Comprehensive notch shape factors for V-notched Brazilian disk specimens loaded under mixed mode I/II from pure opening mode to pure closing mode

Abstract

Stress distribution is analyzed around the tip of V-notches in a disk-type test sample, called V-notched Brazilian disk (V-BD), under in-plane mixed mode loading conditions. The notch stress intensity factors (NSIFs) which are basic parameters in brittle fracture assessment of V-notched components are computed for the V-BD specimen by using the finite element (FE) method for different notch geometries and a wide range of mode mixities from pure positive mode I to pure mode II loading and from pure mode II to pure negative mode I loading conditions. Then, in order to make use of the NSIFs more conveniently in practical applications, the NSIFs are converted to the dimensionless parameters, called the notch shape factors (NSFs). These parameters are useful to compute more rapidly and conveniently the NSIFs in the V-BD specimen for various notch angles and different notch tip radii. The obtained results are partially validated by means of some results reported in the open literature and also by numerical analysis of stress distribution around the V-notch. Very good agreements are shown to exist between the stress distributions obtained from the computed NSIFs and those directly resulted from the FE analysis.

Appendices

Appendix 1

• (a) Functions used in the stress field for blunt V-shaped notches (mode I and mode II) [27]:

  $$\begin{aligned} \begin{array}{l} \left\{ {{\begin{array}{c} {m_{\theta \theta } (\theta )} \\ {m_{rr} (\theta )} \\ {m_{r\theta } (\theta )} \\ \end{array} }} \right\} ^{(\mathrm{I})}=\frac{1}{[1+\lambda _1 +\chi _{b_1 } (1-\lambda _1 )]}\left[ {\left\{ {{\begin{array}{c} {(1+\lambda _1 )\cos (1-\lambda _1 )\theta } \\ {(3-\lambda _1 )\cos (1-\lambda _1 )\theta } \\ {(1-\lambda _1 )\sin (1-\lambda _1 )\theta } \\ \end{array} }} \right\} +\chi _{b_1 } (1-\lambda _1 )\left\{ {{\begin{array}{c} {\cos (1+\lambda _1 )\theta } \\ {-\cos (1+\lambda _1 )\theta } \\ {\sin (1+\lambda _1 )\theta } \\ \end{array} }} \right\} } \right] \\ \left\{ {{\begin{array}{c} {n_{\theta \theta } (\theta )} \\ {n_{rr} (\theta )} \\ {n_{r\theta } (\theta )} \\ \end{array} }} \right\} ^{(\mathrm{I})}=\frac{1}{4(q-1)[1+\lambda _1 +\chi _{b_1 } (1-\lambda _1 )]}\left[ {\chi _{d_1 } \left\{ {{\begin{array}{c} {(1+\mu _1 )\cos (1-\mu _1 )\theta } \\ {(3-\mu _1 )\cos (1-\mu _1 )\theta } \\ {(1-\mu _1 )\sin (1-\mu _1 )\theta } \\ \end{array} }} \right\} +\chi _{c_1 } \left\{ {{\begin{array}{c} {\cos (1+\mu _1 )\theta } \\ {-\cos (1+\mu _1 )\theta } \\ {\sin (1+\mu _1 )\theta } \\ \end{array} }} \right\} } \right] \\ \left\{ {{\begin{array}{c} {m_{\theta \theta } (\theta )} \\ {m_{rr} (\theta )} \\ {m_{r\theta } (\theta )} \\ \end{array} }} \right\} ^{(\mathrm{{II}})}=\frac{1}{[1-\lambda _2 +\chi _{b_2 } (1+\lambda _2 )]}\left[ {\left\{ {{\begin{array}{c} {(1+\lambda _2 )\sin (1-\lambda _2 )\theta } \\ {(3-\lambda _2 )\sin (1-\lambda _2 )\theta } \\ {(1-\lambda _2 )\cos (1-\lambda _2 )\theta } \\ \end{array} }} \right\} +\chi _{b_2 } (1+\lambda _2 )\left\{ {{\begin{array}{c} {\sin (1+\lambda _2 )\theta } \\ {-\sin (1+\lambda _2 )\theta } \\ {\cos (1+\lambda _2 )\theta } \\ \end{array} }} \right\} } \right] \\ \left\{ {{\begin{array}{c} {n_{\theta \theta } (\theta )} \\ {n_{rr} (\theta )} \\ {n_{r\theta } (\theta )} \\ \end{array} }} \right\} ^{(\mathrm{{II}})}=\frac{1}{4(\mu _2 -1)[1-\lambda _2 +\chi _{b_2 } (1+\lambda _2 )]}\left[ {\chi _{d_2 } \left\{ {{\begin{array}{c} {(1+\mu _2 )\sin (1-\mu _2 )\theta } \\ {(3-\mu _2 )\sin (1-\mu _2 )\theta } \\ {(1-\mu _2 )\cos (1-\mu _2 )\theta } \\ \end{array} }} \right\} +\chi _{c_2 } \left\{ {{\begin{array}{c} {-\sin (1+\mu _2 )\theta } \\ {\sin (1+\mu _2 )\theta } \\ {-\cos (1+\mu _2 )\theta } \\ \end{array} }} \right\} } \right] \\ \end{array} \end{aligned}$$

• (b) The values of the parameters $\lambda _i, \mu _i, \chi _{{b_i}}, \chi _{{c_i}}, \chi _{{d_i}}$ for different notch angles [27]:

$2\alpha$ ($^\circ$)	$\lambda _1$	$\mu _1$	$\chi _{{b_1}}$	$\chi _{{c_1}}$	$\chi _{{d_1}}$
0	0.5	$-$0.5	1	4	0
30	0.5014	$-$0.4561	1.0707	3.7907	0.0632
45	0.505	$-$0.4319	1.1656	3.5721	0.0828
60	0.5122	0.4057	1.3123	3.2832	0.096
90	0.5448	0.3449	1.8414	2.5057	0.1046
120	0.6157	$-$0.2678	3.0027	1.515	0.0871
145	0.6736	$-$0.2198	4.153	0.9933	0.0673
150	0.752	$-$0.1624	6.3617	0.5137	0.0413

$2\alpha$ ($^\circ$)	$\lambda _2$	$\mu _2$	$\chi _{{b_2}}$	$\chi _{{c_2}}$	$\chi _{{d_2}}$
0	0.5	$-$0.5	1	$-$12	0
30	0.5982	$-$0.4465	0.9212	$-$11.3503	$-$0.3506
45	0.6597	$-$0.4118	0.814	$-$10.1876	$-$0.451
60	0.7309	0.3731	0.6584	$-$8.3946	$-$0.4788
90	0.9085	0.2882	0.2189	$-$2.9382	$-$0.2436
120	1.1489	$-$0.198	$-$0.3139	4.5604	0.5133
145	1.3021	$-$0.1514	$-$0.5695	8.7371	1.1362
150	1.4858	$-$0.1034	$-$0.7869	12.9161	1.9376

• (c) The expressions for the parameters $\omega _1$ and $\omega _2$ [27]:

  $$\begin{aligned} \omega _1= & {} \frac{q}{4(q-1)}\left[ \frac{\chi _{{d_1}} (1+\mu _1 )+\chi _{{c_1}} }{1+\lambda _1 +\chi _{{b_1}} (1-\lambda _1)}\right] \\ \omega _2= & {} \frac{q}{4(\mu _2 -1)}\left[ \frac{\chi _{{d_2}} (1+\mu _2 )+\chi _{{c_2}}}{1+\lambda _2 +\chi _{{b_2}} (1-\lambda _2)}\right] \end{aligned}$$

$2\alpha$ ($^\circ$)	q
0	2.00
30	1.83
45	1.75
60	1.67
90	1.50
120	1.33
145	1.19
150	1.17
180	1.00

Appendix 2

See Figs. 6, 7, 8 and 9.

Fig. 6

Variations of the mode I and mode II notch shape factors versus the loading angle for RNL  =  0.4 and different values of RNR and $2\alpha$

Fig. 7

Variations of the mode I and mode II notch shape factors versus the loading angle for RNL  =  0.3 and different values of RNR and $2\alpha$

Fig. 8

Variations of the mode I and mode II notch shape factors versus the loading angle for RNL  =  0.2 and different values of RNR and $2\alpha$

Fig. 9

Variations of the mode I and mode II notch shape factors versus the loading angle for RNL  =  0.1 and different values of RNR and $2\alpha$