Elliptical-Arc Surface Cracks in Round Bars

8756-758X/92 $5.00 + 0.00 Fatigue Fracr. Engng Muter. Srruct. Vol. 15, No. 11, pp. 1141-1153, 1992 Printed in Great Britain. All rights reserved Copyright Q 1992 Fatigue of Engineering Materials Ltd

ELLIPTICAL-ARC SURFACE CRACKS IN ROUND BARS

ANDREA CARPINTERI Istituto di Scienza e Tecnica delle Costruzioni, Universita di Padova, Via Marzolo 9,

35131 Padova, Italy

(Received in final form 10 June 1992)

Abstract-The crack aspect ratio influence on the stress-intensity factor of elliptical-arc edge flaws in solid round bars under tension or bending loading is discussed. The relative crack depth 5 = a / D between the depth a of the crack's deepest point and the bar diameter D ranges from 0.1 to 0.6, while the aspect ratio a /b of the flaw is made to vary from 0.0 (straight crack front) to 1.0 (circular-arc crack front). For each value of 5 being considered, the maximum stress-intensity factor is attained in correspondence to the deepest point on the crack front in the case of a / b = 0.0, while it is attained near the external surface in the case of a / b = 1.0. For intermediate aspect ratios, the stress-intensity factor presents the maximum value at the crack's deepest point for 5 C tCR and near the free surface for 5 3 tcR, with the critical relative crack depth tCR getting lower and lower as the parameter a /b is increased.

INTRODUCTION

As is well-known, the stresses near the crack tip for two-dimensional crack configurations are characterized by the stress-intensity factor. This parameter is constant and determines the strength of the stress field square-root singularity in the vicinity of the crack tip.

When part-through cracks are considered, a three-dimensional analysis is necessary to obtain the variation of the stress-intensity factor along the crack front. Both straight-fronted cracks and curved-fronted cracks in round bars (Fig. 1) show a three-dimensional nature[l-15].

The purpose of the present study is to examine the behaviour of elliptical-arc surface cracks in round bars subjected to tension or bending loading (Fig. 2). As a matter of fact, only a few investigators have analysed the influence of the crack aspect ratio alh on the stress-intensity factor for this type of edge flaw[l6-201.

In particular, an elliptical-arc surface crack is assumed to exist in the centre-line cross section of a round bar with diameter D and length 21, subjected to a tensile axial force Facting perpendicularly to the crack plane or to a bending moment A4 about an axis perpendicular to the crack depth a, so that the flaw is in the tension side of the bar (Figs 1 and 2). The relative crack depth 5 = a/D of the crack's deepest point A ranges from 0.1 to 0.6, while the flaw aspect ratio a/b is made to vary from 0.0 (straight crack front) to I .O (circular-arc crack front). The generic point P along the crack front is identified by the normalized coordinate [ / h .

As is well-known, the stress-intensity factor is not defined at the intersection B of the crack front with the free surface (Fig. 2). Therefore, the results near the external surface, which will be presented in the following, are actually related to a point very close to point B ( l / h 2 0. l), but not coincident with it.

It is interesting to examine the transition from the straight-fronted edge crack to the circular-arc edge crack, and the results obtained can be used to study the propagation path of the surface flaws being considered. First of all, it is possible to remark that the maximum stress-intensity factor is attained in correspondence to the deepest point on the crack front for a/b = 0.0, while

I141

I142 ANDREA CAKPINTERI

1 LH

STRAIGHT CRACK FRONT

CURVED CRACK FRONT

SECTION H - H

Fig. 1. Cylindrically shaped structural component with straight-fronted crack or curved-fronted crack.

it is attained near the external surface for alb = 1.0. Therefore, a straight crack front tends to become curved, i.e. the flaw aspect ratio a/b increases, while a circular crack front tends to flatten, i.e. alb decreases.

For intermediate values of aspect ratio, the stress-intensity factor reaches the maximum at point A for 5 < tCR and near point B for 5 2 rCR, tCR being the critical relative crack depth. It can be shown that ecR decreases by increasing the aspect ratio. Moreover, for a given value of a/b, tCR for tension loading is greater than that for bending loading.

For each couple of values assumed for the parameters t and a/b, a three-dimensional finite element analysis is carried out by employing 20-node isoparametric elements. The finite element midside nodes adjacent to the crack front are shifted to quarter-point positions[21] to induce the square-root singularity of the stress field. For each case examined, the stress-intensity factor along

I

b h d

Fig. 2. Elliptical-arc surface crack.

Elliptical-arc surface cracks in round bars I143

the crack front is obtained. Finally, some results of the present study are compared with those deduced by other authors.

STRESS-INTENSITY FACTOR CALCULATION

Since the cylindrically shaped structural component in Figs 1 and 2 presents two planes of symmetry, only one-quarter of the bar needs to be analysed (Fig. 3). Obviously, symmetry conditions have to be prescribed on the portions of the boundary defined by the two symmetry planes. A mesh generation program was developed and a total of 210 finite elements and 1175 nodes were employed to model the body in Fig. 3. The geometry of the mesh satisfies the need to have nodal points lying along straight lines normal to the crack front at each point where the stress-intensity factor has to be calculated.

As previously said, the singular behaviour of the stress field in the vicinity of the crack is induced by shifting the midside nodes near the crack front to quarter-point positions [21]. The ratio between quarter-point element size perpendicular to the crack front and maximum flaw depth a is assumed to be equal to 0.05 +- 0.15 for the different values of 5 .

The stress field is computed for a cracked bar with D = 50 mm and 1 = 200 mm, while elastic modulus and Poisson ratio of the material are assumed to be equal to 206,000 N mm-’ and 0.3, respectively. As an example, the opening normal stress nY for the points along the x-axis ( c / h = 1 .O) against the distance r evaluated perpendicularly to the crack front is shown for a circular-arc surface crack in the case of tension loading with a resultant F = lo3 N [Fig. 4(a)] and in the case

h 4 Fig. 3. Finite element mesh for a round bar with an elliptical-arc surface crack.

1144 ANDREA CARPINTERI

-7 4.0 E E 5. < 3.0 cn" u) w U I-

A U

0

CJ z z w P

u) 2.0

E = 1.0

- * 0.0

I -

0.4

0.3

0.2

0.1

I I I L 0 5 10

(a)

DISTANCE FROM CRACK FRONT, r (mm)

alb = 1.0 (lh = 1.0 E

E

0.0 I I I

0 5 10 h

DISTANCE FROM CRACK FRONT, r (mm)

(b )

Fig. 4. Opening normal stress uy against distance r from crack front for a circular-arc surface crack under: (a) tension and (b) bending.

Elliptical-arc surface cracks in round bars 1145

of bending loading with a resultant moment M = 104N mm [Fig. 4(b)]. It can be remarked that the stress presents the square-root singularity near the crack front, that is, for r+O. An analogous behaviour can be observed for different values of the parameter [ / h .

After obtaining the stress fields for both tension and bending loading by means of the finite element analysis, the dimensionless stress-intensity factors & and are calculated as follows:

where

KI,F = CTy,,(2Zr)Ii2, with Y -0, K~,M = CTy,M(2%T)'iZ, with r +o,

C T ~ , , , C T ~ , ~ = stress by (for tension and bending, respectively) at the distance r, CT, = F/(nD2/4) = applied uniform tensile stress,

C T ~ = M/(zD3/32) = maximum bending stress.

Figure 5 shows the dimensionless stress-intensity factor along the crack front for tension loading in the case of a /b = 1 .O (continuous curves). It can be observed that, for each relative crack depth < being considered, the maximum stress-intensity factor of a circular-arc edge crack is attained near the external surface, that is, for c / h s 0.00. The continuous curves plotted decrease monotonically

N . r h

25

1L. 2 u-

2.0

1.0

.0.5

.0.4 0.3 0.2

0.1

0.0 1 I I I I * 0.00 0.25 0.50 0.75 1.00

NORMALIZED COORDINATE, I h

Fig. 5. Dimensionless stress-intensity factor along the crack front in the case of tension loading, for a /b = 1.0 (continuous curves) and a /b = 0.0 (dashed curves).

FFEMS 15/11-D


by increasing the normalized coordinate, with the slope getting lower and lower as the parameter t is decreased.

The results presented in[22] for a straight-fronted edge crack, i.e. for a /b = 0.0, are also displayed in Fig. 5 (dashed curves). In this case, each curve shows the maximum value for i / h = 1.00, that is, at point A in Figs 2 and 3. Moreover, it can be remarked that, for each relative crack depth being examined, the stress-intensity factor for a straight crack front is greater than that for a circular crack front and this result appears reasonable by considering the smaller crack area for the latter case at the same value of 5.

The dimensionless stress-intensity factor along the crack front for bending loading is plotted in Fig. 6 in the case of flaw aspect ratios equal to 1.0 and 0.0, respectively. These curves are qualitatively similar to those in Fig. 5, that is to say, the stress-intensity factor for a / b = 1.0 presents the maximum near point B ( ( / h z 0.00) and the minimum at point A ( ( / h = 1.00), contrarily to what occurs for a / b = 0.0.

Therefore, for both tension and bending loading, a circular crack front tends to flatten, while a straight crack front tends to become curved, as described by other authors (for example, see[16,23-25]).

= 0.3. It can be remarked that, in the case of tension loading, the slope of the curve & F becomes negative for a /b greater than or equal to about 0.70. In other words, the maximum stress-intensity factor is attained at point A for a /b < 0.70 and near point B for a /b 2 0.70. If the elliptical-arc surface crack is

The transition from a / b = 0.0 to a/h = 1.0 is shown in Fig. 7 for

1.5 I e a Y

a D

r - Y

. r

Y-

T 1 Y-

II

e------ -----4 ./- 0.5

0.4

0.3

0.2

0.1

0.0 I I I I I - 0.00 0.25 0.50 0.75 1.00

NORMALIZED COORDINATE, 4 1 h

Fig. 6. Dimensionless stress-intensity factor along the crack front in the case of bending loading, for a / b = 1.0 (continuous curves) and a / b = 0.0 (dashed curves).


- = 0.8 I 1.0 J

1.0

0.0

0.00 0.25 0.50 0.75 1.00

NORMALIZED COORDINATE, 5 I h

Fig. 7. Transition from straight-fronted crack (a /b = 0.0) to circular-arc crack ( a / b = 1.0): stress-intensity factor along the crack front for both tension and bending loadings, in the case of = 0.3.

subjected to bending loading, an analogous transition can be observed in the curves Kl,M for a / b 2 0.55.

For the other values of relative crack depth, the change of the slope sign can be noticed in correspondence to different aspect ratios, with the a / b transition value for tension greater than that for bending at the same 5 .

In conclusion, the stress-intensity factor variation along the crack front is more pronounced for high values of the parameter 5. Moreover, for each 5 being examined, the variation is remarkable for a / b equal to 0.0 and/or 1.0, while it is less significant for intermediate aspect ratios.

ASPECT RATIO CONSIDERATIONS

Figure 8 shows the dimensionless stress-intensity factor & F at point A (continuous curves) and near point B (dashed curves) against the relative crack depth 5 for different flaw aspect ratios a/b . It can be remarked that, for a / b > 0.4, the two curves related to c / h = 1.0 and c / h z 0.0 intersect in a point, the abscissa of which defines the critical value tCR of the relative crack depth. Therefore, I& presents the maximum at point A for 5 < g,, and near point B for 5 2 gCR. It is possible to notice that SCR decreases by increasing the value of a/b . The results plotted in Fig. 8 are also given in Table 1.

at point A and near point B against the parameter < is displayed in Fig. 9 for different values of a/b. The trends of these curves are analogous to those described for tension loading in Fig. 8, that is, the value of z,,M at point A is greater than that near point

The stress-intensity factor


2.0 ..i 1 .o

1.0 2*oI 3 c5 rn 1.0 2-oE Y' : 'di

lY5 2T 1.0

2*oI 1 .o

2*ot 1.0

Y

\

'L. 1.0

0.0

0.2

0.4

0.5

0.6

0.0 0.2 0.4 0.6

RELATIVE CRACK DEPTH, 5 = a / D

Fig. 8. Dimensionless stress-intensity factor if,.F (tension) at point A ( ( / h = 1.0) and near point B ( ( / h I 0.0) against parameter <, for different values of crack aspect ratio a/b.

B for ( < tCR, with the critical relative crack depth getting lower and lower as the aspect ratio is increased. The results for bending are also presented in Table 2.

The above-defined critical depth tCR is shown in Fig. 10 as a function of the crack aspect ratio. It can be observed that tCR for tension is greater than that for bending at the same value of a / b . Moreover, both curves tend to a vertical asymptote in correspondence to a / b z 0.4 (threshold aspect ratio), since the diagrams in Figs 8 and 9 do not intersect for a / b lower than the mentioned threshold value.

Now the results in Figs 8 and 9 can be used to examine the crack propagation for tension and bending, respectively.

Consider the case of tension loading first. If the initial flaw (5 z 0.0) presents a straight crack front (a /b = O.O), the maximum stress-intensity factor is attained at point A and, therefore, the


Table 1. Dimensionless stress-intensity factor (tension) at point A (</h = 1.0) and near point B (</h ~0.0) for different relative crack depths E, and flaw asDect ratios alb

~~

6 = a / D a /b 0.1 0.2 0.3 0.4 0.5 0.6 0.0 A 1.05 1.23 1.49 1.92 2.70 3.81

B 0.78 0.93 1.19 1.59 2.33 3.35 0.2 A 0.99 1.22 1.48 1.89 2.65 3.66

B 0.72 0.92 1.18 1.58 2.32 3.34 0.4 A 0.95 1.15 1.39 1.77 2.46 3.50

B 0.72 0.91 1.17 1.58 2.31 3.33 0.5 A 0.91 1.10 1.32 1.66 2.30 3.29

B 0.72 0.91 1.17 1.57 2.31 3.32 0.6 A 0.87 1.04 1.25 1.56 2.14 3.00

B 0.72 0.91 1.17 1.56 2.26 3.30 0.8 A 0.79 0.93 1.07 1.30 1.75 2.34

B 0.72 0.90 1.13 1.46 2.12 3.02 1.0 A 0.71 0.82 0.93 1.08 1.35 1.67

B 0.71 0.86 1.03 1.30 1.84 2.52

parameters a / b and { tend to increase. When a / b reaches a value for which the maximum &F is attained near point B, the flaw aspect ratio tends to decrease in the following propagation stages until the stress-intensity factor shows the maximum at point A again. At that moment, a /b increases and so on.

< 0.6, the above-described phenomenon of crack growth stabilizes for a / b equal to a value between 0.5 and 0.6. On the basis of a similar reasoning, the same conclusion can be drawn for an initial flaw with the generic aspect ratio.

Analogously, if we consider the diagram in Fig. 9, the aspect ratio of the surface crack under bending tends to stabilize at a value in the range 0.4-0.5 for 0.5 <

Caspers et al.[23,24] have shown that the aspect ratio of an initially circular-arc flaw tends to approach 0.78 in tension and 0.71 in bending, but they have not specified at what value of 5 that

It is possible to qualitatively deduce from Fig. 8 that, for 0.5 <

< 0.6.

Table 2. Dimensionless stress-intensity factor $,M (bending) at point A (@ = 1 .O) and near point B ((fh s 0.0) for different relative crack depths 6 and flaw aspect ratios a/b

alb 0.1 0.2 0.3 0.4 0.5 0.6 0.0 A 0.86 0.87 0.89 0.96 1.19 1.47

B 0.63 0.61 0.65 0.75 0.97 1.20 0.2 A 0.83 0.85 0.87 0.94 1.16 1.39

B 0.62 0.61 0.66 0.76 0.97 1.20 0.4 A 0.78 0.80 0.82 0.91 1.08 1.36

B 0.61 0.66 0.71 0.82 1.03 1.31 0.5 A 0.75 0.76 0.78 0.83 1.01 1.28

B 0.64 0.68 0.74 0.83 1.05 1.40 0.6 A 0.71 0.72 0.73 0.78 0.92 1.14

B 0.65 0.71 0.77 0.86 1.06 1.32 0.8 A 0.64 0.63 0.62 0.62 0.74 0.84

B 0.64 0.72 0.17 0.86 1.07 1.27 1.0 A 0.58 0.55 0.50 0.50 0.52 0.55

B 0.63 0.69 0.74 0.82 0.98 1.14


1 .o

0.5

0.5

0.5

0.5

0.5

a / b

0.0

0.2

0.4

0.5

0.6

0.8

1 .o

L I I I I I I b

0.0 0.2 0.4 0.6

RELATIVE CRACK DEPTH, 6 = 0 1 D

Fig. 9. Dimensionless stress-intensity factor $,,, (bending) at point A ( [ / h = 1.0) and near point B ( [ / h z 0.0) against parameter t, for different values of crack aspect ratio a/b .

occurs. In a similar way, Athanassiadis et al.[16] have predicted the stabilization of the crack front shape at the value of a/b = 0.78 for both tension and bending loading.

Forman and Shivakumar [25] proposed some equations for straight-fronted or circular-arc cracks to simulate the propagation path “preferred” by such surface flaws. It results from these equations that the aspect ratios for both straight front and circular-arc front with centre beyond the bar circumference converge to a / h E 0.63 at 5 = 0.5, the above-mentioned value of aspect ratio being the same in the case of both tension and bending.

Finally, some results of the present study are compared with those obtained by other authors. The comparison between the stress-intensity factors proposed in the literature is difficult because of different definition and measurement of crack shapes.

As an example, the case of a/h = 0.6 is considered in Fig. 11, since several data are available for this value of flaw aspect ratio[l6-18]. It can be remarked that the agreement between the results

Elliptical-arc surface cracks in round bars

0.6

0.4

0.2

-

-

-

-

-

-

0.0 I I I I I I =-. 0.0 0.2 0.4 0.6 0.8 1.0

CRACK ASPECT RATIO, a I b

Fig. 10. Critical relative crack depth tCR against flaw aspect ratio a /b for both tension and bending loading.

Point 0 : .I I" Athanassiadis et al. { Shiratori et al. { IL)) Present study ----- 1

Paint A : v -

Athanassladis et el.

Shlratorl et al. { Present study -

0.0 1 I I I I I

RELATIVE CRACK DEPTH, 5 a I D

- 0.0 0.1 0.2 0.3 0.4 0.5

Fig. 11. Dimensionless stress-intensity factors i?',,F and at point A and near point B for an elliptical-arc surface crack with a / b = 0.6 (results taken from Refs [16-181).


plotted is quite satisfactory for both tension and bending. More precisely, the values deduced from the present study vary by approx. 10-1 5% from the other values shown in Fig. 11.

It can be verified that, for each value of a/b, the stress-intensity factors presented in this paper are quite close to those obtained by other investigators.

CONCLUSIONS

The stress-intensity factor variation along the front of elliptical-arc surface cracks in round bars under tension or bending loading has been determined for flaw aspect ratio a / b ranging from 0.0 (straight crack front) to 1.0 (circular-arc crack front), with the relative crack depth 5 = a / D made to vary from 0.1 to 0.6.

The following conclusions can be drawn from the present study:

(1) For a given relative depth [, the maximum stress-intensity factor is attained in correspondence to the deepest point on the crack front (point A) for a /b lower than or equal to a value depending on 5 , while it is attained near the external surface (i.e. near point B) for a/b greater than or equal to the above-mentioned value. The aspect ratio of transition is higher in tension than in bending.

(2) When a / b gO.4 (threshold value), the stress-intensity factor presents the maximum at point A. For alb 5 0.4, the maximum is attained at point A for [ < tCR and near point B for 5 3 tCR, with the critical depth cCR getting lower and lower as the parameter a / b is increased. Moreover, tCR for tension is greater than that for bending at the same value of ajb.

(3) It is possible to qualitatively deduce that, for 0.5 < 5 c 0.6, the phenomenon of crack propagation stabilizes for a /b equal to a value included in the range 0.54.6 for tension and in the range 0.4-0.5 for bending.

(4) The results of the present study are quite close to those obtained by other authors for both tension and bending.

Acknowledgements-The author gratefully acknowledges the research support for this work provided by the Italian Ministry for University and Technological and Scientific Research (MURST) and the Italian National Research Council (CNR).

REFERENCES

1. A. J. Bush (1976) Experimentally determined stress-intensity factors for single-edge-crack round bars loaded in bending. Exp. Mech. 16, 249-257.

2. W. S. Blackburn (1976) Calculation of stress intensity factors for straight cracks in grooved and ungrooved shafts. Engng Fract. Mech. 8, 731-736.

3 . 0. E. K. Daoud, D. J. Cartwright and M. Carney (1978) Strain-energy release rate for a single-edge- cracked circular bar in tension. J. Strain Anal. 13, 83-89.

4. M. A. Astiz and M. Elices (1980) On the application of the stiffness derivative method to two and three-dimensional fracture problems. In Proceedings of the 2nd Int. Con$ on Numerical Methods in Fracture Mechanics, pp. 93-106.

5. D. Wilhem, J. FitzGerald and D. Dittmer (1981) An empirical approach to determining K for surface cracks. In Advances in Fracture Research (Edited by D. FranGois et aZ.), Vol. 1, pp. 11-21. Pergamon Press, Oxford.

6. A. S. Salah el din and J. M. Lovegrove (1981) Stress intensity factors for fatigue cracking of round bars. Int. J. Fatigue 3, 1 17-123.

7. A. J. Bush (1981) Stress intensity factors for single-edge-crack solid and hollow round bars loaded in tension. J. Test. Eval. 9, 216-223.

Elliptical-arc surface cracks in round bars 1 IS3

8. F. Ouchterlony (1981) Extension of the compliance and stress intensity formulas for the single edge crack round bar in bending. In Fracture Mechanics for Ceramics, Rocks and Concrete, ASTM STP 745, pp. 237-256. ASTM, Philadelphia, Pa.

9. G. G. Trantina, H. G. de Lorenzi and W. W. Wilkening (1983) Three-dimensional elastic-plastic finite element analysis of smalI surface cracks. Engng Fract. Mech. 18, 925-938.

10. T. L. Mackay and B. J. Alperin (1985) Stress intensity factors for fatigue cracking in high-strength bolts. Engng Fract. Mech. 21, 391-397.

11. 0. E. K. Daoud and D. J. Cartwright (1985) Strain energy release rate for a circular-arc edge crack in a bar under tension or bending. J. Strain Anal. 20, 53-58.

12. T. Lorentzen, N. E. Kjaer and T. Henriksen (1986) The application of fracture mechanics to surface cracks in shafts. Engng Fract. Mech. 23, 1005-1014.

13. E. Hojfeldt and C. B. Ostervig (1986) Fatigue crack propagation in shafts with shoulder fillets. Engng Fract. Mech. 25, 421-427.

14. C . K. Ng and D. N. Fenner (1988) Stress intensity factors for an edge cracked circular bar in tension and bending. Int. J. Fract. 36, 291-303.

15. L. A. James and W. J. Mills (1988) Review and synthesis of stress intensity factor solutions applicable to cracks in bolts. Engng Fract. Mech. 30, 641-654.

16. A. Athanassiadis, J. M. Boissenot, P. Brevet, D. FranGois and A. Raharinaivo (1981) Linear elastic fracture mechanics computations of cracked cylindrical tensioned bodies. Int. J . Fract. 17, 553-566.

17. M. Shiratori, T. Miyoshi, Y. Sakai and G. R. Zhang (1986) Analysis of stress intensity factors for surface cracks subjected to arbitrarily distributed surface stresses. Trans. Jap. SOC. Mech. Engrs.

18. Y. Murakami (1987) Stress Zntensigy Factors Handbook. Pergamon Press, Oxford. 19. K. J. Nord and T. J. Chung (1986) Fracture and surface flaws in smooth and threaded round bars.

Int. J . Fract. 30, 47-55. 20. I. S. Raju and J. C. Newmaq(1986) Stress intensity factors for circumferential surface cracks in pipes

and rods. In Fracture Mechanics: Seventeenth Volume, ASTM STP 905, pp. 789-805. ASTM, Philadelphia, Pa.

21. R. S. Barsoum (1976) On the use of isoparametric finite elements in linear fracture mechanics. Int. J. numer. Meth. Engng 10, 25-37.

22. A. Carpinteri (1992) Stress-intensity factors for straight-fronted edge cracks in round bars. Engng Fract. Mech. 42, 1035-1040.

23. M. Caspers, C. Mattheck and D. Munz (1986) Fatigue crack propagation in cylindrical bars. Z. Wkstofftech. 17, 327-333.

24. M. Caspers and C. Mattheck (1987) Weighted averaged stress intensity factors of circular-fronted cracks in cylindrical bars. Fatigue Fract. Engng Muter. Struct. 9, 329-341.

25. R. G. Forman and V. Shivakumar (1986) Growth behavior of surface cracks in the circumferential plane of solid and hollow cylinders. In Fracture Mechanics: Seventeenth Volume, ASTM STP 905, pp. 59-74. ASTM, Philadelphia, Pa.

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Elliptical-Arc Surface Cracks in Round Bars

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