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

Abstract

A model is proposed on a macro scale to predict crack initiation caused by a constant amplitude fatigue load. It predicts the number of cycles (N _i) required to initiate the crack at the tip of a pre-crack, sharpened with a razor blade, by the relation N _i = 𝛼(K _Imax)^𝛽 where K _Imax is the stress intensity factor corresponding to maximum stress of the fatigue load and 𝛼 and 𝛽 are the material constants to be determined by experiments. The validity of the model was checked on thin sheets of commercial aluminium, copper and mild steel.

Keywords

Critical plastic strain fatigue initiation cycles pre-crack constant amplitude fatigue load crack initiation

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: $\begin{eqnarray}\frac{da}{dN}=C({\Delta}K)^{m}\end{eqnarray}$ where $\frac{da}{dN}$ 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

Orientation of crystallographic planes

Strain localization in persistent slip bands

Dislocation nucleation and motion in the forward loading of a fatigue pulse

Dislocation annihilation during reverse loading of the pulse through cross slip of screw dislocations which may be on different slip planes

Generation of vacancies and interstitials through climb of edge dislocation during reverse loading

Dislocation interaction, forming jogs, nodes and locks

Intrusion and extrusion at free surface and at grain boundaries

Hardening within the persistent slip bands

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: $\begin{eqnarray}\displaystyle N_{f}=N_{i}+N_{p}. & & \displaystyle\end{eqnarray}$ (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]: $\begin{eqnarray}\displaystyle \frac{d{\sigma}_{t}}{d{\varepsilon}_{t}}={\sigma}_{t}. & & \displaystyle\end{eqnarray}$ (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, $\begin{eqnarray}\displaystyle {\epsilon}_{pc}={\eta}({\Delta}K_{I})^{{\gamma}}(N_{i})^{{\lambda}} & & \displaystyle\end{eqnarray}$ (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: $\begin{eqnarray}\displaystyle N_{i}=\frac{({\epsilon}_{pc})^{1/{\lambda}}}{{\eta}^{1/{\lambda}}(1-R)^{{\gamma}/{\lambda}}(K_{Imax})^{{\gamma}/{\lambda}}}.& & \displaystyle\end{eqnarray}$ (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: $\begin{eqnarray}\displaystyle N_{i}={\alpha}(K_{Imax})^{{\beta}}. & & \displaystyle\end{eqnarray}$ (5) Taking natural logarithm on both sides of Eq. (5) yields: $\begin{eqnarray}\displaystyle \ln \left(N_{i}\right)=\ln {\alpha}+{\beta}\ln (K_{Imax}) & & \displaystyle\end{eqnarray}$ (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.

3. Specimen preparation and experimental technique

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.

3.1. Preparation of the pre-crack

A centre crack was initially machined cut through a wire cut EDM of length 2 mm less than the required crack length. The width of crack was about 2 mm (Fig. 1). The crack was sharpened using two kinds of blades, surgical blades and razor blades. A suitable blade holder was designed to move a surgical blade of 0.38 mm thickness to extend the crack length by about 0.75 mm. Then a fresh razor blade of 70 μm thickness, applied through an appropriately designed blade holder, made the crack tip very sharp. The V-shaped edge of the razor blade sharpened the crack tip to the radius of the order 15 μm. The length of this pre-crack was measured accurately using a microscope. Figure 2 shows the pre-crack in the specimen.

The specimen was loaded in a tension-tension Fatigue Machine which was custom made by a vendor based on the specifications prescribed. The fatigue machine has maximum loading capacity of 40 KN and maximum operating frequency up to 10 Hz. The upper end of the specimen was fixed and the lower end was pulled by a hydraulic power pack.

The upper limit of the load was chosen and the lower limit of the load was set as 10% of the upper limit. During the testing, cycles to initiate a crack (N _i) were measured. The crack tip was monitored with a light weight and very compact USB digital microscope focused on the specimen. The digital microscope of magnification range 50X-500X was equipped with eight lights emitting diodes (LED) and was connected to a computer through a USB cable.

The adjustable stand, supplied with the USB microscope, was not suitable for this study. A separate metal stand was designed with a mounting platform having capabilities of moving the digital microscope in both X and Y directions of the horizontal plane. Thus, with fine displacement control using screw gauges, the digital microscope could be brought very close to the specimen surface. The stand was not attached to any part of the fatigue machine to avoid the vibrations of the microscope.

Tests were conducted on various specimens to monitor number of cycles required to initiate the crack so that the dependence of ln(N _i) on ln(K _Imax) could be explored.

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]: $\begin{eqnarray}\displaystyle K_{I}=\left[{\sigma}\times \sqrt{{\pi}a}\right]f\left(\frac{a}{W}\right) & & \displaystyle\end{eqnarray}$ (7) where $\begin{eqnarray}\displaystyle f\left(\frac{a}{W}\right)=1.0+0.128\left(\frac{a}{W}\right)-0.288\left(\frac{a}{W}\right)^{2}+1.523\left(\frac{a}{W}\right)^{3}\quad \text{and}\quad a/W\lt 0.7. & & \displaystyle \nonumber\end{eqnarray}$ In each case, it was ascertained that the applied K _I was greater than the threshold stress intensity factor K _th [15–17].

4.1. Copper specimens

The results obtained for tension-tension fatigue tests for the copper sheets are listed in Table 2. The crack length was varied between 2a = 10 mm and 2a = 30 mm and 𝜎_max of the fatigue load was varied between 45 MPa and 90 MPa. The relation between ln(N _i) and ln(K _Imax) is shown in Fig. 3. The relation looks to be linear and the best fit straight line based on least square method is $ln(N_i )=-5.84ln(K_Imax )+25.2$ Thus, the experimental data follows the relation of $N_i= α (K_Imax)^β$ where $K_Imax$ is in $MPa√m$ units. The material constants for the copper specimens are: $α =87×10^9$ and $β=-5.85$ . Thus, $N_i= 87×10^9 (K_Imax)^{-5.85}$ The scatter of the data from the best fit line is determined. The magnitude of the distance $∆ln⁡(N_i)$ of a data point from the best fit line is found out and then standard deviation is evaluated. The standard deviation is ±0.84 which is quite small when it is compared with the value of $ln⁡(N_i)$ (Fig. 3).

4.2. Aluminium specimens

For aluminium thin plates, the crack length was varied between 2a = 10 mm and 2a = 30 mm and 𝜎_max of the fatigue load was varied between 25 MPa and 45 MPa. The relation between ln(N _i) and ln(K _Imax) is also shown in Fig. 4.

The material constants obtained for the aluminium sheet are: 𝛼 =17 ×10⁶ and 𝛽 = −3.26. With respect to the linear relation, the standard deviation of ln(N _i) for aluminium is ±0.507. Thus, $\begin{eqnarray}N_{i}=17\times 10^{6}(K_{Imax})^{-3.26}\end{eqnarray}$ For the low yield stress (𝜎_ys) of 70.7 MPa of this commercially purchased aluminium sheet, the scatter was found to be large for higher values of K _Imax. The plastic zone size was quite large in comparison to the specimen thickness. For example, the plastic zone size (r _y), using the Irwin’s correction factor formula $r_{y}=\frac{1}{{\pi}}\frac{K_{I}^{2}}{{\sigma}_{ys}^{2}}$ , was 2.91 mm for K _max = 7. 17 MPa√m. It is substantially larger than the thickness of the sheet.Thus, the linear elastic fracture mechanism no longer works well for higher value of K _Imax and the scatter is more.

4.3. Mild steel specimens

For mild steel thin plates of yield stress 207.7 MPa, the crack length was varied between 2a = 10 mm and 2a = 30 mm and applied far field stress between 50 MPa and 150 MPa.

The relation between ln(N _i) and ln(K _Imax) is shown in Fig. 5. The relation also looks to be linear and the best fit straight line based on least square method is $\begin{eqnarray}\text{ln}\left(N_{i}\right)=-2.64\text{ln}\left(K_{Imax}\right)+18.15.\end{eqnarray}$ For this material, the experimental data also follows the relation N _i = 𝛼(K _Imax)^𝛽 . The material constants obtained for mild steel specimens are: 𝛼 = 76 ×10⁶ and 𝛽 = −2.64 where K _Imax is in $\text{MPa}\sqrt{\text{m}}$ units. Thus, $\begin{eqnarray}N_{i}=76\times 10^{6}(K_{Imax})^{-2.64}.\end{eqnarray}$ The standard deviation of ln(N _i) with respect to the linear relation for mild steel is ±0.7 which is also quite small in comparison to the value of ln(N _i).

The material constants 𝛼 and 𝛽 for copper, aluminium and mild steel are listed in Table 3.

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.

Footnotes

Conflict of interest

None to report.

References

Gdoutos

E.E.

, Fracture Mechanics- An Introduction, 2nd edn, Springer, Dordrecht, (2005).

Yokobori

, Creep fracture of copper as nucleation process, Journal of Physics, Society of Japan 7 (1952), 48–51.

Yokobori

, Fatigue fracture from the standpoint of the stochastic theory, Journal of Physics, Society of Japan 8 (1953), 265–268.

Yokobori

, The Strength, Fracture, and Fatigue of Materials, P. Noordhoff, Netherlands, (1965).

Barsomand

J.M.

and McNicol

R.C.

, Effect of Stress Concentration on Fatigue Crack Initiation in HY-130 Steel, ASTM STP 559, American Society for Testing and Materials, Philadelphia, (1974), pp. 183–204.

Pearson

, Initiation of fatigue cracks in commercial aluminium alloys and the subsequent propagation of very short cracks, Engineering Fracture Mechanics 7(2) (1975), 235–240.

Manonukul

and Dunne

F.P.E.

, High-and low-cycle fatigue crack initiation using polycrystal plasticity, The Royal Society of London A: Mathematical, Physical and Engineering Sciences 460(2047) (2004), 1881–1903.

Aubin

Rey

and Bompard

, Microstructural modeling of fatigue crack initiation in austenitic steel 304L, International Conference on Advances in Computational Modeling and Simulation (2012), 541–549.

Jha

S.K.

Reji

and Larsen

J.M.

, Incorporating small fatigue crack growth in probabilistic life prediction: Effect of stress ratio in Ti-6Al-2Sn-4Zr-6Mo, International Journal of Fatigue (2013), 83–95.

10.

Sangid

M.D.

, The physics of fatigue crack initiation, International Journal of Fatigue (2013), 58–72.

11.

Polak

and Man

, Fatigue crack initiation: The role of point defects, International Journal of Fatigue (2014), 18–27.

12.

Giang

N.A.

Ozden

U.A.

and Bezold

, A model for predicting crack initiation in forged M3:2 tool steel under high cycle fatigue, International Journal of Fracture 187 (2014), 145–158.

13.

Chakrabarty

, Theory of Plasticity, International edn, Tata McGraw-Hill, Singapore, (1987).

14.

Vaid

B.L.

, Vaid Engineering Industries, www.vaidengineering.com, private communication, (2017).

15.

Pippan

, The effective threshold of fatigue crack propagation in aluminium alloys. I. The influence of yield stress and chemical composition, J. Philosophical Magazine A 77(4) (1998), 861–873.

16.

Liaw

P.K.

Leax

T.R.

Williams

R.S.

and Peck

M.G.

, Near-threshold fatigue crack growth behavior in copper, Metallurgical Transactions A 13(9) (1982), 1671–1618.

17.

Pook

L.P.

and Smith

R.A.

, Fracture Mechanics: Current Status, Future Prospect, Proceeding of Conference, Cambridge (1979), pp. 29–67.