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The wedge splitting test, a new method of performing stable fracture mechanics tests
Engineering Fructure Mechanics Vol. 35, No. 1/2/3, pp. 117-125, 1990 Mnted in Great Britain.
~13-7~~~ $3.00 + 0.00 Fergamon Press pk.
THE WEDGE SPLITTING TEST, A NEW METHOD OF PERFORMING STABLE FRACTURE MECHANICS TESTS E. BROHWILER
and F. H. WITTMANN
Swiss Federal Institute of Technology, Laboratory for Building Materials, Lausanne, Switzerland Ahatraet-The wedge splitting test is a new test method to perform stable fracture mechanics tests on concrete and concrete-like materials. Specific fracture energy Gr as well as fracture toughness K, are determined using simple specimens Iike cubes or cylinders. The main features of the wedge splitting test are described and compared to other tests methods. Identical G,-values are found irrespective of the test method and the specimen shape. The significance of the interaction between the testing machine, the test controller, the test method and the material properties for the performance and interpretation of stable fracture tests is outlined.
1. INTRODUCTION STABLE fracture mechanics tests, i.e. with a descending branch in the overall load-deformation diagram, are needed for the determination of fracture mechanics properties such as the specific fracture energy Gr and the strain softening diagram. The most direct way to determine these material properties would be a deformation controlled, uniaxial tensile test. It is difficult, however, to carry out such uniaxial tensile tests, because of the small deformations at rupture and the extreme stiffness of concrete specimens. Therefore, this test is not suitable as a standard test. It is easier to perform stable tests on notched specimens subjected to bending. RILEM Technical Committee SO-FMC has elaborated a proposal for a RILEM r~ommendation for the dete~ination of the specific fracture energy Gr[l]. This r~ommendation specifies a testing method consisting of stable three point bending tests (TPBT) on notched beams, Recently, the compact tension test (CT-test) was used for the determination of the specific fracture energy Gr of concrete[2]. Both test methods however have drawbacks with regard to future use as standard tests. The aim of this article is to present the wedge splitting test, a new method of performing stable fracture mechanics tests on concrete and concrete-like materials[3]. First, the proposed test method is described and spe&l features are outlined. Then, Gr-values as determined by means of different test methods are reported, and finally, the stability of fracture tests is discussed.
2. THE WEDGE SPLITTING
TEST
2.1. description of the test method The set-up of the wedge splitting test is presented schematically in Fig. 1. First, a specimen is prepared by sawing or casting a groove and a notch. This specimen is placed on a linear support, which is fixed on the lower plate of the testing machine (Fig. la). Two massive loading devices both equipped with rollers, are placed on the top of the specimen (as shown in Fig. lb). A stiff steel profile with two identical wedges is fixed at the upper plate of the testing machine. The actuator is now moved so that the wedges enter between the rollers on each side (Fig. 1~). The test can now be performed. The dimensions of the groove and the notch are chosen so that the crack propagates in the vertical direction; the specimen is split into two halves. The fracture section is essentially subjected to a bending moment. During the test, the applied load F, (vertical component) and the crack opening displacement (COD) are measured. The splitting force F, is the horizontal component of the force acting on the rollers. It is calculated taking the wedge angle tl into consideration (Fig. 2): F, = FJ(2* tgu).
(1)
The COD is measured by means of transducers or a clip gauge; they should be fixed at the level where the splitting force acts on the specimen, i.e. at the axles of the rollers. EFM 35.1/S-H
117
E. BROHWILERand F. H. WITTMANN
118
COD
(Clip
b)
Fig. I. Principleof the wedgesplittingtest:(a) test specimenon a linearsupport;(b) placingof two loading devices with rollers; and (c) the wedges are pressed between rollers in order to split the specimen into two halves.
In a closed-loop servo-hydraulic testing machine, the test is controlled by means of the crack opening displacement (COD). Stable wedge splitting tests can also be performed under actuator displacement control if certain criteria are respected (see Section 4). The aim of the described test is to measure the amount of energy necessary to split the specimen into two halves. This energy is represented by the area under the F,-COD-curve. This fracture energy, divided by the projected fracture area (ligament length h x specimen width d), is assumed to be the specific fracture energy Gr. The influence of the dead weight on the fracture energy is negligible. This is an important advantage compared to the TPBT, where the fracture energy part due to the dead weight of the beam may amount to 4060% of the total fracture energy. The tensiie strain-softening diagram needed for the application of process zone models in computerized structural analysis, is evaluated using an appropriate software module. Such a numerical method was proposed in[4] and extended in[2]; it is based on the finite element method and the fictitious crack model of Hillerborg et al.[5]. A fracture toughness value can be dete~ined, e.g. according to the two parameter model of Jenq and Shah[6]. For that purpose, unloading/reloading cycles must be performed during the test.
a:
b:
l- ----- i
Fig. 2. Statical system of the wedge splitting test: (a) frontal view; and (b) side-view.
Fig. 3. Wedge splitting specimen shapes.
The wedge sphtting test
Fig. 4. Wedge splitting test on a cylindrical specimen (4 = 30 cm) drilled from a dam.
119
The wedge splitting test
121
Remark. The fracture energy may also be determined as the area under the F,-u-curve. However, this energy value was found to be bigger than the fracture energy as determined from the F,-COD-curve. This difference can be attributed mainly to displacements when the loading device is “pressed” into its optimal postion during loading. As a consequence, deformations for the determination of fracture energy must always be measured by means of transducers, strain gauges, etc., fixed on the specimen.
2.2. Specimen shapes The wedge splitting was previously proposed by Linsbauer and Tschegg[l who developed a similar splitting test. The specimen shape is characterized by a groove and a starter notch. They can both easily be moulded or sawn. Four different specimen shapes were used to perform wedge splitting tests (Fig. 3): The cubical specimen A is suitable for freshly poured concrete and is fabricated in standard moulds. In order to examine drilled cores from existing structures, the cylindrical specimens B and C are proposed. The main drawback of specimen shape C is that it needs either a deep notch or a longitudinal reinforcement so as to prevent shear failure of the cantilevers. In principle, specimens can be fabricated from concrete or rock blocks of any shape; in this case, two plain parallel surfaces are polished, and subsequently the groove and the notch are sawn (shape D). Compared to TPBT-beams or CT-specimens, wedge splitting specimens are simple; they can be easily fabricated on-site (cubes) or drilled from existing structures (cylinders). Furthermore, the wedge splitting specimens are easy to handle and there is no risk of breaking them during handling. The fracture area of wedge splitting specimens is large compared to its weight. In order to illustrate this feature, a wedge splitting cube having a side length of 20 cm is compared to the TPBT-beam with a span of 80 cm and dimensions according to the RILEM recommendation[l]. Both specimens have about the same weight (20 kg). However, the fracture area of the wedge splitting cube, generally with a ligament length h of 13 cm, is 5.2 times larger than the fracture area of the beam. The ratio of fracture area to specimen weight is important with regard to testing of concrete with large aggregates, i.e. concrete used for the construction of dams (Fig. 4). For usual concrete with a maximum aggregate size up to 32 mm, the ligament length and the width of the fracture area can easily be kept greater than three times the maximum grain size. This may explain the small scatter of test results. 2.3. Loading device The main feature of the loading device is the use of wedges and rollers. The same principle was previously used by Hillemeier and Hilsdorf[8] who performed tests on wedge loaded CT-specimens; they stated that the use of rollers significantly improved the accuracy and reproducibility of the obtained test results. Another advantage of rollers is in regard to fretting forces. In wedge loading tests where the wedge acts on a skew plane, high fretting forces may occur. In contrast, however, fretting forces of rollers are small, i.e. smaller than about 1% of the force acting on the ball bearings, and can therefore be neglected. The advantage of the chosen test set-up consists in the statical system since it is isostatic (Fig. 2): half of the load is first transferred to each wedge from which each roller is equally loaded. Furthermore, the set-up is mechanically stable because of the cross-wise arrangement of the upper and lower linear supports. (The loading device and the profile with the two wedges can be designed so that adjustments necessary for different specimen widths are easily practicable.) 2.4.
Wedge angle
The performance
of stable fracture mechanics tests on concrete specimens is difficult because
Of:
(i) the small deformations at rupture of concrete; and (ii) the extreme stiffness of concrete specimens compared machine.
to the stiffness of the testing
122
E. BRtiHW~LER
and F. H. WI~MANN
The present wedge splitting test overcomes these difficulties thanks to the use of wedges: (i) In the wedge splitting test, only the vertical load F, deforms the frame of the testing machine (Fig. 2). By the choice of a small wedge angle, Fv is reduced (according to eq. 1) relative to the splitting force Fs for a given specimen; less reversible energy is hence stored in the loading frame compared to the elastic energy stored in the specimen. Consequently, the stiffness of the testing machine is “artificially” increased compared to the specimen stiffness by the use of a small wedge angle; and (ii) The actuator displacement which is perpendicular to the specimen deformation (COD), is increased with respect to the COD, if a small wedge angle is chosen. As a consequence, a wedge angle as small as possible should be used. However, a wedge angle smaller than 5” should not be chosen because of the practical performance of the test. Generally a wedge angle tl of 15” was used in our tests. In fracture tests at high “loading rates”, the actuator needs a certain displacement in order to achieve its full velocity. Therefore, high rate fracture tests can be performed more easily if the actuator displacements are large compared to the deformations at rupture of the specimen. As the actuator displacement can be increased with respect to COD by means of a small wedge angle, the wedge splitting test is well suited to perform stable fracture tests at high COD-rates[3].
3. DETERMINATION
OF GF USING DIFFERENT
TEST METHODS
The specific fracture energy GF was determined using four different test methods, viz. the wedge splitting test (WST), the TPBT, the CT-test and the uniaxial tensile test, and the results compared. Four different concretes were tested using at least two different specimen shapes: In a first test series, a concrete with a maximum grain size of 16 mm was tested by means of the TPBT and the WST. A slightly different concrete mix was used in series 2 in order to carry out TPBT, CT-tests as well as WST. In series 3, WST were performed on cubical and cylindrical specimens of concrete fabricated with aggregates of another origin than the aggregates used for the concrete of the previous two series. In the last series, dam concrete cylinders (maximum grain size of 80 mm) were tested by uniaxial tensile tests and WST using different specimen shapes. The test results showing Gr as a function of the ligament length h of the specimens are presented in Fig. 5. It can be stated that: (i) the specific fracture energy is influenced by the type of concrete and by the ligament length of the tested specimen, i.e. Gr grows with increasing ligament length; and (ii) the specific fracture energy depends neither on the test method nor on the specimen shape. 4. SIGNIFICANCE
OF THE CONTROL
PARAMETER
IN STABLE FRACTURE
TESTS
A test is stable if no sudden drop of the load occurs; this can be checked by means of a load-time-plot. Stable fracture tests show an overall load-deformation diagram with a descending branch after peak load is reached. Such tests are carried out by monotonically increasing either a displacement of the actuator (or the crosshead) of the testing machine or a specimen deformation (e.g. COD, tensile strain) with time. (Load controlled tests cannot be maintained stable once peak load is attained.) Not only the test control parameter (displacement or deformation), but also the stiffness of the testing machine, the stiffness of the specimen, the test method itself and the material properties influence the post-peak stability of a fracture test. They must therefore be considered in order to get stable fracture tests. If this interaction is not respected, unstable fracture tests, i.e. violent failure at peak load, occur. Such unstable fracture tests cannot be attributed to “brittle” material behaviour or “unstable” crack propagation. The significance of this interaction between the testing machine, the test method and the material may be illustrated by the following example: During a wedge splitting test, not only the splitting force F,COD-curve, but also the vertical force F,-(vertical) actuator displacement u-curve can be recorded. These two types of curves
The wedge splitting test
0
100
123
200
300
iid
Fig. 5. Specific fracture energy G, as a function of ligament length h for different specimens and test methods. (In series 4, a G,-value of 160 N/m has been obtained from uniaxial tensile tests.)
obtained from stable tests on splitting cubes of different ligament length, viz. 70 and 130 mm for T70 and T130, respectively, are shown in Fig. 6. Both tests were deformation controlled by means of COD. In the &-u-diagram, a positive slope of the descending branch of test T130 can be observed, whereas the descending branch of T70 has a negative slope (Fig. 6b). Had the tests been performed under actuator displacement control, T130 would not have been stable, and a large portion of the descending branch (drop indicated by a dotted line) would not have been recorded. On the other hand, test T70 with the short ligament length (and hence low specimen stiffness) would have been stable under constant actuator displacement rate. This experimental observation is explained as follows. On loading, elastic energy is stored both in the specimen and in the testing machine (with the loading device). Once maximum load is reached, this stored energy is released and becomes available for the formation of fracture area in the specimen. In a displacement controlled test, this reversible energy consists of the energy stored in the specimen and in the testing machine. In a deformation controlled test, only the elastic a)
F, A Liti 10.130mm 70 mm
5--
I 0
.3
.2
.l
&hlj
*COD
b)
2.1hw; 0
.5
1
1.6
In
Fig. 6. Two loaddeformation-curves from two wedge splitting specimens of differing ligament length: (a) splitting force-crack opening displacement curve; and (b) vertical force-actuator displacement curve.
124
E. BRijHWILER
and
F. H. WITTMANN
energy stored within the “deformation controlled” region of the specimen (e.g. the volume of a tensile specimen within the base length of transducers) is released in the post-peak region. The energy stored in the testing machine is of no importance for the stability of the deformation controlled test, because the (closed-loop) testing machine must perform an exact deformation on the specimen irrespective of the energy stored in the machine. If the stored energy is larger than the fracture energy, it is removed from the machine by a backward movement of the actuator. In test T130, the stored energy at peak load was larger than the fracture energy needed; the actuator moved back (leading in the F,-u-diagram to a descending branch with a positive slope). As a consequence, fracture tests cannot be controlled, if, at peak load, the elastic energy stored in the “test system” is greater than the fracture energy (Grefracture area), necessary to fracture the specimen. This experimental observation is now used in order to develop criteria for the design of stable fracture tests. These criteria take into consideration the interaction between the control parameter chosen, the stiffness of the testing machine kH , the specimen stiffness kp and the specimen geometry, as well as the material properties. On the basis of some simplifying assumptions, equations describing the condition for stability of each fracture mechanics test discussed can be derived. These equations can be generalized, leading to the following conditions for stability on tests performed by means of displacement and deformation control: displacement control: lch> K. L .((k,/k,) deformation
control: f,, > K. L
+ 1)
(2) (3)
where &( = E .Gp/f :) is the characteristic length of the material, and K is a constant depending on the specimen geometry. In a uniaxial tensile test, L is the specimen length (displacement control) or the base length of the transducer (deformation control). For bending specimens, L is the cantilever (CT-tests and WST) or the span (TPBT). Although these criteria are based on simplifying assumptions, they were applied successfully for the design of stable fracture mechanics tests. From eqs (2) and (3) as well as from the experience gained with fracture mechanics tests, the following conclusions can be drawn: (i) Stable fracture tests can be performed on all materials, except “completely brittle” ones; (ii) deformation control is the theoretical limit case of an infinite stiffness of the testing machine; (iii) depending on the test method chosen, unstable fracture tests may occur even under deformation control; (iv) a “very stiff’ testing machine does not guarantee stable fracture tests; and (v) concrete specimens (especially tensile specimens) can be several times stiffer than usual testing machines. 5. CONCLUSIONS (1) The wedge splitting test is a new, promising test method for the determination of fracture mechanics parameters such as the specific fracture energy Gr and the fracture toughness. Simple specimens, like cubes or cylinders, with a large fracture area compared to the specimen weight can be tested by means of this method. The test is stable under both deformation and actuator control if certain conditions are respected. The test is suitable for the performance of stable fracture tests at high deformation rates. (2) Irrespective of the specimen shape, the four test methods discussed, viz. the wedge splitting test, the three point bending test, the compact tension test and the uniaxial tensile test, yield identical G,-values, if the ligament length as well as the concrete are the same. (3) There exists a strong interaction between the material properties, the test method, the specimen shape, the test control parameter and the testing machine. This interaction must be taken into account for the design, performance and interpretation of stable fracture mechanics tests. In the present work of RILEM committee 89-FMT (subcommittee B), wedge splitting tests are performed in view of a possible future recommended test method for the determination of both Gr- and fracture toughness values.
The wedge spfitting test
125
REFERENCES [1] RILEM Draft recommendation (SO-FMC), Determination of the fracture energy of mortar and concrete by means of three-point bend tests on notched beams. Marer. Struct, 18, 287-290 (1985). [2] F. H. Wittmann, K. Rokugo, E. Brilhwiler, H. Mihashi and Ph. Simonin, Fracture energy and strain softening of concrete as determined by compact tension specimens. Mater. Strut. 21, 21-32 (1988). [3] E. Briihwiler, Fracture mechanics of dam concrete subjected to quasi-static and seismic loading conditions. Doctoral Thesis, Laboratory for Building Materials, Swiss Federal Institute of Technology, Lausanne (1988) (in German with an extended summary in English). [4] P. E. Roelfstra and F. H. Wittmann, Numerical methods to link strain softening with failure of concrete, in Fracture Toughness and Fracture Energy (Edited by F. H. Wittmann), pp. 163-175. Elsevier, Amsterdam (1986). [5] A. Hillerborg, M. Modeer and P. E. Petersson, Analysis of formation and crack growth in concrete by means of fracture mechanics and finite elements. Cem. Concr. Res. 6, 773-782 (1976). [6] Y. Jenq and S. P. Shah, Two parameter model for concrete. J. Engng Mech. 111, 1227-1241 (1985). ]7] H. N. Linsbauer and E. K. Tschegg, Fracture energy determination of concrete with cube-shaped specimens. Zement und &ton 31, 38-40 (1986) (in German). [8] B. Hillemeier and H. K. Hilsdorf, Fracture mechanics studies on concrete compounds. Gem. Concr. Res. 7, 523-536 (1977). (Received for p~bliea~ion 16 November 1988)
~13-7~~~ $3.00 + 0.00 Fergamon Press pk.
THE WEDGE SPLITTING TEST, A NEW METHOD OF PERFORMING STABLE FRACTURE MECHANICS TESTS E. BROHWILER
and F. H. WITTMANN
Swiss Federal Institute of Technology, Laboratory for Building Materials, Lausanne, Switzerland Ahatraet-The wedge splitting test is a new test method to perform stable fracture mechanics tests on concrete and concrete-like materials. Specific fracture energy Gr as well as fracture toughness K, are determined using simple specimens Iike cubes or cylinders. The main features of the wedge splitting test are described and compared to other tests methods. Identical G,-values are found irrespective of the test method and the specimen shape. The significance of the interaction between the testing machine, the test controller, the test method and the material properties for the performance and interpretation of stable fracture tests is outlined.
1. INTRODUCTION STABLE fracture mechanics tests, i.e. with a descending branch in the overall load-deformation diagram, are needed for the determination of fracture mechanics properties such as the specific fracture energy Gr and the strain softening diagram. The most direct way to determine these material properties would be a deformation controlled, uniaxial tensile test. It is difficult, however, to carry out such uniaxial tensile tests, because of the small deformations at rupture and the extreme stiffness of concrete specimens. Therefore, this test is not suitable as a standard test. It is easier to perform stable tests on notched specimens subjected to bending. RILEM Technical Committee SO-FMC has elaborated a proposal for a RILEM r~ommendation for the dete~ination of the specific fracture energy Gr[l]. This r~ommendation specifies a testing method consisting of stable three point bending tests (TPBT) on notched beams, Recently, the compact tension test (CT-test) was used for the determination of the specific fracture energy Gr of concrete[2]. Both test methods however have drawbacks with regard to future use as standard tests. The aim of this article is to present the wedge splitting test, a new method of performing stable fracture mechanics tests on concrete and concrete-like materials[3]. First, the proposed test method is described and spe&l features are outlined. Then, Gr-values as determined by means of different test methods are reported, and finally, the stability of fracture tests is discussed.
2. THE WEDGE SPLITTING
TEST
2.1. description of the test method The set-up of the wedge splitting test is presented schematically in Fig. 1. First, a specimen is prepared by sawing or casting a groove and a notch. This specimen is placed on a linear support, which is fixed on the lower plate of the testing machine (Fig. la). Two massive loading devices both equipped with rollers, are placed on the top of the specimen (as shown in Fig. lb). A stiff steel profile with two identical wedges is fixed at the upper plate of the testing machine. The actuator is now moved so that the wedges enter between the rollers on each side (Fig. 1~). The test can now be performed. The dimensions of the groove and the notch are chosen so that the crack propagates in the vertical direction; the specimen is split into two halves. The fracture section is essentially subjected to a bending moment. During the test, the applied load F, (vertical component) and the crack opening displacement (COD) are measured. The splitting force F, is the horizontal component of the force acting on the rollers. It is calculated taking the wedge angle tl into consideration (Fig. 2): F, = FJ(2* tgu).
(1)
The COD is measured by means of transducers or a clip gauge; they should be fixed at the level where the splitting force acts on the specimen, i.e. at the axles of the rollers. EFM 35.1/S-H
117
E. BROHWILERand F. H. WITTMANN
118
COD
(Clip
b)
Fig. I. Principleof the wedgesplittingtest:(a) test specimenon a linearsupport;(b) placingof two loading devices with rollers; and (c) the wedges are pressed between rollers in order to split the specimen into two halves.
In a closed-loop servo-hydraulic testing machine, the test is controlled by means of the crack opening displacement (COD). Stable wedge splitting tests can also be performed under actuator displacement control if certain criteria are respected (see Section 4). The aim of the described test is to measure the amount of energy necessary to split the specimen into two halves. This energy is represented by the area under the F,-COD-curve. This fracture energy, divided by the projected fracture area (ligament length h x specimen width d), is assumed to be the specific fracture energy Gr. The influence of the dead weight on the fracture energy is negligible. This is an important advantage compared to the TPBT, where the fracture energy part due to the dead weight of the beam may amount to 4060% of the total fracture energy. The tensiie strain-softening diagram needed for the application of process zone models in computerized structural analysis, is evaluated using an appropriate software module. Such a numerical method was proposed in[4] and extended in[2]; it is based on the finite element method and the fictitious crack model of Hillerborg et al.[5]. A fracture toughness value can be dete~ined, e.g. according to the two parameter model of Jenq and Shah[6]. For that purpose, unloading/reloading cycles must be performed during the test.
a:
b:
l- ----- i
Fig. 2. Statical system of the wedge splitting test: (a) frontal view; and (b) side-view.
Fig. 3. Wedge splitting specimen shapes.
The wedge sphtting test
Fig. 4. Wedge splitting test on a cylindrical specimen (4 = 30 cm) drilled from a dam.
119
The wedge splitting test
121
Remark. The fracture energy may also be determined as the area under the F,-u-curve. However, this energy value was found to be bigger than the fracture energy as determined from the F,-COD-curve. This difference can be attributed mainly to displacements when the loading device is “pressed” into its optimal postion during loading. As a consequence, deformations for the determination of fracture energy must always be measured by means of transducers, strain gauges, etc., fixed on the specimen.
2.2. Specimen shapes The wedge splitting was previously proposed by Linsbauer and Tschegg[l who developed a similar splitting test. The specimen shape is characterized by a groove and a starter notch. They can both easily be moulded or sawn. Four different specimen shapes were used to perform wedge splitting tests (Fig. 3): The cubical specimen A is suitable for freshly poured concrete and is fabricated in standard moulds. In order to examine drilled cores from existing structures, the cylindrical specimens B and C are proposed. The main drawback of specimen shape C is that it needs either a deep notch or a longitudinal reinforcement so as to prevent shear failure of the cantilevers. In principle, specimens can be fabricated from concrete or rock blocks of any shape; in this case, two plain parallel surfaces are polished, and subsequently the groove and the notch are sawn (shape D). Compared to TPBT-beams or CT-specimens, wedge splitting specimens are simple; they can be easily fabricated on-site (cubes) or drilled from existing structures (cylinders). Furthermore, the wedge splitting specimens are easy to handle and there is no risk of breaking them during handling. The fracture area of wedge splitting specimens is large compared to its weight. In order to illustrate this feature, a wedge splitting cube having a side length of 20 cm is compared to the TPBT-beam with a span of 80 cm and dimensions according to the RILEM recommendation[l]. Both specimens have about the same weight (20 kg). However, the fracture area of the wedge splitting cube, generally with a ligament length h of 13 cm, is 5.2 times larger than the fracture area of the beam. The ratio of fracture area to specimen weight is important with regard to testing of concrete with large aggregates, i.e. concrete used for the construction of dams (Fig. 4). For usual concrete with a maximum aggregate size up to 32 mm, the ligament length and the width of the fracture area can easily be kept greater than three times the maximum grain size. This may explain the small scatter of test results. 2.3. Loading device The main feature of the loading device is the use of wedges and rollers. The same principle was previously used by Hillemeier and Hilsdorf[8] who performed tests on wedge loaded CT-specimens; they stated that the use of rollers significantly improved the accuracy and reproducibility of the obtained test results. Another advantage of rollers is in regard to fretting forces. In wedge loading tests where the wedge acts on a skew plane, high fretting forces may occur. In contrast, however, fretting forces of rollers are small, i.e. smaller than about 1% of the force acting on the ball bearings, and can therefore be neglected. The advantage of the chosen test set-up consists in the statical system since it is isostatic (Fig. 2): half of the load is first transferred to each wedge from which each roller is equally loaded. Furthermore, the set-up is mechanically stable because of the cross-wise arrangement of the upper and lower linear supports. (The loading device and the profile with the two wedges can be designed so that adjustments necessary for different specimen widths are easily practicable.) 2.4.
Wedge angle
The performance
of stable fracture mechanics tests on concrete specimens is difficult because
Of:
(i) the small deformations at rupture of concrete; and (ii) the extreme stiffness of concrete specimens compared machine.
to the stiffness of the testing
122
E. BRtiHW~LER
and F. H. WI~MANN
The present wedge splitting test overcomes these difficulties thanks to the use of wedges: (i) In the wedge splitting test, only the vertical load F, deforms the frame of the testing machine (Fig. 2). By the choice of a small wedge angle, Fv is reduced (according to eq. 1) relative to the splitting force Fs for a given specimen; less reversible energy is hence stored in the loading frame compared to the elastic energy stored in the specimen. Consequently, the stiffness of the testing machine is “artificially” increased compared to the specimen stiffness by the use of a small wedge angle; and (ii) The actuator displacement which is perpendicular to the specimen deformation (COD), is increased with respect to the COD, if a small wedge angle is chosen. As a consequence, a wedge angle as small as possible should be used. However, a wedge angle smaller than 5” should not be chosen because of the practical performance of the test. Generally a wedge angle tl of 15” was used in our tests. In fracture tests at high “loading rates”, the actuator needs a certain displacement in order to achieve its full velocity. Therefore, high rate fracture tests can be performed more easily if the actuator displacements are large compared to the deformations at rupture of the specimen. As the actuator displacement can be increased with respect to COD by means of a small wedge angle, the wedge splitting test is well suited to perform stable fracture tests at high COD-rates[3].
3. DETERMINATION
OF GF USING DIFFERENT
TEST METHODS
The specific fracture energy GF was determined using four different test methods, viz. the wedge splitting test (WST), the TPBT, the CT-test and the uniaxial tensile test, and the results compared. Four different concretes were tested using at least two different specimen shapes: In a first test series, a concrete with a maximum grain size of 16 mm was tested by means of the TPBT and the WST. A slightly different concrete mix was used in series 2 in order to carry out TPBT, CT-tests as well as WST. In series 3, WST were performed on cubical and cylindrical specimens of concrete fabricated with aggregates of another origin than the aggregates used for the concrete of the previous two series. In the last series, dam concrete cylinders (maximum grain size of 80 mm) were tested by uniaxial tensile tests and WST using different specimen shapes. The test results showing Gr as a function of the ligament length h of the specimens are presented in Fig. 5. It can be stated that: (i) the specific fracture energy is influenced by the type of concrete and by the ligament length of the tested specimen, i.e. Gr grows with increasing ligament length; and (ii) the specific fracture energy depends neither on the test method nor on the specimen shape. 4. SIGNIFICANCE
OF THE CONTROL
PARAMETER
IN STABLE FRACTURE
TESTS
A test is stable if no sudden drop of the load occurs; this can be checked by means of a load-time-plot. Stable fracture tests show an overall load-deformation diagram with a descending branch after peak load is reached. Such tests are carried out by monotonically increasing either a displacement of the actuator (or the crosshead) of the testing machine or a specimen deformation (e.g. COD, tensile strain) with time. (Load controlled tests cannot be maintained stable once peak load is attained.) Not only the test control parameter (displacement or deformation), but also the stiffness of the testing machine, the stiffness of the specimen, the test method itself and the material properties influence the post-peak stability of a fracture test. They must therefore be considered in order to get stable fracture tests. If this interaction is not respected, unstable fracture tests, i.e. violent failure at peak load, occur. Such unstable fracture tests cannot be attributed to “brittle” material behaviour or “unstable” crack propagation. The significance of this interaction between the testing machine, the test method and the material may be illustrated by the following example: During a wedge splitting test, not only the splitting force F,COD-curve, but also the vertical force F,-(vertical) actuator displacement u-curve can be recorded. These two types of curves
The wedge splitting test
0
100
123
200
300
iid
Fig. 5. Specific fracture energy G, as a function of ligament length h for different specimens and test methods. (In series 4, a G,-value of 160 N/m has been obtained from uniaxial tensile tests.)
obtained from stable tests on splitting cubes of different ligament length, viz. 70 and 130 mm for T70 and T130, respectively, are shown in Fig. 6. Both tests were deformation controlled by means of COD. In the &-u-diagram, a positive slope of the descending branch of test T130 can be observed, whereas the descending branch of T70 has a negative slope (Fig. 6b). Had the tests been performed under actuator displacement control, T130 would not have been stable, and a large portion of the descending branch (drop indicated by a dotted line) would not have been recorded. On the other hand, test T70 with the short ligament length (and hence low specimen stiffness) would have been stable under constant actuator displacement rate. This experimental observation is explained as follows. On loading, elastic energy is stored both in the specimen and in the testing machine (with the loading device). Once maximum load is reached, this stored energy is released and becomes available for the formation of fracture area in the specimen. In a displacement controlled test, this reversible energy consists of the energy stored in the specimen and in the testing machine. In a deformation controlled test, only the elastic a)
F, A Liti 10.130mm 70 mm
5--
I 0
.3
.2
.l
&hlj
*COD
b)
2.1hw; 0
.5
1
1.6
In
Fig. 6. Two loaddeformation-curves from two wedge splitting specimens of differing ligament length: (a) splitting force-crack opening displacement curve; and (b) vertical force-actuator displacement curve.
124
E. BRijHWILER
and
F. H. WITTMANN
energy stored within the “deformation controlled” region of the specimen (e.g. the volume of a tensile specimen within the base length of transducers) is released in the post-peak region. The energy stored in the testing machine is of no importance for the stability of the deformation controlled test, because the (closed-loop) testing machine must perform an exact deformation on the specimen irrespective of the energy stored in the machine. If the stored energy is larger than the fracture energy, it is removed from the machine by a backward movement of the actuator. In test T130, the stored energy at peak load was larger than the fracture energy needed; the actuator moved back (leading in the F,-u-diagram to a descending branch with a positive slope). As a consequence, fracture tests cannot be controlled, if, at peak load, the elastic energy stored in the “test system” is greater than the fracture energy (Grefracture area), necessary to fracture the specimen. This experimental observation is now used in order to develop criteria for the design of stable fracture tests. These criteria take into consideration the interaction between the control parameter chosen, the stiffness of the testing machine kH , the specimen stiffness kp and the specimen geometry, as well as the material properties. On the basis of some simplifying assumptions, equations describing the condition for stability of each fracture mechanics test discussed can be derived. These equations can be generalized, leading to the following conditions for stability on tests performed by means of displacement and deformation control: displacement control: lch> K. L .((k,/k,) deformation
control: f,, > K. L
+ 1)
(2) (3)
where &( = E .Gp/f :) is the characteristic length of the material, and K is a constant depending on the specimen geometry. In a uniaxial tensile test, L is the specimen length (displacement control) or the base length of the transducer (deformation control). For bending specimens, L is the cantilever (CT-tests and WST) or the span (TPBT). Although these criteria are based on simplifying assumptions, they were applied successfully for the design of stable fracture mechanics tests. From eqs (2) and (3) as well as from the experience gained with fracture mechanics tests, the following conclusions can be drawn: (i) Stable fracture tests can be performed on all materials, except “completely brittle” ones; (ii) deformation control is the theoretical limit case of an infinite stiffness of the testing machine; (iii) depending on the test method chosen, unstable fracture tests may occur even under deformation control; (iv) a “very stiff’ testing machine does not guarantee stable fracture tests; and (v) concrete specimens (especially tensile specimens) can be several times stiffer than usual testing machines. 5. CONCLUSIONS (1) The wedge splitting test is a new, promising test method for the determination of fracture mechanics parameters such as the specific fracture energy Gr and the fracture toughness. Simple specimens, like cubes or cylinders, with a large fracture area compared to the specimen weight can be tested by means of this method. The test is stable under both deformation and actuator control if certain conditions are respected. The test is suitable for the performance of stable fracture tests at high deformation rates. (2) Irrespective of the specimen shape, the four test methods discussed, viz. the wedge splitting test, the three point bending test, the compact tension test and the uniaxial tensile test, yield identical G,-values, if the ligament length as well as the concrete are the same. (3) There exists a strong interaction between the material properties, the test method, the specimen shape, the test control parameter and the testing machine. This interaction must be taken into account for the design, performance and interpretation of stable fracture mechanics tests. In the present work of RILEM committee 89-FMT (subcommittee B), wedge splitting tests are performed in view of a possible future recommended test method for the determination of both Gr- and fracture toughness values.
The wedge spfitting test
125
REFERENCES [1] RILEM Draft recommendation (SO-FMC), Determination of the fracture energy of mortar and concrete by means of three-point bend tests on notched beams. Marer. Struct, 18, 287-290 (1985). [2] F. H. Wittmann, K. Rokugo, E. Brilhwiler, H. Mihashi and Ph. Simonin, Fracture energy and strain softening of concrete as determined by compact tension specimens. Mater. Strut. 21, 21-32 (1988). [3] E. Briihwiler, Fracture mechanics of dam concrete subjected to quasi-static and seismic loading conditions. Doctoral Thesis, Laboratory for Building Materials, Swiss Federal Institute of Technology, Lausanne (1988) (in German with an extended summary in English). [4] P. E. Roelfstra and F. H. Wittmann, Numerical methods to link strain softening with failure of concrete, in Fracture Toughness and Fracture Energy (Edited by F. H. Wittmann), pp. 163-175. Elsevier, Amsterdam (1986). [5] A. Hillerborg, M. Modeer and P. E. Petersson, Analysis of formation and crack growth in concrete by means of fracture mechanics and finite elements. Cem. Concr. Res. 6, 773-782 (1976). [6] Y. Jenq and S. P. Shah, Two parameter model for concrete. J. Engng Mech. 111, 1227-1241 (1985). ]7] H. N. Linsbauer and E. K. Tschegg, Fracture energy determination of concrete with cube-shaped specimens. Zement und &ton 31, 38-40 (1986) (in German). [8] B. Hillemeier and H. K. Hilsdorf, Fracture mechanics studies on concrete compounds. Gem. Concr. Res. 7, 523-536 (1977). (Received for p~bliea~ion 16 November 1988)