Flexural/Tensile Strength Ratio in Rock-like Materials - Springer Link

Rock Mech. Rock Engng. (2001) 34 (3), 217±233

Rock Mechanics and Rock Engineering : Springer-Verlag 2001 Printed in Austria

Flexural/Tensile Strength Ratio in Rock-like Materials By

L. Biolzi, S. Cattaneo, and G. Rosati Dipartimento di Ingegneria Strutturale, Politecnico di Milano, Milano, Italy.

Summary Results are presented and discussed from a testing program to study the ¯exural strength of a rock-like material. In order to investigate the size eëcts, experiments were performed using specimens of a medium grained size granite. For geometrically similar beams of different sizes, this paper presents and discusses experimental evidence from interferometric measurements (ESPI) and locations of acoustic emissions (AE) of the damage zone development, at the peak load, in terms of shape and size. The bending strengths are compared with the direct tensile strength obtained with double-edge-notched specimens. The experimental strengths are interpreted with a stress analysis in the critical cross-section that takes into account the in¯uence of the localized region of microcracks arising when peak load is approached.

1. Introduction It is well established that the tensile strength of a rock-like material can be evaluated in several ways (see for instance, Brook, 1993): direct tensile test, Brazilian test, bending test, etc. The direct tensile test appears to be the natural way to evaluate the strength or to obtain a complete stress-strain curve. However, the results of a direct tensile test are often di½cult to interpret and may be associated with some problems. Indeed, direct tensile tests are complicated to set up, with the specimen grips introducing a perturbation in the uniaxial stress ®eld that, usually, cannot be ignored. Even in an ideal apparatus, slight imperfections in sample preparation and material inhomogeneity can produce non-uniform tensile stress ®elds in the specimen. If the objective of direct tensile test is to obtain a complete stress-strain curve, the problem can be very severe due to unstable response in the post-peak region (Hudson et al., 1972). In this context, two methods (Zongjin, 1996; Gopalaratnam and Shah, 1985) have been used to obtain a stable response in the post-peak region: one uses an elastic load-sharing system parallel to the specimen, the other uses a closed-loop control and introduces a notch in the specimen (used in this study). The tensile strength can be estimated with indirect tests, such as beams in ¯exure (three or four point loading), which are easier tests to conduct and to con-

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trol. The strength, in this case, is expressed in terms of the modulus of rupture, which is the maximum stress at rupture captured from the ¯exure formula: sN

Mmax ; S

1

where Mmax is the maximum moment at the peak load and S is the elastic section modulus. Usually it is asserted in the literature that the results from modulus of rupture tests tend to overestimate the tensile strength as much as 100 percent, mainly because the estimation process assumes the stress-strain relationship linear throughout the critical cross section of the specimen (Jaeger and Cook, 1976). Bending tests also represent an attractive alternative for the determination of the uniaxial stress-strain relationships (Laws, 1981; Mayville and Finnie, 1982) and the simplest method of studying time dependent behavior of rock (creep) (Price, 1964). Nevertheless, these evaluations appear to be inaccurate in the large strain range similarly to the direct tensile tests because they seem to be sensitive to the boundary conditions (Cattaneo and Rosati, 1999). In addition to the di½culties mentioned, it is recognized that the dependence of the strength on the size or the scale is important in rock-like (quasi-brittle) materials such as rock, concrete, ceramics and glass (Bazant and Kazemi, 1990; Biolzi and Labuz, 1993; Labuz and Biolzi, 1991). When laboratory data, usually obtained on small specimens, are used in a structural prediction neglecting this phenomenon, reduced safety factors may be produced. The size dependence seems to be more noticeable where a stress gradient exists (Paterson, 1978). Accordingly, whatever the specimen geometry and the loading condition, the tensile strength evaluated in an experimental test requires a thorough interpretation. The size eëct is generally studied through a comparison of geometrically similar structures of diërent sizes, and is conveniently characterized by load parameters (such as the nominal strength) and overall structural response (see, for instance, Bazant and Planas, 1998). The objective of this paper is to present and to discuss experiments on a medium-grained granite for the study of structural size eëcts. In order to isolate these phenomena, geometrically similar three-point bend beams were considered. The size and shape of the localized damage zone were identi®ed through the locations of acoustic emissions. The tests were monitored with an interferometric technique, referred to as electronic speckle pattern interferometry, that provides maps on displacement ®elds and cracks (Packman, 1975). During the tests, the observation of fringes in real time on a monitor allowed a qualitative monitoring of the crack length at failure. 2. Experimental Techniques 2.1 Testing Machine The testing system consisted of a closed-loop electromechanical Instron load frame with a maximum capacity of 100 kN. The main characteristics are as follows:

Flexural/Tensile Strength Ratio in Rock-like Materials

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a) electromechanical controls with a minimum speed of 2 mm/hour; b) three control channels, one of which can be external (giving the possibility to choose the feedback signal that allows a stable test control); c) closed-loop control with integral and derivative gain (in order to remove the eëct of the ®nite sti¨ness of the machine). It is well known that strain softening structures can be studied in the laboratory only within a closed-loop testing system. In fact, to avoid unstable failure, strain control tests must be performed with an appropriate choice of the feedback signal to the servo-controller. The output of the transducers, including the load cell and the strain gages, were automatically recorded using a data acquisition system. Furthermore, the load-deformation data were continuously displayed through the use of a personal computer. 2.2 Test Specimens Tensile and ¯exural tests were performed on White Montorfano granite, a whitegray medium-grained rock with an average grain size of 6 mm and composed of feldspars, quartz and mica. Young's modulus E and Poisson's ratio n in compression were found to be 20 GPa and 0.16, respectively. The tensile tests were conducted using the double-edge-notched specimen (Fig. 1a) under displacement control in order to impose a more uniform crack

Fig. 1. a) Specimen geometry for tension tests, b) testing machine with ESPI monitoring

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Fig. 2. Three point bent specimens geometry

opening at failure in the critical cross section. The cross section of the specimens was 60 20 mm and the length 150 mm. A nominal notch length of 8 mm, with a width of 2 mm, was considered. The specimens were glued to the platens of the testing machine (Fig. 1b). The specimen geometry for the bending tests was held constant at a span-toheight L=H ratio of 5.7. The dimensions of the beams in mm (span height) were 2400 400, 1200 200, 480 80, and 240 40, thus resulting in a size range 1 : 10, with a thickness T of 30 or 60 mm (Fig. 2). The beams were loaded in three-point bending. The specimens contained a sawn notch, one tenth of the height, to facilitate post-peak control and to provide a predetermined site to observe the damage zone. Three specimens were tested for each geometry considered. The standard deviations on the observed strengths were very small (between 0.10 and 0.17 MPa), denoting excellent repeatability of the results.

2.3 Electronic Speckle Pattern Interferometry (ESPI) ESPI is an interferometric technique used to measure a deformation ®eld of diffusely scattering objects on the surface of the specimen (Jones and Wykes, 1989). In the present tests, the following components were used: a Melles-Griot He-Ne laser (wavelength, l 632 nm, power, P 30 mW), a Panasonic WV BP310/G CCD camera, and a DT-2861 frame grabber. In ESPI, by superimposing a re¯ected light to a reference beam, an interference phenomenon is produced. The resulting interference fringes measure the light path diërence between the two beams as multiples of the wavelength, l. In this way, by comparing two recorded interference patterns before and after an object displacement, the deformation can be evaluated. The displacement vector ®eld is evaluated by illuminating the specimen from diërent directions. The image of the object is acquired by a TV camera and transmitted to a monitor through a frame grabber image processing board controlled by a 486 IBM compatible personal computer. Fig. 1b shows the experimental arrangement for the ESPI measurements.


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2.4 Acoustic Emission The acoustic emission signals generated in laboratory specimens are captured using piezoelectric (PZT-5A) transducers attached to the specimen surface, and preampli®ed before recording (Fig. 3). The sensors have a reasonably ¯at frequency response from 0.1 to 1 MHz and a sensor diameter of about 3 mm. They are mounted directly to the material with a methyl-cyanoacrylate glue and catalyst. Preampli®ers (40 dB gain) and ®lters (bandpass from 0.1 to 1.2 MHz) were chosen to maximize ampli®cation, minimize noise, and assure matched frequency response. The data acquisition system consists of four, two-channel digitizers with a sampling rate of 20 million samples per second per channel (50 nanoseconds between two consecutive samples) and 8-bit resolution. The controller interfaces with a personal computer via a GPIB cable and an AT-GPIB card. The digitizers are equipped with an internal trigger that is activated whenever an AE signal exceeds the preset value. This threshold of amplitude must be set so that environmental noise does not trigger the system. The trigger-out signal of the ®rst digitizer is fanned out to the other three digitizers for the acquisition to begin simultaneously at all the digitizers. One of the trigger-out signals is also sent to the loaddisplacement data acquisition system to correlate AE with the loading history. In general, source location techniques involve a network of AE sensors, positioned at diërent points on the specimen. Microseismic activity due to a change in stress is detected at each sensor at a given time. By knowing the relative arrival times of the P-wave, which is the component of the signal that arrives ®rst, the P-

Fig. 3. Schemes for direct tensile tests

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wave velocity of the material, and the coordinates of each receiver, the event hypocenter can be estimated with a minimum of four sensors. (The problem contains four unknowns: the spatial coordinates of the event and the time at which the event occurred; a ®fth sensor is sometimes needed to remove ambiguities arising from the quadratic nature of the distance equation when the sensors are positioned poorly.) Because some error is associated with arrival-time detection (it is not always clear when the signal arrives) and with the P-wave velocity measurement (as damage accumulates material properties may change or become anisotropic), the number of sensors should be increased so that the location problem becomes over-determined. In the present tests, eight piezoelectric transducers (Physical Acoustic model S9225) were arranged so that the fracture process zone was covered. Four sensors were located on the back face and four sensors were mounted on the front face of specimens. Then, a solution scheme can be developed whereby the error is minimized to obtain a ``best ®t'' type of solution, and statistical methods can be used to evaluate the goodness of the ®t (Labuz and Biolzi, 1998). 3. Experimental Results 3.1 Direct Tensile Tests Tensile tests can be performed by imposing an axial displacement to the specimen with hinges as constraints at its ends (Fig. 3a). In this case the peak load is reached with localization of deformation only on one side of the specimen. In Fig. 4 the AE locations in a test performed on a unnotched specimen under this loading system are shown. It may be noticed that up to a loading ratio equal to 0.8 (de®ned as the ratio between the current and the peak load), AE events were randomly distributed in the specimen. As the load was approaching the peak load, a localization of AE events on one side of the specimen was detected. A single crack was formed in this localized region and observed extending from one side across the specimen. As a result, a bending of the specimen must have existed, leading to a strong perturbation in the homogeneous stress ®eld. This suggest that a prismatic specimen subjected to centric load tends to show bending that increases with crack growth. With a ®xed platens loading system (Fig. 3b) it is possible to impose a more regular crack propagation; the crack usually starts on one side, but a kinematic condition imposes the growth of a second crack on the opposite side. By continuous displacement control, the two cracks advance simultaneously across the specimen until complete failure occurs. The improved fracture behavior that can be obtained with a ®xed platens loading system allows larger peak loads (strengths 10±15% for larger specimen) (Rosati, 1989; Cattaneo and Rosati, 1999). Indeed, compared with the rotating platens test, with an imposed uniform crack opening, the ®xed platens experiment is characterized by a more regular strain distribution. This fact is demonstrated by the interferometric measurements that, in this case, clearly showed the absence of bending (see also, Cattaneo and Rosati, 1999). Therefore, the stress distribution on the critical cross section is not aëcted by


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Fig. 4. Locations of acoustic emission: a) before peak load; b) at the peak load

bending and the maximum tensile stress is reached on the two opposite sides of the critical cross section where the crack ®rst appeared and then developed. The consequences are larger average stresses at the peak loads and, therefore, larger apparent strength with a ®xed platens test. The diërences in strength appeared more evident for larger specimens where, for a given specimen rotation, the bending produced by the crack caused strong deformations at the two sides of the specimen.

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a)

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b)

Fig. 5. Digitized images of the fringe pattern of a tensile test

These results imply that control is favored by introducing two small notches that de®ne the critical cross section where the failure occurs. The notches, in a specimen under a tensile loading, produce a perturbation in the stress ®eld acting in particular as a strain concentrator near the notch roots. Figure 5a shows this phenomenon exhibited by the fringe patters observed at the beginning of a test in the pre-peak loading range. In particular, Labuz et al. (1985) observed that the average failure stresses were not aëcted by a small notch in the specimen, denoting the same failure mechanism for specimens with or without a small notch under direct tension. The symmetry, in the case with ®xed platens, is forced and maintained along the overall loading process without any lateral ¯exing of the tensile specimen (Fig. 5b). The two digitized images of Figs. 5 correspond to a loading level equal to 30 percent of the peak load in the elastic range and equal to 90 percent of the peak load in the post-peak range, respectively. The average tensile strength of the specimens was 4.5 MPa, diërent from the average value of 4.0 MPa obtained with freely rotating platens. 3.2 Bending Tests For beams, the nominal strength sn of the material, that is the maximum allowed stress evaluated according to the elementary methods of classical beam theory, can be de®ned by relation (1). Figure 6 shows the nominal strength as a decreasing function of the size. This is in agreement with other published data (for instance, Bazant and Kazemi, 1990). From the ESPI fringe patterns, it is possible to identify the crack paths by the discontinuities present in the fringe pattern: cracks passing through those points where a fringe breaks o¨ bluntly or presents an angular bent (Fig. 7) (Jacquot and Rastogi, 1983). The ``crack tip'' appears at the end of this region. In this way, it is possible to obtain load-crack length curves that appear as in Fig. 8 (beam C). It may be interesting to compare the crack length at the peak load with the corresponding AE locations (Fig. 9). This results in the AE locations appearing more deeply diüsed in the beams, denoting a large process zone ahead of the crack tip.


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Fig. 6. Eëct of beam depth on ¯exural strength and model prediction

It may be observed that by combining the two techniques, one has diërent, but in a certain sense complementary, information: with the AE locations, the geometrical features of the process zone are identi®ed, whereas with the ESPI fringe patterns it is possible to identify the crack tip. This information is useful in explaining size eëct: if, for a given loading con®guration, the geometrical features of the process zone are a material characteristic and do not depend on the size of the structure, then it may be shown that the nominal stress at failure will decrease with size and reach a limiting value (Labuz and Biolzi, 1998). Furthermore, when the size increases, the post-peak response tends to be more brittle (Fig. 10). This qualitative behavior of specimens of diërent sizes may be observed with normalized load-displacement curves. Dividing the loads and displacements by the corresponding values at peak produces a comparison in the overall response. Figure 11 shows the normalized load-displacement curves for the diërent beam sizes considered. In the pre-peak branch, the shapes of the curves are similar and no signi®cant diërences are observed. Conversely, in the post-peak part of the normalized load-displacement curves, the in¯uence of the specimen size is apparent: the brittleness increases considerably with increasing specimen size. 3.3 Bending Strength To ®nd a solution that holds at peak stress and that satis®es equilibrium, a decomposition of the problem was considered. If uniform traction within the process zone can be assumed, then the solution is obtained with a superposition of the following three problems (Fig. 12): (1) a notched beam with a bending moment, M, without cohesive interaction along the notch (the fundamental stress ®eld); (2) a uniform tensile traction, equal to the theoretical tensile strength of the material,

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Fig. 7. Digitized images of the fringe pattern in a bending test

imposed on the boundary of the notched beam and equilibrated with a tensile force, (3) a notched beam with a compressive force N that balances the tensile force. In this way, the original problem is statically equivalent to the addition of the three considered problems. Note that conditions (2) and (3) represent the perturbation of the fundamental stress ®eld due to the intrinsic process zone. As already stated, the tensile stress at the notch tip is assumed to be equal to the theoretical tensile strength of the material st at peak load: st s1 s2 s3;

2


227

Fig. 7. (continued)

where s1; s2, and s3 are the horizontal normal stresses at the notch tip for the three conditions. For the considered decomposition, s2 st ;

3

N st HT;

4

where T and H are the thickness and the height of the beam.

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Fig. 8. Relation between load and crack length (beam C)

Fig. 9. Locations of acoustic emission at the peak load in bending (beam C)

Therefore, Eq. (2) reduces to: s1 s3 0:

5

It may be shown (Neuber, 1937) assuming a shallow notch of hyperbolic shape, that the stress at the tip of the notch s1 produced by the bending moment M may be written: s1 M=S f1 h=r;

6


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Fig. 10. Load-displacement curves

Fig. 11. Normalized load-displacement curves

where f1 h=r is a function of the ratio between the ligament height h and the radius of curvature of the notch tip r, taken as one-half the process-zone width. Moreover, the stress s3, produced by the compressive force N, is (Neuber, 1937): s3

N=hT f2 h=r;

where f2 h=r is a diërent function of h=r.

7

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Fig. 12. Model problem for the beam at the peak load

Fig. 13. Reduction factor

Considering Eq. (5), it follows that: s3

st H=h f2 h=r:

8

From Eqs. (6) and (8), Eq. (5) can be written: sN st H=hF h=r;

9

where sN is the ¯exural strength de®ned by Eq. (1) and F h=r is the ratio f2 =f1 . If b h=r, then the function F b is given by: F b

3fb 1=2 1

3b1 b 1=2 2b

bfb 1=2 1

1

b

b 2 arctan b 1=2 g

1 b 1=2 1 b arctan b 1=2 g

: 10


231

Figure 13 is a plot of relation (10): F b is 1 when b tends to zero (no notch in the beam) and for values of b greater than 10, the function reaches an asymptotic value approximately equal to 0.74. It may be interesting to observe that if the size of the microcraked zone does not increase, for large specimens the ratio H=h tends to 1 and F b is about 0.74; the limit strength in bending thus appears to be 0.74 the true tensile strength of the material. 4. Discussion From the size eëct Eq. (9), it is possible to observe that the ¯exural strength depends not only on the theoretical tensile strength of the material but also on the cohesive interaction (as measured by the ratio between the beam height H and the undamaged ligament h) and the notch eëct induced by the intrinsic process zone (as measured by the function F that depends on the undamaged ligament and the radius of the notch tip). Indeed, the two factors compete in de®ning the ¯exural strength of a specimen composed of a quasi-brittle material; the ratio H=h is the ampli®cation factor due to the cohesive interaction of the process zone, whereas the function F is the reduction coe½cient that represents the perturbation in the stress ®eld due to the shape and size of the undamaged volume. The ratio H=h, which is always greater than one, is the measure of the positive contribution of the process zone to the ¯exural strength. The width of the microcracked zone was similar for the specimens of diërent size (that is, independent of the beam size). From the experiments monitored with the acoustic emissions technique it was estimated that r 15 mm (one half of the width of the microcracked zone) and H h 60 mm. Therefore, the experimental evidence showed that the peak load was reached with localization almost independent from the beam size. Hence, the ¯exural/tensile strength ratio, de®ned by relation (9) appears to be a function of the microstructure of the material, which de®nes the geometrical features of the microcracked volume at the peak load, and the size of the specimen. The theoretical tensile strength estimated from the analysis is about 5.8 MPa. Assuming that the lower limit of the ¯exural strength under pure bending is 75% of the theoretical strength st of the material, the size eëct Eq. (9) can then be compared with the experimental results in Fig. 7, and reasonable agreement between theory and experimental data is shown (Fig. 6). It is very interesting to compare relation (9) with an empirical expression proposed by the Deutscher Ausschuss fuÈr Stahlbeton (Heilmann, 1976): sN st 0:8 0:26=a 0:6 ;

11

where st is the tensile strength and a, measured in centimeters, is the depth of the tensile zone. Relation (11), for deep beams gives sN G 0:8 st very close to the limit value given by (9). As a ®nal remark, the problem of the evaluation of the ``true'' tensile strength may be construed as an inverse problem when relation (9) is solved with respect to the tensile strength. In this case it is required, however, to ®nd the depth of the tensile zone. Thus, adequate test monitoring is needed to characterize the process zone development, at maximum stress, in terms of size and shape.

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5. Conclusions Bending tests, similar to direct tensile tests, do not always provide a correct evaluation of the true tensile strength of a given material. The bending strength, aëcted by a strong size eëct, usually appears larger than the true tensile strength. However, this diërence decreases with increasing size and, for large specimens, the former may result lower than the latter. In large structures the limit value of bending strength is about 75% of the theoretical one. For proper characterization of the material it may be suggested to conduct tests on specimens of diërent sizes or to perform tests with a satisfactory monitoring that allows to evaluate the damage due to microcracks and fracture. With the electronic speckle pattern technique (ESPI), it is possible to identify the crack tip at failure as the point where the fringes appear sharply broken. Combining the ESPI measurements and acoustic emission (AE) locations, two diërent complementary pieces of information are obtained: with the AE locations, the geometrical features of the process zone are identi®ed, whereas, with the ESPI fringe pattern, the crack paths are detected by the discontinuities present in the fringe pattern (cracks passing through those points where a fringe breaks o¨ bluntly or presents an angular bent).

Acknowledgments The European Union under the Standard Measurements & Testing Program, Contract N. SMT4-CT96-2139, is gratefully acknowledged. The authors are grateful to referees for their valuable comments.

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Jacquot, P., Rastogi, P. K. (1983): Speckle metrology and holographic interferometry applied to the study of cracks in concrete. In: F. H. Wittmann, (ed.) Fracture mechanics of concrete. Elsevier, Amsterdam. Jaeger, J. C., Cook, N. G. W. (1976): Fundamental of rock mechanics. 2nd edn. Chapman and Hall, London. Jones, R., Wykes, C. (1989): Holographic and speckle interferometry. 2nd edn. Cambridge University Press, Cambridge. Labuz, J. F., Biolzi, L. (1991): Class I vs class II stability: a demonstration of size eëct. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 28 (2/3), 198±205. Labuz, J. F., Biolzi, L. (1998): Characteristic strength of quasi-brittle materials. Int. J. Solids Struct. 35 (31±32), 4191±4204. Labuz, J. F., Shah, S. P., Dowding, C. H. (1985): Experimental analysis of crack propagation in granite. Int. J. Rock Mech. Min. Sci. 22 (1), 85±98. Laws, V. (1981): Derivation of the tensile stress-strain curve from bending data. J. Mater. Sci. 16 (6), 1299±1304. Mayvill, R. A., Finnie, I. (1982): Uniaxial stress-strain curve from a bending tests. Exp. Mech. 22 (6), 196±201. Neuber, H. (1937): Kerbspannungslehre. Springer, Berlin. Packman, P. F. (1975): Experimental techniques in fracture mechanics. Vol. 2. Iowa State University Press, Ames, Iowa. Paterson, M. S. (1978): Experimental rock deformation ± the brittle ®eld. Springer, Berlin Heidelberg New York Tokyo. Price, N. J. (1964): A study of the time-strain behavior of coal-measure rocks. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 1 (3), 277±303. Rosati, G. (1989): La resistenza residua a trazione del calcestruzzo fessurato e le sue implicazioni nel problema dell'aderenza. Ph.D. Thesis, Politecnico di Milano. Zongjin, L. (1996): Microcrack characterization in concrete under uniaxial tension. Mag. Conc. Res. 48 (5), 219±228. Authors' address: Dr. Luigi Biolzi, Politecnico di Milano, Dipartimento di Ingegneria Strutturale, Piazza Leonardo da Vinci 32, I-20133 Milano, Italy.