A model to predict fatigue crack initiation in plates with a pre-crack

1. Introduction

Fatigue failure takes place in two stages; (i) fatigue crack initiation, and (ii) fatigue crack propagation. Both stages are important because each one takes a large number of cycles to occur. For a constant amplitude fatigue load, the crack propagation for a known initial crack is addressed by the Paris' law [1] as: d a d N = C ( Δ K ) m where d a d N is the crack growth per cycle, ΔK is the difference between the maximum stress intensity factor (K _max ) and minimum stress intensity factor (K _min ) and C and m are the material constants. The life of a component, determined using this relation, depends heavily on the initial crack length. Usually, the initial crack length is taken through an experimental measurement which, in turn, depends much on the capability of the experimental technique. If a satisfactory model is available to predict the number of cycles required to initiate a crack, the Paris' law can be used more effectively.

In several early investigations, Yokobori [2–4] conducted experiments under the reversed tension and compression fatigue tests and found to have marked characteristics of basic features of fatigue fracture. The stochastic process model was applied and was found to be in good agreement with experiments.

Barsomand and McNicol [5] investigated the fatigue crack initiation life in HY-130 steel by testing specimens having widely varying notch acuities. The variation in notch acuity covered the range from fatigue cracked specimens to polished, un-notched specimens; the fatigue crack initiation data were obtained in the range 10³ to 10⁶ cycles. The data were analysed by using linear elastic fracture mechanics concepts and the theory of stress concentration in notched specimens. Pearson [6] conducted experiments on two commercial aluminium alloys to observe the initiation of fatigue crack at a plane polished surface with no prior crack. He found that crack was initiated at surface inclusions and observed growth for very short cracks ranging from 0.006 mm to 0.5 mm. Manonukul and Dunne [7] developed a polycrystal plasticity finite-element model to predict crack initiation using crystal plasticity, grain morphology and size, crystallographic orientation for face-centered cubic nickel alloy C263. The model is based on two material properties – the initial resolved shear stress and the critical accumulated plastic slip. Li et al. [8] studied the crack nucleation in austenitic steel 304C subjected to a constant stress amplitude fatigue load. He looked into micro aspects such as crystallographic orientation of grains, 12 slip system and dislocation densities. A numerical model was developed using the nonlinear kinematic hardening rule. Jha et al. [9] studied the effect of stress ratio on the statistical aspects of small fatigue crack growth behaviour in a duplex microstructure of Ti-6Al-2Sn-4Zr-6Mo at 260 °C. They used a focused ion beam (FIB) to machine micro-notches of depth in the range of 5–15 μm in the specimen which served as a site for crack initiation. Change in the stress ratio produced almost negligible influence on the small-crack growth behaviour.

In his review paper, Sangid [10] tried to explain the fatigue crack initiation through micro mechanisms of the material. He stated that slip irreversibilities exist in a metal and accumulate during fatigue loading. At the defect level, irreversibilities are due to several aspects of dislocations such as annihilating, cross-slipping, penetrating precipitates, transmission through grain boundaries, and piling-up. These slip irreversibilities are the early signs of damage during cyclic loading. The dislocations subsequently form low-energy stable structures as a means to accommodate the irreversible slip processes and increasing dislocation density during cyclic forward and reverse loading. The result is strain localization in a small region within the materials which causes crack initiation. Polak and Man [11] analysed the role of point defects in the formation of surface relief and in initiation of fatigue crack in crystalline materials. He proposed that the continuous formation, annihilation and migration of the point defects are responsible for redistribution within the persistent slip bands (PSBs). These stresses are relaxed by dislocation movement within PSB and leads to formation of surface relief in the form of extrusions and intrusions. Giang et al. [12] used McDowell’s model to develop his model for fatigue initiation mechanism of forged M3:2 tool steel. According to him, the fracture crack initiation of this material depends on micro structural features, shape, volume fraction and load ratio.

In the last 2–3 decades, many investigations have been carried to explore the crack nucleation and the growth of micro scale. Only a few representative ones are included in the literature survey. They are based on micro-mechanisms such as:

Grain boundaries

The strain developed through the micro-mechanisms is heterogeneous with comparatively much higher strain within persistent slip bands. The important realization emerged out of these works on micro-mechanisms is that the annihilation of dislocations during reversed loading is not complete as screw dislocations slip on different planes and the edge dislocations of positive and negative Burger Vectors may not be on the same slip plane. Consequently, each cycle of fatigue load causes an incremental plastic deformation even if the applied fatigue load is of constant magnitude. The parameters are too many and the quantitative behaviour of the various mechanisms is still not known well. Some investigators used sophisticated finite element analysis or finite element simulation to predict the initiation of the fatigue crack.

For predicting the fatigue life of a finite size crack, the Paris' law works fairly well without looking into how the various micro-mechanisms operate. However, it is felt that a similar model may be explored on macro level to predict the number of cycles required to initiate a fatigue crack.

In structural components, fatigue failure is the dominant mechanism of failure. In most of them, a fatigue crack is initiated on a surface with no prior crack, taking a large number of cycles. However, fatigue cracks are also found to be initiated at some existing discontinuities in a structural member. These discontinuities are of two kinds; (i) manufacturing defects such as voids, slag inclusion or inclusion of other kinds of second phase particles and (ii) design features like small holes, notches, grooves, etc. Because of these discontinuities, the stress concentration exists which, in turn, nucleate fatigue cracks. In comparison to crack nucleation at a surface, these cracks take comparatively less number of cycles to initiate the crack growth. However, these cases are of importance too because, in certain cases especially with small stress intensity factor, the number of cycles required to initiate a crack is significant.

In this study, a model is developed to predict the number of cycles of a constant magnitude fatigue load required to initiate a crack at the sharp tip of a pre-crack. The model is explored on a macro scale, keeping in view the various micro-mechanisms of the deformation at the crack tip. The validity of the model is checked on pre-cracked thin panels of commercially available copper, aluminium and mild steel.

2. A model to predict crack initiation

It usually takes a large number of cycles for a pre-crack to start growing under a fatigue load. This is known as cycles required to initiate the crack (N _i ). After initiation, it takes additional cycles (N _p ) for the crack to grow till failure. Thus, the total number of cycles to failure (N _f ) is given by: (1) As stated earlier, Np is usually determined by the Paris' law. A suitable model is required to predict N _i . The plastic deformation zone at the crack tip plays a vital role in fracture mechanics. In metals, plastic deformation is caused by dislocations within the persistent slip bands and each cycle develops irreversible incremental plastic deformation. If the maximum stress of the fatigue load is considerably smaller than the yield stress, the expected increment of plastic deformation per cycle is small.

Since the plastic zone size at the crack tip of an existing crack is given by the stress intensity factor (K), the incremental increase in plastic deformation per cycle is taken to be governed by ΔK of the applied load. Thus, the cumulative plastic deformation depends on the number of cycles and the stress intensity factor of the crack.

A ductile material is capable of having a very large plastic deformation addressed as critical plastic deformation (𝜖 _pc ) in this work. In fact, with the increasing plastic deformation, a stage is reached beyond which no further plastic deformation is possible. At this stage, there are too many dislocations which criss-cross each other and lock them; so much so that further motion and nucleation of the dislocations is not possible and material at the tip cracks, causing crack initiation.

The conventionally used tensile strength is not really a material property. An unstable failure mechanism occurs with the initiation of a neck. This instability condition is similar to the instability of a column due to buckling under compression loading. The neck appears when the true stress (𝜎 _t ) and the true strain (𝜀 _t ) approach the relation [13]: (2) For a ductile material, this condition is achieved at relatively low plastic strain. Once the neck is initiated, it is not a state of equilibrium. The specimen fails without the further growth of the plastic deformation in the region outside the necked portion.

Some people advocate that the stress-strain behaviour should be determined though a compression test which does not allow formation of neck [13]. The compression test has its own problems. Usually friction at the load surfaces exists and specimen takes the shape of barrel and then the experiment is no longer controlled as a uniaxial state of stress. If Teflon tapes are used at the loading surfaces, there are chances of bollarding. A right nature of friction should be created to avoid barrelling and bollarding. Thus, it is a difficult task to conduct compression tests. The authors feel that suitable test techniques should be developed which allows inhibited plastic deformation to the maximum capability. Various mechanisms like buckling, neck formation, barrelling, bollarding, etc. should not impede the plastic deformation to its maximum extent under appropriate controlled conditions. A test technique based on rolling may be worth developing.

The extent of critical plastic strain, which a material is capable to undergo without any cracking, can be assessed from rolling of metallic sheets through a rolling mill. A commercial aluminium can be reduced to 5% of its original thickness from the annealed state. A mild steel with carbon less than 0.12% can be reduced to 30% of its initial thickness. Similarly an annealed copper sheet can be reduced to 5% of its original thickness and a brass sheet to 60%. The data are provided by a rolling mill expert [14]. The reduction is carried out in small steps in a rolling mill; that is, a sheet is passed multiple times. Each pass is given small reduction in thickness to avoid local cracking and allow the expansion of the sheet in the lateral direction. When the thickness of a sheet is reduced to the maximum possible extent, further reduction is carried out by annealing the sheet to bring it to the virgin state. Thus, an annealed ductile metal is capable of taking a very large strain. In the proposed model, the critical plastic strain 𝜖 _pc is expressed as, (3) where ΔK _I = K _Imax − RK _Imax = (1 − R)K _Imax , and 𝜂, 𝜆, 𝛾 are material constants and R is stress ratio.

Rearranging Eq. (3), one obtains: (4) For a given material in annealed state, 𝜖 _pc , 𝜂, 𝜆 and 𝛾 are material constants. Grouping the four material constants into two material constants 𝛼 and 𝛽, and considering cases with no variation in R, we obtain: (5) Taking natural logarithm on both sides of Eq. (5)  yields: (6) The model of Eq. (5) is valid only for the materials which follow linear relationship between ln(N _i ) and ln(K _Imax ) for a stress ratio R. The material constants 𝛼 and 𝛽 are obtained through experiments.

Testing was carried out on thin specimens with a pre-crack to explore whether the linear relationship of Eq. (6) between ln(N _i ) and ln(K _Imax ) is followed by several commonly used materials. Tests to determine N _i were conducted on thin sheets of three commercially available materials, aluminium, copper and mild steel through a tension-tension fatigue load of constant amplitude.

Commercially available large sheets of size 2.44 m × 1.22 m and of thickness close to 1 mm were purchased from the local market. Tensile test coupons were prepared from these sheets and tested for their properties and they are listed in Table 1. For fatigue testing, the sheets were cut on a shearing machine to external dimensions of 400 mm × 60 mm as shown in Fig. 1. The specimens were oriented to have the rolling direction along the length.

4. Experimental validation of the model

The fatigue load was applied with frequency of 7 Hz on the pre-cracked specimens with stress ratio of 0.1. Cycles required to cause crack initiation N _i were monitored and the relation was determined between ln(N _i ) and ln(K _Imax ) for thin sheets of aluminium, copper and mild steel.

The crack initiation was in the form of a crevice developed on the surface of the tip of the pre-crack. A magnified view was closely observed on the computer monitor. The crack was taken to be nucleated when the crevice was between 0.05 mm and 0.10 mm length. The experiment was continued further to make sure that the crevice continues to grow as a full-grown fatigue crack.

The stress intensity factor K _Imax corresponding to maximum stress 𝜎 _max for the centre-cracked plate of width 2W subjected to uniform tension is given by [1]: (7) where In each case, it was ascertained that the applied K _I   was greater than the threshold stress intensity factor K _th [15–17].

5. Conclusion

A model on macroscale is proposed to predict the number of cycles required to initiate crack growth from the tip of an pre-existing crack sharpened through a sharp razor blade. Each cycle develops an incremental plastic deformation at the crack tip and when the cumulative plastic deformation reaches the critical plastic strain, the crack initiation occurs. The plastic deformation per cycle is expressed in terms of maximum stress intensity factor K _Imax . The model predicts the crack initiation cycles N _i as N _i = 𝛼(K _Imax )^𝛽 where 𝛼 and 𝛽 are material constants.

The model has been tested so far on a thin plate with a pre-existing centre crack of commercially available aluminium, mild steel and copper sheets. The model was found to work well when the thin speciemns of commercial copper, aluminium and mild steel were loaded with a constant amplitude fatigue load of stress ratio 0.1.