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Fracture toughness estimation for high-strength rail steels using indentation test
Accepted Manuscript Fracture toughness estimation for high-strength rail steels using indentation test Feng Yu, P.-Y. Ben Jar, Michael T. Hendry, Chester Jar, Kukkadapu Nishanth PII: DOI: Reference:
S0013-7944(18)30686-6 https://doi.org/10.1016/j.engfracmech.2018.10.030 EFM 6204
To appear in:
Engineering Fracture Mechanics
Received Date: Revised Date: Accepted Date:
7 July 2018 24 October 2018 26 October 2018
Please cite this article as: Yu, F., Ben Jar, P.-Y., Hendry, M.T., Jar, C., Nishanth, K., Fracture toughness estimation for high-strength rail steels using indentation test, Engineering Fracture Mechanics (2018), doi: https://doi.org/ 10.1016/j.engfracmech.2018.10.030
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Fracture toughness estimation for high-strength rail steels using indentation test Feng Yua,d, P.-Y. Ben Jara,b, Michael T. Hendrya,c, Chester Jarb, and Kukkadapu Nishanthe a
Canadian Rail Research Laboratory, bDepartment of Mechanical Engineering, cDepartment of Civil and Environmental Engineering, University of Alberta, Edmonton, Alberta, Canada d School of Mechanical Engineering and Mechanics, Ningbo University, Ningbo 315211, PR China1 e Department of Mechanical Engineering, National Institute of Technology, Rourkela, Odisha, India
Abstract In this paper, tensile properties and mode I critical stress intensity factor (KIc) are measured on the railhead of six types of high-strength rail steels using standard testing methods. An instrumented ball indentation testing method is then developed to estimate fracture toughness for the six rail steels based on continuum damage mechanics. Critical damage parameter, determined by combining repeated loadingunloading tensile tests and a ductile damage model, is used to characterize a critical contact depth under indentation that is used to calculate the specific indentation energy, and thus the fracture toughness for indentation (KInd). Comparison between KInd and KIc for the six high-strength rail steels suggests that the current indentation testing method can successfully rank the six high-strength rail steels for their fracture toughness. Keywords: Fracture toughness; Indentation test; Critical damage parameter; High-strength rail steels
1. Introduction Fracture toughness is one of the most important mechanical properties for assessing the integrity of rail steels. High fracture toughness allows rails to sustain a large crack size before fracture [1], thus increasing the possibility of detecting cracks in time, which is essential for improving safety and efficiency for railway transportation [2]. In the past four decades, research and development have been focused on producing premium, head-hardened pearlitic rail steels with high fracture toughness [3-6]. In addition, low-carbon bainitic rail steels have been developed, which show much higher fracture toughness but lower wear resistance than the pearlitic rail steels at the same characteristic strength level [7-9]. As a result, even though there expects a limit in the fine grain size for the pearlitic steel, efforts have been made to develop new pearlitic rail steels with optimal strength and fracture toughness. Similarly, development of 1
The corresponding author’s current email address: [email protected] 1
bainitic steel with improved wear resistance is expected to provide a promising alternative for the next generation of rail steels. For both types of newly developed rail steels, fracture toughness needs to be characterized first in laboratory and then evaluated for in-field applications. Standard testing methods are available for evaluating fracture toughness [10, 11], however, test specimens require specific geometries and dimensions as well as fatigue precracking that is very time-consuming and labor-intensive. Moreover, the standard tests can neither be applied to newly developed rail steels with small volume nor in-service rail tracks because of the destructive nature. As a result, many alternative testing methods have been developed to estimate fracture toughness [12-20], among which the instrumented indentation test shows its potential to provide a non-destructive means to measure fracture toughness with the minimum need for sample preparation. Note that indentation tests have been used to predict fracture toughness for brittle materials, in view that indentation generates cracks in brittle materials [21-28]. For ductile materials, cracks are generally known to be suppressed under compressive loading. To define a fracture initiation point under indentation, the attainment of critical fracture stress or strain under indentation was initially assumed as the fictitious fracture initiation point [18, 20, 29]. An important theoretical basis was also proposed in these studies that the stress state beneath an indenter tip is similar to that in front of a crack tip [30]. This enables the use of indentation test to estimate fracture toughness. Recently, with the development of damage theories [31-33], Lee et al. [19] showed that during an indentation test, voids were generated beneath an indenter and therefore, the fracture initiation point under indentation can be quantified in terms of critical damage parameter. Since then, many studies have applied indentation techniques to prediction of fracture toughness for metals based on the concept of continuum damage mechanics (CDM) [34-37]. However, an important issue was not properly addressed in the previous studies. That is, the critical damage parameter after necking cannot be directly obtained from repeated loading-unloading tensile tests. This is because damage measured from such tests is only valid for uniform deformation [38]. In the present work, an instrumented ball indentation test is developed to estimate fracture toughness for six types of high-strength rail steels based on CDM. Monotonic tensile, three-point bending, and repeated loading-unloading tensile tests are carried out for the six rail steels to measure their tensile properties, fracture toughness (KIc), and deterioration of elastic 2
modulus, respectively. The critical damage parameter is determined through extrapolation of experimentally quantified deterioration of elastic modulus to the strain level for fracture according to a ductile damage model. Indentation tests are performed to a critical contact depth at which the accumulated damage under indentation reaches the same level as that for the critical damage parameter. The critical contact depth value is then used to calculate the specific indentation energy, and thus the fracture toughness for indentation (KInd). The correlation of KInd and KIc for the six rail steels is discussed.
2. Theoretical background 2.1. Specific indentation energy Previous studies showed that the degree of constraint in the deformed volume under a ball indenter was similar to that ahead of the crack tip, though the former is compressive and the latter is tensile [18, 19, 29]. In this work, the fracture energy is interpreted as the deformation capability of the material under a highly constrained stress field. Based on this concept, the specific work of fracture at the crack tip can be replaced by the specific indentation energy in Eq. (1), which is applied to estimate fracture toughness of ductile materials using the ball indentation test [19]. (1) where,
is the fracture toughness estimated through the indentation test,
modulus, and 2
the Young’s
the specific indentation energy determined using Eq. (2) below. (2)
where,
is the contact depth under indentation,
the critical value of
to generate the same
level of damage under indentation as the critical damage parameter for tensile specimens, indentation load which is a function of indenter tip with radius R,
, and
the
the projected indentation area. For a spherical
can be expressed as, (3)
Note that determination of
should include both pile-up and elastic deflection in indentation
tests [39-42], that is, (4)
3
where,
is the height of pile-up,
the consideration of pile-up, and
the maximum value of indentation depth (h) without
a constant set to be 0.75 for the ball indenter tip [41], and
the maximum indentation load and unloading contact stiffness at
,
respectively.
2.2. Determination of critical contact depth (hcr) As defined in CDM [33], damage can be expressed in terms of the change of elastic modulus using the following expression. (5) where, and
is the damage parameter,
the elastic modulus of virgin material (Young’s modulus),
the degraded elastic modulus due to the presence of damage. Change of E is commonly measured using the conventional repeated loading-unloading
tensile tests [38]. To ensure accuracy of damage measurement, Eq. (5) is normally applied to results from smooth tensile specimens that are subjected to uniform deformation. This approach, however, limits the change of E values to the plane-stress condition, i.e., before neck formation which starts at a low level of stress triaxiality (defined as the ratio of hydrostatic stress to von Mises stress) of around 0.33. For rail steels considered in the current study, standard smooth specimens often fracture after significant neck formation. Therefore, critical damage parameters for these rail steels have to be determined through an extrapolation to a strain level at the onset of fracture. In this study, expression for the extrapolation is given below, which was modified from Bonora’s original work for a nonlinear ductile damage model [43]. (6) where,
is an adjusting parameter,
the equivalent plastic strain at which E is measured,
is the plastic logarithmic area strain at fracture (
and
, respectively, denote
the initial and minimum transversal areas of round tensile specimens at the necking zone), the threshold strain for damage initiation which for the high-strength rail steels, is regarded as the yield strain of 0.002 [44], and
defined as [45] (7)
4
where
is the stress triaxiality as a function of , which is determined at the centre of smooth
tensile specimen by using FE modelling. Expression for
in Eq. (6) is given below [33]. (8)
being the Poisson’s ratio. Therefore,
with
is the only unknown in Eq. (6), and can be
determined uniquely by fitting Eq. (6) to the experimental data of deterioration of elastic modulus. Once
is determined, the critical value for elastic modulus, Ecr, can be determined at
. Damage accumulation under indentation, on the other hand, has been shown to increase with the increase of indentation depth, and the change of elastic modulus, E, was calculated using Eq. (9), as given below [19], which is based on the Hertzian contact law [46] and Sneddon’s elastic punch theory [47], (9) where, and
is the effective modulus for the contact system between specimen and indenter, are the Poisson’s ratio and Young’s modulus, respectively, for the ball indenter. At the very right expression of Eq. (9), E is expressed as a function of
expression for
in Eq. (3), a relationship between
. In view of the
and E can be established from Eq. (9).
Base on the assumption that Ecr is a material parameter and thus independent of loading conditions, the critical contact depth under indentation,
, can thus be determined by setting E
in Eq. (9) equal to Ecr.
3. Experiments 3.1. Materials and testing devices Six types of high-strength rail steels used in this test program were supplied by Transportation Technology Centre Inc. (TTCI). Due to confidential requirements, information about the material compositions and vendors cannot be provided. Therefore, the six rails are listed using a numbering system shown in Table 1, together with their microstructural features for the railhead.
5
An Instron hydraulic universal testing machine, with a load capacity of 222 kN, was used for tensile and three-point bending tests. An MTS 44-kN hydraulic testing system was used for the instrumented indentation tests. All tests were conducted at room temperature.
3.2. Tensile tests Fig. 1(a) labels locations of tensile specimens on the cross section of railhead, and Fig. 1(b) their geometry and dimensions. These tensile specimens were machined with the longitudinal direction parallel to the rolling direction. Geometry and dimensions of the tensile specimens conform to those specified in ASTM E8/E8M [48]. The tensile specimens labelled A, B, D, E, F, I, and J were used for monotonic tensile tests, while the remaining for repeated loading-unloading tensile tests. Both monotonic and repeated loading-unloading tensile tests were performed at a crosshead speed of 8.5x10-3 mm/s. An axial extensometer with the initial gauge length of 12.5 mm was placed in the middle of reduced gauge section to measure global elongation. Also, a diametric extensometer was mounted at the centre of tensile specimens to capture local diameter changes. In repeated loading-unloading tensile tests, the unloading phase was triggered based on a load increment of 0.8 kN before reaching the ultimate tensile strength (UTS), while after the UTS an increment of 0.255 mm in elongation was used which corresponded to a loading period of 30 seconds.
3.3. Three-point bending tests Single-edge-notched bend (SENB) specimens were used to measure KIc values for the railhead region, where locations for the SENB specimens are shown in Fig. 2(a). Longitudinal direction of the SENB specimens is along the rolling direction of the rails. Configurations and dimensional ratios of the SENB specimens follow those specified in ASTM E399 [10], with thickness (B), width (W) and span length (L) being 7.7, 13.4 and 60 mm, respectively, as shown in Fig. 2(b). Edge notch of the SENB specimens was first cut using abrasive waterjet, and further sharpened using electric-discharge machining (EDM). Pre-cracking was later introduced at the notch tip through fatigue loading. Total crack length was controlled to be within the range of 0.5 to 0.53 of W. Crosshead speed for the three-point bending test was set at 2x10-3 mm/s, and a clip-
6
on extensometer was used to measure the crack mouth opening displacement (CMOD), δ. The following equations were used to determine the conditional stress intensify factor KQ [10]. (10) (11) where
is a conditional load, corresponding to the load at the intersection between the
experimental curve of load versus CMOD and a secant line with 95% of the tangent slope, as illustrated in Fig. 3, and
the averaged initial pre-crack length measured along the crack front
after the specimen was fractured.
3.4. Instrumented indentation test A lab-scaled indentation testing system was developed in-house for the purpose of estimating fracture toughness for rail steels. Fig. 4(a) shows the schematic drawing of the indentation testing system with a ball indenter, same as that used in our previous publication [49]. The specimen is one half of railhead with 10 mm in thickness. Surface of the specimen was gradually polished to a mirror-like finish using a 0.05-μm grade alpha alumina polishing powder at the final polishing step. The indenter consists of O1 tool steel with proper heat treatment and a tungsten carbide ball with diameter of 1.19 mm, Ei of 480 GPa, and
of 0.28. The ball was
compression-fit into a pre-machined semi-spherical cavity at the end of a truncated-cone indenter. A clip-on extensometer was used to measure the indentation depth. Fig. 4(b) presents a typical curve from an indentation loading-unloading test, illustrating the measurement of the unloading contact stiffness (S), the maximum indentation depth (hmax), the corresponding load (Fmax), and the residual depth (hf) after the indenter is removed. To generate damage under indentation, loading-unloading indentation tests were conducted at ten indentation depths of 0.04, 0.06, 0.09, 0.12, 0.18, 0.24, 0.3, 0.36, 0.42, and 0.48 mm. Five tests were conducted at each of the above indentation depths and the average value was obtained for calculation. Indentation speed of 0.1 mm/min was used at both loading and unloading phases. The indentation load was recorded using an MTS load cell with a resolution of 0.01 N, and the indentation depth measured using the clip-on extensometer with a resolution of 3 μm. Note that height of the pile-up generated during the indentation loading, around the indenter, 7
was directly measured using a digital optical microscope with a magnification of 320X and a digital dial indicator with a resolution of 2.54 μm. Details of the test procedure are described as follows. Firstly, the original, flat specimen surface near an indent was focused under the microscope and its vertical position recorded using the digital dial indicator, to serve as the reference. The pile-up region at the edge of the indent was then examined, and change in the dial indicator reading from the reference in order to focus on the peak of the pile-up represented the local height of the pile-up. Five measurements of the pile-up height were taken around each indent, and their average value was used as the pile-up height.
4. Finite element modelling Finite element (FE) modelling of monotonic tensile and indentation tests was carried out using ABAQUS/CAE 6.13. Fig. 5(a) shows the axisymmetric FE model of smooth tensile specimen, following the dimensions given in Fig. 1(b). The model consists of 4,260 linear quadrilateral elements of type CAX4R and 4,657 nodes. The FE model was evenly discretized in the reduced gauge section with element size of 0.125 mm and aspect ratio of 1. Boundary conditions were set to be the same as those used in the experimental testing, i.e., with one end fixed and the other end moving at a specified displacement rate. Fig. 5(b) depicts an axisymmetric FE model of indentation test. The specimen is modelled as a cylinder with radius of 12.5 mm and height of 7 mm. It consists of 22,001 linear quadrilateral elements of type CAX4R and 22,262 nodes. The minimum element size at the contact region is 3 µm. The ball indenter was modelled as an analytical rigid body and its radius can be varied to investigate the effect of ball size on
. A contact surface was placed
between the ball indenter and specimen in the condition of small sliding, “hard” normal contact, and tangential friction of 0.2 [42]. The boundary conditions were set to be the same as those for the testing, i.e., with the bottom of the specimen supported and the indenter moving down at a specified displacement rate.
8
5. Results 5.1. Tensile properties Fig. 6(a) illustrates the variation of engineering stress versus engineering strain curves on railhead for Rail#control. It shows a decreasing trend of tensile strength from the top of the railhead to the core region. This phenomenon has also been observed for the other five types of rail steels as all of the rails are head-hardened but using different heat treatment methods. Fig. 6(b) summarizes the mean values of UTS and yield stress for the six rail steels, together with their corresponding maximum and minimum values. FE modelling of smooth tensile specimens was performed to establish constitutive equations for the six rail steels, according to the approach proposed by Zhang and Li [50] and successively formalized by Ling [51]. According to their work, the equivalent stress-strain curve was established by regenerating the experimental data of load-elongation curve and cross-section reduction through an iterative procedure based on the finite element analyses, as illustrated in Fig. 7(a) for tensile specimen A of Rail#control. This approach was applied to each of the six rail steels, and equivalent stress-strain curves were determined for all tensile specimens labelled A, B, D, E, F, I, and J. For the sake of conciseness, Fig. 7(b) compares the mean value of equivalent stress-strain curves for the six rail steels. It is shown that the Rail#2 shows the minimum ductility. Rail#2, Rail#3, and Rail#5 have similar magnitudes of stress at fracture. Rail#6 has the smallest stress at fracture but the best ductility. Rail#4 and Rail#control appear to be identical except that the latter is more ductile than the former.
5.2. KIc Values of mode I critical stress intensity factor, KIc, for SENB specimens on railhead, are summarized in Table 2 for the six rail steels. Table 2 also lists details for the pre-crack length (a0), specimen width (W), mean value of yield stress
, and ratio of PQ to Pmax, which are
required to validate KQ for KIc. According to ASTM E399 [10], to ensure validity of KIc measurements, value for (a0-W) should be bigger than 2.5(KQ/
) and PQ/Pmax less than 1.10.
Both conditions must be satisfied at the same time. Therefore, the experimental data that violate these two requirements are excluded, as shown in Table 2. By using the remaining experimental data, the average KIc value and standard deviation are calculated for each of the six rail steels. It is shown in Table 2 that the KIc values for Rail#control and Rail#4 are ranked as the first class 9
while Rail#5 and Rail#6 the second. However, difference in KIc values for these four types of rails is small, which is within 1 MPa.m0.5. KIc value for Rail#2 is bigger than that for Rail#3, but both show smaller KIc values than the other four rails.
5.3. Identification of hcr To determine hcr for each of the six rail steels, both repeated loading-unloading tensile and instrumented indentation tests were carried out for the damage measurement [19]. Fig. 8(a) illustrates the repeated loading-unloading tensile tests for Rail#control. For the accuracy of damage measurement, the loading-unloading tensile tests were repeated to the logarithmic area strain (
) of about 20%, where the neck formation was negligible. Damage parameter, in
terms of the deterioration of elastic modulus, was determined by measuring the slope of linear portion of the unloading true stress-logarithmic area strain curves [52, 53]. In addition, the effect of Poisson’s ratio has been taken into account to convert the measured unloading slopes to elastic modulus [54]. Fig. 8(b), on the other hand, shows instrumented ball indentation tests at ten different indentation depths for Rail#control. Previous study noted that to consider the influence of pre-existing damage and plastic hardening on the indentation response multiple loadingunloading indentation tests should be performed in a single location [55]. In view of good repeatability for the experimental data shown in Fig. 8(b) and following the works conducted by previous researchers [34, 35, 37], the indentation loading-unloading curves in Fig. 8(b) were obtained from multiple locations on the surface of test specimens. Degradation of elastic modulus under indentation was then measured by the changes in stiffness of indentation unloading curves [19]. Also, the pile-up after each indentation test was measured using the optical microscope. Due to the nonlinearity of indentation unloading curves [56], a power-law function is used to fit the unloading curves, also shown in Fig. 8(b). Two unknowns, K and m, were determined by using the least square method to best fit each of the unloading curves. First derivative of the power-law function was used to determine the unloading contact stiffness at
, i.e.,
that is one of the essential parameters needed for calculating elastic
modulus in Eq. (9). Fig. 9(a) shows the experimental data of deterioration of elastic modulus from repeated loading-unloading tensile tests for Rail#control. To determine the critical value for elastic 10
modulus (Ecr), Eq. (6) is used to fit the experimental data. As mentioned earlier,
is the only
adjustable variable in Eq. (6) and therefore, the solid line in Fig. 9(a) can be uniquely determined by applying the least square method to best fit the experimental data. Ecr is then determined based on an extrapolation of the solid line to
. The same approach was applied to the other five
types of rail steels. Fig. 9(b) presents both damage evolution and Ecr for each of the six rail steels. Damage evolution in the six rail steels, characterized using the indentation tests, is presented in Fig. 10, in which elastic modulus is plotted as a function of contact depth (hc). Linear functions are used to fit the relationship between elastic modulus and hc for all of the six rail steels. Young’s modulus was determined using the mean value of the measured elastic moduli at the first three depths. Value of hcr for each of the six rail steels was determined by matching hc to the value of Ecr from Fig. 9(b), and indicated in Fig. 10. 5.4. Fracture toughness estimation In addition to hcr and Ac, indentation load, F, is needed to be expressed as a function of hc in Eq. (2) to calculate the specific indentation energy, 2
Fig. 11 summarizes the typical
relationship between F and hc for the six rail steels. A second order polynomial function was then determined to fit each of the six curves, and applied to Eq. (2) to calculate the values for
. With
value for each of the six rail steels determined, fracture toughness under
indentation, KInd, is then calculated and compared with the corresponding KIc value. Fig. 12(a) shows that the ranking among the six rail steels using KInd is consistent with that using KInd, however, the KInd value for each of the six rails is about one order of magnitude bigger than the corresponding KIc value. Fig. 12(b) compares the normalized values of KInd with those of KIc for the six rail steels. Good correlation between the two parameters is shown, which indicates a potential capability for using KInd to correlate the KIc.
6. Discussion In view of different magnitudes of toughness values between KInd and KIc, two considerations are discussed in this section. For the first scenario, it may be attributed to the difference of localized deformation volume between the crack tip and the indenter tip. Eq. (12) is
11
used to calculate the critical crack tip opening displacement (CTOD) based on fracture mechanics [57], (12) where, KIc,
represents the critical value of CTOD at crack initiation. According to the values of , and
for Rail#control, i.e., 42.34 MPa.m0.5, 903 MPa, and 200 GPa, respectively, the
corresponding
value is 12.64 µm. In view of the deformation volume generated by the
current ball indenter (1.19 mm in diameter), there exists a huge difference in the plastic deformation zone sizes between the indenter tip and the crack tip. FE modelling of indentation test with 1.19-mm ball indenter was first carried out to compare the simulation results with the experimental data. FE modelling of indentation test shown in Fig. 13 was based on the equivalent stress-strain curve for Rail#control. Fig. 13(a) and (b) present the full distribution of stress triaxiality and its enlarged view under the ball tip, respectively. It is observed that the variation of stress triaxiality under the current ball indenter varies from 0.4 to -5.4, and there exists a region where the magnitude of stress triaxiality is at about 3.0, which is similar to that ahead of the crack tip [19, 49]. Although the degree of constrain under indentation is compressive and that around the crack tip tensile, it is generally accepted that the compressive and tensile deformations are equivalent if the deformation is not cyclic [29]. Fig. 13(c) plots the simulation result of indentation load versus contact depth, and compares with results from the experimental testing. The good agreement between the two sets of results suggests that the 2
value at hcr can be well predicted using the FE simulation.
A series of FE modelling was then performed using ball indenters of different diameters at a reduction factor of 0.5 each time. For each ball indenter size, the critical contact depth (
)
is chosen to be the value that generates the same magnitude of plastic strain under indentation, calculated using Eq. (13) below [58, 59]. (13) where,
is the critical representative plastic strain for indentation, =0.14 [58], R the radius of
ball indenter, and
which represents the critical diameter of the projected
area under indentation. The application of Eq. (13) is based on the assumption that damage is proportional to the plastic strain. That is, for the same
value, the critical damage parameters
generated by ball indenters of different sizes also have the same value. 12
As a result, Fig. 14 plots the simulation results of
for Rail#control under different
ball indenter sizes, which shows a linear relationship between the
values and ball sizes. It
is indicated that in order to approach a reasonable magnitude for the KIc value, the size of the ball indenter has to be in the range of tens of micrometers. Such a small ball indenter would cause a strong strain gradient hardening effect, as pointed out by many researchers, i.e., Swadener et al. [60]. and Sun et al. [61]. Within this consideration, an appropriate diameter of ball indenter to determine the correct fracture toughness value may be a few micrometers. However, for engineering application, a 1.19-mm ball indenter would be more practical and economical than a micro-sized ball indenter though only a ranking order among rail steels can be distinguished by the current indentation testing method, as illustrated in Fig. 12. Another challenging is to accurately determine damage parameter under indentation. According to Tasan et al.’s study [55], they suggested that the measurement of damage parameter under indentation should consider material microstructures and textures because micro-voids are generated in a few microns and their volume may only reach up to approximately 1-2%. Under compressive loading condition, the closure effect on damage measurement should also be considered [38]. In view of these important issues, further investigations are still required to correctly determine the KIc value by using the indentation technique.
7. Conclusions Mechanical properties, including tensile properties and mode I critical stress intensity factor (KIc), have been characterized for the railhead region of six types of high-strength rail steels using standard testing methods. Using the above test results, a lab-scaled, instrumented ball indentation test is developed based on CDM, and applied to estimate fracture toughness for the six rail steels. Both repeated loading-unloading tensile and indentation tests were performed to measure damage. The former was used, together with a ductile damage model, to determine the critical damage parameter, and the latter to identify the critical contact depth to generate the accumulated indentation damage that is equivalent to the critical damage parameter. The critical contact depth was used to calculate the specific indentation energy (2
) and thus the fracture toughness under
indentation (KInd). The results show a clear correlation between KInd and KIc values. However, 13
KInd determined using the 1.19-mm ball indenter has one order of magnitude bigger than that of KIc measured using pre-cracked SENB specimens. To correctly determine the KIc by using the indentation technique, further investigation is still required.
Acknowledgements This work was sponsored by the Canadian Rail Research Laboratory (CaRRL). CaRRL is funded by the Canadian National Railway, the Natural Sciences and Engineering Research Council of Canada (NSERC), Transport Canada, the National Research Council and Alberta Innovates – Technology Futures.
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[18] Haggag FM, Byun T-S, Hong JH, Miraglia PQ, Murty KL. Indentation-energy-to-fracture (IEF) parameter for characterization of DBTT in carbon steels using nondestructive automated ball indentation (ABI) technique. Scripta Materialia. 1998;38:645-51. [19] Lee J-S, Jang J-i, Lee B-W, Choi Y, Lee SG, Kwon D. An instrumented indentation technique for estimating fracture toughness of ductile materials: A critical indentation energy model based on continuum damage mechanics. Acta Materialia. 2006;54:1101-9. [20] Jeon SW, Lee KW, Kim JY, Kim WJ, Park CP, Kwon D. Estimation of Fracture Toughness of Metallic Materials Using Instrumented Indentation: Critical Indentation Stress and Strain Model. Experimental Mechanics. 2016. [21] Lawn B, Wilshaw R. Indentation fracture: principles and applications. Journal of Materials Science. 1975;10:1049-81. [22] Evans A, Charles E. Fracture toughness determinations by indentation. Journal of the American Ceramic society. 1976;59:371-2. [23] Niihara K. A fracture mechanics analysis of indentation-induced Palmqvist crack in ceramics. Journal of Materials Science Letters. 1983;2:221-3. [24] Loubet J, Georges J, Marchesini O, Meille G. Vickers indentation curves of magnesium oxide (MgO). Journal of Tribology. 1984;106:43-8. [25] Miyoshi T, Sagawa N, Sassa T. Study on fracture toughness evaluation for structural ceramics. Trans Jap Soc Mech Eng. 1985;51:2487-9. [26] Ostojic P, McPherson R. A review of indentation fracture theory: its development, principles and limitations. International Journal of Fracture. 1987;33:297-312. [27] Nose T, Fujii T. Evaluation of Fracture Toughness for Ceramic Materials by a Single‐ Edge‐ Precracked‐ Beam Method. Journal of the American Ceramic Society. 1988;71:328-33. [28] Quinn GD, Bradt RC. On the Vickers Indentation Fracture Toughness Test. Journal of the American Ceramic Society. 2007;90:673-80. [29] Byun TS, Kim JW, Hong JH. A theoretical model for determination of fracture toughness of reactor pressure vessel steels in the transition region from automated ball indentation test. Journal of Nuclear Materials. 1998;252:187-94. [30] Byun TS, Kim SH, Lee BS, Kim IS, Hong JH. Estimation of fracture toughness transition curves of RPV steels from ball indentation and tensile test data. Journal of Nuclear Materials. 2000;277:263-73. [31] Gurson AL. Continuum theory of ductile rupture by void nucleation and growth: Part I—Yield criteria and flow rules for porous ductile media. Journal of engineering materials and technology. 1977;99:2-15. [32] Tvergaard V, Needleman A. Analysis of the cup-cone fracture in a round tensile bar. Acta metallurgica. 1984;32:157-69. [33] Lemaitre J. A Continuous Damage Mechanics Model for Ductile Fracture. Journal of Engineering Materials and Technology. 1985;107:83-9. [34] Ghosh S, Das G. Effect of pre-strain on the indentation fracture toughness of high strength low alloy steel by means of continuum damage mechanics. Engineering Fracture Mechanics. 2012;79:126-37. [35] Li J, Li F, He M, Xue F, Zhang M, Wang C. Indentation technique for estimating the fracture toughness of 7050 aluminum alloy with the Berkovich indenter. Materials & Design. 2012;40:176-84. [36] Amiri S, Lecis N, Manes A, Giglio M. A study of a micro-indentation technique for estimating the fracture toughness of Al6061-T6. Mechanics Research Communications. 2014;58:10-6. [37] Li J, Li F, Ma X, Wang Q, Dong J, Yuan Z. A strain-dependent ductile damage model and its application in the derivation of fracture toughness by micro-indentation. Materials & Design. 2015;67:623-30. [38] Lemaitre J. A course on damage mechanics: Springer Science & Business Media; 2012. [39] Das G, Ghosh S, Ghosh S, Ghosh RN. Materials characterization and classification on the basis of materials pile-up surrounding the indentation. Materials Science and Engineering: A. 2005;408:158-64. [40] Matthews J. Indentation hardness and hot pressing. Acta Metallurgica. 1980;28:311-8. [41] Oliver WC, Pharr GM. Measurement of hardness and elastic modulus by instrumented indentation: Advances in understanding and refinements to methodology. Journal of Materials Research. 2004;19:3-20. [42] Taljat B, Pharr GM. Development of pile-up during spherical indentation of elastic–plastic solids. International Journal of Solids and Structures. 2004;41:3891-904. [43] Bonora N. A nonlinear CDM model for ductile failure. Engineering Fracture Mechanics. 1997;58:11-28. [44] Yu F, Jar P-YB, Hendry M. Fracture behaviour at the sharp notch tip of high strength rail steels–Influence of stress triaxiality. Engineering Fracture Mechanics. 2017;178:184-200.
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[45] Bao Y, Wierzbicki T. On fracture locus in the equivalent strain and stress triaxiality space. International Journal of Mechanical Sciences. 2004;46:81-98. [46] Hertz H. Hertz’s Miscellaneous Papers; Chapters 5 and 6. ed: Macmillan, London, UK. 1896. [47] Sneddon IN. The relation between load and penetration in the axisymmetric Boussinesq problem for a punch of arbitrary profile. International Journal of Engineering Science. 1965;3:47-57. [48] Standard ASTM. E8/E8M Standard test methods for tension testing of metallic materials. ASTM International, West Conshohocken, PA. 2011. [49] Yu F, Jar P-YB, Hendry MT. Indentation for fracture toughness estimation of high-strength rail steels based on a stress triaxiality-dependent ductile damage model. Theoretical and Applied Fracture Mechanics. 2018. [50] Zhang K, Li Z. Numerical analysis of the stress-strain curve and fracture initiation for ductile material. Engineering Fracture Mechanics. 1994;49:235-41. [51] Ling Y. Uniaxial true stress-strain after necking. AMP Journal of Technology. 1996;5:37-48. [52] Alves Ml, Yu J, Jones N. On the elastic modulus degradation in continuum damage mechanics. Computers & Structures. 2000;76:703-12. [53] Lemaitre J, Dufailly J. Damage measurements. Engineering Fracture Mechanics. 1987;28:643-61. [54] Celentano DJ, Chaboche J-L. Experimental and numerical characterization of damage evolution in steels. International Journal of Plasticity. 2007;23:1739-62. [55] Tasan CC, Hoefnagels JPM, Geers MGD. A critical assessment of indentation-based ductile damage quantification. Acta Materialia. 2009;57:4957-66. [56] Zeng K, Chiu C-H. An analysis of load-penetration curves from instrumented indentation. Acta materialia. 2001;49:3539-51. [57] Anderson TL. Fracture Mechanics: Fundamentals and Applications: CRC Press/Taylor & Francis; 2017. [58] Kim J-Y, Lee K-W, Lee J-S, Kwon D. Determination of tensile properties by instrumented indentation technique: Representative stress and strain approach. Surface and Coatings Technology. 2006;201:4278-83. [59] Kang S-K, Kim J-Y, Kang I, Kwon D. Effective indenter radius and frame compliance in instrumented indentation testing using a spherical indenter. Journal of Materials Research. 2011;24:2965-73. [60] Swadener JG, George EP, Pharr GM. The correlation of the indentation size effect measured with indenters of various shapes. Journal of the Mechanics and Physics of Solids. 2002;50:681-94. [61] Sun Z, Li F, Cao J, Ma X, Li J. A method for determination of intrinsic material length base on strain gradient study in spherical indentation. International Journal of Mechanical Sciences. 2017;134:253-62.
16
Tables Table 1: Rail samples and their microstructures Identification number
Microstructures
Rail#2
Hypereutectoid
Rail#3
Hypereutectoid
Rail#4
Head hardened pearlite
Rail#5
Head hardened pearlite
Rail#6
Micro head hardened
Rail#control
Deep head hardened
Table 2: Analysis of KIc for six types of high-strength rail steels Rail types
Label
a0 mm
W mm
a0-W mm
PQ kN
Pmax kN
KQ . 1/2 MPa m
MPa
2.5(KQ/ mm
)
1/2
Pmax/P Q
KIc . 1/2 MPa m
Average KIc MPa.m (standard deviation)
Rail#2
A B C D E F G H
7.02 6.93 6.95 6.98 6.90 6.82 6.91 6.84
13.36 13.36 13.37 13.37 13.33 13.32 13.32 13.33
6.34 6.43 6.42 6.39 6.43 6.50 6.41 6.49
3.05 3.03 2.58 2.86 3.10 3.18 3.15 2.89
3.05 3.19 2.94 2.97 3.10 3.45 3.15 3.10
39.61 38.41 32.86 36.52 39.28 39.84 40.17 36.14
914 914 914 914 914 914 914 914
4.69 4.41 3.23 3.99 4.61 4.75 4.83 3.91
1.00 1.05 1.14 1.04 1.00 1.08 1.00 1.07
39.61 38.41 32.86 36.52 39.28 39.84 40.17 36.14
37.85 (2.52)
Rail#3
A B C D E F G H
7.01 6.97 6.94 6.78 6.92 6.75 6.89 6.82
13.35 13.33 13.35 13.35 13.35 13.34 13.35 13.35
6.34 6.36 6.41 6.57 6.43 6.59 6.46 6.53
2.55 2.80 2.77 3.06 2.50 2.58 2.63 2.96
2.79 2.88 3.12 3.36 2.97 3.06 2.95 3.38
33.09 36.06 35.28 37.44 31.70 31.41 33.09 36.70
781 781 781 781 781 781 781 781
4.49 5.34 5.11 5.75 4.12 4.05 4.49 5.52
1.10 1.03 1.13 1.10 1.19 1.19 1.13 1.14
33.09 36.06 35.28 37.44 31.70 31.41 33.09 36.70
35.28 (1.84)
Rail#4
A B C D E F G H
7.05 6.84 7.01 6.99 6.93 6.99 7.03 7.15
13.44 13.45 13.43 13.45 13.44 13.45 13.45 13.44
6.39 6.61 6.42 6.46 6.51 6.46 6.42 6.29
3.11 3.48 3.41 3.34 3.69 3.30 3.31 2.90
3.43 3.67 3.41 4.08 3.69 3.58 3.70 3.62
40.19 42.78 43.82 42.59 46.46 42.09 42.52 38.43
877 877 877 877 877 877 877 877
5.26 5.95 6.25 5.90 7.03 5.77 5.88 4.80
1.10 1.06 1.00 1.22 1.00 1.09 1.12 1.25
40.19 42.78 43.82 42.59 46.46 42.09 42.52 38.43
42.28 (1.33)
Rail#5
A B C D E F G H
6.85 6.83 6.86 6.77 6.90 6.95 6.91 6.95
13.32 13.33 13.33 13.33 13.33 13.31 13.32 13.32
6.47 6.50 6.47 6.56 6.43 6.36 6.41 6.37
3.01 3.82 3.54 3.38 3.03 3.19 3.79 3.13
3.58 3.82 3.70 3.67 3.31 3.62 3.79 3.56
37.82 47.63 44.44 41.63 38.41 41.14 48.37 40.16
923 923 923 923 923 923 923 923
4.20 6.66 5.80 5.09 4.33 4.97 6.87 4.74
1.19 1.00 1.05 1.09 1.09 1.13 1.00 1.14
37.82 47.63 44.44 41.63 38.41 41.14 48.37 40.16
41.16 (2.21)
Rail#6
A B C D E F G H
6.95 6.95 6.96 6.93 7.15 6.94 6.92 6.93
13.45 13.45 13.46 13.46 13.48 13.45 13.44 13.46
6.50 6.50 6.50 6.53 6.33 6.51 6.52 6.54
3.45 3.72 3.53 3.38 3.08 3.00 3.06 3.50
3.45 3.72 4.01 3.60 3.21 3.37 3.47 3.50
43.32 46.84 44.25 42.16 40.46 37.74 38.19 43.66
878 878 878 878 878 878 878 878
6.08 7.11 6.35 5.76 5.31 4.62 4.73 6.18
1.00 1.00 1.14 1.06 1.04 1.12 1.14 1.00
43.32 46.84 44.25 42.16 40.46 37.74 38.19 43.66
41.40 (2.65)
Rail#control
A B C D E F G H
6.95 6.95 6.89 7.09 6.87 6.93 6.88 6.94
13.41 13.43 13.45 13.45 13.43 13.45 13.40 13.44
6.46 6.49 6.56 6.36 6.56 6.52 6.52 6.50
3.60 3.59 3.88 3.75 3.44 3.06 2.70 3.13
3.60 3.69 3.88 3.75 3.76 3.49 3.31 3.57
45.58 45.54 48.29 48.94 42.74 38.36 33.74 39.47
903 903 903 903 903 903 903 903
6.38 6.36 7.16 7.35 5.61 4.52 3.49 4.78
1.00 1.03 1.00 1.00 1.09 1.14 1.23 1.14
45.58 45.54 48.29 48.94 42.74 38.36 33.74 39.47
42.34 (3.35)
17
Figures (b)
(a)
Figure 1: Tensile test specimens: (a) sampling locations on the cross section of railhead, and (b) dimensions and geometry of the tensile specimen
(a)
(b)
Figure 2: Single-edge-notched bend (SENB) specimens: (a) sampling locations on the cross section of railhead, and (b) dimensions and geometry of the SENB specimen
Figure 3: Representative curve of load versus crack mouth opening displacement (CMOD, δ) from threepoint bending tests
18
(a)
(b)
Extensometer
Indenter Specimen
Figure 4: Instrumented indentation test [49]: (a) schematic presentation of the test setup and ball indenter, and (b) a typical indentation loading-unloading curve and depiction of the measurement of the unloading stiffness S
(a) R
(b) Figure 5: Finite element modelling of: (a) smooth tensile specimen, and (b) indentation tests with different sizes of ball indenters
19
Figure 6: Experimental data of smooth tensile specimens: (a) illustration of engineering stress-strain curves on railhead of Rail#control, and (b) summary of UTS and yield stress ( ) for six types of highstrength rail steels
Figure 7: FE modelling of smooth tensile specimens: (a) illustration for regenerating experimental data for Rail#control-A, and (b) comparison of mean equivalent stress-strain curves for six types of highstrength rail steels
20
Figure 8: Illustration of damage measurement for Rail#control: (a) repeated loading-unloading tensile tests, and (b) indentation loading-unloading tests
Figure 9: Damage evolution of smooth tensile specimen: (a) determination of the critical value of elastic modulus (Ecr) at fracture initiation for Rail#control, and (b) damage evolution and values of Ecr for six types of high-strength rail steels
Figure 10: Damage evolution under indentation and determination of critical indentation depth (hcr) for each of the six high-strength rail steels by using the corresponding value of Ecr in Fig. 9
21
Figure 11: Indentation load-contact depth curves for six types of high-strength rail steels
Figure 12: Comparison of (a) fracture toughness determined from SENB tests, KIc, with that from indentation tests, KInd, and (b) normalized KIc and normalized KInd, for six types of high-strength rail steels
22
(a)
(b)
(c) Figure 13: FE modelling of ball indentation test with ball diameter of 1.19 mm for Rail#control: (a) full contour plot of stress triaxiality, (b) enlarged view of stress triaxiality under the ball tip, and (c) comparison of experimental data with those from simulation and the critical value of hc (hcr) to calculate
2
0.5
9.19 MP.m
23
Figure 14: Illustration of specific indentation energy (2 ) for Rail#control as a function of indenter diameter (2R), and the KInd value for 2R in the vicinity of critical value of CTOD (dcr)
24
Highlights 1. Develop an indentation test to estimate fracture toughness of rail steels 2. Determine critical damage parameter by using a ductile damage model 3. Investigate the influence of ball indenter size on fracture toughness estimation 4. Characterize tensile properties and KIc for six types of high-strength rail steels
25
S0013-7944(18)30686-6 https://doi.org/10.1016/j.engfracmech.2018.10.030 EFM 6204
To appear in:
Engineering Fracture Mechanics
Received Date: Revised Date: Accepted Date:
7 July 2018 24 October 2018 26 October 2018
Please cite this article as: Yu, F., Ben Jar, P.-Y., Hendry, M.T., Jar, C., Nishanth, K., Fracture toughness estimation for high-strength rail steels using indentation test, Engineering Fracture Mechanics (2018), doi: https://doi.org/ 10.1016/j.engfracmech.2018.10.030
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Fracture toughness estimation for high-strength rail steels using indentation test Feng Yua,d, P.-Y. Ben Jara,b, Michael T. Hendrya,c, Chester Jarb, and Kukkadapu Nishanthe a
Canadian Rail Research Laboratory, bDepartment of Mechanical Engineering, cDepartment of Civil and Environmental Engineering, University of Alberta, Edmonton, Alberta, Canada d School of Mechanical Engineering and Mechanics, Ningbo University, Ningbo 315211, PR China1 e Department of Mechanical Engineering, National Institute of Technology, Rourkela, Odisha, India
Abstract In this paper, tensile properties and mode I critical stress intensity factor (KIc) are measured on the railhead of six types of high-strength rail steels using standard testing methods. An instrumented ball indentation testing method is then developed to estimate fracture toughness for the six rail steels based on continuum damage mechanics. Critical damage parameter, determined by combining repeated loadingunloading tensile tests and a ductile damage model, is used to characterize a critical contact depth under indentation that is used to calculate the specific indentation energy, and thus the fracture toughness for indentation (KInd). Comparison between KInd and KIc for the six high-strength rail steels suggests that the current indentation testing method can successfully rank the six high-strength rail steels for their fracture toughness. Keywords: Fracture toughness; Indentation test; Critical damage parameter; High-strength rail steels
1. Introduction Fracture toughness is one of the most important mechanical properties for assessing the integrity of rail steels. High fracture toughness allows rails to sustain a large crack size before fracture [1], thus increasing the possibility of detecting cracks in time, which is essential for improving safety and efficiency for railway transportation [2]. In the past four decades, research and development have been focused on producing premium, head-hardened pearlitic rail steels with high fracture toughness [3-6]. In addition, low-carbon bainitic rail steels have been developed, which show much higher fracture toughness but lower wear resistance than the pearlitic rail steels at the same characteristic strength level [7-9]. As a result, even though there expects a limit in the fine grain size for the pearlitic steel, efforts have been made to develop new pearlitic rail steels with optimal strength and fracture toughness. Similarly, development of 1
The corresponding author’s current email address: [email protected] 1
bainitic steel with improved wear resistance is expected to provide a promising alternative for the next generation of rail steels. For both types of newly developed rail steels, fracture toughness needs to be characterized first in laboratory and then evaluated for in-field applications. Standard testing methods are available for evaluating fracture toughness [10, 11], however, test specimens require specific geometries and dimensions as well as fatigue precracking that is very time-consuming and labor-intensive. Moreover, the standard tests can neither be applied to newly developed rail steels with small volume nor in-service rail tracks because of the destructive nature. As a result, many alternative testing methods have been developed to estimate fracture toughness [12-20], among which the instrumented indentation test shows its potential to provide a non-destructive means to measure fracture toughness with the minimum need for sample preparation. Note that indentation tests have been used to predict fracture toughness for brittle materials, in view that indentation generates cracks in brittle materials [21-28]. For ductile materials, cracks are generally known to be suppressed under compressive loading. To define a fracture initiation point under indentation, the attainment of critical fracture stress or strain under indentation was initially assumed as the fictitious fracture initiation point [18, 20, 29]. An important theoretical basis was also proposed in these studies that the stress state beneath an indenter tip is similar to that in front of a crack tip [30]. This enables the use of indentation test to estimate fracture toughness. Recently, with the development of damage theories [31-33], Lee et al. [19] showed that during an indentation test, voids were generated beneath an indenter and therefore, the fracture initiation point under indentation can be quantified in terms of critical damage parameter. Since then, many studies have applied indentation techniques to prediction of fracture toughness for metals based on the concept of continuum damage mechanics (CDM) [34-37]. However, an important issue was not properly addressed in the previous studies. That is, the critical damage parameter after necking cannot be directly obtained from repeated loading-unloading tensile tests. This is because damage measured from such tests is only valid for uniform deformation [38]. In the present work, an instrumented ball indentation test is developed to estimate fracture toughness for six types of high-strength rail steels based on CDM. Monotonic tensile, three-point bending, and repeated loading-unloading tensile tests are carried out for the six rail steels to measure their tensile properties, fracture toughness (KIc), and deterioration of elastic 2
modulus, respectively. The critical damage parameter is determined through extrapolation of experimentally quantified deterioration of elastic modulus to the strain level for fracture according to a ductile damage model. Indentation tests are performed to a critical contact depth at which the accumulated damage under indentation reaches the same level as that for the critical damage parameter. The critical contact depth value is then used to calculate the specific indentation energy, and thus the fracture toughness for indentation (KInd). The correlation of KInd and KIc for the six rail steels is discussed.
2. Theoretical background 2.1. Specific indentation energy Previous studies showed that the degree of constraint in the deformed volume under a ball indenter was similar to that ahead of the crack tip, though the former is compressive and the latter is tensile [18, 19, 29]. In this work, the fracture energy is interpreted as the deformation capability of the material under a highly constrained stress field. Based on this concept, the specific work of fracture at the crack tip can be replaced by the specific indentation energy in Eq. (1), which is applied to estimate fracture toughness of ductile materials using the ball indentation test [19]. (1) where,
is the fracture toughness estimated through the indentation test,
modulus, and 2
the Young’s
the specific indentation energy determined using Eq. (2) below. (2)
where,
is the contact depth under indentation,
the critical value of
to generate the same
level of damage under indentation as the critical damage parameter for tensile specimens, indentation load which is a function of indenter tip with radius R,
, and
the
the projected indentation area. For a spherical
can be expressed as, (3)
Note that determination of
should include both pile-up and elastic deflection in indentation
tests [39-42], that is, (4)
3
where,
is the height of pile-up,
the consideration of pile-up, and
the maximum value of indentation depth (h) without
a constant set to be 0.75 for the ball indenter tip [41], and
the maximum indentation load and unloading contact stiffness at
,
respectively.
2.2. Determination of critical contact depth (hcr) As defined in CDM [33], damage can be expressed in terms of the change of elastic modulus using the following expression. (5) where, and
is the damage parameter,
the elastic modulus of virgin material (Young’s modulus),
the degraded elastic modulus due to the presence of damage. Change of E is commonly measured using the conventional repeated loading-unloading
tensile tests [38]. To ensure accuracy of damage measurement, Eq. (5) is normally applied to results from smooth tensile specimens that are subjected to uniform deformation. This approach, however, limits the change of E values to the plane-stress condition, i.e., before neck formation which starts at a low level of stress triaxiality (defined as the ratio of hydrostatic stress to von Mises stress) of around 0.33. For rail steels considered in the current study, standard smooth specimens often fracture after significant neck formation. Therefore, critical damage parameters for these rail steels have to be determined through an extrapolation to a strain level at the onset of fracture. In this study, expression for the extrapolation is given below, which was modified from Bonora’s original work for a nonlinear ductile damage model [43]. (6) where,
is an adjusting parameter,
the equivalent plastic strain at which E is measured,
is the plastic logarithmic area strain at fracture (
and
, respectively, denote
the initial and minimum transversal areas of round tensile specimens at the necking zone), the threshold strain for damage initiation which for the high-strength rail steels, is regarded as the yield strain of 0.002 [44], and
defined as [45] (7)
4
where
is the stress triaxiality as a function of , which is determined at the centre of smooth
tensile specimen by using FE modelling. Expression for
in Eq. (6) is given below [33]. (8)
being the Poisson’s ratio. Therefore,
with
is the only unknown in Eq. (6), and can be
determined uniquely by fitting Eq. (6) to the experimental data of deterioration of elastic modulus. Once
is determined, the critical value for elastic modulus, Ecr, can be determined at
. Damage accumulation under indentation, on the other hand, has been shown to increase with the increase of indentation depth, and the change of elastic modulus, E, was calculated using Eq. (9), as given below [19], which is based on the Hertzian contact law [46] and Sneddon’s elastic punch theory [47], (9) where, and
is the effective modulus for the contact system between specimen and indenter, are the Poisson’s ratio and Young’s modulus, respectively, for the ball indenter. At the very right expression of Eq. (9), E is expressed as a function of
expression for
in Eq. (3), a relationship between
. In view of the
and E can be established from Eq. (9).
Base on the assumption that Ecr is a material parameter and thus independent of loading conditions, the critical contact depth under indentation,
, can thus be determined by setting E
in Eq. (9) equal to Ecr.
3. Experiments 3.1. Materials and testing devices Six types of high-strength rail steels used in this test program were supplied by Transportation Technology Centre Inc. (TTCI). Due to confidential requirements, information about the material compositions and vendors cannot be provided. Therefore, the six rails are listed using a numbering system shown in Table 1, together with their microstructural features for the railhead.
5
An Instron hydraulic universal testing machine, with a load capacity of 222 kN, was used for tensile and three-point bending tests. An MTS 44-kN hydraulic testing system was used for the instrumented indentation tests. All tests were conducted at room temperature.
3.2. Tensile tests Fig. 1(a) labels locations of tensile specimens on the cross section of railhead, and Fig. 1(b) their geometry and dimensions. These tensile specimens were machined with the longitudinal direction parallel to the rolling direction. Geometry and dimensions of the tensile specimens conform to those specified in ASTM E8/E8M [48]. The tensile specimens labelled A, B, D, E, F, I, and J were used for monotonic tensile tests, while the remaining for repeated loading-unloading tensile tests. Both monotonic and repeated loading-unloading tensile tests were performed at a crosshead speed of 8.5x10-3 mm/s. An axial extensometer with the initial gauge length of 12.5 mm was placed in the middle of reduced gauge section to measure global elongation. Also, a diametric extensometer was mounted at the centre of tensile specimens to capture local diameter changes. In repeated loading-unloading tensile tests, the unloading phase was triggered based on a load increment of 0.8 kN before reaching the ultimate tensile strength (UTS), while after the UTS an increment of 0.255 mm in elongation was used which corresponded to a loading period of 30 seconds.
3.3. Three-point bending tests Single-edge-notched bend (SENB) specimens were used to measure KIc values for the railhead region, where locations for the SENB specimens are shown in Fig. 2(a). Longitudinal direction of the SENB specimens is along the rolling direction of the rails. Configurations and dimensional ratios of the SENB specimens follow those specified in ASTM E399 [10], with thickness (B), width (W) and span length (L) being 7.7, 13.4 and 60 mm, respectively, as shown in Fig. 2(b). Edge notch of the SENB specimens was first cut using abrasive waterjet, and further sharpened using electric-discharge machining (EDM). Pre-cracking was later introduced at the notch tip through fatigue loading. Total crack length was controlled to be within the range of 0.5 to 0.53 of W. Crosshead speed for the three-point bending test was set at 2x10-3 mm/s, and a clip-
6
on extensometer was used to measure the crack mouth opening displacement (CMOD), δ. The following equations were used to determine the conditional stress intensify factor KQ [10]. (10) (11) where
is a conditional load, corresponding to the load at the intersection between the
experimental curve of load versus CMOD and a secant line with 95% of the tangent slope, as illustrated in Fig. 3, and
the averaged initial pre-crack length measured along the crack front
after the specimen was fractured.
3.4. Instrumented indentation test A lab-scaled indentation testing system was developed in-house for the purpose of estimating fracture toughness for rail steels. Fig. 4(a) shows the schematic drawing of the indentation testing system with a ball indenter, same as that used in our previous publication [49]. The specimen is one half of railhead with 10 mm in thickness. Surface of the specimen was gradually polished to a mirror-like finish using a 0.05-μm grade alpha alumina polishing powder at the final polishing step. The indenter consists of O1 tool steel with proper heat treatment and a tungsten carbide ball with diameter of 1.19 mm, Ei of 480 GPa, and
of 0.28. The ball was
compression-fit into a pre-machined semi-spherical cavity at the end of a truncated-cone indenter. A clip-on extensometer was used to measure the indentation depth. Fig. 4(b) presents a typical curve from an indentation loading-unloading test, illustrating the measurement of the unloading contact stiffness (S), the maximum indentation depth (hmax), the corresponding load (Fmax), and the residual depth (hf) after the indenter is removed. To generate damage under indentation, loading-unloading indentation tests were conducted at ten indentation depths of 0.04, 0.06, 0.09, 0.12, 0.18, 0.24, 0.3, 0.36, 0.42, and 0.48 mm. Five tests were conducted at each of the above indentation depths and the average value was obtained for calculation. Indentation speed of 0.1 mm/min was used at both loading and unloading phases. The indentation load was recorded using an MTS load cell with a resolution of 0.01 N, and the indentation depth measured using the clip-on extensometer with a resolution of 3 μm. Note that height of the pile-up generated during the indentation loading, around the indenter, 7
was directly measured using a digital optical microscope with a magnification of 320X and a digital dial indicator with a resolution of 2.54 μm. Details of the test procedure are described as follows. Firstly, the original, flat specimen surface near an indent was focused under the microscope and its vertical position recorded using the digital dial indicator, to serve as the reference. The pile-up region at the edge of the indent was then examined, and change in the dial indicator reading from the reference in order to focus on the peak of the pile-up represented the local height of the pile-up. Five measurements of the pile-up height were taken around each indent, and their average value was used as the pile-up height.
4. Finite element modelling Finite element (FE) modelling of monotonic tensile and indentation tests was carried out using ABAQUS/CAE 6.13. Fig. 5(a) shows the axisymmetric FE model of smooth tensile specimen, following the dimensions given in Fig. 1(b). The model consists of 4,260 linear quadrilateral elements of type CAX4R and 4,657 nodes. The FE model was evenly discretized in the reduced gauge section with element size of 0.125 mm and aspect ratio of 1. Boundary conditions were set to be the same as those used in the experimental testing, i.e., with one end fixed and the other end moving at a specified displacement rate. Fig. 5(b) depicts an axisymmetric FE model of indentation test. The specimen is modelled as a cylinder with radius of 12.5 mm and height of 7 mm. It consists of 22,001 linear quadrilateral elements of type CAX4R and 22,262 nodes. The minimum element size at the contact region is 3 µm. The ball indenter was modelled as an analytical rigid body and its radius can be varied to investigate the effect of ball size on
. A contact surface was placed
between the ball indenter and specimen in the condition of small sliding, “hard” normal contact, and tangential friction of 0.2 [42]. The boundary conditions were set to be the same as those for the testing, i.e., with the bottom of the specimen supported and the indenter moving down at a specified displacement rate.
8
5. Results 5.1. Tensile properties Fig. 6(a) illustrates the variation of engineering stress versus engineering strain curves on railhead for Rail#control. It shows a decreasing trend of tensile strength from the top of the railhead to the core region. This phenomenon has also been observed for the other five types of rail steels as all of the rails are head-hardened but using different heat treatment methods. Fig. 6(b) summarizes the mean values of UTS and yield stress for the six rail steels, together with their corresponding maximum and minimum values. FE modelling of smooth tensile specimens was performed to establish constitutive equations for the six rail steels, according to the approach proposed by Zhang and Li [50] and successively formalized by Ling [51]. According to their work, the equivalent stress-strain curve was established by regenerating the experimental data of load-elongation curve and cross-section reduction through an iterative procedure based on the finite element analyses, as illustrated in Fig. 7(a) for tensile specimen A of Rail#control. This approach was applied to each of the six rail steels, and equivalent stress-strain curves were determined for all tensile specimens labelled A, B, D, E, F, I, and J. For the sake of conciseness, Fig. 7(b) compares the mean value of equivalent stress-strain curves for the six rail steels. It is shown that the Rail#2 shows the minimum ductility. Rail#2, Rail#3, and Rail#5 have similar magnitudes of stress at fracture. Rail#6 has the smallest stress at fracture but the best ductility. Rail#4 and Rail#control appear to be identical except that the latter is more ductile than the former.
5.2. KIc Values of mode I critical stress intensity factor, KIc, for SENB specimens on railhead, are summarized in Table 2 for the six rail steels. Table 2 also lists details for the pre-crack length (a0), specimen width (W), mean value of yield stress
, and ratio of PQ to Pmax, which are
required to validate KQ for KIc. According to ASTM E399 [10], to ensure validity of KIc measurements, value for (a0-W) should be bigger than 2.5(KQ/
) and PQ/Pmax less than 1.10.
Both conditions must be satisfied at the same time. Therefore, the experimental data that violate these two requirements are excluded, as shown in Table 2. By using the remaining experimental data, the average KIc value and standard deviation are calculated for each of the six rail steels. It is shown in Table 2 that the KIc values for Rail#control and Rail#4 are ranked as the first class 9
while Rail#5 and Rail#6 the second. However, difference in KIc values for these four types of rails is small, which is within 1 MPa.m0.5. KIc value for Rail#2 is bigger than that for Rail#3, but both show smaller KIc values than the other four rails.
5.3. Identification of hcr To determine hcr for each of the six rail steels, both repeated loading-unloading tensile and instrumented indentation tests were carried out for the damage measurement [19]. Fig. 8(a) illustrates the repeated loading-unloading tensile tests for Rail#control. For the accuracy of damage measurement, the loading-unloading tensile tests were repeated to the logarithmic area strain (
) of about 20%, where the neck formation was negligible. Damage parameter, in
terms of the deterioration of elastic modulus, was determined by measuring the slope of linear portion of the unloading true stress-logarithmic area strain curves [52, 53]. In addition, the effect of Poisson’s ratio has been taken into account to convert the measured unloading slopes to elastic modulus [54]. Fig. 8(b), on the other hand, shows instrumented ball indentation tests at ten different indentation depths for Rail#control. Previous study noted that to consider the influence of pre-existing damage and plastic hardening on the indentation response multiple loadingunloading indentation tests should be performed in a single location [55]. In view of good repeatability for the experimental data shown in Fig. 8(b) and following the works conducted by previous researchers [34, 35, 37], the indentation loading-unloading curves in Fig. 8(b) were obtained from multiple locations on the surface of test specimens. Degradation of elastic modulus under indentation was then measured by the changes in stiffness of indentation unloading curves [19]. Also, the pile-up after each indentation test was measured using the optical microscope. Due to the nonlinearity of indentation unloading curves [56], a power-law function is used to fit the unloading curves, also shown in Fig. 8(b). Two unknowns, K and m, were determined by using the least square method to best fit each of the unloading curves. First derivative of the power-law function was used to determine the unloading contact stiffness at
, i.e.,
that is one of the essential parameters needed for calculating elastic
modulus in Eq. (9). Fig. 9(a) shows the experimental data of deterioration of elastic modulus from repeated loading-unloading tensile tests for Rail#control. To determine the critical value for elastic 10
modulus (Ecr), Eq. (6) is used to fit the experimental data. As mentioned earlier,
is the only
adjustable variable in Eq. (6) and therefore, the solid line in Fig. 9(a) can be uniquely determined by applying the least square method to best fit the experimental data. Ecr is then determined based on an extrapolation of the solid line to
. The same approach was applied to the other five
types of rail steels. Fig. 9(b) presents both damage evolution and Ecr for each of the six rail steels. Damage evolution in the six rail steels, characterized using the indentation tests, is presented in Fig. 10, in which elastic modulus is plotted as a function of contact depth (hc). Linear functions are used to fit the relationship between elastic modulus and hc for all of the six rail steels. Young’s modulus was determined using the mean value of the measured elastic moduli at the first three depths. Value of hcr for each of the six rail steels was determined by matching hc to the value of Ecr from Fig. 9(b), and indicated in Fig. 10. 5.4. Fracture toughness estimation In addition to hcr and Ac, indentation load, F, is needed to be expressed as a function of hc in Eq. (2) to calculate the specific indentation energy, 2
Fig. 11 summarizes the typical
relationship between F and hc for the six rail steels. A second order polynomial function was then determined to fit each of the six curves, and applied to Eq. (2) to calculate the values for
. With
value for each of the six rail steels determined, fracture toughness under
indentation, KInd, is then calculated and compared with the corresponding KIc value. Fig. 12(a) shows that the ranking among the six rail steels using KInd is consistent with that using KInd, however, the KInd value for each of the six rails is about one order of magnitude bigger than the corresponding KIc value. Fig. 12(b) compares the normalized values of KInd with those of KIc for the six rail steels. Good correlation between the two parameters is shown, which indicates a potential capability for using KInd to correlate the KIc.
6. Discussion In view of different magnitudes of toughness values between KInd and KIc, two considerations are discussed in this section. For the first scenario, it may be attributed to the difference of localized deformation volume between the crack tip and the indenter tip. Eq. (12) is
11
used to calculate the critical crack tip opening displacement (CTOD) based on fracture mechanics [57], (12) where, KIc,
represents the critical value of CTOD at crack initiation. According to the values of , and
for Rail#control, i.e., 42.34 MPa.m0.5, 903 MPa, and 200 GPa, respectively, the
corresponding
value is 12.64 µm. In view of the deformation volume generated by the
current ball indenter (1.19 mm in diameter), there exists a huge difference in the plastic deformation zone sizes between the indenter tip and the crack tip. FE modelling of indentation test with 1.19-mm ball indenter was first carried out to compare the simulation results with the experimental data. FE modelling of indentation test shown in Fig. 13 was based on the equivalent stress-strain curve for Rail#control. Fig. 13(a) and (b) present the full distribution of stress triaxiality and its enlarged view under the ball tip, respectively. It is observed that the variation of stress triaxiality under the current ball indenter varies from 0.4 to -5.4, and there exists a region where the magnitude of stress triaxiality is at about 3.0, which is similar to that ahead of the crack tip [19, 49]. Although the degree of constrain under indentation is compressive and that around the crack tip tensile, it is generally accepted that the compressive and tensile deformations are equivalent if the deformation is not cyclic [29]. Fig. 13(c) plots the simulation result of indentation load versus contact depth, and compares with results from the experimental testing. The good agreement between the two sets of results suggests that the 2
value at hcr can be well predicted using the FE simulation.
A series of FE modelling was then performed using ball indenters of different diameters at a reduction factor of 0.5 each time. For each ball indenter size, the critical contact depth (
)
is chosen to be the value that generates the same magnitude of plastic strain under indentation, calculated using Eq. (13) below [58, 59]. (13) where,
is the critical representative plastic strain for indentation, =0.14 [58], R the radius of
ball indenter, and
which represents the critical diameter of the projected
area under indentation. The application of Eq. (13) is based on the assumption that damage is proportional to the plastic strain. That is, for the same
value, the critical damage parameters
generated by ball indenters of different sizes also have the same value. 12
As a result, Fig. 14 plots the simulation results of
for Rail#control under different
ball indenter sizes, which shows a linear relationship between the
values and ball sizes. It
is indicated that in order to approach a reasonable magnitude for the KIc value, the size of the ball indenter has to be in the range of tens of micrometers. Such a small ball indenter would cause a strong strain gradient hardening effect, as pointed out by many researchers, i.e., Swadener et al. [60]. and Sun et al. [61]. Within this consideration, an appropriate diameter of ball indenter to determine the correct fracture toughness value may be a few micrometers. However, for engineering application, a 1.19-mm ball indenter would be more practical and economical than a micro-sized ball indenter though only a ranking order among rail steels can be distinguished by the current indentation testing method, as illustrated in Fig. 12. Another challenging is to accurately determine damage parameter under indentation. According to Tasan et al.’s study [55], they suggested that the measurement of damage parameter under indentation should consider material microstructures and textures because micro-voids are generated in a few microns and their volume may only reach up to approximately 1-2%. Under compressive loading condition, the closure effect on damage measurement should also be considered [38]. In view of these important issues, further investigations are still required to correctly determine the KIc value by using the indentation technique.
7. Conclusions Mechanical properties, including tensile properties and mode I critical stress intensity factor (KIc), have been characterized for the railhead region of six types of high-strength rail steels using standard testing methods. Using the above test results, a lab-scaled, instrumented ball indentation test is developed based on CDM, and applied to estimate fracture toughness for the six rail steels. Both repeated loading-unloading tensile and indentation tests were performed to measure damage. The former was used, together with a ductile damage model, to determine the critical damage parameter, and the latter to identify the critical contact depth to generate the accumulated indentation damage that is equivalent to the critical damage parameter. The critical contact depth was used to calculate the specific indentation energy (2
) and thus the fracture toughness under
indentation (KInd). The results show a clear correlation between KInd and KIc values. However, 13
KInd determined using the 1.19-mm ball indenter has one order of magnitude bigger than that of KIc measured using pre-cracked SENB specimens. To correctly determine the KIc by using the indentation technique, further investigation is still required.
Acknowledgements This work was sponsored by the Canadian Rail Research Laboratory (CaRRL). CaRRL is funded by the Canadian National Railway, the Natural Sciences and Engineering Research Council of Canada (NSERC), Transport Canada, the National Research Council and Alberta Innovates – Technology Futures.
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16
Tables Table 1: Rail samples and their microstructures Identification number
Microstructures
Rail#2
Hypereutectoid
Rail#3
Hypereutectoid
Rail#4
Head hardened pearlite
Rail#5
Head hardened pearlite
Rail#6
Micro head hardened
Rail#control
Deep head hardened
Table 2: Analysis of KIc for six types of high-strength rail steels Rail types
Label
a0 mm
W mm
a0-W mm
PQ kN
Pmax kN
KQ . 1/2 MPa m
MPa
2.5(KQ/ mm
)
1/2
Pmax/P Q
KIc . 1/2 MPa m
Average KIc MPa.m (standard deviation)
Rail#2
A B C D E F G H
7.02 6.93 6.95 6.98 6.90 6.82 6.91 6.84
13.36 13.36 13.37 13.37 13.33 13.32 13.32 13.33
6.34 6.43 6.42 6.39 6.43 6.50 6.41 6.49
3.05 3.03 2.58 2.86 3.10 3.18 3.15 2.89
3.05 3.19 2.94 2.97 3.10 3.45 3.15 3.10
39.61 38.41 32.86 36.52 39.28 39.84 40.17 36.14
914 914 914 914 914 914 914 914
4.69 4.41 3.23 3.99 4.61 4.75 4.83 3.91
1.00 1.05 1.14 1.04 1.00 1.08 1.00 1.07
39.61 38.41 32.86 36.52 39.28 39.84 40.17 36.14
37.85 (2.52)
Rail#3
A B C D E F G H
7.01 6.97 6.94 6.78 6.92 6.75 6.89 6.82
13.35 13.33 13.35 13.35 13.35 13.34 13.35 13.35
6.34 6.36 6.41 6.57 6.43 6.59 6.46 6.53
2.55 2.80 2.77 3.06 2.50 2.58 2.63 2.96
2.79 2.88 3.12 3.36 2.97 3.06 2.95 3.38
33.09 36.06 35.28 37.44 31.70 31.41 33.09 36.70
781 781 781 781 781 781 781 781
4.49 5.34 5.11 5.75 4.12 4.05 4.49 5.52
1.10 1.03 1.13 1.10 1.19 1.19 1.13 1.14
33.09 36.06 35.28 37.44 31.70 31.41 33.09 36.70
35.28 (1.84)
Rail#4
A B C D E F G H
7.05 6.84 7.01 6.99 6.93 6.99 7.03 7.15
13.44 13.45 13.43 13.45 13.44 13.45 13.45 13.44
6.39 6.61 6.42 6.46 6.51 6.46 6.42 6.29
3.11 3.48 3.41 3.34 3.69 3.30 3.31 2.90
3.43 3.67 3.41 4.08 3.69 3.58 3.70 3.62
40.19 42.78 43.82 42.59 46.46 42.09 42.52 38.43
877 877 877 877 877 877 877 877
5.26 5.95 6.25 5.90 7.03 5.77 5.88 4.80
1.10 1.06 1.00 1.22 1.00 1.09 1.12 1.25
40.19 42.78 43.82 42.59 46.46 42.09 42.52 38.43
42.28 (1.33)
Rail#5
A B C D E F G H
6.85 6.83 6.86 6.77 6.90 6.95 6.91 6.95
13.32 13.33 13.33 13.33 13.33 13.31 13.32 13.32
6.47 6.50 6.47 6.56 6.43 6.36 6.41 6.37
3.01 3.82 3.54 3.38 3.03 3.19 3.79 3.13
3.58 3.82 3.70 3.67 3.31 3.62 3.79 3.56
37.82 47.63 44.44 41.63 38.41 41.14 48.37 40.16
923 923 923 923 923 923 923 923
4.20 6.66 5.80 5.09 4.33 4.97 6.87 4.74
1.19 1.00 1.05 1.09 1.09 1.13 1.00 1.14
37.82 47.63 44.44 41.63 38.41 41.14 48.37 40.16
41.16 (2.21)
Rail#6
A B C D E F G H
6.95 6.95 6.96 6.93 7.15 6.94 6.92 6.93
13.45 13.45 13.46 13.46 13.48 13.45 13.44 13.46
6.50 6.50 6.50 6.53 6.33 6.51 6.52 6.54
3.45 3.72 3.53 3.38 3.08 3.00 3.06 3.50
3.45 3.72 4.01 3.60 3.21 3.37 3.47 3.50
43.32 46.84 44.25 42.16 40.46 37.74 38.19 43.66
878 878 878 878 878 878 878 878
6.08 7.11 6.35 5.76 5.31 4.62 4.73 6.18
1.00 1.00 1.14 1.06 1.04 1.12 1.14 1.00
43.32 46.84 44.25 42.16 40.46 37.74 38.19 43.66
41.40 (2.65)
Rail#control
A B C D E F G H
6.95 6.95 6.89 7.09 6.87 6.93 6.88 6.94
13.41 13.43 13.45 13.45 13.43 13.45 13.40 13.44
6.46 6.49 6.56 6.36 6.56 6.52 6.52 6.50
3.60 3.59 3.88 3.75 3.44 3.06 2.70 3.13
3.60 3.69 3.88 3.75 3.76 3.49 3.31 3.57
45.58 45.54 48.29 48.94 42.74 38.36 33.74 39.47
903 903 903 903 903 903 903 903
6.38 6.36 7.16 7.35 5.61 4.52 3.49 4.78
1.00 1.03 1.00 1.00 1.09 1.14 1.23 1.14
45.58 45.54 48.29 48.94 42.74 38.36 33.74 39.47
42.34 (3.35)
17
Figures (b)
(a)
Figure 1: Tensile test specimens: (a) sampling locations on the cross section of railhead, and (b) dimensions and geometry of the tensile specimen
(a)
(b)
Figure 2: Single-edge-notched bend (SENB) specimens: (a) sampling locations on the cross section of railhead, and (b) dimensions and geometry of the SENB specimen
Figure 3: Representative curve of load versus crack mouth opening displacement (CMOD, δ) from threepoint bending tests
18
(a)
(b)
Extensometer
Indenter Specimen
Figure 4: Instrumented indentation test [49]: (a) schematic presentation of the test setup and ball indenter, and (b) a typical indentation loading-unloading curve and depiction of the measurement of the unloading stiffness S
(a) R
(b) Figure 5: Finite element modelling of: (a) smooth tensile specimen, and (b) indentation tests with different sizes of ball indenters
19
Figure 6: Experimental data of smooth tensile specimens: (a) illustration of engineering stress-strain curves on railhead of Rail#control, and (b) summary of UTS and yield stress ( ) for six types of highstrength rail steels
Figure 7: FE modelling of smooth tensile specimens: (a) illustration for regenerating experimental data for Rail#control-A, and (b) comparison of mean equivalent stress-strain curves for six types of highstrength rail steels
20
Figure 8: Illustration of damage measurement for Rail#control: (a) repeated loading-unloading tensile tests, and (b) indentation loading-unloading tests
Figure 9: Damage evolution of smooth tensile specimen: (a) determination of the critical value of elastic modulus (Ecr) at fracture initiation for Rail#control, and (b) damage evolution and values of Ecr for six types of high-strength rail steels
Figure 10: Damage evolution under indentation and determination of critical indentation depth (hcr) for each of the six high-strength rail steels by using the corresponding value of Ecr in Fig. 9
21
Figure 11: Indentation load-contact depth curves for six types of high-strength rail steels
Figure 12: Comparison of (a) fracture toughness determined from SENB tests, KIc, with that from indentation tests, KInd, and (b) normalized KIc and normalized KInd, for six types of high-strength rail steels
22
(a)
(b)
(c) Figure 13: FE modelling of ball indentation test with ball diameter of 1.19 mm for Rail#control: (a) full contour plot of stress triaxiality, (b) enlarged view of stress triaxiality under the ball tip, and (c) comparison of experimental data with those from simulation and the critical value of hc (hcr) to calculate
2
0.5
9.19 MP.m
23
Figure 14: Illustration of specific indentation energy (2 ) for Rail#control as a function of indenter diameter (2R), and the KInd value for 2R in the vicinity of critical value of CTOD (dcr)
24
Highlights 1. Develop an indentation test to estimate fracture toughness of rail steels 2. Determine critical damage parameter by using a ductile damage model 3. Investigate the influence of ball indenter size on fracture toughness estimation 4. Characterize tensile properties and KIc for six types of high-strength rail steels
25