E.E. Gdoutos Democritus University of Thrace, School of Engineering Xanthi, Greece

Transcription

1 International Journal of Fracture 27 (1985) R23-P, Martinus Nijhoff Publishers, Dordrecht. Printed in The Netherlands. R23 DISUSSION: "OMPARISON OF THE RITERIA FOR MIXED MODE BRITTLE FRATURE BASED ON THE PREINSTABILITY STRESS-STRAIN FIELD. PART I: SLIT AND ELLIP- TIAL RAKS UNDER UNIAXIAL TENSILE LOADING. PART II: PURE SHEARAND UNI- AXIAL OMPRESSIVE LOADING," by S.K. Maiti and R.A. Smith* E.E. Gdoutos Democritus University of Thrace, School of Engineering Xanthi, Greece The problem of an angle slit and elliptical crack in an uniaxial tensile, compressive or pure shear stress field is considered by the authors [1,2]. They used four different fracture criteria in order to assess their predictive capabilities and limitations. The angle of initial crack extension, the critical load at incipient fracture, and the crack path are determined for each of the four criteria from the stress state of the initially straight crack. The results obtained are then compared with the available experimental data. The criteria of maximum tangential stress (MTS), maximum tangential principal stress (MTPS), maximum tangential strain (MTSN), and strain energy density (SED) are mentioned. In most of the cases, the predictions based on (MTS), (MTPS), and (MTSN) are nearly the same while differing from those based on the (SED) criterion. This was the underlying philosophy used by the authors to establish the credibility of the (SED) criterion in relation to the others. It should be pointed out that such a philosophy for evaluating the validity of a theory has no scientific ground and even less logic. The merit of a theory cannot be judged on the basis that it disagrees with others. When all theories agree, there are the two distinct possibilities that either they are all correct or all wrong. It is fitting here to quote from the great Greek philosopher, Socrates, who said "do not follow the many but the one, the expert." It is not surprising that the results obtained from (MTS), (MTPS), and (MTSN) are close to one another because they are essentially the same within the framework of linear elasticity. The maximum tangential stress (MTS), the maximum tangential principal stress (MTPS) and the maximum tangential strain (MTSN) are, in fact, very closely related as the crack tip is approached. Therefore, they are almost identical and share the same limitations. Only one out of the six stress or strain components was considered. Since the stress state ahead of the crack is multiaxial, it would be incomplete to ignore the influence of the remaining ones. Another serious disadvantage of the single stress or strain component criterion is that stress or strain alone cannot serve as a measure of the fracture toughness of the material. Hence, no predictive capability could be established. The (SED) criterion does not only predict the direction of crack initiation but it can also be used as a measure of fracture toughness by including the contribution of all six components of stress and strain. Eve~L more absurd and inconsistent are the modified (MTSN) and (SED) criteria contribed by the authors who claim that they lead to more accurate predictions of the critical load. The modifications actually involved the combination of several criteria. The locations of crack extension were predicted from the directions of maximum tangential strain and minimum *International Journal of Fracture 23 (1983),24 (1984)

2 R24 strain energy density, while the critical loads were halculated from the tangential stresses at the critical locations. In another [3] of the series of papers published recently, the same authors again attempt to discredit the (SEN) criterion by employing the same technique of comparing predictions based on those criteria mentioned earlier. Their claims were that they could not find stationary values of the (SED) for an angle crack under unequal biaxial loading in the range of -90 deg < 90 deg. They further showed overlap of the crack trajectories emanating from an elliptical cavity. All these false claims were removed in a recent discussion by Sih and Tzou [4] who showed convincingly that (SED) gave the correct predictions as observed physically. It was the other criteria that failed to predict the phenomenon of crack trajectory curvature reversal for failure initiating from an elliptical cavity. The authors not only failed to recognize this well-known physical phenomenon, but made several serious errors in the stress analysis* that led to their misconceptions and misinterpretations of the results. In addition to the above comments, the present discussion attempts to further clarify the authors' wide-sweeping statements [2] with regard to predicting crack path from stress state referred to the original crack configuration. The argument that the crack runs so fast that there is no time for stress redistribution does not always hold. It is fortunate that most structures today are not designed after the Griffith crack loading configuration where crack initiation is closely followed by catastrophic failure. The knowledge of crack path in this case serves little or no useful purpose in fracture prevention. rack growth, in general, is a highly load time history dependent process and it has a significant influence on the critical loads that trigger globa] instability. Indeed, the merit of any criterion should be tested by its versatility and usefulness for predicting material or structure failure and not by limiting the discussion to a few simple trivial problems. In order to exhibit the load time history dependence of the crack growth process, the angle crack problem in Fig. l(a) will be considered. rack growth is predicted by the strain energy density criterion in steps for a piecewise increasing load. The direction of the first segment of crack extension AO 1 (Fig. l(a)) is determined from the minimum value of the strain energy density function dw/dv along a circle of radius r. rack initiation is assumed to start when dw/dv reaches a critical O value (dw/dv). The new crack is P~BOAOI. The next segment of crack ex- I i tension can be determined from the minimum value of dw/dv on a circle centered at 04. Figure l(b) shows the segment 0]0_ corresponding to a load incremen~ Aa. Its length can be found from t~e strain energy density criterion: S 1 S 2 S. S _j _ c (i) (dw/dv)c = r I r 2 r. r j c The quantity S. in (i) is defined by 3 S. = (dw/dv)r. (2) ] ] *The stresses in (4), (5) and (6) of the authors' paper [i] contain numerous errors and/or misprints.

3 R25 where r. represents the jth segment of crack growth. from th~ equation [5] S can be calculated 2 2 S = allk I + 2al2klk 2 + a22k 2 (3) with 16Vall = (l+cos0) (K-cos0) 16~a12 = [2cos0 -(K-l)]sine (4) 16Ba22 = (K+I) (l-cose) + (l+cose) (3cos0-i) where ~ is the shear modulus of rigidity, K = 3-4~ or (3-v)/(l+~) for plane strain or generalized plane stress conditions and k.,k 2 are the opening-mode and sliding-mode stress intensity factors. ~or the straight crack AB, k I and k 2 are given by kl = a~a-- sin2bo ' k2 = acr--sinbc SBo (5) The calculation of k. and k 2 for the bent crack PIBOAO_ or~oab can be i approximated by introduclng an equivalent straight crack o~ length 2a 1 oriented at an angle B 1 such that Aalsin0 O S 1 = g + ao+aalcoso (6) and Aal+aoOSe o a I = a o + ao+aalc Seo A~l (7) Hence, (5) can be used for a bent crack simply by replacing a and B with O a I and By. The same procedure can be repeated by inserting a2, B2, ' etc., into k I ~nd k 2 as the crack grows in segments. The process of stable crack growth is analyzed by increasing the applied stress s in constant increments of As and calculating the corresponding crack growth increments r.. The stress o. at crack initiation is determined from the condition tha~ the minimum straln energy density function along a circle of radius r centered at the crack tip becomes critical, i.e., (dw/dv)mi n = (dw/dv). Referring to Fig. l(a) and (b) for an applled stress oi = s. + As, the crack grows from its tip A to the point i 1 01 by an increment r~ that makes an angle 0 with the initial crack direction BA. Increasingly the stress 01 to the value 02 = 01 + As, the crack

4 R26 BAO 1 extends further to 02 by an increment (O107) = r? making an angle e 1 witn AO I. The value of r9 is determined from (T). TNis process is continued Nntil the last inc~ement r~ becomes equal to the critical size r corresponding to global instability. Thus, the critical stress at instability o is obtained. c Numerical results are obtained for a steel with the following properties: E = 3 x 107 psi; v = 0.3; (dw/dv) = 26,684 in-lb/in 3", S = 77 ib/in The value of r is calculated ~s r = S /(dw/dv) = x 10-3 in, while --J r is taken equal to 1.2 x i0 in~ ThE value o~ the applied stress o is.o increased at the intervals Ao = 0.5, 1.0 and 2.0 ksi. Figure 2(a) presents the variation of the stress ~. versus B while in Fig. 2(b), the variation of a versus B for Ao = 0.5,11.0 and 2.0 ksi is plotted. Higher stress increments correspond to higher loading rates. It is observed that o decreases as the stress increment increases, which is in accordance withexperimental observation. Finally, Fig. 3 presents crack growth patterns for Ao = 0.5 ksi (a) and Ao = 2.0 ksi (b) and various values of the angle B. Note that the extent of crack growth decreases as the stress increment step Ao increases, or in other words, the amount of slow crack growth can be increased by lowering the rate of loading. This is why at higher loading rates, materials tend to behave in a more brittle fashion. Figures 2 and 3 establish the dependence of crack extension on the loading rate, which should be taken into account when comparing theoretical predictions with experimental results. REFERENES [i] S.K. Maiti and R.A. Smith, fnternational Journal of Fracture 23 (1983) [2] S.K. Maiti and R.A. Smith, International Journal of Fracture 24 (1984) [3] S.K. Maiti and R.A. Smith, Engineering Fracture Mechanics ) ~ ] G.. Sih and D.Y. Tzou, Engineering Fracvure Mechanics (to be published). [5] G.. Sih, fnternational Journal of Fracture i0 (1974) August 1984

5 t t t I or R27,~ ~~oo, PI (a) O" / / O~ (b) Figure i. rack geometry (a) and incremental crack growth (b).

6 R O3 O3 W rr k- (/) i 15 IO I I l, 30 o ( o ) J~-RAK ANGLE I OO L~o- ( ksi ) = ~ 150 I-- I ff I00 50 I I I I 15 o (b) B-RAK ANGLE )O o Figure 2. ritical stress at crack initiation o. (a) and unstable crack growth (b) for three d~fferent loading step increments.

7 R ~ \ \ A ~ 0.09 OA ~2x(in ) o o,f ~ y(in) Act = 0.5 ksi (a) ~=15 \ ~oo \ \ -- \ \\ o ~ o o~ oo~ ~ ly(in) /So- =2ksi 0,12.L,.. x(in) (b) Figure 3. Paths of crack growth for Ao = 0.5 ksi (a) and Ao = 2.0 ksi (b).