## 40

The effect of thickness on plane stress fracture toughness a. Al-Cu-Mg alloy 9 b. Al-Zn-Mg alloy 4 (courtesy ASTM) solid line C-W6). This is a well-known effect 7 it is reflected in the intersection of the apparent Ku curves lor the 2024 and 7075 alloys in figure 8.2. Hence, for application in narrow panels or structures (e.g. stringers) a low toughness material may perform better than a high toughness material. The latter, however, will probably have lower fatigue crack propagation...

## 9 Elasticplastic fracture

Linear elastic fracture mechanics (LEFM) can be usefully applied as long as the plate zone is small compared to the crack size. This is usually the case in materials where fracture occurs at stresses appreciably below the yield stress and under conditions of plane strain. In such circumstances the fracture can be characterized by K'k or Glc. When plane stress prevails the crack tip plastic zone is larger than in the case of plane strain. If fracture still takes place at stresses which are low...

## 30

Effect of grain size on toughness 34 noted that if grain size is varied by changing the alloy content, the variations in properties will be different from those occurring when the grain size is varied by a change of recrystallization time and temperature. At present there is only little quantitative information available on whether the residual strength of a cracked component depends very much upon grain size. In general, the processing steps causing changes in...

## 149 Service failure analysis

Despite our increasing knowledge of fracture and fracture behaviour, service failures will continue to occur. A proper analysis of the circumstanccs under which the event took place may yield valuable information for the prevention of future incidents. An inventory of the environmental circumstances, loads and stresses, is of paramount importance. The fracture surface may reveal sufficient evidence as to the nature of the defect that initiated crack growth and fracture. In the case where a...

## U

Van der Veer, I. 433 Van der Vet, W.J. 65(2) Van Dijk, G.M. 435,454 Van Elst, H.C. 404 Van Leeuwen, H.P. 60,66(2), 140, 261, 285(2), 307, 346, 358 Van Oosten-Slingeland, G.L. 375 Vasoukis, G. 407 Veerman, C.C. 227, 228, 248 Vlieger, H. 113, 201, 204, 217, 226, 248, 351, 375, 409(2), 418(2), 432(2), 433 Walker, E.K. 212, 216, 254(2), 284, 285 Wells, A.A. 18, 23(2), 146, 149, 168, 169, 221, 222, 247(2), 248, 249, 376 Wells, O.C. 63 Wells, R.H. 405 Wessel, E.T. 284,375,407 Westergaard, H.M. 23,...

## 14 Practical problems

This chapter treats a number of detailed problems arising in the technical application of fracture mechanics. Problems of largely different natures are considered. Therefore, the chapter as a whole lacks consistency in the subject of the discussions. Instead, it is consistent in variation. The first three sections treat the problems of cracks emanating from holes, and of the crack arrest capacity of holes. Later sections deal with mixed mode loading, the fracture toughness of weldments, the...

## O

Several steps of the linear integration are shown in detail in table 17.2 top. The first cycle produces a stress intensity of AK i.5j5n 53.5 kg mm1. Thus, da dN 3 x 10 10 x (53.5)4 2.45 x 103 mm cycles, so that the next crack size is 5 + 0.00245 5.00245 mm. These steps are repeated for the other cyclcs of the block. After one block, the crack size is 5.1471 mm. Hence the crack extension is 3 percent which means that the block size is not too large. Repetition of this process gives the crack...

## W

Filed of element size on accuracy of K for compact tension specimen 15 with a boundary collocation solution. It turns out that the curves reach a constant slope at smaller values of r W if the crack tip elements are further refined. The stress intensity following from extrapolation to r 0 for the coarse mesh (area of crack tip element 3.1 10 4cr2) deviates about 11 per cent from the boundary collocation solution. The stress intensity resulting from the finest mesh (area of crack...

## 56 Tearing modulus

The crack resistance curve was discussed in section 5.3 in association with the plane stress behaviour of materials showing essentially linear elastic fracture. Materials exhibiting appreciable plasticity at fracture usually also show slow and stable crack growth before fracture even if the crack tip state of stress is essentiallyplane strain. Thus, the crack will start stable growth at the critical value J c, but further increase of the stress is required to maintain crack growth. Apparently,...

## Plane stress and transitional behaviour

A generally accepted method for plane stress toughness testing and presentation of results does not exist. This is due to difficulties in understanding the observed phenomena. However, many structures, especially in aircraft, are built out of sheet, and consequently the plane stress problem is of great practical importance. A useful engineering solution for the plane stress problem is available. The residual strength of a stiffened sheet structure with a crack can be predicted (chapter 16) on...

## L

K curve concept for stiffened panel. Short crack with arrest at C R curve (same curve asin figure 16 14 Figure 16.15. K curve concept for stiffened panel. Short crack with arrest at C The situation is more complicated for a short crack in a stiffened panel, depicted in figure 16.15. Slow stable crack growth starts at a stress a,. The part OA of the curve Gf is still straight, since the stringer is remote. This means that slow growth commences at the same stress < rf as in the...

## Mbl

In the first cycle subsequent to the overload, a is still equal to a . Hence Knus.rcq would be equal to the stress intensity of the overload, as should be expected. Wilienborg et al. make the rather odd assumption (hat Kactually occurring at the current crack length ti,, will be effectively reduced by an amount KrL.d, given by The residual compressive stresses introduced by an overload reduce the effective stress at the crack tip. This implies that the effective stress is the difference between...

## 5 The energy principle

The Griffith energy criterion for fracture 1,2 states crack growth can occur if the energy required to form an additional crack of size do can just be delivered by the system. The case of a plate with fixed ends was discussed in chapter 1. Due to the fixed ends the external load cannot do work. The energy required for crack growth must then be delivered as a release of elastic energy. If the ends of the plate are free to move during crack extension, work is done by the external load. In this...

## A C

Crack growth in tapered cantilever beam specimen The tapered cantilever beam specimen can be used to study crack growth at constant K as outlined in figure 5.15. The specimen is loaded to P, along OA, where the stress intensity (and the energy release rate) for crack growth is reached. The crack extends a little, which causes the load to drop. Reloading to the same load P is necessary to restart crack growth, since at Pj the same K value obtains. (Note that e.g. in an edge cracked...

## Tfw

Mixed mode cyclic histories a. History used in most tests b. Realistic service history close agreement with the principal stress criterion. These data are shown in figure 14.14. A finite element model was used which followed the curved crack growth path as actually observed in the tests. It turned out that Kn immediately dropped to zero. Thus, the experiments were actually in mode I, apart from the first crack growth increment. It is apparently so that fatigue crack growth tends...

## 80 70 60 50 40 30 20

I aligue crack growth in stiffened panel 4 (courtesy ASTM) Figure 16.5. Fatigue crack propagation through two bays of stiffened panel 4J (courtesy ASTM) Figure 16.5. Fatigue crack propagation through two bays of stiffened panel 4J (courtesy ASTM) Figure 16.6. Fatigue crack propagation in integrally stiffened panel 4 (courtesy ASTM) Figure 16.6. Fatigue crack propagation in integrally stiffened panel 4 (courtesy ASTM) obtained by integration of Poe's da dn data. It shows how crack...

## Preface to the first edition

When asked to start teaching a course on engineering fracture mechanics, I realized that a concise textbook, giving a general oversight of the field, did not exist. The explanation is undoubtedly that the subject is still in a stage of early development, and that the methodologies have still a very limited applicability. It is not possible to give rules for general application of fracture mechanics concepts. Yet our comprehension of cracking and fracture behaviour of materials and structures is...

## Tod

Application of dual clip gauge a. Dual clip gauge b. Slot c. Displacements a fatigue crack originating from a fairly deep machined notch that can accommodate a dual clip gauge (figure 9.7). In this case it is not necessary to know the rotational factor, since CTOD follows directly from the measurement of COD at two locations ' (COD)f- )(COD)g (COD)p (COD)y a Also the rotational factor can be determined from the two COD records obtained with the dual clip gauge. Veerman and Muller...

## I

Finite element idealization of centre cracked panel by Watwood 14 470 the crack tip. The method requires small elements in the crack tip region and a large storage capacity of the computer. Verification of the applicability of this procedure is obtained by solving a simple case for which the solution is known. Watwood 14 made the analysis for a centre-cracked panel by using the finite element idealization shown in figure 13.1. Because of symmetry, only one quarter of the panel...

## Lhb

Kalthoff, J.F. 153,169 Kanazawa, T. 169,248,313 168(2), 169(2), 406(3), 407 Karel, V. 64 Kaufman, J.G. 313,407 Kearny, V.E. 285 Kendall, D.P. 112 Kerlins, V. 63 Kibler, J.J. 367,368,376 Kiefner, J.F. 379,405(2) Kies, J.A. 376 Kihara, H. 405 King, I.P. 345 Knott, J.F. 248(2) Knowles, J.K. 140 Kobayashi, A.S. 84, 90(2), 136, 140, 339, 345, 353, 367, 375(2), 376, 389, 405 Koda, S. 64 Koiter, W.T. 73,89 Krafft, J.M. 63, 124, 139, 169(2), 195, 198,217,312(2) Krautkr mer, H. 327 Krautkr mer, J. 327...

## 2

COD E V2 y2 + < CTOD 2' < 9'8) According to eq (9.7) the CTOD can be determined indirectly from a measurement of COD (e.g. at v t), the center of the crack) without any assumptions about the size of the plastic zone correction. The COD can readily be measured by means of a clip gauge (chapter 7). Alternatively, use is often made of the equations for crack tip opening that follow from the Dugdale approach (chapter 4). It turns out 3 that Scries expansion of the log sec yields 8fT < f I 1 (...

## D

The energy concept in plane stress Eq (8.9) is a useful fracture criterion if an analytical relation for R is available, apart from the relation G na2u E. Otherwise the equation cannot be evaluated. Raju 19 and Wnuk 20 have attempted to derive such an expression on the basis of plasticity theory, by calculating the rate of plastic energy consumption in the plastic zone ahead of the crack. Krafft et ul. 21 have proposed that R is a function of A a only, independent of< 0. Then the...

## C p Wb

It follows that the energy release rate is given by IB Ca Eh* B2 and the stress intensity factor by Eq (5.26) is only a rough approximation to the stress intensity factor. Discrepancies occur due to the fact that the beams also undergo a shear deformation (this can be accounted for in the derivation of the compliance) and due to the fact that the beams are not rigidly fixed at their ends, but supported by an elastic hinge. It follows from eq (5.26) that the stress intensity factor for a...

## 97 Use of the J integral

Where r is an arbitrary contour around the crack tip, beginning and ending at the (opposite) crack faces. The contour may be arbitrary because the J integral is path independent. Therefore, it is permissible to take a circular contour at a radius r from the crack tip. For such a contour eq (9.26) can be rewritten as where 6 is the polar angle. W, T and du dx depend upon r and 0. The expression for W is < 7yd so that the dimension of W is er > ,,. The traction T has the dimension of stress...

## Uai

Sheared single crystals of pure copper (courtesy Weiner) Sometimes, however, particles of this size may be produced on purpose, as in the case of carbides in certain steels. b. Intermediate particles, only visible by means of the electron microscope. Their size is in the order of 500-5000 Angstrom units. These particles may also consist of complex compounds of the various alloying Figure 2.16. Cracking of large particles in Al-Cu-Mg-alloy. A. 3 per cent strain B. 6 per cent strain...

## 2 1

Plastic zone correction proposed by Hahn et al. 31 a ays the factor (j> can be approximated to by a step function. Then the fracture criterion reduces to which is equivalent to eq (9.19) apart from the size correction. It appears that the failure of pipelines and pressure vessels built of high toughness materials could be predicted 31 by means of eq (9.22). These results are discussed in chapter 15. The J integral offers potentials for application to fracture problems where the...

## 125 W

Bend specimen b. Compact tension specimen The specimens have to be provided with a fatigue crack. In order to ensure that cracking occurs at the right place, the specimens contain a starter notch. In thick members fatigue cracks usually start at a corner (figure 7.2a). Such cracking behaviour results in an irreproducible, curved crack-front, not suitable for a standard test. It can be avoided by providing the specimens with a chevron notch (figure 7.2b). This...

## 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...