Chapter 2

2 "Crack like to blunt" notch S/N curve model

Introduction

In this Chapter the infinite life design philosophy of notched specimens is analyzed, then the same concepts are specialized to finite life design concepts in the context of the “Theory of the Critical Distances” to analyze the transition from “crack-like to blunt” notch. It is shown that a notched specimen behaves very similarly to a crack up to a certain number of fatigue cycles, then its fatigue behavior can be approximated with a plain specimen reduced by the effective stress concentration factor. Accordingly, some fast assessment methods have been suggested: (i) a crack like notch could be replaced by a crack for which there is a wide amount of solutions in Literature and (ii) a blunt notch could be treated through infinite life design concepts only. To verify the analytical method, the SAE Keyhole test program constant amplitude fatigue test data have been used and predictions are in good agreement with experiments, also compared with other methods as strain-life approaches through Neuber’s rule.

2.1 Infinite life design

The effect of notches in infinite life fatigue design can be simply estimated through the concept of crack like notch introduced by Smith and Miller [1] which allows to obtain characteristic diagrams [2]–[5] under the assumption that the governing factor for infinite life modeling is the threshold stress intensity factor range ΔKth, value below which a so-called long crack should not propagate consistently with Paris’ law [6]. Indeed, infinite life fatigue design is governed by the threshold stress intensity factor range for long cracks, and by the fatigue limit stress range ΔSe for short cracks [7]; hence in order to identify the order of magnitude of the critical length of transition from short to long crack, El-Haddad’s intrinsic defect size a0 can be introduced

a0 = 1 / π · (ΔKth / ΔSe)2

2.1

The equation is given for a prescribed load ratio R=S_min/S_max. Such transition has been studied in principle experimentally in 1976 by Kitagawa and Takahashi [8] which plotted stress range vs. crack size (ΔS-a) diagrams confirming the validity of the relation connecting the stress intensity factor range ΔK and the stress range ΔS

ΔK = f · ΔSth · √(π a)

2.2

Where f is a geometric factor. Then, in 1977 Smith evidenced that Equation (2.2) is valid only beyond a specific value of crack size and later, in 1980, this limit was overcome by El Haddad et al. [9] who proposed the empirical formula

ΔSe = ΔKth / √[π · (a + a0]

2.3

In which the intrinsic defect size has been simply added to Equation (2.2) and the geometric factor f has been set to 1 for simplicity. The idea behind Equation (2.3) is one aspect of the “Theory of the Critical Distances” whose ancestor, as summarized by Taylor [10], and Yao et al. [11], can be identified in the effective stress concentration factor K_f proposed by Neuber [12], Kuhn and Hardrath [13] who in the early ‘50s assumed that the notched specimen fails if the averaged stress over the distance A_KH ahead of the notch root is equal to the fatigue limit Se of the plain specimen. From this hypothesis, Kf was calculated as

Kf = 1 + (Kt - 1) / [1 + π/(π - ω) · √(AKH / ρ)]

2.4

Where K_t is the theoretical stress concentration factor (e.g. 3 for a circular hole in an infinite plate), ω is the notch flank angle, ρ is the notch root radius and the distance A_KH depends on the material under exam. Later the equation has been modified as

Kf = 1 + (Kt - 1) / (1 + aN / ρ)

2.5

a_N is known as Neuber’s material constant.
Few years later Peterson [14] derived an equation based on the hypotheses that (i) the notched material fails if the point stress at a distance d0 away from the notch root is at least equal to the fatigue strength of the plain specimen and (ii) the stress ahead of the notch root drops linearly up to d₀ obtaining

Kf = 1 + (Kt - 1) / (1 + aP / ρ)

2.6

Where a_P is Peterson’s material constant. Notwithstanding, as also confirmed by Topper et al. [15], as ρ increases the fatigue limit is actually fully controlled by the theoretical stress concentration factor, thus K_f→K_t since a_P/ρ→0 and the notch is addressed as blunt notch. From a mathematical point of view, a notch can be effectively be addressed as blunt when its characteristic size a^* is [2]–[5]

a* ≥ Kt2 · a0

2.7

Hence, for instance for a hole in an infinite plate (K_t=3) this occurs when the circle radius is almost one order of magnitude larger than El Haddad’s intrinsic defect size. For aluminum alloys and steels, typically a₀ is of the order of 10 μm and 100 μm respectively implying a^* of the order of 0.1 mm and 1 mm respectively which are typical orders of magnitude of drilled holes for riveting. Therefore, Atzori and Lazzarin suggested the following infinite life design criterion for notched components

Kf = min⁡[√(1 + a / a0), Kt ]

2.8

Combining Equation (2.7) with Equation (2.8) implies that below a^* notches behave like cracks, and above a^* they behave as blunt notches. In 2004, with the purpose of collecting and comparing exhaustively the existing notch sensitivity estimation models for infinite life design, Ciavarella and Meneghetti [16] reviewed a series of classical and modern approaches to the stress concentration factor estimate and concluded that Neuber’s method [12] is the most conservative and accurate amongst the “classical” approaches whilst the Atzori-Lazzarin criterion is the most conservative yet easy-to-use between the “modern” ones. Moreover, they proposed the following modification to the Atzori-Lazzarin criterion of Equation (2.8) in order to make it consistent with Lukáš and Klesnil [17] discussion which can be interpreted as a modification of Neuber’s rule including the effect of cyclic plasticity.

Kf = min⁡{[1 + (a / a0)r]0.5/r, Kt }    0 < r ≤ 1

2.9

Equation (2.9) for r=1 obviously returns the Atzori-Lazzarin criterion, while for r=0.5 gives Lukáš-Klesnil criterion. Bazant [18] has shown in detail that the expression (1+(a/a₀)^r)^1/2r corresponds to an asymptotic matching with truncation at the first order between the large-size (a ≫ a₀) and the short-size (a ≪ a₀) expansions of the crack propagation criterion in terms of stress intensity factor and concluded that the El-Haddad equation can be seen as a “matching asymptotics” solution for the transition between fatigue limit towards fatigue threshold dominated threshold. Nevertheless, Ciavarella and Meneghetti in their review did not discuss the implications of applying the “Theory of the Critical Distances”[10], [19]–[22] (TCD) in its point variant (TCD-P); this shall be done here since the transition from infinite to finite life design here proposed is based on the TCD-P. Anyway, since it has already been demonstrated that Neuber’s (2.5) and Peterson’s (2.6) equations are prototypical critical distance approaches, a similar result with respect to Ciavarella and Meneghetti’s is expected. Indeed, the basic hypothesis of the TCD-P is that the material fails when the stress at a distance a₀/2 from the notch root/crack tip reaches the fatigue strength of the plain material. For instance, considering the Kirsch solution [23] for an infinite plate with a circular hole under uniform remote tension S_nom and imposing that the stress S_Y parallel to S_nom at distance r=a+a₀/2 from the center of the hole is equal to the alternate fatigue limit S_e leads to

Kf = SY (a + a0 / 2)) / Snom = Se / Snom = 0.5 ⋅ {2 + [a / (a + a0 / 2)]2 + 3 ⋅ [a / (a + a0 / 2)]4}

2.10

Returning the well-known K_t = 3 for a ≫ a₀. Similarly, considering Westergaard [24] solution for a crack immersed into an infinite plate and loaded perpendicularly to the crack flanks it results, at a distance a0/2 from the crack tip

Kf = SY (a0 / 2) / Snom = Se / Snom = [1 + (a0 / 2) / a] / [√(a0 / 2 / a) ⋅ √(2 + a0 / 2 / a) ]

2.11

Which goes to infinite for a ≫ a₀. In Figure 2.1 on the left it is shown the normalized stress component S_Y/S_nom ahead of the crack tip/notch root for the Kirsch and Westergaard problems, while on the right there is a comparison between stress concentration factors according to different criteria. Atzori-Lazzarin criterion is extremely close to the TCD-P applied to Westergaard problem for a<a^* and then it becomes constant for a ≥ a^* and at the same time it is comparable with the TCD-P applied to the Kirsch solution within the entire domain. However, there is a region where the TCD-P applied to Kirsch solution is more conservative than both Westergaard solution and the Atzori-Lazzarin criterion. This may sound counterintuitive, but comparing the stress gradient ahead of a hole (low) with the stress gradient ahead of a crack (high) provides a simple explanation; indeed, even if the stress immediately ahead of a hole is lower than the stress ahead of a crack, then at a certain distance the situation has to reverse to respect the global equilibrium, as evident from Figure 2.1 (Left).

Figure 2.1: (Left) Stress field for Kirsch, Westergard and Westergaard for a ≫ a₀; (Right) 1/K_f as a function of the notch/crack size. El Haddad’s equation is nearly indistinguishable from Westergaard TCD. The Atzori-Lazzarin criterion for a hole is equal to El Haddad’s equation up to a^*, beyond which it holds 1/K_t and it is comparable to Kirsch TCD

The infinite life design approach is useful when there are no other design constraints such as weight limits, but it becomes too limiting in aerospace applications, where the conflicting requirements of lightweight and durable structures must be simultaneously met. For this reason, it is convenient to extend the TCD-P design philosophy to finite life.

2.2 Infinite life design

2.12

2.13

2.14

2.15

2.16

2.17

2.18

2.******

References

[1]    R. A. Smith and K. J. Miller, ‘Prediction of fatigue regimes in notched components’, Int. J. Mech. Sci., vol. 20, no. 4, pp. 201–206, 1978.

[2]    B. Atzori and P. Lazzarin, ‘Notch Sensitivity and Defect Sensitivity under Fatigue Loading: Two Sides of the Same Medal’, Int. J. Fract., vol. 107, no. 1, pp. 1–8, Jan. 2001.

[3]    B. Atzori and P. Lazzarin, ‘A three-dimensional graphical aid to analyze fatigue crack nucleation and propagation phases under fatigue limit conditions’, Int. J. Fract., vol. 118, no. 3, pp. 271–284, Dec. 2002.

[4]    B. Atzori, P. Lazzarin, and G. Meneghetti, ‘Fracture mechanics and notch sensitivity’, Fatigue Fract. Eng. Mater. Struct., vol. 26, no. 3, pp. 257–267, 2003.

[5]    B. Atzori, P. Lazzarin, and G. Meneghetti, ‘A unified treatment of the mode I fatigue limit of components containing notches or defects’, Int. J. Fract., vol. 133, no. 1, pp. 61–87, Jan. 2005.

[6]    P. C. Paris and F. Erdogan, A Critical Analysis of Crack Propagation Laws. ASME, 1963.

[7]    T. Nicholas, High Cycle Fatigue: A Mechanics of Materials Perspective. Elsevier Science, 2006.

[8]    H. Kitagawa and S. Takahashi, ‘Applicability of fracture mechanics to very small cracks or the cracks in the early stage’, in Proc. of 2nd ICM, Cleveland, 1976, 1976, pp. 627–631.

[9]    M. H. El Haddad, N. E. Dowling, T. H. Topper, and K. N. Smith, ‘J integral applications for short fatigue cracks at notches’, Int. J. Fract., vol. 16, no. 1, pp. 15–30, 1980.

[10]    D. Taylor, ‘The theory of critical distances’, Eng. Fract. Mech., vol. 75, no. 7, pp. 1696–1705, 2008.

[11]    W. Yao, K. Xia, and Y. Gu, ‘On the fatigue notch factor, Kf’, Int. J. Fatigue, vol. 17, no. 4, pp. 245–251, May 1995.

[12]    H. Neuber, Theory of notch stresses: Principles for exact stress calculation, vol. 74. JW Edwards, 1946.

[13]    P. Kuhn and H. F. Hardrath, ‘An engineering method for estimating notch-size effect in fatigue tests on steel’, National Advisory Committee for Aeronautics, Langley Field, Va, NACA Technical Note NACA-TR-2805, 1952.

[14]    R. Peterson, Ed., Manual on Fatigue Testing. West Conshohocken, PA: ASTM International, 1949.

[15]    T. Topper, R. M. Wetzel, and J. Morrow, ‘Neuber’s rule applied to fatigue of notched specimens’, ILLINOIS UNIV AT URBANA DEPT OF THEORETICAL AND APPLIED MECHANICS, 1967.

[16]    M. Ciavarella and G. Meneghetti, ‘On fatigue limit in the presence of notches: classical vs. recent unified formulations’, Int. J. Fatigue, vol. 26, no. 3, pp. 289–298, Mar. 2004.

[17]    P. Lukáš and M. Klesnil, ‘Fatigue limit of notched bodies’, Mater. Sci. Eng., vol. 34, no. 1, pp. 61–66, Jun. 1978.

[18]    Z. P. Bazant, ‘Scaling of quasibrittle fracture : asymptotic analysis’, Scaling Quasibrittle Fract. Asymptot. Anal., vol. 83, no. 1, pp. 19–40, 1997.

[19]    L. Susmel and D. Taylor, ‘A novel formulation of the theory of critical distances to estimate lifetime of notched components in the medium-cycle fatigue regime’, Fatigue Fract. Eng. Mater. Struct., vol. 30, no. 7, pp. 567–581, 2007.

[20]    L. Susmel and D. Taylor, ‘On the use of the Theory of Critical Distances to predict static failures in ductile metallic materials containing different geometrical features’, Eng. Fract. Mech., vol. 75, no. 15, pp. 4410–4421, Oct. 2008.

[21]    L. Susmel and D. Taylor, ‘The Theory of Critical Distances to estimate lifetime of notched components subjected to variable amplitude uniaxial fatigue loading’, Int. J. Fatigue, vol. 33, no. 7, pp. 900–911, Jul. 2011.

[22]    L. Susmel and D. Taylor, ‘A critical distance/plane method to estimate finite life of notched components under variable amplitude uniaxial/multiaxial fatigue loading’, Int. J. Fatigue, vol. 38, pp. 7–24, May 2012.

[23]    C. Kirsch, ‘Die theorie der elastizitat und die bedurfnisse der festigkeitslehre’, Z. Vereines Dtsch. Ingenieure, vol. 42, pp. 797–807, 1898.

[24]    H. M. Westergaard, ‘Bearing pressures and cracks’, Trans AIME J Appl Mech, vol. 6, pp. 49–53, 1939.

[25]    M. Ciavarella and F. Monno, ‘On the possible generalizations of the Kitagawa–Takahashi diagram and of the El Haddad equation to finite life’, Int. J. Fatigue, vol. 28, no. 12, pp. 1826–1837, Dec. 2006.

[26]    N. Pugno, M. Ciavarella, P. Cornetti, and A. Carpinteri, ‘A generalized Paris’ law for fatigue crack growth’, J. Mech. Phys. Solids, vol. 54, no. 7, pp. 1333–1349, Jul. 2006.

[27]    M. Ciavarella, ‘Crack propagation laws corresponding to a generalized El Haddad equation’, Int. J. Aerosp. Lightweight Struct. IJALS, vol. 1, no. 1, 2011.

[28]    M. Ciavarella, ‘A simple approximate expression for finite life fatigue behaviour in the presence of “crack-like” or “blunt” notches’, Fatigue Fract. Eng. Mater. Struct., vol. 35, no. 3, pp. 247–256, 2012.

[29]    H. O. Fuchs and R. I. Stephens, Metal fatigue in engineering. Wiley, 1980.

[30]    R. I. Stephens, A. Fatemi, R. R. Stephens, and H. O. Fuchs, Metal Fatigue in Engineering. John Wiley & Sons, 2000.

[31]    B. Atzori and P. Lazzarin, ‘Analisi delle problematiche connesse con la valutazione numerica della resistenza a fatica’, in AIAS National Conference, Lucca Italy, also Quaderno AIAS, 2000, pp. 33–50.

[32]    B. Atzori, G. Meneghetti, and M. Ricotta, ‘Fatigue and Notch Mechanics’, in Applied Mechanics, Behavior of Materials, and Engineering Systems, 2017, pp. 9–23.

[33]    B. S. Company, Bethlehem RQC-80, RQC-90, RQC-100, Roller Quenched and Tempered Carbon Steel Plate: Fabrication and Welding Data. Bethlehem Steel, 1972.

[34]    AISC, Ed., Hot Rolled Steel Shapes and Plates, Second. United States Steel - USS, 1963.

[35]    H. M. Westergaard, ‘Stresses At A Crack, Size Of The Crack, And The Bending Of Reinforced Concrete’, J. Proc., vol. 30, no. 11, pp. 93–102, Nov. 1933.

[36]    L. Tucker and S. Bussa, ‘The SAE Cumulative Fatigue Damage Test Program’, SAE Trans., vol. 84, pp. 198–248, 1975.

[37]    R. W. Landgraf, F. D. Richards, and N. R. LaPointe, ‘Fatigue Life Predictions for a Notched Member Under Complex Load Histories’, SAE Trans., vol. 84, pp. 249–259, 1975.

[38]    J. M. Potter, ‘Spectrum Fatigue Life Predictions for Typical Automotive Load Histories and Materials Using the Sequence Accountable Fatigue Analysis’, SAE Trans., vol. 84, pp. 260–269, 1975.

[39]    H. Neuber, ‘Theory of stress concentration for shear-strained prismatical bodies with arbitrary nonlinear stress-strain law’, J. Appl. Mech., vol. 28, no. 4, pp. 544–550, 1961.

[40]    D. V. Nelson and H. O. Fuchs, ‘Predictions of Cumulative Fatigue Damage Using Condensed Load Histories’, SAE Trans., vol. 84, pp. 276–299, 1975.