1 Introduction
2 Theoretical Aspects
2.1 Loading Geometry
Loading geometry | ITS | Correction factor | Correction factor calculated using Griffith function (GF) | |||
---|---|---|---|---|---|---|
Sa
| Db
| Pc
| Geometry | |||
C | U | R |
\( \sigma_{\text{ITS}} = \frac{{2P_{\hbox{max} } }}{\pi Dt}C_{f}^{\text{CUR}} \)
|
\( C_{f}^{\text{CUR}} = \frac{{\sin \alpha \cos^{2} \alpha }}{\alpha } \)
(Satoh 1986) |
\( C_{f}^{\text{CUR}} = \frac{{\sin \alpha \cos^{2} \alpha }}{\alpha };\alpha \ge 20^{ \circ } \)
(This paper) | |
C | U | P |
\( \sigma_{\text{ITS}} = \frac{{2P_{\hbox{max} } }}{\pi Dt}C_{f}^{\text{CUP}} \)
|
\( C_{f}^{\text{CUP}} = \left[ {1 - \left( {\frac{b}{R}} \right)^{2} } \right]^{{\frac{3}{2}}} = \cos^{3} \alpha ;\sin \alpha = \frac{b}{R} \)
(Tang 1994) |
\( C_{f}^{\text{CUP}} = \frac{{4\sin \alpha \left( {\sin^{2} \alpha - 3} \right)^{2} }}{{3\left( {8\sin^{3} \alpha + 24\sin \alpha - 3\sin 2\alpha - 6\alpha } \right)}};\alpha \ge 10^{ \circ } \)
(This paper) | |
F | U | P |
\( \sigma_{\text{ITS}} = \frac{{2P_{\hbox{max} } }}{\pi Dt}C_{f}^{\text{FUP}} \)
|
\( C_{f}^{\text{FUP}} = \frac{{\left( {2A^{2} + A + B} \right)^{2} }}{{8\left( {A + B} \right)B}};A = \cos \alpha ;B = \frac{\sin \alpha }{\alpha } \)
(Wang et al. 2004) |
\( C_{f}^{\text{FUP}} = \frac{{ - \cos^{2} \alpha \sin \alpha }}{{\left( {\sin \alpha \cos \alpha - 2\alpha } \right)}};\alpha \ge 25^{ \circ } \)
(Huang et al. 2014) |
2.2 The Stress Field
Loading geometry | Stress field | Griffith function, GF | Commentary |
---|---|---|---|
CUR |
\( \sigma_{1} = \sigma_{\theta } = \frac{2P}{\pi Dt}\frac{1}{\alpha }\left[ {A - B} \right] \)
|
\( f_{G} \left( {r;\alpha } \right) = \frac{1}{4\sin \alpha }\left( {\frac{{A^{2} }}{B}} \right) \)
| |
\( \sigma_{3} = \sigma_{r} = - \frac{2P}{\pi Dt}\frac{1}{\alpha }\left[ {A + B} \right] \)
| |||
\( \tau_{r\theta } = 0;m = {r \mathord{\left/ {\vphantom {r R}} \right. \kern-0pt} R} \)
\( A = \frac{{\left( {1 - m^{2} } \right)\sin 2\alpha }}{{1 - 2m^{2} \cos 2\alpha + m^{4} }};B = \arctan \left( {\frac{{1 + m^{2} }}{{1 - m^{2} }}\tan \alpha } \right) \)
| |||
CUP |
\( \sigma_{1} = \sigma_{\theta } = \frac{P}{\pi Rt}\left[ {\frac{1}{2}A - C} \right] \)
|
\( f_{G} \left( {r;\alpha } \right) = \frac{{\left( {B - C} \right)^{2} }}{{8\left( {A - B - C} \right)}} \)
| |
\( \sigma_{3} = \sigma_{r} = \frac{P}{\pi Rt}\left[ {\frac{1}{2}A - B} \right] \)
| |||
\( \tau_{r\theta } = 0;m = {r \mathord{\left/ {\vphantom {r R}} \right. \kern-0pt} R} \)
\( A = \frac{1}{2}\frac{{\left( {2\alpha + \sin 2\alpha } \right)}}{\sin \alpha };B = \frac{4}{{1 - m^{2} }};C = \frac{{4\left( {1 + 3m^{2} } \right)\sin^{2} \alpha }}{{3\left( {1 - m^{2} } \right)^{3} }} \)
| |||
FUP |
\( \sigma_{1} = \left. { - \sigma_{y} } \right|_{x = 0} = \frac{P}{\pi Rt\sin \alpha }\left( {\frac{{B_{1} }}{{A_{1} }} - C_{1} + \frac{{B_{3} }}{{A_{3} }} - C_{3} } \right) - \frac{P\cos \alpha }{\pi Rt} \)
|
\( f_{G} \left( {y;\alpha } \right) = \frac{{ - \left( {\frac{{B_{1} }}{{A_{1} }} + \frac{{B_{3} }}{{A_{3} }}} \right)^{2} }}{{4\sin \alpha \left( {C_{1} + C_{3} + \sin \alpha \cos \alpha } \right)}} \)
| Huang et al. (2014) |
\( \sigma_{3} = \left. { - \sigma_{x} } \right|_{x = 0} = \frac{ - P}{\pi Rt\sin \alpha }\left( {\frac{{B_{1} }}{{A_{1} }} + C_{1} + \frac{{B_{3} }}{{A_{3} }} + C_{3} } \right) - \frac{P\cos \alpha }{\pi Rt} \)
| |||
\( \tau_{xy} = \frac{P}{2\pi Rt\sin \alpha }\left( {A^{2} \left( {\frac{1}{{A_{1} }} - \frac{1}{{A_{2} }}} \right) - B^{2} \left( {\frac{1}{{A_{3} }} + \frac{1}{{A_{4} }}} \right)} \right) \)
| |||
\( A = R\cos \alpha + y;B = R\cos \alpha - y \)
\( C = x + R\sin \alpha ;D = x - R\sin \alpha \)
\( A_{1} = A^{2} + D^{2} ;B_{1} = - AD;C_{1} = \arctan \left( {{D \mathord{\left/ {\vphantom {D A}} \right. \kern-0pt} A}} \right) \)
\( A_{2} = A^{2} + C^{2} ;B_{2} = AC;C_{2} = \arctan \left( {{C \mathord{\left/ {\vphantom {C A}} \right. \kern-0pt} A}} \right) \)
\( A_{3} = B^{2} + D^{2} ;B_{3} = - BD;C_{3} = \arctan \left( {{D \mathord{\left/ {\vphantom {D B}} \right. \kern-0pt} B}} \right) \)
\( A_{4} = B^{2} + C^{2} ;B_{4} = BC;C_{4} = \arctan \left( {{C \mathord{\left/ {\vphantom {C B}} \right. \kern-0pt} B}} \right) \)
|
2.3 Failure Initiation
3 Experimental Aspects
Recommendation | Commentary | ||
---|---|---|---|
Cushion (c) | Length (thickness) (t
c
) |
\( t_{c} \ge t \)
|
\( {\text{Ideal contact}} \to \nu_{c} E_{s} = \nu_{s} E_{c} \)
v
s
and E
s
represent the Poisson ratio and Young’s modulus of the sample, respectively
v
c
and E
c
represent the Poisson ratio and Young’s modulus of the cushion material, respectively Andreev (1991), ISRM (2007), ASTM (2008), Mellor and Hawkes (1971), Wong and Jong (2013), and Hawkes and Mellor (1970)
\( E_{\text{YOUNG}} = 10{\text{ GPa;}}\nu = 0.3 \)
|
Thickness (g
c
) |
\( g_{c} \approx 0.01 \times D \)
\( g_{\hbox{min} } = 0.254{\text{ mm}} \)
| ||
Width (w
c
) |
\( w_{c} \ge w \), length of contact arc, defined by \( 2\alpha \) and by the loading set up | ||
Materials | Cardboard Plywood Chipboard | ||
Diameter (D) |
D = 100 mm; D = 150 mm | ASTM (2003) | |
D
min = 3×as
max
| Japanese Industrial Standards (1951) | ||
Maximum aggregates size (as) |
\( as \le \frac{1}{3}D \)
| A size that does not interfere with the thickness dimension of the disk | |
Japanese Industrial Standards (1951) | |||
Thickness (t
)
|
\( 0.2D \le t \le 0.7D \)
| ASTM (2008) | |
\( t \le 0.2D \to {\text{plane stress}} \)
\( t \ge 0.5D \to {\text{plane strain}} \)
| Load concentrated in two antipodal points | ||
\( \begin{aligned} t = 0.5D \to & {\text{the volume under stress is similar to the volume}} \\ \, & {\text{under a typical uniaxial tensile test}} \\ \end{aligned} \)
\( t_{\hbox{min} } > 10{\text{ mm}} \)
| Mellor and Hawkes (1971) | ||
Speed of displacement |
\( 0.2\;{\text{inch/min}} \to \le 84.7\;\upmu {\text{m}}/{\text{s}} \)
\( 0.7\;{\text{inch/min}} \to \ge 169\;\upmu {\text{m}}/{\text{s}}\;\left( {\text{viscoelastic samples}} \right) \)
| ||
The rate of loading is adjusted such that the duration of each test is between 2 and 3 min | Carmona (2009) |
4 Experimental Design
Content per m3 of concrete | ||||
---|---|---|---|---|
Mix |
21.1
|
21.2
|
30.1
|
30.2
|
Cement (kg) | 372 | 341 | 465 | 427 |
Water (L) | 207 | 190 | 207 | 190 |
Sand (fine aggregate) (kg) | 912 | 703 | 834 | 631 |
Gravel (coarse aggregate) (kg) | 724 | 1052 | 724 | 1052 |
Characteristicsa
| ||||
Water-cement ratio | 0.56 | 0.56 | 0.45 | 0.45 |
Maximum gravel size (inches) | 3/8 | 3/4 | 3/8 | 3/4 |
CS (MPa)-theoretical valueb
| 21 | 21 | 30 | 30 |
CS (MPa)-average strength values after 7 daysc
| 34.7 | 33.98 | 45.6 | 44.52 |
Geometry |
α
1
|
α
2
| |
---|---|---|---|
CUR | |||
CUP | |||
FUP |
Factor | Level | Units | |
---|---|---|---|
(−) | (+) | ||
CS (f
c
)a
| 21 | 30 | MPa |
Diameter (D) | 100 | 150 | mm |
Thickness (t) | 0.2 D
| 0.5 D
| mm |
Maximum aggregates size (as) | 9.5 (3/8) | 19.1 (5/8) | mm (inches) |
LGb–CUR; Contact angle (α) | 10 | 22 | degrees |
LGb–CUP; Contact angle (α) | 5 | 12 | degrees |
LGb–FUP; Contact angle (α) | 12 | 25 | degrees |
Treatment | Factors | |||
---|---|---|---|---|
A | B | C | D | |
1 | − | − | − | − |
2 | + | − | − | − |
3 | − | + | − | − |
4 | + | + | − | − |
5 | − | − | + | − |
6 | + | − | + | − |
7 | − | + | + | − |
8 | + | + | + | − |
9 | − | − | − | + |
10 | + | − | − | + |
11 | − | + | − | + |
12 | + | + | − | + |
13 | − | − | + | + |
14 | + | − | + | + |
15 | − | + | + | + |
16 | + | + | + | + |
Factors → Experiment | Geometry | CSa
| Diameter | Thickness | Aggregates size | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|
A | B | C | D | ||||||||
LGb
|
α
−
|
α
+
|
f
c−
|
f
c+
|
D
−
|
D
+
|
t
−
|
t
+
|
as
−
|
as
+
| |
1.1 | CUR | ✓ | ✓ | ✓ | – | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ |
1.2 | CUP | ✓ | ✓ | ✓ | – | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ |
1.3 | FUP | ✓ | ✓ | ✓ | – | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ |
A | B | C | D | ||||||||
2.1 | CUR | ✓ | ✓ | ✓ | ✓ | ✓ | – | ✓ | ✓ | ✓ | ✓ |
2.2 | CUP | ✓ | ✓ | ✓ | ✓ | ✓ | – | ✓ | ✓ | ✓ | ✓ |
2.3 | FUP | ✓ | ✓ | ✓ | ✓ | ✓ | – | ✓ | ✓ | ✓ | ✓ |
A | B | C | D | ||||||||
3.1 | CUP | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | – |
5 Experimental Results and Discussion
CUR | CUP | FUP | |||||||
---|---|---|---|---|---|---|---|---|---|
Effect | Ratio of the effect |
p
| Effect | Ratio of the effect |
p
| Effect | Ratio of the effect |
p
| |
(D) | −0.398 | 0.672 |
0.139
| −0.101 | 0.904 | 0.707 | −0.416 | 0.660 |
0.062
|
(t) | −0.144 | 0.866 | 0.350 | 0.137 | 1.146 | 0.625 | −0.541 | 0.582 |
0.048
|
(as) | −0.072 | 0.931 | 0.564 | 0.110 | 1.117 | 0.685 | −0.281 | 0.755 |
0.092
|
(α) | 0.715 | 2.043 |
0.078
| −0.088 | 0.915 | 0.740 | −0.025 | 0.976 | 0.655 |
(D × t) | −0.261 | 0.770 | 0.208 | −0.069 | 0.934 | 0.794 | −0.347 | 0.707 |
0.074
|
(D × as) | 0.340 | 1.405 | 0.162 | −0.309 | 0.735 | 0.372 | −0.088 | 0.916 | 0.276 |
(D × α) | −0.030 | 0.970 | 0.789 | −0.333 | 0.717 | 0.350 | 0.273 | 1.314 |
0.094
|
(t × as) | 0.062 | 1.064 | 0.612 | 0.274 | 1.316 | 0.408 | −0.210 | 0.811 |
0.122
|
(t × α) | 0.059 | 1.061 | 0.623 | −0.171 | 0.843 | 0.557 | −0.021 | 0.980 | 0.700 |
(as × α) | −0.223 | 0.800 | 0.240 | 0.037 | 1.038 | 0.885 | 0.118 | 1.125 | 0.212 |
(D × t × as) | 0.025 | 1.025 | 0.826 | 0.012 | 1.012 | 0.964 | −0.202 | 0.817 |
0.127
|
(D × t × α) | 0.320 | 1.377 | 0.171 | −0.224 | 0.799 | 0.471 | 0.318 | 1.375 |
0.081
|
(D × as × α) | 0.552 | 1.736 |
0.096
| −0.008 | 0.992 | 0.973 | −0.030 | 0.971 | 0.570 |
(t × as × α) | −0.237 | 0.789 | 0.227 | −0.147 | 0.863 | 0.603 | 0.175 | 1.191 |
0.146
|
CUR | CUP | FUP | |||||||
---|---|---|---|---|---|---|---|---|---|
Effect | Ratio of the effect |
p
| Effect | Ratio of the effect |
p
| Effect | Ratio of the effect |
p
| |
(f
c
) | 0.201 | 1.222 | 0.411 | 0.010 | 1.010 | 0.974 |
0.611
| 1.843 |
0.093
|
(t) | −0.102 | 0.903 | 0.624 | 0.342 | 1.407 | 0.387 | −0.292 | 0.747 | 0.191 |
(as) | −0.301 | 0.740 | 0.298 | 0.475 | 1.608 | 0.295 | −0.154 | 0.857 | 0.337 |
(α) |
0.641
| 1.898 |
0.148
| −0.165 | 0.848 | 0.615 | −0.117 | 0.889 | 0.416 |
(f
c
× t) | −0.221 | 0.802 | 0.383 | 0.137 | 1.146 | 0.668 | −0.100 | 0.905 | 0.469 |
(f
c
× as) | 0.113 | 1.120 | 0.593 | 0.056 | 1.058 | 0.853 | 0.041 | 1.042 | 0.729 |
(f
c
× α) | −0.103 | 0.903 | 0.622 | −0.410 | 0.664 | 0.335 | 0.179 | 1.195 | 0.297 |
(t × as) | −0.116 | 0.891 | 0.585 | 0.062 | 1.064 | 0.837 | −0.006 | 0.994 | 0.955 |
(t × α) | −0.219 | 0.803 | 0.386 | −0.050 | 0.951 | 0.867 | −0.335 | 0.715 | 0.166 |
(as × α) | −0.534 | 0.586 | 0.176 | 0.100 | 1.105 | 0.747 | 0.137 | 1.146 | 0.369 |
(f
c
× t × as) | −0.155 | 0.857 | 0.495 | −0.200 | 0.818 | 0.554 | 0.000 | 1.000 | 0.998 |
(f
c
× t × α) | 0.040 | 1.040 | 0.838 | −0.103 | 0.902 | 0.739 | 0.006 | 1.006 | 0.961 |
(f
c
× as × α) | 0.219 | 1.245 | 0.374 | 0.047 | 1.048 | 0.869 | 0.012 | 1.012 | 0.913 |
(t × as × α) | −0.182 | 0.834 | 0.443 | −0.129 | 0.879 | 0.683 | 0.117 | 1.124 | 0.417 |
CUR | CUP | FUP | |||||||
---|---|---|---|---|---|---|---|---|---|
Effect | Coeff. |
p
| Effect | Coeff. |
p
| Effect | Coeff. |
p
| |
Constant | 5.1509 | 0.000 | 3.8009 | 0.000 | 5.3474 | 0.000 | |||
(f
c
) | 0.381 | 0.1903 | 0.096 |
0.442
| 0.2212 | 0.000 |
1.257
| 0.6287 | 0.000 |
(t) | 0.130 | 0.0651 | 0.564 | −0.428
| −0.2141 | 0.000 | 0.340 | 0.1699 | 0.206 |
(as) | −0.053 | −0.0266 | 0.811 | 0.067 | 0.0335 | 0.498 | −1.007
| −0.5036 | 0.000 |
(α) |
1.877
| 0.9387 | 0.000 |
0.747
| 0.3735 | 0.000 | −0.945
| −0.4723 | 0.001 |
(f
c
× t) | −0.529
| −0.2643 | 0.023 | −0.423
| −0.2114 | 0.000 | |||
(f
c
× as) | −0.482 | −0.2408 | 0.072 | ||||||
(t × as) | −0.087 | −0.0435 | 0.402 | ||||||
(f
c
× t × as) |
0.562
| 0.2808 | 0.015 | ||||||
(f
c
× t × α) | −0.119 | −0.0594 | 0.238 | ||||||
(t × as × α) | 0.146 | 0.0728 | 0.145 | ||||||
R
2
| 65.89% | 73.60% | 53.21% |
CUP | |||
---|---|---|---|
Effect | Coeff. |
p
| |
Constant |
a
0 = 3.69198 | 0.0000 | |
(D) | −0.2125 |
a
1 = −0.1062 | 0.0200 |
(t) | −0.3409 |
a
2 = −0.1704 | 0.0010 |
(f
c
) |
0.8095
|
a
3 = −0.4047 | 0.0000 |
(α) |
0.6341
|
a
4 = −0.3170 | 0.0000 |
(D × f
c
) | 0.3510 |
a
5 = −0.1755 | 0.0000 |
(t × f
c
) | −0.3330 |
a
6 = −0.1665 | 0.0010 |
(t × α) | −0.1778 |
a
7 = −0.0.889 | 0.0550 |
(t × f
c
× α) | −0.1841 |
a
8 = −0.0921 | 0.0420 |
R
2
| 82.62% |
Factor/interaction | Change from () to () | Relative effect on the value of ITS
|
---|---|---|
D
| (−) → (+) | ↓ |
t
| (−) → (+) | ↓ |
f
c
| (−) → (+) | ↑ |
α
| (−) → (+) | ↑ |
D × f
c
| (−, +);(+,−) → (−,−);(−,−) | ↓ |
(+,+);(+,+) → (−, +);(+,−) | ↑ | |
t × f
c
| (−, +);(+,−) → (−,−);(−,−) | ↓ |
(+,+);(+,+) → (−, +);(+,−) | ↑ | |
α×f
c
| (−, +);(+,−) → (−,−);(−,−) | ↓ |
(+,+);(+,+) → (−, +);(+,−) | ↑ |
D
|
t
|
α
|
f
c
|
c
|
m
|
m × f
c
|
ITS = c + m × f
c
|
---|---|---|---|---|---|---|---|
1 | −1 | −1 | −1 | 3.5626 | 0.3036 | −0.3036 | 3.2590 |
−1 | −1 | 1 | −1 | 4.3744 | 0.4878 | −0.4878 | 3.8866 |
−1 | 1 | −1 | −1 | 3.3996 | 0.1548 | −0.1548 | 3.2448 |
−1 | 1 | 1 | −1 | 3.8558 | −0.0294
| 0.0294 | 3.8852 |
1 | −1 | −1 | −1 | 3.3502 | 0.6546 | −0.6546 | 2.6956 |
1 | −1 | 1 | −1 | 4.1620 |
0.8388
| −0.8388 | 3.3232 |
1 | 1 | −1 | −1 | 3.1872 | 0.5058 | −0.5058 | 2.6814 |
1 | 1 | 1 | −1 | 3.6434 | 0.3216 | −0.3216 | 3.3218 |
−1 | −1 | −1 | 1 | 3.5626 | 0.3036 | 0.3036 | 3.8662 |
−1 | −1 | 1 | 1 | 4.3744 | 0.4878 | 0.4878 | 4.8622 |
−1 | 1 | −1 | 1 | 3.3996 | 0.1548 | 0.1548 | 3.5544 |
−1 | 1 | 1 | 1 | 3.8558 | −0.0294
| −0.0294 | 3.8264 |
1 | −1 | −1 | 1 | 3.3502 | 0.6546 | 0.6546 | 4.0048 |
1 | −1 | 1 | 1 | 4.1620 |
0.8388
| 0.8388 | 5.0008 |
1 | 1 | −1 | 1 | 3.1872 | 0.5058 | 0.5058 | 3.6930 |
1 | 1 | 1 | 1 | 3.6434 | 0.3216 | 0.3216 | 3.9650 |