Crack Propagation in Layered Plates-an Experimental Study by P.S. Theocaris and H.G. Georgiadis
ABSTRACT--The effects of the layer height and the material properties of a layered medium on the propagation mode of a crack are discussed. CracK-propagation velocities and dynamic stress-intensity factors were evaluated during fracture and the corresponding curves were plotted. The medium consisted of two identical external layers and a middle layer of softer or harder material. This layer contained an initial crack which ran parallel to the interfaces of the ~ayers. All phases of the specimens were fabricatea from epoxy ,oolymers. After the preparation of the middle layer, the surrounding layers were successively cast along the middle layer's straight boundaries without applying any Kind of adhesive. High-speed photography and aynamJc caustics were utilizea to establish the cnanges in crack speeds and intensities of the stress field. The influence of geometry and especially of the height of the middle layer and tne mechanical properties of the layerea material was studied: interesting results were derived.
Introduction The wide use of composite materials in structures and machines, as well as the necessity of laboratory studies in the area of geodynamics and seismology, provided the impetus for the present investigation. As is well known, the capacity of a fibrous composite to carry loads is much greater than that of a matrix material of the same dimensions. Moreover, another reason for the use of composites in technology is that, under certain circumstances, their inclusions can work as buffers, and stop or decelerate propagating cracks caused by initial flaws or debonding processes. A number of theoretical I as well as experimental 2-7 studies on the behavior of a crack against an inclusion or an interface have appeared in the literature. It is shown, in all the experimental studies, that the interface of an inclusion, or even a hole or a longitudinal flaw in the matrix, may be considered an a t t e n u a t o r and a barrier, causing crack deceleration and sometimes arrest. On the other hand, a rather limited number of papers consider the problem of stationary or running cracks lying on the interface between dissimilar media? The case of a crack in the middle of a layer in a three-layer medium has been treated theoretically only by Hilton and Sih 9 and Sih and Chen '~ for stationary and dynamic cracks, respectively. A somewhat related problem is the P.S. Theocaris (SEM Fellow) and H.G. Georgiadis are Professor and Research Engineer, respectively, The National Technical University of Athens, Department of Theoretical and Applied Mechanics, 5, Heroes of Polytechnion Avenue, Zographou, GR-157 73, Athens, Greece. Original manuscript submitted: February 24, 1984. Final manuscript received: October 18, 1984.
case of a cracked strip with finite, dimensions transverse to the crack axis. This problem has been treated theoretically''-'4 and experimentally ,s,'6 both for static or for running cracks. However, the cracked strip is simply a particular example of the general case of the multilayered medium. The propagation of a crack m a generic strip of a layered composite, or a stratified rock mass, raises an interesting question about the effect of the height of the strip and the difference between the elastic properties of the successive layers on the intensity of the local dynamic stresses and the velocity of the running crack. The answer to this question is sought here in experimental terms. To the best of our knowledge, such experimental study has not appeared hitherto in the literature. Our study makes use of high-speed photography and interprets the experimental data by means of dynamic caustics. The experimental analysis shows that when the crack was propagating in the soft material, the K, factor decreased with the decreasing height of the strip. Exactly the opposite phenomenon was observed when the dynamic crack was lying inside a harder central layer with respect to the surrounding material. Moreover, the cracktip velocities increased in the first case, whereas they fairly decreased in the second case.
Experimental Procedure Figure l(a) presents the shape and dimensions of the specimens utilized in the tests. They were about 3-ram thick. The upper and lower layers of the three-layer sandwich plate had identical elastic properties (shear modulus # and Poisson's ratio p) that differed from those of the central layer. In what follows, the subscripts 1 and 2 refer to the center layer and the surrounding layers, respectively. The height 2h of the central strip varied between 10 mm and 80 ram. The center strip was also cracked at its middle height. This initial slit had a constant length of 20 mm for all tests and a distance between its lips, when the crack was unloaded, less than 500/an. Thus, the crack propagation was taking place along a path parallel to the interfaces of the sandwich plate, the loading being perpendicular to them so that Mode I deformation always existed. Because of the symmetry in the loading and geometry Of the specimens and the subsequent symmetry in the mode of fracture, it always happened that, while the crack was propagating the ratio h / a ( t ) [ a ( t ) being the instantaneous crack length] was continuously decreasing over a wide interval of such ratios. The specimens Were prepared from a pure cold-setting commercial epoxy prepolymer (Epicote 828). They were polymerized by the addition of eight-percent triethylene-
Experimental Mechanics ~ 85
tetramine (TETA) hardener per weight of the epoxy prepolymer. Added to the prepolymer was a plasticizer consisting of a polysulfide polymer Thiokol LP3 in the amounts: 0, 10 and 50 percent of the weight of the epoxy prepolymer. The epoxy-plasticizer mixtures were thoroughly mixed in an open cup before the hardener was added. The mixture was then degassed in a vacuum chamber for 10 rain to remove all bubbles. The casting was made in orthogonal molds coated with a suitable inhibiting oil to eliminate any adhesion of the mixture with the mold. Plates with highly reflective surfaces can be prepared in this manner. Each plate was cured by heating at 100~ and then slowly cooling to ambient temperature. The duration of each curing cycle lasted approximately seven days in order to insure complete polymerization of each plate without the introduction of residual and shrinkage stresses. 17 Moreover, Shrinkage stresses along the interfaces were low because the surrounding layers were being cast along straight boundaries. Indeed, Fig. 2(a) shows photoelasticpolariscope photographs of specimens before fracture without any loading, in which residual stresses are almost absent. Two cases involving the differences in the mechanical properties of the layers were considered. (1) Pure epoxy resins (0-percent plasticizer) or epoxy resins with 10percent plasticizer used as a central layer (Phase 1), and epoxy resins with 50-percent plasticizer as outer-layer (Phase 2) plates, to provide a ductile-brittle-ductile plate. (2) Epoxy resins with 50-percent plasticizer as central-layer (Phase 1) plates and epoxy resin with 10-percent placticizer
as outer layers (Phase 2), to provide a brittle-ductile-brittle plate. Moreover, another series of specimens were prepared with CR-39 as the central layer and a 50-percent plasticized epoxy resin as the outer layers, to provide another three-layer plate with different ratios in g2//~, and ~2/~, for the case of ductile-brittle-ductile plates. Special care was given during the manufacture of the sandwich plates to have clear interfaces before casting the surrounding material in the mold with the pre-existing central layer. Preliminary tests with uncracked-layered tension specimens showed that the strength of the interface region was greater than that of the individual phases (layers), and that rupture of the three-layer specimen had always started from points far away from the interfaces. Table 1 illustrates the mechanical and optical constants of the materials utilized in the experimentsJ 8 A Cranz-Schardin high-speed camera with 24 sparks and a maximum frequency of 106 frames/s was used to study the dynamic crack propagation. The synchronization of the fracture process with the high-speed camera was achieved by means of a conventional silver-contact circuit, which triggered the first spark of the series with the initiation of the crack propagation. The specimens were loaded in all tests quasistatically with a strain rate ~ = 4s -~. The optical part of the experimental arrangement was typical for photographing transmitted caustics (e.g., Refs. 2 and 3). The light beam from each spark was first reflected on a spherical mirror of a high reflectivity with a diameter of 500 mm and a focal distance of 3.50 mm. After passing "through the specimen, the light beam was focused on each respective camera lens.
TABLE 1--MECHANICAL AND OPTICAL PROPERTIESOF THE MATERIALS UTILIZED IN THE TESTS i
i
Shear Modulus f f ( x 10s Nm-2)
Material
Epoxy resin with 0% plasticizer Epoxy resin with 10% plasticizer Epoxy resin with 50% plasticizer CR-39
12.33 12.03 4.39 8.94
Stress-Optical Poisson's Ratio Constant ~, ct(x 10-1~ mZN-') 0.338 0,338 0.480 0.443
2.181 2.485 5,446 1,200
O.OZ,. v, KI
% KI '
_.K,
|.I~"
-222"E_-'2_I (#,v, p)
"t
f
,411, {a)
86 9 March 1985
1786.4 1764.6 1202.9 1642.4
1027.7 1015.2 613.3 866.7
Fig. 1--Geometry of the specimens and characteristic behavior of the crack velocity and the dynamic SIF
0.08
O.09m
Shear-Wave Velocity cz (ms-1)
0.01 0.02
2h
z
Longitudinal-Wave Velocity c, (ms-1)
(b)
Theoretical Considerations and Interpretation of Experimental Data It is well known that the presence of a boundary or a discontinuity near a crack, either stationary or running, greatly affects the intensity and spatial dependence of the stress field. ' ''' The presence of an inclusion of different materials '~ has a similar effect. For the case of a three-layer centrally cracked medium, Sih and coworkers 9,1~ point out that the singular part of the stress field (r- and S-dependence) remains unchanged with respect to an infinite uniform cracked plate; only the intensity of the stresses changes between these two cases. In our experiments with layered plates, this fact was always confirmed. The form of the caustics generated by the high-stress concentration around the crack tips was exactly the same as the typical one corresponding to a uniform plate. Moreover, the photoelastic experiments utilizing SEN sandwich plates [see Fig.. 2(b)] showed that the distribution of the near-tip r,,ax stresses corresponds essentially to the singular part of the stress field created in a uniform cracked plate, whereas the far-tip distribution is quite different, especially near the interfaces. This experimental far-tip stress distribution Closely resembles the theoretical one given by Knauss ~1 for an infinite strip containing a semi-infinite crack. Fortunately, the caustics are created from an infinitesimal region around the crack tip, where the elastic-stress singularity completely dominates the stress field. The optical system used in the present study was designed so that the radius of the generatrix (initial) curve of caustics is always of the order of the plate thickness, i.e., about 3 ram. Therefore, the classical analysis of the method
still can be applied in our case. The basic relation, which connects the dynamic SIF with the geometrical characteristics of the caustic, is also valid: ~ Other experimental methods, such as photoelasticity, interferometry and moire, are needed to make measurements outside the near-tip region for obtaining readable and reliable data. To this end, higher-orde r terms must be introduced in the series expansion of the stress field. 21.~2 The data analysis requires computer processing and therefore is time consuming. Thus, for the specimen geometry under consideration, anyone trying to analyze experimental data using any one o f t h e preceding methods will be forced to derive first the remaining nonsingular part of the stress field from the theoretical solution. The singular stress field created around a Mode I dynamic crack, either in uniform or layered plates, has the form: ,o.23,~, a~-
KI( t)B~
(2x),,2
[(1+23~-fl~) 4flA32 (1 + (3#)
K,(t)B,
ay =
rl/2
(2 ~r)1/2
"
[ _ (1 +fi#) cos (0,/2) ry 2
4fi,/32
cos (02/2).]
1 + ~)
-K,(t)B~
r:,2
cos_ (OJ2).]
(27r),,~ +
rxy --
cos (0,/2)
(:)
r "~
sin (05/2) ]
[2fi, sin (8,/2) r)/2
2[3,
r~ :~
where f17 = 1 - ( v / c j ) 2, v being the crack velocity, cj being the longitudinal ( j = 1) and shear ( j = 2) wave
Fig. 2--(a) Photographs of layered specimens with no loading in a photoelastic polariscope. (b) Dark-field isochromatics created around the tips of stationary cracks in SEN-layered plates
Experimental Mechanics 9 87
where f [ a ( t ) , A , B ] is a correction factor greater than unity, resulting from the finite boundaries of the plate and g(v/cR) is a velocity correction, first derived by Freund.26.27 This velocity-correction factor is given by:
velocities and 1 + fl~ B, = 4 / 3 , f l ~ - ( 1 + fi#)2
(2)
The scaled r: and 0j- coordinates, introduced for convenience in the above relations, are related with the r and O coordinates in the physical plane as follows:
g(v/c,) ~ 1 - 0 . 8 v_%for v_ < 0.6 CR
tan Oi = ~3j tan 0
rj = r[cos 2 0 + ~) sin 2 01 ':2
CR
(5)
where cR is the Rayleigh-wave velocity. It can be seen from eqs (4) and (5) that the dynamic SIF decreases with the crack speed and reaches zero when the crack speed coincides with the speed of Rayleigh waves. Finally, note that, although all the above relations have been derived for cracks moving under constant velocity, the polar-angle distribution of the singularity is also valid for nonuniformly moving cracks, provided that we insert the instantaneous velocity instead of the steady-state one. 2s The magnitude of the singularity is determined by the stress-intensity factor which, of course, generally depends on time and on crack acceleration. The method of caustics is based on the principles of geometric optics and has been successfully applied to cases of both transparent and opaque solids. Referring to in-plane modes of loading of the specimen, the stress intensification in the region surrounding the crack tip produces a reduction in the thickness of the plate because of the lateral-contraction effect and/or a variation of the refractive index along the same direction. As a consequence, the incident light rays in the vicinity of the crack tip are deviated and form the caustic envelope. In order to calculate the caustic for a given problem, the function As(x, y), expressing the relative optical-path retardation for this problem, must be first determined. For the dynamic problem considered it is valid that:
(3)
In eqs (1) the scalar quantity K,(t) is the time-dependent stress-intensity factor, which generally is an unknown, provided only by experimental methods. This K,(t) factor takes definite values when one considers some of the idealized crack-propagation models, e.g,, the yoffd, Craggs, or Broberg models. Note that the Yoff~ and Broberg models have identical singular stress distributions, whereas the stress distribution of the Craggs model is given in closed form? ~ It is common in theoretical studies to work with one of the latter simplified models. Following this practice, Sih and Chen ~~ treated the problem of a three-layer cracked plate by considering Yoff~'s constant-crack-length model. In this case, one does not have the whole history of the crack propagation and the variation of the SIF vs. time. Indeed, there the SIF is taken as a time-independent quantity and therefore is of a highly idealized nature. However, with the experimental methods--using only the stress distribution from the respective theoretical solutions, and also taking the amplitude of the stress field (i,e., the dynamic SIF) as an unknown which needs to be determined--one can correlate the actual timedependent SIF with certain geometrical characteristics of the patterns taken from the high-speed photographs. In other words, for a complete determination of the stress field around a dynamically propagating crack, it is necessary to use both theory and experiments. For a quasistatic loading ooo(t) at the transverse boundaries of a finite plate with dimensions A • B and a continuously increasing crack length a(t), the dynamic SIF is given 6y:
As(r1,0,)
=
dG(o,+o-2) =
K,( t) B,
d G ( ~
2 ( i l l - fl~) cos ( 0 , / 2 )
(6)
where d is the thickness of the specimen and c, is the transmission stress-optical constant (see Table 1) of the material. Without going into further details (e.g., Refs. 20 and 21), the correlation of the dynamic SIF, K,(t), with the
Ki(t) = f [ a ( t ) , A , B ] g(v/c,)a=(t) [~ra(t)] '/2 (4)
3.4
f
/
/i
3.3
~o
Fig. 3 - - C o r r e c t i o n f a c t o r ~m.x vs. (v/cl) f o r interpreting d y n a m i c c a u s t i c s to
3 . 1 - -
3.0 . 0
.
. 0.1
.
. O.Z
0.3 ulc 1
88 9 March 1985
0.~,
0.5
0.6
maximum transverse diameter of the caustic D,""x is given by the expression:
2(2~) ,,2 K,( t) -
(D, ~~ ),,2
3Tood~q,
(7)
where zo is the distance of the reference plane (which is photographed by the high-speed camera) from the specimen, X is a magnification factOr depending on the optical setup, and 6,,ax is a correction factor depending only on the crack velocity v for a given material and is diagrammed in Fig. 3. Thus, with one simple measurement of the maximum transverse diameter of the dynamic caustic and by using the diagram of Fig. 3 and eq (7), one may evaluate the dynamic SIF of a propagating crack in an elastic and isotropic medium.
Results and Discussion Figures 4, 5 and 6 illustrate crack propagations in the brittle layer of the sandwich plate. In Fig. 4 the stratification is CR-39 as the central layer (Phase 1) with 2h = 10 mm and 50-percent plasticized epoxy resin as the outer layers (Phase 2); in Figs. 5 and 6 pure epoxy resin as the
central layer (Phase 1) with heights 2h = 20 mm and 40 mm respectively and 50-percent plasticized epoxy resin as the Outer layerS (Phase 2). Figure.7 shows high,speed photographs from an experiment with the opposite composition, i.e., ductile central layer (epoxy resin with 50-percent plasticizer) and brittle outer layers (epoxy resin with 10-Percent plasticizer). It is well known that in SEN long and uniform speci-" mens with remotely applied stresses at their transverse boundaries, the crack velocity and the dynamic stressintensity factor are increasing and sharply increasing functions, respectively, in terms of the increasing crack length (e.g., Ref. 29). As a general result of our experimental 9analysis, it can be concluded that the latter characteristic behavior of the moving cracks does not prevail in SEN layered specimens with heights for the centralcracked layer comparable to the crack lengths, Both the crack velocity and the dynamic SIF show an almost insignificant variation with increasing crack lengths and thus their curves vs. time are nearly7 parallel to the time-axis abscissas. T'his observation is presented more expressively in the sketches of Fig. 1. Since the preceding crack behavior is typical for fixed displacement conditions in strip-like specimens, 3~ it is valid that the layered plate is closer to the latter configuration than to long specimens with remotely applied stresses. It
Fig. 4--Photographs of crack propagation in a layered plate: ductile (50-percent plasticized epoxy resin)-brittle (CR-39)ductile composition with 2h = 10 mm
Experimental Mechanics 9 89
does not matter if the crack velocity and the dynamic SIF in the layered specimen are larger than in the corresponding uniform plate; this 'bounded' behavior is always observed. It is shown in subsequent diagrams. Since the crack propagated in the brittle phase (hard central layer) our tests indicate that, with the decreasing height of the internal strip, the crack speed decreases, whereas t h e K~(t) factor increases. A comparison of these fracture characteristics for a uniform plate and a layered plate, with its central layer made of the same material as that of the uniform plate and with an arbitrary height (10 mm<_2h___80 ram), always shows greater crack velocities and lower dynamic SIFs during fracture for the uniform plates than those for the layered plates. These characteristic phenomena are shown in the diagrams of v-t and K,-t in Figs. 8-10 for several configurations. The fact that the stress-intensity factor increases with the decreasing height of the central cracked layer is in qualitative agreement with the theory. 9,'~ However, as has been already discussed, the elastodynamic theory fails to give the velocity characteristics and its variations, as well as the history of the dynamic SIF and the connection between the latter and the crack speed during fracture. On the other hand, the limiting case of an infinite layered plate with a hard material as central layer and
softer material as outer layers, i.e., with g2/g~ < 1, is the case of an infinite strip containing a line crack parallel to the upper and lower sides of the strip. In this case #2/#i = 0; therefore Phase 2 (outer layers) does not exist. Theoretical studies of this configuration show that the stress intensity increases with the decreasing height of the strip. 1'-~4.~5 This is in fair agreement with our experimental findings. Exactly the opposite situation, but less amplified, prevails in the inverse configuration of the layered plates. When the crack propagates in a ductile central layer (Phase 1) surrounded by brittle outer layers (Phase 2), the crack-tip velocity v(t) increases slightly with the decreasing height 2h of the central layer, whereas the timedependent K,(t) factor slightly decreases. Figures 9 and 11 show the respective v-t and K,-t diagrams from experiments with the latter arrangement. In all tests with the hard-soft-hard layer arrangement, only small differences were observed in velocities and stress intensities, either by changing the height of the central layer or by comparing the crack behavior in a uniform soft plate (epoxy resin, 50-percent plasticized) and in a layered plate of an arbitrary central-layer height. This phenomenon may be attributed to the fact that the medium in which the crack is moving is very soft and not perfectly elastic. Therefore, it is reasonable to have
Fig. 5--Photographs of crack propagation in a layered plate: ductile (50-percent plasticized epoxy resin)-brittle (pure epoxy resin)-ductile composition with 2h = 20 mm
90 9 March t985
quantitative divergences from the theory which, for the hard-soft-hard combination, predicts again an analogous spectacular behavior, as in the case of soft-hard-soft stratification. Qualitatively, the decrease of the stressintensity factor with the decreasing height of the central cracked layer, demonstrated by our experiments, is again in agreement with the work of Hilton and Sih 9 and Sih and Chert.' ~ Finally, another aspect of dynamic fracture may be discussed, namely the emission of stress waves from a moving crack tip. The following types of stress waves can be generated by a fast-running crack tip: 32-3' longitudinal and shear waves, generated by the abrupt change of the stress-intensity factor during propagation, and Rayleigh (surface) waves, either generated by the triaxial state of stresses at the vicinity of the crack tip and propagating along the lateral surfaces of the plate or those developed at the lips of the crack, as a result of a Lamb impact. Practically, only large and abrupt changes of K factor, such as in the case of starting or arrested cracks, may develop' body waves. 32 On the other hand, Rayleigh waves in two dimensions are visible only when specimens with a well polished surface are utilized in conjunction with optical arrangements for transmitted or refected caustics. 33-35 In the experiments reported here, such waves were not observed, because the specimens utilized were made from epoxy resins which do not have the necessary
high-quality smoothness and reflectivity of plates made of PMMA (Plexiglas) and PCBA (Lexan). Note that, in dynamic photoelastic experiments, fracture-Rayleigh wavefronts are not visible, a fact which demonstrates the negligible contribution of these waves to the whole stress field in a fracturing plate. In view of the above discussion, any explanation of fracture phenomena based on the effect of stress waves emitted by a moving crack tip, is of a highly speculative nature, except perhaps in regard to Rayleigh waves propagated along the lips of the crack. 36 But, the latter waves appear in all configurations, either in uniform or layered plates; therefore their presence is common in all the cases that have been compared in our study. Thus, these one-dimensional (edge) surface waves cannot be considered as an extra parameter of the problem.
Conclusions The characteristics of the phenomenon of a fast-running crack moving parallel to the interfaces of layered strips are studied in this paper. The analysis of the experimental results by the optical method of dynamic caustics corroborates the existing theoretical results on the problem of a cracked layered plate. Moreover, the interesting velocity variations during the crack propagation reported in this study, and established
Fig. 6--Photographs of crack propagation in a layered plate: ductile (50-percent plasticized epoxy resin)-brittle (pure epoxy resin):ductile composition with 2h = 40 mm
Experimental Mechanics ~ 91
Fig. 7--Photographs of crack propagation in a layered plate: brittle (lO-percent plasticized epoxy resin)-ductile (50,percent plasticized epoxy resin)-brittle composition with 2h = 40 mm
600 [
~
m
e Fig. 8--Crack velocity/time diagrams corresponding to the composition: ductile (50,percent plasticized epoxy resin)-brittle (pure epoxy resin or CR-39)ductile layers
,ooi 20O
,'
I -200~ 0
92 9 March 1985
.L r
phose 1: epoxy resin 0% plasf. phase 2: epoxy resin 50% piasf. J
J
phase 1:CR-39 phase 2: epoxy resin 50% ptost.
L L 80 120 Time ! ps)
160
200
for the first time by experiment, indicate the weakness of the respective theory in detecting such important phenomena. Notably, the case in which the central layer was brittle and the surrounding material was ductile yielded more spectacular changes in the crack-propagation speed and the dynamic stress-intensity factor.
Acknowledgments The research work contained in this paper was partly supported by the Hellenic Aluminium Company. The authors, especially the second one, are indebted to this institution for its financial support.
L
) I phase 1: epoxy resin 10% piosf. phase 2: elxeoj resin 500 ptost. 2h =O.02m ~L,O
|
.L
~0
o
v 9
200
I I phase 1: epoxy resin 50% plasL phase 2: epmy resin 10% plast.
6O0 T.I/I
9~
L-o
2h =0.02 m \
400
Fig. 9--Crack velocity/time diagrams corresponding to the composition: ductile (50-percent plasticized epoxy resin)-brittle (lO-percent plasticized epoxy resin)-ductile layers, and inversely
0
~ ~ = O . 0 4 m P U
2OO 0
40
80 Time
120
60
I
1
200
(ps)
1.5
phase 1: epoxy resin 0% p[asf. phase 2: epoxy resin 50% plasf.
M't
X
2h=O.~m x
. 10
I
~
Zh--= ~.08m 2h
:=E
Fig. lO--Dynamic-SIF time diagrams corresponding to the composition: ductile (50-percent plasticized epoxy resin)-brittte (pure epoxy resin or lO-percent plasticized epoxy resin)-ductile layers
0.7 ~" N
1.3
U'I
.;.
2h=O.i j
,-,r
..)~...X-,')<
t:
m
x
phase 1: epoxy resin 10% p[asf. phase 2: epoxy resin 50% plasf,
::Z O.=. 0
J
40
80 Time
120 II~s) - - ~
I
160
200
Experimental Mechanics ,,
93
References 1. Theocaris, P.M., "Stress and Displacement Singularities Near Corners, '" J. Appl. Math. and Phys. (ZAMP), 26, 77-98 (1975). 2. Theocaris, P.M. and Milios, J., "'Dynamic. Crack Propagation in Composites," Int. J. Frac., 16, 31-51 (1080). 3. Theocaris, P.M. and Milios, J . , "Crack-Arrest at a Bimaterial Interface, ""Int. J. Solids Struc., 17, 217-223 (1981). 4. Theocaris, P.M. and Pazis, D., "'Crack Deceleration and Arrest Phenomena at an Oblique Bimaterial Interface, "" 1bid., 19 (7), 611-623 (1983). 5. Dally, J.W. and Kobayashi, T., "'Crack Arrest in Duplex Specimens, '" Ibid., 14, 121-129 (1978). 6. Shukla, A. and Dally, d. W., "Influence of Late-breaking Ligaments on Crack Propagation in Compact Specimens- A Photoelastic Study," EXPERIMENTAL MECHANICS, 23 (3), 298-303 (1983). 7. Kobayashi, A.S., Johnson, B. and Wade, B.G., "Crack Approaching a Hole, "' Fracture Analysis, ASTM STP 560, 35-68 (1974). 8. Theoearis, P.S., "Partly Unbonded Interfaces between Dissimilar Materials Under Normal and Shear Loading, "' Act. Mech., 24, 99-115 H976). 9. Hilton, P.D. and Sih, G,C., "A Sandwiched Layer of Dissimilar Material Weakened by Crack-Like Imperfections,, Developments in Theor. and Appl. Mech., 5 (Proc. V Southeastern Conf. on Theor. and Appl. Mech.), ed. G.L. Rogers, S.C. Krane and E.G. Henneke, 123-149 (1970). 10. Sih, G.C. and Chen, E.P., "Moving Cracks in Layered Composites, '" Int. J. Eng. Sci., 20 (1), 1181-1192 (1982). 11. Knauss, W.G., "'Stress in an Infinite Strip Containing a SemiInfinite Crack," J. Appl. Mech., 33, 356-366 (1966); see also Rice, J.R., Discussion, s Appl. Mech., 34, 248-249 (1967). 12. Sih, G.C. and Chert, E.P., "Moving Crack in a Finite Strip under Tearing Action, "' J. Franklin Inst., 290 (1), 25-35 (1970). 13. Sih, G.C. and Chert, E.P., "'Crack Propagation in a Strip of Material under Plane Extension, ""Int. J. Eng. Sci., 10, 537-551 (1972). 14. Nilsson, F., "Dynamic Stress-lntensity Factors for Finite Strip Problems, '" Int. F. Frac., 8 (4), 403-411 (1972). 15. Nilsson, F, "Crack Propagation Experiments on Strip Specimens, "' Eng. Fract. Mech., 6, 397-403 (1974). 16. Parson, T.L. and Lucas, R.A., "'An Experimental Investigation of the Velocity Characteristics of a Fixed Boundary Fracture Model," Dynamic Crack Propagation, ed. G.C. Sih, Noordhoff Int. Publ., 415-426 (1973). 17. Theocaris, P.S., "Viscoelastic Properties of Epoxy Resins Derived from Creep and Relaxation Tests at Different Temperatures," Rheol. Acta, 2 (2), 92-97 (1962). 18. Theocaris, P.S. and Prassianakis, J., "'Interrelation of Mechanical
and Optical Properties of Plasticized Epoxy Polymers," J. Appl. Pol. Sci., 22, 1725-1734 (1978). 19. Tamate, 0., "The Effect of a Circular Inclusion on the Stresses Around a Line Crack in a Sheet Under Tension, "" Int. J. Fract., 4, 257-266 (1968). 20. Theocaris, P.M. and Papadopoulos, G.A., "Elastodynamic Forms of Caustics for Running Cracks under Constant Velocity, "" Eng. Fract. Mech., 13, 683-698 (1980). 21. Theocaris, P.M., "'Experimental Methods for Determining Stress Intensity Factors," Plenary Lecture, 6th Int. Conf. on Frae., New Delhi, India (1984). 22. Dally, J. IV., "'Dynamic Photoelastic Studies of Fracture, "" EXPERIMENTAL MECHANICS, 19 (10), 349-361 (1978). 23. Rice, J.R., "'Mathematical Analysis in the Mechanics of Fracture, "" Fracture, 2, ed. H. Liebowitz, 191-311 (1968). 24. Eringen, A.C. and Suhubi, E.S., "'Elastodynamics, "' 2, Academic Press, 584-591 (1975). 25. Georgiadis, H.G. and Theocaris, P.M., "'On the Solution of SteadyState Elastodynamic Crack Problems by Using Complex Variable Methods, "" ZAMP, in press (1985). 26. Freund, L.B., "'Crack Propagation in an Elastic Solid Subjected to General Loading- L Uniform Rate of Extension, "" J. Mech. Phys. SoL, 20, 129-140 (1972). 27. Sih, G.C. and Chen, E.P,, "'Cracks Moving at Constant Velocity and Acceleration, "' Elastodynamic Crack Problems, ed. G.C. Sih, Noordhoff lnt. Publ., 59-118 (1977). 28. Nilsson, F., "'A Note on the Stress Singularity at a Non-uniformly Moving Crack Tip, "",L Elasticity, 4 (1), 73-75 (1974). 29. Rossmanith, H.P., "'Modelling of Fracture Process Zones and Singularity Dominated Zones, "Eng. Frac. Mech., 17 (6), 509-525 (1983). 30. Carlsson, J., Dahlberg, L. and Nilsson, F., "'Experimental Studies of the Unstable Phase of Crack Propagation in Metals and Polymers, "" Dynamic Crack Propagation, ed. G.C. Sih, Noordhoff Int. Publ., 165-181 (1973). 31. Erdogan, F., "'Fracture Problems in Composite Materials," Eng. Frac. Mech., 4, 811-840 (1972). 32, Rose, L.R.F., "The Stress Wave Radiation from Growing Cracks," lnt. J. Fracture, 17, 45-60 (1981). 33. Theocaris, P.S. and Georgiadis, H.G., "Emission of Stress Waves During Fracture, "' J. Sound Vib., 92 (4), 517-528 (1984), 34. Theocaris, P.M. and Georgiadis, H.G., "Rayleigh Waves Emitted by a Propagating Crack in a Strain-Rate Dependent Elastic Medium," J. Mech, Phys. Sol., in press (1985). 35. Rossrnanith, H.P. and Fourney, W.L., "'Determination of Crackspeed History and Tip Locations for Cracks Moving with Nonuniform Velocity, "' EXPERIMENTAL MECHANICS, 22 (3), 111-116 (1982). 36. Shmuely, M., Peretz, D. and Perl, M., "Effect of Rayleigh Waves in Dynamic Fracture Mechanics, ""Int. J. Frac., 14 (2), R69-R72 (1978).
1.5
t Fig. 11--Dynamic-SIF time diagrams corresponding to the composition: brittle (10percent plasticized epoxy resin)-ductile (50-percent plasticized epoxy resin)brittle layers
Zh :O.04m , x . . ~ . ~
x
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I
0.5
phase I: epoxy resin 50% plast. phase 2: epoxy resin 10% ptast. 0
94 9 March 1985
0
4O
80 Time
I
120 (tzs) - - ~
160
200