## 1 1 1 J I I

Infinite plate with collinear cracks If the plate is cut along the lines AB and CD one obtains a strip of finite width W, containing a central crack 2a. It is likely that the solution of eq (3.28) is approximately valid for the strip. In the case of the collinear cracks a strip of width W bears stresses (note that shear stresses are zero because of symmetry) along its edges AB and CD (figure 3.4). whereas the edges of a plate of finite size are stress free. Supposedly, the stresses...

## 01 02 03 04 05 06 07 08 09

Evaluation of leak-beforc-hrcak criterion There is a slight complication in the use of figure 15.11. As outlined in chapter 11. the critical stress intensity for instability of surface flaws. Kl . is somewhere between K,c and Klcr The ratio Klc Klf should replace KJKle, in figure 15.11. For shallow flaws Kif will be close to KtLl, which means that the left part of figure 15.11 remains valid. Semi-circular flaws must be expected to start propagation if Kif is close to Kw. which...

## Pr

Bulging of cracked area 120 ksi) a plastic zone correction was applied to the K expression of eq 15.1. In their earlier work 20 , they used the Dugdale plastic zone 21 as a plastic zone correction (chapter 4) In order to account for the effect of work hardening, 2ays was replaced by ayS + au- I 'ater work they applied the plastic zone correction proposed by Hahn et al. 22 , discussed in chapter 9. By noting that and by taking the Dugdale solution for the CTOD an expression for the...

## 11 Introduction

Through the ages the application of materials in engineering design has posed difficult problems to mankind. In the Stone Age the problems were mainly in the shaping of the material. In the early days of the Bronze Age and the Iron Age the difficulties were both in production and shaping. For many centuries metal-working was laborious and extremely costly. Estimates go that the equipment of a knight and horse in the thirteenth century was of the equivalent price of a Centurion tank in World War...

## Civil Mechanics

Crack growth prior to fracture is extremely difficult and therefore CTODc is usually determined at maximum load. There are various geometrical parameters that may affect the result of a COD test. These are the crack size i H 3, 17, 20, 21 , test piece thickness 12,22 . the ligament depth 23 , notch acuity 20,22 and machinc stiffness 22 , Some test results 3, 20 are presented in figure 9.9. It seems that the critical CTOD reaches a fairly constant value for a W 0.2. Figure 9.9 Ffleet of geometry...

## Fracture Mechanics Habibie

Habibie, B.J. 268, 286 Hahn, G.T. 63, 100, 101, 112(2), 113(2), 158, 159,160-161,168,169(3), 234, 249, 284, 287, 294, 312(3), 313, 380, 405, 407 375(2), 405 Hall, W.J. 405 Halmanov, H. 287 Haidrath, H.F. 286, 393, 406, 428(2), 433(2), 447,454 Harris, D. 327,404 Harrison, J.C. 218 Harrison, T.C. 248 Hartman, A. 61,62,66,257,285(2) Haitranft, R.J. 113, 168, 203, 204, 205, 217 Havner, K.S. 140 Hayes, D.J. 337, 345 Henri, G. 64 Hertzberg, R.W. 286(3) Heyer, R.H. 198,217(2) HUI, P.W. 113 169(3)...

## 205 225

W The application of electron Iractography to fatigue studies. ASTM STP. 436 (1968) pp. 89 123. Von E uw, E. F. J., Hertzberg. R. W. mid Roberts. R., Dcluy cffccts in futiguc crack propagation. ASTM STP 513. (1972) pp. 230 259. Corbly. D. M. and Packman. P. F. On the influence of single and multiple peak overloads on fatigue crack propagation in 7075-T651I aluminum. Eny. Fracture Mechanics. 5 (1973) pp. 479-497. Schijve. J. and De Rijk. P., The effect of...

## Emn

Universal representation of energy criterion F igure 5.5. Universal representation of energy criterion Figure 5.6. Crack growth under constant stress and under fixed grip conditions Figure 5.6. Crack growth under constant stress and under fixed grip conditions of crack extension under constant stress and under fixed grips. However, this is only so for the onset of crack extension. During crack growth it is not true any more. If the crack extends under constant stress, G develops as...

## 20

Kobayashi correction (MK) for proximity of front free-surface Figure 3.13. Kobayashi correction (MK) for proximity of front free-surface Figure 3.14. Stress intensity for surface flaws 18 (courtesy ASME) a. Tension b. Bending Figure 3.14. Stress intensity for surface flaws 18 (courtesy ASME) a. Tension b. Bending maximum stress intensity for a surface flaw becomes where MK is the front free-surface correction of figure 3.13. For the case where a semi-elliptical flaw extends deep...

## 12 A crack in a structure

Consider a structure in which a crack develops. Due to the application of repeated loads or due to a combination of loads and environmental attack this crack will grow with time. The longer the crack, the higher the stress concentration induced by it. This implies that the rate of crack propagation will increase with time. The crack propagation as a function of time can be represented by a rising curve as in figure 1.1a. Due to the presence of the crack the strength of the structure is...

## 11 Fracture resistance of materials

This section will be concerned with the case of an existing crack the conditions that led to the initiation of this crack are not of interest for the discussion. For a perfectly brittle extension of this crack by cleavage, the criterion for crack propagation seems fairly easy. Cleavage failure occurs by the breaking of atomic bonds consequently cleavage crack propagation can take place when the stresses at the very crack tip exceed the interatomic cohesive forces. An estimate of the atomic bond...

## 352 353

Stress analysis of cracks. ISI publication. 121 (1968) pp. 13-48. 6 Irwin. G. R Fracture dynamics. Fracturing of metals, pp. 147-166. ASM publ. (1948). 7 Orowan. E, Energy criteria of fracture, IVeUling Journal. 34 (1955) pp. I57s-160s. 8 Wnuk. M. P Subcritical growth of fracture. Int. J Fracture Meeh 7 (1971) pp. 383-407. 9 Raju. k N Oil the calculation of plastic energy dissipation rate during stable crack growth, hit J. Fracture Meeh 5 (1969) pp. 101 112. 10 Broek. D , The...

## 5 Dynamics and crack arrest

So far, the discussions were limited to the problerfi of slow crack growth and to the onset of fracture instability. This chapter deals with post-instability behaviour. Fracture instability occurs, when upon crack extension, the elastic energy release rate G remains larger than the crack resistance R. The surplus of released energy, (G R), can be converted into kinetic energy. This kinetic energy is associated with the rapid movement of the material at each side of the crack path, during the...

## X

Planes of maximum shear stress for 0 close to zero a. Plane stress b. Plane strain Figure 4.12. Planes of maximum shear stress for 0 close to zero a. Plane stress b. Plane strain stresses and o2. The transverse stress txz is always the principal stress < r3. In the case of plane stress, the maximum shear stress, Tm ix, is at planes rotated over angles of 45 from the directions of a, and ct3. If < 7, ay and a3 az 0 (plane stress, 0 0) these are planes through the A' axis...

## Info

On the basis of eqs (4.21) the Tresca plastic zone is of the shape as shown in figure 4.5b. The Tresca zones are slightly larger and of a slightly different shape than the Von Mises zones. Similar analyses can be made for modes II and III cracks. Plastic zone shapes for these modes are shown in figure 4.6 on the basis of a Von Mises yield criterion 8 , In deriving the plastic zone boundaries depicted in figure 4.5 the same Figure 4.6. Plastic zone shapes for modes II and III 8 (courtesy ASTM)...

## 16 Stiffened sheet structures

Bccause of the requirements for sufficient stiffness, sheet structures usually consist of stringer-stiffened panels. The most prominent examples can be found in aircraft structures, viz. the wing and fuselage skin panels. The skin material is a relatively thin sheet, to which evenly spaced stringers are attached by means of riveting or adhesive bonding. When considering crack propagation and fracture of thin sheets, it is necessary to take into account the effect of the stiffening elements if...

## 103 Factors affecting crack propagation

When predictions of crack propagation have to be made, data should be available relevant to the conditions prevailing in service. Such data may 0.3 Factors affecting crack propagation be hard to find. Fatigue eraek propagation is affected by an endless number of parameters, and the circumstances during the test will seldom be the same as in service. The influence of the environment is the most conspicuous. Tests are usually not performed under controlled environmental conditions and part of the...

## 106 Similitude

All fracture mechanics analysis is based on the concept of similitude equal stress intensity (or J) (and therefore equal plastic zone size) will have equal consequences. Similitude concepts are used extensively in science and engineering, and there is no objection against using them, provided the similitude requirements are indeed satisfied. Engineering design analysis has employed the concept in static strength analysis by calculating a fracture load for the condition that the stress in the...

## References

1 Broek, D., The residual strength of light alloy sheets containing fatigue cracks, Aerospace Proceedings, 1966, pp. 811-835, McMillan, London 1966. 2 Walker, E. K A study of the influence of geometry on the strength of fatigue cracked panels, AKFDL-TR-66-92 (1966). 3 Christensen, R. H. and Denke, P. H Crack strength and crack propagation characteristics of high strength materials, ASD-TR-61-207 (1962). 4 Allen, F. C., Effect of thickness on the fracture toughness of 7075 aluminium in the T6...

## 167 Crack arrest

Crack arrest has two important aspects. The first is arrest of a fatigue crack which, after a dormant period, may reinitiate and continue propagation. The second is arrest of a rapidly growing unstable crack which would have caused catastrophic failure if no arrest had occurred. Both aspects of crack arrest bear largely upon the same principles. The theoretical background is treated in chapter 6. As shown in this chapter, crack arrest is an essential behaviour in stiffened sheet structures....

## J5

Failure assessment diagram 37, 38) Figure 15.21. Failure assessment diagram 37, 38) 30 Hardrath, H. F. A., A unified technology plan for fatigue and fracture design, NASA paper presented to ICAF (1973). 31 Schra et al., Private communication. 32 Zahoor, A. and Abou-Sayed, I. S., Prediction of stable crack growth in type 304 stainless steel, Synip. on computational methods in non-linear structural and solid mechanics (1980) Arlinton, VA. 33 Zahoor, A. and Kanninen, M. F A plastic...

## 4

The plastic hinge Bottom plastic deformation spreading to top surface (courtesy Robinson and Tetelman) Figure 9.1 I. The plastic hinge Bottom plastic deformation spreading to top surface (courtesy Robinson and Tetelman) of the plastic zone in steels containing more than 0.005 per cent nitrogen can be delineated by means of an etching technique 25 . Results of this procedure 12 are shown in figure 9.11. Under certain conditions plasticity may spread to the other specimen surface,...

## Hcv Civil Engineering

ffu v, , K r' . (13.6) The finite element method provides a stress and displacement distribution. By taking the stress, calculated for a particular element not too far from the crack tip, a value of Kt can be determined by substituting the r and 6 of the element in eq (13.6). Similarly, a value of Kt can be obtained from the displacement. The same can be done for a number of elements. This yields a series of values for Ku which ideally should all come out equal. Since the analytical eqs (13.6)...

## 173 Approximation of the stress spectrum

Once the load spectrum is established (figure 17.2a), it has to be converted into a stress spectrum and then into a representative stress history. The conversion into a stress spectrum presents no great difficulties if loads and stresses are proportional. In the event they are nonpoportional, a systems analysis may be necessary to arrive at the stresses. For the purpose of the present discussion, proportionality will be assumed, so that the stress spectrum follows from the load spectrum by the...

## Ttt

A critical value. (It is discussed in chapters 2 and 11 that there is a critical combination of stress and strain required for fracture.) Assuming negligible strain hardening, the stress at the crack tip hardly increases after general yield and the fracturc condition is reached upon the occurrence of a sufficiently large strain. A measure for the plastic strain at the crack tip is the crack tip opening displacement (CTOD). Hence, it is conceivable that fracture takes place at the excecdance of...

## Nrl

Crack growth rate for wedge opening loaded and uniformly loaded specimens 26 (courtesy ASTM) a. Crack growth rate b. K histories of specimens in left diagram A relation between the fatigue crack propagation rate and the stress intensity factor is useful, becausc the stress intensity factor can be calculated for many different design-geometries. Once a plot of the type of figures 10.1 through 10.4 is obtained for a particular configuration, it is possible to predict fatigue crack...

## 02 04 06 08 10 12 14

Comparison of engineering solutions for quarter circular (law 7 Figure 14.10. Calculated crack growth lives for quarter circular crack emanating from a hole 7 Figure 14.10. Calculated crack growth lives for quarter circular crack emanating from a hole 7 If corner cracks at holes are likely to occur, it is important to obtain an accurate estimate of the stress intensity factor. From the large differences in crack growth lives predicted by the two methods, it must be concluded that...

## 10

Regimes of growth rates 64 but then slowing down and becoming dormant. Upon their initiation, these cracks are in a reversed strain field (as in figure 10.23a). When the crack tip plastic zone begins to approach the boundary of the notch plastic zone, the crack tip will gradually begin to feel the same effect as in an overload plastic zone. The strain return point moves back from A to B (in figure 10.23) and may eventually reach point C where straining remains elastic and no...

## Applications

Fail-safety and damage tolerance 317 12.2 Means to provide fail-safety 318 12.3 Required information for fracture mechanics approach 323 Determination of stress intensity factors 328 13.2 Analytical and numerical methods 330 13.3 Finite element methods 330 13.4 Experimental methods 338 14.2 Through cracks emanating from holes 347 14.3 Corner cracks at holes 352 14.4 Cracks approaching holes 356 14.6 Fatigue crack growth under mixed mode loading 366 14.8 Fracture toughness of weldments 371 14.9...

## 24 Fatigue cracking

Under the action of cyclic loads cracks can be initiated as a result of cyclic plastic deformation 38, 39 , Even if the nominal stresses are well below the elastic limit, locally the stresses may be above yield due to stress concentrations at inclusions or mechanical notches. Consequently, plastic deformation occurs locally on a microscale, but it is insufficient to show in engineering terms. Figure 2.24. Void coalescence by slip Figure 2.24. Void coalescence by slip Several equivalent models...

## Aiaa Paper 71-113

L The effect of a stringer on the stress in it cracked sheet. Harvard University TR 18 1963 . 2 Vlieger. H. and Broek, D Residual strength of cracked stiffened panels. Built up shed .srnicnires. AGARD Fracture Mechanics Survey 1974 . 3 Vlieger. H Residual strength of cracked stiffened panels. Eng. Fracture Mechanics. 5 1973 pp. 447-478. 4 Poe. C. C Fatigue crack propagation in stiffened panels. ASTM ST P. 1971 pp. 79 97. 5 Poe. C. C The effect of riveted and...

## 2 A

Note that division by B is necessary because the energy release rate is always given per unit thickness. Eq 9.40 was derived for the case that i i is small, but this restriction is not necessary, since an equation similar to eq 9.34 can be written for pct 41 which leaves the derivation of eq 9.40 essentially Figure 9.16. Procedure for experimental measurement of J c unchanged. Thus eq 9.40 is a general expression for specimens with ligaments in bending. The elastic part can be separated out to...

## 22 Cleavage fracture

Brittle-ductile transition of steel F igure 2.6. Brittle-ductile transition of steel graphic plane. Iron for example cleaves along the cube planes 100 of its unit cell. This causes the relative flatness of a cleavage crack within one grain, as indicated in figure 2.7. Since neighbouring grains will have slightly different orientations, the cleavage crack changes direction at a grain boundary to continue propagation on the preferred cleavage plane. The flat cleavage facets through...

## 18 da E

Per unit plate thickness, where E is Young's modulus. Usually dU da is replaced by which is the so called elastic energy release rate per crack tip. G is also called the crack driving force its dimensions of energy per unit plate thickness and per unit crack extension are also the dimensions of force per unit crack extension. The energy consumed in crack propagation is denoted by R dW da which is called the crack resistance. To a first approximation it can be assumed that the energy required to...

## 14 The Griffith criterion

Although fracture mechanics have been developed mainly in the last two decades, one of the basic equations was established already in 1921 by Griffith 9, 10 . Consider an infinite cracked plate of unit thickness with a central transverse crack of length 2a. The plate is stressed to a stress a Figure I X. The Griffith criterion for fixed grips a. Cracked plate with fixed ends b. Elastic energy and fixed at its ends as in figure 1.8a. The load displacement diagram is given in figure 1.8b. The...