Explanation of size effect in concrete fracture using non-uniform energy distribution

Materials and Structures/Mat6riaux et Constructions, Vol. 35, July 22, pp 326-331 Explanation of size effect in concrete fracture using non-uniform energy distribution K. Duan 1, X.-Z. Hu I and F. H. Wittmann 2 (1) Department of Mechanical and Materials Engineering, University of Western Australia, 35 Stirling Highway, Crawley, WA 69, Australia (2) Institute for Building Materials, Swiss Federal Institute of Technology, CH-893, Zi#ich, Switzerland Paper received: September 13, 21; Paper accepted: October 25, 21 A B S T R A C T R I~ S U M I~ A local fracture energy model originally proposed to explain the influence of fracture process zone (FPZ) on fracture energy of cementitious materials is further developed in this study. By assuming a bilinear distribution for the fracture energy distribution, the ligament-dependent fracture energy Gf is obtained. The analytical expression of GF contains two important parameters: the intrinsic size-independent fracture energy Gp and a reference ligament size al* which determines the intersection of the two linear fracture energy functions. It is shown that the ligament-dependent GF approaches the size-independent G r asymptotically. Asia result, G F can be determined from the ligament-dependent Gf results. It is also found that while the reference ligament size at* is influenced by the specimen geometry, size and loading conditions, the derived fracture energy G F is virtually constant. The present local fracture energy distribution model is also discussed and compared with the original local fracture energy model. Cette 4rude pr&ente un developpement plus approfondi d'un modhle de l'&ergie locale de rupture, originairement propos~ pour expliquer l' effet de la taille de la zone d' endommagement sur I'&ergie de rupture des mat&iaux a base de ciment. En supposant une &tribution bilin~aire de I'&ergie locale de rupture, on peut obtenir I'&ergie de rupture Gf d~pendante de la longueur du ligament. L' expression analytiqffe de I'&ergie de rupture Gf contient deux param~tres importants: l'&ergie de rupture GF, ind~pendante de la taille, et une valeur de r~&ence de la longueur du ligament, al*, qui d&ermine l'intersection de la distribution bilin&ire de l'&ergie locale de rupture. On montre que I'&ergie de rupture ddpendante du ligament G t- tend asymptotiquement vers la valeur de l'&ergie de rupture nogddpendante de la taille, GF. Ainsi, G v peut &e d~duite de l'~nergie d~pendante du ligament, Gf On a aussi remarqu~ que l'&ergie de rupture GF, ainsi d~dutte, demeure virtuellement constante bien que la longueur de r~f&ence du ligament, at*, puisse 8tre influence'e par la g&m&ie et la taille de l'~prouvette et les conditions de souicitations. Ce movie de dstribution d'&ergie locale de rupture est discut~ en d~tail dans cette contribution et est compar~ avec le modhle original de l'~nergie locale de rupture. 1. INTRODUCTION The size effect on fracture properties of concrete has been widely reported over the past four decades [1-17]. Numerous experiments have shown that for geometrically similar concrete specimens, increasing specimen size usually leads to increasing fracture energy and toughness, and decreasing strength, and that for specimens of identical size, fracture energy and toughness vary with different crack/width ratios. The size-dependence of fracture properties decreases with increasing ratio of specimen size to the maximum grain/aggregate size, and is negligible when specimen size is substantially larger than the maximum grain size [12-17]. Therefore, very large specimens, particularly for coarse grain concrete, are needed to measure the size-independent fracture properties. Unfortunately, the specimen size necessary to achieve the size-independent value is often beyond the capability of most laboratories. The size effect on fracture properties of concrete is also an important issue as reflected by the well-received fictitious crack model [9, 18-2]. According to the fictitious crack model, the specific fracture energy defined as energy necessary to create a unit crack area is consumed within the fracture process zone (FPZ), a region where tensile strain softening occurs [18-2]. The model, 1359-5997/2 9 RILEM 326

Duan, Hu, Wittmann although it significantly improved the understanding of concrete fracture, cannot provide a satisfactory explanation for size effects on fracture toughness and energy because a unique tensile softening curve and thus, a sizeindependent fracture energy is assumed for concrete. Getting used to a line crack model such as the fictitious crack model with a fixed tensile softening relation, the role of FPZ width in determining the tensile softening behavior may be overlooked. In fact, both experimental and theoretical studies have shown that FPZ in concrete has to have a certain width to allow various mechanisms such as multiple-cracking, aggregate-interlocking and interface grain bridging in the width direction to create strain-softening [1, 11, 19, 2, 35, 37]. There is no tensile softening if the FPZ width is zero. The first significant size effect model is Ba~ant's sizeeffect law (SEL), developed from a dimensional analysis of the fracture energy released during crack propagating in geometrically similar specimens [4-5]. According to the SEL, material strength depends on specimen size, and cannot be determined by a single strength criterion. For concrete, the linear elastic fracture mechanics (LEFM) criterion determines the strength of large specimens, and the maximum tensile stress condition dominates the failure of small specimens. Carpinteri and his colleagues analysed the fracta]ity of fracture surface and proposed a multi-fractal scaling law (MFSL) to explain the size effect on concrete fracture [21-22]. Hu and Wittmann examined the fracture of a large plate with an edge crack and pointed out that the failure transition from the strength-dominated failure to the LEFM-controlled fracture was due to the interactions between the crack, FPZ and front boundary, which leads to the common size effects [23-27]. They proposed an asymptotic approach to the strength-crack size relationship for the large plate. This asymptotic approach has been extended to concrete specimens with a finite size [28], which provides accurate predictions for the sizedependent fracture properties. The size effect on the specific fracture energy has also been related to the influence of specimen ligament size on FPZ [1-11]. Hu and Wittmann performed a saw-cutting experiment on mortar and showed that the development of FPZ in a wedge splitting specimen was severely limited when specimen size or ligament length was reduced [1, 11]. The small FPZ associated with small specimens or short ligaments led to a lower specific fracture energy. Considered the important role of varied FPZ size/width in fracture energy dissipation, a local fracture energy model was proposed, which assumed different fracture energy dissipations occurred at different positions along the crack path [1-11]. The non-constant "local fracture energy distribution" has been shown to be the main reason for the size effect in the RILEM defined specific fracture energy averaged over an entire fracture area [1-11]. The present paper aims at further extension of the local fracture energy model [1-11]. By using a bilinear function to approximate energy distributions along the ligament of a notched specimen, the relationship between the specific fracture energy and specimen geometry and dimensions is established. The derived analytical expression is then applied to published experimental data to study the size effects, and to work out the size-independent fracture energy. 2. FRACTURE ENERGY DEFINITIONS 2.1 Averaged and local fracture energy The widely used RILEM specific fracture energy Gf is defined as the energy necessary to create one unit projected crack area [29]. It is averaged over an entire fracture area, and has frequently been observed to be size/ligament dependeat because of its "averaged" nature. To avoid a W confusion in the following analysis, Gf is used if the RILEM specific fracture energy is size d@endent, and G E is used if the RILEM specific fracture energy is size-independent. G f is the asymptotic value of Gf when specimen size is very large. According to the RILEM recommendation [29], Gf for a specimen with an initial notch/crack a, width Wand thickness B is given by: G f(a) = 1 I Pal8 (W- a)b (1) where P is applied load and 8, displacement at loading point. Gf(a) is used here because the RILEM specific fracture energy can be ligament/crack-size dependent even for a fixed W. As most energy required for crack growth is consumed in FPZ, a region where concrete experiences tensile strain softening [18-2], the fracture energy Gf can also be determined by the area under the tensile softdning or cohesive stress-crack opening (%-w) curve, w c Gf = 1%dw (2) where w c is the critical crack opening. If a size/ligament dependent Gf(a) is observed, Equation (2) implies different tensile softening relation (%-w) may have to be considered along the crack path. Examining the definitions of Gf(a) or @ as given by Equations (1) and (2), the size/ligament effects on G((a)is not accidental. Gf(a)in Equation (1) is a very rough measurement of fracture energy as it is averaged over the entire fracture area. "Gf in Equation (2)" in fact is related to "local" tensile softening, i.e. the %-w relation in FPZ. If the ob-w relation does not vary along the crack path (i.e. "Gf in Equation (2) -- constant), a size independent'g; must be observed. Likewise, if the %-w relation varies along the crack path, a size/ligament dependent Gf(a) is inevitable. Can the %-w relation vary along a drack path? Both experimental observations and theoretical analysis show that the existence of tensile softening requires a certain FPZ width, and w c is directly proportional to the FPZ width [1, 11, 19, 2, 35]. A wider FPZ allows more extensive aggregate interlocking and multiple cracking, resulting in a bigger w c and higher Gf. A FPZ with a near zero width will have extremely 5mited aggregate interlocking and will only have 'single plane' cracking, resulting in a much reduced w c and thus smaller Gf The stress gradient and loading condition, crack-tip location, 327

Materials and Structures/Mat4riaux et Constructions, Vol. 35, July 22 the specimen size and the physical limitation of a remaining ligament can all influence the size/width of FPZ, which thus becomes position dependent. This location-dependent fracture energy because of the location-dependent FPZ is referred to as the local fracture energy, denoted as g~ Let x be the position parameter along the crack path, the local fracture energy &(x) can be related to the local critical crack opening wc~(x) and FPZ width by [1, 11]: w c(x ) oc FPZ width By integrating the local fracture energy&(x) over the entire fracture area, the RILEM specific frdcture energy Gf(a) as defined in Equation (1) can be rewritten as: m-a Gs(a )- (wl_a)! g/(x)dx (4) We should emphasize that the introduction of the local fracture energy &(x) is not contradictory to the RILEM definition of fracture energy. In fact, the local fracture energygf:f(x) represents an average of fracture energy over a small segment of a ligament while the RILEM fracture energy is averaged over the entire ligament. If&(x) is constant, it is equal to G F, and size/ligament effects do not exist. Ifgf(x) is not constant, size/ligament effects are inevitable gnd Gf(a) is only the average measurement of the local fracture energy distribution. 2.2 Boundary influence and inner/outer regions Fracture Mechanics of metals has been well studied and understood, which should shed some light on the size/ligament effects on concrete fracture. LEFM applies only if a crack-tip plastic zone is small (in comparison with specimen size) and away from all boundaries specified by the front and back surfaces, and the top and bottom surfaces [3]. The specimen has to be thick enough to minimize the influence of two side surfaces. During crack growth, the influence of the front and back surfaces will vary. LEFM is applicable in the inner region where the crack-tip plastic zone is away from front and back boundaries, and LEFM fails when the crack is approaching to the back surface or when the crack is in the boundary/outer region. Similarly, LEFM applies ira FPZ in a concrete specimen is small and away from the front and back surfaces. When the crack is approaching to the back surface during fracture, LEFM fails because the FPZ is not small in comparison with the remaining ligament and the FPZ development with crack growth is influenced by the back surface. Therefore, there are also inner and outer regions for a concrete specimen. Instead of emphasizing the fracture energy in the inner region, the 1KILEM defined specific fracture energy averaged over the entire fracture area includes the energy dissipations in both the inner and outer regions. By adopting such a definition, Gf can be size/ligament independent only if the inner region is dominant in comparison with the outer region. Under such a condition G r approaches to G F asymptotically with increasing sp&imen size as the maximum outer/boundary region has already been established and only the inner region is being increased. The only difference between metal and concrete is that in the outer region a metal specimen experiences large-scale plastic yielding (LEFM does not apply), and a concrete specimen while still fracturing in a brittle manner has an ever reducing FPZ when the crack is approaching to the back boundary (LEFM does not apply, because locally smeared&(x) is not constant). Knowing the regions where LEFM does and does not apply, we can work out the size-independent G F from the size-dependent Gf measurements. In the inner region, a constant local frdcture energy exists (gf = GF). In the outer region, the local fracture energy &(x) is location dependent. The size/ligament-depe~ndent RILEM specific fracture energy Gf(a) contains local energy dissipations in both the inner find outer regions. 2.3 Bilinear simplification of local fracture energy distribution The previous studies [1, 11, 31, 32] have shown that FPZ width decreases dramatically when the FPZ approaches to the back surface of a notched concrete specimen. The local fracture energy &(x) reflects this feature as shown in experiments [1, 11]. To characterize this feature and keep the modeling as simple as possible, a simple bilinear function consisting of a horizontal line with the value of G F for the inner region, and a descending line that reduces to zero at back surface for the outer region is used here to approximate the &(x) distribution as shown in Fig. 1. The intersectiofi of these two straight lines is referred to as the transition ligament length at* that indicates the transition ofgc from the inner to outer regions. Like FPZ, al* is infldenced by specimen geometry/size and loading condition. As shown in Fig. la, the gf(x) defined by the bilinear function can be written as: ( )/, *, (5) gf x = G~ W-a-x a I x>-w-a-a l for a specimen with a ligament size (W-a) larger than the transition ligament al*. For (W-a) < al*, the first function in Equation (5) disappears. Substitute Equation (5) into (4), the size-dependent specific fracture energy is obtained as: G s G~ 1- al W-a> a; W-a<_a; (6) 328

Duan, Hu, Wittmann g: GF 5O 9 a;/w=.22 ~2 8 1 * : ~' W J (a) (b) Fig. 1 - The distribution of fracture energy (Gfandgf) along the ligament of a fracture mechanics specimen. In a self-similar form, one has: x Gf fl -~ 1 l_-ta/w a;/w 1- a/w > a 7/W -GT- O-a/W) 1- a/w <- a 7 /W (6') The fracture energy Gf based on Equation (6) is schematically illustrated in Fig. la by a dashed horizontal line. The ligament effect is further illustrated in Fig. lb where the local fracture energy gf from Equation (5) and the specific fracture energy Gifrom Equation (6) are schematically illustrated. For a ~pecimen with a fixed W and loading condition, the measured Gf is decreased with an increasing initial crack size a (the local fracture energy distribution gf(x) is not influenced by the initial crack length), as observed in many experiments [e.g. 3, 1, 11]. 3. APPLICATIONS OF LOCAL FRACTURE ENERGY MODEL Equation (6) correlates the size-dependent fracture energy Gf and size-independent fracture energy Gl~, and therefore provides a means to estimate G F using sizedependent fracture energy G: results. Specimens of a fixed size but different initi:/1 crack/notch lengths are needed to establish the inner and outer regions, and the parameters G F and al* can be estimated. Fig. 2 displays the fracture energy data of a mortar with the maximum grain size of 1 mm using the wedge splitting test method [1, 11, 33, 34]. Firstly, an al*/wratio can be i W I.3 a; /W =/126 "~ t I I I I I.4.5.6.7.8.9 1 a/w Fig. 2 - The comparison of the predictions from Equation (6) and the best fit with the measured fracture energy data given in refs. [1, 11]. Table 1 - Estimated specific fracture energy G/: and ligament transition length al* from [1] by different pre-set al*/w at* (assumed) GF(N/m) al*/w (calculated) at* (mm).34 38.95.11 19.8.23 45.71.26 46..15 43.6.22 4. assumed so that Equation (6) can be applied to analyze the experimental data. After the curve fitting, the assumed and estimated a/*s are compared. Three different trials have been done and the results are listed in Table 1. The predictions using the parameters given in Table 1 are plotted in Fig. 2 together with the experimental results. The second set of data not only gives a good prediction, but also satisfies the pre-set curve-fitting assumption. The sizeindependent specific fracture energy from this analysis is 45.71 N/m, which is in a good agreement with 43.88 N/m given in the earlier reports [1, 11]. The transition ligament length predicted using this model is 46 ram. The fracture energy of a concrete with maximum grain size of 32 mm is plotted in Fig. 3 [12, 13]. The wedge splitting tests were completed using specimens of 5 x 5 x 1, 2 x 2 x 6 and 15 x 15 x 6 mm. The results of 5 x 5 x 1 are not plotted in Fig. 3 as the specimen thickness is different from the other two 12 1 8o ~4 2.3 15x15x6 r ~. ~ -% I I I I I I.4.5,6.7 1).8.9 1, ojw Fig. 3 - The comparison of the predictions from Equation (6) with the measured fracture energy data given in refs. [12, 13]. 329

Materials and Structures/Mat6riaux et Constructions, Vol. 35, July 22 1 8O 6 4 2.1 24 mmv/ 2 mm "/ 14 mm ~ I I I.2.3.4.5.6.7 a/w Fig. 4 - The comparison of the predictions from Equation (6) with the measured fracture energy data given in refs. [7, 8]. specimens and the thickness of all three specimens is less than the critical value that will eliminate thickness effect [13]. For these two smaller specimens, it can be assumed that the changes in fracture energy Gf due to limited thickness are identical so that Equation~ (6) is applicable without considering the thickness effect [36]. Applying Equation (6) to these two smaller specimens, the sizeindependent fracture energy G F is obtained as 12.1 and 15.8 N/m, and the transition ligament sizes, 3. and 32.5 ram, respectively for the specimens of 15 x 15 x 6 and 2 x 2 x 6. The predictions from these parameters are plotted in Fig. 3 together with the measured values, showing a good agreement. It should be noted here that the predicted G F is for the thickness of 6 mm only, and it could be higher for thicker specimens because thickness effect is not considered here. Nallathambi et al. investigated the influence of various factors including specimen size, crack length, span, aggregate texture and size, and water/cement ratio on fracture energy of concrete using three-point bending tests [7, 8]. Their experiments clearly showed that the ratio of beam depth W to span length L had a significant effect on fracture energy while other conditions are identical. Therefore, a set of fracture energy data of 2 mm mix with identical depth-to-span ratio of 1/6 are chosen to compare with Equation (6) as shown in Fig. 4. The size-independent fracture energy G F and transition ligament length at* calculated from the results of these four groups of specimens with depths of 14, 2, 24 and 3 mm are listed in Table 2. The predictions based on the parameters in Table 2 are also displayed in Fig. 4, and obviously, they are in a good agreement with the experimental data. The results in Table 2 are plotted in Fig. 5 to show the influence of specimen size on the fracture parameters in Equation (6). A constant fracture energy Gi~, 133. N/m on average, is obtained from Equation (6) although the transition ligament length ar is increased with the increasing specimen depth W. The change in at* with W reflects how a fully developed FPZ interacts with boundaries for give specimen size and loading condition, at* will only be stabilized for very large specimens as its trend in Fig. 5 suggested. The Gfresults in Fig. 4 also suggest the specimens are far too sn'fill for the inner region to be dominant and for the outer region to be fully developed. 16 ~12 z 8 4 1..., _G r_,....... I I I I 15 2 25 3 35 w (ram) 16 12 8 ~ Fig. 5 - The size-independent fracture energy G F and the transition ligament al* as function of specimen depth W, predicted using Equation (6). 4. DISCUSSION AND CONCLUSIONS It is realized that the RILEM defined fracture energy Gfis an average measurement over an entire fracture area. The tensile strain-softening relation used for a fictitious crack model should be related to "local" fracture features along the crack path. If the locally smeared fracture energy gr is always constant, a size independent G F must be obse'rved. If G r is size/ligament dependent, the gf distribution cannot bdconstant. It is also realized that LEFM must apply if a specimen is large enough and if FPZ is far away from any boundaries. Hence, an inner region where LEFM applies should be separated from the boundary/outer region where LEFM does not apply. A simplified bilmear function has been used to describe the local fracture energy gf distributions in the inner and outer regions of a concrete specimen, leading to the derivation of Equation (6). The usefulness of the bilinear assumption for the local fi:acture energy distributions is that a constant fracture energy Gp in the inner region can be determined even though the RILEM defined fracture energy G cis severely size-dependent. To do that, specimens of a fixea size but with different crack/notch length are required. This is different from most size-effect models that require geometrically similar specimens of different sizes [e.g. 4, 5]. The averaged G o which contains the energy dissipations in both the inner ~d outer/boundary regions, can only be size independent if the inner region where LEFM applies is dominant. The separation of the inner and outer region indicated by the transition ligament length al* can vary with specimen size and loading condition as it reflects how FPZ is influenced by these conditions. 4 Table 2 - Estimated specific fracture energy G F and ligament transition length al* from [7] for the beams with different depths W(mm) GF(N/m) al* (ram) 14 129.5 8.6 2 139.6 16.7 24 134.1 117.9 3 129. 131.4 33

Duan, Hu, Wittmann The boundary effect on Gs has also been studied through the interaction betweefi FPZ and front surface [23-28]. In this case, the crack length is also the distance of FPZ to the front boundary. A reference crack a* (similar to at* ) defined by the intersection of the strength and toughness criteria was used to separate the inner and outer/boundary regions. An asymptotic function was then proposed to approximate the transition from the strength to toughness criterion. The influence of such a transition on Gfis shown to be equivalent to Equation (6). T hat is the same conclusion can be obtained on boundary effects from either the front or back surface analysis although they are based on two different approaches. ACKNOWLEDGEMENT The financial support from the Australian Research Council is acknowledged. REFERENCES [1] Kaplan, M. F., 'Crack propagation and the fracture of concrete', ACIJouma158 (1961) 591-61. [2] Walsh, P. 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