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Mixed mode crack propagation in quasi-brittle materials
Engineering Fracture Mechanics
Vol. 38, No. 213,pp. 129-145,
1991
0013-7944/91
$3.00 + 0.00
0 1991 Pergamon Press plc.
Printed in Great Britain.
MIXED
MODE CRACK PROPAGATION QUASI-BRITTLE MATERIALS
IN
A. K. MAJIt, M. A. TASDEMIRS and S. P. SHAHS tuniversity
of New Mexico, Albuquerque, New Mexico, U.S.A. SIstanbul Technical University, Turkey #NSF Science Technology Center for Advanced Cement Based Materials, Northwestern University, Evanston, Illinois, U.S.A. Abstract-Crack propagation in rectangular blocks of mortar containing a central notch and subjected to uniaxial compression were studied. Four different notch orientation angles (18”, 36”, 54” and 72”) with respect to the loading direction were used. Holographic Interferometry was used to observe crack initiation and propagation. It was possible to detect crack extensions during the experiments, hence the load vs crack extension curve for each notch orientation angles could be obtained. A separate Holographic Interferometry method was used to measure crack opening and sliding displacements by making four independent observations of holographic fringes from the same holographic plate. Crack initiation theories were employed to study their relative merits for predicting crack initiation angles and loads. A Finite Element Method (FEM) using quarter point singular crack tip element was used to calculate Stress Intensity Factors (SIF) and crack surface displacements for different inclinations and extensions of the propagating cracks. It was found that while crack initiation was predicted well by some of the theories, it was necessary to account for the traction forces on the crack surface before any propagation criterion could be identified. Opening and sliding of the crack faces determined by Holographic Interferometry (HI) and clip gage measurements were used to find the normal and shear traction applied to the propagating crack faces. The SIFs for the traction free cracks were corrected by taking into account normal and shear tractions along the propagating crack. It was concluded that the Maximum Hoop Stress Criterion was reasonable and K, stress intensity factor at the tip of the propagating crack was dominant in the failure mechanism.
INTRODUCTION LINEAR Elastic Fracture Mechanics (LEFM) has been used to study the quasi-brittle behavior of materials such as concrete, rocks and ceramics. It has been reported that the measured fracture toughness values (K,,) varied with the experimental method, specimen size and specimen geometry[l]. In a typical mode I type of test, the fracture toughness value increases as the crack progresses leading to a R-curve type of behavior. This additional toughness has been attributed partly to the ligaments behind the crack tip which have to be broken before the crack can progress. Experimental observations of acoustic emission source locations in a uniaxial tension test showed that significant acoustic emission activities occur behind the crack tip (Fig. 1) indicating ligament connection[2]. This may mean that the actual crack is not traction free and the forces transmitted across the crack faces (by ligament connections etc.) affect the stress intensity factors (SIF) at the crack tip. This could possibly be accounted for in a cohesive crack type model (Fig. 2) where forces transmitted across a crack are included in the fracture toughness calculations. The application of such a model for mode I fracture is beginning to be understood[3-81. However, many structural applications involve a mixed mode loading situation. It is necessary to see if an extension of the mode I model could possibly be used for this case where tensile opening of the crack faces is also accompanied by sliding. A limited set of studies have been conducted on mixed mode loading on concrete and in other brittle materials[9-191. Most of this work has been done in tension-shear condition and there is a lack of information about mixed mode fracture under compression-shear loading. This information is necessary for finite element modeling and for a better understanding of material behavior.
DETERMINATION
OF CRACK-TIP
STRESS
FIELD
Consider an infinite plate under uniaxial uniform compressive load with a central crack of length 2a and inclined by B to the direction of applied load as described in Fig. 3. In this case, 129
130
A. K. MAJI et al.
AE sources
a*:: t
I .*
-
Crack
Ligament
,
Elastic
connection
i Fig. 1. Localization of acoustic emission (AE) sources.
Fig. 2. LEFM model with a cohesive crack.
the crack is subjected to a far field stress condition of au normal to the crack, 0, parallel and a shear stress of fXy. These stresses can be written as [19]: oY= --asin*/?,
crX= -aces*/?,
rX,,= -osinficos/I
(1)
where CJ> 0 for compressive stress. If the tip is ideally sharp (zero curvature), ox does not effect the near crack tip singular stresses. The SIFs K, in mode I and K,, in mode II can be expressed as follows[l9]: K,=
-aJ’&sin*fi
K,, = -0
,/&
sin /I cos /I
(2.1) (cr > 0).
(2.2)
If the crack closes depending on load configuration, K, becomes negative. The negative sign of K,, implies an opposite direction of the shear stress component. In the case of the crack tip having a finite curvature (p), cr, produces tensile stresses near the crack tip[l9]. For a slender elliptical slit where 2a and 26 describe the major and minor axis, the maximum hoop stress (G,,,) at the crack front is equal to the parallel stress CY~ (Fig. 3), i.e. (3)
Enlorgsmsnt of crack tip
Fig. 3. Inclined crack in an infinite plate.
Mixed mode crack propagation in quasi-brittle materials
131
For this case, the opening mode associated with K, is given as:
from equations (I), (3) and (4), it follows that: K,= 1/2Jpla*a&cos2/3.
(5)
By combining the stress intensity factors from eqs (2) and (5), the final expressions for K, and K,, imply the following: K, = Q fi (l/2 fi cos2 j!l - sin2 j?) (6.1) K,, = --d fi
sin j? cos j?.
(6.2)
Based on these values of SIFs, different fracture theories can be considered: (a) Maximum Hoop Stress Theory[20], (b) Strain Energy Release Rate Theory[21], and (c) Strain Energy Density Theory[22]. FINITE
ELEMENT
MODELING
An interactive Finite Element Program (FRANC) using quarter point singular elements[23] was used. A program was written to generate the input meshes corresponding to different values of the crack inclination angles (8). A typical mesh for fi = 36” is shown in Fig. 4. The K, and K,, values were also obtained from analysis of sharp cracks by the program. These values were checked against available solutions obtained by Murakami[24] and against the solutions for the infinite plate (eqs 6.1 and 6.2). The results are shown in Fig. 5. The FEM computations were within 5% of the theoretical values. HOLOGRAPHIC
INTERFEROMETRY
(HI)
Various techniques using Holographic Interferometry have been utilized[25]. The primary advantages of the technique are as follows: (a) The high sensitivity of crack detection (0.1 micron). (b) The technique is noninvasive and the specimen is not disturbed. (c) Whole field detection makes it useful where crack path is not known a priori and to detect cracking in unexpected places.
m 0
l Finite element 0 Murakomi l Int inits plate (rqn. 2.1)
l
- 600
0 I -
Y
-400
-6’k
0
-2tLL18
Fig. 5. Comparison Fig. 4. Starting FEM mesh with slit crack.
36
54
72
of K, from different techniques for sharp crack.
132
A. K. MAJI et al.
Double-exposure holograms were used because of their brightness. In this technique, two holographic images are formed at two stages of loading. Displacement between the two loading stages can then be observed as fringes running across the specimen. These fringes correspond to the component of the displacement along the sensitivity vector which will be defined later. This process forms a permanent recording of displacements between two stages of loading which could be viewed at any later time[26]. Cracking is visible as discontinuities in the otherwise continuous fringes.
QUANTITATIVE
HOLOGRAPHY
Although Holographic Interferometry is a common inspection tool, the measurement of displacements by this technique is difficult due to the necessity of separating the three individual displacement components[27]. A holographic interference pattern contains information about the component of displacement along a sensitivity vector. This sensitivity vector is the bisector of the illumination and observation directions for any object point. Hence, a minimum of three different observation directions are necessary to get three different sensitivity vectors. This would provide a set of three equations involving the three displacement components. If the three directions are close, the equations are similar and the process loses some of its sensitivity. The individual viewing directions have to be as widely separated as possible to give maximum sensitivity. In the experimental technique used, the holographic plate was positioned very close to the inspection area (Fig. 6). The plate was placed 45 in front of the specimen. Since the plate dimensions are 4” x 5”, this made it possible to observe the crack from four corners of the same plate hence providing four separate viewing directions. Double exposure holograms were made with this setup at different loading stages so that each hologram contained displacement information between two subsequent loading stages. These were inspected at the end of the test to obtain crack opening and sliding data. It was difficult to observe the fringes and count them while looking at the virtual image of the specimen through specific points on the holographic plate. This is because the image is some distance away from the plate and there is no way of touching it. Also, it is difficult to position the eye at specific points near the plate. To circumvent this problem, a point by point filtering technique was adopted using an unexpanded laser light (Fig. 7). The laser beam was passed through one corner of the developed plate at a time. Proper allignment of the plate to the laser beam formed a real image of the specimen with the displacement fringes on a screen placed behind the plate. Allignment of the plate to get the brightest image could be determined from the initial reference beam angle used to make the holograms. Difference in fringe counts on either side of the cracks could be easily noted from the images projected on the screen.
1
He-Ne
laser
2
Manual
shutter
3
Boom-elevator
4
Beam-splitter
5
Reference
6
Object
7
Specimen
6
Cardboard
9
Holographic
pin-hole
pin-hole
block plate
Fig. 6(a). Holographic interferometry setup for quantitative measurements.
133
Mixed mode crack propagation in quasi-brittle materials (b)
Magnrtlc MagnWc
bar*
baee Fig. 6(b). Position of plate in front of specimen.
The four different fringe counts obtained by looking through the four corners provided four equations with the three unknown displacement components. The equations involved the four sensitivity vectors which were obtained from measuring the coordinates of the optical components and the object. A computer program was then implemented to find the vectors, set up the equations and solve the set of four equations by a least squares approach to find the displacement components. EXPERIMENTAL STUDY Test specimtw Specimens were made of mortar with a 3 : 1: 0.6 mix by weight of sand, cement and water. Type I cement was used. Four types of specimen were tested with varying angles of the initial crack (fl = 18”, 36”, 54” and 72”). Specimens were 3” thick, 6” wide and 11” high. (Fig. 6b). Separate end Screen
Plate
holder
Fig. 7. Setup to inspect real image and count fringes from holograms.
A. K. MAJI
134
et al.
blocks were cast to eliminate end effects[28]. Molds were made of plexiglass and the notches were cast by 0.078” (1.98 mm) thick and 2” long steel blades. The blade could be rotated to generate the four different inclination angles. Plexiglass separators l/6” thick were used to cast the end blocks. The molds were removed after 24 hours and the concrete was cured in water at room temperature thereafter. The steel blade was removed 4; hours after casting for easy removal. The specimens were tested approximately 60 days after casting. Test procedure
Specimens were tested in a 120 Kip capacity MTS closed loop machine. A quick-setting cement was used for capping. Surfaces joining the end blocks to the central block were greased to distribute the load and reduce friction. Axial deformation was measured by Linear Variable Displacement Transducers (LVDT). Specially designed clip-gages were used to measure opening and sliding of the crack faces close to the crack tip. The direction of crack propagation was observed from testing one specimen from each group and the clip-gages were thereafter placed perpendicular and parallel to that direction to measure the normal and shear displacements respectively. The aluminum clip-gage mounts were attached to the mortar by superglue adhesive and epoxy. Data acquisition was through a Nicolet 4094 Digital Oscilloscope. Two types of Holographic Interferometry tests were done. Two specimens of each group were tested by a Holomatic 6000 system to study crack propagation[26]. This process used only one viewing direction and could not measure displacements. However a large number of holograms could be made because the process is automated by the Holomatic 6000 system and the film used is more economical. The load at different crack extension lengths were recorded. The third specimens were tested using holographic plates and the process described earlier (Figs 6 and 7). This test made it possible to measure crack opening and sliding displacements.
EXPERIMENTAL OBSERVATIONS The initiation of the crack from the precast notch and the propagation of the kink-crack were observed with the aid of holographic interferometry. Various stages of kink-crack development for a specimen with the notch inclined at angle /I = 36” are shown in Fig. 8. Some typical fringe patterns for other cases are illustrated in Fig. 9. From such observations, kink initiation stress and stresses for various kink lengths were calculated and are plotted in Fig. 10. The crack opening and crack sliding displacements measured by clip-gages are illustrated in Fig. 11. For brevity, only the results for p = 18” and /I = 54” are given. The other details are reported in [29]. The holographically measured kink profiles for various kink lengths for /I = 36” and /? = 72” are plotted in Figs 12 and 13. Comparisons of the measured values with those predicted using LEFM (analytical-infinite plate and FEM-finite plate) are described below. ac r
60
20
I+
1”
08
Kink
Fig. 10. Load
1 I/;
extension
vs kink extension
lengths.
2”
Mixed mode crack
propagation
Fig. 8. Crack
in quasi-brittle
development
for /I = 36”.
materials
135
136
A. K. MAJI
Fig. 9(c-d).
Holographic
intcrferometry
et al.
fringes
for B = 54” and 72’ specimens
Mixed mode crack propagation in quasi-brittle materials
137
18O
60
Fig. 11. Comparison of opening and sliding displacements at the crack mouth from clip gage and FEM for 18” and 54” specimens.
KINK INITIATION DIRECTION
The direction of kink-crack initiation can be predicted from the knowledge of the stress-intensity factors (K, and K,,) at the tip of the notch and using one of the following three theories. The maximum hoop stress theory
According to this criteria given by Erdogan and Sih[20], the crack propagates in a radial direction from its tip and the direction of crack growth is normal to the maximum tangential stress (also called hoop stress or circumferential stress), cO. The hoop stress can be given by: 0 cos t%=s
1
K, cos* i - i K,, sin 8 - c cos* B sin* 8. [
(7)
The equation shown above is based on William’s analysis which contains the additional nonsingular term for crack-tip curvature[30]. From eqs (6) and (7) and maximising erg,the direction of 8 can be calculated as shown in Fig. 14. The direction of crack propagation 8,, is given by: aa&?
= 0
and
a2o0j&32< 0.
The values of p and a in our experiment were 1 mm and 25.4 mm respectively, resulting in p/a = 0.039.
The mode I SIF in direction 0 is related to this hoop stress rrOas:
The K,(8) criterion and crBcriterion are hence analogous.
A. K. MAJI et
138
al.
800
-
Holographic
----
FEM
profilm
profile
600 0
.c
b .g
400
5 0” xx,
0
400 .c 0 b 0, .E 0 7 v)
-
Holographicprofile
----
FEM
proflle
200
0
1 ”
II+”
extension
length
‘4’
Crock
Fig. 12. Profiles of crack opening and sliding for 36” specimen.
.c
-
Holographic
----
FEM
profile
profile
-
Holographic
----
FEM
profile
profile
‘f s ? z ._ CG I 1”
0
Crock
2”
extension length
Fig. 13. Profiles of crack opening and sliding for 72” specimen.
Mixed mode crack propagation in quasi-brittle materials
139
For +=0.039 Criteria , 1400
I-
criterion
/
Prrsrnt 0
PMMAlTirosh
I 20°
0
S,,,
:
Notch
rxpsrimmts
:+
ond Cot2 (19)):
+
I
I
I
40’=
60°
60*
orientation
angle
(fi
-0.039 -4031
1
Fig. 14. Comparison between theoretical and experimental crack initiation angles.
The energy release rate theory
This theory, used by Hussain et a1.[21] evaluates the energy release rate, G, at the kinked crack which propagates from the main crack in the direction of 8. The criteria for crack propagation are: (i) The crack will propagate in the direction which causes the maximum energy release rate to occur and (ii) Propagation occurs when the release rate in that direction reaches a certain value. The strain energy release rate is given as:
e/n
G(@=!
’
[( 1 + 3 cos* 8)K:-
E (3 + cos* 0)’
8K,K,, sin 8 cos 8 + (9 - 5 COS*e)K;,]. (8)
By substituting K, and K,, from eq. (6) into (8) and maximising eq. (8), i.e. for aG/ae = 0 and a*G/@ < 0, the 8 vs p relation can be obtained as shown in Fig. 14. The strain energy density theory
In this theory developed by Sih[22], the strain energy density factor, S can be written as:
s(e) = a,,Kf
+ 2 a,,K,K,,
+ a,,Ki,
in which a,, = [(K - cos e)(i + cos 0)]/167rp a,* = [(2 cos 8 - K + l)]sin 8/167rp a,,=[(rc
+ l)(l -cose)+(i
+c0se)(3c0se
-
1)]/167tp
where IC= 3 - 4v for plane strain and K = (3 - v)/(l + v) for plane stress. p = E[2(1 + v)] (Shear Modulus), where E is Modulus of Elasticity and v is Poisson’s Ratio. EFM
38-213-C
140
A. K. MAJI
et al.
Sih proposed the following criterion: (i) The crack initiation direction is obtained from the minimum strain energy density factor, S, as: aS/ae = 0 and a2SlLY12> 0 for crack initiation angle. (ii) The crack starts to propagate when S reaches a critical value, S, at 19= 19,,,the crack initiation angle. Substituting K, and K,, from eq. (6) into eq. (9) and considering the plane strain case where K = 3 - 4v and v = 0.21, the relationship between 8, and fl can be obtained as shown in Fig. 14. The results determined by the three crack initation criteria are plotted in the same figure for comparison with the experimental observations. Experimental initiation angle was obtained by joining the initial notch tip to a point on the crack l/2” away. It is seen in this figure that the hoop stress theory gave a better fit than the other two criteria for smaller values of the inclination angle /I. But this theory overestimated experimental results after /I = 40” when the G criterion shows a better match with the experiments. Incidentally, a similar trend was obtained by Tirosh and Catz[l9] for the material PMMA (p/a = 0.031) as depicted in Fig. 14. Kink initiation stress
If kink initiates when either cB (or equivalently, K,), G(B) S(e) reaches a critical value as postulated by the three theories discussed, then it is possible to predict the stress at which the kink-crack starts propagating from the main-crack (precast-notch). The value of this initiation stress (gi) is given by the following three equations for each of the three theories. These three equations were based on respectively maximizing eqs (7) and (8) and minimizing eq. (9). These equations are valid for infinite plate geometry. Thus the kink initiation stress (ai) can be written as follows: For maximum hoop stress theory,
8
8
cos2 fl cos3 z - 2 sin2 /I cos3 z + 3 sin /I cos /I
For maximum energy release rate theory, un
l-t! ~;=~~(~+cos~s)~
l-8 i
+ 2( i fi
?t
I
[(l+ 3cos2e)(~Jpiacos2p
-sin2fl)=
7t
cos2 fi - sin’ &sin 2/I sin 28 + (9 - 5 cos2 8)sin2 /3 cos2 /.I)]- I. (11)
For maximum strain energy density theory,
+ a2*sin’ p cos’/?-‘.
(12)
In the above equations 8 = 8,,, the crack initiation angle (Fig. 14). The above equations are plotted in Fig. 15 with the theoretical and experimental data obtained by using the theoretical and experimental 8,, values. Since the specific critical values of K,,, G, and S, are not known, the values of initial stress were normalized by dividing each value with that for fl = 54”. The comparison between the theory and experiments is perhaps qualitatively acceptable. Prediction of crack initiation stress in a finite sized specimen based on infinite plate theories is not very accurate. Kink propagation criteria
The actual finite specimen with a blunt notch-tip was studied by finite element method. The experimentally observed kink direction and kink length were used in the finite element analysis. The holographically observed kink path was approximated by a series of l/4” straight lines. For
141
Mixed mode crack propagation in quasi-brittle materials
(a) Maximum hoop stress theory : Experiment
I
-
+
(b)
50.039 Eq.10
_--J=
D
0
._
C0 3
I I
b
Strain cnsrgy release rate theory
%3 e
Experiment
4
: ---
=0.039
p =o
Ea.11
Cl 1
%
E
2
.o Y .c
.
2
‘\ \ \ \ \
‘0
2 j_ 0
1
E z I 20’
Notch
I 400
I 600
I
60°
orientation
angle
(fi
0
)
Fig. 15(a). Relation between the normalized initiation stress and crack orientation angle (co,, criterion).
--__A
400
200
Notch
.
orientation
60°
angle
(p
60.
)
Fig. 15(b). Relation between the normalized initiation stress and crack orientation angle (G,, criterion).
(cl Stroin density
6‘;;
3-
Experiment
E t; g 3 .;
energy theory
:
-7 ---
J
‘0.039
4
=O
Eq.12
Cl I
.
2
z._ P .N OlE 0’ z
0
I
I
I
I
I
200
400
60°
600
Notch
orientation
angle
(B 1
Fig. 15(c). Relation between the normalized initiation stress and crack orientation angle (S,, criterion).
each step of kink extension, the stress intensity factor corresponding to unit load was calculated. The actual SIFs for kink propagation were then obtained by multiplying these by the experimentally observed crack propagation stress using holography. The calculated values of K, (critical) and K,, are plotted in Fig. 16. It can be seen that the value of K,, decreases with kink extension, perhaps indicating that kink crack propagates in the direction of the critical value of K,. However, the value of K,(critical) is different for different values of B and for different values of crack extension. Thus K,(critical) as determined above can not be regarded as a valid crack propagation criterion. Note that the values plotted for the zero kink extension in Fig. 16 corresponds to sharp crack rather than the actual blunt crack. The calculated values of crack opening and sliding displacements calculated using finite element method are compared with the data in Figs 12 and 13. It can be observed that the openings observed experimentally for the 18” and 36” specimen were much larger and those for the 72” specimen much smaller than the FEM predictions.
142
A. K. MAJI et
al.
600
lh"
l+"
1I+”
1 ‘I
Crack
2"
length
Fig. 16(a). K,_ at crack propagation stages from FEM. 500
720 400
549 36O 160
300 : 3r Y
200 A 0 A
100
0 '
0 0
A
. .
.
Q
9 1”
‘/z I1
Crack
0
09 1ti
I
2"
lengths
Fig. 16(b). KI,at crack propagation stages from FEM.
EXTENSION OF COHESIVE CRACK MODEL The difference in the K, values for different types of specimens as observed in Fig. (16) and the discrepancy in displacement measurements (Figs 12 and 13) motivated further investigation into the nature of kind cracks. It is possible that an extension of the ‘Cohesive Crack Model’ (Fig. 2) is necessary to incorporate tractional forces on the crack face in the mixed mode conditions. The cohesive crack models used for mode I crack propagation usually consist of applying normal tractions across the crack surface (Hillerborg, Kobayashi, Jenq, Ingraffea etc. [4]). The magnitude of such normal closing pressures is inversely proportional to the crack opening displacement. The relationship between closing pressure and opening displacement are considered material property and can be determined experimentally[31]. For the mixed mode case, cracks open as well as slide and thus one must include both the normal and shear tractions. The relationship between normal traction, shear traction and the corresponding opening and sliding displacements include complex coupling and are not easy to measure experimentally[32-381. When a shear stress is applied to a crack in dilating materials like concrete and rock, sliding is accompanied by opening. The relationship between the sliding and
Mixed mode crack propagation in quasi-brittle materials
143
opening displacement depends on applied shear traction, possible normal traction and the initial shape of the crack. It was concluded from the review of the available texts cited above that the discrepancy between the measured and the (LEFM) calculated opening and sliding displacement can be accounted for by the addition of shear and normal tractions. To account for unknown interface traction forces, additional FEM based study was done. Unit opening and sliding forces were placed on intermediate nodes on the crack faces in the FEM mesh. The opening and sliding displacements corresponding to these unit forces were determined from analysis. This yielded a stiffness matrix S which contained nodal opening and sliding for unit opening or sliding loads. A different matrix D was set up which contained the difference in nodal displacements between the FEM analysis and experimental observations (Figs 12 and 13). For the j3 = 36” and 72” specimens, a set of equations were used to match the FEM and experimental crack profiles:
Kl PI = PI
(13)
where S and D are the matrices described above. The matrix L is determined by solving the above equation. This matrix contains the nodal opening and sliding loads which should be applied so as to make the FEM and experimental crack profiles the same. For a crack length with four intermediate nodes, the L matrix would have 4 x 2 = 8 terms. These will be the opening and sliding loads at those four nodes. The value of K,(critical) shown in Fig. (16) was corrected by adding the Ki values obtained from these nodal loads. The corrected values of K,(critical) and K,, shown in Fig. (17) seem to be within a reasonably narrow band. The ratio of K,critical/KIIis always greater than 1.0. This shows the possibility of using a K, criterion to predict crack growth where the crack face tractions can be accurately known from previous experiments on their constitutive behavior. For the /I = 18” and 54” specimens, only the clip gage location measurements were matched assuming uniform distribution of nodal forces to get Ki (Fig. 17). Table 1 shows some applied nodal forces for a crack length with four intermediate nodes. The crack length in this case was 1.9” long. So the length of crack per node was approximately l/2”. The stresses on the crack surface at the nodes were obtained by dividing the nodal forces by 0.5” and were in the range of 48 to 166 psi. In a study of interface behavior conducted by Van Mier[37],
500 r
(a)
0
.
A 300 -
.
:: 0 . ,.
l
0
A 0 0
A 0
1
1 ‘4”
0
200 -
54’ 100 -
A
36O . 160 0
$2”
II
2”
Crack length Fig. 17(a). Kcn,,c, after matching crack surface displacements.
A. K. MAJI et al.
144 lb) 300r
200
72O0 54’ 36O
A 0
160
0
0
-
IOO-
. 0
0
A 0 0
8
0
0
I+”
1
0
II
4
I
1’4”
2”
Crack length Fig. 17(b). I,, after matching displacements by nodal forces in FEM.
a shear load was applied to a crack which was under tensile stress. A shear load of 7.5 kN (corresponding to about 150 psi over the specimen cross section area) was found to cause a drop in the tensile stress of 5 N (corresponding to about 100 psi) when the crack opening was held constant at about 450 p in. It must be noted that 300 ~1in was a typical crack opening displacement in our experiments. Hence, a tensile pressure of 48 or 166 psi could have been caused by dilatancy due to shear stress on a crack surface. SUMMARY
OF RESULTS
(1) The Maximum Hoop Stress (MHS) or K,,, criterion and the Strain Energy Release Rate or G criterion can predict crack initiation angle and stress for compression-shear failure. Crack propagation is in tension-shear once the kink crack initiates. (2) Experimental Normalized Stress Ratio at kink initiation is well predicted by the MHS and Strain Energy Density (SED) theories. (3) Although the MHS criterion based on a traction free crack could predict crack initiation, it could not predict propagation load because of the coupling between sliding and opening displacement intensified by granularity or mortar. (4) The MHS criterion was reasonably good if the FEM predicted crack face displacements were corrected by applying normal and shear tractions to match those observed experimentally at the different propagation stages. The K, Stress Intensity Factor was dominant in the failure mechanism. While the basic physical arguments for such corrections were presented, it proves the necessity of conducting investigations into constitutive relations for compression-sliding of quasi-brittle materials in order to predict crack propagation under mixed mode conditions. (5) Holographic interferometry has been an excellent tool to detect cracks during the tests in real time. The high sensitivity of the order of 0.1 microns is a unique method of studying fracture process and provides improved crack tip location. (6) A single plate holographic interferometry could be used for quantitative measurement of crack surface opening and sliding. This process is relatively inexpensive and useful in studying features of the process zone behind the crack tip. Table 1. Nodal forces required to match crack profiles for B = 36” (Crack extension length was 1.9” for this case) Node’s distance from tip of kink 1.5” 1.1” 0.7” 0.3”
Nodal force (lbs) 29 24 31 83
Normal stress on interface (psi) 58 48 62 166
Crack opening p IN.
Crack sliding /I IN.
580 390 280 140
130 20 10 10
Mixed mode crack propagation in quasi-brittle materials
145
Acknowledgements-The research reported here is supported by the U.S. Air Force Office of Scientific Research through a grant (AFOSR-85-0261) to Northwestern University. The helpful suggestion of the AFOSR program manager Dr Spencer T. Wu is gratefully acknowledged.
REFERENCES PI S. P. Shah, Fracture toughness of concrete, in Fractureof Concrete und Rock (Edited by S. P. Shah and S. E. Swartz), pp. l-12. Springer (1988). PI A. K. Maji and S. P. Shah, Process zone and acoustic emission measurements in concrete. Experimental Mechanics, March (1988). I31 F. H. Wittman (Editor), Fracture Mechanics of Concrete, p. 680. Elsevier, Amsterdam (1983). S. P. Shah (Editor), Application ofFracture Mechanics to Cementitious Composites. Martinus Nijhoff, The Hague (1985). ;; G. C. Sih and A. Ditommasd (Editors), Fracture Mechanics of Concrete: Structural Application and Numerical Calculation, D. 276. Martinus Niihoff, The Haaue (1985). [61 A. Carpinte; and A. R. Ingraffea (Editors), Ft&ture Mechanics of Concrete: Material Characterization and Testing, p. 202. Martinus Nijhoff, Dordrecht, (1984). International Conference on Fracture of Concrete and Rock, I71 S. P. Shah and S. E. Swartz (Editors), SEM/RILEM Houston, TX (1987). 181Z. P. Bazant and L. Cedolin, Fracture mechanics of reinforced concrete. ASCE J. of EMD. 106, 1287-1306, December (1980). [91 Y. S. Jenq and S. P. Shah, Mixed mode fracture of concrete. Int. .I. of Fracture 38, 123-142 (1988). PO1 Z. P. Bazant and P. A. Pfeiffer, Shear fracture tests in concrete. Muter. Structures, RILEM 19 110, 111-121 (1986). IllI S. Huang, K. Long and L. Zhang, Experimental investigation on the fracture criterion of mixed mode I and II for V-Notch in Concrete. SEMIRILEM Int. Conf on Fracture of Concrete and Rock (Edited by S. P. Shah and S. E. Swartz). Houston, Texas (1987). [121 S. E. Swartz, L. W. Lu and T. L. Tang, Mixed-mode Fracture Toughness Testing of Concrete Beams in Three Point Bending, p. 20. Kansas State University, Manhattan, Kansas (1987). 1131 M. Arrea and A. R. Ingraffea, Mixed-mode Crack Propagation in Mortar and Concrete. Report No. 81-13, p, 143. Department of Structural Engineering, Cornell University (1982). 1141P. Desayi, Fracture of concrete in compression. Muteriaux et Constructions, 10 (57), 139-144 (1977). 1151J. Glucklich, Fracture of plain concrete. J. Engng Mechanics Division ASCE, 127-137 (December 1963). 1161S. S. Diaz and H. K. Hilsdorf, Fracture mechanism of concrete under comnressive loads. Cement and Concrete Research 3, 363-388 (1973). nonplaner crack extension with application to splitting 1171 S. Nemat-Nasser and H. Horii, Compression-Induced exfoliation, and rockburst. J. Geophys. Rec. 87, 6805-6821 (1982). 1181H. Horri and S. Nemat-Naser, Brittle-failure in compression: splitting, faulting and brittle-ductile transition, Phi. Trans. R. Sot. London, A319, 339-374
(1986).
1191J. Tirosh and E. Catz, Mixed mode fracture angle and fracture locus of materials subjected to compressive loading. Engng Fracture Mech. 14, 27-38 (1981). PO1 F. Erdogan and G. C. Sih, On the crack extension in plates under plane loading and transverse shear. J. Busic Engng 85, 519-527 (1963). I211 M. A. Hussain, S. L. Pu and J. H. Underwood, Strain energy release rate for a crack under combined mode I and mode II. Fracture Analysis, ASTM STP 560, 2-28 (1974). G. C. Sih, Strain energy density factor applied to mixed-mode crack problems. Int. J. Fracture, 10, 305-321 (1974). t::; P. A. Wawrzynek, Interactive finite element analysis of fracture processes: an integrated approach. Struct. Engng Report No. 87-1 (Supervised by A. Ingraffea), Cornell University (1987). Y. Murakami, Stress Intensity Factors Handbook, pp. 909-911. Pergamon Press, New York (1987). t;z; Nils Abramson, The Making and Evaluation of Holograms. Academic Press (1981). P61 A. K. Maji and S. P. Shah, Application of Acoustic Emission and Laser Holography to Study Microfracture in Concrete. ACI SP-112, Nondestructive Testing (1988). 1271 R. A. Miller, S. P. Shah and H. I. Bjelkhagen, Crack profiles in mortar measured by holographic interferometry. Exp. Mech. 338-393 (Dec. 1988). 1281 S. P. Shah and R. Sankar, Internal cracking and strain softening response of concrete under uniaxial compression. ACI Muter. J. 84 (3), 200, (May-June 1987). 1291A. K. Maji, Study of Concrete Fracture Using Acoustic Emission and Laser Holography. Ph.D. dissertation submitted to Northwestern University, Evanston, IL (August 1988). M. L. Williams, On the stress distribution at the base of a stationary crack. J. uppl. Mech. 24, 109 (1957). t:;; V. S. Gopalaratanam and S. P. Shah, Softening response of plain concrete in direct tension, ACI I, 3 10-323, May-June (1985). 1321 &I. P.‘Divakar, A. Fafitis, S. P. Shah, A constitutive model for shear transfer in cracked concrete. ASCE J. Structural Engng 113, (5) May (1987). [331 A. R. Ingraffea and W. H. Gerstle, Non-linear fracture models for discrete crack propagation. Proc. of NATO Advanced Reseurch Workshov. (Edited bv S. P. Shah.) Sent. (1984). [341 R. C. Fenwick andT.‘Paulay, Mechanics of shea; resistance of concrete beams. ASCE I. Structures Division 94 (STlO), 2325-2350 (1968). [351 Z. P. Bazant and T. Tsubaki, Slip-dilatancy model for cracked reinforced concrete, ASCE I. Structures Division, 1947-1966 (Sept. 1980). I361 M. Hassanzadeh, A. Hillberborg and Z. Ping, Tests of material properties in mixed mode I and II. Proc. SEM-RILEM Int. Conf Fracture Concrete Rock (June 1987). I371 J. G. M. Van Mier, Fracture study of concrete specimens subjected to combined tensile and shear loading. In?. Conf Measurement Testing Civil Engng, Lyon (Sept. 1988). [381 Z. P. Bazant and P. Gambarova, Crack shear in concrete: crack band microplane model. J. Structure Division, ASCE, 110, 2015-2035 (Sept. 1984). (Received 10 November 1989)
Vol. 38, No. 213,pp. 129-145,
1991
0013-7944/91
$3.00 + 0.00
0 1991 Pergamon Press plc.
Printed in Great Britain.
MIXED
MODE CRACK PROPAGATION QUASI-BRITTLE MATERIALS
IN
A. K. MAJIt, M. A. TASDEMIRS and S. P. SHAHS tuniversity
of New Mexico, Albuquerque, New Mexico, U.S.A. SIstanbul Technical University, Turkey #NSF Science Technology Center for Advanced Cement Based Materials, Northwestern University, Evanston, Illinois, U.S.A. Abstract-Crack propagation in rectangular blocks of mortar containing a central notch and subjected to uniaxial compression were studied. Four different notch orientation angles (18”, 36”, 54” and 72”) with respect to the loading direction were used. Holographic Interferometry was used to observe crack initiation and propagation. It was possible to detect crack extensions during the experiments, hence the load vs crack extension curve for each notch orientation angles could be obtained. A separate Holographic Interferometry method was used to measure crack opening and sliding displacements by making four independent observations of holographic fringes from the same holographic plate. Crack initiation theories were employed to study their relative merits for predicting crack initiation angles and loads. A Finite Element Method (FEM) using quarter point singular crack tip element was used to calculate Stress Intensity Factors (SIF) and crack surface displacements for different inclinations and extensions of the propagating cracks. It was found that while crack initiation was predicted well by some of the theories, it was necessary to account for the traction forces on the crack surface before any propagation criterion could be identified. Opening and sliding of the crack faces determined by Holographic Interferometry (HI) and clip gage measurements were used to find the normal and shear traction applied to the propagating crack faces. The SIFs for the traction free cracks were corrected by taking into account normal and shear tractions along the propagating crack. It was concluded that the Maximum Hoop Stress Criterion was reasonable and K, stress intensity factor at the tip of the propagating crack was dominant in the failure mechanism.
INTRODUCTION LINEAR Elastic Fracture Mechanics (LEFM) has been used to study the quasi-brittle behavior of materials such as concrete, rocks and ceramics. It has been reported that the measured fracture toughness values (K,,) varied with the experimental method, specimen size and specimen geometry[l]. In a typical mode I type of test, the fracture toughness value increases as the crack progresses leading to a R-curve type of behavior. This additional toughness has been attributed partly to the ligaments behind the crack tip which have to be broken before the crack can progress. Experimental observations of acoustic emission source locations in a uniaxial tension test showed that significant acoustic emission activities occur behind the crack tip (Fig. 1) indicating ligament connection[2]. This may mean that the actual crack is not traction free and the forces transmitted across the crack faces (by ligament connections etc.) affect the stress intensity factors (SIF) at the crack tip. This could possibly be accounted for in a cohesive crack type model (Fig. 2) where forces transmitted across a crack are included in the fracture toughness calculations. The application of such a model for mode I fracture is beginning to be understood[3-81. However, many structural applications involve a mixed mode loading situation. It is necessary to see if an extension of the mode I model could possibly be used for this case where tensile opening of the crack faces is also accompanied by sliding. A limited set of studies have been conducted on mixed mode loading on concrete and in other brittle materials[9-191. Most of this work has been done in tension-shear condition and there is a lack of information about mixed mode fracture under compression-shear loading. This information is necessary for finite element modeling and for a better understanding of material behavior.
DETERMINATION
OF CRACK-TIP
STRESS
FIELD
Consider an infinite plate under uniaxial uniform compressive load with a central crack of length 2a and inclined by B to the direction of applied load as described in Fig. 3. In this case, 129
130
A. K. MAJI et al.
AE sources
a*:: t
I .*
-
Crack
Ligament
,
Elastic
connection
i Fig. 1. Localization of acoustic emission (AE) sources.
Fig. 2. LEFM model with a cohesive crack.
the crack is subjected to a far field stress condition of au normal to the crack, 0, parallel and a shear stress of fXy. These stresses can be written as [19]: oY= --asin*/?,
crX= -aces*/?,
rX,,= -osinficos/I
(1)
where CJ> 0 for compressive stress. If the tip is ideally sharp (zero curvature), ox does not effect the near crack tip singular stresses. The SIFs K, in mode I and K,, in mode II can be expressed as follows[l9]: K,=
-aJ’&sin*fi
K,, = -0
,/&
sin /I cos /I
(2.1) (cr > 0).
(2.2)
If the crack closes depending on load configuration, K, becomes negative. The negative sign of K,, implies an opposite direction of the shear stress component. In the case of the crack tip having a finite curvature (p), cr, produces tensile stresses near the crack tip[l9]. For a slender elliptical slit where 2a and 26 describe the major and minor axis, the maximum hoop stress (G,,,) at the crack front is equal to the parallel stress CY~ (Fig. 3), i.e. (3)
Enlorgsmsnt of crack tip
Fig. 3. Inclined crack in an infinite plate.
Mixed mode crack propagation in quasi-brittle materials
131
For this case, the opening mode associated with K, is given as:
from equations (I), (3) and (4), it follows that: K,= 1/2Jpla*a&cos2/3.
(5)
By combining the stress intensity factors from eqs (2) and (5), the final expressions for K, and K,, imply the following: K, = Q fi (l/2 fi cos2 j!l - sin2 j?) (6.1) K,, = --d fi
sin j? cos j?.
(6.2)
Based on these values of SIFs, different fracture theories can be considered: (a) Maximum Hoop Stress Theory[20], (b) Strain Energy Release Rate Theory[21], and (c) Strain Energy Density Theory[22]. FINITE
ELEMENT
MODELING
An interactive Finite Element Program (FRANC) using quarter point singular elements[23] was used. A program was written to generate the input meshes corresponding to different values of the crack inclination angles (8). A typical mesh for fi = 36” is shown in Fig. 4. The K, and K,, values were also obtained from analysis of sharp cracks by the program. These values were checked against available solutions obtained by Murakami[24] and against the solutions for the infinite plate (eqs 6.1 and 6.2). The results are shown in Fig. 5. The FEM computations were within 5% of the theoretical values. HOLOGRAPHIC
INTERFEROMETRY
(HI)
Various techniques using Holographic Interferometry have been utilized[25]. The primary advantages of the technique are as follows: (a) The high sensitivity of crack detection (0.1 micron). (b) The technique is noninvasive and the specimen is not disturbed. (c) Whole field detection makes it useful where crack path is not known a priori and to detect cracking in unexpected places.
m 0
l Finite element 0 Murakomi l Int inits plate (rqn. 2.1)
l
- 600
0 I -
Y
-400
-6’k
0
-2tLL18
Fig. 5. Comparison Fig. 4. Starting FEM mesh with slit crack.
36
54
72
of K, from different techniques for sharp crack.
132
A. K. MAJI et al.
Double-exposure holograms were used because of their brightness. In this technique, two holographic images are formed at two stages of loading. Displacement between the two loading stages can then be observed as fringes running across the specimen. These fringes correspond to the component of the displacement along the sensitivity vector which will be defined later. This process forms a permanent recording of displacements between two stages of loading which could be viewed at any later time[26]. Cracking is visible as discontinuities in the otherwise continuous fringes.
QUANTITATIVE
HOLOGRAPHY
Although Holographic Interferometry is a common inspection tool, the measurement of displacements by this technique is difficult due to the necessity of separating the three individual displacement components[27]. A holographic interference pattern contains information about the component of displacement along a sensitivity vector. This sensitivity vector is the bisector of the illumination and observation directions for any object point. Hence, a minimum of three different observation directions are necessary to get three different sensitivity vectors. This would provide a set of three equations involving the three displacement components. If the three directions are close, the equations are similar and the process loses some of its sensitivity. The individual viewing directions have to be as widely separated as possible to give maximum sensitivity. In the experimental technique used, the holographic plate was positioned very close to the inspection area (Fig. 6). The plate was placed 45 in front of the specimen. Since the plate dimensions are 4” x 5”, this made it possible to observe the crack from four corners of the same plate hence providing four separate viewing directions. Double exposure holograms were made with this setup at different loading stages so that each hologram contained displacement information between two subsequent loading stages. These were inspected at the end of the test to obtain crack opening and sliding data. It was difficult to observe the fringes and count them while looking at the virtual image of the specimen through specific points on the holographic plate. This is because the image is some distance away from the plate and there is no way of touching it. Also, it is difficult to position the eye at specific points near the plate. To circumvent this problem, a point by point filtering technique was adopted using an unexpanded laser light (Fig. 7). The laser beam was passed through one corner of the developed plate at a time. Proper allignment of the plate to the laser beam formed a real image of the specimen with the displacement fringes on a screen placed behind the plate. Allignment of the plate to get the brightest image could be determined from the initial reference beam angle used to make the holograms. Difference in fringe counts on either side of the cracks could be easily noted from the images projected on the screen.
1
He-Ne
laser
2
Manual
shutter
3
Boom-elevator
4
Beam-splitter
5
Reference
6
Object
7
Specimen
6
Cardboard
9
Holographic
pin-hole
pin-hole
block plate
Fig. 6(a). Holographic interferometry setup for quantitative measurements.
133
Mixed mode crack propagation in quasi-brittle materials (b)
Magnrtlc MagnWc
bar*
baee Fig. 6(b). Position of plate in front of specimen.
The four different fringe counts obtained by looking through the four corners provided four equations with the three unknown displacement components. The equations involved the four sensitivity vectors which were obtained from measuring the coordinates of the optical components and the object. A computer program was then implemented to find the vectors, set up the equations and solve the set of four equations by a least squares approach to find the displacement components. EXPERIMENTAL STUDY Test specimtw Specimens were made of mortar with a 3 : 1: 0.6 mix by weight of sand, cement and water. Type I cement was used. Four types of specimen were tested with varying angles of the initial crack (fl = 18”, 36”, 54” and 72”). Specimens were 3” thick, 6” wide and 11” high. (Fig. 6b). Separate end Screen
Plate
holder
Fig. 7. Setup to inspect real image and count fringes from holograms.
A. K. MAJI
134
et al.
blocks were cast to eliminate end effects[28]. Molds were made of plexiglass and the notches were cast by 0.078” (1.98 mm) thick and 2” long steel blades. The blade could be rotated to generate the four different inclination angles. Plexiglass separators l/6” thick were used to cast the end blocks. The molds were removed after 24 hours and the concrete was cured in water at room temperature thereafter. The steel blade was removed 4; hours after casting for easy removal. The specimens were tested approximately 60 days after casting. Test procedure
Specimens were tested in a 120 Kip capacity MTS closed loop machine. A quick-setting cement was used for capping. Surfaces joining the end blocks to the central block were greased to distribute the load and reduce friction. Axial deformation was measured by Linear Variable Displacement Transducers (LVDT). Specially designed clip-gages were used to measure opening and sliding of the crack faces close to the crack tip. The direction of crack propagation was observed from testing one specimen from each group and the clip-gages were thereafter placed perpendicular and parallel to that direction to measure the normal and shear displacements respectively. The aluminum clip-gage mounts were attached to the mortar by superglue adhesive and epoxy. Data acquisition was through a Nicolet 4094 Digital Oscilloscope. Two types of Holographic Interferometry tests were done. Two specimens of each group were tested by a Holomatic 6000 system to study crack propagation[26]. This process used only one viewing direction and could not measure displacements. However a large number of holograms could be made because the process is automated by the Holomatic 6000 system and the film used is more economical. The load at different crack extension lengths were recorded. The third specimens were tested using holographic plates and the process described earlier (Figs 6 and 7). This test made it possible to measure crack opening and sliding displacements.
EXPERIMENTAL OBSERVATIONS The initiation of the crack from the precast notch and the propagation of the kink-crack were observed with the aid of holographic interferometry. Various stages of kink-crack development for a specimen with the notch inclined at angle /I = 36” are shown in Fig. 8. Some typical fringe patterns for other cases are illustrated in Fig. 9. From such observations, kink initiation stress and stresses for various kink lengths were calculated and are plotted in Fig. 10. The crack opening and crack sliding displacements measured by clip-gages are illustrated in Fig. 11. For brevity, only the results for p = 18” and /I = 54” are given. The other details are reported in [29]. The holographically measured kink profiles for various kink lengths for /I = 36” and /? = 72” are plotted in Figs 12 and 13. Comparisons of the measured values with those predicted using LEFM (analytical-infinite plate and FEM-finite plate) are described below. ac r
60
20
I+
1”
08
Kink
Fig. 10. Load
1 I/;
extension
vs kink extension
lengths.
2”
Mixed mode crack
propagation
Fig. 8. Crack
in quasi-brittle
development
for /I = 36”.
materials
135
136
A. K. MAJI
Fig. 9(c-d).
Holographic
intcrferometry
et al.
fringes
for B = 54” and 72’ specimens
Mixed mode crack propagation in quasi-brittle materials
137
18O
60
Fig. 11. Comparison of opening and sliding displacements at the crack mouth from clip gage and FEM for 18” and 54” specimens.
KINK INITIATION DIRECTION
The direction of kink-crack initiation can be predicted from the knowledge of the stress-intensity factors (K, and K,,) at the tip of the notch and using one of the following three theories. The maximum hoop stress theory
According to this criteria given by Erdogan and Sih[20], the crack propagates in a radial direction from its tip and the direction of crack growth is normal to the maximum tangential stress (also called hoop stress or circumferential stress), cO. The hoop stress can be given by: 0 cos t%=s
1
K, cos* i - i K,, sin 8 - c cos* B sin* 8. [
(7)
The equation shown above is based on William’s analysis which contains the additional nonsingular term for crack-tip curvature[30]. From eqs (6) and (7) and maximising erg,the direction of 8 can be calculated as shown in Fig. 14. The direction of crack propagation 8,, is given by: aa&?
= 0
and
a2o0j&32< 0.
The values of p and a in our experiment were 1 mm and 25.4 mm respectively, resulting in p/a = 0.039.
The mode I SIF in direction 0 is related to this hoop stress rrOas:
The K,(8) criterion and crBcriterion are hence analogous.
A. K. MAJI et
138
al.
800
-
Holographic
----
FEM
profilm
profile
600 0
.c
b .g
400
5 0” xx,
0
400 .c 0 b 0, .E 0 7 v)
-
Holographicprofile
----
FEM
proflle
200
0
1 ”
II+”
extension
length
‘4’
Crock
Fig. 12. Profiles of crack opening and sliding for 36” specimen.
.c
-
Holographic
----
FEM
profile
profile
-
Holographic
----
FEM
profile
profile
‘f s ? z ._ CG I 1”
0
Crock
2”
extension length
Fig. 13. Profiles of crack opening and sliding for 72” specimen.
Mixed mode crack propagation in quasi-brittle materials
139
For +=0.039 Criteria , 1400
I-
criterion
/
Prrsrnt 0
PMMAlTirosh
I 20°
0
S,,,
:
Notch
rxpsrimmts
:+
ond Cot2 (19)):
+
I
I
I
40’=
60°
60*
orientation
angle
(fi
-0.039 -4031
1
Fig. 14. Comparison between theoretical and experimental crack initiation angles.
The energy release rate theory
This theory, used by Hussain et a1.[21] evaluates the energy release rate, G, at the kinked crack which propagates from the main crack in the direction of 8. The criteria for crack propagation are: (i) The crack will propagate in the direction which causes the maximum energy release rate to occur and (ii) Propagation occurs when the release rate in that direction reaches a certain value. The strain energy release rate is given as:
e/n
G(@=!
’
[( 1 + 3 cos* 8)K:-
E (3 + cos* 0)’
8K,K,, sin 8 cos 8 + (9 - 5 COS*e)K;,]. (8)
By substituting K, and K,, from eq. (6) into (8) and maximising eq. (8), i.e. for aG/ae = 0 and a*G/@ < 0, the 8 vs p relation can be obtained as shown in Fig. 14. The strain energy density theory
In this theory developed by Sih[22], the strain energy density factor, S can be written as:
s(e) = a,,Kf
+ 2 a,,K,K,,
+ a,,Ki,
in which a,, = [(K - cos e)(i + cos 0)]/167rp a,* = [(2 cos 8 - K + l)]sin 8/167rp a,,=[(rc
+ l)(l -cose)+(i
+c0se)(3c0se
-
1)]/167tp
where IC= 3 - 4v for plane strain and K = (3 - v)/(l + v) for plane stress. p = E[2(1 + v)] (Shear Modulus), where E is Modulus of Elasticity and v is Poisson’s Ratio. EFM
38-213-C
140
A. K. MAJI
et al.
Sih proposed the following criterion: (i) The crack initiation direction is obtained from the minimum strain energy density factor, S, as: aS/ae = 0 and a2SlLY12> 0 for crack initiation angle. (ii) The crack starts to propagate when S reaches a critical value, S, at 19= 19,,,the crack initiation angle. Substituting K, and K,, from eq. (6) into eq. (9) and considering the plane strain case where K = 3 - 4v and v = 0.21, the relationship between 8, and fl can be obtained as shown in Fig. 14. The results determined by the three crack initation criteria are plotted in the same figure for comparison with the experimental observations. Experimental initiation angle was obtained by joining the initial notch tip to a point on the crack l/2” away. It is seen in this figure that the hoop stress theory gave a better fit than the other two criteria for smaller values of the inclination angle /I. But this theory overestimated experimental results after /I = 40” when the G criterion shows a better match with the experiments. Incidentally, a similar trend was obtained by Tirosh and Catz[l9] for the material PMMA (p/a = 0.031) as depicted in Fig. 14. Kink initiation stress
If kink initiates when either cB (or equivalently, K,), G(B) S(e) reaches a critical value as postulated by the three theories discussed, then it is possible to predict the stress at which the kink-crack starts propagating from the main-crack (precast-notch). The value of this initiation stress (gi) is given by the following three equations for each of the three theories. These three equations were based on respectively maximizing eqs (7) and (8) and minimizing eq. (9). These equations are valid for infinite plate geometry. Thus the kink initiation stress (ai) can be written as follows: For maximum hoop stress theory,
8
8
cos2 fl cos3 z - 2 sin2 /I cos3 z + 3 sin /I cos /I
For maximum energy release rate theory, un
l-t! ~;=~~(~+cos~s)~
l-8 i
+ 2( i fi
?t
I
[(l+ 3cos2e)(~Jpiacos2p
-sin2fl)=
7t
cos2 fi - sin’ &sin 2/I sin 28 + (9 - 5 cos2 8)sin2 /3 cos2 /.I)]- I. (11)
For maximum strain energy density theory,
+ a2*sin’ p cos’/?-‘.
(12)
In the above equations 8 = 8,,, the crack initiation angle (Fig. 14). The above equations are plotted in Fig. 15 with the theoretical and experimental data obtained by using the theoretical and experimental 8,, values. Since the specific critical values of K,,, G, and S, are not known, the values of initial stress were normalized by dividing each value with that for fl = 54”. The comparison between the theory and experiments is perhaps qualitatively acceptable. Prediction of crack initiation stress in a finite sized specimen based on infinite plate theories is not very accurate. Kink propagation criteria
The actual finite specimen with a blunt notch-tip was studied by finite element method. The experimentally observed kink direction and kink length were used in the finite element analysis. The holographically observed kink path was approximated by a series of l/4” straight lines. For
141
Mixed mode crack propagation in quasi-brittle materials
(a) Maximum hoop stress theory : Experiment
I
-
+
(b)
50.039 Eq.10
_--J=
D
0
._
C0 3
I I
b
Strain cnsrgy release rate theory
%3 e
Experiment
4
: ---
=0.039
p =o
Ea.11
Cl 1
%
E
2
.o Y .c
.
2
‘\ \ \ \ \
‘0
2 j_ 0
1
E z I 20’
Notch
I 400
I 600
I
60°
orientation
angle
(fi
0
)
Fig. 15(a). Relation between the normalized initiation stress and crack orientation angle (co,, criterion).
--__A
400
200
Notch
.
orientation
60°
angle
(p
60.
)
Fig. 15(b). Relation between the normalized initiation stress and crack orientation angle (G,, criterion).
(cl Stroin density
6‘;;
3-
Experiment
E t; g 3 .;
energy theory
:
-7 ---
J
‘0.039
4
=O
Eq.12
Cl I
.
2
z._ P .N OlE 0’ z
0
I
I
I
I
I
200
400
60°
600
Notch
orientation
angle
(B 1
Fig. 15(c). Relation between the normalized initiation stress and crack orientation angle (S,, criterion).
each step of kink extension, the stress intensity factor corresponding to unit load was calculated. The actual SIFs for kink propagation were then obtained by multiplying these by the experimentally observed crack propagation stress using holography. The calculated values of K, (critical) and K,, are plotted in Fig. 16. It can be seen that the value of K,, decreases with kink extension, perhaps indicating that kink crack propagates in the direction of the critical value of K,. However, the value of K,(critical) is different for different values of B and for different values of crack extension. Thus K,(critical) as determined above can not be regarded as a valid crack propagation criterion. Note that the values plotted for the zero kink extension in Fig. 16 corresponds to sharp crack rather than the actual blunt crack. The calculated values of crack opening and sliding displacements calculated using finite element method are compared with the data in Figs 12 and 13. It can be observed that the openings observed experimentally for the 18” and 36” specimen were much larger and those for the 72” specimen much smaller than the FEM predictions.
142
A. K. MAJI et
al.
600
lh"
l+"
1I+”
1 ‘I
Crack
2"
length
Fig. 16(a). K,_ at crack propagation stages from FEM. 500
720 400
549 36O 160
300 : 3r Y
200 A 0 A
100
0 '
0 0
A
. .
.
Q
9 1”
‘/z I1
Crack
0
09 1ti
I
2"
lengths
Fig. 16(b). KI,at crack propagation stages from FEM.
EXTENSION OF COHESIVE CRACK MODEL The difference in the K, values for different types of specimens as observed in Fig. (16) and the discrepancy in displacement measurements (Figs 12 and 13) motivated further investigation into the nature of kind cracks. It is possible that an extension of the ‘Cohesive Crack Model’ (Fig. 2) is necessary to incorporate tractional forces on the crack face in the mixed mode conditions. The cohesive crack models used for mode I crack propagation usually consist of applying normal tractions across the crack surface (Hillerborg, Kobayashi, Jenq, Ingraffea etc. [4]). The magnitude of such normal closing pressures is inversely proportional to the crack opening displacement. The relationship between closing pressure and opening displacement are considered material property and can be determined experimentally[31]. For the mixed mode case, cracks open as well as slide and thus one must include both the normal and shear tractions. The relationship between normal traction, shear traction and the corresponding opening and sliding displacements include complex coupling and are not easy to measure experimentally[32-381. When a shear stress is applied to a crack in dilating materials like concrete and rock, sliding is accompanied by opening. The relationship between the sliding and
Mixed mode crack propagation in quasi-brittle materials
143
opening displacement depends on applied shear traction, possible normal traction and the initial shape of the crack. It was concluded from the review of the available texts cited above that the discrepancy between the measured and the (LEFM) calculated opening and sliding displacement can be accounted for by the addition of shear and normal tractions. To account for unknown interface traction forces, additional FEM based study was done. Unit opening and sliding forces were placed on intermediate nodes on the crack faces in the FEM mesh. The opening and sliding displacements corresponding to these unit forces were determined from analysis. This yielded a stiffness matrix S which contained nodal opening and sliding for unit opening or sliding loads. A different matrix D was set up which contained the difference in nodal displacements between the FEM analysis and experimental observations (Figs 12 and 13). For the j3 = 36” and 72” specimens, a set of equations were used to match the FEM and experimental crack profiles:
Kl PI = PI
(13)
where S and D are the matrices described above. The matrix L is determined by solving the above equation. This matrix contains the nodal opening and sliding loads which should be applied so as to make the FEM and experimental crack profiles the same. For a crack length with four intermediate nodes, the L matrix would have 4 x 2 = 8 terms. These will be the opening and sliding loads at those four nodes. The value of K,(critical) shown in Fig. (16) was corrected by adding the Ki values obtained from these nodal loads. The corrected values of K,(critical) and K,, shown in Fig. (17) seem to be within a reasonably narrow band. The ratio of K,critical/KIIis always greater than 1.0. This shows the possibility of using a K, criterion to predict crack growth where the crack face tractions can be accurately known from previous experiments on their constitutive behavior. For the /I = 18” and 54” specimens, only the clip gage location measurements were matched assuming uniform distribution of nodal forces to get Ki (Fig. 17). Table 1 shows some applied nodal forces for a crack length with four intermediate nodes. The crack length in this case was 1.9” long. So the length of crack per node was approximately l/2”. The stresses on the crack surface at the nodes were obtained by dividing the nodal forces by 0.5” and were in the range of 48 to 166 psi. In a study of interface behavior conducted by Van Mier[37],
500 r
(a)
0
.
A 300 -
.
:: 0 . ,.
l
0
A 0 0
A 0
1
1 ‘4”
0
200 -
54’ 100 -
A
36O . 160 0
$2”
II
2”
Crack length Fig. 17(a). Kcn,,c, after matching crack surface displacements.
A. K. MAJI et al.
144 lb) 300r
200
72O0 54’ 36O
A 0
160
0
0
-
IOO-
. 0
0
A 0 0
8
0
0
I+”
1
0
II
4
I
1’4”
2”
Crack length Fig. 17(b). I,, after matching displacements by nodal forces in FEM.
a shear load was applied to a crack which was under tensile stress. A shear load of 7.5 kN (corresponding to about 150 psi over the specimen cross section area) was found to cause a drop in the tensile stress of 5 N (corresponding to about 100 psi) when the crack opening was held constant at about 450 p in. It must be noted that 300 ~1in was a typical crack opening displacement in our experiments. Hence, a tensile pressure of 48 or 166 psi could have been caused by dilatancy due to shear stress on a crack surface. SUMMARY
OF RESULTS
(1) The Maximum Hoop Stress (MHS) or K,,, criterion and the Strain Energy Release Rate or G criterion can predict crack initiation angle and stress for compression-shear failure. Crack propagation is in tension-shear once the kink crack initiates. (2) Experimental Normalized Stress Ratio at kink initiation is well predicted by the MHS and Strain Energy Density (SED) theories. (3) Although the MHS criterion based on a traction free crack could predict crack initiation, it could not predict propagation load because of the coupling between sliding and opening displacement intensified by granularity or mortar. (4) The MHS criterion was reasonably good if the FEM predicted crack face displacements were corrected by applying normal and shear tractions to match those observed experimentally at the different propagation stages. The K, Stress Intensity Factor was dominant in the failure mechanism. While the basic physical arguments for such corrections were presented, it proves the necessity of conducting investigations into constitutive relations for compression-sliding of quasi-brittle materials in order to predict crack propagation under mixed mode conditions. (5) Holographic interferometry has been an excellent tool to detect cracks during the tests in real time. The high sensitivity of the order of 0.1 microns is a unique method of studying fracture process and provides improved crack tip location. (6) A single plate holographic interferometry could be used for quantitative measurement of crack surface opening and sliding. This process is relatively inexpensive and useful in studying features of the process zone behind the crack tip. Table 1. Nodal forces required to match crack profiles for B = 36” (Crack extension length was 1.9” for this case) Node’s distance from tip of kink 1.5” 1.1” 0.7” 0.3”
Nodal force (lbs) 29 24 31 83
Normal stress on interface (psi) 58 48 62 166
Crack opening p IN.
Crack sliding /I IN.
580 390 280 140
130 20 10 10
Mixed mode crack propagation in quasi-brittle materials
145
Acknowledgements-The research reported here is supported by the U.S. Air Force Office of Scientific Research through a grant (AFOSR-85-0261) to Northwestern University. The helpful suggestion of the AFOSR program manager Dr Spencer T. Wu is gratefully acknowledged.
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1191J. Tirosh and E. Catz, Mixed mode fracture angle and fracture locus of materials subjected to compressive loading. Engng Fracture Mech. 14, 27-38 (1981). PO1 F. Erdogan and G. C. Sih, On the crack extension in plates under plane loading and transverse shear. J. Busic Engng 85, 519-527 (1963). I211 M. A. Hussain, S. L. Pu and J. H. Underwood, Strain energy release rate for a crack under combined mode I and mode II. Fracture Analysis, ASTM STP 560, 2-28 (1974). G. C. Sih, Strain energy density factor applied to mixed-mode crack problems. Int. J. Fracture, 10, 305-321 (1974). t::; P. A. Wawrzynek, Interactive finite element analysis of fracture processes: an integrated approach. Struct. Engng Report No. 87-1 (Supervised by A. Ingraffea), Cornell University (1987). Y. Murakami, Stress Intensity Factors Handbook, pp. 909-911. Pergamon Press, New York (1987). t;z; Nils Abramson, The Making and Evaluation of Holograms. Academic Press (1981). P61 A. K. Maji and S. P. Shah, Application of Acoustic Emission and Laser Holography to Study Microfracture in Concrete. ACI SP-112, Nondestructive Testing (1988). 1271 R. A. Miller, S. P. Shah and H. I. Bjelkhagen, Crack profiles in mortar measured by holographic interferometry. Exp. Mech. 338-393 (Dec. 1988). 1281 S. P. Shah and R. Sankar, Internal cracking and strain softening response of concrete under uniaxial compression. ACI Muter. J. 84 (3), 200, (May-June 1987). 1291A. K. Maji, Study of Concrete Fracture Using Acoustic Emission and Laser Holography. Ph.D. dissertation submitted to Northwestern University, Evanston, IL (August 1988). M. L. Williams, On the stress distribution at the base of a stationary crack. J. uppl. Mech. 24, 109 (1957). t:;; V. S. Gopalaratanam and S. P. Shah, Softening response of plain concrete in direct tension, ACI I, 3 10-323, May-June (1985). 1321 &I. P.‘Divakar, A. Fafitis, S. P. Shah, A constitutive model for shear transfer in cracked concrete. ASCE J. Structural Engng 113, (5) May (1987). [331 A. R. Ingraffea and W. H. Gerstle, Non-linear fracture models for discrete crack propagation. Proc. of NATO Advanced Reseurch Workshov. (Edited bv S. P. Shah.) Sent. (1984). [341 R. C. Fenwick andT.‘Paulay, Mechanics of shea; resistance of concrete beams. ASCE I. Structures Division 94 (STlO), 2325-2350 (1968). [351 Z. P. Bazant and T. Tsubaki, Slip-dilatancy model for cracked reinforced concrete, ASCE I. Structures Division, 1947-1966 (Sept. 1980). I361 M. Hassanzadeh, A. Hillberborg and Z. Ping, Tests of material properties in mixed mode I and II. Proc. SEM-RILEM Int. Conf Fracture Concrete Rock (June 1987). I371 J. G. M. Van Mier, Fracture study of concrete specimens subjected to combined tensile and shear loading. In?. Conf Measurement Testing Civil Engng, Lyon (Sept. 1988). [381 Z. P. Bazant and P. Gambarova, Crack shear in concrete: crack band microplane model. J. Structure Division, ASCE, 110, 2015-2035 (Sept. 1984). (Received 10 November 1989)