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Determination of unconfined compressive strength and Young's modulus of porous materials by indentation tests
Engineering Geology 59 (2001) 267±280
www.elsevier.nl/locate/enggeo
Determination of uncon®ned compressive strength and Young's modulus of porous materials by indentation tests M.H. Leite a,*, F. Ferland b a
Department of Civil, Geological and Mining Engineering, EÂcole Polytechnique, MontreÂal, C.P. 6079, Succ. Centre-Ville, QueÂbec, Canada H3C 3A7 b Tecsult, MontreÂal, QueÂbec, Canada Received 24 July 2000; accepted for publication 29 November 2000
Abstract The formation of a compacted zone under the indenter seems to be the major factor controlling the indentation process in porous rocks. In the case of very porous materials, where the pore structure fails and deformation (by structural collapse) proceeds with almost no increase in the applied load and with very limited damage to the surrounding material, no chipping is observed. The extent of the compacted zone is controlled by the porosity of the material and by the strength of its porous structure. This paper presents an interpretation model developed by the authors to obtain the uniaxial compressive strength of porous materials from the results of indentation tests. It is based on the model proposed by Wilson et al. (Int. J. Mech. Sci., 17, 1975, 457) for the interpretation of indentation tests on compressible foams and on an estimation by the authors of the extent of the compacted zone under the indenter. The results of indentation tests can also be used to obtain the Young's modulus of the material with a model proposed by Gill et al. (Proceedings of the 13th Canadian Symposium on Rock Mechanics, 1980, 1103). Uniaxial compression and indentation tests have been performed on arti®cial porous materials showing porosities varying between 44 and 68%. The uniaxial compressive strength values obtained from both types of test show a very good agreement. For the Young's modulus, the values obtained from the two types of test are different but the variation of the moduli with porosity is the same. Finally, a parameter called permanent penetration modulus is proposed as a means of characterizing the uniaxial compressive strength of porous materials. q 2001 Elsevier Science B.V. All rights reserved. Keywords: Indentation tests; Porous rocks; Uniaxial compressive strength; Young's modulus
1. Introduction Rock indentation has been the subject of a great number of studies for more than three decades for two main reasons: ®rst, during the last few decades, the need for underground space (tunnels, storage, mining) has constantly increased and mechanized excavation of rock has shown a remarkable develop* Corresponding author. Fax: 11-514-340-5841. E-mail address: [email protected] (M.H. Leite).
ment. The performance of full-face rock boring machines depends mainly on its penetration rate, which can be predicted by many methods. Some of these methods are based on the results of indentation tests on rock specimens. The second reason for taking a closer look at indentation tests is that they can, in principle, be used to characterize the mechanical behavior of rocks. One of the major challenges of rock engineers is the determination of signi®cant mechanical properties for a rock mass due to the intrinsic spatial variability of rock properties. The planning of an economical test program requires careful choice of
0013-7952/01/$ - see front matter q 2001 Elsevier Science B.V. All rights reserved. PII: S 0013-795 2(00)00081-8
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the specimens to be tested so the mechanical properties obtained from laboratory or in situ tests can be considered representative of the rock mass relevant to the project. Since indentation tests are relatively simple to perform, they can be used to characterize the mechanical behavior of rocks by obtaining a greater amount of information than would be possible with the same budget using conventional tests. Hence, when used for site characterization, index indentation tests can cover larger areas than it would be possible with conventional laboratory tests at the same cost. For this reason they are very helpful tools in establishing a site zoning which will help to plan the sampling of specimens to be tested by conventional laboratory or in situ tests to obtain the deformability and strength parameters of the mass. They can also be used to determine the best location for stress measurements based on the spatial variability of the Young's modulus of the material so that a clear image of the in situ stress state can be obtained with a minimum number of tests. A review of the existing literature on rock indentation shows that excellent reports on the subject are available (Maurer, 1967; Cheatham and Gnirk, 1967; Simon, 1967; Mishnaevsky, 1995; Alehossein and Hood, 1996). Some of the publications concentrate on a qualitative description of the indentation process; others propose force±penetration relationships based on mathematical models relying on some hypotheses about material behavior and physical mechanisms under the indenter. Such relationships are then compared to experimental results obtained for different types of indenter (wedge, cone, sphere, pyramid) or disc cutters and for different rock types (Paul and Siskarskie, 1965; Miller and Siskarskie, 1968; Lundberg, 1974; Gill et al., 1980; Kou et al., 1995). Finally, some authors try to establish a correlation between some kind of index obtained from indentation tests to mechanical properties (uniaxial compressive strength, in general) obtained from conventional tests. For this kind of application, the indentation tests have been performed either in the laboratory (Szlavin, 1974; Swedzcki and Donald, 1996) or inside boreholes with borehole indenters (Muromachi et al., 1988; Leite et al., 1997). This paper shows that the majority of the studies concern brittle dense rocks and that much less attention has been paid to the problem of indentation of porous
rocks. The following section will concentrate on this subject. The objective of this paper is to evaluate the potential of indentation tests for the determination of the Young's modulus, E, and of the uniaxial compressive strength, C0, of porous rocks. To do so, indentation and conventional compression tests have been preformed on arti®cial porous materials in the laboratory. From these test results values of E and C0 have been obtained and compared. The experimental procedures as well as the models used are presented. For the calculation of the Young's modulus, the calculation model proposed by Gill et al. (1980) has been applied. It has been found that, although both tests give results that differ, the values vary almost identically with porosity. A preliminary model developed by the authors for the calculation of the uniaxial compressive strength of porous materials from the results of indentation tests is presented. The basis of the model as well as the considerations involved on the estimation of the compacted zone under the indenter are presented. It is shown that the values of C0 obtained from indentation tests are in very good agreement with the values of C0 obtained from conventional uniaxial compressive tests. Finally, it is shown that a penetration index called PPM, based on the permanent penetration of the indenter (the permanent penetration modulus), can be used to characterize the strength of porous materials. 2. Indentation process of porous materials It is generally accepted that at the very beginning of the indentation process of brittle dense rocks, the rock under the indenter will undergo elastic deformations after some crushing of surface asperities. As the load increases, an increasing crushed zone will be formed under the indenter. After a certain load level, this crushed and recompacted region begins to act as a rigid material which transfers the stresses to surrounding intact rock (Dutta, 1972; Gill et al., 1980; Kou et al., 1995) and, if the load is increased, cracks will develop around the crushed zone, as shown in Fig. 1. Some cracks will develop up to the surface and chipping will occur around the indenter. This can be identi®ed on a load±penetration curve by a sudden load drop, as shown in Fig. 2. If the loading process
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269
Fig. 2. Load±penetration curve for both dense and porous rocks. Fig. 1. Schematic representation of the indentation process of porous rocks.
continues, successive chipping events will be observed. It seems that, in the case of porous rocks, the formation of a compaction zone under the indenter is the major factor controlling the indentation process, resulting in a different pattern from the one described above. Zaman et al. (1994) developed a constitutive model to describe the behavior of weakly cemented porous rocks under reservoirs conditions. These authors show that the response of such rocks may comprise three phases: an elastic phase, a porecollapse phase and a post-pore-collapse phase. During an indentation test, such rocks will deform elastically under the indenter at the beginning of the process. As the load increases, the pore structure begins to fail and deformation (by structural collapse) will proceed with almost no increase in the applied load and with very limited damage to the surrounding material (Ladanyi, 1967). In the extreme case no chipping can be observed around the indenter and the load±penetration curve will not show sudden drops as in the case of brittle dense rocks (Fig. 2). Thiercelin and Cook (1988) performed scanning electron microscope studies on two porous rocks indented by a sharp wedge. They showed that a large range of discontinuous processes such as tensile cracks and shear bands can be found and that their occurrence depends on the rock tested. They have shown that under the indenter, Portland limestone (porosity of 26%) underwent a loss of cohesion due to the breakage of grains and grain boundaries, greatly reducing the porosity of this zone.
SuaÂrez-Rivera et al. (1990) tested two porous rocks, Berea sandstone (21% porosity) and Indiana limestone (10.3% porosity) and a refractory brick (65% porosity) using different types of axisymmetrical indenter and showed that the process of rock compaction under the indenter controls the behavior of these porous materials under indentation. After the tests, the specimens have been cut and examined under an optical microscope. SuaÂrez-Rivera et al. (1991) point out that an adequate model for the interpretation of indentation tests on porous materials must incorporate parameters that characterize the failure by compaction, de®ned as the mechanism by which the rock accommodates the material displaced by the indenter and suppresses the formation of chips. The extent of the compacted zone is controlled by the porosity of the material and by the strength of its porous structure. As the compacted region grows during the indentation process, localized zones of tensile stresses develop creating conditions that may lead to tensile cracks that will progress away from the compacted material. In the case of the highly porous refractory brick specimens, SuaÂrez-Rivera et al. observed that pore collapse was the only mode of failure with no chipping. In this material, the plastic zone (pore collapse zone) expands away from the indenter in a radial manner, constrained by the surrounding elastic material. Visual observation of the cut specimens after indentation clearly shows a uniformly compacted zone, approximately spherical in shape where the original porous structure of the material has disappeared. Near its external boundary, the porosity increased gradually to its original value. Moreover,
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as the ratio between the indentation load and the projected loaded area, is approximately equal to the yield strength of the foam, s y. A series of indentation tests on two different types of foam showed good agreement with the proposed model. 3. The experimental program
Fig. 3. Idealized stress±strain curve on which the model of Wilson et al. (1975) is based.
once the compaction zone was fully developed, the shape of the indenter was no longer relevant to the indentation process, which was in fact, totally controlled by the size of the compaction zone. While the physical processes undergone by porous materials during indentation seem to be at least, qualitatively understood, no analytical model is available to the authors' knowledge, to relate indentation results to mechanical properties obtained by conventional tests (uniaxial compressive strength and Young's modulus). Wilson et al. (1975) developed a model for the interpretation of indentation tests on lowdensity compressible foams by cylindrical and spherical indenters based on an idealized stress±strain behavior for the low-density foam in compression showing the following stages (Fig. 3): ² A linear elastic regime with small strains. ² Yielding when the major principal stress reaches a critical value, s 1 s y : Strains are not in¯uenced by the intermediate and by the minor stress components, s 2 and s 3, respectively. At this point, under the indenter, the structure collapses in a way which is hardly in¯uenced by the surrounding material. During this phase, the strains increase while the stress stays approximately constant. ² Signi®cant strain hardening when the collapsed material is completely compacted under the indenter, corresponding to a strain e m. The authors develop an analytical model based on the stress distribution under the indenter given by Timoshenko and Goodier (1951) for an elastic material to show that the indentation hardness (H), de®ned
Two main types of test have been performed at the geotechnical laboratory of Ecole Polytechnique: indentation tests with a sphericonical indenter and uniaxial compression tests. Although the objective of the work is to develop an interpretation model for indentation tests in porous rocks, an arti®cial material has been selected for the tests, rather than natural porous rocks. The reason behind this choice is the requirement to have control over the porosity ratio of the test material and also to enable the repeatability of the test results without the in¯uence of the heterogeneity common to natural rocks. The arti®cial rock consists of a mixture of industrial plaster, sand, water and polystyrene spheres. By trial and error and based on the results of preliminary uniaxial compression tests, a ratio of 1.5 between the mass of sand and the mass of plaster has been chosen so that the uniaxial compressive strength of the material would be less than 10 MPa. The amount of polystyrene spheres could be adjusted to obtain a given porosity. In calculating the porosity it was assumed that polystyrene spheres were equivalent to voids in the material, which, of course is not true for two reasons: ®rst, the compressibility of the polystyrene spheres, although very low, is greater than the compressibility of air. Secondly, if the voids were ®lled with air, instead of being ®lled with polystyrene spheres, the crushed material resulting from a pore structure collapse would be able to ®ll in the voids completely while the crushed material can only ®ll in partially the initial ªvoidsº, the rest of the space being occupied by the compacted polystyrene spheres. Specimens with porosity ratios varying between 44 and 68% were obtained by changing the amount of polystyrene spheres in the mixtures. Fig. 4 shows a 60% porosity specimen where the polystyrene spheres are clearly identi®able. For the uniaxial compression tests, cylindrical specimens of approximately 50 mm diameter and
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271
Fig. 4. Specimens used for the tests (photos with different degrees of contrast to show the polystyrene spheres).
122 mm height were prepared while for the indentation tests, moulds of 350 £ 63.5 £ 32 mm 3 were used to cast the blocks to be tested. All the specimens were cured in a humid chamber for 2 weeks before being tested. Preliminary uniaxial compression tests performed at different curing periods showed that after 2 weeks, no signi®cant increase in strength could be observed. A total of 34 uniaxial compression tests were performed following the ASTM D2938-86 standard procedure and under a constant displacement rate of 0.02 mm/min. Two DCDT were used to measure longitudinal displacements while transversal displacements were measured by a DCDT ®xed to a chain mounted around the specimen. During the tests, an unload±reload cycle was performed so that the Young's modulus and Poisson's ratio of the material could be determined from the unload portion of the stress±strain curve. The uniaxial compressive strength was calculated as the maximum stress attained during the test. For the 214 indentation tests performed, load± unload cycles were applied to the indenter mounted on an INSTRON 1350 press (90 kN capacity). The displacement of the indenter was measured by a DCDT with a 0.003 mm precision and a 4 mm course. During the tests, load±unload cycles were imposed to the material up to a total indenter displacement of 3 mm. In order to have a signi®cant number of cycles for all the blocks tested, the maximum load applied to the indenter was increased from one cycle to the other by 100 N for the specimens showing a porosity up to
55% and by 50 N for the specimens showing porosity ratios greater than 55%. The tungsten carbide sphericonical indenter used is composed of a spherical tip with a radius (R in Fig. 5) of 1.2 mm, an apex angle of 1208 (2a in Fig. 5) and a total displacement stroke of 3 mm. 4. Interpretation of the indentation tests From the results of the indentation tests, three parameters were calculated: the Young's modulus (E), a parameter called PPM and the uniaxial compressive strength (C0). The procedure used to obtain each parameter will be explained in the following paragraphs. 4.1. Determination of the Young's modulus (E) The elastic displacement under a circular ¯at rigid indenter is given by (Timoshenko and Goodier, 1951): de
F 1 2 n2 2Ea
1
where de is the elastic displacement under the indenter, F is the load applied to the indenter, E is the Young's modulus of the material, n is the Poisson's ratio of the material and a is the radius of loaded area (radius of the ¯at cylindrical indenter). The elastic displacement of the indenter can be obtained from an indentation test in which load±unload cycles are performed, as shown in Fig. 6. This equation has been modi®ed by Gill et al. (1980) to take into account the fact that, for a sphericonical indenter, the
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Fig. 6. General load±penetration curve showing the elastic and permanent parts of the penetration.
Fig. 5. Determination of the radius of the loaded area for: (a) the spherical part of the indenter; and (b) the conical part of the indenter.
radius of the projected loaded area (ai) will change as the indenter penetrates the material. For a test where n cycles are performed, the Young's modulus can be calculated as: E
1 X 0:5Fi 1 2 n2 n dei ai
2
where Fi and dei are, respectively, the maximum load applied and the elastic displacement for the ith cycle. At the beginning of the test, only the spherical part of the indenter is in contact with the material (Fig. 5a) and the projected loaded area is given by: s R 2 dpi 2 3 ai R 1 2 R where R is the radius of the spherical part of the
indenter and dpi is the permanent displacement for the ith cycle. If dpi $ R 1 2 sin a; the conical part of the indenter will get in contact with the material (Fig. 5b) and the projected loaded area will be given by: 1 2 sin a 4 1 dpi tan a ai R sin a In calculating Young's modulus, Rancourt (1996) suggests that the ®rst three cycles should be neglected so that the initial contact effects will not introduce large variations in the results. Eq. (2) shows that the Young's modulus of the material can be obtained from an indentation test as long as the value of the Poisson's ratio is determined independently or assumed. It can be shown (Cruz, 1989) that the error introduced by assuming a wrong value for the Poisson's ratio is at most 9% for the range of Poisson's ratio values commonly found for rocks (0.10±0.35). 4.2. Determination of the PPM The PPM is de®ned as a deformability modulus related to the permanent displacement of the indenter
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273
Fig. 7. Stress distribution under the indenter in the model of Wilsea et al. (1975).
instead of the elastic displacement: PPM
1 X 0:5Fi 1 2 n2 n dpi ai
5
This parameter has been introduced by Leite et al. (1997) as a means of studying the variability of the in situ strength of a damaged concrete structure by indentation tests performed inside boreholes. Very few concrete cores could be retrieved from the more severely damaged zones of the structure where most of the indentation tests were performed. As a consequence, although a good correlation R2 0:815 has been found between the PPM and the uniaxial compressive strength values, such correlation was based on a limited number of uniaxial tests results. One of the objectives of the experimental program presented in the paper was to verify the existence of a good correlation between PPM and C0 for porous materials. 4.3. Determination of the uniaxial compressive strength (C0) For the determination of C0, the authors have modi®ed the model presented by Wilson et al. (1975),
developed for the interpretation of indentation tests on low-density compressible foams by cylindrical and spherical indenters. The modi®cations introduced to the model are based on experimental observations of the shape and extent of the compaction zone under an indenter during the indentation of porous materials. Wilson et al. (1975) proposed a model based on the idealized behavior presented in Fig. 3. For this type of material, a structural collapse with little lateral deformation occurs when the major principal stress reaches a value s y. For a cylindrical indenter of radius R and length L, Timoshenko and Goodier (1951) show for a perfectly elastic semi-in®nite medium, contours of the major principal stress are circles passing through A and B (Fig. 7). The major principal stress is given by:
s1
P a 1 sin a 2paL
6
Assuming that the indentation is shallow and the elastic components of strains are small, the geometry of Fig. 7 allows to express the strain of material elements lying along a principal stress trajectory PQS as:
e
SQ CD d ù ; SP CE h
7
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At the elastic±plastic boundary, s 1 s y is given by: ! hy =a3 P sy 12 12 1 1 hy =a2 3=2 pa2
Fig. 8. Typical stress±strain curve for the specimens tested.
Moreover, for shallow indentations: dù
a2 2R
and
8
h acot
a 2
9
If a y is the value of a for which s 1 s y ; then beneath this arc, the material remains elastic s 1 , s y ; while above this arc, the material is fully compressed to a strain e m. Provided the elastic strain is small compared to e m, one can write:
em
d ù hy
a a 2Rcot 2
10
The authors extended these equations to a spherical indenter by considering that the constant principal stress surfaces are very close to spherical caps passing through the periphery of the loaded circle and that plastic deformation will be con®ned to a spherical cap passing beneath the indenter. In this case, one can write:
em
d a2 ù 2Rhy hy
11
From Eqs. (11) and (12), it can be seen that, for a given load P applied to the indenter, the value of s y can be obtained as long as the depth of the plastic region (hy) is known. In order to apply the model of Wilson et al. model to porous materials, it is ®rst assumed that pore collapse will start when s 1 s y C0 (Ladanyi, 1968). This is true for very porous materials showing a stress±strain relationship similar to that shown in Fig. 3. Fig. 8 shows stress±strain curves representative of the results obtained in uniaxial compression for the more porous specimens. Although in this case, a yield plateau can be approximately de®ned, no strain hardening phase could be observed allowing to evaluate e m. Hence, the second assumption concerns the extent of the compacted zone under the indenter. The compacted zone under the indenter is assumed to be a spherical cap with a radius equal to the projected loaded area or, in other words, hy a in Fig. 7. This assumption is sustained by visual observations of the indented blocks after the tests (Fig. 9) as well as by the experimental observations and numerical results of other researchers. Fig. 9 shows a photograph of one block that has been cut after indentation. Although the quality of the photograph is not entirely satisfactory, a zone of compacted material can be identi®ed under the indenter in which the texture and the color of the material have changed while beyond this zone, intact sand grains and polystyrene spheres can be identi®ed. The sketch superposed to the photograph shows that the assumption of hy a is quite satisfactory. Yoffe (1982) studied the stress ®elds caused by indentation in brittle materials based on elastic solutions and on observations of silicate glass indentation. He shows that the shape of the yielded zone under the indenter resembles part of a sphere that intersects the surface plane on the circle of contact between the indenter and the material. For materials of open structure like porous ceramics, Yoffe (1982) suggests that the shape is a shallow segment of a sphere of depth
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275
Fig. 9. Photograph of a specimen after an indentation test showing the compacted zone under the indenter.
less than the radius of the loaded area, a, but can be deeper for less porous materials. Other authors suggest that the compacted zone under the indenter is approximately a spherical cap centered at the indenter tip with a radius equal to the distance r in Fig. 10, or, in other words, hy r 1 dp : SuaÂrez-Rivera et al. (1990) present photographs showing that the compacted zone under conical indenters on refractory bricks agrees very well with this assumption. Fleck et al. (1992a) presented the results of a study on the in¯uence of porosity upon indenta-
tion resistance by both a ®nite strain ®nite element calculation and by a cavity expansion model. Two material models are used: the Gurson model, for lower porosities (up to 30%) and a particle yield model by Fleck et al. (1992b) for porous materials consisting of spherical parts joined by discrete necks. These authors show that compaction occurs in a plastic zone of roughly hemispherical shape surrounding the conical indenter. The porosity contours under the indenter obtained from the numerical analyses show that the contour corresponding to
Fig. 10. Shape and extent of the compacted zone under the indenter from visual observations and numerical analyses by various researchers.
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Fig. 11. E versus porosity for uniaxial compression and indenter tests.
the initial porosity of the material agrees well with the assumption that hy r 1 dp : Based on the considerations presented above and replacing hy a in Eq. (12), one gets: P 1 1 2 13 C0 pa2 23=2 If n cycles of loading±unloading cycles are performed during a test, the value of C0 can be obtained by averaging the values calculated for the 4th to the nth cycle: n X 1 Pi 1 1 2 3=2 14 C0 n 2 3 i4 pa2i 2 where Pi is the maximum load at the ith cycle and ai is the radius of the projected loaded area at the ith cycle. 5. Experimental results 5.1. Young's modulus (E) Fig. 11 shows the variation of E with porosity for both uniaxial compression and indentation tests. For
the indentation test, each point on Fig. 11 corresponds to the average of 10 tests performed on a same block, each one with a series of load±unload cycles. As expected, the greater the porosity, the lower the Young's modulus. As it has already been pointed out in Section 4.1, the Young's modulus of the material can be obtained from an indentation test as long as the value of the Poisson's ratio is determined independently or assumed. In the case of the results shown in this paper, the average Poisson's ratio obtained from the uniaxial compression tests n 0:15 has been used for the calculation of E from the indentation tests. It should be noted that from the 34 uniaxial compression tests performed, only 24 allowed the determination of the Poisson's ratio of the material because of problems with the lateral DCDT in 15 tests. Fig. 11 shows that for a given porosity ratio, the value of the Young's modulus obtained from indentation tests is approximately half the value obtained from conventional uniaxial compression tests. This result is similar to what has been found for dense brittle rocks by Cruz (1989) with an indenter similar to the one used for our tests. and by Wagner and
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277
Fig. 12. Uniaxial compressive strength (C0) as a function of porosity from indentation and compression uniaxial tests.
SchuÈmann (1971), Serata et al. (1983) and Cook et al. (1984) with ¯at indenters of different diameters. 5.2. Uniaxial compressive strength (C0) Values of C0 obtained from both uniaxial compression and indentation tests are shown on Fig. 12. It can be seen that the agreement between the C0 values obtained from the two types of test is quite good for the porosity ratios studied. The calculation of C0 for the indentation tests rely on the model proposed by Wilson et al. (1975) for the idealized behavior shown in Fig. 3. However, even if in some cases a large amount of axial strain occurred under an almost constant stress level, neither of the stress±strain curves corresponded the idealized behavior of Fig. 3 and in some cases even a stress drop has been observed. However, a volume decrease has been observed during the majority of the tests which is in agreement with the assumption of the development of a compacted region under the indenter during the indentation tests. The assumed extent of this zone for a given load also seems to give satisfactory results.
5.3. Permanent penetration modulus (PPM) Based on a very limited number of tests, Leite et al. (1997) suggested that the PPM could be used to characterize the resistance of porous materials. In order to con®rm this statement, PPM values have been calculated for the indentation tests and are shown in Fig. 13 as a function of porosity together with the C0 values obtained from the conventional tests. It can be seen from this ®gure that the PPM is a very good index to assess the variability of strength of porous materials since, although the value of the PPM is approximately 10 times greater than the value of C0 for a given porosity ratio, the variation of both with porosity is by all practical means the same.
6. Discussion The range of porosity ratios considered in the present study (45±67%) is relatively high if compared to those of porous rocks, seldom greater than 30%. Such high porosities have been deliberately chosen in order to avoid chipping during indentation and to
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favor the creation of a compacted zone by pore collapse. However, as already mentioned, the assumption that the polystyrene spheres are equivalent to voids, used to calculate the porosity of the specimens, does not correspond to reality. In fact, if the voids were ®lled with air, instead of being ®lled with polystyrene spheres, the crushed material resulting from a pore structure collapse would be able to ®ll in the voids completely while the crushed material can only ®ll in partially the initial ªvoidsº, the rest of the space being occupied by the compacted polystyrene spheres. Hence, the real porosity of the specimens, which is lower than the calculated porosity, could have been evaluated if the volume of the polystyrene spheres after compaction was known. Since the purpose of the present study was to compare the mechanical properties of porous materials obtained by conventional and indentation tests, an approximate calculation of the porosity can be used as long as it is used for all the specimens. The second point that needs to be emphasized is that the shape and extent of the compacted zone under the indenter have been assumed on the basis of limited visual observations obtained by the authors by the time this study was performed. Visual observations and numerical studies performed by other researchers seem to indicate that hy is comprised between the values of hy a and hy r 1 dp (see Fig. 10) resulting in a difference of about 20% in the volume of the compacted zone. Further studies on the shape and extent of the compacted zone are needed to substantiate the proposed model. The model of Wilson et al. (1975) on which our calculation method is inspired, was developed for a spherical indenter. However, SuaÂrez-Rivera et al. (1990) showed that once the compacted zone fully develops, the shape of the indenter is no longer relevant to the indentation process. As can be seen from Figs. 11 and 12, much higher R 2 coef®cients have been observed for the values of C0 than for the values of E obtained from the indentation tests (R2 0:9133 for C0 and R2 0:6990 for E). This is probably due to the fact that C0 is calculated from the plastic penetration of the indenter at each load level (dpi) which is much more important than the elastic penetration (dei) used for the calculation of the Young's modulus. Consequently, given an absolute error on the measurement of the displacement, the
scatter of Young's moduli will be more important than that of C0. Moreover, the higher values of E obtained from uniaxial compression tests than the ones obtained from indentation tests suggest that the elastic displacements under the indenter (dei) are overestimated. This difference may be explained by the fact that when E is determined from uniaxial compression tests at low stress levels, there is probably no porous collapse while, for the indentation tests, pore collapse occurs at the very beginning of the indentation process. Thus, in the latter case, the elastic response is that of a damaged material which is usually more deformable than the intact material as observed in cyclic uniaxial compression tests. Further work is required to validate this assessment. 7. Conclusions Indentation tests using a sphericonical indenter have been performed on arti®cial porous materials showing porosity ratios between 45 and 67%. From these tests, three parameters have been calculated: Young's modulus (E), uniaxial compressive strength (C0) and an index called the PPM. Values of E and C0 obtained from the indentation tests have been compared to those obtained from conventional uniaxial compression tests. It is shown that Young's moduli obtained from uniaxial compression tests are approximately twice those obtained from indentation for a given porosity. As for the values of C0, the agreement between the values obtained from both tests is very good. A calculation model used to obtain C0 from the indentation tests was developed based on the model of Wilson et al. (1975) for indentation tests on porous foams and on visual observations of the compacted zone under the indenter. The agreement between the values of C0 calculated with this method and those obtained from uniaxial compression tests is excellent. Moreover, the PPM index proposed by Leite et al. (1997) was found to be an ef®cient way of characterizing the strength of the material from the results of indentation tests. Finally, although the preliminary model developed by the authors for the calculation of C0 seems very promising, the validity of the assumption about the compacted zone under the indenter is the subject of
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Fig. 13. PPM and C0 values obtained from indentation tests for various porosity rates.
ongoing studies. Further developments also include the application of the model to the interpretation of indentation tests on natural porous rocks. Acknowledgements The authors would like to express their gratitude to Fondation de l'EÂcole Polytechnique for the ®nancial support received and their appreciation to Prof. D.E. Gill of EÂcole Polytechnique for sharing with them his point of view and knowledge on the problem of indentation of rocks. References Alehossein, H., Hood, M., 1996. State-of-the-art review of rock models for disc roller cutters. In: Proceedings of the 2nd North American Rock Mechanics Symposium, Montreal, vol. 1, pp. 693±700. Cheatham, J.B., Gnirk, P.F., 1967. The mechanics of rock failure
associated with drilling at depth. In Proceedings of the 8th US Symposium on Rock Mechanics, pp. 410±439. Cook, N.G.W., Hood, M., Tsai, F., 1984. Observation of crack growth in hard rock loaded by an indentor. Int. J. Rock Mech. Min. Sci. 21 (2), 97±107. Cruz, B., 1989. Le poincËonnement des roches fragiles non poreuses en tant qu'essai in situ palliatif. MSc thesis, EÂcole Polytechnique, MontreÂal, Canada. Dutta, P.K., 1972. A theory of percussive drill bit penetration. Int. J. Rock Mech. Min. Sci. 9 (4), 543±567. Fleck, N.A., Otoyo, H., Needlemen, A., 1992a. Indentation of porous solids. Int. J. Solids Struct. 29 (13), 1613±1636. Fleck, N.A., Kuhn, L.T., McMeeking, R.M., 1992b. Yielding of metal powder bonded by isolated contacts. J. Mech. Phys. Solids 40 (5), 1139±1162. Gill, D.E., Pichette, C., Rochon, P., DubeÂ, P.B., 1980. Relation between some of the methods for predicting the penetration rate of full-face boring machines. In: Proceedings of the 13th Canadian Symposium Rock Mechanics, CIM Sp. 22, pp. 1103± 1110. Kou, S.-Q., Huang, Y., Tan, X.-C., Lindqvist, P.-A., 1995. Identi®cation of the governing parameters related to rock indentation depth by using similarity analysis. Engng Geol. 49, 261±269.
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Ladanyi, B., 1967. Expansion of cavities in brittle media. Int. J. Rock Mech. Min. Sci. 4 (3), 301±328. Ladanyi, B., 1968. Rock failure under concentrated loading. In: Proceedings of the 10th US Symposium on Rock Mechanics, Austin, TX, pp. 363±387. Leite, M.H., CortheÂsy, R., Gill, D.E., 1997. Essais de poincËonnement de trous de forage et essays de compression uniaxiale dans le beÂton des structures de la retenue d'eau de la centrale du barrage Rapide-des-Quinze. C.D.T. Repport, EÂcole Polytechnique, MontreÂal. Lundberg, B., 1974. Penetration of rock by conical indenters. Int. J. Rock Mech. Min. Sci. 11 (6), 209±214. Maurer,W.O., 1967. The state of rock mechanics knowledge in drilling. In: Fairhurst (Ed.), Proceedings of the 8th US Symposium on Rock Mechanics, pp. 355±395. Mishnaevsky Jr., L.L., 1995. Physical mechanisms of hard rock fragmentation under mechanical loading: a review. Int. J. Rock Mech. Min. Sci. 32 (8), 763±766. Miller, M.H., Siskarskie, D.L., 1968. On the penetration of rock by three-dimensional indenters. Int. J. Rock Mech. Min. Sci. 5 (5), 375±398. Muromachi, T., Sakai, Y., Tsuchiya, H., Yamamoto, Y., 1988. In: De Ruiter (Ed.), Borehole cone apparatus for weak rocks. In International Symposium on Penetration Testing, pp. 875±880. Paul, B., Siskarskie, D.L., 1965. A preliminary theory of static penetration by a rigid wedge into a brittle material. Trans. Soc. Min. Engng (232), 373±383. Rancourt, A., 1996. CaracteÂrisation de la variabilite des proprieÂteÂs meÂcaniques des roches au moyen d'un essai index in situ. MSc thesis, EÂcole Polytechnique, MontreÂal. Serata, S., Tsai, F., McNamara, J.F., 1983. Field experience with a
multiple-piston loading probe. In Proceedings of the 14th US Symposium on Rock Mechanics, pp. 359±369. Simon, R., 1967. Rock fragmentation by concentrated loading. In: Fairhurst (Ed.), Proceedings of the 8th US Symposium on Rock Mechanics, pp. 440±454. SuaÂrez-Rivera, F.R., Cook, N.G., Cooper, G.A., Zheng, Z., 1990. Indentation by pore collapse in porous rocks. In: Proceedings of the 31st US Symposium on Rock Mechanics, pp. 671±678. SuaÂrez-Rivera, F.R., Cook, P.J., Cook, N.G.W., Myer, L.R., 1991. The role of wetting ¯uids during the indentation of porous rocks. In: Roegiers (Ed.), Proceedings of the 32nd US Symposium on Rock Mechanics, pp. 683±692. Swedzcki, T., Donald, D., 1996. Assessment of mechanical parameters of rock using an indentation test. In Proceedings of AusIMM Annual Conference, Perth, pp. 33±36. Szlavin, J., 1974. Relationships between some physical properties of rock determined by laboratory tests. Int. J. Rock Mech. Min. Sci. 11 (2), 57±66. Thiercelin, M., Cook, J., et al., 1988. Failure mechanisms induced by indentation of porous rocks. In: Cundall, et al. (Eds.), Proceedings of the 29th US Symposium, pp. 135±142. Timoshenko, S., Goodier, J.N., 1951. Theory of Elasticity. McGraw-Hill, New York. Wagner, H., SchuÈmann, E.H.R., 1971. The stamp-load bearing strength of rock. Rock Mech. 3, 185±207. Wilson, M., Johnson, K.L., Ashby, M.F., 1975. Indentation of foamed plastics. Int. J. Mech. Sci. 17, 457±460. Yoffe, E.H., 1982. Elastic stress ®elds caused by indenting materials. Philos. Mag. A 46 (4), 617±628. Zaman, M., Roegiers, J.-C., Abdulraheem, A., Azeemuddin, M., 1994. Pore collapse in weakly cemented and porous rocks. J. Energy Resour. Technol. 116, 97±103.
www.elsevier.nl/locate/enggeo
Determination of uncon®ned compressive strength and Young's modulus of porous materials by indentation tests M.H. Leite a,*, F. Ferland b a
Department of Civil, Geological and Mining Engineering, EÂcole Polytechnique, MontreÂal, C.P. 6079, Succ. Centre-Ville, QueÂbec, Canada H3C 3A7 b Tecsult, MontreÂal, QueÂbec, Canada Received 24 July 2000; accepted for publication 29 November 2000
Abstract The formation of a compacted zone under the indenter seems to be the major factor controlling the indentation process in porous rocks. In the case of very porous materials, where the pore structure fails and deformation (by structural collapse) proceeds with almost no increase in the applied load and with very limited damage to the surrounding material, no chipping is observed. The extent of the compacted zone is controlled by the porosity of the material and by the strength of its porous structure. This paper presents an interpretation model developed by the authors to obtain the uniaxial compressive strength of porous materials from the results of indentation tests. It is based on the model proposed by Wilson et al. (Int. J. Mech. Sci., 17, 1975, 457) for the interpretation of indentation tests on compressible foams and on an estimation by the authors of the extent of the compacted zone under the indenter. The results of indentation tests can also be used to obtain the Young's modulus of the material with a model proposed by Gill et al. (Proceedings of the 13th Canadian Symposium on Rock Mechanics, 1980, 1103). Uniaxial compression and indentation tests have been performed on arti®cial porous materials showing porosities varying between 44 and 68%. The uniaxial compressive strength values obtained from both types of test show a very good agreement. For the Young's modulus, the values obtained from the two types of test are different but the variation of the moduli with porosity is the same. Finally, a parameter called permanent penetration modulus is proposed as a means of characterizing the uniaxial compressive strength of porous materials. q 2001 Elsevier Science B.V. All rights reserved. Keywords: Indentation tests; Porous rocks; Uniaxial compressive strength; Young's modulus
1. Introduction Rock indentation has been the subject of a great number of studies for more than three decades for two main reasons: ®rst, during the last few decades, the need for underground space (tunnels, storage, mining) has constantly increased and mechanized excavation of rock has shown a remarkable develop* Corresponding author. Fax: 11-514-340-5841. E-mail address: [email protected] (M.H. Leite).
ment. The performance of full-face rock boring machines depends mainly on its penetration rate, which can be predicted by many methods. Some of these methods are based on the results of indentation tests on rock specimens. The second reason for taking a closer look at indentation tests is that they can, in principle, be used to characterize the mechanical behavior of rocks. One of the major challenges of rock engineers is the determination of signi®cant mechanical properties for a rock mass due to the intrinsic spatial variability of rock properties. The planning of an economical test program requires careful choice of
0013-7952/01/$ - see front matter q 2001 Elsevier Science B.V. All rights reserved. PII: S 0013-795 2(00)00081-8
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the specimens to be tested so the mechanical properties obtained from laboratory or in situ tests can be considered representative of the rock mass relevant to the project. Since indentation tests are relatively simple to perform, they can be used to characterize the mechanical behavior of rocks by obtaining a greater amount of information than would be possible with the same budget using conventional tests. Hence, when used for site characterization, index indentation tests can cover larger areas than it would be possible with conventional laboratory tests at the same cost. For this reason they are very helpful tools in establishing a site zoning which will help to plan the sampling of specimens to be tested by conventional laboratory or in situ tests to obtain the deformability and strength parameters of the mass. They can also be used to determine the best location for stress measurements based on the spatial variability of the Young's modulus of the material so that a clear image of the in situ stress state can be obtained with a minimum number of tests. A review of the existing literature on rock indentation shows that excellent reports on the subject are available (Maurer, 1967; Cheatham and Gnirk, 1967; Simon, 1967; Mishnaevsky, 1995; Alehossein and Hood, 1996). Some of the publications concentrate on a qualitative description of the indentation process; others propose force±penetration relationships based on mathematical models relying on some hypotheses about material behavior and physical mechanisms under the indenter. Such relationships are then compared to experimental results obtained for different types of indenter (wedge, cone, sphere, pyramid) or disc cutters and for different rock types (Paul and Siskarskie, 1965; Miller and Siskarskie, 1968; Lundberg, 1974; Gill et al., 1980; Kou et al., 1995). Finally, some authors try to establish a correlation between some kind of index obtained from indentation tests to mechanical properties (uniaxial compressive strength, in general) obtained from conventional tests. For this kind of application, the indentation tests have been performed either in the laboratory (Szlavin, 1974; Swedzcki and Donald, 1996) or inside boreholes with borehole indenters (Muromachi et al., 1988; Leite et al., 1997). This paper shows that the majority of the studies concern brittle dense rocks and that much less attention has been paid to the problem of indentation of porous
rocks. The following section will concentrate on this subject. The objective of this paper is to evaluate the potential of indentation tests for the determination of the Young's modulus, E, and of the uniaxial compressive strength, C0, of porous rocks. To do so, indentation and conventional compression tests have been preformed on arti®cial porous materials in the laboratory. From these test results values of E and C0 have been obtained and compared. The experimental procedures as well as the models used are presented. For the calculation of the Young's modulus, the calculation model proposed by Gill et al. (1980) has been applied. It has been found that, although both tests give results that differ, the values vary almost identically with porosity. A preliminary model developed by the authors for the calculation of the uniaxial compressive strength of porous materials from the results of indentation tests is presented. The basis of the model as well as the considerations involved on the estimation of the compacted zone under the indenter are presented. It is shown that the values of C0 obtained from indentation tests are in very good agreement with the values of C0 obtained from conventional uniaxial compressive tests. Finally, it is shown that a penetration index called PPM, based on the permanent penetration of the indenter (the permanent penetration modulus), can be used to characterize the strength of porous materials. 2. Indentation process of porous materials It is generally accepted that at the very beginning of the indentation process of brittle dense rocks, the rock under the indenter will undergo elastic deformations after some crushing of surface asperities. As the load increases, an increasing crushed zone will be formed under the indenter. After a certain load level, this crushed and recompacted region begins to act as a rigid material which transfers the stresses to surrounding intact rock (Dutta, 1972; Gill et al., 1980; Kou et al., 1995) and, if the load is increased, cracks will develop around the crushed zone, as shown in Fig. 1. Some cracks will develop up to the surface and chipping will occur around the indenter. This can be identi®ed on a load±penetration curve by a sudden load drop, as shown in Fig. 2. If the loading process
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269
Fig. 2. Load±penetration curve for both dense and porous rocks. Fig. 1. Schematic representation of the indentation process of porous rocks.
continues, successive chipping events will be observed. It seems that, in the case of porous rocks, the formation of a compaction zone under the indenter is the major factor controlling the indentation process, resulting in a different pattern from the one described above. Zaman et al. (1994) developed a constitutive model to describe the behavior of weakly cemented porous rocks under reservoirs conditions. These authors show that the response of such rocks may comprise three phases: an elastic phase, a porecollapse phase and a post-pore-collapse phase. During an indentation test, such rocks will deform elastically under the indenter at the beginning of the process. As the load increases, the pore structure begins to fail and deformation (by structural collapse) will proceed with almost no increase in the applied load and with very limited damage to the surrounding material (Ladanyi, 1967). In the extreme case no chipping can be observed around the indenter and the load±penetration curve will not show sudden drops as in the case of brittle dense rocks (Fig. 2). Thiercelin and Cook (1988) performed scanning electron microscope studies on two porous rocks indented by a sharp wedge. They showed that a large range of discontinuous processes such as tensile cracks and shear bands can be found and that their occurrence depends on the rock tested. They have shown that under the indenter, Portland limestone (porosity of 26%) underwent a loss of cohesion due to the breakage of grains and grain boundaries, greatly reducing the porosity of this zone.
SuaÂrez-Rivera et al. (1990) tested two porous rocks, Berea sandstone (21% porosity) and Indiana limestone (10.3% porosity) and a refractory brick (65% porosity) using different types of axisymmetrical indenter and showed that the process of rock compaction under the indenter controls the behavior of these porous materials under indentation. After the tests, the specimens have been cut and examined under an optical microscope. SuaÂrez-Rivera et al. (1991) point out that an adequate model for the interpretation of indentation tests on porous materials must incorporate parameters that characterize the failure by compaction, de®ned as the mechanism by which the rock accommodates the material displaced by the indenter and suppresses the formation of chips. The extent of the compacted zone is controlled by the porosity of the material and by the strength of its porous structure. As the compacted region grows during the indentation process, localized zones of tensile stresses develop creating conditions that may lead to tensile cracks that will progress away from the compacted material. In the case of the highly porous refractory brick specimens, SuaÂrez-Rivera et al. observed that pore collapse was the only mode of failure with no chipping. In this material, the plastic zone (pore collapse zone) expands away from the indenter in a radial manner, constrained by the surrounding elastic material. Visual observation of the cut specimens after indentation clearly shows a uniformly compacted zone, approximately spherical in shape where the original porous structure of the material has disappeared. Near its external boundary, the porosity increased gradually to its original value. Moreover,
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as the ratio between the indentation load and the projected loaded area, is approximately equal to the yield strength of the foam, s y. A series of indentation tests on two different types of foam showed good agreement with the proposed model. 3. The experimental program
Fig. 3. Idealized stress±strain curve on which the model of Wilson et al. (1975) is based.
once the compaction zone was fully developed, the shape of the indenter was no longer relevant to the indentation process, which was in fact, totally controlled by the size of the compaction zone. While the physical processes undergone by porous materials during indentation seem to be at least, qualitatively understood, no analytical model is available to the authors' knowledge, to relate indentation results to mechanical properties obtained by conventional tests (uniaxial compressive strength and Young's modulus). Wilson et al. (1975) developed a model for the interpretation of indentation tests on lowdensity compressible foams by cylindrical and spherical indenters based on an idealized stress±strain behavior for the low-density foam in compression showing the following stages (Fig. 3): ² A linear elastic regime with small strains. ² Yielding when the major principal stress reaches a critical value, s 1 s y : Strains are not in¯uenced by the intermediate and by the minor stress components, s 2 and s 3, respectively. At this point, under the indenter, the structure collapses in a way which is hardly in¯uenced by the surrounding material. During this phase, the strains increase while the stress stays approximately constant. ² Signi®cant strain hardening when the collapsed material is completely compacted under the indenter, corresponding to a strain e m. The authors develop an analytical model based on the stress distribution under the indenter given by Timoshenko and Goodier (1951) for an elastic material to show that the indentation hardness (H), de®ned
Two main types of test have been performed at the geotechnical laboratory of Ecole Polytechnique: indentation tests with a sphericonical indenter and uniaxial compression tests. Although the objective of the work is to develop an interpretation model for indentation tests in porous rocks, an arti®cial material has been selected for the tests, rather than natural porous rocks. The reason behind this choice is the requirement to have control over the porosity ratio of the test material and also to enable the repeatability of the test results without the in¯uence of the heterogeneity common to natural rocks. The arti®cial rock consists of a mixture of industrial plaster, sand, water and polystyrene spheres. By trial and error and based on the results of preliminary uniaxial compression tests, a ratio of 1.5 between the mass of sand and the mass of plaster has been chosen so that the uniaxial compressive strength of the material would be less than 10 MPa. The amount of polystyrene spheres could be adjusted to obtain a given porosity. In calculating the porosity it was assumed that polystyrene spheres were equivalent to voids in the material, which, of course is not true for two reasons: ®rst, the compressibility of the polystyrene spheres, although very low, is greater than the compressibility of air. Secondly, if the voids were ®lled with air, instead of being ®lled with polystyrene spheres, the crushed material resulting from a pore structure collapse would be able to ®ll in the voids completely while the crushed material can only ®ll in partially the initial ªvoidsº, the rest of the space being occupied by the compacted polystyrene spheres. Specimens with porosity ratios varying between 44 and 68% were obtained by changing the amount of polystyrene spheres in the mixtures. Fig. 4 shows a 60% porosity specimen where the polystyrene spheres are clearly identi®able. For the uniaxial compression tests, cylindrical specimens of approximately 50 mm diameter and
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271
Fig. 4. Specimens used for the tests (photos with different degrees of contrast to show the polystyrene spheres).
122 mm height were prepared while for the indentation tests, moulds of 350 £ 63.5 £ 32 mm 3 were used to cast the blocks to be tested. All the specimens were cured in a humid chamber for 2 weeks before being tested. Preliminary uniaxial compression tests performed at different curing periods showed that after 2 weeks, no signi®cant increase in strength could be observed. A total of 34 uniaxial compression tests were performed following the ASTM D2938-86 standard procedure and under a constant displacement rate of 0.02 mm/min. Two DCDT were used to measure longitudinal displacements while transversal displacements were measured by a DCDT ®xed to a chain mounted around the specimen. During the tests, an unload±reload cycle was performed so that the Young's modulus and Poisson's ratio of the material could be determined from the unload portion of the stress±strain curve. The uniaxial compressive strength was calculated as the maximum stress attained during the test. For the 214 indentation tests performed, load± unload cycles were applied to the indenter mounted on an INSTRON 1350 press (90 kN capacity). The displacement of the indenter was measured by a DCDT with a 0.003 mm precision and a 4 mm course. During the tests, load±unload cycles were imposed to the material up to a total indenter displacement of 3 mm. In order to have a signi®cant number of cycles for all the blocks tested, the maximum load applied to the indenter was increased from one cycle to the other by 100 N for the specimens showing a porosity up to
55% and by 50 N for the specimens showing porosity ratios greater than 55%. The tungsten carbide sphericonical indenter used is composed of a spherical tip with a radius (R in Fig. 5) of 1.2 mm, an apex angle of 1208 (2a in Fig. 5) and a total displacement stroke of 3 mm. 4. Interpretation of the indentation tests From the results of the indentation tests, three parameters were calculated: the Young's modulus (E), a parameter called PPM and the uniaxial compressive strength (C0). The procedure used to obtain each parameter will be explained in the following paragraphs. 4.1. Determination of the Young's modulus (E) The elastic displacement under a circular ¯at rigid indenter is given by (Timoshenko and Goodier, 1951): de
F 1 2 n2 2Ea
1
where de is the elastic displacement under the indenter, F is the load applied to the indenter, E is the Young's modulus of the material, n is the Poisson's ratio of the material and a is the radius of loaded area (radius of the ¯at cylindrical indenter). The elastic displacement of the indenter can be obtained from an indentation test in which load±unload cycles are performed, as shown in Fig. 6. This equation has been modi®ed by Gill et al. (1980) to take into account the fact that, for a sphericonical indenter, the
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Fig. 6. General load±penetration curve showing the elastic and permanent parts of the penetration.
Fig. 5. Determination of the radius of the loaded area for: (a) the spherical part of the indenter; and (b) the conical part of the indenter.
radius of the projected loaded area (ai) will change as the indenter penetrates the material. For a test where n cycles are performed, the Young's modulus can be calculated as: E
1 X 0:5Fi 1 2 n2 n dei ai
2
where Fi and dei are, respectively, the maximum load applied and the elastic displacement for the ith cycle. At the beginning of the test, only the spherical part of the indenter is in contact with the material (Fig. 5a) and the projected loaded area is given by: s R 2 dpi 2 3 ai R 1 2 R where R is the radius of the spherical part of the
indenter and dpi is the permanent displacement for the ith cycle. If dpi $ R 1 2 sin a; the conical part of the indenter will get in contact with the material (Fig. 5b) and the projected loaded area will be given by: 1 2 sin a 4 1 dpi tan a ai R sin a In calculating Young's modulus, Rancourt (1996) suggests that the ®rst three cycles should be neglected so that the initial contact effects will not introduce large variations in the results. Eq. (2) shows that the Young's modulus of the material can be obtained from an indentation test as long as the value of the Poisson's ratio is determined independently or assumed. It can be shown (Cruz, 1989) that the error introduced by assuming a wrong value for the Poisson's ratio is at most 9% for the range of Poisson's ratio values commonly found for rocks (0.10±0.35). 4.2. Determination of the PPM The PPM is de®ned as a deformability modulus related to the permanent displacement of the indenter
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273
Fig. 7. Stress distribution under the indenter in the model of Wilsea et al. (1975).
instead of the elastic displacement: PPM
1 X 0:5Fi 1 2 n2 n dpi ai
5
This parameter has been introduced by Leite et al. (1997) as a means of studying the variability of the in situ strength of a damaged concrete structure by indentation tests performed inside boreholes. Very few concrete cores could be retrieved from the more severely damaged zones of the structure where most of the indentation tests were performed. As a consequence, although a good correlation R2 0:815 has been found between the PPM and the uniaxial compressive strength values, such correlation was based on a limited number of uniaxial tests results. One of the objectives of the experimental program presented in the paper was to verify the existence of a good correlation between PPM and C0 for porous materials. 4.3. Determination of the uniaxial compressive strength (C0) For the determination of C0, the authors have modi®ed the model presented by Wilson et al. (1975),
developed for the interpretation of indentation tests on low-density compressible foams by cylindrical and spherical indenters. The modi®cations introduced to the model are based on experimental observations of the shape and extent of the compaction zone under an indenter during the indentation of porous materials. Wilson et al. (1975) proposed a model based on the idealized behavior presented in Fig. 3. For this type of material, a structural collapse with little lateral deformation occurs when the major principal stress reaches a value s y. For a cylindrical indenter of radius R and length L, Timoshenko and Goodier (1951) show for a perfectly elastic semi-in®nite medium, contours of the major principal stress are circles passing through A and B (Fig. 7). The major principal stress is given by:
s1
P a 1 sin a 2paL
6
Assuming that the indentation is shallow and the elastic components of strains are small, the geometry of Fig. 7 allows to express the strain of material elements lying along a principal stress trajectory PQS as:
e
SQ CD d ù ; SP CE h
7
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At the elastic±plastic boundary, s 1 s y is given by: ! hy =a3 P sy 12 12 1 1 hy =a2 3=2 pa2
Fig. 8. Typical stress±strain curve for the specimens tested.
Moreover, for shallow indentations: dù
a2 2R
and
8
h acot
a 2
9
If a y is the value of a for which s 1 s y ; then beneath this arc, the material remains elastic s 1 , s y ; while above this arc, the material is fully compressed to a strain e m. Provided the elastic strain is small compared to e m, one can write:
em
d ù hy
a a 2Rcot 2
10
The authors extended these equations to a spherical indenter by considering that the constant principal stress surfaces are very close to spherical caps passing through the periphery of the loaded circle and that plastic deformation will be con®ned to a spherical cap passing beneath the indenter. In this case, one can write:
em
d a2 ù 2Rhy hy
11
From Eqs. (11) and (12), it can be seen that, for a given load P applied to the indenter, the value of s y can be obtained as long as the depth of the plastic region (hy) is known. In order to apply the model of Wilson et al. model to porous materials, it is ®rst assumed that pore collapse will start when s 1 s y C0 (Ladanyi, 1968). This is true for very porous materials showing a stress±strain relationship similar to that shown in Fig. 3. Fig. 8 shows stress±strain curves representative of the results obtained in uniaxial compression for the more porous specimens. Although in this case, a yield plateau can be approximately de®ned, no strain hardening phase could be observed allowing to evaluate e m. Hence, the second assumption concerns the extent of the compacted zone under the indenter. The compacted zone under the indenter is assumed to be a spherical cap with a radius equal to the projected loaded area or, in other words, hy a in Fig. 7. This assumption is sustained by visual observations of the indented blocks after the tests (Fig. 9) as well as by the experimental observations and numerical results of other researchers. Fig. 9 shows a photograph of one block that has been cut after indentation. Although the quality of the photograph is not entirely satisfactory, a zone of compacted material can be identi®ed under the indenter in which the texture and the color of the material have changed while beyond this zone, intact sand grains and polystyrene spheres can be identi®ed. The sketch superposed to the photograph shows that the assumption of hy a is quite satisfactory. Yoffe (1982) studied the stress ®elds caused by indentation in brittle materials based on elastic solutions and on observations of silicate glass indentation. He shows that the shape of the yielded zone under the indenter resembles part of a sphere that intersects the surface plane on the circle of contact between the indenter and the material. For materials of open structure like porous ceramics, Yoffe (1982) suggests that the shape is a shallow segment of a sphere of depth
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Fig. 9. Photograph of a specimen after an indentation test showing the compacted zone under the indenter.
less than the radius of the loaded area, a, but can be deeper for less porous materials. Other authors suggest that the compacted zone under the indenter is approximately a spherical cap centered at the indenter tip with a radius equal to the distance r in Fig. 10, or, in other words, hy r 1 dp : SuaÂrez-Rivera et al. (1990) present photographs showing that the compacted zone under conical indenters on refractory bricks agrees very well with this assumption. Fleck et al. (1992a) presented the results of a study on the in¯uence of porosity upon indenta-
tion resistance by both a ®nite strain ®nite element calculation and by a cavity expansion model. Two material models are used: the Gurson model, for lower porosities (up to 30%) and a particle yield model by Fleck et al. (1992b) for porous materials consisting of spherical parts joined by discrete necks. These authors show that compaction occurs in a plastic zone of roughly hemispherical shape surrounding the conical indenter. The porosity contours under the indenter obtained from the numerical analyses show that the contour corresponding to
Fig. 10. Shape and extent of the compacted zone under the indenter from visual observations and numerical analyses by various researchers.
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Fig. 11. E versus porosity for uniaxial compression and indenter tests.
the initial porosity of the material agrees well with the assumption that hy r 1 dp : Based on the considerations presented above and replacing hy a in Eq. (12), one gets: P 1 1 2 13 C0 pa2 23=2 If n cycles of loading±unloading cycles are performed during a test, the value of C0 can be obtained by averaging the values calculated for the 4th to the nth cycle: n X 1 Pi 1 1 2 3=2 14 C0 n 2 3 i4 pa2i 2 where Pi is the maximum load at the ith cycle and ai is the radius of the projected loaded area at the ith cycle. 5. Experimental results 5.1. Young's modulus (E) Fig. 11 shows the variation of E with porosity for both uniaxial compression and indentation tests. For
the indentation test, each point on Fig. 11 corresponds to the average of 10 tests performed on a same block, each one with a series of load±unload cycles. As expected, the greater the porosity, the lower the Young's modulus. As it has already been pointed out in Section 4.1, the Young's modulus of the material can be obtained from an indentation test as long as the value of the Poisson's ratio is determined independently or assumed. In the case of the results shown in this paper, the average Poisson's ratio obtained from the uniaxial compression tests n 0:15 has been used for the calculation of E from the indentation tests. It should be noted that from the 34 uniaxial compression tests performed, only 24 allowed the determination of the Poisson's ratio of the material because of problems with the lateral DCDT in 15 tests. Fig. 11 shows that for a given porosity ratio, the value of the Young's modulus obtained from indentation tests is approximately half the value obtained from conventional uniaxial compression tests. This result is similar to what has been found for dense brittle rocks by Cruz (1989) with an indenter similar to the one used for our tests. and by Wagner and
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Fig. 12. Uniaxial compressive strength (C0) as a function of porosity from indentation and compression uniaxial tests.
SchuÈmann (1971), Serata et al. (1983) and Cook et al. (1984) with ¯at indenters of different diameters. 5.2. Uniaxial compressive strength (C0) Values of C0 obtained from both uniaxial compression and indentation tests are shown on Fig. 12. It can be seen that the agreement between the C0 values obtained from the two types of test is quite good for the porosity ratios studied. The calculation of C0 for the indentation tests rely on the model proposed by Wilson et al. (1975) for the idealized behavior shown in Fig. 3. However, even if in some cases a large amount of axial strain occurred under an almost constant stress level, neither of the stress±strain curves corresponded the idealized behavior of Fig. 3 and in some cases even a stress drop has been observed. However, a volume decrease has been observed during the majority of the tests which is in agreement with the assumption of the development of a compacted region under the indenter during the indentation tests. The assumed extent of this zone for a given load also seems to give satisfactory results.
5.3. Permanent penetration modulus (PPM) Based on a very limited number of tests, Leite et al. (1997) suggested that the PPM could be used to characterize the resistance of porous materials. In order to con®rm this statement, PPM values have been calculated for the indentation tests and are shown in Fig. 13 as a function of porosity together with the C0 values obtained from the conventional tests. It can be seen from this ®gure that the PPM is a very good index to assess the variability of strength of porous materials since, although the value of the PPM is approximately 10 times greater than the value of C0 for a given porosity ratio, the variation of both with porosity is by all practical means the same.
6. Discussion The range of porosity ratios considered in the present study (45±67%) is relatively high if compared to those of porous rocks, seldom greater than 30%. Such high porosities have been deliberately chosen in order to avoid chipping during indentation and to
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favor the creation of a compacted zone by pore collapse. However, as already mentioned, the assumption that the polystyrene spheres are equivalent to voids, used to calculate the porosity of the specimens, does not correspond to reality. In fact, if the voids were ®lled with air, instead of being ®lled with polystyrene spheres, the crushed material resulting from a pore structure collapse would be able to ®ll in the voids completely while the crushed material can only ®ll in partially the initial ªvoidsº, the rest of the space being occupied by the compacted polystyrene spheres. Hence, the real porosity of the specimens, which is lower than the calculated porosity, could have been evaluated if the volume of the polystyrene spheres after compaction was known. Since the purpose of the present study was to compare the mechanical properties of porous materials obtained by conventional and indentation tests, an approximate calculation of the porosity can be used as long as it is used for all the specimens. The second point that needs to be emphasized is that the shape and extent of the compacted zone under the indenter have been assumed on the basis of limited visual observations obtained by the authors by the time this study was performed. Visual observations and numerical studies performed by other researchers seem to indicate that hy is comprised between the values of hy a and hy r 1 dp (see Fig. 10) resulting in a difference of about 20% in the volume of the compacted zone. Further studies on the shape and extent of the compacted zone are needed to substantiate the proposed model. The model of Wilson et al. (1975) on which our calculation method is inspired, was developed for a spherical indenter. However, SuaÂrez-Rivera et al. (1990) showed that once the compacted zone fully develops, the shape of the indenter is no longer relevant to the indentation process. As can be seen from Figs. 11 and 12, much higher R 2 coef®cients have been observed for the values of C0 than for the values of E obtained from the indentation tests (R2 0:9133 for C0 and R2 0:6990 for E). This is probably due to the fact that C0 is calculated from the plastic penetration of the indenter at each load level (dpi) which is much more important than the elastic penetration (dei) used for the calculation of the Young's modulus. Consequently, given an absolute error on the measurement of the displacement, the
scatter of Young's moduli will be more important than that of C0. Moreover, the higher values of E obtained from uniaxial compression tests than the ones obtained from indentation tests suggest that the elastic displacements under the indenter (dei) are overestimated. This difference may be explained by the fact that when E is determined from uniaxial compression tests at low stress levels, there is probably no porous collapse while, for the indentation tests, pore collapse occurs at the very beginning of the indentation process. Thus, in the latter case, the elastic response is that of a damaged material which is usually more deformable than the intact material as observed in cyclic uniaxial compression tests. Further work is required to validate this assessment. 7. Conclusions Indentation tests using a sphericonical indenter have been performed on arti®cial porous materials showing porosity ratios between 45 and 67%. From these tests, three parameters have been calculated: Young's modulus (E), uniaxial compressive strength (C0) and an index called the PPM. Values of E and C0 obtained from the indentation tests have been compared to those obtained from conventional uniaxial compression tests. It is shown that Young's moduli obtained from uniaxial compression tests are approximately twice those obtained from indentation for a given porosity. As for the values of C0, the agreement between the values obtained from both tests is very good. A calculation model used to obtain C0 from the indentation tests was developed based on the model of Wilson et al. (1975) for indentation tests on porous foams and on visual observations of the compacted zone under the indenter. The agreement between the values of C0 calculated with this method and those obtained from uniaxial compression tests is excellent. Moreover, the PPM index proposed by Leite et al. (1997) was found to be an ef®cient way of characterizing the strength of the material from the results of indentation tests. Finally, although the preliminary model developed by the authors for the calculation of C0 seems very promising, the validity of the assumption about the compacted zone under the indenter is the subject of
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Fig. 13. PPM and C0 values obtained from indentation tests for various porosity rates.
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