ISSN 1063-7850, Technical Physics Letters, 2017, Vol. 43, No. 1, pp. 130–132. © Pleiades Publishing, Ltd., 2017. Original Russian Text © A.M. Bragov, A.K. Lomunov, A.Yu. Konstantinov, D.A. Lamzin, 2017, published in Pis’ma v Zhurnal Tekhnicheskoi Fiziki, 2017, Vol. 43, No. 2, pp. 92–97.
A Modified Kolsky Method for Determining the Shear Strength of Brittle Materials A. M. Bragova*, A. K. Lomunova,b, A. Yu. Konstantinova, and D. A. Lamzina a
Research Institute of Mechanics, Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod, 603600 Russia b Nizhny Novgorod State University of Architecture and Civil Engineering, Nizhny Novgorod, 603950 Russia *e-mail:
[email protected] Received March 31, 2016
Abstract—A new modification of the Kolsky method is proposed, according to which a loaded sample is arranged in an obliquely cut tube casing. Using this configuration, it is possible to determine the shear stress of low-plasticity materials. DOI: 10.1134/S1063785017010175
“Pure” shear is separation of a material element into two parts along a section to which shearing forces are applied. There are several main schemes for material testing with respect to cut (shear) under dynamic loading conditions, according to which: (i) a sample shaped as a rectangular parallelepiped is arranged in a split Hopkinson pressure-bar (SHPB) system between the edges of specially configured gauge bars operating as punch and die [1]; (ii) a plate sample is arranged between the loading pressure bar and base tube with a diameter slightly exceeding that of the bar [2]; and (iii) samples of special configurations are arranged between the edges of gauge bars in the SHPB system [3, 4], which encounters some difficulties in manufacturing samples. In the first scheme, the sample is shear strained along two parallel planes; in the latter two schemes, the shear proceeds via a ring surface and plane, respectively. In the present work, we have developed a simple and effective technique for determining the dynamic shear strength of brittle materials such as concretes, rocks, ceramics, and composites, which is based on the Kolsky method employing a split Hopkinson pressure bar (SHPB) [5]. According to the proposed and implemented modification of this method (Fig. 1), sample 5 of a material studied is arranged in a hard (steel) tube casing cut into two parts (3, 4) at angle α relative to the horizon. In order to ensure that the sample would be loaded by pure shear without compression, the sample length must be smaller than that of the cut tube. The tube with sample is placed between the loading 1 and base 2 gauge bars. This SHPB assembly is loaded by the impact of a steel striker
accelerated in a gas gun channel. The striker and 1.5-m-long gauge bars had a diameter of 20 mm, and their yield strength amounted to 2000 MPa. Since the yield strength of tube casing is at least ten times as large as that of a sample, the casing can be considered undeformable. The compression impulse generated by the striker in the loading bar produces a small shift of two parts of the cut tube along the shear plane, which leads to deformation and breakage of a brittle sample. Force P(t) acting upon the tube can be expressed via deformation impulse εT(t) measured in the base bar:
130
P (t ) = EAεT (t ),
(1)
6 3
5 Pn
P
Pτ
α 1 5
4
3
2 5
4 4 Fig. 1. Schematic diagram of shear stress testing: (1) loading bar, (2) base bar and gauges, (3, 4) parts of cut tube casing, (5) sample, and (6) shear plane.
A MODIFIED KOLSKY METHOD
131
60 50 τ, MPa
40 30 20 10
20 mm
0
Fig. 2. Cut-tube casing with fiber-reinforced concrete sample broken during the shear strength test.
30
2 (3) AS = πR , sin α where R is the inner radius of the tube. Using relations (1)–(3), the temporal variation of shear stress in the sample can be calculated as follows:
Pτ(t ) EA cos α sin α T = ε (t ). AS πR 2 The displacements of points in the shear plane can be determined by considering the motion of edges of the gauge bars. Total displacement U1(t) of the leftτ(t ) =
hand edge represents a sum of displacement U 1I (t ) caused by propagation of incident impulse εI(t) and displacement U 1R (t ) caused by propagation of reflected impulse εR(t): t
∫
U 1(t ) = C (ε I (t ) − ε R (t ))dt. 0
Displacement U2(t) of the right-hand edge is caused by transmitted impulse εT(t): t
∫
U 2(t ) = C εT (t )dt. 0
Vol. 43
No. 1
40 t, μs
50
60
70
Then, the average horizontal displacement of points in the cut tube is t
∫
Δ l(t ) = U 1(t ) − U 2 (t ) = C (ε I (t ) − ε R (t ) − εT (t ))dt. 0
(2)
The shear stress acting on the sample can be calculated as the ratio of Pτ(t) to area AS of the sample cross section (ellipse) in the shear plane:
TECHNICAL PHYSICS LETTERS
20
Fig. 3. Dynamic diagram obtained during shear strength testing of fiber-reinforced concrete.
where E is the Young’s modulus, and A is the crosssectional area of the base bar. This force can be separated into two components: tangential force Pτ(t) parallel to the shear plane and normal force Pn(t) perpendicular to the plane, so that:
Pτ(t ) = P cos α, Pn(t ) = P sin α.
10
2017
Assuming that the hardness of the cut tube casing is much greater than that of a sample and ignoring deformation of the tube, the displacement of points in the oblique cross-section plane of the sample can be determined as t
Δ l(t ) Δ l τ(t ) = = C (ε I (t ) − ε R (t ) − εT (t ))dt, cos α cos α
∫ 0
and the relative displacement of these points can be expressed as t
Δ l (t ) ε τ(t ) = τ = C tan α (ε I (t ) − ε R (t ) − εT (t ))dt, l0 2R
∫ 0
where l0 is the double major semiaxis of elliptic section in the shear plane. Thus, using the proposed method, it is possible to determine the temporal variation of shear stress or the displacement of points in the sample cross section by measuring impulses of gauge bar deformation. For reliable determination of displacements, the deformation impulses of gauge bars should be strictly synchronized in time. In order to provide for this, strain gauges should be fixed on bars at the same distance from sample edges in order to ensure simultaneous arrival of the reflected and transmitted impulses to the cross sections of loading and base bars in which the deformation is monitored. The proposed method has been verified by testing some brittle media including concretes, rocks, and ceramics. Figure 2 shows a photograph of a broken sample of fine-grained fiber-reinforced concrete. Fig-
132
BRAGOV et al.
ure 3 presents the corresponding temporal variation of the average shear stress with 95% confidence intervals. The stress growth rate was controlled by varying the striker speed and amounted to about 3600 GPa/s in this experiment. It can be suggested that samples tested by this method were broken almost precisely along the shear plane. In concluding, we have developed a simple modification of the Kolsky method that allows the shear strength—an important characteristic of brittle materials—to be reliably determined at high deformation rates. Acknowledgements. This study was supported in part by the Russian Foundation for Basic Research, project nos. 16-38-60122_mol_a_dk and 15-0805517_a.
REFERENCES 1. M. V. Hosur, IslamS. M. Waliul, U. K. Vaidya, et al., Compos. Struct. 70, 295 (2005). 2. L. Ren, B. A. Gama, J. W. Gillespie, Jr., and ChianFong Yen, Final Report, Contract DAAD19-01-2-0005 (2004). 3. S. Nemat-Nasser, J. B. Isaacs, and M. Liu, Acta Mater. 46, 1307 (1998). 4. G. T. Gray III, K. S. Vecchio, and V. Livescu, Acta Mater. 103, 12 (2016). 5. A. M. Bragov and A. K. Lomunov, Int. J. Impact Eng. 16 (2), 321 (1995).
TECHNICAL PHYSICS LETTERS
Translated by P. Pozdeev
Vol. 43
No. 1
2017
