High Speed Crack Propagation and Branching Under Uniaxial and Biaxial

A photoelastic investigation of high speed crack propagation and branching in a brittle polyester material called Homalite 100 was conducted for both uniaxial and biaxial loading conditions. Cross type specimens were loaded in a specially designed loading fixture where loads perpendicular and parallel to the crack could be controlled independently. The photoelastic data obtained were analyzed to get the stress intensity factor K, the crack tip position a, the crack velocity a and other non-singular stress field coefficients including the stress acting parallel to the crack, a . ox It was observed that K showed an increasing trend as the crack propagated through the specimens. Tensile stress parallel to the crack gave rise to higher stress intensity factors, compared to the compressive case for the same crack tip positions and .initial normal loads. It was also found that tensile parallel remote stress enhanced branching and branching angle. While branching angles of 22° to 29° were encountered in uniaxial normal loading and biaxial tension-compression loading, angles as high as 45° and 73° were obtained in the tensiontension case. It was concluded in this study that any crack branching criterion has to take into account the nature of the remote stress parallel to the crack and that crack branching is strongly influenced by non-singular stress field coefficients.

High speed crack propagation and branching are of interest in several fields like mining and aircraft industries and large structures like nuclear reactors, pipelines, ships, etc. In the mining industry, one wants to optimize fragmentation and thus reduce the cost of mining operations by controlled fracturing. On the other hand, in structures like ships, bridges, pressure vessels etc., one wants to know if the crack will arrest after initiation or will go through the structure.
Dynamic fracture has always been one of the most important and difficult problems in mechanics. The existence of singularity at the crack tip and the inertia effects make it extremely difficult to obtain a solution by either analytical or numerical techniques.
Researchers [1,2,3,4] have been attempting to characterize dynamic fracture in terms of a relation between. the stress intensity factor, K and crack velocity a for different materials. However, till now, no study has been conducted to investigate the influence of non singular stress field coefficients on the a-K behavior.
In the experimental studies of Fracture Mechanics, photoelasticity is a widely used technique to determine the stress intensity factor K at the crack tip. Though Post [5] and Wells and Post [6] applied photo-elasticity to study the static and dynamic aspects of fracture and  11] have. also employed dynamic photoelasticity in their studies of propagating cracks.
In photoelastic crack tip stress analysis, it is not advisable to use data from near the crack tip for practical reasons like fringe clarity, light scattering from the caustic at the crack tip, an unknown degree of plane strain constraint and so on [12]. In order to perform meaningful analysis from a larger and more desirable area in the stress field, additional non-singular terms have to be included. In this research, the equations derived by Irwin [13] were used and the series coefficients, including the stress intensity factor, were obtained using a multipoint over-deterministic technique developed by Sanford and Dally [14]. A tota l of six stress field coefficients were included for the analysis and a computer program was written in BASIC language 3 to obtain the values of the coefficients.
The results showed that the stress intensity factor, K and the farfield stress parallel to the crack, a ox, varied systematically as the crack propagated across the specimen and higher order terms showed an oscillatory behavior. The value of the stress intensity factor at branching seemed to vary slightly depending on the initial normal load applied. The branching angle depended on the nature of the far-field stress. si"ty factor should reach a critical value , Kb (or the strain energy 1 nten release rate should reach a critical value, Gb) for branching to occur at terminal velocity. Kalthoff [19] found that there is a characteristic forking angle in which case the branches neither repel nor attract each other. The corresponding branching angle was about 30°. Kobayashi et al [4] observed that the branching stress intensity factor for Homalite 100 was approximately three times the fracture toughness. K values of the order of 1.54 and 1.98 MPaliil were encountered for 3/8" and 1/8 11 thick sheets at branching. These values were equal to the maximum K observed in fractured plates without branches. The results indicated that the dynamic stress intensity factor was a necessary but not sufficient condition for branching. The combination of excessively large strain energy release rate, shown by the large static stress intensity factor, available at the time when the running crack tip is subjected to a maximum dynamic stress intensity factor, could be a plausible cause for the crack to branch. Dally et al [2] obtained crack branching at branching stress intensity factor, Kib ranging from 3.3 to 3.8 times Klm' the arrest toughness when the crack moved at terminal velocity in Homalite 100.

Research by Dally, Kobayashi and others pointed to the fact that Kib
is not a material property, contrary to results obtained by Congleton,Anthony [20,21] and Doll (22]. Kirchner [23] found that Klb and Young's Modulus had a strong correlation and developed a strain intensity criterion for crack branching in ceramics. Ramulu et al [15] have developed a branching criterion which contends that a necessary and sufficient condition for branching is a crack branching stress intensity facto r, Klb' accompanied by a minimum characteristic distance H.P. Rossmanith and G.R. Irwin [3] suggested that non -singular ro ; rc. . stress field coefficients may influence a-K behavior in the terminal velocity region. Till now, no research has been conducted on high velocity crack propagation and branching by systematically varying the non-singular coefficients. This is the objective of this project.

THE MULTIPLE SPARK CAMERA AND ITS APPLICATION TO DYNAMIC PHOTOELASTICITY
Crack propagation studies need a high speed recording system to take pictures of the transient crack tip stress patterns in dynamic photoelastic stress analysis. One of the most frequently used recording systems is the Multiple Spark Camera, originally designed by Cranz and Schardin [25]. A camera similar to this was designed and built at URI and is housed in the Photomechanics Laboratory. This chapter will briefly discuss the method of photoelasticity and the components of the Spark gap Camera.
Many transparent noncrystalline materials that are optically isotropic when free of stress become optically anisotropic and display characteristics similar to crystals when they are stressed. These characteristics persist while loads on the material are maintained but disappear when the loads are removed. This behavior is called temporary double refraction and the method of photoelasticity is based on this physical behavior of transparent noncrystall i ne materials [26].
For experimentation, the model is fabricated from a polymeric, transparent, birefringent material. When circularly polarized light passes through the stressed model and then through another circular polari zer, an optical interference occurs due to stress-induced birefringence in the model. This optical interference produces a series of light and dark bands which are termed isochromatic fringe patterns.
The stress optic law, which relates the stress state of the model to 8 the order of the associated interference pattern is given by where 01 and o 2 are the in-plane principal stresses, T the maximum m in-plane shear stress, N the fringe order, f 0 the material fringe value and h the thickness of the model. In dynamic photoelasticity, the fringes move at high velocities and so a high speed recording system like the Multiple Spark Camera is used for purposes of stress analysis.

Description of the Camera
The camera consists of three main subdivisions, viz., the spark gap circuit, the optical arrangement and the control circuit.
3.1.l The Spark Circuit -The Multiple Mach spark circuit is shown in Fig. 2. In the camera that was used in this project, there are twenty spark gaps (SG), each of them connected to L-C circuits in series.
TSG is the trigger spark gap. In operation the condensers are charged to about 15 KV and the circuit L 1 c 1 closed by applying a trigger pulse to the spark gap TSG. The firing sequence is initiated at a predetermined time after the crack initiation· by applying a 30 KV pulse to the trigger gap. When the trigger gap is fired, the capacitor cl discharges to below the ground potential. When the voltage on c 1 becomes sufficiently negative, a spark occurs at gap SG 1 and capacitor * cl discharges, producing a short, intense flash of light.
The tim~ng between the first and second sparks depends on the in-* ductance L 2 in the c 1 L 2 c 2 loop. When the gap SGl fires, the voltage on C 2 decays with time and the gap SG2 fires when the voltage on c 2 is suffici ently negative. Likewise, a ll the twenty gaps fi re . The light from the spark gaps is led out of the camera by fibre optics.
2 Optical Arrangement -The optical setup is shown in Fig. 3. 3.1.
-Two circular polarizers are kept on each side of the specimen. As an example, light from the spark SGl passes through a field lens, the first polarizer, the specimen, the second polarizer and the second field lens onto the camera lens L 1 . In a similar manner, light from spark Gaps SG2, SG3, SG4 etc. is focused on the corresponding camera lenses L2, L3, L 4 etc.
The spark gaps, the field lenses and the camera lenses are so placed that the light from one particular spark falls on one particular camera lens so that the image from one camera lens is due to one spark only. The analysis shall make use of Irwin's crack tip stress function [28] and shall assume plane strain (sz = 0). The leading edge of the crack will be taken as the positive x-axis and the crack tip as ~he origin of the coordinate systel]l, as shown in Fig. 5.
For crack propagation in the x-direction and assuming that the stress field does not change with time with respect to the moving coordinate system, the following equations apply: where u and v are the displacements in the x and y directions and c the crack velocity.
Equilibrium conditions· in the x and y directions lead to the following results: where a , a and T are the stress components and p the density of the x y xy body. Equations(4.14)and(4.1S)can be simplified in appearance in the following ways. In equation (4.14), we substitute y 1 = Aly where In equat i on (4.15), The resulting pair of equations is where z 1 = f'(z 1 ), z 2 = f'(z 2 ), z 1 = x+iy 1 and z 2 = x+iy 2 .
In addition it is hel pful to recognize that the displacements can be divided into non rotational (u 1 , v 1 ) and nondilatational (u 2 , v 2 ) components. Thus v = v 1 +v 2 where aul avl 2 Since the shear stress on the crack faces is zero, L = 0 on y = 0 xy for x < 0. Along this line f' (z 1 ) = f' (z 2 ) ·= f' (x). One finds that on y = 0 (4. 28) Since Irn[f'(x)] on y = 0 is not zero along the entire x axis, we must assume . 2Al ( 4. 29) In order to detennine A, consider next that   The first three terms in each of the series Z and Y shall be ineluded in the analysis. So z 1 and z 2 can be expressed as follows: Wh ere A /2Tf = K, the stress intensity factor and remote stress cr = 2B . ReY 2 B 2 = 0 + B 1 p 2 cos¢ 2 + B 2 p 2 cos2¢ 2 (4.58) lmY 2 = B 1 p 2 sin¢ 2 + B 2 p 2 2 sin2¢ 2 ( 4. 59) and where 21 From Fig. 6, it is evident that pressions for and T in terms of the series constants and the xy polar coordinates rand e.

Application of Photoelasticity to the Dynamic Equations:
The stress optic law, which relates the optical properties of the material to its stress state, is given by where Tm is the maximum in plane shear stress, fCJ the material fringe value, h the thickness of the model and N the order of the fringe in consideration.
It is known that  at the Kth data point.
( 4.67) A combination of least squares and Newton-Raphson techniques is applied to this function as follows.

The Overdetermin is ti c Method:
The approach used is that suggested by Sanford and Dally [14].
The series constants have to be determined. to make GK = O. Though equation (4.67) can be solv·ed in closed form, the algebra becomes quite involved and a simpler numerical method based on the Newton-Raphson technique is employed.
In the overdeterministic method, the function GK is evaluated at a large number of data points in the stress field. If initial estimates are given for the series constants in eqn. ( 4. 67), GK f 0, since the initial estimates will usually be in error. To correct the estimates, a series of iterative equations based on a Taylor series expansion of G are written as where the subscript i refers to the ith iteration step and 6A 0 , 6A 1 , etc. are corrections for the previous estimates of A 0 , A 1 etc.
The corrections are determined so that (GK)i+l = 0 and thus eqn.  where N is the total number of data points considered. The correction factors are given by ( 4. 71) The iterative procedure is employed till the series constants are determined to obtain a close fit of the function G to the N data points.
The differentials of G with respect to the series coefficients are obtained as follows: Differentiating eqn. is the specified (input) fringe order for a data point, n the where ni c fringe order calculated from the computed set of coefficients and N* the total number of data points used [12]. Typically, errors of one tenth of a fringe order or less should indicate that the constants are accurate [12].
The error analysis has also been included in the computer program in the appendix.

S. EXPERIMENTAL PROCEDURE AND RESULTS
Cross type models were used in this study. The geometry of the specimen is shown in Fig. 7. The specimen length and width were made fairly large to insure that the boundaries were far away from the crack tip as the crack propagated. The starter crack is made with a band saw and the crack tip blunted with a fine file. The model is mounted on the loading frame, shown in Fig. 8, which was so designed that it could be used for both uniaxial and biaxial loading. Loading was applied using In Experiment 1, a uniaxial normal stress of 233 psi (1.61 MPa) was applied. The variation of crack tip position with respect to time is shown in Fig. 9. A constant crack velocity of 383 m/s was obtained. 27 The stress intensity factor, as shown in Fig. 10 and Table 1, showed · g trend, values ranging from 1 MPalill to 1.55 MPalill. The an increasin t ess a varied as in Fig. 11 and Table 1, with values osremote s r ox cillating about a mean of -2 MPa.
In Experiment 2, a higher uniaxial normal stress of 558 psi (3.85 MPa] was applied. A crack velocity of 400 m/s was obtained, as shown in Fig. 12. The stress intensity factor showed a sharply increasing trend, as in Fig. 13 and   Table 2.
To study the influence of far-field stresses parallel to the crack, a series of biaxial tension-compression and tension-tension experiments were conducted.
Typical isochromatics for tension-compression loading are shown in Fig. 15. It is seen that the fringe loops exhibit a strong forward tilt for this kind of loading, which indicates the nature of the remote parallel stress.
In Experiment 3, a tensile normal stress of 443 psi [3.05 MPa] and a compressive parallel remote stress of 197 psi [1.36 MPa] were applied.
The dynamic K was not enough at any stage to produce branching. A con-t crack velocity of 368 m/s was obtained, as shown in Fig. 16. stan Again, K showed an increasing trend as in Fig. 17 and Table 3, with values going up to 1.81 MPav'ill and a was oscillating about a mild ox negative mean of -2.l MPa, as in Fig. 18 and Table 3 velocity was a constant 385 m/s, as in Fig. 22. a values were osox cillating about a negative mean of -3.7 MPa as in Fig. 24 and Table 5. It seems that remote compressive stress parallel to the crack tends to lower the stress intensity factor and hence suppresses branching.
In Experiment 5, the crack made an attempt to branch -in fact, it almost branched at a crack jump distance of 7.2 cm, though actual branching occurred at a crack jump distance of 11.8 cm. In Experiment 6, the crack travelled 9.8 cm before it branched. This evidence adds to the fact that compressive remote parallel stress suppresses branching. This was also suggested by Dally et al [2] and Ravichandar [29]. It was also observed in experiments 5 and 6 that the fracture surface roughness was very high either when an attempt was made to branch or when actual branching occurred. Again, the branching angle in Experiment 6 was 24.5° as compared to 29° obtained in Experiment 5 , indicating that compressive parallel remote stress affects the branching angle, though not considerably. The case of compressive a 0 x has to be studied more by going in for very high compressive parallel loads.
Next, a series of biaxial tension-tension experiments were carried out. Typical isochromatics for this case are shown in Fig. 28 though the normal stress in this experiment was only slightly higher than that in Experiment 6, the K values in this experiment were much higher than the previous experiment for the same crack tip positions.
For example at a*/w = .38, K was 1.18 MPaliil in Experiment 6 compared to 1.82 MPalffi in Experiment 7. In fact, the stress intensity factor was so high that the crack branched very early. K varied with crack tip position as in Fig. 30 Fig. 37 and Table 9, because of the considerable backward tilt of the fringe loops due to the high parallel stress applied. The branching angle was 73°, which was much higher than the compressive a tests. One more interesting ox comparison can be made between Experiment 9 (normal stress 538 psi and parallel tensile stress 924 psi) and Experiment 2 (uniaxial normal stress 553 psi). In spite of the fact that the applied normal stress was less in Experiment 9, the K values for the same crack tip locations were much higher in that experiment than . Experiment 2, as shown in Tables 2 and 9. As an example, for a*/w = .39 , K was 1.56 MPaliil in Experiment 9, compared to 1. 28 Table 10. Kbr seems to vary v ery slightly with different magnitudes of applied stresses and probably depends on initial KQ. More experimentation has to be performed to confirm this. The variation of branching angle and the pre-branching fracture area with respect to the ratio of the applied normal stress to the applied parallel stress (a /a ) is shown in Figures 38 and 39 respectively, just to highlight y x the facts that branching occurs earlier in the case of positive oox' even with lower normal stresses, compared to the uniaxial and the tensile-compressive cases and that branching angles are much higher in the tensile-tensile case.
The isochromatics for uniaxial, tensile-compressive and tensiletensile cases are compared in Fig. 40   In this work, r was calculated for the .various branching situations 0 and the values, which ranged from 1.1 to 8 .1 mm, wer e found to differ considerably from the r value of 1.3 mm, obtained by Ramulu and coc workers. The r values are shown in Table 11. Also, the branching 0 angle was found to be dependent on the sign of 0 i.e. the nature of ox' the remote parallel stress, in contradiction to the equation (6.2), used by Ramulu et al, which seems to signify that the angle is independent of the sign of a ox This study .concludes that high velocity crack propagation and crack branching are considerably influenced by non-singular stress field co -efficients. More investigation has to be carried out to formulate an l ·rical branching criterion in terms of K, the crack velocity a and emp higher order stress field coefficints.
In this juncture, it has to be mentioned that the branching angles obtained were the macroscopic angles measured just at the crack tip.
The branches tend to attract or repel each other, as was observed by Kalthoff [19], which makes it difficult to measure the exact macroscopic branching angle.
It was also observed that the results obtained by the crack tip stress analysis procedure, outlined in Chapter 4, are sensitive to all the input parameters including the radius, theta and the fringe order of the data points. For some unknown reason, there was a convergence problem in the six parameter model when the remote parallel stress was compressive. This was not the case when it was tensile or when uniaxial normal loads were applied.

59
-  ay/a as a Function of Branching Angle.