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Crack velocity in concrete
0013-7944/90 $3.00 + 0.00 Pergamon Press pk.
Engineering Fracture Mechanics Vol. 35, No. l/2/3, pp. 321-326, 1990
Printed in Great Britain.
CRACK VELOCITY
IN CONCRETE
M. CURBACH and J. EIBL Institut filr Massivbau und Baustofftechnologie, Universitlt D-7500 Karlsruhe 1, F.R.G.
Karlsruhe, Postfach 6980,
Abstract-An experimental investigation to determine the crack velocity in concrete is discussed. Apart from the description of the equipment and the measurement technique, main attention is paid to the results and their dependence on strain rate. Finite element calculations which were carried out using a special constitutive model for the descending branch of concrete under tensile loading show that the maximum value of crack velocity is strongly affected by the shape of the stress-strain diagram.
1. INTRODUCTION THE PROCESS of fracture is closely connected to the formation and propagation of cracks. Therefore the application of fracture mechanics has increased the knowledge about concrete behaviour, especially under high tensile loading rates. Although it has been known since 1917 that concrete strength increases with an increasing loading rate[l] attempts to explain the underlying physical reasons have only been carried out for a few years. So, e.g., explanations have been published by Reinhardt[2] and Curbach[3] which depend to a high extent on the velocity of cracks in concrete, but crack velocity data for concrete are very scarce. So Bhargava and Rhenstrom[4] described experiments where concrete specimens have been loaded by an explosion at their upper end. The propagation of the vertical crack was recorded by photographing at different stages with a time increment of At = 50 ,us. Thereof a crack velocity of about vR = 180 m/s has been estimated. Mindess et al.[5] used high speed motion picture photography with about 10,000 frames per second, i.e. with a time increment of At = 100 ps, to measure the crack velocity in reinforced concrete beams and beams made of hardened cement paste. Their average crack velocities were in the range of about vR= 75 m/s to vR= 115 m/s. Shah and John[6] tested plain concrete and plain mortar beams. The result of their investigation is that-within the range of applied strain rates-the logarithm of crack velocity increases linearly with the logarithm of strain rates. A maximum value of crack velocity in concrete of about vR = 80 m/s has been observed. Muria Vila and Hamelin[ir] also used high speed motion picture photography but with 3600 frames per second. This results in time increments of about At = 278 ps. Applying strain rates with a magnitude of i - 10 l/s crack velocity values between vR= 800 m/s and vR= 1200 m/s have been observed. These values lead to an extension of the range of linear relation as given by PI. It should be mentioned that with such given values of At = 278 ps and vR= 800m/s to vR= 1200 m/s[7] the crack propagation from one frame to the next should be in the range of about 220 mm to 330 mm which could only be reached by an appropriate test specimen, the details of which are not reported. Additionally, in Watson et a1.[8] only one value for the crack velocity is given: OR= 2100 m/s. This value is approximately identical to the theoretical value of maximum crack velocity which was calculated by Broberg[9] for linear elastic-brittle materials and which has not yet been watched in any material[lO]. Studying this literature one always has to distinguish between stable and unstable crack growth. If Shah and John[6] report on a crack velocity of VR- 10e3 m/s due to strain rate ; - 1O-5 l/s, only a stable crack growth after the beginning of crack spreading can be assumed which could be reached, e.g. with beams in deformation controlled tests. 321
M. CURBACH and J. EIBL
322
al Fig. I. (a) Experimental setup for the determination
of crack velocity. (b) Dimensions of specimen.
To contribute to this discussion on crack propagation an experimental program as well as calculation using the finite element technique have been carried out by the authors, the results of which are given in the following. 2. EXPERIMENTAL
INVESTIGATION
Experiments were carried out using the impact equiment at the authors’ institute which mainly consists of a vertical tube to guide a steel cylinder with a mass of about 200 kg. This loading system was combined with a special arrangement to load a notched concrete specimen in high speed tension. As Fig. l(a) shows, the impact cylinder hits a steel beam which causes a stress wave in the concrete specimen. Two conditions determined the shape of the test specimen (Fig. lb). First an unstable crack growth had to be reached, secondly stability of crack direction had to be gained both by means as proposed by Williams[ 1l] and Cotterell[lZ]. All specimens had been cast using the same concrete mixture to obtain comparable material properties. Mixture and properties are given in Tables 1 and 2. The measurement technique for the crack velocity, i.d. the detection of the crack tip at several time increments is equivalent to Erdogan’s proposal[ 131but liquid silver instead of graphite barriers have been used. The silver barriers were connected within an electrical circuit with electrical resistances as shown in Fig. 2. During the crack propagation these bariers were destroyed, the electrical resistances activated and the change of electrical voltage has been registered. A typical result of the behaviour of the voltage vs time is shown in Fig. 3. Every “jump” in the voltage line indicates failure of a barrier. As the distance between the silver barriers is known-15 barriers with a spacing of 15 mm were used-the velocity of the crack between two barriers can be calculated. In the diagram in Fig. 3 the dotted line indicates the development of the crack velocity during crack propagation. It can be seen that the crack first started with a rather low velocity and reaches then a maximum in the second part of the crack ligament, the maximum value being related to the measured strain rate before crack spreading started.
Table 1. Matrix proportions Cement Aggregates Water w/c-ratio Grading curve act. to DIN 1045
of used concrete 212.0 kg/m’ 1957.0 kg/m3 170.0 I/m’ 0.80 B16
Table 2. Material properties Compressive strength Flexural strength Splitting tensile strength Axial tensile strength Young’s modulus
f, = 1;. ,, = f;, ,~ = 1; = E, =
22.2 N/mm2 3.5 N/mm* 2.5 N/mm’ 2.0 N/mm* 30100 N/mm*
Crack velocity in concrete
323
Fig. 2. Electrical circuit with liquid silver barriers and resistances.
Altogether seventeen specimens were tested with different strain rates before failure, two of them quasistatic. The measured crack velocities and the corresponding strain rates are summarized in Fig. 4. Here also the results of other authors as mentioned in Section 1 are included. To mark the theoretical maximum crack velocity the Rayleigh-wave speed act. to Grafftl4] is calculated: 0.87+ 1.12~ CRay= With the experimental
l+v
1 ~- E J-- 2(1 + v) p’
(11
values E, = 30,100 MN/m2
v = 0.2 p = 2330 kg/m3 = 2.33. 10e3 MN s2/m4 it can be obtained c Ray=
2115 m/s.
This value is given as a dotted line in Fig. 4.
U [mVl -
-
v,fm/sl
x50
600
3000
050
2250
300
%Gu
ls0
750
0
0 4.6
5.0
5.4
5.6
Fig. 3. Electric voltage vs time.
62
66 t Imsl
324
M. CURBACH and J. EIBL 3ooo # Im/sl 1000
50
A CALCULATIONS MURIA VILA I HAMELIN MINOESS ET AL.
10 WI-*
10-l
loo
i IlM
10’
Fig. 4. Crack velocity vs strain rate.
3. FINITE ELEMENT APPROACH To investigate the physical background of crack propagation in concrete, several finite element calculations have been carried out. Main attention was paid to different constitutive laws used to describe the behaviour of concrete. Figure 5 shows the discretization of the test specimen (Fig. 1b) which has been used for the estimation of the crack velocity. The calculations were carried out with a finite element code which considers inertia effects as well as linear or nonlinear material behaviour. A detailed description of this code is given by Schliiter[ 151. The different constitutive laws which were studied are given in Fig. 6. Fig. 6(a) represents a linear elastic brittle material, Fig. 6(b) simulates an elastoplasic behaviour and Fig. 6(c) describes the real behaviour of concrete under tension considering the descending branch.
Fig. 5. Dimensions, boundary conditions and discretization of the specimen.
325
Crack velocity in concrete
E
al Fig. 6. Different constitutive
cl
E
I
laws used in the finite element calculations. (b) elastoplastic, (c) concrete.
s (a) Linear elastic brittle,
In all three cases the crack tip has to be determined. Using law I this is quite simple and self-evident; in case II the fictitious crack tip follows the end of the elastic behaviour, while in case III the crack tip corresponds to the maximum stress c,,,. One example of crack propagation with a constitutive law III is shown in Fig. 7. After quasistatic loading and crack initiation several stress distributions are plotted. The movement of the stress maximum from right to left marks the propagation of a fictitious crack tip. Since time increment and distance are known crack velocity can be calculated. Using the linear elastic-brittle constitutive law I the calculated crack velocity reaches values[ 161 which are in the range of the Rayleigh-wave speed. This is consistent to the value given by Broberg[9]. Supposing an elastoplastic behaviour as in case II the rate of propagation of the observed point is considerably smaller. The highest values of crack velocity reached in calculations are in the range of about uRN 1000 m/s. Taking the real behaviour of concrete into account as in III, i.e. using a constitutive law with descending branch as is shown in Fig. 6(c) and which had been developed in [2] and has also been published by Brameshuber and Hilsdorf[l7] the crack velocity reaches values up to the range of about vRN 700 m/s. Several results obtained by the realistic model are given also in Fig. 4.
4. SUMMARY AND CONCLUSIONS The behaviour of cracks during their propagation is important for the explanation of, e.g., fracture parameters in dynamic cases and the strength increase due to increasing loading rate. To improve knowledge about crack velocity in concrete an experimental investigation as well as computer simulations were carried out. The measurement technique consisted of conductive liquid silver barriers which were painted perpendicular to the expected crack and electrically conducted. During crack propagation these barriers were destroyed and a change of the electrical signal can be registered. In addition finite element calculations were carried out using different constitutive models. To describe the concrete behaviour in a realistic way a new constitutive law was developed and implemented in a finite element code.
G
CRACK
PRDWGKTION
G mox
[mm1
Fig. 7. Stress distribution during crack propagation at several moments.
M. CURBACH and J. EIBL
326
From the investigations
the following conclusions may be drawn:
-A distinction must be made between stable and unstable crack growth. The values in this paper represent unstable crack growth. -Crack velocity in concrete is not constant during propagation, It accelerates to a maximum value and decreases afterwards. -The maximum value of unstable crack velocity seems to be in the magnitude of about vRN 500 m/s to vRN 700 m/s, that is about 0.20 vRayto 0.30 vRay; vRaybeing the Rayleigh-wave velocity. -Calculations with different constitutive laws show that the relations of crack velocity to Rayleigh-wave velocity are different with different materials. Further investigations are necessary, of course, and are partly under progress at the authors’ institute. REFERENCES 111D. A. Abrams, Effect of rate of application of load on the compressive strength of concrete. Am. Sot. Testing Materials, Proc. 20th Ann. Meeting 17, Part II, Technical Papers, 364-377 (1917).
PI H. W. Reinhardt, Strain rate effects on the tensile strength of concrete as predicted by thermodynamic and fracture
mechanics models, in Cement-Based Composites: Strain Rate Effects on Fracture (Edited by S. Mindess and S. P. Shah), Muter. Res. Sot. Syrnpo. Proc. 64, 1-13, Pittsburgh (1986). [31 M. Curbach, Festigkeitssteigerung von Beton bei hohen Blastungsgeschwindigkeiten. Dissertation und Heft 1 der Schriftenreihe des Instituts ftir Massivbau und Baustofftechnologie, Universitit Karlsruhe (1987). 141J. Bhargava and A. Rehnstriim, High-speed photography for fracture studies of concrete. Cement Concr. Res. 5, 239-248 (1975). [51 S. Mindess, N. P. Banthia, A. Ritter and J. P. Skalny, Crack development in cementitious materials under impact loading in Cement-Bused Composites: Strain Rate Effects on Fracture (Edited by S. Mindess and S. P. Shah), Muter. Res. Sot. Symp. Proc. 64, 217-223, Pittsburgh (1986). VI S. P. Shah and R. John, Strain rate effects in mode I crack propagation in concrete, in Fracture Toughness and Fracture Energy of Concrete (Edited by F. H. Wittmann). Elsevier, Amsterdam (1986). [71 D. Muria Vila and P. Hamelin, Comportement au choc des b&ton et mortiers a matrices hydrauliques, 1st International RILEM Congress 2, in Combining Materials: Design, Production and Properties (Edited by J. C. Maso), pp. 725-732. Chapman and Hall, London (1987). A. J. Watson, W. F. Anderson and B. Archer, Hypervelocity impact of concrete, in Concrete Structures under Impact and Impulsive Loading, RILEM CEB . IABSE . IASS Symposium Proceeding. BAM, Berlin (1982). B. Broberg, On the speed of a brittle crack. J. uppl. Mech. 31, 546547 (1964). D. Broek, Elementary Engineering Fracture Mechanics, 3rd Ed. Martinus Nijhoff, The Hague (1984). M. L. Williams, On the stress distribution at the base of a stationary crack. J. uppl. Mech. 24, 109-l 14 (1957). B. Cotterell, Notes on the path and stability of cracks. Znt. J. Fracture Mech. 2, 526-533 (1966). F. Erdogan, Crack-propagation theories, in Fracfure-An Advanced Treatise (Edited by H. Liebowitz), pp. 498-590. Academic Press, New York (1968). K. F. Graff, Wave Motion in Elastic Solids. Clarendon Press, Oxford (1975). F.-H. Schlilter, Dicke Stahlbetonplatten unter stogartiger Belastung-Flugzeugabsturz-Dissertation und Heft 2 der Schriftenreihe des Instituts fur Massivbau und BaustotTtechnologie, Universitlt Karlsruhe (1987). J. Eibl, P. Godde and M. Curbach, Untersuchung der Betonzugfestigkeit unter hoher BelastungsgeschwindigkeitRiDgeschwindigkeit-Final report of research project IV B 5-FA 9293 of Nordrhein-Westfalen, Institut fur Massivbau und Baustofftechnologie, Karlsruhe (1986). 1171W. Brameshuber and H. K. Hilsdorf, Influence of ligament length and stress state on facture energy of concrete. Engng Fracture Mech. 35, 95106 (1989).
(Received for publication 16 November 1988)
Engineering Fracture Mechanics Vol. 35, No. l/2/3, pp. 321-326, 1990
Printed in Great Britain.
CRACK VELOCITY
IN CONCRETE
M. CURBACH and J. EIBL Institut filr Massivbau und Baustofftechnologie, Universitlt D-7500 Karlsruhe 1, F.R.G.
Karlsruhe, Postfach 6980,
Abstract-An experimental investigation to determine the crack velocity in concrete is discussed. Apart from the description of the equipment and the measurement technique, main attention is paid to the results and their dependence on strain rate. Finite element calculations which were carried out using a special constitutive model for the descending branch of concrete under tensile loading show that the maximum value of crack velocity is strongly affected by the shape of the stress-strain diagram.
1. INTRODUCTION THE PROCESS of fracture is closely connected to the formation and propagation of cracks. Therefore the application of fracture mechanics has increased the knowledge about concrete behaviour, especially under high tensile loading rates. Although it has been known since 1917 that concrete strength increases with an increasing loading rate[l] attempts to explain the underlying physical reasons have only been carried out for a few years. So, e.g., explanations have been published by Reinhardt[2] and Curbach[3] which depend to a high extent on the velocity of cracks in concrete, but crack velocity data for concrete are very scarce. So Bhargava and Rhenstrom[4] described experiments where concrete specimens have been loaded by an explosion at their upper end. The propagation of the vertical crack was recorded by photographing at different stages with a time increment of At = 50 ,us. Thereof a crack velocity of about vR = 180 m/s has been estimated. Mindess et al.[5] used high speed motion picture photography with about 10,000 frames per second, i.e. with a time increment of At = 100 ps, to measure the crack velocity in reinforced concrete beams and beams made of hardened cement paste. Their average crack velocities were in the range of about vR= 75 m/s to vR= 115 m/s. Shah and John[6] tested plain concrete and plain mortar beams. The result of their investigation is that-within the range of applied strain rates-the logarithm of crack velocity increases linearly with the logarithm of strain rates. A maximum value of crack velocity in concrete of about vR = 80 m/s has been observed. Muria Vila and Hamelin[ir] also used high speed motion picture photography but with 3600 frames per second. This results in time increments of about At = 278 ps. Applying strain rates with a magnitude of i - 10 l/s crack velocity values between vR= 800 m/s and vR= 1200 m/s have been observed. These values lead to an extension of the range of linear relation as given by PI. It should be mentioned that with such given values of At = 278 ps and vR= 800m/s to vR= 1200 m/s[7] the crack propagation from one frame to the next should be in the range of about 220 mm to 330 mm which could only be reached by an appropriate test specimen, the details of which are not reported. Additionally, in Watson et a1.[8] only one value for the crack velocity is given: OR= 2100 m/s. This value is approximately identical to the theoretical value of maximum crack velocity which was calculated by Broberg[9] for linear elastic-brittle materials and which has not yet been watched in any material[lO]. Studying this literature one always has to distinguish between stable and unstable crack growth. If Shah and John[6] report on a crack velocity of VR- 10e3 m/s due to strain rate ; - 1O-5 l/s, only a stable crack growth after the beginning of crack spreading can be assumed which could be reached, e.g. with beams in deformation controlled tests. 321
M. CURBACH and J. EIBL
322
al Fig. I. (a) Experimental setup for the determination
of crack velocity. (b) Dimensions of specimen.
To contribute to this discussion on crack propagation an experimental program as well as calculation using the finite element technique have been carried out by the authors, the results of which are given in the following. 2. EXPERIMENTAL
INVESTIGATION
Experiments were carried out using the impact equiment at the authors’ institute which mainly consists of a vertical tube to guide a steel cylinder with a mass of about 200 kg. This loading system was combined with a special arrangement to load a notched concrete specimen in high speed tension. As Fig. l(a) shows, the impact cylinder hits a steel beam which causes a stress wave in the concrete specimen. Two conditions determined the shape of the test specimen (Fig. lb). First an unstable crack growth had to be reached, secondly stability of crack direction had to be gained both by means as proposed by Williams[ 1l] and Cotterell[lZ]. All specimens had been cast using the same concrete mixture to obtain comparable material properties. Mixture and properties are given in Tables 1 and 2. The measurement technique for the crack velocity, i.d. the detection of the crack tip at several time increments is equivalent to Erdogan’s proposal[ 131but liquid silver instead of graphite barriers have been used. The silver barriers were connected within an electrical circuit with electrical resistances as shown in Fig. 2. During the crack propagation these bariers were destroyed, the electrical resistances activated and the change of electrical voltage has been registered. A typical result of the behaviour of the voltage vs time is shown in Fig. 3. Every “jump” in the voltage line indicates failure of a barrier. As the distance between the silver barriers is known-15 barriers with a spacing of 15 mm were used-the velocity of the crack between two barriers can be calculated. In the diagram in Fig. 3 the dotted line indicates the development of the crack velocity during crack propagation. It can be seen that the crack first started with a rather low velocity and reaches then a maximum in the second part of the crack ligament, the maximum value being related to the measured strain rate before crack spreading started.
Table 1. Matrix proportions Cement Aggregates Water w/c-ratio Grading curve act. to DIN 1045
of used concrete 212.0 kg/m’ 1957.0 kg/m3 170.0 I/m’ 0.80 B16
Table 2. Material properties Compressive strength Flexural strength Splitting tensile strength Axial tensile strength Young’s modulus
f, = 1;. ,, = f;, ,~ = 1; = E, =
22.2 N/mm2 3.5 N/mm* 2.5 N/mm’ 2.0 N/mm* 30100 N/mm*
Crack velocity in concrete
323
Fig. 2. Electrical circuit with liquid silver barriers and resistances.
Altogether seventeen specimens were tested with different strain rates before failure, two of them quasistatic. The measured crack velocities and the corresponding strain rates are summarized in Fig. 4. Here also the results of other authors as mentioned in Section 1 are included. To mark the theoretical maximum crack velocity the Rayleigh-wave speed act. to Grafftl4] is calculated: 0.87+ 1.12~ CRay= With the experimental
l+v
1 ~- E J-- 2(1 + v) p’
(11
values E, = 30,100 MN/m2
v = 0.2 p = 2330 kg/m3 = 2.33. 10e3 MN s2/m4 it can be obtained c Ray=
2115 m/s.
This value is given as a dotted line in Fig. 4.
U [mVl -
-
v,fm/sl
x50
600
3000
050
2250
300
%Gu
ls0
750
0
0 4.6
5.0
5.4
5.6
Fig. 3. Electric voltage vs time.
62
66 t Imsl
324
M. CURBACH and J. EIBL 3ooo # Im/sl 1000
50
A CALCULATIONS MURIA VILA I HAMELIN MINOESS ET AL.
10 WI-*
10-l
loo
i IlM
10’
Fig. 4. Crack velocity vs strain rate.
3. FINITE ELEMENT APPROACH To investigate the physical background of crack propagation in concrete, several finite element calculations have been carried out. Main attention was paid to different constitutive laws used to describe the behaviour of concrete. Figure 5 shows the discretization of the test specimen (Fig. 1b) which has been used for the estimation of the crack velocity. The calculations were carried out with a finite element code which considers inertia effects as well as linear or nonlinear material behaviour. A detailed description of this code is given by Schliiter[ 151. The different constitutive laws which were studied are given in Fig. 6. Fig. 6(a) represents a linear elastic brittle material, Fig. 6(b) simulates an elastoplasic behaviour and Fig. 6(c) describes the real behaviour of concrete under tension considering the descending branch.
Fig. 5. Dimensions, boundary conditions and discretization of the specimen.
325
Crack velocity in concrete
E
al Fig. 6. Different constitutive
cl
E
I
laws used in the finite element calculations. (b) elastoplastic, (c) concrete.
s (a) Linear elastic brittle,
In all three cases the crack tip has to be determined. Using law I this is quite simple and self-evident; in case II the fictitious crack tip follows the end of the elastic behaviour, while in case III the crack tip corresponds to the maximum stress c,,,. One example of crack propagation with a constitutive law III is shown in Fig. 7. After quasistatic loading and crack initiation several stress distributions are plotted. The movement of the stress maximum from right to left marks the propagation of a fictitious crack tip. Since time increment and distance are known crack velocity can be calculated. Using the linear elastic-brittle constitutive law I the calculated crack velocity reaches values[ 161 which are in the range of the Rayleigh-wave speed. This is consistent to the value given by Broberg[9]. Supposing an elastoplastic behaviour as in case II the rate of propagation of the observed point is considerably smaller. The highest values of crack velocity reached in calculations are in the range of about uRN 1000 m/s. Taking the real behaviour of concrete into account as in III, i.e. using a constitutive law with descending branch as is shown in Fig. 6(c) and which had been developed in [2] and has also been published by Brameshuber and Hilsdorf[l7] the crack velocity reaches values up to the range of about vRN 700 m/s. Several results obtained by the realistic model are given also in Fig. 4.
4. SUMMARY AND CONCLUSIONS The behaviour of cracks during their propagation is important for the explanation of, e.g., fracture parameters in dynamic cases and the strength increase due to increasing loading rate. To improve knowledge about crack velocity in concrete an experimental investigation as well as computer simulations were carried out. The measurement technique consisted of conductive liquid silver barriers which were painted perpendicular to the expected crack and electrically conducted. During crack propagation these barriers were destroyed and a change of the electrical signal can be registered. In addition finite element calculations were carried out using different constitutive models. To describe the concrete behaviour in a realistic way a new constitutive law was developed and implemented in a finite element code.
G
CRACK
PRDWGKTION
G mox
[mm1
Fig. 7. Stress distribution during crack propagation at several moments.
M. CURBACH and J. EIBL
326
From the investigations
the following conclusions may be drawn:
-A distinction must be made between stable and unstable crack growth. The values in this paper represent unstable crack growth. -Crack velocity in concrete is not constant during propagation, It accelerates to a maximum value and decreases afterwards. -The maximum value of unstable crack velocity seems to be in the magnitude of about vRN 500 m/s to vRN 700 m/s, that is about 0.20 vRayto 0.30 vRay; vRaybeing the Rayleigh-wave velocity. -Calculations with different constitutive laws show that the relations of crack velocity to Rayleigh-wave velocity are different with different materials. Further investigations are necessary, of course, and are partly under progress at the authors’ institute. REFERENCES 111D. A. Abrams, Effect of rate of application of load on the compressive strength of concrete. Am. Sot. Testing Materials, Proc. 20th Ann. Meeting 17, Part II, Technical Papers, 364-377 (1917).
PI H. W. Reinhardt, Strain rate effects on the tensile strength of concrete as predicted by thermodynamic and fracture
mechanics models, in Cement-Based Composites: Strain Rate Effects on Fracture (Edited by S. Mindess and S. P. Shah), Muter. Res. Sot. Syrnpo. Proc. 64, 1-13, Pittsburgh (1986). [31 M. Curbach, Festigkeitssteigerung von Beton bei hohen Blastungsgeschwindigkeiten. Dissertation und Heft 1 der Schriftenreihe des Instituts ftir Massivbau und Baustofftechnologie, Universitit Karlsruhe (1987). 141J. Bhargava and A. Rehnstriim, High-speed photography for fracture studies of concrete. Cement Concr. Res. 5, 239-248 (1975). [51 S. Mindess, N. P. Banthia, A. Ritter and J. P. Skalny, Crack development in cementitious materials under impact loading in Cement-Bused Composites: Strain Rate Effects on Fracture (Edited by S. Mindess and S. P. Shah), Muter. Res. Sot. Symp. Proc. 64, 217-223, Pittsburgh (1986). VI S. P. Shah and R. John, Strain rate effects in mode I crack propagation in concrete, in Fracture Toughness and Fracture Energy of Concrete (Edited by F. H. Wittmann). Elsevier, Amsterdam (1986). [71 D. Muria Vila and P. Hamelin, Comportement au choc des b&ton et mortiers a matrices hydrauliques, 1st International RILEM Congress 2, in Combining Materials: Design, Production and Properties (Edited by J. C. Maso), pp. 725-732. Chapman and Hall, London (1987). A. J. Watson, W. F. Anderson and B. Archer, Hypervelocity impact of concrete, in Concrete Structures under Impact and Impulsive Loading, RILEM CEB . IABSE . IASS Symposium Proceeding. BAM, Berlin (1982). B. Broberg, On the speed of a brittle crack. J. uppl. Mech. 31, 546547 (1964). D. Broek, Elementary Engineering Fracture Mechanics, 3rd Ed. Martinus Nijhoff, The Hague (1984). M. L. Williams, On the stress distribution at the base of a stationary crack. J. uppl. Mech. 24, 109-l 14 (1957). B. Cotterell, Notes on the path and stability of cracks. Znt. J. Fracture Mech. 2, 526-533 (1966). F. Erdogan, Crack-propagation theories, in Fracfure-An Advanced Treatise (Edited by H. Liebowitz), pp. 498-590. Academic Press, New York (1968). K. F. Graff, Wave Motion in Elastic Solids. Clarendon Press, Oxford (1975). F.-H. Schlilter, Dicke Stahlbetonplatten unter stogartiger Belastung-Flugzeugabsturz-Dissertation und Heft 2 der Schriftenreihe des Instituts fur Massivbau und BaustotTtechnologie, Universitlt Karlsruhe (1987). J. Eibl, P. Godde and M. Curbach, Untersuchung der Betonzugfestigkeit unter hoher BelastungsgeschwindigkeitRiDgeschwindigkeit-Final report of research project IV B 5-FA 9293 of Nordrhein-Westfalen, Institut fur Massivbau und Baustofftechnologie, Karlsruhe (1986). 1171W. Brameshuber and H. K. Hilsdorf, Influence of ligament length and stress state on facture energy of concrete. Engng Fracture Mech. 35, 95106 (1989).
(Received for publication 16 November 1988)