mu_{1} - \mu_{2} }}} \right)$$

(A10)

$$\hat{f}_{22} (\theta ) = {\text{Re}} \left( {\frac{{\mu_{1} \nu_{2} - \mu_{2} \nu_{1} )}}{{\mu_{1} - \mu_{2} }}} \right)$$

(A11)

$$\hat{g}_{22} (\theta ) = {\text{Re}} \left( {\frac{{\nu_{2} - \nu_{1} )}}{{\mu_{1} - \mu_{2} }}} \right)$$

(A12)

$\mu_{2}$ and $\mu_{1}$ coefficients are the roots of the following equation:

$$S_{11} \mu^{4} - 2S_{16} \mu^{3} + (2S_{12} + S_{66} )\mu^{2} - 2S_{26} \mu + S_{22} = 0$$

(A13)

$S_{ij}$ coefficients are the components of the compliance matrix as follows:

$$\left( {\begin{array}{*{20}c} {\varepsilon_{11} } \\ {\varepsilon_{22} } \\ {\varepsilon_{12} } \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {S_{11} } & {S_{12} } & {S_{16} } \\ {S_{21} } & {S_{22} } & {S_{26} } \\ {S_{61} } & {S_{62} } & {S_{66} } \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {\sigma_{11} } \\ {\sigma_{22} } \\ {\sigma_{12} } \\ \end{array} } \right)$$

(A14)

$\nu_{1}$ and $\nu_{2}$ coefficients are defined as follows:

$$\nu_{1} = \frac{1}{{\sqrt {\cos \theta + \mu_{1} \sin \theta } }}$$

(A15)

$$\nu_{2} = \frac{1}{{\sqrt {\cos \theta + \mu_{2} \sin \theta } }}$$

(A16)