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FRACTURE ANALYSIS OF MORTAR-AGGREGATE INTERFACES IN CONCRETE
By Kwang Myong Lee,1 Student Member, ASCE, Oral Buyukozturk,2
Member, ASCE, and Ayad Chimera3
ABSTRACT: Deformation and failure behavior of concrete is influenced by the characteristics of mortar-aggregate interfaces. Recently, the interest in the study of the role of mortar-aggregate interfaces in material behavior has increased particularly because of the need for the development of high-performance cementitious materials. Presently, there is a need for the development of a fracture mechanics-based approach for the characterization of interface fracture properties. In this paper, novel fracture models arc presented to quantify the fracture properties of the mortar-aggregate interfaces. Sandwich specimens are used to develop the fracture toughness curves of the mortar-aggregate interfaces in concrete. Also, a criterion based on energy release-rate concepts is considered to study the crack-penetration-versus-crack-deflection scenarios in the interface regions of concrete modeled as a two-phase composite. The tests indicate that the interface fracture toughness is markedly increased with the increase in the effects of shear loading relative to that of tensile loading. The interface fracture toughness values determined from the sandwich models are correlated with the results from a numerical-experimental study of a concrete composite consisting of an aggregate inclusion embedded in a mortar matrix. Finally, recommendations for further study are given.
INTRODUCTION
Over the past 20 years, considerable research has been conducted to study the microcrack development, the nonlinear deformation behavior and the failure mechanisms of concrete. It has been generally established that in normal-strength concrete the development of bond cracks at the mortar-aggregate interfaces plays a significant role in the inelastic deformation behavior and that final failure occurs through the formation of continuous cracks in mortar, bridging the bond cracks (Hsu et al. 1963; Shah and Winter 1966; Buyukozturk et al. 1971, 1972; Liu et al. 1972; Struble et al. 1980). Currently, little information is available on the nature of progressive failure of high-strength concrete in which cracks through aggregates were observed by some researchers (Gerstle 1979; Carrasquillo et al. 1981; Zaitsev 1983), indicating a less pronounced effect of crack arrest by the aggregates. The interest in the study of mortar-aggregate interfaces has recently increased, in particular, in view of the efforts to develop high-performance concrete materials with improved mechanical properties and durability.
Concrete is a heterogeneous material, and its behavior is complicated by the interaction of many internal elements that constitute the material. Therefore, for any rigorous study of the effects of internal events on the overall mechanical behavior of concrete, idealizations must be made and relatively simple models that capture the essential elements of the behavior must be
'Grad. Student, Dept. of Civ. Engrg., Massachusetts Inst, of Tech., Cambridge, MA 02139.
2Prof., Dept. of Civ. Engrg., Massachusetts Inst, of Tech., Cambridge, MA. -'Struct. Engr., T. Y. Lin Int. , Alexandria, V A 22304. Note. Discussion open until March 1,1993. To extend the closing date one month,
a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on December 12, 1991. This paper is part of the Journal of Engineering Mechanics, Vol. 118, No. 10, October, 1992. ©ASCE, ISSN 0733-9399/92/0010-2031/$l.00 + $.15 per page. Paper No. 3074.
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constructed. Buyukozturk, et al. (1971, 1972) modeled concrete as a two-phase composite consisting of mortar and aggregate inclusions and investigated the cracking problem through laboratory testing of physical models and numerical analysis using the finite element method. Recently, following the work of Buyukozturk et al. (1971, 1972), Yamaguchi and Chen (1991) predicted the propagation of microcracks in two-phase concrete models by the finite element analysis incorporating the smeared cracking model for mortar, and using interface finite elements for bond coupled with the generalized plasticity concept. In these studies, a Mohr-Coulomb-type failure criterion was employed for cracking at the interface regions. The approach developed by these previous studies led to some very useful information with respect to the understanding of global concrete behavior as affected by bond cracking. However, the results obtained in this way are limited due to the phenomenological nature in which the interface-bond property was assumed, and the bond-cracking criterion was established. Hence, there is a need to develop methodologies for defining and quantifying explicitly the mortar-aggregate interface characteristics such that fundamental knowledge essential in engineering high-performance concrete materials can be derived. Such a methodology would involve both testing of physical laboratory models and employment of interface fracture mechanics concepts.
An early experimental study on the measurement of the fracture toughness of mortar-aggregate interfaces in concrete was performed by Hillemeier and Hilsdorf (1977). They reported limited test results that involved mode-I loading conditions only. Some experimental problems associated with their tests were discussed by Ziegeldorf (1983). However, cracking of the mortar-aggregate interfaces involves mixed-mode fracture effects due to the differences in the properties of individual materials when concrete is modeled as a two-phase material and due to loading conditions. Furthermore, criteria are needed for the crack, once initiated in one of those materials in the interface region, to propagate along the interface versus to penetrate into the second material, since this is essential to optimize the material behavior.
In this paper, a combined analytical-experimental methodology is given for the assessment of mortar-aggregate interface fracture toughness. For this, sandwich specimens are used and cracking at the interfaces is investigated through interface fracture mechanics concepts, which are briefly discussed in the beginning section of this paper. A criterion based on energy release-rate considerations is discussed for the study of crack propagation in interface regions. Finally, a study is presented on composite models consisting of mortar and aggregate inclusions, to investigate the interface-crack propagation. Numerical analyses and experimental work performed on these models are briefly described.
BASIC INTERFACE FRACTURE MECHANICS CONCEPTS FOR BIMATERIALS
The objective of interface fracture mechanics is to define a property, i.e., toughness, that characterizes the fracture resistance of interfaces. Solutions to bimaterial interface-crack problems were presented in the earliest papers by England (1965), Erdogan (1965), and Rice and Sih (1965). Williams (1959) investigated the singular crack-tip fields. Recently, a number of studies on the interface fracture mechanics of bimaterials have been made by Rice (1988), Suo (1989), and Hutchinson (1990).
A bimaterial is a composite of two homogeneous materials with continuity of traction and displacement across interfaces maintained. Consider a semi-
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#1 r
Xl
§2
FIG. 1. Geometry of Interface Crack
infinite free crack lying along the interface between two homogeneous isotropic half-planes, with material 1 above and material 2 below (Fig. 1). Here, plane strain deformation in isotropic bimaterial is considered. With the interface on the x raxis, let Eu \ilf and v1 be the Young's modulus, shear modulus, and Poisson's ratio, respectively, of material 1. Similar quantities, E2, |x2, and v2, for material 2 are also defined. Dundurs (1969) has observed that a wide class of plane problems of elasticity for bimaterials depends on two nondimensional combinations of the elastic moduli. For plane strain, the moduli mismatch parameters of Dundurs are
__ £i - E2 _ 1 M l - 2v2) - fe(l - 2vJ a E, + E2'
P 2 ^ ( 1 - v2) + ^ ( 1 - Vl) {i)
where E = £7(1 — v2) = 2|x/(l — v). The parameter a measures the relative stiffness of the two materials. The parameter (3 causes the linear crack-tip stress and displacement fields to oscillate (Rice 1988). Note that both a and (3 vanish when dissimilarity between the elastic properties of the materials is absent, and a and (3 change signs when the materials are switched. For convenience, an average stiffness, E* is defined as
1 1 / 1 1 Y^2\T, + W2J W
For the plane problems, the normal and shear stresses of the singular field acting on the interface at distance r ahead of the tip can be written in the compact complex form
is
cr22 + ia12 = —=== riB (3)
where K = Kx + iK2 = the stress intensity factor at the interface, / = ( —1)1/2; and the oscillation index e depends on p according to
e = _L l n ( i ^ | ) (4)
The parameter 6 brings in some complications that are not present in the elastic fracture mechanics of homogeneous solids. The associated crack-face displacements a distance r behind the crack tip, 8, = ut(r, 0 = IT) - u^r, 6 = — ir), are given by
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82 + ** = £*(1 + 2f)cosh OT ( V ^ J r'E '-(5)
For the case when (3 =£ 0 (i.e., E + 0), one can define
KLic = \K\e^ (6)
Note that \Lie\ = 1,̂ \KLie\ = \K\, and ij) = the real phase angle of the complex quantity KLie. When p = 0, Ku and K2 measure the normal and shear traction singularities on the interface ahead of the crack tip with the standard definition for the stress intensity factors. With (3 = 0, the phase angle 41 is defined as
* = ton-1© ••• (?) Here, ij/ measures the relative proportion of the effect of mode II to mode I on the interface. The case in which i|i = 0° corresponds to pure mode I, and <Ji = 90° corresponds to pure mode-II conditions.
The energy release rate G per unit length of extension of the crack at an interface is related to the stress intensity factors with an Irwin-type relation
\K\2
E* C0sh2TT8
where \K\2 = K\ + K\; and cosh2irE = 1/(1 - 02). If both ^ a n d G are defined, one can establish a fracture toughness curve, or Gc — ij/ curve, for an interface. In linear elastic fracture mechanics, Gc — \\> curve is the property of the given interface in the sense that, in principle, it is independent of the specimen geometry and the loading system. However, in general, the toughness curve is clearly dependent on the nature of the interface such as the roughness of the free surfaces before bonding and the bonding history, and the testing environment such as the temperature.
For any interfacial crack problem, the complex K will have the form of
KU" = YTVZe1* (9)
where T = an applied traction on the structure; L = a relevant length describing the geometry; Y = a dimensionless real positive number; and \\i = the phase angle of KLie. Both Y and v|i depend on the geometric and loading details of the structure. The effort to establish the relationship between the applied load and K given in (9) is referred to as a K-calibration for the structure (Wang and Suo 1990). With a ^-calibration for a given structure, the loading amplitude G can be calculated from (8). From (6) and (9), the loading phase angle i}» may be calibrated by
4. = * + e In (j) (10)
where L = the length used in the iC-calibration; and L = the fixed length, used in defining the loading phase angleifi. Here, the selection of L is somewhat arbitrary (Rice 1988), as long as a fixed length is used for reporting data in conjunction with the toughness curve.
The preceding discussion provides a theoretical insight into the interface fracture problem and defines the major parameters involved. For a quan-
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titative assessment of the fracture toughness of a given interface, the development of appropriate physical models is necessary.
ASSESSMENT OF INTERFACE FRACTURE TOUGHNESS
In the study of concrete fracture, characterization of the interfacial fracture toughness as a property of the mortar-aggregate interface is essential. Material data on the relative magnitude of the fracture properties of mortar and aggregate, and the characteristics of the mortar-aggregate interface play a major role in studying the various scenarios of crack growth and its effects on durability and failure mode of the composite material. While various methods and standards exist for the measurement of fracture toughnesses of the constituent materials such as mortar and aggregate, at present no established method is present for the characterization of the mortar-aggregate interface.
In what follows, we introduce the sandwich model for fracture characterization of the interfaces in concrete composite systems. This will be followed by the description of the specimens used in characterizing the fracture toughness properties of mortar-aggregate interfaces in concrete, i.e., for the investigation of fracture energy trends at these interfaces. The combined analytical-experimental methodology described herein represents an application of the interface fracture mechanics concepts given in the preceding section.
Interface Crack Model: Sandwiches An interface-crack model including a thin layer of material 2 sandwiched
in a homogeneous body of material 1 is shown in Fig. 2. Each material is assumed to be isotropic and linearly elastic. Here, mortar and aggregate are referred to as material 1 and 2, respectively. The crack lies along one of the interfaces coincident with the x raxis with the tip at the origin. If the thickness of the sandwich layer h is small compared to the crack length and to all other relevant in-plane length quantities, a universal asymptotic relation exists between the interface intensity factors, Kt and K2, and the stress intensity factors, Kx and Ku, for the homogeneous problem (Suo and Hutchinson 1989).
The far field for the asymptotic problem in Fig. 2 is characterized by mode-I and mode-II stress intensity factors, i.e., Kx and ^ n , respectively, for the homogeneous body of material 1 in the absence of the middle layer. Then, with K°° = Ki + iKn as apparent stress intensity factors, the traction at distance r far ahead of the crack tip is given by
h I • # 2
FIG. 2. Interface Crack Problem in Sandwiched Layer Model
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(11)
The energy release rate in the far field is
G = i | f f f (12)
where Ex is defined for material 1 and |^°°|2 = Kf + K&. By elementary energy arguments, or by application of the /-integral, the
energy release rate in (8) and (12) must be equal, and, thus, the asymptotic relation between the interface stress intensities and the applied stress intensities is obtained as
Kh" = / * _ " XV<--» (13)
where \hie\ — 1; and the real function w(a, p) = the shift in phase of the interface stress intensity factors relative to the applied stress intensity factors. Here, one also defines
Khh = \K\e^\ K~ = \K'\e^ (14)
where i|i* = the real phase angle of Khh; and § = t a n - 1 ^ / ^ ) . Then, the universal relation shown in (13) can be expressed as
V = <f> + w(a, 0) (15)
Numerical values of « corresponding to the different Dundurs' (1969) parameters, which show a weak dependence on p values, are provided by Suo and Hutchinson (1989).
Sandwich Specimens In this study, two types of sandwich specimens are used as physical lab
oratory models to assess the mortar-aggregate interface toughness. These are the sandwiched beam specimen and the sandwiched Brazilian disk specimen, from which the interface fracture toughness curves can be established in the full range of mixed-mode effects. The universal relation given in (13) may be applied to these sandwich specimens. Residual stresses in the layer do not contribute to K in the sandwich specimen, and, in calibrating such a specimen, one needs to take account of the external loading only. Then, the apparent stress intensity factors, Kt and Ku, are calculated from the measured failure load as if the specimen were homogeneous. The actual interface stress-intensity factors are calculated from the apparent stress-intensity factors through the universal relationship between them, given in (13).
First, consider the four-point bending specimen with a sandwiched aggregate layer shown in Fig. 3. To apply the universal relation, the thickness h should be small compared with the crack length. For this four-point pure bending specimen, the apparent intensity factor, Ku for the homogeneous body is simply calculated by the finite element scheme or by the following equation (Tada et al. 1985):
Ki = / I ( J , V O T (16)
where a r = 6M/bd2; a = the crack length; M = the applied moment; b =
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mortar matrix
• aggregate layer
width : b
FIG. 3. Sandwiched Four-Point Bending Specimen
aggregate layer
thickness: t
FIG. 4. Sandwiched Brazilian Disk Specimen
the width; d = the height; and/i = a geometrical correction factor for the four-point pure bending specimen. From (13), the interface stress-intensity factors are given by
K = Ki + iK2 = 1 - p=
k-'^f^rY (17)
When h is very thin, the energy release rate can be calculated from (12). In fact, in a four-point bending specimen with a sandwich layer, an interface crack tip is not in mode-I state. However, since the shift to is not large (Suo and Hutchinson 1989), this specimen can be considered to be essentially in mode I.
Next, consider the Brazilian disk specimen shown in Fig. 4, often called a Brazil-nut specimen consisting of a thin layer of aggregate bonded between two halves of mortars. The Brazilian disk specimen has been used for the mixed-mode fracture testing of polycrystalline ceramics (Singh and Shetty 1989) and for the fracture toughness testing of steel-epoxy interface (Wang and Suo 1990). It was also suggested by Ojdrovic and Petroski (1987) for use in measuring the fracture toughness of concrete. Stress-intensity factors for the inclined cracks in the Brazilian disk specimens can be calculated by the finite element scheme or the series solutions given by Atkinson et al. (1982). In general, mode-I and mode-II stress-intensity factors in the homogeneous body are obtained from the initial precracked length a, the fracture load P, and the crack inclination angle 0, as
PVa PVa
*' = V ^ ' *«= ±V^2 (18)
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where the plus sign in the equation for Kn is for tip A, and the minus sign for tip B as shown in Fig. 4. R and t are the disk radius and thickness, respectively. N1 and N2 are nondimensional coefficients that are functions of relative crack length (a/R) and the angle of the inclination (6,). In the sandwiched Brazilian disk specimen mixed-mode stress states ranging from pure mode I to pure mode II can be achieved by selecting the angle of inclination of the central through-crack with respect to the diametral line of compression loading.
By substituting Kt and Kn given in (18) into (13), the interface fracture intensity factors Kt and K2 are calculated. From (12), the energy release rate is given by
G = ^ ^ + N^ <19>
where Ex is defined for mortar. Let L = h and <j> = ± tan ' 1 (N2/N1) and by substituting (15) into (10), the interfacial loading phases at tips A and B are given by
4. = i t a n - 1 (^J + a, + E In (~J (20)
Thus, by measuring the critical loadAP and the inclination angle 9, of the specimen, and by specifying a length L, one can calculate G and \\> according to (19) and (20). When the crack is trapped at the interface, the fracture toughness of the interface can be measured as a function of ij;, as described previously. Otherwise, the crack moves out of the interface, in which case, the measured energy refers either to the mortar or to the aggregate.
Testing of Sandwich Specimens To generate the fracture toughness curves of the mortar-aggregate inter
faces, two types of sandwich specimens presented in the preceding section were tested. These include the sandwiched four-point bending specimen for mode-I loading test and the sandwiched Brazilian disk specimens for the mixed-mode loading test. The dimensions of the sandwiched beam specimen were 152.0 mm (length) x 50.8 mm (height) x 38.1 mm (thickness). The radius (R) and the thickness (t) of the sandwiched Brazilian disk specimen were 38.1 mm and 25.4 mm, respectively. The thickness of the aggregate layer, h, was 2.54 ± 0.127 mm for both specimens. The relative crack size (a/R) in the Brazilian disk specimen was fixed to be 0.25 and the relative crack size (aid) in the beam specimen was 0.375.
One mortar mix with an average 28-day compressive strength of 42.5 MPa, and two types of aggregates, granite and limestone, were used. The ratio of cement:sand:water in the mortar mix was 1:2:0.5, by weight. Type-I Portland cement with no admixtures was used. Sands passing sieve number 8 were used. Granite was selected as a strong aggregate, limestone as a weak one. From each batch of mortar, sandwiched beam and Brazilian disk specimens were cast along with mortar cylinders of diameter 76.2 mm and length 152.0 mm for compressive strength measurements. The surfaces of aggregate layers were adjusted to have the same roughness. To make a sharp precrack, a notch plate made of thin plastic with the thickness of 0.1 mm was attached to one side of the aggregate layer using paraffin.
The specimens, after pouring the fresh mortar into the molds, were covered with plastic for 24 hr. They were placed in water, after the notch plates
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TABLE 1. Material Properties for Mortar, Granite, and Limestone
Materials 0)
Mortar Granite Limestone
(MPa) (2)
42.5 140.1 57.5
Properties
E (GPa)
(3)
27.8 55.3 34.5
(J/m2) (4)
10.3 17.5 11.8
V
(5)
0.22 0.16 0.18
TABLE 2. a, 0, E, and <•> Values for Two Bimaterial Combinations
Mortar/aggregate combinations
0) Mortar-granite Mortar-limestone
a (2)
-0.320 -0.099
Bimaterial Parameters
0 (3)
-0.099 -0.021
E
(4)
0.032 0.007
<o(°)
(5)
2.8 0.8
were removed. All specimens were tested after 28 days of curing. Average material properties of the tested materials are summarized in Table 1. Table 2 shows Dundurs' parameters a and p, oscillation index e, and shift angle in a sandwich specimen co for the two bimaterial combinations, e.g., mortar and granite, and mortar and limestone.
Diametral compression tests on the Brazilian disk specimens and four-point bending tests on the beam specimens were performed using an Instron machine with a displacement control. Loading rate was 0.00127 mm/s. The inclination angle (9,) of the Brazilian disk specimen was adjusted ranging from 7° (<() = 28.8°) to 25° (<J> = 83.3°) with the deviation of ±0.5°, to achieve the mixed-mode loading condition. During the testing, the load and displacement signals from the Instron were recorded through a Fluke data-acquisition system connected to an IBM PC.
Fig. 5 shows a typical load versus load-line displacement curve for the sandwiched Brazilian disk specimens. Little microcracking was detected, and cracks advanced very rapidly. Fig. 6(a) shows the observed failure mode of the beam specimens. Fig. 6(b) shows the six different failure types observed for the Brazilian disk specimens. When the loading phase is small, most specimens failed in type 1, although some specimens showed type 2 or 3 failure. In type-2 failure, after interface cracking started at tip B, at tip A the crack ran along the interface for a short distance, and then kinked to the other interface. When the loading phase is large (<j» > 70°), some specimens failed in type 4, showing mortar cracking at tip A and aggregate cracking at tip B, in which case, the measured values would be interpreted as lower-bound values of interface fracture toughnesses. Occasionally, type-5 failure occurred when the aggregate was limestone, and type-6 failure occurred when the aggregate was granite.
By using the measured values of the critical load, P, the fracture energies of the two mortar-aggregate interfaces were calculated. For the sandwiched Brazilian disk specimens, the loading phase angles were calculated from (20). The fixed length L was selected to be 2.54 mm, the thickness of the aggregate layer. For the mortar-granite system, the total phase shift is about
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1.5
1.25
1
I 0.75
I 0.5
0.25
0 0 0.002 0.004 0.006 0.008 0.01 0.012
Displacement (in)
FIG. 5. A Typical Load versus Displacement Curve for Sandwiched Specimens
•
r , | , , , | , , ,
O 6 6j= 10 I
/ -^
\^/
<
;
;
; :
|
':
(a)
Type 1 Type 2
Type 4
Type 3
(D ® QD FIG. 6. Failure Types for Sandwiched Specimens: (a) Failure Type of Beam Specimens; and (b) Failure Types of Brazilian Disk Specimens
2.8°. Almost no phase shift is required for the mortar-limestone system. The interfacial toughness curves for the two mortar-aggregate combinations are plotted in Fig. 7. It is observed from Fig. 7 that the fracture energy markedly increases as the loading phase increases. For the mortar-granite interface [Fig. 7(a)] the fracture toughness for \\i = 0° is 1.2 J/m2, while it is 7.0 J/m2 for ij» = 45° and 18.0 J/m2 for <|< = 70°. As shown in Fig. 7(b), the fracture toughness curve for the mortar-limestone interface shows a similar trend as that for the mortar-granite interface.
For the sandwich specimen with a mortar-aggregate interface, the applicability of linear elastic fracture mechanics (LEFM) has not been theoretically established. The small-scale yielding condition at the crack tip should be satisfied. However, considering the load-displacement curve shown in Fig. 5, it can be assumed that the applicability of the LEFM to the fracture analysis of mortar-aggregate interfaces is not unrealistic. This point, together with that of the possible dependence of the interface fracture toughness on size, should be further investigated.
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^ 40 C— I _ — ?—'—I—'—T~~*—I—•—I—r t\j ! ! ! !
0 10 20 30 40 50 60 70 80 90 Phase Angle y (degree)
4 0 L i | • | i | i | i | i | i
0 10 20 30 40 50 60 70 80 90 Phase Angle \p (degree)
FIG. 7. Fracture Toughness Curves for Tested Mortar-Aggregate Interfaces: (a) Mortar-Granite Interface; and (b) Mortar-Limestone Interface
CRACK PROPAGATION AT MORTAR-AGGREGATE INTERFACE
In elastically homogeneous brittle solids, cracks are generally found to follow a local trajectory for which Ku = 0. This crack-path criterion is clearly not valid when the crack advances at an interface because, in this case, the consideration of the relative magnitudes of the fracture energy between the interface and those of the aggregate or the mortar are also involved. In this section, the problem of interfacial crack propagation is studied by the use of combined numerical-experimental approach.
Crack Penetration versus Deflection at an Interface In concrete, a crack impinging a mortar-aggregate interface may advance
by either penetrating into the aggregate or deflecting along the interface. Fig. 8 shows the deflection and penetration configurations at the interface. Let T, be the toughness of the interface as a function of 4< and let I \ be the mode-I toughness of material 1. The impinging crack is likely to be deflected if
I \ G™ K '
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y A
#1
UiilL
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#2 // #2
(a) (b)
FIG. 8. Crack Geometry: (a) Crack penetration; and (b) Crack Deflection
Thickness: 25.4
Thickness: 25.4
FIG. 9. Two-Phase Concrete Composite Models: (a) Model 1 ; and (b) Model 2
where I \ and T, = material properties, which can be measured by fracture testing; Gd = the energy release rate of the deflected crack; and G™ax = the maximum energy release rate of the penetrated crack. When in Fig. 8, the hitting angle y2 is 90°, the penetration angle yu which maximizes G„, is 90°. For this case will be denoted by Gp for simplification. For complex geometries, the ratio GJGP can be calculated using a numerical analysis scheme, as described in the next section. He and Hutchinson (1989) computed the ratio GJGP by solving the semi-infinite crack problem.
Application: Two-Phase Composite Models Composite models shown in Fig. 9 are developed to study the crack
penetration versus deflection at interface. The model consisting of mortar and a slab inclusion is referred to as model 1, shown in Fig. 9(a), and the model with a circular inclusion is referred to as model 2, shown in Fig. 9(b). A precrack extending from the top to the aggregate is introduced in each model that is subjected to wedge-type loading. One can achieve the different hitting angle in model 1 by inclining the aggregate inclusion embedded in the model. The energy release rates, Gd and Gp in (21), can be computed
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by the finite element analysis using the LEFM approach. The energy release rate is defined as the potential energy (II) change per unit crack advance:
Crack propagation is simulated by analyzing three different cases of the crack position when the hitting angle is 90°. In the first case, the crack tip is assumed to be in contact with the aggregate. In the second, the crack tip is assumed to have penetrated a small distance (Aa) through the aggregate. Finally, the crack tip is assumed to have deflected at the interface a small distance (Aa). For all three cases, the load-versus-displacement curves are obtained from finite element analysis. The energy release rates, Gp and Gd, are determined by computing the difference in the work done by the applied loads between any two consecutive positions of the crack tip, divided by the area of crack extension. This difference represents the energy release rate for the crack to extend from the first position to the next one.
A finite element program was used for computing Gd and Gp for model 1. The aggregate and the mortar matrix that constitute the model were assumed to be linear and elastic materials. The finite element meshes were generated using two-dimensional isoparametric eight-node elements. Loading configuration used in the finite element model is shown in Fig. 9(a). The crack increment Aa was 1.27 mm for the models of both the crack penetration and crack deflection. Five different mortar-aggregate combinations have been considered in the finite element analysis. The Young's modulus of the mortar matrix was taken to be 28.0 GPa, and that of the aggregate was varied between 28.0 GPa and 140.0 GPa. Poisson's ratios were 0.22 for both the matrix and the aggregate. Thus, a values considered in the analysis were 0.0, 0.2, 0.33, 0.5, and 0.66.
For each case, three separate runs corresponding to the three crack tip positions were performed. From each run, the horizontal displacements of the points where the loads were applied were obtained, and the values of Gp and Gd, and the ratio GJGP were computed. The computed ratio GJ G„ and the corresponding a and |3 values are shown in Table 3. Comparison of these results with those found by He and Hutchinson (1989) for the semi-infinite problem, is made in Fig. 10. He and Hutchinson (1989) found that with a = 0.0 and y2
= 90°, GJGp is equal to approximately 1/4, meaning that the crack will deflect if the interface toughness is less than a quarter of the toughness of the material ahead of the crack. It is seen from Fig. 10
TABLE 3. GJGP Values by Finite Element Analysis and He and Hutchinson (1989) with y2 = 90°
Case (1) 1 2 3 4 5
E, (GPa)
(2)
28.0 42.0 56.0 84.0
140.0
Vl
(3)
0.22 0.22 0.22 0.22 0.22
E2
(GPa) (4)
28.0 28.0 28.0 28.0 28.0
v2
(5) 0.22 0.22 0.22 0.22 0.22
a (6)
0.0 0.20 0.33 0.50 0.66
P (7)
0.0 0.07 0.11 0.18 0.24
Finite element analysis
(8)
0.38 0.41 0.44 0.49 0.59
He and Hutchinson
(1989) (p = 0.0)
(9) 0.25 0.30 0.35 0.47 0.65
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-
:
r~""
- r - i i , - r -r• , - |
• Finite Element Analysis -«^> . - - He and Hutchinson (1989)
--.-." .--^*
•
.» :
• •
0 0.2 0.4 0.6 0.8 a
FIG. 10. GJGP versus a with y2 - 90°
that the computed values by the proposed method in this paper are somewhat higher than those by He and Hutchinson (1989) when a is less than 0.5, although both predictions show a consistent increase of GJG with an increase in a. The variation between the two curves can be attributed to the differences in the initial assumptions adopted for geometry and loading conditions. The assumption made by He and Hutchinson (1989) in their analysis for computing the ratio GJGp is that the two materials constituting the interface form two semi-infinite planes that meet at the interface. Numerical analysis reported herein accounts for the finite geometry and the (3 effect.
Model 2, shown in Fig. 9(b), was tested to investigate the crack propagation at the mortar-aggregate interface, i.e., the crack penetration versus crack deflection. This specimen consisted of mortar block with an embedded aggregate inclusion. The dimensions of the block were 76.2 mm (width) x 101.6 mm (height) x 25.4 mm (thickness). The diameter of the aggregate inclusion was 38.1 mm. The precrack length, a, was 31.75 mm. Mortar had the same properties as those for that used in the sandwich specimens, and granite was chosen for making the aggregate inclusion. For the granite-mortar combination, the mismatch parameter a is 0.32 as given in Table 2.
The wedge-loading tests of the specimens were performed using an Instron machine with a displacement control at a loading rate of 0.002 mm/s. Load and displacement values were recorded. Fig. 9(b) shows a failed specimen with a maximum load 8.45 kN. It was observed that in the wedge specimen embedded with a granite inclusion, the crack went around the granite, i.e., the interface cracking occurred. When the hitting angle is 90° and a is 0.32, the phase angle <{/ (with the fixed length L = 2.54 mm) for the deflected crack is predicted to be 35°. The interface fracture toughness, T,(35) can be semiempirically estimated by the criterion in (21) as
r , ( 3 5 ) < r g ^ (23) p
where the mode-I fracture toughness of granite, Tg, is 17.5 J/m2 from Table 1. For the tested model, the ratio GJGP is 0.43 as predicted by finite element analysis. By substituting Tg and the GdIGp values into (23), r,(35) is estimated to be less than 7.5 J/m2. From Fig. 7(A), T;(35) for the mortar-granite interface is approximately 5.0 J/m2, and is less than 7.5 J/m2, as stated by
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(23). Thus, it can be concluded that the established interface fracture tough-' ness values are consistent with the findings of the tested composite model, and the cracking criterion given by (21) is valid providing a reasonable range of interface fracture toughness values.
SUMMARY AND CONCLUSION
In this paper, novel fracture specimens are presented and a combined analytical-experimental methodology is described to quantify the fracture properties of mortar-aggregate interfaces. Sandwich specimens are proposed and used to measure the fracture toughness of mortar-aggregate interfaces in concrete under mixed-mode loading conditions involving tensile and shear loadings. A study is then presented on the two-phase composite models of concrete, consisting of a mortar matrix and an aggregate inclusion, to investigate the crack propagation in the interface regions. A criterion based on energy release rate is considered to study the crack-penetration-versus-crack-deflection scenarios at the interfaces. The tests indicate that the interface fracture toughness is markedly increased with the increase in the effects of the shear loading relative to that of the tensile loading. The interface fracture toughness values determined from the sandwich models are compared with the results from a limited numerical-experimental study of the concrete composite models, and consistent results are found.
It is concluded that the described fracture mechanics-based methodology can be applied to the study of the mortar-aggregate interface behavior of concrete composites. The present approach is based on linear elastic interface fracture mechanics concepts and is considered to be appropriate due to the observed linear elastic behavior and brittle failure of the interfaces. This aspect needs to be further studied. Furthermore, experimental work should be extended to study the effects of roughness variations of aggregate surfaces, the size effects of the test specimens, and the loading rate effects. Finally, more comprehensive composite models for better representation of the concrete cracking behavior should be developed.
ACKNOWLEDGMENT
Support of this work was provided by the National Science Foundation (NSF) through Grant number MSM-9016814. The cognizant NSF program official was K. P. Chong; his support is gratefully acknowledged. The writers would like to thank J. W. Hutchinson of Harvard University, and C. K. Y. Leung of Massachusetts Institute of Technology (MIT) for their valuable comments. Appreciation is extended to W. R. Grace & Co., in Cambridge, Mass., and to N. Berke, for supplying the test materials and helpful discussions.
APPENDIX I. REFERENCES
Atkinson, C, Smelser, R. E., and Sanchez, J. (1982). "Combined mode fracture via the cracked Brazilian disk test." Int. J. Fract,, 18(4), 279-291.
Buyukozturk, O., Nilson, A. H., and Slate, F. O. (1971). "Stress-strain response and fracture of a concrete model in biaxial loading." ACIJ., 68(8), 590-599.
Buyukozturk, O., Nilson, A. H., and Slate, F. O. (1972). "Deformation and fracture of particulate composite." J. Engrg. Mech., ASCE, 98(6), 581-593.
Carrasquillo, R. L., Slate, F. O., and Nilson, A. H. (1981). "Microcracking and
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behavior of high-strength concrete subject to short-term loading." ACI J., 78(3), 179-186.
Dundurs, J. (1969). "Edge-bonded dissimilar orthogonal elastic wedges." J. Appl. Mech. Trans. ASME, 36, 650-652.
England, A. H. (1965). "A crack between dissimilar media." J. Appl. Mech. Trans. ASME, 32, 400-402.
Erdogan, F. (1965). "Stress distribution in bonded dissimilar materials with cracks." /. Appl. Mech. Trans. ASME, 32, 403-410.
Gerstle, K. (1979). "Material behavior under various types of loading." Proc. of a Workshop on High Strength Concrete, Nat. Sci. Foundation, 43-78.
He, M.-Y., and Hutchinson, J. W. (1989). "Crack deflection at an interface between dissimilar elastic materials." Int. J. Solids Struct., 25(9), 1053-1067.
Hillemeier, B., and Hilsdorf, H. K. (1977). "Fracture mechanics studies on concrete compounds." Cent. Concr. Res., 7(5), 523-535.
Hsu, T. T. C , Slate, F. O., Sturman, G. M., and Winter, G. (1963). "Microcracking of plain concrete and the shape of the stress-strain curve." ACI J., 60(2), 209-223.
Hutchinson, J. W. (1990). "Mixed mode fracture mechanics of interfaces." Metal-ceramic interfaces, M. Ruhle, A. G. Evans, M. F. Ashby, and J. P. Hirth, eds., Pergamon Press, New York, N.Y., 295-306.
Liu, T. C. Y., Nilson, A. H., and Slate, F. O. (1972). "Stress-strain response and fracture of concrete in uniaxial and biaxial compression." ACI J., 69(5), 291-295.
Ojdrovic, R. P., and Petroski, H. J. (1987). "Fracture behavior of notched concrete cylinder." /. Engrg. Mech., ASCE, 113(10), 1551-1564.
Rice, J. R. (1988). "Elastic fracture concepts for interfacial cracks." /. Appl. Mech. Trans. ASME, 55(1), 98-103.
Rice, J. R., and Sih, G. C. (1965). "Plane problems of cracks in dissimilar media." I. Appl. Mech. Trans. ASME, 32, 418-423.
Shah, S. P., and Winter, G. (1966). "Inelastic behavior and fracture of concrete." Causes, mechanism, and control of cracking in concrete, SP-20, American Concr. Inst. (ACI), 5-28.
Singh, D., and Shetty, D. K. (1989). "Fracture toughness of polycrystalline ceramics in combined mode I and mode II loading." J. Am. Ceram. Soc, 72(1), 78-84.
Struble, L., Skalny, J., and Mindess, S. (1980). "A review of the cement-aggregate bond." Cem. Concr. Res., 10(2), 277-286.
Suo, Z. (1989). "Interface fracture mechanics," PhD thesis, Harvard Univ., Cambridge, Mass.
Suo, Z., and Hutchinson, J. W. (1989). "Sandwich test specimens for measuring interface crack toughness." Mater. Sci. Engrg., A107, 135-143.
Tada, H., Paris, P. C , and Irwin, G. R. (1985). The stress analysis of cracks handbook. 2nd Ed., Paris Productions Inc., St. Louis, Mo.
Wang, J.-S., and Suo, Z. (1990). "Experimental determination of interfacial toughness curves using Brazil-nut-sandwiches." Acta metall. mater., 38(7), 1279-1290.
Williams, M. L. (1959). "The stress around a fault or crack in dissimilar media." Bull. Seismol. Soc. Am., 49, 199-204.
Yamaguchi, E., and Chen, W.-F. (1991). "Microcrack propagation study of concrete under compression." J. Engrg. Mech., ASCE, 117(3), 653-673.
Zaitsev, Y. (1983). "Crack propagation in a composite material." Fracture mechanics of concrete, F. H. Wittmann, ed., Elsevier Science Publishers, Amsterdam, Netherlands, 251-299.
Ziegeldorf, S. (1983). "Fracture mechanics parameters of hardened cement paste, aggregates, and interfaces." Fracture mechanics of concrete, F. H. Wittmann, ed., Elsevier Science Publishers, Amsterdam, Netherlands, 371-409.
APPENDIX II. NOTATION
The following symbols are used in this paper:
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a = crack length; b, d = width and height of beam specimen, respectively;
E„ (JL„ v, = Young's modulus, shear modulus, and Poisson's ratio of material i, respectively;
Et = E,/(l — vj) = plane strain tensile modulus of material i; HE* = 112(11 Et + IIE2);
/ t = geometrical correction factor for four-point bending specimen;
G = energy release rate; Gd = energy release rate of deflected crack; Gp = energy release rate of penetrated crack with y2 =
90°; G™a* = maximum energy release rate of penetrated crack;
h = thickness of sandwich layer; K = Kx + iK2 = interface stress-intensity factor;
K* = Kt + iKu = stress intensity factor; L = structure size; L = fixed length for reporting data; M = applied moment;
NUN2 = coefficients for Brazilian disk specimen; P = failure load for Brazilian disk specimen;
R, t = disk radius and thickness of Brazilian disk specimen, respectively;
T = applied traction on structure; Y = dimensionless real positive number;
a, (3 = Dundurs' mismatch parameters; I \ = mode-I toughness of material 1; F,- = fracture toughness of interface; Tg = mode-I toughness of granite; y1 = crack-penetration angle; 72 = crack-hitting angle; Aa = crack increment;
8l5 82 = crack face displacements distances behind crack tip; e = oscillation index; 6, = inclination angle of Brazilian disk specimen; II = potential energy;
°22> o"i2 = normal stress and shear stress, respectively; ov = 6M/bd2 = tensile stress at center of four-point bending speci
men; 4> = t a n - W ^ i ) ; ij; = phase angle of KL" »ff = phase angle of KL'e;
\\i* = phase angle of Khh; and o)(a, p) = phase shift in sandwich specimen.
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![Case Studies in Construction Materials · blast furnace slag as addition or aggregate in mortar [1–13]. Mortar is a workable paste typically made from a mixture of fine aggregate,](https://img.pdfslide.us/doc/110x75/606079ca1575eb1dd2120308/case-studies-in-construction-materials-blast-furnace-slag-as-addition-or-aggregate.jpg)
