31

International Journal of Fracture, Vol. 16, No. I, February 1980 © 1980 Sijthoff & Noordhoff International Publishers Alphen aan den Rijn, The Netherlands

Dynamic crack propagation in composites P.S. T H E O C A R I S

a n d J. M I L I O S

Department of Theoretical and Applied Mechanics, The National Technical University, Athens (625), Greece

(Received June 23, 1978; in revised form March 23, 1979)

ABSTRACT A study of the behaviour of cracked composite specimens under dynamic tensile load was undertaken. The crack propagation in the two-phase epoxy resin specimens was studied by the method of high speed photography along with the optical method of caustics. Our investigation was concentrated both on the dependence of the maximum crack propagation velocity and the stress intensity factor at the crack tip upon the different material combinations of the composite, as well as on the role of the interface again in regard to the crack propagation and the singular stress field concentrations at the crack tip. The results show that, under a given value of the applied dynamic load and given notch dimensions, the stress intensity factor at the crack tip and the crack propagation velocity in each phase of the composite is highly dependent on the material characteristics of each phase and on the existence of a stable interface between the two phases. More concretely, it was proved that the interface plays the role of a "barrier" to the crack propagation. Indeed, the crack propagates with a certain maximum velocity in the first phase of the composite and then stops momentarily when it reaches the interface, thus attaining later in the second phase a new maximum velocity. The maximum velocity and the stress intensity factor in the second phase of the composite specimens strongly depend on the material characteristics of the first (notched) phase and are also highly influenced by the crack arrest process itself. The crack propagation and the stress field concentrations at the crack tip in the first phase of the composite specimens is mainly independent from the material characteristics of the second phase of the composite specimens.

I. Introduction A s w a s a l r e a d y k n o w n , t h e p r o p a g a t i o n o f a c r a c k in a b r i t t l e m a t e r i a l b e c o m e s unstable under usual load conditions reaching under certain circumstances a velocity o f 60 p e r c e n t o f t h e v e l o c i t y o f t h e R a y l e i g h w a v e s f o r t h e specific m a t e r i a l . T h e r e a r e m a n y p a p e r s r e l a t e d to this s u b j e c t c o n c e r n i n g n o t o n l y s t a t i c l o a d c o n d i t i o n s , b u t also dynamic ones. The study of dynamic crack propagation has recently become a subject of great interest. The problem of what the dynamic case concerns, has been theoretically app r o a c h e d e i t h e r b y m e a n s o f t h e Griffith e n e r g y c o n c e p t o r b y m e a n s o f d y n a m i c e l a s t i c i t y . T h e e x p e r i m e n t a l a p p r o a c h e s , o n the o t h e r h a n d , a r e m a i n l y b a s e d o n the m e t h o d o f high s p e e d p h o t o g r a p h y in r e l a t i o n w i t h o n e o f t h e k n o w n e x p e r i m e n t a l m e t h o d s o f s t r e s s a n a l y s i s , s u c h as p h o t o e l a s t i c i t y o r t h e m e t h o d o f c a u s t i c s . E a r l y e x p e r i m e n t a l i n v e s t i g a t i o n s h a v e s h o w n t h e e x i s t e n c e o f a limiting v e l o c i t y o f c r a c k p r o p a g a t i o n a n d h a v e d e t e r m i n e d t h e m a g n i t u d e o f this v e l o c i t y f o r v a r i o u s m a t e r i a l s [1]. L a t e r on, M a n o g s [2] i n t r o d u c e d an e l e m e n t a r y t h e o r y c o n c e r n i n g t h e 0376--94291801010031-21500.20/0

Int. Journ. of Fracture, 16 (1980) 31-51

32

P.S. Theocaris a n d J. Milios

method of transmitted caustics for the simple case of an edge transverse crack in an infinite plate to study crack propagation phenomena in plexiglas and CR-39 plates under static and dynamic loadings. The general problem of an arbitrarily oriented crack in a transparent or opaque material with mechanical properties either elastic or elastoplastic was first encountered by Theocaris who introduced the theory of reflected caustics. These caustics by reflection yield the only possibility for studying the stress fields in opaque materials, which are the most interesting ones in constructions. For a review of the general theory of reflected caustics see one of the review papers contained in [3]. These early results were further developed later on. Using the method of caustics Katsamanis et al. [4] have shown for a dynamic loading in PMMAspecimens that the constant velocity of crack propagation increases with fracture stress, the magnitude of this increase depending on the load conditions. This maximum velocity reached a limiting value for stresses higher than a certain level. More recently Theocaris et al. [5] investigating the case of a compressive impact load, which has transmitted longitudinal-stress waves in a PMMA specimen, have shown that a crack propagation takes place only under the influence of an tensile pulse, while it remains inert under the influence of a compressive pulse. The transmitted initial compressive pulse was reflected on the opposite transverse boundary of the specimen and so was converted into a tensile pulse, which was again transformed into a compressive one when it was reflected at the other transverse boundary. The crack propagated by steps under the influence of only the tensile pulses with a velocity and a step length which depended on the magnitude of the stress pulse and on the existing crack length before each specific pulse. Lastly Kalthoff et al. [6], using the method of transmitted caustics, have studied the influence of dynamic effects on the crack arrest process on wedge-loaded DCB (Araldite-B) specimens. The stress distribution around the crack tip in terms of the stress intensity factor at crack initiation and crack arrest under dynamic loading were investigated and compared with the corresponding static values. The behaviour of the dynamic stress-intensity factor after crack arrest was also investigated, as well as the dynamic crack arrest toughness. The problem of crack propagation through definite discontinuities was first studied by Kobayashi et al [7] who used the method of dynamic photoelasticity, to investigate the crack extention through a hole lying ahead of the propagating crack in Homalite-100 specimens. They also made comparisons of the elasto-dynamic state of stress with the corresponding elasto-static state of stress. In a similar experimental work Ishikawa et al. [8] studied the behaviour of a rapidly moving crack near a small hole in PMMA-specimens and determined that the time interval of crack arrest at the hole diminishes with diminishing hole diameter and increasing load. They also concluded that the crack propagation velocity increases after the crack arrest, that is after passing through the hole. The problem of crack propagation through a rigid interface was first approached by Dally et al. [9] who investigated, by means of dynamic photoelasticity, the process of crack propagation in bimaterial plates which had a brittle material (Homalite 100) as first (notched) phase and a ductile material (epoxy resin) as phase II. The two materials were joined with a rather thick and tough high-shear strength epoxy layer. The results of this investigation showed that the crack stops abruptly as it reaches the adhesive layer between the two materials to penetrate later on, after a short crack arrest time, into the second material of the composite specimen. During the crack arrest period the value of the stress intensity factor K~ at the crack tip increases constantly until it reaches a sufficiently high value, corresponding to the toughness of Int. Journ. of Fracture, 16 (1980) 31-51

Dynamic crack propagation in composites

33

the adhesive joint by which the crack can be reinitiated in the second phase of the composite. As the crack starts propagating in the second phase of the composite the high K-value decreases. If the required high value of K is not achieved, the crack remains stationary while the K field at the crack tip oscillates. The problem of crack propagation through bimaterial plates without any intermediate adhesive layer between the two phases was first approached by Sereda et al. [10] again by means of dynamic photoelastisity. These authors used specimens composed of three layers of equal dimensions all cast from the same epoxy-resin, but containing different amounts of plasticizer. The results show that as the crack approaches the interface from the side of the material with a higher modulus of elastisity (EdE2 = 4) the stress intensity increases monotonically until the crack crosses the interface. After crossing the interface the stress intensity appears markedly decreased and subsequently, with the increasing distance from the crack tip to the interface, starts to grow again, stabilizing at last at a certain constant value. For the case of a crack approaching the interface from the side of a material with a lower modulus of elasticity the stress intensity decreases until the crack crosses the interface. After crossing the interface the stress intensity appears increased and subsequently continues growing as the crack propagates in the second phase of the composite, to stabilize again at last at a certain constant value. On the other hand, several theoretical investigations have dealt with the stress field configurations and singularities on bimaterial plates. Bogy [I1] was the first who studied the stress singularities on a perfectly edge bonded bimaterial wedge. On the other hand, Theocaris [12] has shown in a later paper that the optical method of reflected caustics can be used to calculate the order of singularity for the case of an entire plane consisting of two materially dissimilar half planes bonded together along their common interface and with a crack running along the one phase with its crack tip reaching the interface at any angle. In the present work the dynamic crack propagation in bimaterial specimens was investigated. Our interest was focussed on the process of crack propagation through the interface (stress intensity factors, crack propagation velocities) as well as the effects that the different material combinations cause on this process. The two phases of the composite specimens were made out of pure-epoxy polymers to which different amounts of plasticizer were added. Conclusions were derived concerning the crack propagation velocities in the two phases of the composite specimens, the stress intensity factors at the tip of the propagating crack and the role of the interface. 2. The method of transmitted caustics applied to crack propagation

In our experiments the stress intensity factor in the composites was determined by the optical method of caustics. According to this method a convergent or divergent light beam impinges on the specimen at the close vicinity of the crack tip and the transmitted rays are recieved on a reference plane, parallel to the plane of the specimen. The thickness of the specimen in the neighborhood of the crack tip is reduced due to the Poisson's ratio effect, because of the stress concentration at the vicinity of the crack tip. Simultaneously, the refractive index of the material was changed in the same region. As a result of these phenomena the transmitted rays from the neighborhood of the crack tip are deflected and they are concentrated along a strongly illuminated curve (caustic) on the reference plane, while the area enveloped by the caustic on the reference plane becomes a dark shadow. It has been shown [2, 3] that the equations of the caustic for the case of a straight Int. Journ. o[ Fracture, 16 (1980) 31-51

P.S. Theocaris and J. Milios

34

transverse edge crack in a tensile field are given in a parametric form by: xt

(

= Amr0\cos 0 + ~ cos

y ' = Amr0(sin 0 + ~ s i n ~ 0-)

(1)

with: r°=

-X--~J " C=ZotCtk,/k/~-~,

Am-

zi

(2)

ZO jr. Zi

where z0 denotes the distance between the specimen and the reference screen (Fig. 1), t is the thickness of the specimen, ct is the stress optical coefficient related to the transmitted light rays and zi is the distance between the reference plane and the focus of the light beam, which coincides with the focus of the multilens camera.

v3 ..j~; ~

Spark Gaps

Specimen //

1,I I

r"

i

I

'' i

+

)

I

L

Concave

0

reflector

0

Zi

"

Reference plane

Figure 1. Optical part of the experimental arrangement.

From eqn. (1) we obtain by differentiation that: dy__j'=d0Amro(cosO+cos~O). This expression becomes zero for 0 = 0max= 72 °. For 0 = 0max ( l ) becomes: x' = 0.1Amr0,

y' = (3.17)A,,r0

So, for the maximum transverse diameter of the caustic (/9, = 2y') we have: Dt = 3,17Amr0

(3)

By substituting the value of r0 in relation (2) we may derive for the stress intensity factor that: K~

1,671 × 1 ( D r ~5/2 = zotc, ~ \3,17]

Int. Journ. of Fracture, 16 (1980) 31-51

(4)

Dynamic crack propagation in composites

35

3. Experimental arrangement For the present experimental investigation we used a series of composite epoxy-resin specimens with a length of 300 mm, a width of 100 mm and a thickness of 2 mm. The specimens were composed of two full length phases of equal width, (300 mm x 50 mm) bonded together along their common interface. The mechanical properties of the two phases of the composite were varied by adding different amounts of plasticizer (Thiokol LP3) to the pure epoxy prepolymer (Epikote 828, Shell Co.). The two phases were bonded together without any adhesive joint. This was achieved by casting the material of phase II along the longitudinal boundary of the phase I strip which was already prepared and appropriately placed in the mould. After consolidation of phase II the composite specimen obtained was thermically treated for one week, so that both total polymerization of the plates and stress free boundaries at the interfaces were assured. Phase I always contained an initial transverse slit of constant length a0 -- 20 mm which had a maximum distance between its adjacent lips less than 0, 3 mm. This slit was always used to initiate the crack at the same position of phase (I). Twenty different complex specimens were prepared by using five different types of compositions in all possible combinations with one another as phase I, or phase II. These different compositions contained amounts of plasticizer increasing by l0 percent for each, from zero percent up to 40 percent. Thus, five groups of composite specimens were studied, each of which had phase I (slitted) with a constant percentage C plasticizer and phase II (plain strip) with varying amounts of plasticizer except the percentate used for phase I. Furthermore, two more tests were executed with both phases of the same composition (zero percent and ten percent of plasticizer) in order to reveal the influence of an interface in similar materials. This series of tests with phase I always being of the same composition (indentical phase I arrangement) yielded results related to the influence of the mechanical properties of the yet uncracked phase II on the mechanism of the crossing of the crack through the interface of a composite. Another arrangement of the same specimens, with the phase II always having the same mechanical properties and phase | having varying properties also yielded interesting results related to the influence of the properties of the already cracked phase on the transversing of the crack through the interface (identical phase II arrangement). Since addition of plasticizer in the epoxy prepolymer results in a reduction of the modulus of elasticity of the substance and a change of the polymer to a more rubbery one (and therefore ductile) interesting results may be derived by arranging the tests into two groups i.e. those for which the crack propagates from a more brittle to a more ductile phase and those where the crack moves from the rubberier phase I to the more brittle phase II. Table I gives the static mechanical and optical properties of the various plasticized polymers used in the tests. The applied dynamic load was of the type of falling weight (see Fig. 2) with a constant load rate, creating a stress rate ~ = 1 x 104N/cm 2 sec. The load was recorded with the help of an electrical dynamometric unit connected in series with the specimen, the signal of which was recorded by means of a storage oscilloscope. For the study of the crack propagation process we used a Cranz-Schardin high-speed camera, disposing 24 sparks with a maximum frequency of 10 6 frames per second. With the help of an external generator the sparks can be set in work with any desired frequency. The regulation of the spark frequency can also be made according to spark groups, which enables us to better study the fracture process at certain positions of

Int. Journ. of Fracture, 16 (1980) 31-51

P.S. Theocaris and J. Milios

36 TABLE I

Plasticizer C%

Young's modulus E [N/cm 2]

Poisson's ratio

Stress optical coefficient C[m"N -1. 10

0 10 20 30 40

324,000 316,000 292,000 245,00 179,000

0.338 0.338 0.340 0.358 0.43

17.03 20.03 20.79 21.66 30.78

\

// '~ Quartz force

=!

./

/ /

e

[53

IS:]

,.triggercircuit

/

24 spark gaps

specimen

motor

/! \\\\\\\\\\\\\\

=

high speed

rl \\\\\\\\\\\\\\\\\\

oscilloscope

'amplifier

transducer

I

II

trigger circuit

D.C. supply

'~

=]

camera \\\\~\\

I=, .......

I

/

generator

controller

Figure 2. Block diagram of the experimental set-up.

the specimen. The synchronisation of the fracture process with the high speed camera is achieved by means of a proper silver contact circuit, which triggers the sparks with the initiation of the crack propagation (Fig. 2). The optical part of the experimental arrangement is shown in Fig. 1. The light beam from each spark reflects on a spherical mirror of great reflective ability, with a diameter of 50cm and a focal distance of 700cm, and after passing through the specimen is focused on the respective lens of the camera. The type of film used was Kodak TRI-X-PAN, which combines both sensibility and enlargement ability of the photographs taken. 4. Results The first phenomenon to notice is the crack arrest at the interface. The crack propagates in phase I of the composite with a constant maximum velocity (Fig. 3). This constant velocity is maintained until a very short distance from the interface. Within this very short distance from the interface the crack starts decelarating and Int. Journ. of Fracture, 16 (1980) 31-51

Dynamic crack propagation in composites

37

Figure 3. Series of photographs showing the crack propagation in a composite. (Phase I 20% plasticizer, phase II 10% plasticizer).

Int. Journ. o[ Fracture, 16 (1980) 31-51

38

P.S. Theocaris and J. Milios

finally completely stops at the interface. The process of crack deceleration, crack arrest at the interface and crack acceleration in the second phase of the composite is characterized by the continual disappearance of the caustic in phase I and the subsequent appearance of it in phase II at the close vicinity of the interface. As the crack tip and therefore the caustics are approaching the interface and the forefront of the caustic is continuously cut off, the rest of the shape of the caustic remains unaffected preserving its initial shape of the generalized epicycloid. This process also continues after the crack arrest, that is after the crack tip, which lies approximately at the center of the caustic, has reached the interface, until the whole area covered by the caustic disappears, submerged at the interface. Furthermore, during this progressive disappearance of the caustic in phase I there are no signs that a new caustic is in the process of development in phase II. Subsequently and while the crack remains at the arrest condition a new caustic is continually appearing in phase II. The new caustic has initially the form of an arc of a circle, to emerge in a few microseconds as a full semicircle with its diameter at an angle with the interface. In Fig. 3 a series of photographs is presented which show the crack propagations in a composite with a more ductile phase I (20% plasticized) and a more brittle phase II (10% plasticizer). The process of crack propagation through the interface is illustrated in more detail in Fig. 4, which presents a series of photographs showing the crack propagation through a specimen with an unplasticized phase I and a more ductile (20% plasticized) phase II. Deviations from the above-described rule of crack crossing the interface, that is preliminary crack arrest in phase II with caustics which do not correspond to arcs of the final semicircle, occur very seldom, as for example in Fig. 3 (frame 15) and are possibly due to local imperfections of the interface. Notice that the caustic does not appear in the second phase of the specimen during the whole period of crack deceleration and in the first period of crack arrest although that the crack tip practically "touches" the interface. The final semicircular from of the caustic at crack arrest coincides with the form of caustics determined and calculated by Theocaris [12] for the static case of a crack reaching the interface of two dissimilar media bonded together. It can therefore be calculated accordingly. For the preliminary arc forms we assume that they correspond to semicircular caustics with a transverse diameter Dt = 2liar where l is the length of the arc. It is worth noticing here that the semi-circular caustic which appears at the last period of crack arrest depends only on the crack arrest process itself and not on the path that the crack follows after reinitiation. This is shown for example in Fig. 10, which illustrates the crack propagation and bifurcation on a specimen with a ductile phase I. The semicircular caustic was built up in phase II during the last period of crack arrest, although the crack did not subsequently enter phase II, but moved first along the interface. All these phenomena during crack arrest, the continual disappearance of the caustic from phase I at the interface, the subsequent emergence of it etc., show that the caustic and therefore the stress-strain field at the crack tip propagate through the interface in a wave-like way. This observation enables us to suggest that the stress field at the crack tip is directly correlated with the stress waves which emanate from the tip of the propagating crack and which are observed by means of high speed photography in all experiments performed with completely transparent materials such as plexiglas or polycarbonate. It has also often been observed in our experiments that these stress waves emanating from the crack tip are reflected at the free boundary of a polycarInt. Journ. of Fracture, 16 (1980) 31-51

Dynamic crack propagation in composites

39

Figure 4. Series of photographs showing the crack propagation through the interface of a composite. (Phase I 0% plasticizer, phase II 20% plasticizer).

bonate specimen when the crack reaches this b o u n d a r y - t h u s rupturing completely the s p e c i m e n - a n d creating a new caustic which travels back along the lips of the already existing crack together with the reflected wave. The same phenomenon can be observed in the experiments performed by Takahashi in plexiglas specimens ([13], Fig. 17).

Int. Journ. o[ Fracture, 16 (1980) 31-51

40

P.S. Theocaris and J. MUios

Taking all these phenomena into account we may assume that the disappearance and reappearance of the caustic at the interface is due to the distortion of the stress field in the close vicinity of the crack tip due to the reflection and refraction of the stress waves at the rigid interface. This means that the stress-strain field is not transmitted uniformly from one phase to the other. After the reflection or refraction of it at the interface it is subsequently reconstructed in phase II under the influence of the external load. This transfer of the stress-strain field from phase I to phase II of the composite is not observed by Dally et al. [9] who claim a simultaneous appearance of stress-strain fields at both sides of the arrested crack in the intermediate layer. These different results of the above authors are mainly due to the existence of the tough adhesive layer between the two materials. The adhesive layer was not only considerably tougher than both phases, but also significantly thicker as the interface of our case, which consisted of the areas of adsorption interaction between the two compositeswhich of course contain also the areas of mechanical imperfections such as voids microcracks etc. For a study of the interface phenomena in polymers see [14]. In the experiments of Dally et al. the crack stops in the intermediate layer, having practically left behind phase I and without touching the boundary of phase II. It is obvious that in this intermediate material there exists a possibility for a rather symmetrical built-up of the stress strain field around the crack tip. As soon as the crack is initiated in phase II, it accelerates and propagates in the composite and here again we have the typical process of crack acceleration, up to a new maximum constant propagation velocity. The constant velocity of the propagating crack in phase II of the composite takes in general a different value than the constant crack propagation velocity in phase I and it depends mainly on two factors. The first factor that determines the value of the crack propagation velocity in phase II is the crack arrest process itself. After the arrest, the crack reaches a higher constant velocity value than before crack arrest. This effect is always valid, independent of the material characteristics of the two phases, since it is also present in fractures of specimens composed of phases made of the same material. Figure 5 6.0

'

o

t

F

10% p l a s t i c i z e r

L

,..,,

4.5 m-

E

o

'7

3.0 !l'

~1

'

'O

II 1.5

i 0

2

4

Crack

6

tength

8

10

(cm) ~

Figure 5. Crack propagation velocity as a function of the crack length for a composite having both phases made of the same material (10% of plasticizer). Notice the effect of crack arrest.

Int. Journ. o[ Fracture, 16 (1980) 31-51

Dynamic crack propagation in composites

41

shows the crack propagation velocity in a composite where both phases are made of the same material, that is an epoxy polymer plasticized with 10%. Note in this figure the effect of the crack arrest resulting in an increase of the crack velocity in phase II. An explanation may be given that a new fracture process begins after the crack arrest period and under the same load conditions as those prevailing during the first initiation of crack propagation in phase I. However, this new fracture process takes place with a much longer preexisting initial slit than the initial slit of the composite before the load application. As already has been noticed, other authors have shown before that a longer initial notch under the same dynamic-load conditions results in a higher maximum fracture velocity [4]. Similar to our experiments is the case of crack arrest in a small hole studied by Ishikawa et al. [8], who also came to similar conclusions. Furthermore, in [5], where the case of a successive crack propagation and crack arrest was studied in a tensile notched specimen submitted to a succession of stress pulses created from an air gun, it can be readily concluded that the constant crack velocity for the second or third step of pulse was always noticeably greater than the constant velocities of their previous steps of crack propagation. Again, the load conditions were the same, since it was assumed that no attenuation happened in the reflected initial pulse, the preexisting notch being of course each time longer for a later step than its length in the previous step of loading. The second factor which determines the crack propagation velocity is the material characteristics. The effect of the material properties of the composite superimposes with the effect due to the crack arrest. When phase II is made out of a more brittle material than the phase I, then the whole process results in a further increase of the crack propagation velocity in phase II. The opposite result is observed when the crack starts from a more brittle material and afterwards enters in a more ductile phase. In this case the material effect tends to diminish the crack-velocity increase, due to the crack arrest. In those cases where the brittleness of phase I is significantly greater than that of phase II the effect due to the mechanical properties of phases counterbalances partially or completely the effect of the crack arrest and hence the velocity diminishes slightly in phase II. For example, in the case of a specimen, whose phase I was made of an unplasticized epoxy polymer and phase II was made of a 40% plasticized epoxy polymer we have the following results: The crack propagation velocity in phase I was found to be v ° = 362 m/s., while in phase II it was v~ -- 314 m/s, while for the opposite case these velocities become: v~ = 283 m/s and v°1 = 494 m/s. 0 4 0 1.15 for the Indeed, while for the initial arrangement of phases the ratio vl/vn inverse arrangement this ratio becomes v~/v°~ = 0,57. Furthermore, the ratio of the crack propagation velocities in phase I had only reached the value v°/v~ = 1.28 while the ratio of the crack propagation velocities in phase II for a similar but inverse change was increased by the factor vn/vn 0 40= 1.57. For the respective case of the specimen with 0% and 30% plasticized epoxy polymers for materials in the two phases respectively we have that v°=365m/s, and v ~ = 3 1 8 m / s for the one arrangement of the phases, while for the inverse arrangement of phases we have that v~ = 307 m/s. and v°i = 427 m/s, and consequently vt/vtl 0 3o= 1.15, while v~/v°1= 0,72 for the inverse arrangement of phases. We were able to study this effect more systematically by taking into account the phase classification of the specimens that was proposed before. For the identical phase 11 classification, that is for the specimen groups were we keep the phase I1 with the same material, we have determined, for all groups, that the stress intensity factor Int. Journ. of Fracture, 16 (1980) 31-51

42

P.S. Theocads and J. Milios

in the second phase increases continuously with increasing ductility of the first phase. The same is valid for the crack propagation velocity. As the ductility of phase I increases from specimen to specimen within each group, the corresponding crack propagation velocity in phase II increases. On the other hand, for the identical phase I classification we could not determine any definite change of the crack velocity or of the stress intensity factor in phase I of the composites in each group with changing ductility of the phase II materials. The crack propagation velocity of each material when bonded as phase I of a composite converges to a certain mean value, characteristic for each material. Figures 6 and 7 present the dependence of the crack length on time, the slopes of the straight lines being the crack propagation velocities for both cases of a brittleductile and a ductile-brittle bimaterial specimen. The above discussed dependence of the crack propagation velocity in phase II of the composites on the material characteristics of phase I is shown in Fig. 8. The velocities in phase II of all specimen groups having the same material as phase II (indentical phase II classification) is plotted as a function of the modulus of elasticity of phase I. As the diagram shows, the crack propagation velocity in phase II decreases linearly with increasing elastic modulus (and brittleness) of phase I. The groups of specimens with a brittle phase II lie on the diagram always above the respective groups with a ductiler one. That is for increasing brittleness of phase II, the material characteristics of phase I being the same, the velocity in phase II tends to increase. The material combinations in the bimaterial specimens determines also the crack-arrest time and the way that the crack propagates through the interface. For the case of a crack propagation from a more brittle to a more ductile phase 10

, 40%

, plasticizer

/

/

i

, + o

E "" z:

Phase

/ +

I

Phas e II

g6 I

u

o

........~...................

i n t e r f a c e (-X-)

4

0

48

96 Time

144

192

240

(l~S) - - - ~

Figure 6. Diagram of the crack length versus the crack propagation time for a composite with a more ductile phase I (phase I contains 40% plasticizer, phase II contains 10% plasticizer). Notice the crack arrest time (=8/xsec). For our case we have: vr = 314 m/s, vlt = 461 m/s.

Int. Journ. of Fracture, 16 (1980) 31-51

Dynamic crack propagation in composites 10

I

o

I

43

r

0%

plasticizer

20%

,,

8 E U

cCn

c

6

u L.)

2 f

J

0

48

t

96

144

192

2/.0

T i m e (ITS) " ~ Figure 7. Diagram of the crack length versus the crack propagation time for a specimen with phase I containing zero percent plasticizer and phase II containing 20% plasticizer (for this case we have vl = 331 m / s , vu = 350 m / s ) .

amount

of p l a s t i c i z e r

o '

o

500~,...,......,...,~

300 - -

200 18

+ zx x o

10"/* 20*/* 30*/, 40*/,

CI

o '

oo ''J

,o

J 23 28 E l ( N/cm 2) . 104

33

Figure 8. The dependence of the crack propagation velocities in phase II of composites containing the same material in phase II, on the elastic modulus of phase I. Int. Journ. of Fracture, 16 (1980) 3 1 - 5 1

44

P.S. Theocaris and J. Milios

we have arrest times between 20 and 50/zs., while for the inverse case we have shorter times of crack arrest, that is between 5 and 16 p~s. (see Figs. 6, 7 and 11). Again, when phase I of the composite is more brittle than phase II, the crack passes through the interface without any change in direction. Figure 9a shows a typical case for the path of the crack through the composite specimen. (Phase I not plasticized, phase II 40% plasticized). For the inverse arrangement when the crack starts from the ductile material and later enters the more brittle one, the crack deviates from its original direction, as it enters phase II of the composite, thus forming an angle of about 600-70 ° with the interface. After entering the second phase and at a relatively short distance from the interface it again takes the initial vertical to the interface direction (Fig. 9b). This phenomenon shows that in the inverse case, that is when the crack starting from the ductiler enters the more brittle material, the shear stresses at the interface reach a much higher value. Furthermore, the increased shear stresses when the crack crosses the interface from the side of the more ductile material, caused in some cases, the disruption of the interface at the region near the area of crack propagation, which resulted in a bifurcation of the crack into two branches when it entered phase II.

Figure 9. Shapes of cracks in three composite specimens after fracture, a) for a more brittle phase I (ph. 1 20% pl., ph. II 40% pl) b) for a more ductile phase I (ph. I 10% pl., ph. II 0% pl.,) c) for a more ductile phase I with bifurcation (ph. I 30% pl., ph. II 0% pl.)

Int. Journ. of Fracture, 16 (1980) 31-51

Dynamic crack propagation in composites

45

Figure 10 shows the fracture process of a composite specimen (phase I with 30% of plasticizer, phase II with a pure epoxy polymer). The crack propagated in phase I with a constant velocity and was arrested at the interface. The increased shear stress at the interface and, of course, some eventual additional imperfections at this region (for instance non perfect bondage) made the crack propagate along t h e interface before entering phase II. The appearance of two caustics in phase II surrounding the tips of the propagating crack along the interface are typical in Fig. 10, which clearly shows the splitting of the unique crack in phase I into two branches starting at the interface and propagating along it. The two branches, after meeting some points of a stronger interface bondage which create insuperable obstacles to the crack propagation along the interface, and because of the new redistribution of stresses in the composite where phase I resumes less and less of the load as the crack propagates along the interface, stop to propagate along the interface as they enter phase II and progressively are oriented in a direction normal to the interface direction. Figure 9c shows the path of the cracks in the composite specimen. It is again worthwhile noticing that the angle subtended by the crack-axes and the interface when the two crack branches entered phase II was at the beginning always equal to 70° and then progressively increased to 90 degrees. Another important remark is that when the crack splits into two branches the energy release per crack surface is split into two parts and therefore the crack velocities of the branches are of lower values than the one expected for the case of a single crack propagation. This phenomenon is related to the stress distribution around the two crack tips. More concretely, the crack propagation velocity in phase I for the specimen with 30% of plasticizer was v~ = 262 m/s and it took in the second phase the values v°t = 298 m/sec for the first crack and v~ = 257 m/s. for the second crack. Figure 11 shows the diagram of crack length as a function of time. At this point it is reasonable to discuss the shear stresses which act along the interface of the composite specimen and tend to tear it apart at this region, that is in a direction vertical to the initial direction of the crack propagation. The reason for the development of these shear stresses along the interface is the unequal distribution of the tensile stresses in the two phases of the composite. Assuming a composite specimen with a perfect interface bondage, that is with no relative motion between the two different phases, we can easily conclude that the relation of the stresses in the two parts of the composite is directly proportional to the relation of the Young's moduli of the two materials. More concretely for our case we have: For E I = • I I = const, it is valid that: trt/E~ = o'nlEn = const., therefore: Et[Et~ = o'1/crn, where the index I characterizes phase I and the index H phase II of the composite. The difference Ao- between trt and o-n results in a shear stress along the interface (~-= Act/4). These shear stresses superimpose with the singular stresses around the crack tip when it reaches the interface. At this instant the danger appears for the interface to be disrupted. So the phenomena described above can be explained by considering the stress distributions in the two phases as functions of the elastic characteristics of the different materials. Since all composite specimens were subjected during the tests to a uniform elongation along their transverse boundaries, it was noted in the case when a ductile material is in phase I (which is always slotted), the elongation at fracture of the composite is higher than in the case when phase I is occupied by a brittle material. This is because the elastic modulus of the ductile material is lower than that of a brittle one (for the epoxy polymers, see table I), while, on the other hand, the stable interface bondage requires an equal elongation and strain on both phases. Therefore, if a brittle material occupies phase II, it is forced to sustain a higher elongation and strain than Int. Journ. of Fracture, 16 (1980) 31-51

46

P.S. Theocaris and J. Milios

Figure 10. Series of photographs showing the crack bifurcation in the specimen of Fig. 9c.

Int. Journ. of Fracture, 16 (1980) 31-51

Dynamic crack propagation in composites

10[

; o

t

]

47

T

30% plasticizer 0% .,

8

E u IO~

c 6

P

Phase II

Phase I

(D

(o11

x)

4

0

48

96 144 Time (ITS)

192

240

Figure ] 1. Diagram of the crack length versus the crack propagation time for the specimen shown in Fig. 9c. Notice the two crack arrest processes when reaching the interface (before bifurcation) and before entering phase [I (after bifurcation).

when it is placed as phase I. This larger strain corresponds to a larger value of fracture stress for this material. The increased stresses in the brittle material when it occupies phase II of the composite explains the tendency for a further increase of the crack propagation velocity (intensification phenomenon due to the crack arrest effect) and also explains the results determined by the study of the specimens according to their phase classifications. The increased stress intensification of a brittle phase II means also a higher absolute value of the difference of the phase stresses at the interface, that is a higher shear stress at the interface. The relation of the phase stresses for two given materials (excluding the effects due to the crack tip singularity) are always the same independent of which material is used as phase I. This means that the ratio trb[Crd Of the stresses when the brittle material is used as phase I is equal to the ratio tr~ltr~ when the ductile material is used as phase I, where trb and ~a denote the fracture stresses for the case of the brittle phase I and o'~,, o'~ denote the fracture stresses for the case that the ductile material occupies phase I. It can be shown with the arguments developed previously that it is always valid that o-~ > trb and tr~ > trd SO that (tr~ - try) > (trb - trd). Thus, in the case of a more ductile phase I the increased shear stress at the interface causes a local deviation of the direction of the crack axis or a bifurcation of the propagating crack. In the case of a brittle phase I the effect of the shear stresses at the interface is much less notable. The shortening of the crack arrest time before the crack starts to propagate in phase II in the case of a more brittle phase II can be explained by means of the increased elastic energy stored in this phase of the specimen. This effect was also determined by Ishikawa et al [8] for the case of a crack arrest at a small hole lying ahead of the propagating crack. Int. Journ. of Fracture, 16 (1980) 31-51

48

P.S. Theocaris and J. Milios

It is worthwhile to study the behaviour of the stress intensity factor at the tip of the propagating crack in both phases. The stress intensity factor remains constant after the crack initiation in phase I and within a certain distance from the interface. As the crack reaches a point lying at a distance of approximate 0,2 of the width of phase I from the interface the stress intensity factor starts increasing, thus reaching at the interface a value of 30--50% higher than its constant value far away from the interface. This increase is due to the existence of both the interface, which plays the role of a transverse boundary, and the phase II with its different material characteristics. Although the contribution of phase II in the increase of the value of K, is considerable, much higher than the influence of interface, its mechanical properties seem not to influence the variation of Kr and therefore phase II may be assumed as a reinforcing element of the interface. This leads us to the conclusion that the increased values of the stress intensity factor depend mainly on the stable interface. The effect due to the mechanical properties of the phases reported by Sereda et al. [10], that is a decrease of KI as the crack approaches the interface coming from the side of the more ductile material, may be noticable only for large differences in the elastic moduli of the two phases. (In the above mentioned paper it was taken ERIE,, = 1/4) As it has already been mentioned, the singular stress field enters phase II of the composite during the period of the crack arrest, that is before the crack starts propagating in phase II. Shortly after the crack initiation in phase II the stress intensity factor takes values considerably larger than the value of /(i in phase II during crack initiation, this increase being of the order of 20 to 50% of the initial value

1.5

[

!

~O

=~1.0

O

,.¢.

.,¢,

0.5 z~ K[Ii/K~0

10% plasticizer

o KII[i/Krll:0 20%

,,

1 2

4

6 Crack length (cm)

8

10

Figure 12. The variation of the stress intensity factor at each phase of a composite having a more ductile phase I, normalized to its initial value at the initiation of the crack at each phase (K~0, K10), u as a function of the crack length. Notice the increased normalized value of KI in phase I (10% plasticizer) when the crack reaches the interface and the decreased value of KI immediately after passing through the interface (phase II, 20% plasticizer).

Int. Journ. of Fracture, 16 (1980) 31-51

Dynamic crack propagation in composites 1.5

1.5

t

49

Y

1.0

f

1.0

q

I

0.5

z~ KIi 0

IKIo

KIIIi /" ~rl "I0

0.5

40"1, plasticizer 10%

1 2.0

"

,

4.0 6.0 8.0 Crack length (cm) ---~

10.0

Figure 13. The variation of the normalized value of the stress intensity factor at each phase of a composite having a brittle phase I as a function of the crack length (phase I: 40% plasticizer, phase II: 10% plasticizer).

of K~ in phase II. This phenomenon of the progressive increase of /(i as the crack propagates in phase II is apparent from the successive size of caustics. These caustics reach a final stable size at the end. Figures 12 and 13 show the variation of the stress intensity factor at each phase of the composite, normalized to its initial value at the initiation of the crack at each phase, as a function of the crack length. This variation of K after the crack has crossed the interface is in agreement with the results of Sereda et ai. [10]. Totally opposite to both Sereda's [10] and to our results are the data presented by Dally et al who studied the crack propagation in specimens with a brittle phase I and a ductile phase II. They predicted a decrease of /(i before the crack reaches the interface, an increased value of K~ immediately after crack arrest and a subsequent monotonic decrease of the Krvalue as the crack propagated in phase II. These results by Dally are mainly due to the existence of the tough adhesive layer between the two phases of the composite which strongly modifies the process of crack propagation through the interface. The propagating crack in phase I approaches practically not a more ductile but a much tougher material (adhesive layer) and stops in this material without touching the transverse boundary of phase II. As mentioned before, this results in a significantly different process of stress-strain transfer from phase I to phase II. As mentioned before, the final stable values of the stress intensity factor in phase II of the composite are also influenced by the mechanical properties of phase I. Figure 14 shows the variation of the final stable values of Kr in phase II for the case of specimens with identical phase II and varying phase I as a function of the elastic modulus of phase I. To show the relative changes of the final stable /(i values in phase II of each group, the values were normalized to the final stable value of /(i in phase II in the case when a pure epoxy polymer was used as phase I. Int. Journ. of Fracture, 16 (1980) 31-51

50

P.S. Theocaris and J. Milios amount o

of

ptasticizer

o

2'011.5

I

CI

o

~

oo

I

I.

t I , - - i i,=,,~

1.0 .... 0 + ,',

× n

0 "/.plasticizer 10% 20 % ,, 30 % ,, 40 % ,,

0.5

18

-~

L

23 28 E l (Nlcm2) x104

33

Figure 14. The variation of the final stable values of Kt in phase II (K~) for the case of specimens with identical phase II and varying phase I, as a function of the elastic modulus of phase I. All given values of KI~ of each group of specimens were normalized to the final stable K~ value in phase II of the group specimen with a pure epoxy polymer as phase I (KI0)."

5.Conclusions From the whole study it can be concluded that: i) The propagating crack reaches in both phases of the composite specimen a limiting stable velocity value, ii) The value of this stable and almost constant velocity in phase I (slitted) of the composite is a function mainly of the load conditions and the characteristics of the material used in this phase. A dependence on the material characteristics of phase II was not detected, iii) An arrest of the propagating crack occurred at the interface. During the crack arrest time the singular stress distribution is also established beyond the interface which caused the further propagation of the crack in phase II of the composite, iv) The crack arrest phenomenon imposes an increased crack propagation velocity in phase II of the composite, because the preexisting crack behaves like an initial notch in the second step of crack propagation, v) when the phase I (slitted) is composed of a more ductile material than phase II, then an increased stress intensification in phase II of the composite occurs, which results in increased values of the crack propagation velocity and the stress intensity factor in phase II. This increased stress intensification results also in a diminishing of the crack arrest time at the interface and in increased values of the shear stresses at this area. vi) The increased shear stresses at the interface result in local deviations of the propagating crack from the vertical direction as it intersects the interface and, even, in crack bifurcations at this area. The opposite effects were determined for the case of specimens with increased brittleness of phase I relative to phase II. Int. Journ. of Fracture, 16 (1980) 31-51

D y n a m i c crack propagation in composites

51

Acknowledgment T h e r e s e a r c h w o r k c o n t a i n e d in t h i s p a p e r w a s p a r t l y s u p p o r t e d b y U n i v e r s i t y f u n d s . It is a p a r t o f t h e t h e s i s o f D o c t o r E n g i n e e r o f t h e s e c o n d a u t h o r at t h e N a t i o n a l Technical University of Athens. REFERENCES [1] H. Schardin, "Velocity Effects in Fracture", J. Wiley and Son (1959) 297. [2] P. Manogg, "Anwendung der Schattenoptik zur Untersuchung des Zerreissvorgangs von Platten", Dissertation 4/64, Universitaet Freiburg (1964). [3] P.S. Theocaris, "Developments in Stress Analysis I", edited by G. Holister, (1979) Chap. 2, 27. P.S. Theocaris, "Mechanics of Fracture" edited by G. Sih, Vol. VII, (1979) (in print). [4] F. Katsamanis, D. Raftopoulos and P.S. Theocaris, Experimental Mechanics, 17 (1977) 128. [5] P.S. Theocaris and F. Katsamanis, Engineering Fracture Mechanics, 10 (1978) 197. [6] J.F. Kalthoff, J. Beinert and S. Winkler, submitted to the "Symposium on Fast Fracture and Crack Arrest" ASTM Committee E-24 on Fracture Testing of Metals, Chicago, Illinois, June 28--30 (1976). [7] A.S. Kobayashi, B.G. Wade and N.E. Naiden, Experimental Mechanics, 12 (1972) 32. [8] K. Ishikawa, A.K. Green and P.L. Pratt, Journal of Strain Analysis, 9 (1974) 233. [9] J.W. Dally, T. Kobayashi, International Journal of Solids and Structures, 14 (1978) 121. [10] V.E. Sereda and V.M. Finkel, Tambov Institute of Chemical Engineering, UDC 620 192.45, translated from Problemi Prochnosti, 12 (1977) 24. [11] D.B. Bogy, Journal of Applied Mechanics, 38 (1971) 337. [12] P.S. Theocaris, Journal of Applied Mechanics and Physics (ZAMP), 26 (1975) 77. [13] K. Takahashi, Bericht W 5/1977, Fraunhofergesellschaft, Institut fiir FestkSrpermechanik, Freiburg. [14] G.C. Papanicolaou, S.A. Paipetis and P.S. Theocaris, Colloid and Polymer Science, 7 (1978) 43. RI~SUMI~ On a entrepris une 6tude sur le comportement d'6prouvettes composites figur6es sous contrainte de traction dynamique. La propagation de la fissure dans des 6prouvettes de r6sine Epoxy ~ 2 phases a 6t6 6tudi6e par la m6thode de photographie en grande vitesse ainsi qu'en utilisant la m6thode optique des caustiques. L'6tude a 6t6 concentr6e h la fois sur la d6pendance de la vitesse maximum de propagation d'une fissure ainsi que du facteur d'intensit~ d'entaille ~ rextr6mit6 d'une fissure sur les diff6rentes combinaisons de mat6riaux pouvant constituer le composite et sur le r61e de l'interface existant entre la propagation de la fissure et les concentrations de champs de contrainte ~ I'extr6mit~ de la fissure. Les r6sultats montrent que, sous des valeurs d6termin6es de la charge dynamique appliqu6e et pour des dimensions d'entaille donn6e, le facteur d'intensit6 des contraintes ~ l'extr6mit6 de la fissure et la vitesse de propagation de la fissure dans chacune des phases du composite d6pend dans une large mesure des caract6ristiques du mat6riau de chaque phase et de l'existence d'un interface stable entre les 2 phases. Concr6tement parlant, on a pu 6tablir que l'interface joue le r61e d'une barri~re ~ la propagation de la fissure. En effet, la fissure se propage avec une certaine vitesse maximum dans la premiere phase du composite et ensuite s'arr~te momentan6ment lorsqu'elle atteint l'interface jusqu'h reprendre sa course dans la deuxi~me phase du composite avec une nouvelle vitesse maximum. La vitesse maximum et le facteur d'intensit6 des contraintes dans la seconde phase de l'6prouvette composite d6pend fortement des caract6ristiques du mat6riau de la premiere phase (la phase entaill6e) et sont 6galement largement influenc6s par le processus d'arr~t des fissures lui-m6me. La propagation des fissures et les concentrations du champ de contrainte ~ l'extr6mit6 de la fissure dans la premiere phase d'une 6prouvette composite est g6n6ralement ind6pendant des caract6ristiques du mat6riau de la deuxi~me phase de ce m~me 6chantillon.

Int. Journ. of Fracture, 16 (1980) 31-51