Notes on Fracture Mechanics

Notes on Fracture Mechanics

APPENDIX NOTES ON FRACTURE MECHANICS 7 A7.1 INTRODUCTION The field of fracture mechanics has progressed a long way since the first study by A.A. Gr...

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A7.1 INTRODUCTION The field of fracture mechanics has progressed a long way since the first study by A.A. Griffith (18931963). It is useful to recall the simple yet brilliant logic behind them. Fundamentally Griffith (Ewing and Hill, 1967) understood that as a crack propagated in a stressed material an energy exchange took place. On one hand, the crack propagation required energy for the creation of further fracture surfaces in front of the crack point and, on the other hand, energy was released by the zone of material which was unloaded by the propagation itself. Fig. A7.1 illustrates this phenomenon and the concept of “critical crack length.” Curve A represents the energy necessary to create rupture surfaces corresponding to a certain crack length L. The curve is substantially a straight line as the area of the rupture surfaces is proportional to the crack length and the rupture energy is proportional to this area. Curve B represents the energy released for the extension of the crack from zero length up to length L. This curve has a parabolic shape as the energy released is proportional to the volume of material unloaded by the propagation (indicated around the crack in the left part of the diagram), which in turn is roughly proportional to the square of the crack length. The third curve represents the difference between released energy and rupture energy for the various lengths of crack; the quantity Lg represents the critical crack length, that is, the value for which the increase of length of the crack releases more energy than is consumed in the creation of new rupture surfaces. In analytical terms, Griffith arrived at the conclusion shown in Eq. (A7.1): Lg 5

1 Rupture work for unit area 2GE 3 5 ; π Deformation energy for unit volume πs2


where Lg is the crack length (m) (with reference to the geometry depicted in Fig. A7.1), G is the energy needed for a unit increase of the crack surface (J/m2), E is Young’s modulus (N/m2), and s is the tension in the plate (N/m2).




Fracture Mechanics — Griffth (1920) P B: Released energy




Crack length A: Absorbed energy


FIGURE A7.1 Energy balance in crack propagation.

G has the order of magnitude of 12 3 105 J/m2 for construction steels and s is usually in the range of 70150 3 106 N/m22, so for a construction steel plate stressed at 150 3 106 N/m2, the following result is obtained: Lg 5

2 3 ð1:5 3 105 Þ 3 E 5 0:91 m πð150 3 106 Þ2

Among other things, Griffith’s energy formulation gives a logical explanation to the fact that, notwithstanding the very high stresses present at the crack tip, the resistance to its propagation is high for ductile materials.

A7.2 CURRENT PRACTICE Two of today’s approaches to fracture study are summarized here. The first is based on the use of the stress intensity factor K. The second is based on the J integral. The latter approach is suitable for situations of ductile fracture with strong deformations (ductile materials, low stress triaxiality, and so on). The approach based on the K factor is based on the possibility of representing the stress field around the crack tip by, precisely, a stress intensity factor K, which in turn is dependent on the way the crack is invited to propagate, on the mode of application of the load, on the level and variation of the stress in the material far from the crack tip and, finally, on the type of crack (thickness, elliptical, or with constant depth, etc.). The three stress modes usually considered are shown in Fig. A7.2. The various load application modes are shown in Fig. A7.3. The coordinate system generally adopted to describe the stress field around the crack is shown in Fig. A7.4.



Mode I

Mode II

Mode III

FIGURE A7.2 Modes of crack stressing (KI, KII, KIII).

An example of the distribution of stresses around the crack in biaxial geometry is given in Fig. A7.5. The expressions for O(r) in Fig. A7.5 represent distributions of stresses in the zones far removed from the crack tip and dependent on the complete stress state of the structure. Fig. A7.6 shows KI for the case of a longitudinal crack of various depths in a cylinder wall (e.g, in a pipe or the reactor vessel). A variety of already calculated cases exists for the distribution of stresses around a crack tip for various types of cracks and of loading conditions. Guidance on this can be found in specialist texts on fracture mechanics (Anderson, 2013; Milella, 1999, 2013; Miannay, 1997; Wilkowski et al., 1997). Fig. A7.7 shows the material properties KIC and KIA (intensity factors for crack initiation and for crack arrest of a propagating crack), with reference to a typical pressure vessel steel.



FIGURE A7.3 Modes of load application.

FIGURE A7.4 Coordinate system.

The temperature RTndt is the transition temperature between brittle and ductile rupture. It can be determined by tests on specific toughness specimens or it can be correlated (for increased easiness) with an energy value absorbed in the common Charpy V test (generally 5.1 3 105 or 8.7 3 105 J/ m2, corresponding to 30 or 50 ft/lb, respectively. The way in which the various types of data are used is generally the following one: • •

KI is determined for the crack to be studied. KIC is determined for the material corresponding to the conditions at the crack tip. The comparison between this value and KI indicates whether the crack will start to propagate in an unstable way or not.



FIGURE A7.5 Stresses around the crack tip.

If it can be controlled, then the possibility exists that the crack which started to propagate is arrested at a certain point. For this investigation, KI, corresponding to various stages of extension of the crack has to be again determined. These values have to be compared with the corresponding KIA. If, for a certain stage of crack propagation it is found that KI is lower than KIA, then the crack will stop at that point.

In the case of a reactor pressure vessel the crack may stop because, with its extension, it arrives to zones of the material which are less embrittled than the one from where the crack has started. In other cases the arrest may occur because the material reached during the propagation is less stressed than the initial one. It is useful to remember the existence of the phenomenon of “warm prestress” according to which, in general terms, if a component containing a crack is loaded in warm conditions (ie, in conditions of good ductility), it is not susceptible to unstable crack propagation for lower load conditions, even if correspondingly the temperature and ductility are lower. This principle, which finds its evident logical basis in the effect of “protective” plasticization at the crack tip, is usually



FIGURE A7.6 KI for a longitudinal crack in the wall of a cylinder.

accepted in the following, less ample, formulation: “after an initial preload, no unstable crack propagation will occur if the stress intensity factor is constant or decreasing.” The J integral method is more widely used especially in cases of strong plasticization of the material during its rupture. This method substantially follows the K factor approach with the difference that the parameter to be evaluated is, now, a special integral operator, called the J integral (Rice, 1968). The integral is defined in Eq. (A7.2) with the symbols indicated in Fig. A7.8 ð J5

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Wdy 2 T




where T is the stress vector (kg/m2), u is the displacement (m), and W is the strain energy density (J/m3).



Critical toughness Toughness (kg/m1.5)

6.00E+07 5.00E+07 4.00E+07 KIc



2.00E+07 1.00E+07 0.00E+00 –50 0 RTndt


(T-RTndt) (°C) 50


FIGURE A7.7 Critical toughness and arrest toughness of a construction steel as a function of temperature (relative to the transition one).











nx= cosθ =

dy ds

ny= sinθ =−

dx ds

θ dy



ds dx x

FIGURE A7.8 Definition of the symbols used in the expression of the J integral.



The integral is calculated along any path Γ which includes the crack tip as indicated in the figure. It is invariant of the specific path chosen. The value of J that is critical for the material is measured on special samples. In order to clarify the physical meaning of K and, above all, of J, these quantities can be simply related to each other and with a concept already used by Griffith, that is with the specific potential energy related with the crack area, GR [see Eq. (A7.3)] GR 5 2

@U @A


assuming a small plastic area at the crack tip, and where R is the specific potential energy related to crack area (J/m2), U is the potential energy (J), and A is the crack area (m2). GR is then the variation of the elastic potential energy of deformation of the material corresponding to the unit variation of the crack area. The following relationships hold: J 5 GR; for plane problems


K12 5 GR 3 E; for plane stress states


GR 3 E ; for plane strain states 1 2 v2


(where E is Young’s modulus) K12 5

(where v is the Poisson modulus)

REFERENCES Anderson, T.L., 2013. Fracture Mechanics, third ed. CRC Taylor & Francis. Ewing, D.J.F., Hill, R.J., 1967. The plastic constraint of V-notched tension bars. J. Mech. Phys. Solids 15, 115. Miannay, D.P., 1997. Fracture Mechanics. Springer. Milella, P.P., 1999. Meccanica della frattura. Ansaldo Nucleare, Corso Perrone, 25, Genova. Milella, P.P., 2013. Fatigue and Corrosion in Metals. Springer. Rice, J.R., 1968. A path independent integral and the approximate analysis of strain concentration by notches and cracks. J. Appl. Mech. 15, 379386. Wilkowski, G.M., et al., 1997. State-of-the-art Report on Piping Fracture Mechanics. NUREG/CR-6540; BMI 2196.