Experimental investigation and numerical simulations of U-notch specimens under mixed mode loading by the conventional and extended finite element methods

Abstract

In brittle or quasi-brittle materials, mechanical fracture phenomenon occurs suddenly and without any warning. Therefore, prediction of brittle materials failure is an essential challenge confronting design engineers. In this research, using the conventional finite element method (CFEM) and extended finite element method (XFEM) based on linear elastic fracture mechanics, rupture behavior of U-notch specimens under mixed mode loadings are numerically and practically studied. As the main contribution and objective of the current study, two different fracture criteria established on CFEM and six various criteria founded on XFEM are employed to numerically predict load carrying capacity and crack initiation angle of the U-notch samples. Also, the load carrying capacity and crack initiation angle are experimentally obtained from tensile tests of the U-notch instances under planar mixed mode loading to verify the simulation results. The empirical results are compared with the corresponding estimated values achieved by CFEM and XFEM methods which permit to assess the accuracy of the mentioned criteria in predicting the load carrying capacity and crack initiation angle of U-notch coupons subjected to mixed mode loadings, as the novelty of the investigation. The comparison shows that although both the CFEM and XFEM can properly predict the load carrying capacity and crack initiation angle, applying the XFEM in addition to reduce the computational costs and mesh sensitivity is more precise. Besides, a comparison between the XFEM results denotes that stress-based models are significantly more accurate than strain-based types in predicting the load carrying capacity and crack initiation angle of the U-notch instances under mixed mode loading.

Appendix 1

Functions used in the tangential stress formula [Eq. (23)] for rounded-tip V-shaped notches (mode I + II) [28]:

$$\begin{aligned} m_{\theta \theta }^{(\mathrm{I})}= & {} \frac{1}{(1+\lambda _1 +\chi _{b1} (1-\lambda _1 ))}\left[ {(1+\lambda _1 )\cos ((1-\lambda _1 )\theta )+\chi _{b1} (1-\lambda _1 )\cos ((1+\lambda _1 )\theta )} \right] \\ n_{\theta \theta }^{(\mathrm{I})}= & {} \frac{q}{4(q-1)(1+\lambda _1 +\chi _{b1} (1-\lambda _1 ))}\left[ {\chi _{d1} (1+\mu _1 )\cos ((1-\mu _1 )\theta )+\chi _{c1} \cos ((1+\mu _1 )\theta )} \right] \\ m_{\theta \theta }^{(\mathrm{II})}= & {} \frac{1}{(1+\lambda _2 +\chi _{b2} (1+\lambda _2 ))}\left[ {(1+\lambda _2 )\sin ((1-\lambda _2 )\theta )+\chi _{b1} (1+\lambda _2 )\sin ((1+\lambda _2 )\theta )} \right] \\ n_{\theta \theta }^{(\mathrm{II})}= & {} \frac{q}{4(\mu _2 -1)(1+\lambda _2 +\chi _{b2} (1+\lambda _1 ))}\left[ {\chi _{d2} (1+\mu _2 )\sin ((1-\mu _1 )\theta )-\chi _{c2} \sin ((1+\mu _2 )\theta )} \right] \\ \end{aligned}$$

Eigenvalues applied in the tangential stress formula [Eq. (23)] for rounded-tip V-shaped notches (mode I + II) [28]:

$2\alpha \;(^{\circ })$	$\lambda _1$	$\lambda _2$	$\mu _1$	$\mu _2$
0	0.5	0.5	−0.5	−0.5
30	0.5014	0.5982	−0.4561	−0.4118
60	0.5122	0.7309	−0.4057	−0.3731
90	0.5448	0.9085	−0.3449	−0.2882
120	0.6157	1.1489	−0.2678	−0.1980
135	0.6736	1.3021	−0.2198	−0.1514

Values of the parameters q, $\chi _{b1}$, $\chi _{b2}$, $\chi _{c1}$, $\chi _{c2}$, $\chi _{d1}$, and $\chi _{d2}$ have been reported in [28] for various notch angles.