Fatigue and fracture emanating from notch; the use of the notch stress intensity factor

doi:10.1016/S0029-5493(98)00183-6

Nuclear Engineering and Design

Volume 185, Issues 2–3, October 1998, Pages 173-184

https://doi.org/10.1016/S0029-5493(98)00183-6 Get rights and content

Abstract

The analysis of the stress distribution at notch tip shows a pseudo-stress singularity characterised by the notch stress intensity factor (NSIF) K_ρ. The critical value of this parameter K_ρ^ccan be used to determine the fracture toughness of very brittle materials from notched specimens. The range of the notch stress intensity factor ΔK_ρ plays an important role in initiation of fatigue emanating from notches and in the notch fatigue sensitivity index.

Introduction

The introduction of a defect into a component is mechanically worse than the consequence of the net section reduction and the increase of the load bearing. This effect is usually named notch effect. It can be easily shown in a graph: critical gross stress versus non-dimensional defect size, usually known as the Feddersen diagram. This diagram is plotted on Fig. 1 for the case of a simple plate loaded to a uniform stress and having a central notch of length 2a.

The load bearing capacity of the ligament area is equal to the ultimate strength of the material. This leads to a linear decreasing of the critical gross stress according to: $σ_{g}^{c} =R_{m} · 1−a/W$

When a notch is present, the critical gross stress varies according to a non-linear relationship and exhibits a value smaller than those obtained from Eq. (1) except for very small and large defects. The differences between the experimental value of the critical gross stress and those from Eq. (1) characterises the so-called notch effect.

Notch nocivity is strongly related not only to the notch dimensions but also to other geometrical parameters like notch radius ρ and notch angle ψ. Values for these two parameters have the following classification:

•
crack: ρ=0 and ψ=0,
•
infinite sharp notch: ρ=0 and ψ≠0,
•
simple notch: ρ≠0 and ψ≠0

Crack is the most dangerous for the majority of notch. For brittle fracture, linear fracture mechanics indicates that the product of the square root of the notch size by the critical gross stress is equal to a constant (Griffith, 1920): $σ_{g}^{c} · a = cst_{1}$

The nocivity of a simple notch is less than that of a crack and Eq. (2) is modified according to: $σ_{g}^{c} · a^{β} =cst_{2}$ where β is a constant. This notch effect is related to the physical nature of the fracture process. This process needs a given volume. The stress gradient at notch tip cannot be too severe in order to allow mean stress in this volume to be high enough.

This is the reason why the hot spot concept, where the maximum local stress is assumed to play the major role in fracture, cannot be applied to verify experimental results.

The role of stress gradient in the fracture process has been established by Irwin (1958) who introduced the concept of stress intensity factor to describe the distribution of stress in the fracture process zone at a crack tip.

Similarly, the stress gradient plays an important role on the fatigue crack initiation process emanating from a notch. The analysis of Wöhler curves shows that those from notch specimens cannot be derived simply from those obtained with smooth specimens and the value of the maximum elastic stress at notch tip calculated with the elastic stress concentration factor; the stress gradient also plays a major role in the fatigue process as has been suggested by Brandt and Sutterlin (1980) and Buch (1974).

The process volume for fatigue and fracture is generally and simply assumed having a cylindrical shape with a thickness equal to the structured one. Probabilistic approaches allow direct computing of this volume.

The diameter of this cylindrical process volume is assimilated to the process distance (X_pz). This distance is strongly connected to microstructure for small values of the process volume associated with high stress concentration. For low values of stress concentration, the microscopic investigation of initiation sites shows the important role of the notch radius and location of the maximum triaxiality on the process distance.

In the present paper, we show how we can obtain a description of the stress distribution at notch tip and determine the process distance (called the effective distance by many authors). This can easily be done with the concept of notch stress intensity factor (NSIF). We also show how to apply this concept to determine the fracture toughness of brittle material or the fatigue notch sensitivity index on the basis of tests carried out with notched specimens.

Finally we discuss the possibility of considering the classical linear fracture mechanic as a particular case of the notch fracture mechanic (NFM).

Section snippets

Stress distribution at notch tip

The general trend of the stress distribution normal–notch plane is shown on Fig. 2 relative to a beam specimen loaded by a three point bending loading mode. The stress distributions have been computed by finite element methods and for different notch radii. Graphical representation has been made with non-dimensional stress (local stress divided by net stress σ_N) versus non-dimensional distance (distance divided by notch length a). This distribution can be characterised by two elements: the

Notch stress intensity factor

The stress distribution at notch tip can be presented in another form. In a bilogarithmic graph, we can plot normal stress–notch plane divided by nominal stress σ_yy versus distance divided by the ligament size (B).

Example of such a diagram is given on Fig. 3a relative to specimen loaded by three point bending. This distribution is schematically described in Fig. 3b and can be divided into three zones (Niu et al., 1994):

•
zone I, very close to the notch tip. The non-dimensional stress is

Brittle fracture criteria for notched components

From the stress distribution at notch tip, two types of fracture criteria can be used: a global and a local fracture criterion.

In the case of a notch there is no stress singularity at crack tip but a maximum stress followed by a pseudo-singularity in which the stress distribution is governed by a NSIF. This is the basic method for NFM and for a determination of fracture toughness.

Application to the measurement of the fracture toughness of a brittle material: the glass ‘float’

Fracture toughness determination of the glass ‘float’ was made on three point bending specimens. These specimens have the same notch length but a different notch radii (ρ=0.125, 0.4, 0.5, 0.6, and 0.75 mm). In this case, notch specimens are preferred to pre-cracked specimens because it is not very easy to pre-crack very brittle materials like ceramics or high-strength steels. Up to now, the determination of fracture toughness on notched specimens has been made using the ASTM standard for

Notch effect in fatigue

Fatigue crack initiation is sensitive to notch effect as can be seen in Fig. 7 where reference Wöhler’s curve is compared to the fatigue resistance curve of notch specimen. For the same fatigue life duration the stress range amplitude is reduced by an important factor which characterises the fatigue notch effect (Frost et al., 1974).

To explain this phenomena, let’s consider that the reference Wöhler’s curve can be intrinsic to the material, and eventually to the loading mode and specimen size

Conclusion

The phenomena of fatigue and fracture are not governed by the local maximum stress (or strain or strain energy density). The use of the assumption called the hot spot approach always leads to an overestimation of the fracture or fatigue stress.

The physical reason is the fact that these two phenomena need a physical volume to take place. In this process zone, the stress distribution exhibits a strong gradient which plays an important role.

For a crack, this stress distribution is an inverse

References (25)

T. Boukharouba et al.
The use of notch stress intensity factor as a fatigue crack initiation parameter
Eng. Fract. Mech.
(1995)
A. Buch
Analytical approach to size and notch-size effect in fatigue of aircraft material specimens
Mater. Sci. Eng.
(1974)
G. Glinka et al.
Int. J. Fatigue
(1987)
P. Lukas et al.
Fatigue limit of notched bodies
Mater. Sci. Eng.
(1978)
S. Bhattacharya et al.
Rotational factor using bending moment approach under elasto-plastic situation in 3PB notch geometry
Eng. Fract. Mech.
(1995)
Brandt A., Sutterlin R., 1980, Calcul des pièces à la fatigue; méthode du gradient,...
C.C. Chen et al.
M Creager et al.
Elastic field equations for blunt cracks with reference to stress corrosion cracking
Int. J. Fract. Mech.
(1967)
N.E Frost et al.
A.A Griffith
Phil. Trans. R. Soc. Lond. A
(1920)

Harris W.J., 1961. Metallic fatigue, International Series of Monographs in Aeronautics and Astronautics, Pergamon...

R.B. Heywood

Designing by Photo-Elasticity

(1952)

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Fatigue and fracture emanating from notch; the use of the notch stress intensity factor

Abstract

Introduction

Section snippets

Stress distribution at notch tip

Notch stress intensity factor

Brittle fracture criteria for notched components

Application to the measurement of the fracture toughness of a brittle material: the glass ‘float’

Notch effect in fatigue

Conclusion

Eng. Fract. Mech.

Mater. Sci. Eng.

Int. J. Fatigue

Mater. Sci. Eng.

Rotational factor using bending moment approach under elasto-plastic situation in 3PB notch geometry

Eng. Fract. Mech.

Elastic field equations for blunt cracks with reference to stress corrosion cracking

Int. J. Fract. Mech.

Phil. Trans. R. Soc. Lond. A

Designing by Photo-Elasticity