- Email: [email protected]
Three-dimensional work of fracture for mortar in compression
Engineering Fracture Mechanics 65 (2000) 223±234
www.elsevier.com/locate/engfracmech
Three-dimensional work of fracture for mortar in compression Eric N. Landis a,*, Edwin N. Nagy b, 1 a
Department of Civil and Environmental Engineering, University of Maine, Orono, 04469 Maine, USA b Shepard Ð Wesnitzer, Inc. AZ 86340 Sedona, USA
Abstract A high resolution three-dimensional scanning technique called X-ray microtomography was used to measure internal crack growth in small mortar cylinders loaded in uniaxial compression. Tomographic scans were made at dierent load increments in the same specimen. Three-dimensional image analysis was used to measure internal crack growth during each load increment. Load±deformation curves were used to measure the corresponding work of the external load on the specimen. Fracture energy was calculated using a linear elastic fracture mechanics approach, but using the actual surface area of internal cracks created. Preliminary results indicate fracture energies in the same range as those measured using traditional techniques. # 2000 Elsevier Science Ltd. All rights reserved. Keywords: X-ray; Tomography; Three-dimensional analysis; Mortar; Compression fracture
1. Introduction Portland cement concrete is among the most widely used materials in the world because of its good strength and durability relative to its cost. Because of the huge worldwide investment in concrete infrastructure a substantial research eort has gone into understanding fracture and failure of this material. Much of the research has been summarized in a number of recent textbooks [1±4]. As summarized in these text, concrete is a `quasibrittle' material in which friction and other mechanisms increase the energy requirement for crack propagation. Some of the other `toughening mechanisms' have been identi®ed and include crack bridging, microcracking, and particle interlocking. The cumulative eects of these dierent mechanisms is well documented. However, not so much work has gone into * Corresponding author. Tel.: +1-207-581-2173; fax: +1-207-581-3888. E-mail address: [email protected] (E.N. Landis). 1 Formerly Department of Civil and Environmental Engineering, University of Maine. 0013-7944/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 3 - 7 9 4 4 ( 9 9 ) 0 0 1 2 4 - 1
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isolating the individual eects (for example how much energy is dissipated due to crack bridging). The reason for this is simply the diculties we have characterizing a complex heterogeneous microstructure. Indeed, elementary linear elastic fracture mechanics models are introduced in textbooks [5,6] as twodimensional because three-dimensional problems are analytically more dicult, and their parameters are more dicult to determine experimentally. For metals and materials of a relatively homogeneous microstructure a two-dimensional model is a reasonable approximation. The approach that has typically been taken in concrete is to treat the fracture problem as a twodimensional crack problem with appropriate modi®cations to account for observed toughening behavior. For example, the popular ®ctitious crack model of Hillerborg [7] a tension-softening function is used to account for the nonlinear eects. In the two-parameter model of Jenq and Shah [8] an `eective' crack length is used. While these models are very useful and can be very eective at predicting fracture behavior, they tend to discount the complexity of the physical phenomena of fracture. In more recent work, researchers have examined the complexity of fracture by examining fractal and roughness properties of fracture surfaces [9,10]. Carpinteri and his colleagues have taken this work a step further and used the fractal nature of cracks in concrete to develop a multifractal scaling law, which has been used to explain size eect in fracture energy of concrete structures [11,12]. While there is much interest in the role of disorder and complexity in fracture [13], Bazant [14] argues that much of this work does not have a thermodynamic basis. What we seek to do in the research described in this paper is to take a step back, and look at fracture again at a very elementary level. The motivation for doing this is the availability of a tool that allows us to examine internal crack growth and fracture of specimens in three dimensions. Using this tool, we can revisit some of the original fracture theories of Grith and Irwin, and re-examine the applicability of their work to heterogeneous materials such as concrete. The goal of our work is to measure fracture energy in three dimensions, and to consider to what extent the quasibrittle behavior can be traced simply to the complexity of the crack systems observed. That is, we measure work of fracture, dW=dA, where dW is the incremental energy dissipated by crack growth, and dA is the area of the new surfaces created by fracture. This surface area is not limited to planar surfaces, but can include (within our measurement resolution) all the tortuosity that characterizes fracture of concrete. We can restate this goal as the following hypothesis to be tested: the majority of the nonlinear eects we observe in concrete fracture are in fact not nonlinearities, but are rather due to the way we traditionally de®ne crack area. If a better measure of crack area were possible, we would perhaps consider linear elastic fracture mechanics (LEFM) an appropriate model for concrete. We can test this hypothesis to some degree by measuring work of fracture using our basic de®nition described below, and comparing our measurements to results of others using nonlinear models. Con®rmation or rejection of our hypothesis presents a dicult problem, and will not be attempted in this preliminary work. As a ®rst pass however, we will consider how well our measurements compare with others. Our starting point is the testing of small mortar cylinders loaded in uniaxial compression. As detailed below, the specimen parameters were based on available scanning equipment as well as ease of mechanical testing. 2. Experimental technique Our method of measuring internal crack area starts with a scanning technique called X-ray microtomography [15,16]. The technique is similar in practice to conventional medical CAT-scans in that we use hundreds of through-transmission radiographs to create a tomographic reconstruction of the internal structure of the object being scanned. The dierences are that the microtomography system described here utilizes extremely bright synchrotron radiation as the X-ray source, and a high resolution
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CCD is the receiver. Upon tomographic reconstruction of several hundred projection images, the result is three-dimensional digital image data which has high spatial resolution (<2 mm) and pixel intensity resolution. These three-dimensional images allow us to view the internal structure of the object being scanned. The experiments described here were conducted at beamline X2B of the National Synchrotron Light Source at Brookhaven National Laboratory. Beamline X2B was built by Exxon Research and Engineering for microtomography of geologic materials [16]. Fig. 1 illustrates the components of the experimental setup. Details of the experiments are presented in Refs. [17,18], or [19], but our approach can be summarized as follows. An undamaged specimen is mounted in a small loading frame that can be placed on the rotation stage in the X-ray path shown in Fig. 1. The small loading frame was built speci®cally for in situ scanning of specimens under load. This frame was capable of continuous real-time load and platen to platen displacement monitoring. An initial tomographic scan was made of the undamaged specimen. Then a compressive load was applied to the specimen, and a second scan was taken. After the tomographic scan was completed the specimen was unloaded to measure unloading compliance and reloaded to a higher platen displacement. This cycle was repeated several times until the specimen was extensively damaged. Load and displacement data were continuously recorded by a digital data acquisition system. A typical load±displacement curve is presented in Fig. 2. The X-ray scans and tomographic reconstruction produce volume data representing the internal structure of the specimen. This volume data may be presented in dierent ways, but is most commonly presented in the form of cross-sectional `slices', where each slice represents a particular horizontal crosssection located at a dierent vertical positions. In our work, several hundred slices were produced from a single scan. These slices were sucient to reproduce the entire internal structure of the specimen being scanned. Fig. 3 illustrates how we are able to use the microtomography system coupled with the in situ loading frame. The ®gure shows roughly the same cross-section in a single specimen at six dierent displacement levels. (The cross-sections are not exactly the same due to a non uniform deformation ®eld within the specimen, an inevitable response for a heterogeneous material). The cylindrical specimen was loaded along an axis normal to plane of the images. The stable growth of cracks in the specimen is clearly illustrated. We see no cracks in the initial scan. In the second scan we clearly see a crack has formed in the left-hand portion of the specimen. The subsequent images illustrate additional crack networks that develop as the stress is continuously being redistributed through the cross section. We can see here the
Fig. 1. Illustration of experimental setup for microtomography.
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growth of cracks both in length and thickness, and we can observe how the cracks meander in and around aggregates, illustrating interactions between cracks and aggregates. In order that we may emphasize the three-dimensional nature of this data, three-dimensional renderings are shown in Fig. 4. These renderings were created by `stacking up' hundreds of cross-sectional slices in a commercially available software package. The cutaway section illustrates the vertical crack patterns in the specimen. While there is much qualitative information that can be inferred from this sequence of images, by themselves they do not advance our knowledge beyond what was done using more conventional techniques over 35 years ago (e.g. [20]). As is described in the sections below, what is new is how we can combine the recorded load±displacement data with a quantitative analysis of this three-dimensional data. We should note here that a signi®cant limitation of our experimental technique and the subsequent data analysis is that we are limited to very small specimens. At the energy with which we were working (30 keV), the X-ray beam had a usable size of about 6 mm wide by 5 mm high. Further tomographic requirements with respect to our loading frame limited us to 4 mm diameter by 4 mm high cylinders. Because of the specimen size limitation we focused our attention towards what can be described as the mortar phase of the concrete. At an intermediate scale of observation concrete may be thought of as a two-phase material consisting of aggregates embedded in a mortar matrix. As such, our measurements of fracture energy are not truly valid for what we would typically refer to as `concrete', but rather they
Fig. 2. Typical load±displacement plot for specimen.
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Fig. 3. Tomographic images of a single specimen plane at six dierent levels of damage. The image numbers correspond to the scan numbers shown in Fig. 2.
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will be valid for one phase of that concrete. The specimens were 4 mm diameter by 4 mm high cylinders having a mix proportion of 1 to 2 to 0.6 parts by weight of type I portland cement to sand (0.425 mm maximum size) to water. The mix was proportioned to correspond roughly to the mortar phase of a conventional normal strength concrete mix. Scale eects are beyond the scope of this work, but are being considered in future work. 3. Fracture analysis As hinted above, the approach taken for our fracture analysis is quite elementary. We merely equate the change in potential energy of our system with the corresponding change in measured internal crack surface area. The idea may be summarized as follows. As a starting point, we will use a basic Grithtype energy balance [5,6]. That is, the total potential energy, P, of the specimen load system may be written as: PUÿFW
1
where U is the internal strain energy, F the work of the external load, and W the energy dissipated by with crack growth. The condition of crack growth is: dP 0 dA
2
dW d F ÿ U dA dA
3
or,
where dA is an incremental change in crack area, and dW=dA is the work-of-fracture, or fracture
Fig. 4. Three-dimensional renderings of tomographic data. Specemen is 4 mm diameter by 4 mm high.
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energy, Gc , such that: Gc
dW : dA
4
Note that at this point we have made no assumptions about specimen or crack geometry. In order to evaluate fracture energy we simply need to evaluate the terms on the right-hand side of Eq. (3). Of these F and U are determined from our load±displacement data: F being the area under the load± displacement curve increment and U being the area under the unloading curve. dA is determined through an analysis of our microtomography data as detailed in the image analysis section below. Before we could apply Eq. (3) to our data we had to make a few modi®cations based on the limitations of our experiments. First, we had to recognize that the derivatives of Eqs. (3) and (4) must be modi®ed to re¯ect our discrete rather than in®nitesimal load±displacement increments. To do this we rewrite the equations as: Gci
DWi D Fi ÿ Ui DAi DAi
5
Here Gci is the fracture energy dissipated over loading increment i. Fi and Ui are total and unloading areas, respectively. An additional experimental issue that requires us to modify our analysis was the appearance of measurable creep deformation during the tomographic scans. Each tomographic scan took between 2 and 4 h depending on the current in the synchrotron ring. The reason for this time is that an accurate tomographic reconstruction requires 720 projection images, each taking several seconds to expose and record. The load relaxation due to creep was on the order of 3±5%, which corresponded to a strain relaxation that was well below the spatial resolution of the microtomography system. The reconstructed images were therefore not aected by this deformation. The relaxation was enough, however, to introduce error into our work-of-fracture calculation as de®ned by Eq. (5). (Both load and deformation are aected because our loading device is neither pure displacement nor load control.) For our analysis, we assume creep is an inelastic deformation that is not related to crack growth. This is a reasonable assumption based on the generally accepted microstructural mechanism for creep of gel pore water transport [21]. We can then de®ne a quantity Ci that de®nes the change in potential energy in the material due to creep deformation. This change is re¯ected in the modi®cation of Eq. (5): Gci
DWi D Fi ÿ Ui ÿ Ci DAi DAi
6
Again, Ci is subtracted from Fi because it is a deformation that is not re¯ected in crack growth. Experimentally, Ci is de®ned as the area under the load±displacement curve between the start and end of the scan. It is illustrated along with the Fi and Ui in Fig. 5. Through the use of Eq. (6) we are implying crack growth is the only energy dissipation mechanism, which is of course not the case. However, in our hypothesis we are assuming that other mechanisms are small in comparison. The only non-crack energy dissipation mechanism we will speci®cally address here is the friction between the specimen and the loading platens. We assume this to be small because of con®nement eects. Inspection of reconstructed specimen slices located close to the platens showed little or no lateral expansion compared to slices located near the middle of the specimens [18]. We therefore assume this friction term to be negligible. The Fi and Ui terms are relatively easy to determine given a load±displacement curve such as illustrated in Fig. 2. Simple numerical integration is satisfactory. The DAi term requires a bit more sophisticated analysis. A summary of this analysis follows.
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4. Image analysis Each tomographic scan produced at least 500 slice images of 1024 by 1024 two byte pixels each. Our job was to extract crack surface area measurements from these images for comparison with subsequent scans. A summary of the process is presented here. The complete analysis process is presented in Ref. [17]. The basic steps of our crack measurement scheme are ®rst, separation of void space from solid; second, isolation and identi®cation of individual crack or void features; and third, measurement of those features. The thresholding was accomplished by selecting a particular pixel intensity that represents the transition from solid to void. Given that the intensity of a pixel value directly corresponds to a speci®c X-ray absorptivity, this is a perfectly valid operation. Error results from the fact that this X-ray absorptivity value is an average over the whole area of the pixel. Thus, some amount of judgement is involved in selecting the threshold level. Although the absolute value of the crack area measured in any one scan is quite sensitive to the choice of threshold value, our investigations suggest that the change in the crack area measure between loads is not. An illustration of a thresholded slice is shown in Fig. 6. In this ®gure black represents solid while white represents void space, or air. Now that the cracks and voids have been isolated from the solid, they can be labeled and measured. To do this, a three-dimensional connected-components routine ®nds each individual `blob' of void space. It does this by scanning through the current slice looking for an unlabeled void voxel (a voxel is a three-dimensional volume element as opposed to a two-dimensional pixel, or picture element). When it ®nds one, it gives it a label (colors it) and then looks at its neighbors in all three dimensions. Any of its neighbors which is both unlabeled and void colored gets the same label and is put in a list. After it looks at all ®ve neighbors (four in the plane and one above), it goes to the ®rst element in the list, and looks at its neighbors. It continues to go through the list until the list is empty, and then looks for the next unlabeled air voxel in the plane. When it gets to the end of the plane, it starts in the top left corner of the next plane in the vertical stack. In this way, all the cracks and voids in the cylinder are distinguished. Once found, the surface area and volume of each blob can be measured. For this study, the volume was measured by counting the number of voxels that make up the blob, and multiplying by the unit volume of a single voxel. The surface area is measured by counting the number of free voxel
Fig. 5. Illustration of terms used to determine nonrecoverable work-of-load.
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faces, and multiplying by the unit area of a single voxel face. For the scans described in this study the voxels are cubes measuring 9.6 mm on a side. Thus the unit voxel volume is (9.6 mm)3 =885 mm3, while the unit voxel face area is (9.6 mm)2 = 92 mm2. Again, we should emphasize that the analysis is done on a three-dimensional data set, so although we have presented examples of cross-sectional slices, when we analyze the data we eectively stack up all the slices of a scan into a single volumetric data set. The accuracy of the measured crack area is dicult to assess because of both spatial discretization and intensity discretization errors. With respect to spatial discretization errors we can consider a lower bound crack surface area increment as one p that extends from one voxel edge diagonally to another. Such a surface would have an area of a2 2, where a is the length of the voxel edge. Our measurement however would be the area of the two voxel faces, or 2a2 : Thus, we would over estimate the area of the surface by 41%. An upper bound for the surface measurements is more dicult to estimate for it is really a function of how we de®ne a surface, and the resolution of our measurement device. For this
Fig. 6. Example of image slice thresholded to isolate cracks and voids from solid.
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work, as a preliminary estimate we will assume that an upper bound error is on the same order as our lower bound error. Once the total void surface area is measured, we can calculate the change in void surface area between successive tomographic scans. This incremental change in surface area is the dAi term of Eq. (6). We assume for this work that the increase in void surface area is due exclusively to crack growth. We believe this to be a valid assumption. Although there is surely a change in pore volume due to the compressive deformation throughout the specimen, these are going to be extremely small compared to the increase in void volume due to cracking. 5. Experimental results and discussion Fracture energy was computed using Eq. (6) for several load increments in each of two dierent test specimens. A plot of fracture energy versus crack area is plotted in Fig. 7. Obviously, the fracture response is quite dierent in the two specimens although a qualitative inspection of the tomographic images showed no such dierences. However, what is quite remarkable is that in both cases fracture energy calculated using a completely dierent approach than has been done in the past led to values of Gc in quite close agreement with published results using for example notched beam specimens [22]. (See Ref. [3] for numerous other published values). The two results shown in Fig. 7 are interesting examples because they illustrate two dierent types of fracture behavior. Specimen 1 shows a relatively constant work of fracture, while specimen 2 varies, but ultimately increases with crack area. Specimen 1 represents the behavior we would expect to see from a
Fig. 7. Plot of calculated fracture energy versus crack area for two dierent specimens.
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true brittle material: a constant Gc : While specimen 2 represents behavior we would expect to see from a quasibrittle material: an increasing toughness. We refrain from using the term R-curve here because its de®nition based on a single critical crack, while we are dealing here with a branching multiple crack system. However, specimen 2 does demonstrate (ignoring the second data point) what we would normally think of as rising R-curve behavior. Since rising R-curves are a commonly accepted trait of quasibrittle materials, this result is perhaps less interesting than specimen 1, where a constant work of fracture was observed. The constant Gc observed in specimen 1 forces us to challenge our de®nition of a quasibrittle material to some extent. An inspection of the microtomographic images (for example Fig. 3), clearly indicates stable crack growth between the scans. Stable crack growth is not something we should expect to observe in a material of constant Gc (unless there is a falling strain energy release rate, which we will not consider here). Regardless, what we do have is a multicrack network. Therefore, while individual crack fronts may be aected by the usual crack stopping mechanisms, collectively they tend to average out due to the heterogeneous microstructure and all of its in¯uence. Now, back to whether we should accept our original hypothesis: the majority of the nonlinear eects we observe in concrete fracture are in fact not nonlinearities, but are rather due to the way we traditionally de®ne crack area. It seems we are not yet in a position to accept or reject because our sample number was small, and because the results were not necessarily consistent. Apart from fracture size-scale eects, our small specimen size is problematic due to the wide experimental scatter it is likely to produce. Such a small specimen is certainly going to be much more sensitive to defects than a larger specimen. Nevertheless, the observed constant Gc of specimen 1 does present the possibility that we could apply linear elastic fracture mechanics to cement-based materials if we use actual surface area to de®ne our crack rather than a planar projection or approximation.
6. Conclusions No de®nitive conclusions about the fracture properties of mortar are made here due to the relative infancy of the experimental technique, and due to the limited number of experiments conducted to date. The principal conclusions of our work to date are simply that X-ray microtomography can be a powerful tool for investigating fracture energy in complex materials such as concrete, and the threedimensional nature of the data we generate allows us to ask questions we were heretofore prevented from asking. We have shown that using an experimental setup where tomographic scans are made of a specimen under load, changes in internal damage can measured using three-dimensional image analysis techniques. When load and deformation data are available, we can make a work of fracture calculation based on the tortuous crack surfaces that arise in the fracture of heterogeneous materials. Finally we have shown that for a small mortar specimen loaded in compression, we can have a relatively constant work of fracture if we consider the total area of all the crack systems involved.
Acknowledgements This work is supported by the National Science Foundation (CMS-9733769) under a program directed by Dr. John Scalzi. Parts of this research was conducted at the National Synchrotron Light Source, Brookhaven National Laboratory, which is supported by the U.S. Department of Energy, Division of Materials Sciences and Division of Chemical Sciences (DOE contract number DE-AC0276CH00016).
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References [1] Bazant ZP, Planas J. Fracture mechanics and size eect in concrete and other quasibrittle materials. New York: CRC Press, 1998. [2] Karihaloo BL. Fracture mechanics and structural concrete. New York: Longman, 1995. [3] Shah SP, Swartz SE, Ouyang C. Fracture mechanics of concrete: applications of fracture mechanics to concrete, rock, and other quasi-brittle materials. New York: Wiley, 1995. [4] van Mier JGM. Fracture processes of concrete. New York: CRC Press, 1997. [5] Broek D. Elementary engineering fracture mechanics, 4th ed. Boston: Martinus Nijho, 1986. [6] Anderson TL. Fracture mechanics: fundamentals and applications. Boca Raton: CRC Press, 1995. [7] Hillerborg A, Modeer M, Petersson PE. Analysis of crack formation and crack growth in concrete by means of fracture mechanics and ®nite elements. Cement and Concrete Research 1976;6:773±82. [8] Jenq YS, Shah SP. A fracture tougness criterion for concrete. Engineering Fracture Mechanics 1985;21:1055±69. [9] Saouma VE, Barton CC, Gamaleldin NA. Fractal characterization of fracture surfaces in concrete. Engineering Fracture Mechanics 1990;35:47±53. [10] Lange DA, Jennings HM, Shah SP. Relationship between fracture surface roughness and fracture behavior of cement paste and mortar. Journal of the American Ceramic Society 1993;76:589±97. [11] Carpinteri A. Scaling laws and renormalization groups for strength and toughness of disordered materials. International Journal of Solids and Structures 1994;31:291±302. [12] Carpinteri A, Chiaia B. Multifractal nature of concrete fracture surfaces and size eects on nominal fracture energy. Materials and Structures 1995;28:435±43. [13] Carpinteri A, editor. Size-scale eects in the failure mechanisms of materials and structures. London: E&FN Spon, 1996. [14] Bazant ZP. Scaling of quasibrittle fracture: hypotheses of invasive and lacunar fractility, their critique and weibull connection. International Journal of Fracture 1997;83:41±65. [15] Flannery BP, Deckman HW, Roberge WG, D'Amico KL. Three-dimensional X-ray microtomography. Science 1987;237:1439±44. [16] Deckman HW, Dunsmuir JH, D'Amico KL, Ferguson SR, Flannery BP. Development of quantitative X-ray microtomography. Materials Research Society Symposium Proceedings 1991;217:97±110. [17] Nagy E. Application of X-ray microtomography to the three-dimensional study of conrete fracture energy, M.S. Thesis, Department of Civil and Environmental Engineering, University of Maine, Orono, 1998. [18] Landis EN, Nagy EN, Keane DT, Nagy G. A technique to measure three-dimensional work-of-fracture of concrete in compression. Journal of Engineering Mechanics 1999;125:599±605. [19] Landis EN, Nagy EN. Work of load versus internal crack growth for mortar in compression. In: Mihashi H, Rokugo K, editors. Third International Conference on Fracture Mechanics of Concrete Structures, Gifu. Freiburg: AEDIFICATIO, 1998. p. 35±46. [20] Hsu TTC, Slate FO, Sturman GM, Winter G. Microcracking of plain concrete and the shape of the stress±strain curve. Journal of the American Concrete Institute, Proceedings 1963;60:209±24. [21] Mindess S, Young JF. Concrete. Englewood Clis, NJ: Prentice-Hall, 1981. [22] Bazant ZP, Pheier PA. Determination of fracture energy from size eect and brittleness number. ACI Materials Journal 1987;84:463±80.
www.elsevier.com/locate/engfracmech
Three-dimensional work of fracture for mortar in compression Eric N. Landis a,*, Edwin N. Nagy b, 1 a
Department of Civil and Environmental Engineering, University of Maine, Orono, 04469 Maine, USA b Shepard Ð Wesnitzer, Inc. AZ 86340 Sedona, USA
Abstract A high resolution three-dimensional scanning technique called X-ray microtomography was used to measure internal crack growth in small mortar cylinders loaded in uniaxial compression. Tomographic scans were made at dierent load increments in the same specimen. Three-dimensional image analysis was used to measure internal crack growth during each load increment. Load±deformation curves were used to measure the corresponding work of the external load on the specimen. Fracture energy was calculated using a linear elastic fracture mechanics approach, but using the actual surface area of internal cracks created. Preliminary results indicate fracture energies in the same range as those measured using traditional techniques. # 2000 Elsevier Science Ltd. All rights reserved. Keywords: X-ray; Tomography; Three-dimensional analysis; Mortar; Compression fracture
1. Introduction Portland cement concrete is among the most widely used materials in the world because of its good strength and durability relative to its cost. Because of the huge worldwide investment in concrete infrastructure a substantial research eort has gone into understanding fracture and failure of this material. Much of the research has been summarized in a number of recent textbooks [1±4]. As summarized in these text, concrete is a `quasibrittle' material in which friction and other mechanisms increase the energy requirement for crack propagation. Some of the other `toughening mechanisms' have been identi®ed and include crack bridging, microcracking, and particle interlocking. The cumulative eects of these dierent mechanisms is well documented. However, not so much work has gone into * Corresponding author. Tel.: +1-207-581-2173; fax: +1-207-581-3888. E-mail address: [email protected] (E.N. Landis). 1 Formerly Department of Civil and Environmental Engineering, University of Maine. 0013-7944/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 3 - 7 9 4 4 ( 9 9 ) 0 0 1 2 4 - 1
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isolating the individual eects (for example how much energy is dissipated due to crack bridging). The reason for this is simply the diculties we have characterizing a complex heterogeneous microstructure. Indeed, elementary linear elastic fracture mechanics models are introduced in textbooks [5,6] as twodimensional because three-dimensional problems are analytically more dicult, and their parameters are more dicult to determine experimentally. For metals and materials of a relatively homogeneous microstructure a two-dimensional model is a reasonable approximation. The approach that has typically been taken in concrete is to treat the fracture problem as a twodimensional crack problem with appropriate modi®cations to account for observed toughening behavior. For example, the popular ®ctitious crack model of Hillerborg [7] a tension-softening function is used to account for the nonlinear eects. In the two-parameter model of Jenq and Shah [8] an `eective' crack length is used. While these models are very useful and can be very eective at predicting fracture behavior, they tend to discount the complexity of the physical phenomena of fracture. In more recent work, researchers have examined the complexity of fracture by examining fractal and roughness properties of fracture surfaces [9,10]. Carpinteri and his colleagues have taken this work a step further and used the fractal nature of cracks in concrete to develop a multifractal scaling law, which has been used to explain size eect in fracture energy of concrete structures [11,12]. While there is much interest in the role of disorder and complexity in fracture [13], Bazant [14] argues that much of this work does not have a thermodynamic basis. What we seek to do in the research described in this paper is to take a step back, and look at fracture again at a very elementary level. The motivation for doing this is the availability of a tool that allows us to examine internal crack growth and fracture of specimens in three dimensions. Using this tool, we can revisit some of the original fracture theories of Grith and Irwin, and re-examine the applicability of their work to heterogeneous materials such as concrete. The goal of our work is to measure fracture energy in three dimensions, and to consider to what extent the quasibrittle behavior can be traced simply to the complexity of the crack systems observed. That is, we measure work of fracture, dW=dA, where dW is the incremental energy dissipated by crack growth, and dA is the area of the new surfaces created by fracture. This surface area is not limited to planar surfaces, but can include (within our measurement resolution) all the tortuosity that characterizes fracture of concrete. We can restate this goal as the following hypothesis to be tested: the majority of the nonlinear eects we observe in concrete fracture are in fact not nonlinearities, but are rather due to the way we traditionally de®ne crack area. If a better measure of crack area were possible, we would perhaps consider linear elastic fracture mechanics (LEFM) an appropriate model for concrete. We can test this hypothesis to some degree by measuring work of fracture using our basic de®nition described below, and comparing our measurements to results of others using nonlinear models. Con®rmation or rejection of our hypothesis presents a dicult problem, and will not be attempted in this preliminary work. As a ®rst pass however, we will consider how well our measurements compare with others. Our starting point is the testing of small mortar cylinders loaded in uniaxial compression. As detailed below, the specimen parameters were based on available scanning equipment as well as ease of mechanical testing. 2. Experimental technique Our method of measuring internal crack area starts with a scanning technique called X-ray microtomography [15,16]. The technique is similar in practice to conventional medical CAT-scans in that we use hundreds of through-transmission radiographs to create a tomographic reconstruction of the internal structure of the object being scanned. The dierences are that the microtomography system described here utilizes extremely bright synchrotron radiation as the X-ray source, and a high resolution
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CCD is the receiver. Upon tomographic reconstruction of several hundred projection images, the result is three-dimensional digital image data which has high spatial resolution (<2 mm) and pixel intensity resolution. These three-dimensional images allow us to view the internal structure of the object being scanned. The experiments described here were conducted at beamline X2B of the National Synchrotron Light Source at Brookhaven National Laboratory. Beamline X2B was built by Exxon Research and Engineering for microtomography of geologic materials [16]. Fig. 1 illustrates the components of the experimental setup. Details of the experiments are presented in Refs. [17,18], or [19], but our approach can be summarized as follows. An undamaged specimen is mounted in a small loading frame that can be placed on the rotation stage in the X-ray path shown in Fig. 1. The small loading frame was built speci®cally for in situ scanning of specimens under load. This frame was capable of continuous real-time load and platen to platen displacement monitoring. An initial tomographic scan was made of the undamaged specimen. Then a compressive load was applied to the specimen, and a second scan was taken. After the tomographic scan was completed the specimen was unloaded to measure unloading compliance and reloaded to a higher platen displacement. This cycle was repeated several times until the specimen was extensively damaged. Load and displacement data were continuously recorded by a digital data acquisition system. A typical load±displacement curve is presented in Fig. 2. The X-ray scans and tomographic reconstruction produce volume data representing the internal structure of the specimen. This volume data may be presented in dierent ways, but is most commonly presented in the form of cross-sectional `slices', where each slice represents a particular horizontal crosssection located at a dierent vertical positions. In our work, several hundred slices were produced from a single scan. These slices were sucient to reproduce the entire internal structure of the specimen being scanned. Fig. 3 illustrates how we are able to use the microtomography system coupled with the in situ loading frame. The ®gure shows roughly the same cross-section in a single specimen at six dierent displacement levels. (The cross-sections are not exactly the same due to a non uniform deformation ®eld within the specimen, an inevitable response for a heterogeneous material). The cylindrical specimen was loaded along an axis normal to plane of the images. The stable growth of cracks in the specimen is clearly illustrated. We see no cracks in the initial scan. In the second scan we clearly see a crack has formed in the left-hand portion of the specimen. The subsequent images illustrate additional crack networks that develop as the stress is continuously being redistributed through the cross section. We can see here the
Fig. 1. Illustration of experimental setup for microtomography.
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growth of cracks both in length and thickness, and we can observe how the cracks meander in and around aggregates, illustrating interactions between cracks and aggregates. In order that we may emphasize the three-dimensional nature of this data, three-dimensional renderings are shown in Fig. 4. These renderings were created by `stacking up' hundreds of cross-sectional slices in a commercially available software package. The cutaway section illustrates the vertical crack patterns in the specimen. While there is much qualitative information that can be inferred from this sequence of images, by themselves they do not advance our knowledge beyond what was done using more conventional techniques over 35 years ago (e.g. [20]). As is described in the sections below, what is new is how we can combine the recorded load±displacement data with a quantitative analysis of this three-dimensional data. We should note here that a signi®cant limitation of our experimental technique and the subsequent data analysis is that we are limited to very small specimens. At the energy with which we were working (30 keV), the X-ray beam had a usable size of about 6 mm wide by 5 mm high. Further tomographic requirements with respect to our loading frame limited us to 4 mm diameter by 4 mm high cylinders. Because of the specimen size limitation we focused our attention towards what can be described as the mortar phase of the concrete. At an intermediate scale of observation concrete may be thought of as a two-phase material consisting of aggregates embedded in a mortar matrix. As such, our measurements of fracture energy are not truly valid for what we would typically refer to as `concrete', but rather they
Fig. 2. Typical load±displacement plot for specimen.
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Fig. 3. Tomographic images of a single specimen plane at six dierent levels of damage. The image numbers correspond to the scan numbers shown in Fig. 2.
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will be valid for one phase of that concrete. The specimens were 4 mm diameter by 4 mm high cylinders having a mix proportion of 1 to 2 to 0.6 parts by weight of type I portland cement to sand (0.425 mm maximum size) to water. The mix was proportioned to correspond roughly to the mortar phase of a conventional normal strength concrete mix. Scale eects are beyond the scope of this work, but are being considered in future work. 3. Fracture analysis As hinted above, the approach taken for our fracture analysis is quite elementary. We merely equate the change in potential energy of our system with the corresponding change in measured internal crack surface area. The idea may be summarized as follows. As a starting point, we will use a basic Grithtype energy balance [5,6]. That is, the total potential energy, P, of the specimen load system may be written as: PUÿFW
1
where U is the internal strain energy, F the work of the external load, and W the energy dissipated by with crack growth. The condition of crack growth is: dP 0 dA
2
dW d F ÿ U dA dA
3
or,
where dA is an incremental change in crack area, and dW=dA is the work-of-fracture, or fracture
Fig. 4. Three-dimensional renderings of tomographic data. Specemen is 4 mm diameter by 4 mm high.
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energy, Gc , such that: Gc
dW : dA
4
Note that at this point we have made no assumptions about specimen or crack geometry. In order to evaluate fracture energy we simply need to evaluate the terms on the right-hand side of Eq. (3). Of these F and U are determined from our load±displacement data: F being the area under the load± displacement curve increment and U being the area under the unloading curve. dA is determined through an analysis of our microtomography data as detailed in the image analysis section below. Before we could apply Eq. (3) to our data we had to make a few modi®cations based on the limitations of our experiments. First, we had to recognize that the derivatives of Eqs. (3) and (4) must be modi®ed to re¯ect our discrete rather than in®nitesimal load±displacement increments. To do this we rewrite the equations as: Gci
DWi D Fi ÿ Ui DAi DAi
5
Here Gci is the fracture energy dissipated over loading increment i. Fi and Ui are total and unloading areas, respectively. An additional experimental issue that requires us to modify our analysis was the appearance of measurable creep deformation during the tomographic scans. Each tomographic scan took between 2 and 4 h depending on the current in the synchrotron ring. The reason for this time is that an accurate tomographic reconstruction requires 720 projection images, each taking several seconds to expose and record. The load relaxation due to creep was on the order of 3±5%, which corresponded to a strain relaxation that was well below the spatial resolution of the microtomography system. The reconstructed images were therefore not aected by this deformation. The relaxation was enough, however, to introduce error into our work-of-fracture calculation as de®ned by Eq. (5). (Both load and deformation are aected because our loading device is neither pure displacement nor load control.) For our analysis, we assume creep is an inelastic deformation that is not related to crack growth. This is a reasonable assumption based on the generally accepted microstructural mechanism for creep of gel pore water transport [21]. We can then de®ne a quantity Ci that de®nes the change in potential energy in the material due to creep deformation. This change is re¯ected in the modi®cation of Eq. (5): Gci
DWi D Fi ÿ Ui ÿ Ci DAi DAi
6
Again, Ci is subtracted from Fi because it is a deformation that is not re¯ected in crack growth. Experimentally, Ci is de®ned as the area under the load±displacement curve between the start and end of the scan. It is illustrated along with the Fi and Ui in Fig. 5. Through the use of Eq. (6) we are implying crack growth is the only energy dissipation mechanism, which is of course not the case. However, in our hypothesis we are assuming that other mechanisms are small in comparison. The only non-crack energy dissipation mechanism we will speci®cally address here is the friction between the specimen and the loading platens. We assume this to be small because of con®nement eects. Inspection of reconstructed specimen slices located close to the platens showed little or no lateral expansion compared to slices located near the middle of the specimens [18]. We therefore assume this friction term to be negligible. The Fi and Ui terms are relatively easy to determine given a load±displacement curve such as illustrated in Fig. 2. Simple numerical integration is satisfactory. The DAi term requires a bit more sophisticated analysis. A summary of this analysis follows.
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4. Image analysis Each tomographic scan produced at least 500 slice images of 1024 by 1024 two byte pixels each. Our job was to extract crack surface area measurements from these images for comparison with subsequent scans. A summary of the process is presented here. The complete analysis process is presented in Ref. [17]. The basic steps of our crack measurement scheme are ®rst, separation of void space from solid; second, isolation and identi®cation of individual crack or void features; and third, measurement of those features. The thresholding was accomplished by selecting a particular pixel intensity that represents the transition from solid to void. Given that the intensity of a pixel value directly corresponds to a speci®c X-ray absorptivity, this is a perfectly valid operation. Error results from the fact that this X-ray absorptivity value is an average over the whole area of the pixel. Thus, some amount of judgement is involved in selecting the threshold level. Although the absolute value of the crack area measured in any one scan is quite sensitive to the choice of threshold value, our investigations suggest that the change in the crack area measure between loads is not. An illustration of a thresholded slice is shown in Fig. 6. In this ®gure black represents solid while white represents void space, or air. Now that the cracks and voids have been isolated from the solid, they can be labeled and measured. To do this, a three-dimensional connected-components routine ®nds each individual `blob' of void space. It does this by scanning through the current slice looking for an unlabeled void voxel (a voxel is a three-dimensional volume element as opposed to a two-dimensional pixel, or picture element). When it ®nds one, it gives it a label (colors it) and then looks at its neighbors in all three dimensions. Any of its neighbors which is both unlabeled and void colored gets the same label and is put in a list. After it looks at all ®ve neighbors (four in the plane and one above), it goes to the ®rst element in the list, and looks at its neighbors. It continues to go through the list until the list is empty, and then looks for the next unlabeled air voxel in the plane. When it gets to the end of the plane, it starts in the top left corner of the next plane in the vertical stack. In this way, all the cracks and voids in the cylinder are distinguished. Once found, the surface area and volume of each blob can be measured. For this study, the volume was measured by counting the number of voxels that make up the blob, and multiplying by the unit volume of a single voxel. The surface area is measured by counting the number of free voxel
Fig. 5. Illustration of terms used to determine nonrecoverable work-of-load.
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faces, and multiplying by the unit area of a single voxel face. For the scans described in this study the voxels are cubes measuring 9.6 mm on a side. Thus the unit voxel volume is (9.6 mm)3 =885 mm3, while the unit voxel face area is (9.6 mm)2 = 92 mm2. Again, we should emphasize that the analysis is done on a three-dimensional data set, so although we have presented examples of cross-sectional slices, when we analyze the data we eectively stack up all the slices of a scan into a single volumetric data set. The accuracy of the measured crack area is dicult to assess because of both spatial discretization and intensity discretization errors. With respect to spatial discretization errors we can consider a lower bound crack surface area increment as one p that extends from one voxel edge diagonally to another. Such a surface would have an area of a2 2, where a is the length of the voxel edge. Our measurement however would be the area of the two voxel faces, or 2a2 : Thus, we would over estimate the area of the surface by 41%. An upper bound for the surface measurements is more dicult to estimate for it is really a function of how we de®ne a surface, and the resolution of our measurement device. For this
Fig. 6. Example of image slice thresholded to isolate cracks and voids from solid.
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work, as a preliminary estimate we will assume that an upper bound error is on the same order as our lower bound error. Once the total void surface area is measured, we can calculate the change in void surface area between successive tomographic scans. This incremental change in surface area is the dAi term of Eq. (6). We assume for this work that the increase in void surface area is due exclusively to crack growth. We believe this to be a valid assumption. Although there is surely a change in pore volume due to the compressive deformation throughout the specimen, these are going to be extremely small compared to the increase in void volume due to cracking. 5. Experimental results and discussion Fracture energy was computed using Eq. (6) for several load increments in each of two dierent test specimens. A plot of fracture energy versus crack area is plotted in Fig. 7. Obviously, the fracture response is quite dierent in the two specimens although a qualitative inspection of the tomographic images showed no such dierences. However, what is quite remarkable is that in both cases fracture energy calculated using a completely dierent approach than has been done in the past led to values of Gc in quite close agreement with published results using for example notched beam specimens [22]. (See Ref. [3] for numerous other published values). The two results shown in Fig. 7 are interesting examples because they illustrate two dierent types of fracture behavior. Specimen 1 shows a relatively constant work of fracture, while specimen 2 varies, but ultimately increases with crack area. Specimen 1 represents the behavior we would expect to see from a
Fig. 7. Plot of calculated fracture energy versus crack area for two dierent specimens.
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true brittle material: a constant Gc : While specimen 2 represents behavior we would expect to see from a quasibrittle material: an increasing toughness. We refrain from using the term R-curve here because its de®nition based on a single critical crack, while we are dealing here with a branching multiple crack system. However, specimen 2 does demonstrate (ignoring the second data point) what we would normally think of as rising R-curve behavior. Since rising R-curves are a commonly accepted trait of quasibrittle materials, this result is perhaps less interesting than specimen 1, where a constant work of fracture was observed. The constant Gc observed in specimen 1 forces us to challenge our de®nition of a quasibrittle material to some extent. An inspection of the microtomographic images (for example Fig. 3), clearly indicates stable crack growth between the scans. Stable crack growth is not something we should expect to observe in a material of constant Gc (unless there is a falling strain energy release rate, which we will not consider here). Regardless, what we do have is a multicrack network. Therefore, while individual crack fronts may be aected by the usual crack stopping mechanisms, collectively they tend to average out due to the heterogeneous microstructure and all of its in¯uence. Now, back to whether we should accept our original hypothesis: the majority of the nonlinear eects we observe in concrete fracture are in fact not nonlinearities, but are rather due to the way we traditionally de®ne crack area. It seems we are not yet in a position to accept or reject because our sample number was small, and because the results were not necessarily consistent. Apart from fracture size-scale eects, our small specimen size is problematic due to the wide experimental scatter it is likely to produce. Such a small specimen is certainly going to be much more sensitive to defects than a larger specimen. Nevertheless, the observed constant Gc of specimen 1 does present the possibility that we could apply linear elastic fracture mechanics to cement-based materials if we use actual surface area to de®ne our crack rather than a planar projection or approximation.
6. Conclusions No de®nitive conclusions about the fracture properties of mortar are made here due to the relative infancy of the experimental technique, and due to the limited number of experiments conducted to date. The principal conclusions of our work to date are simply that X-ray microtomography can be a powerful tool for investigating fracture energy in complex materials such as concrete, and the threedimensional nature of the data we generate allows us to ask questions we were heretofore prevented from asking. We have shown that using an experimental setup where tomographic scans are made of a specimen under load, changes in internal damage can measured using three-dimensional image analysis techniques. When load and deformation data are available, we can make a work of fracture calculation based on the tortuous crack surfaces that arise in the fracture of heterogeneous materials. Finally we have shown that for a small mortar specimen loaded in compression, we can have a relatively constant work of fracture if we consider the total area of all the crack systems involved.
Acknowledgements This work is supported by the National Science Foundation (CMS-9733769) under a program directed by Dr. John Scalzi. Parts of this research was conducted at the National Synchrotron Light Source, Brookhaven National Laboratory, which is supported by the U.S. Department of Energy, Division of Materials Sciences and Division of Chemical Sciences (DOE contract number DE-AC0276CH00016).
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