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Constitutive equations of basalt filament tows under quasi-static and high strain rate tension
Materials Science and Engineering A 527 (2010) 3245–3252
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Materials Science and Engineering A journal homepage: www.elsevier.com/locate/msea
Constitutive equations of basalt filament tows under quasi-static and high strain rate tension Lvtao Zhu a , Baozhong Sun a , Hong Hu b , Bohong Gu a,c,∗ a b c
College of Textiles, Donghua University, Shanghai 201620, China Institute of Textiles and Clothing, The Hong Kong Polytechnic University, Hung Hom Kowloon, Hong Kong Department of Textile Engineering, Zhongyuan Institute of Technology, Zhengzhou, Henan Province 450007, China
a r t i c l e
i n f o
Article history: Received 15 September 2009 Received in revised form 28 January 2010 Accepted 3 February 2010
Keywords: Basalt filament tows Tensile properties Strain rate Weibull distribution function
a b s t r a c t The tensile properties of basalt filament tows were tested at quasi-static (0.001 s−1 ) and high strain rates (up to 3000 s−1 ) with MTS materials tester (MTS 810.23) and split Hopkinson tension bar (SHTB), respectively. Experimental results showed that the mechanical properties of the basalt filament tows were rather sensitive to strain rate. Specifically, the stiffness and failure stress of the basalt filament tows increased distinctly as the strain rate increased, while the failure strain decreased. From scanning electronic microscope (SEM) photographs of the fracture surface, it is indicated that the basalt filament tows failed in a more brittle mode and the fracture surface got more regular as the strain rate increases. The strength distributions of the basalt filament tows have been evaluated by a single Weibull distribution function. The curve predicted from the single Weibull distribution function was in good agreement with the experimental data points. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Basalt is a common volcanic rock that can be found in most countries around the globe and is directly suitable for fiber manufacturing. Its chemical structure is near to glass. The most important components of basalt are SiO2 , Al2 O3 , CaO, MgO, Fe2 O3 and FeO [1–5]. Basalt rocks are molten approximately between 1350 and 1700 ◦ C [6–8]. Basalt fibers are produced in one step, directly from crushed basalt stone. Basalt fibers are more resistant to strong alkalis than glass fibers, but glass can better withstand strong acids. Basalt fibers can be used over a wide range of temperature, from −200 to 600 ◦ C [9–11]. Nowadays basalt fibers have great potential application to composite materials. The idea of using basalt fibers as reinforcement of composite materials first emerged in the former Soviet Union in an aerospace research program. Today most of the continuous basalt fibers are manufactured in Russia and Ukraine [11]. The aim of this study was to evaluate the tensile properties of basalt filament tows under various tensile speeds (corresponding to various strain rates). The rate-sensitivity of the tensile properties would be investigated. The fracture morphologies of the basalt filament tows under different strain rates were photographed with
∗ Corresponding author at: College of Textiles, Donghua University, Shanghai 201620, China. Tel.: +86 21 67792661/371 67698880; fax: +86 21 67792627/371 62506970. E-mail address: [email protected] (B. Gu). 0921-5093/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2010.02.015
scanning electronic microscopy (SEM). The fracture behaviors of the basalt fibers were analyzed from these fractographs. Based on the filament tows model and the statistical theory of fiber strength, a single Weibull statistical model was employed to describe the strength distribution of the basalt filament tows, and the Weibull parameters were obtained by the filament tows testing method. Consistency between the simulated and experimental results indicates that the model and the method are valid and reliable. 2. Experimental 2.1. Basalt fibers The basalt filament tows (as shown in Fig. 1) were manufactured by Hengdian Group Shanghai Russia & Gold Basalt Fiber Co. Ltd. in China. The fineness of the basalt filament tows is 2400tex/21800f (provided by the manufacturer). The diameter of monofilamanet is 7 m. The volume density is 2.6 g/cm3 . The photograph of the bobbin of basalt filament tows is show in Fig. 1. 2.2. Testing The quasi-static tensile tests (with the strain rate of 0.001 s−1 ) and high strain rate tensile tests (impact tensile tests) were performed on a MTS 810.23 materials tester system and a self-designed split Hopkinson tension bar (SHTB) apparatus (as shown in Fig. 2) [12], respectively. The basalt fiber tows were connected with the
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Fig. 1. Bobbin of basalt fiber tow.
SHTB bars as shown in Fig. 3. In Fig. 3, Ls was the testing span (fiber tow length in testing region) and Ls = 10 mm. As shown in Fig. 3(a), before the tensile tests, first the lining blocks were glued on the supplement plate perpendicularly. Then the fiber tows were wound in parallel onto the lining blocks. In the quasi-static test, the fiber tow specimen was glued to the slots of two short metal bars and using high shear strength (>20 MPa) adhesive (Type: WD-1001, made by Shanghai Kangda Chemical Co. Ltd. in China). The short metal bars were clamped in the grips of
Fig. 3. Preparation of fiber tows specimen for tensile test: (a) sketch diagram; (b) lining blocks and supplement plate; (c) basalt fiber tows for test.
MTS 810.23 tester. In the high strain rate tensile tests, the wound fiber specimen shown in Fig. 3(c) was inserted into the slots and then glued with the incident bar and transmission bar with the adhesive. Finally the supplement plate was removed before testing. Fig. 4 shows the fiber tow specimen for tensile testing. The fiber tow specimens were tested at room temperature and approximately 55% relative humidity. The fiber tows were tested at least three times at each strain rate (including quasi-static testing). 3. Results 3.1. Stress–strain curves The stress–strain curves of the basalt fiber tow are shown in Fig. 5. It can be found that the basalt fiber is strain rate sensitive. Fig. 6 shows the relationships between tensile modulus and strain rate. Fig. 7 depicts the relationships between tensile strength, tensile failure strain and strain rate. The tensile modulus and strength both increase with the strain rate, while the failure strain decreases with strain rate. It can be shown in Figs. 6 and 7 that the plots appear approximately linear increase in the whole range of strain rates. However, owing to lack of the test data during the medium strain rates (1–100 s−1 ), the linear regression method was not used for curve fitting. We used the following fitting equations to describe the relationship between mechanical properties and strain rates. Tensile modulus:
E = 105.44 − 43.74 exp −
ε˙ 2283.10
(1)
Tensile strength:
Fig. 2. Self-designed split Hopkinson tension bar (SHTB) apparatus.
max = 2956.59 − 1385.08 exp −
ε˙ 8419.09
(2)
L. Zhu et al. / Materials Science and Engineering A 527 (2010) 3245–3252
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Fig. 5. Stress–strain curves of basalt fiber tows under different strain rates.
Fig. 6. Modulus vs. strain rate curve.
equilibrium state. The tensile failure is controlled by the weak link or defects in the fibers. The weakest fibers fail in one dominant flaw or the fibers fail in the weakest part of the fiber. While under high rate of loading a larger proportion of fibers (with a range of flaw population, rather than a single flaw) get activated providing effective rate dependence. And also, the stress wave propagated in the
Fig. 4. Basalt fiber tows connected with steel bars: (a) before adhesvive consolidation; (b) after adhesvive consolidation: (b-1) front view and (b-2) top view.
Failure strain:
εmax = 2.42 + 0.67 exp −
ε˙ 1025.19
+ 0.16 exp −
ε˙ 37.89
(3)
From Figs. 7 and 8, it is interesting to note the gradual change in the modulus (ultimate stress and failure strain as well) over five orders of magnitude of strain rate (0.001–600 s−1 ) and a rapid change over the remaining one order of magnitude (600–3000 s−1 ). There is the same phenomenon for silicon carbide (SiC) fiber [13] and para-aramid fiber [14,15]. It is likely that under low rate of loading, there is enough time for the stress in the fiber to reach an
Fig. 7. Tensile strength, failure strain vs. strain rate curves.
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theory from Weibull distribution [18–23]. The Weibull distribution is based on the theory in which the fracture is controlled by the weakest defect of all the defects in a fiber, the so-called “weakest link theory” [20–22]. To characterize the strength distribution of the basalt fiber tow, a fiber tow model is presented in this paper, as shown in Fig. 11. In this model the N parallel fibers of the same length L, cross-sectional area A, are securely fixed between the two ends. The fibers are clamped at the ends in such a way that all of the unbroken fibers have the same strain, and such that there’s no interaction between the individual fibers. Each filament of the filament bundle remains completely elastic until it ruptures when the tensile stress in the fiber reaches its rupture strength. The model assumed: (1) The relationship between applied stress f and strain εf for a single fiber follows Hooke’s Law up to failure under tensile test: f = Ef εf
Fig. 8. Comparison of stress–strain curves of SiC, para-aramid and basalt fiber tows.
fiber will lead a un-equilibrium status at the beginning time of tension. The fiber stress equals to the tension stress behind the wave front and zero before the stress wave front. Owing to the shorter length of fiber under the tension state compared with that in quasistatic tension, then the probability of fiber fails at the weakest part decreases and the failure strength, modulus increase. The higher of the strain rate, the higher of the Young’s modulus and strength. Fig. 8 compares the stress–strain curves of silicon carbide (SiC) fiber [13] and para-aramid fiber [14] under quasi-static (0.001 s−1 ) and high strain rate (1200 s−1 ) with that of the basalt fiber. As shown in Fig. 8, both the three fiber tows are sensitive to strain rate. The para-aramid fiber tows have the highest failure stress, strain. The SiC fiber tows have the highest Young’s modulus. The basalt fiber tows have higher Young’s modulus than para-aramid fiber tows under high strain rate.
where Ef is the modulus of fiber. (2) The probability distribution of single fiber strength under tensile impact follows Weibull distribution. Failure probability H( f ) of single fibers under a stress no greater than of single and multi-modal Weibull distribution are as follows: ˇ1 f H(f ) = 1 − [1 − H1 (f )] = 1 − exp − (5) 1 H(f ) = 1 − [1 − H1 (f )][1 − H2 (f )]
ˇ1 f
= 1 − exp −
4. Statistical constitutive model of the basalt fiber tow
1
−
ˇ2 f
(6)
2
H(f ) = 1 − [1 − H1 (f )][1 − H2 (f )][1 − H3 (f )]
ˇ1 f
= 1 − exp −
3.2. Fractographs Fig. 9 shows damage morphology of the basalt fiber under high strain rate tension. The basalt fibers were cut from the fractured fiber tows. The fractographs were observed and photographed under scanning electron microscope (SEM). Fig. 10 displays the fractographs of the basalt fiber under different strain rates. It is seen from the figures that basalt fibers are broken in a brittle manner and the failure propagates along the surface defect, as shown by the white arrows of Fig. 10. Just like the SiC fiber [16], this defect may be a “flaw” type, but the fibers broken by a “pit” type of defect and undetectable defects on the surfaces were sometimes observed in Fig. 10. As the strain rate increases, the “pit” will be open more severely and the crack will propagate to both along cross-section direction. This is main failure mode of the basalt fiber under quasi-static and high strain rate tension, i.e., the fracture is typical of brittle materials – defect induced failure. Contrary to the glass fiber [17], the failure mode of basalt fiber has not the ductile–brittle transition when the strain rate increases. The crack in the basalt fiber will be generated from the surface flaw and then propagates along cross-section direction, finally to the fracture.
(4)
1
−
ˇ2 f
2
−
ˇ3 f
3
(7)
where Eq. (5) is single Weibull distribution, Eq. (6) is two modal Weibull distribution, and Eq. (7) is three-modal Weibull distribution. i (i = 1–3) is the scale parameter, and ˇi (i = 1–3) is the shape parameter. (3) The rupture strength is fiber type dependent. As n fibers break, the load is uniformly borne by (N − n) fibers immediately. From the assumptions, the constitutive equation of the filament bundle is given as:
= Eε 1 − =
n N
= Eε(1 − )
(8)
n = 1 − H(f ) N
(9)
E = Ef ε = εf
(10)
Then the tensile constitutive equation of filament fiber tow could be obtained as:
Eε ˇ1
= Eε exp −
1
Eε ˇ1
4.1. Weibull strength theory of parallel filament fiber tow
= Eε exp −
It is well known that the constitutive equation of parallel filament fiber tow could be characterized with Weibull strength
= Eε exp −
1
Eε ˇ1 1
−
(11)
Eε ˇ2
−
2
Eε ˇ2 2
−
(12)
Eε ˇ3 3
(13)
L. Zhu et al. / Materials Science and Engineering A 527 (2010) 3245–3252
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Fig. 9. Damage of basalt fiber tows under high strain rate tension: (a) strain rate of 600 s−1 ; (b) strain rate of 1200 s−1 ; (c) strain rate of 1800 s−1 ; (d) strain rate of 2400 s−1 ; (e) strain rate of 3000 s−1 .
where E can be estimated from the tensile impact experiment, i (i = 1–3) and ˇi (i = 1–3) can be obtained by nonlinear least squares method using the Levenberg-Marquardt algorithm [24,25]. Eqs. (11)–(13) are statistical constitutive equations for single Weibull distribution, bimodal Weibull and three-modal Weibull distribution, respectively. For single Weibull distribution, we take double logarithms on both sides of Eq. (11) one can obtain:
ln − ln
Eε
= ln
Eε ˇ1 1
(14)
The nonlinear parameters 1 and ˇ1 can be determined simply by regression analysis method. As for 2 and ˇ2 , 3 and ˇ3 in bimodal Weibull and three-modal Weibull distribution, they
can be determined by using the Levenberg-Marquardt algorithm [24,25]. From Eq. (14), the Weibull plot of ln[− ln(/Eε)] against ln[Eε] can be obtained based on the stress–strain curve of fiber bundles. An algorithm for least squares estimation of nonlinear parameters is used to simulate the experimental points and estimate the Weibull parameters 1 and ˇ1 . Thus, the tensile strength distribution of fibers at different strain rate can be determined from fiber tow tensile tests. 4.2. Constitutive equation of basalt fiber tow under various strain rates In the weibull plot curve of the experimental data of the basalt fiber tow under various strain rate (Fig. 12), all the curves are in straight line. Combined with Fig. 10, it is shown that there is
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Fig. 10. Fractographs of basalt fiber under quasi-static and high strain rate tension.
only one tensile failure mechanism for the basalt fiber tow. The failure mechanism is that the crack propagation from the surface flaw along cross-section direction and longitudinal direction. This is quite different with E-glass fiber tows [26] and SiC fiber tows [13,16], although the basalt fiber is an inorganic fiber, just like glass fiber. Then the constitutive equation of the basalt fiber tow could
be obtained from the single Weibull strength distribution theory using Eq. (11). By the least squares estimation, the Weibull distribution parameters in Eq. (11) under various strain rates could be calculated and are listed in Table 1. It could be shown that both the scale parameter 1 and shape parameter ˇ1 increase with the strain rate. As
L. Zhu et al. / Materials Science and Engineering A 527 (2010) 3245–3252
89.702ε 4.60
= 89.702ε exp −
3277
93.898ε 5.10
= 93.898ε exp −
3312
3251
strain rate : 2400 s−1
strain rate : 3000 s−1
The solid lines in Fig. 5 were drawn from the Weilbull distribution constitutive equations, which fitted the experimental data well. 5. Conclusions
Fig. 11. Model of parallel filament fiber tow.
The tensile properties of basalt finer tow have been tested under quasi-static and high strain rate conditions with MTS 810.23 materials tester and self-designed split Hopkinson tension bar (SHTB) apparatus. The stress strain curves under different strain rates were obtained. The fractographs of the basalt fibers were photographed for analyzing the failure mode. Weibull distribution strength theory was employed for deriving the constitutive equation. The following conclusions were drawn: 1. Basalt fiber is a rate-dependent material. The elastic modulus, strength and the failure strain of the basalt fiber tows apparently increase with the strain rate from 0.001 to 3000 s−1 . 2. The crack propagation induced from surface defects is the main failure mode both under quasi-static and high strain rate tension. 3. The statistical results show that the strength distribution of the basalt fiber complies with the single Weibull distribution. The single Weibull constitutive model can describe the stress strain relationship of the fiber bundles under different strain rates. Both the scale parameter and shape parameter increase with the strain rate.
Fig. 12. Weibull plot for basalt fiber tows under different strain rates.
the strain rate increases, the variation of strength also increases because the crack propagation from the surface defect will be not along cross-section direction only. From Table 1, the constitutive equations of the basalt fiber tow under quasi-static and high strain rate testing are as follows:
61.761ε 3.99
= 61.761ε exp −
2840
71.560ε 4.10
= 71.560ε exp −
2998
79.885ε 4.20
= 79.885ε exp −
3123
85.692ε 4.50
= 85.692ε exp −
3236
strain rate : 0.001 s−1
strain rate : 600 s−1
Acknowledgements The authors acknowledge the financial supports from the National Science Foundation of China (grant numbers 10802022 and 10872049) and the Key-grant Project of Chinese Ministry of Education (no. 309014). The sponsors from Shanghai Educational Development Foundation (08CG39) and Shanghai Rising-Star Program (08QA14008) are also gratefully acknowledged. This work is also supported by Program for Innovative Research Team (in Science and Technology) in University of Henan Province (2009HASTIT027 and 2010IRTSTHN007). References
strain rate : 1200 s−1
strain rate : 1800 s−1
Table 1 Weibull distribution parameters of basalt fiber tow at different strain rates (specimen length in testing region is 10 mm). Strain rate ε˙ (s−1 )
E (GPa)
1
ˇ1
0.001 600 1200 1800 2400 3000
61.761 71.560 79.885 85.692 89.702 93.898
2840 2998 3123 3236 3277 3312
3.99 4.10 4.20 4.50 4.60 5.10
[1] T. Deak, T. Czigány, Textile Research Journal 79 (7) (2009) 645–651. [2] J. Militky, V. Kovacic, Textile Research Journal 66 (4) (1996) 225–229. [3] Q. Liu, M.T. Shaw, R.S. Parnas, A.M. McDonnell, Polymer Composites 27 (1) (2006) 41–48. [4] T. Czigány, in: K. Friedrich (Ed.), Polymer Composites: From Nano to Macroscale, Springer Verlag, München, Germany, 2005, pp. 309–328. [5] T. Czigány, Express Polymer Letters 1 (2) (2007) 59. [6] J. Militky, V. Kovacic, J. Rubnerová, Engineering Fracture Mechanics 69 (9) (2002) 1025–1033. [7] M. Botev, A. Betchev, D. Bikiaris, Journal of Applied Polymer Science 74 (3) (1999) 523–531. [8] L.M. Vas, T. Czigány, Journal of Composite Materials 40 (19) (2006) 1735–1762. [9] N.N. Morozov, V.S. Bakunov, E.N. Morozov, L.G. Aslanova, P.A. Granovskii, V.V. Prokshin, A.A. Zemlyanitsyn, Glass and Ceramics 58 (3–4) (2001) 100–104. [10] J. Sim, C. Park, D.Y. Moon, Composites Part B 36 (6–7) (2005) 504–512. [11] F. Fourné, Synthetic Fibers, Carl Hanser Verlag, München, Germany, 1999. [12] B.Z. Sun, F. Liu, B.H. Gu, Composites Part A 36 (11) (2005) 1477–1485. [13] W. Huang, W. Xu, Y.M. Xia, Journal of Materials Science 40 (2) (2005) 465–468. [14] B.H. Gu, Journal of Composite Materials 38 (22) (2004) 2049–2074. [15] Y. Wang, Y.M. Xia, Composites Part A 29 (11) (1998) 1411–1415. [16] K. Goda, H. Fukunaga, Journal of Materials Science 21 (12) (1986) 4475–4480. [17] Y.M. Xia, X. Wang, B.C. Yang, Journal of Materials Science Letters 12 (18) (1993) 1481–1484.
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[18] W. Weibull, Proceedings of Royal Swedish Institute for Engineering Research [(Stockholm: Ingeniors Vetenskaps Akademiens)] 151 (1939) 1–45. [19] W. Weibull, ASME Journal of Applied Mechanics, Transactions of the American Society of Mechanical Engineers 18 (3) (1951) 293–297. [20] F.T. Peirce, Journal of the Textile Institute 17 (7) (1926) T355–T368. [21] H.E. Daniels, Proceedings of the Royal Society of London Series A, Mathematical and Physical Sciences 183 (995) (1945) 405–435.
[22] B.D. Coleman, Journal of the Mechanics and Physics of Solids 7 (1) (1958) 60–70. [23] R.V. Curtis, A.S. Juszczyk, Journal of Materials Science 33 (5) (1998) 1151–1157. [24] K. Feng, Numerical Mathematics and Algorithm, China National Defense Press, Beijing, 1978, pp. 154–162 (in Chinese). [25] D.W. Marquardt, SIAM Journal on Applied Mathematics 11 (2) (1963) 431–441. [26] Z. Wang, Y.M. Xia, Composites Science and Technology 57 (12) (1997) 1599–1607.
Contents lists available at ScienceDirect
Materials Science and Engineering A journal homepage: www.elsevier.com/locate/msea
Constitutive equations of basalt filament tows under quasi-static and high strain rate tension Lvtao Zhu a , Baozhong Sun a , Hong Hu b , Bohong Gu a,c,∗ a b c
College of Textiles, Donghua University, Shanghai 201620, China Institute of Textiles and Clothing, The Hong Kong Polytechnic University, Hung Hom Kowloon, Hong Kong Department of Textile Engineering, Zhongyuan Institute of Technology, Zhengzhou, Henan Province 450007, China
a r t i c l e
i n f o
Article history: Received 15 September 2009 Received in revised form 28 January 2010 Accepted 3 February 2010
Keywords: Basalt filament tows Tensile properties Strain rate Weibull distribution function
a b s t r a c t The tensile properties of basalt filament tows were tested at quasi-static (0.001 s−1 ) and high strain rates (up to 3000 s−1 ) with MTS materials tester (MTS 810.23) and split Hopkinson tension bar (SHTB), respectively. Experimental results showed that the mechanical properties of the basalt filament tows were rather sensitive to strain rate. Specifically, the stiffness and failure stress of the basalt filament tows increased distinctly as the strain rate increased, while the failure strain decreased. From scanning electronic microscope (SEM) photographs of the fracture surface, it is indicated that the basalt filament tows failed in a more brittle mode and the fracture surface got more regular as the strain rate increases. The strength distributions of the basalt filament tows have been evaluated by a single Weibull distribution function. The curve predicted from the single Weibull distribution function was in good agreement with the experimental data points. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Basalt is a common volcanic rock that can be found in most countries around the globe and is directly suitable for fiber manufacturing. Its chemical structure is near to glass. The most important components of basalt are SiO2 , Al2 O3 , CaO, MgO, Fe2 O3 and FeO [1–5]. Basalt rocks are molten approximately between 1350 and 1700 ◦ C [6–8]. Basalt fibers are produced in one step, directly from crushed basalt stone. Basalt fibers are more resistant to strong alkalis than glass fibers, but glass can better withstand strong acids. Basalt fibers can be used over a wide range of temperature, from −200 to 600 ◦ C [9–11]. Nowadays basalt fibers have great potential application to composite materials. The idea of using basalt fibers as reinforcement of composite materials first emerged in the former Soviet Union in an aerospace research program. Today most of the continuous basalt fibers are manufactured in Russia and Ukraine [11]. The aim of this study was to evaluate the tensile properties of basalt filament tows under various tensile speeds (corresponding to various strain rates). The rate-sensitivity of the tensile properties would be investigated. The fracture morphologies of the basalt filament tows under different strain rates were photographed with
∗ Corresponding author at: College of Textiles, Donghua University, Shanghai 201620, China. Tel.: +86 21 67792661/371 67698880; fax: +86 21 67792627/371 62506970. E-mail address: [email protected] (B. Gu). 0921-5093/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2010.02.015
scanning electronic microscopy (SEM). The fracture behaviors of the basalt fibers were analyzed from these fractographs. Based on the filament tows model and the statistical theory of fiber strength, a single Weibull statistical model was employed to describe the strength distribution of the basalt filament tows, and the Weibull parameters were obtained by the filament tows testing method. Consistency between the simulated and experimental results indicates that the model and the method are valid and reliable. 2. Experimental 2.1. Basalt fibers The basalt filament tows (as shown in Fig. 1) were manufactured by Hengdian Group Shanghai Russia & Gold Basalt Fiber Co. Ltd. in China. The fineness of the basalt filament tows is 2400tex/21800f (provided by the manufacturer). The diameter of monofilamanet is 7 m. The volume density is 2.6 g/cm3 . The photograph of the bobbin of basalt filament tows is show in Fig. 1. 2.2. Testing The quasi-static tensile tests (with the strain rate of 0.001 s−1 ) and high strain rate tensile tests (impact tensile tests) were performed on a MTS 810.23 materials tester system and a self-designed split Hopkinson tension bar (SHTB) apparatus (as shown in Fig. 2) [12], respectively. The basalt fiber tows were connected with the
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L. Zhu et al. / Materials Science and Engineering A 527 (2010) 3245–3252
Fig. 1. Bobbin of basalt fiber tow.
SHTB bars as shown in Fig. 3. In Fig. 3, Ls was the testing span (fiber tow length in testing region) and Ls = 10 mm. As shown in Fig. 3(a), before the tensile tests, first the lining blocks were glued on the supplement plate perpendicularly. Then the fiber tows were wound in parallel onto the lining blocks. In the quasi-static test, the fiber tow specimen was glued to the slots of two short metal bars and using high shear strength (>20 MPa) adhesive (Type: WD-1001, made by Shanghai Kangda Chemical Co. Ltd. in China). The short metal bars were clamped in the grips of
Fig. 3. Preparation of fiber tows specimen for tensile test: (a) sketch diagram; (b) lining blocks and supplement plate; (c) basalt fiber tows for test.
MTS 810.23 tester. In the high strain rate tensile tests, the wound fiber specimen shown in Fig. 3(c) was inserted into the slots and then glued with the incident bar and transmission bar with the adhesive. Finally the supplement plate was removed before testing. Fig. 4 shows the fiber tow specimen for tensile testing. The fiber tow specimens were tested at room temperature and approximately 55% relative humidity. The fiber tows were tested at least three times at each strain rate (including quasi-static testing). 3. Results 3.1. Stress–strain curves The stress–strain curves of the basalt fiber tow are shown in Fig. 5. It can be found that the basalt fiber is strain rate sensitive. Fig. 6 shows the relationships between tensile modulus and strain rate. Fig. 7 depicts the relationships between tensile strength, tensile failure strain and strain rate. The tensile modulus and strength both increase with the strain rate, while the failure strain decreases with strain rate. It can be shown in Figs. 6 and 7 that the plots appear approximately linear increase in the whole range of strain rates. However, owing to lack of the test data during the medium strain rates (1–100 s−1 ), the linear regression method was not used for curve fitting. We used the following fitting equations to describe the relationship between mechanical properties and strain rates. Tensile modulus:
E = 105.44 − 43.74 exp −
ε˙ 2283.10
(1)
Tensile strength:
Fig. 2. Self-designed split Hopkinson tension bar (SHTB) apparatus.
max = 2956.59 − 1385.08 exp −
ε˙ 8419.09
(2)
L. Zhu et al. / Materials Science and Engineering A 527 (2010) 3245–3252
3247
Fig. 5. Stress–strain curves of basalt fiber tows under different strain rates.
Fig. 6. Modulus vs. strain rate curve.
equilibrium state. The tensile failure is controlled by the weak link or defects in the fibers. The weakest fibers fail in one dominant flaw or the fibers fail in the weakest part of the fiber. While under high rate of loading a larger proportion of fibers (with a range of flaw population, rather than a single flaw) get activated providing effective rate dependence. And also, the stress wave propagated in the
Fig. 4. Basalt fiber tows connected with steel bars: (a) before adhesvive consolidation; (b) after adhesvive consolidation: (b-1) front view and (b-2) top view.
Failure strain:
εmax = 2.42 + 0.67 exp −
ε˙ 1025.19
+ 0.16 exp −
ε˙ 37.89
(3)
From Figs. 7 and 8, it is interesting to note the gradual change in the modulus (ultimate stress and failure strain as well) over five orders of magnitude of strain rate (0.001–600 s−1 ) and a rapid change over the remaining one order of magnitude (600–3000 s−1 ). There is the same phenomenon for silicon carbide (SiC) fiber [13] and para-aramid fiber [14,15]. It is likely that under low rate of loading, there is enough time for the stress in the fiber to reach an
Fig. 7. Tensile strength, failure strain vs. strain rate curves.
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theory from Weibull distribution [18–23]. The Weibull distribution is based on the theory in which the fracture is controlled by the weakest defect of all the defects in a fiber, the so-called “weakest link theory” [20–22]. To characterize the strength distribution of the basalt fiber tow, a fiber tow model is presented in this paper, as shown in Fig. 11. In this model the N parallel fibers of the same length L, cross-sectional area A, are securely fixed between the two ends. The fibers are clamped at the ends in such a way that all of the unbroken fibers have the same strain, and such that there’s no interaction between the individual fibers. Each filament of the filament bundle remains completely elastic until it ruptures when the tensile stress in the fiber reaches its rupture strength. The model assumed: (1) The relationship between applied stress f and strain εf for a single fiber follows Hooke’s Law up to failure under tensile test: f = Ef εf
Fig. 8. Comparison of stress–strain curves of SiC, para-aramid and basalt fiber tows.
fiber will lead a un-equilibrium status at the beginning time of tension. The fiber stress equals to the tension stress behind the wave front and zero before the stress wave front. Owing to the shorter length of fiber under the tension state compared with that in quasistatic tension, then the probability of fiber fails at the weakest part decreases and the failure strength, modulus increase. The higher of the strain rate, the higher of the Young’s modulus and strength. Fig. 8 compares the stress–strain curves of silicon carbide (SiC) fiber [13] and para-aramid fiber [14] under quasi-static (0.001 s−1 ) and high strain rate (1200 s−1 ) with that of the basalt fiber. As shown in Fig. 8, both the three fiber tows are sensitive to strain rate. The para-aramid fiber tows have the highest failure stress, strain. The SiC fiber tows have the highest Young’s modulus. The basalt fiber tows have higher Young’s modulus than para-aramid fiber tows under high strain rate.
where Ef is the modulus of fiber. (2) The probability distribution of single fiber strength under tensile impact follows Weibull distribution. Failure probability H( f ) of single fibers under a stress no greater than of single and multi-modal Weibull distribution are as follows: ˇ1 f H(f ) = 1 − [1 − H1 (f )] = 1 − exp − (5) 1 H(f ) = 1 − [1 − H1 (f )][1 − H2 (f )]
ˇ1 f
= 1 − exp −
4. Statistical constitutive model of the basalt fiber tow
1
−
ˇ2 f
(6)
2
H(f ) = 1 − [1 − H1 (f )][1 − H2 (f )][1 − H3 (f )]
ˇ1 f
= 1 − exp −
3.2. Fractographs Fig. 9 shows damage morphology of the basalt fiber under high strain rate tension. The basalt fibers were cut from the fractured fiber tows. The fractographs were observed and photographed under scanning electron microscope (SEM). Fig. 10 displays the fractographs of the basalt fiber under different strain rates. It is seen from the figures that basalt fibers are broken in a brittle manner and the failure propagates along the surface defect, as shown by the white arrows of Fig. 10. Just like the SiC fiber [16], this defect may be a “flaw” type, but the fibers broken by a “pit” type of defect and undetectable defects on the surfaces were sometimes observed in Fig. 10. As the strain rate increases, the “pit” will be open more severely and the crack will propagate to both along cross-section direction. This is main failure mode of the basalt fiber under quasi-static and high strain rate tension, i.e., the fracture is typical of brittle materials – defect induced failure. Contrary to the glass fiber [17], the failure mode of basalt fiber has not the ductile–brittle transition when the strain rate increases. The crack in the basalt fiber will be generated from the surface flaw and then propagates along cross-section direction, finally to the fracture.
(4)
1
−
ˇ2 f
2
−
ˇ3 f
3
(7)
where Eq. (5) is single Weibull distribution, Eq. (6) is two modal Weibull distribution, and Eq. (7) is three-modal Weibull distribution. i (i = 1–3) is the scale parameter, and ˇi (i = 1–3) is the shape parameter. (3) The rupture strength is fiber type dependent. As n fibers break, the load is uniformly borne by (N − n) fibers immediately. From the assumptions, the constitutive equation of the filament bundle is given as:
= Eε 1 − =
n N
= Eε(1 − )
(8)
n = 1 − H(f ) N
(9)
E = Ef ε = εf
(10)
Then the tensile constitutive equation of filament fiber tow could be obtained as:
Eε ˇ1
= Eε exp −
1
Eε ˇ1
4.1. Weibull strength theory of parallel filament fiber tow
= Eε exp −
It is well known that the constitutive equation of parallel filament fiber tow could be characterized with Weibull strength
= Eε exp −
1
Eε ˇ1 1
−
(11)
Eε ˇ2
−
2
Eε ˇ2 2
−
(12)
Eε ˇ3 3
(13)
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Fig. 9. Damage of basalt fiber tows under high strain rate tension: (a) strain rate of 600 s−1 ; (b) strain rate of 1200 s−1 ; (c) strain rate of 1800 s−1 ; (d) strain rate of 2400 s−1 ; (e) strain rate of 3000 s−1 .
where E can be estimated from the tensile impact experiment, i (i = 1–3) and ˇi (i = 1–3) can be obtained by nonlinear least squares method using the Levenberg-Marquardt algorithm [24,25]. Eqs. (11)–(13) are statistical constitutive equations for single Weibull distribution, bimodal Weibull and three-modal Weibull distribution, respectively. For single Weibull distribution, we take double logarithms on both sides of Eq. (11) one can obtain:
ln − ln
Eε
= ln
Eε ˇ1 1
(14)
The nonlinear parameters 1 and ˇ1 can be determined simply by regression analysis method. As for 2 and ˇ2 , 3 and ˇ3 in bimodal Weibull and three-modal Weibull distribution, they
can be determined by using the Levenberg-Marquardt algorithm [24,25]. From Eq. (14), the Weibull plot of ln[− ln(/Eε)] against ln[Eε] can be obtained based on the stress–strain curve of fiber bundles. An algorithm for least squares estimation of nonlinear parameters is used to simulate the experimental points and estimate the Weibull parameters 1 and ˇ1 . Thus, the tensile strength distribution of fibers at different strain rate can be determined from fiber tow tensile tests. 4.2. Constitutive equation of basalt fiber tow under various strain rates In the weibull plot curve of the experimental data of the basalt fiber tow under various strain rate (Fig. 12), all the curves are in straight line. Combined with Fig. 10, it is shown that there is
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Fig. 10. Fractographs of basalt fiber under quasi-static and high strain rate tension.
only one tensile failure mechanism for the basalt fiber tow. The failure mechanism is that the crack propagation from the surface flaw along cross-section direction and longitudinal direction. This is quite different with E-glass fiber tows [26] and SiC fiber tows [13,16], although the basalt fiber is an inorganic fiber, just like glass fiber. Then the constitutive equation of the basalt fiber tow could
be obtained from the single Weibull strength distribution theory using Eq. (11). By the least squares estimation, the Weibull distribution parameters in Eq. (11) under various strain rates could be calculated and are listed in Table 1. It could be shown that both the scale parameter 1 and shape parameter ˇ1 increase with the strain rate. As
L. Zhu et al. / Materials Science and Engineering A 527 (2010) 3245–3252
89.702ε 4.60
= 89.702ε exp −
3277
93.898ε 5.10
= 93.898ε exp −
3312
3251
strain rate : 2400 s−1
strain rate : 3000 s−1
The solid lines in Fig. 5 were drawn from the Weilbull distribution constitutive equations, which fitted the experimental data well. 5. Conclusions
Fig. 11. Model of parallel filament fiber tow.
The tensile properties of basalt finer tow have been tested under quasi-static and high strain rate conditions with MTS 810.23 materials tester and self-designed split Hopkinson tension bar (SHTB) apparatus. The stress strain curves under different strain rates were obtained. The fractographs of the basalt fibers were photographed for analyzing the failure mode. Weibull distribution strength theory was employed for deriving the constitutive equation. The following conclusions were drawn: 1. Basalt fiber is a rate-dependent material. The elastic modulus, strength and the failure strain of the basalt fiber tows apparently increase with the strain rate from 0.001 to 3000 s−1 . 2. The crack propagation induced from surface defects is the main failure mode both under quasi-static and high strain rate tension. 3. The statistical results show that the strength distribution of the basalt fiber complies with the single Weibull distribution. The single Weibull constitutive model can describe the stress strain relationship of the fiber bundles under different strain rates. Both the scale parameter and shape parameter increase with the strain rate.
Fig. 12. Weibull plot for basalt fiber tows under different strain rates.
the strain rate increases, the variation of strength also increases because the crack propagation from the surface defect will be not along cross-section direction only. From Table 1, the constitutive equations of the basalt fiber tow under quasi-static and high strain rate testing are as follows:
61.761ε 3.99
= 61.761ε exp −
2840
71.560ε 4.10
= 71.560ε exp −
2998
79.885ε 4.20
= 79.885ε exp −
3123
85.692ε 4.50
= 85.692ε exp −
3236
strain rate : 0.001 s−1
strain rate : 600 s−1
Acknowledgements The authors acknowledge the financial supports from the National Science Foundation of China (grant numbers 10802022 and 10872049) and the Key-grant Project of Chinese Ministry of Education (no. 309014). The sponsors from Shanghai Educational Development Foundation (08CG39) and Shanghai Rising-Star Program (08QA14008) are also gratefully acknowledged. This work is also supported by Program for Innovative Research Team (in Science and Technology) in University of Henan Province (2009HASTIT027 and 2010IRTSTHN007). References
strain rate : 1200 s−1
strain rate : 1800 s−1
Table 1 Weibull distribution parameters of basalt fiber tow at different strain rates (specimen length in testing region is 10 mm). Strain rate ε˙ (s−1 )
E (GPa)
1
ˇ1
0.001 600 1200 1800 2400 3000
61.761 71.560 79.885 85.692 89.702 93.898
2840 2998 3123 3236 3277 3312
3.99 4.10 4.20 4.50 4.60 5.10
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