i~k'7:.1 ;Zlilf if n M42, 20084r-, pp.63-67 Research Reports of the School of Engineering,
Kin ki University No42, 2008, pp.63-67
Comparison between
and some of the
the dynamic crack equation published experiments
Masaaki WATANABE* and Ryon Chol SO**
Abstract
Making use of the data of some of the published paper on dynamic crack experiment, we have examined and discussed whether the experimental results are consistent with the dynamic crack equation or not.
Keywords: dynamic crack propagation, consistency of equation and experimental results
1. Introduction
Much work has been done on dynamic crack propagation in brittle materials and numerous problems have been discussed. In 1980's, for example, the uniqueness of the stress intensity factor - crack velocity relationship has been discussed by Daily et. al.". The idea behind this subject seems to be the following.
When the stress intensity factor, K1, is equal to the fracture toughness of the material, i.e., the condition, K1 = K, , is satisfied, the static crack starts to propagate, which is unique, independent of such experimental conditions as, CT, SEN and other tests.
The dynamic crack equation 2).3j, which corresponds to the condition, K1= K](', in static case, is given as
f=G(I)A(v)~G(!)(1— v/CR) (1)
where T is the fracture energy, which is the energy needed to create a crack of unit length. The quantity, I, is the instantaneous crack length, A(v) is the universal function of the crack velocity, v, and CR , the Rayleigh wave speed. The quantity, GU), is the amount of energy per unit area present at the tip of a static crack of length I and contains all of the effects of the applied stresses and specimen geometry. Equation (1) is the
function of the velocity, and the typical parameters measured in the experiment of dynamic crack propagation are the crack velocity, v, and "the fracture toughness", F. Thus it seems to be natural to expect the relationship, F— v could be obtained in any of the experiments associated with dynamic crack propagation although whether this relationship is unique, independent of the experimental conditions or not is a different problem, and should be examined carefully.
The dynamic crack equation (1) is derived for the crack in an infinitely large elastic plate. Ravi-Chandar and Knauss 41 have simulated this condition by choosing the size of the specimen such that the reflected waves do not interact with the crack tip for the duration of the experiment. They have obtained numerous important results on dynamic crack propagation in the series of the extensive work 5-71. Ma and Freund 8) have examined the dynamic crack equation (1) and compared it with their experiments, and found agreement at the initial stage of dynamic crack propagation when the crack velocity remains small, say, 240 infs. They have shown disagreement with the theory when the velocities of the crack become large, say, 410 m/s or 435m/s in the experiments on Homalite-100. In order to clarify the cause of disagreement, Rosakis and Ravi-Chander 9)
* z"
Department of *Intelligent Mechanical Engineering,
**Information and Systems Engineering , School of Engineering, Kinki University
63
64 `f .1.'! nli{i}pj'li ' No42
have experimentally examined crack-tip stress state associated with three dimensional effect .
Recently, Sharon and Fineberg 1°) have claimed that they have confirmed agreement between the dynamic crack equation (1) with their experiment of dynamic crack propagation. In this experiment they have observed that the micro-branches, which is three dimensional, grow as the crack velocity exceed the critical velocity 11). The authors are not fully convinced with this experiment since the experiment clearly shows growth of three dimensional micro-branches, while the dynamic crack equation (1) is two dimensional theory of dynamic crack.
It is important to compare the dynamic crack equation with the experiment of dynamic crack propagation as discussed above. Among numerous experiments of dynamic crack propagation, we find the experiment of Arakawa and Takahashi 123 is interesting because they measure the values of so many experimental parameters, which enable us to see the consistency of the dynamic crack equation (I) and the experiment
2. Comparison between the dynamic crack equation (1) and the experiment performed by Takahashi and Arakawa 11)
Materials used in the experiment 12) were polymers, PMMA and Epoxy. We first analyze the data for PMMA as shown in Fig. 6 of this work, from which we find the relationship between and a , as shown in Fig.1. We then calculate "the normalized fracture toughness", G0, from the following formulae,
GD= T(v,a)1 Go(2)
where r(v,a) = Id 1E for the case of plane stress.
The quantity, E, is the Young's modulus. The energy release rate at the initiation of crack propagation is obtained as, Go = 542.7 J1m2. The crack velocity,
v0 at the condition a = 0, is found to be v0 = 294.3 in/s, which can be seen from Fig. 1. The Rayleigh wave velocity for PMMA is cR =1309 mls. Making use of these quantities, we define the normalized crack velocity, vl, and the normalized crack acceleration, a,
as
vl = (v— v°)IcR (3a)
a= ad/ Icl(3b)
where Al (= 6.6x 10-1m) is the mesh length which we use to extract various quantities from the data. We then make the table for the normalized quantities, GD, vl and Cc from Fig. 6 of the paper 121. Making use of this table we show the graphs of GD versus vi, and GD versus a, which are shown in Fig. 2 and Fig. 3, respectively. In most experiments the fracture toughness is a function of the crack velocity and the graph GD versus a shown in Fig. 3 seems to be strange, however, Figs. 2 and 3 are specific for SEN experiment. Since the quantity, GD is the function of
vi and a, we expand GD as,
G^
a (x 10 mls`) 1.25•
1•
0.75 • 0.5 -
0.25 s)
-0.25•.
-0.5.
Fig. l I. The crack acceleration, a, versus the velocity of the crack v , for PMMA.
Fig. 2. The normalized fracture toughness versus velocity of the crack for PMMA.
64 `f .1.'! nli{i}pj'li ' No42
have experimentally examined crack-tip stress state associated with three dimensional effect .
Recently, Sharon and Fineberg 1°) have claimed that they have confirmed agreement between the dynamic crack equation (1) with their experiment of dynamic crack propagation. In this experiment they have observed that the micro-branches, which is three dimensional, grow as the crack velocity exceed the critical velocity 11). The authors are not fully convinced with this experiment since the experiment clearly shows growth of three dimensional micro-branches, while the dynamic crack equation (1) is two dimensional theory of dynamic crack.
It is important to compare the dynamic crack equation with the experiment of dynamic crack propagation as discussed above. Among numerous experiments of dynamic crack propagation, we find the experiment of Arakawa and Takahashi 123 is interesting because they measure the values of so many experimental parameters, which enable us to see the consistency of the dynamic crack equation (I) and the experiment
2. Comparison between the dynamic crack equation (1) and the experiment performed by Takahashi and Arakawa 11)
Materials used in the experiment 12) were polymers, PMMA and Epoxy. We first analyze the data for PMMA as shown in Fig. 6 of this work, from which we find the relationship between and a , as shown in Fig.1. We then calculate "the normalized fracture toughness", G0, from the following formulae,
GD= T(v,a)1 Go(2)
where r(v,a) = Id 1E for the case of plane stress.
The quantity, E, is the Young's modulus. The energy release rate at the initiation of crack propagation is obtained as, Go = 542.7 J1m2. The crack velocity,
v0 at the condition a = 0, is found to be v0 = 294.3 in/s, which can be seen from Fig. 1. The Rayleigh wave velocity for PMMA is cR =1309 mls. Making use of these quantities, we define the normalized crack velocity, vl, and the normalized crack acceleration, a,
as
vl = (v— v°)IcR (3a)
a= ad/ Icl(3b)
where Al (= 6.6x 10-1m) is the mesh length which we use to extract various quantities from the data. We then make the table for the normalized quantities, GD, vl and Cc from Fig. 6 of the paper 121. Making use of this table we show the graphs of GD versus vi, and GD versus a, which are shown in Fig. 2 and Fig. 3, respectively. In most experiments the fracture toughness is a function of the crack velocity and the graph GD versus a shown in Fig. 3 seems to be strange, however, Figs. 2 and 3 are specific for SEN experiment. Since the quantity, GD is the function of
vi and a, we expand GD as,
G^
a (x 10 mls`) 1.25•
1•
0.75 • 0.5 -
0.25 s)
-0.25•.
-0.5.
Fig. l I. The crack acceleration, a, versus the velocity of the crack v , for PMMA.
Fig. 2. The normalized fracture toughness versus velocity of the crack for PMMA.
a Fig. 3. The normalized fracture toughness
versus acceleration of the crack for PMMA.
G D = 3.412+ a11 + a2 v12+a(bo + bi v1) + b2a 2 (4)
which is the Maclaurin series in terms of the quantities, and a. The number 3.412 in right hand side of (4) is the value of GD at the condition a = 0 and
= 0 . We have live unknown constants, at , 02, bo , b1 and b2 , which are obtained by substituting the values of G0 , ui and a, from the table described above, into (4). Making use of this table we obtain Fig.4, which shows the behavior of the curve around v9= 294.3 nils in Fig. I in terms of the normalized quantities, vi and a. We note that the normalized crack velocity, vi, shown in Fig. 4, always takes the negative value since we expand the quantity GD at the maximum crack velocity, v = v0 and ã= 0. Since
a
Fig. 4. The graph of a versus the experiment 12)
for PMMA of
have the following equation to obtain the relationship between and a,
b2&2 + a(bo + livi) +ui(ai+G(i)a--0 / Go +0
(6) where we have substituted the equation,
G(/)„.__0 / Go = 4.402, into (6), which is obtained by the following equation,
(1— vo leR)G(1),0 /Go = 3.412 (7) Solving (6) by the formula for the quadratic equation, we find
cx=(-00 bivl)
Substituting the values of the constants, al , a2, bo, b1 and b2, obtained from (6), we obtain the graph of a versus viin Fig. 5, which should be compared with Fig. 4, which shows the direct experimental observation, We find, however, the acceleration, a does not take negative values for the case, Go ? 3.412 as shown in Fig. S. We then find the experimental result for PMMA'2 is not consistent with the dynamic crack equation (1).
Taking the similar procedure we have examined the data for Epoxy from Fig. 7 in the paper 12j and find the
66 iftatle f-.1:'i-gliftjf #e i', No42
a CR1097 m/s, Go-111.2 JIm2' and
G(1)a`Q IGo = 4.853 . Fig. 6 is the graph of a versus i-)1 forEpoxy, which corresponds to Figure 4 for PMIVMA. Fig. 7 is the graph of a versus , which is obtained by (8). In Fig, 6 we find the crack acceleration could take negative value, while we find no real solution for a from (8) for the case, GD ? 3.30.
We again conclude that the data for Epoxy of the experiment 12) is not consistent with the dynamic crack equation (1).
Fig. 6.
a
a
3. DISCUSSION
We have analyzed the experiment 12) and obtained the relation between the crack acceleration and the velocity of the crack as shown in Fig. 1 for PMMA. The reason why we have specifically obtained this graph, a versus v, is based on the general principle that one should show the relation between directly observable quantities as much as possible. The quantity, Kd , is the derived quantity, i.e., the stress intensity factor, Kd , is measured by the method of caustics or any other means, while the velocity and acceleration are not only the basic quantities of mechanics in general but also the observable quantities in the experiment if one could have sufficient resolution for them_ From the point of view of the fracture mechanics, it seems not to be relevant to measure both of the crack acceleration and the stress intensity factor or the fracture toughness, Kd , because the parameters associated with the crack acceleration can only be obtained from the higher orders stress fields 131 when the applied stress is a constant, while the stress intensity factor is the parameter associated with the singular stress field at the tip of the crack. Their experimental apparatus is such that the applied stress increases as the crack propagate through the pin located across the plane of the crack propagation. Thus the data itself is relevant in the sense of the approximation.
Beside we should point out that the boundary condition of the experiment 12) and the dynamic crack equation (1) is different, i.e., the crack in infinitely large elastic plate is assumed in deriving equation (1), while the sound waves reflected from the nearest boundary could interact a few times with the moving crack during the experiment.
The effect of the sound wave on the dynamic fracture toughness, Kd , is clearly demonstrated in the experiment 141. Although we do not know the detailed physical effect of the sound wave on the moving crack, we do know that the reflected sound waves could cause multiple branching of the moving crack st. Strictly
Comparison between the dynamic crack equation and some of the published experiments 67
speaking, one must derive the dynamic crack equation, which corresponds to equation (1), taking account of the boundary condition of this experiment 12), and then compare it with the experimental result although it is very hard in practice.
Keeping these problems in mind we conclude that the experiment of Takahashi and Arakawa is found not be consistent with the dynamic crack equation as discussed in the previous section. The reason for it is probably the interaction between the moving crack and the reflected waves from the boundary of the specimen.
crack acceleration on the dynamic stress intensity factor in Polymers", Exp. Mech.25, pp. 247-262
(1987)
13) Nillson, F. , "A note on the stress singularity at a non-uniformly moving crack !lip" J. of Elasticity 7, pp. 73-75 (1974)
14) KaItho1 ; J.F. , "On the measurement of dynamic fracture toughness- a review of recent work", Int. J.
of Fracture 27, pp. 277-2983. (1986)
4. REFERENCES
1) Daily, J. W. and Fourrney, W.L., "On the uniqueness of the stress intensity factor- crack velocity relationship'', I nt. J. of Fracture 27, pp. 159-168 (1985)
2) Freund, L.13., "Crack propagation in an elastic solid subjected to general loading" J. Mech. Phys. Solids 20, pp.129-162 (1972)
3) Freund, L,B., "Dynamic Fracture Mechanics"
Cambridge Univ. Press, New York (1990)
4) Ravi-Chandar, K. and Knauss, W.G., "An experimental investigation into dynamic fracture. I Crack initiation and arrest ", Int. J. of Fracture 25, pp. 247-262 (1984) 5) Ravi-Chandar, K. and Knauss, W.G., "An experimental
investigation into dynamic fracture. II Microstructural aspect", Int. J. of Fracture 26, pp. 65-80 (1984) 6) Ravi-Chandar, K. and Knauss, W.G., "An experimental
investigation into dynamicfractured!! On steady state crack propagation and crack branching", Int. J. of
Fracture 26, pp. 141-154 (1984)
7) Ravi-Chandar, K. and Knauss, W.G., "An experimental investigation into dynamic fracture. IV On the interaction of stress waves with propagating cracks ",
Int. J. of Fracture 26, pp. 189-200 (1984)
8) Ma, C.C. and Freund, LB., "The extent of the stress intensity factor field during crack growth under dynamic loading condition", ASME J. of Appl. Mech.
53, pp. 303-310 (1986)
9) Rosakis, A j., and Ravi-Chandar, K, "On crack —lip stress state: An experimental evaluation of three dimensional effects", Intl. J. of solids Structure, pp.
121-134 (1986)
10) Sharon, E. and Fineberg, J., "Confirming the continuum theawy of dynamic brittle fracture for fast cracks", Nature 397, pp. 333-335 (1999)
11) Sharon, E. and Fineberg, J., "Micrabrcmching instability and the dynamic fractue of brittle materials", Phys. Rev. B 54, pp. 7128-7139 (1996) 12) Takahashi, K. and Arakawa, K., "Dependence of
