PSI - Issue 68

Kimmo Kärkkäinen et al. / Procedia Structural Integrity 68 (2025) 646–652 K. Ka¨rkka¨inen et al. / Structural Integrity Procedia 00 (2024) 000–000

650

5

Fig. 4. Schematic illustration of the cyclic resistance curve analysis (Ka¨rkka¨inen et al., 2024b). Over- and underloads can be thought to reset the R-curve. A loading history containing three sporadic underloads is considered in this instance. Overloads follow the same principle; the only di ff erence is that an altered R-curve (Fig. 3(b)) is used from the first overload onwards.

(1) transforms into a second order polynomial equation, the tangent condition takes the form

b 2 2 − 4 b 1 b 3 = 0 , where   

b 1 = 4 πσ 2 2 b 2 = ( a 0 + a ∗ + a init − a s ) b 1 − ∆ K 2 th , lc b 3 = ( a 0 + a ∗ − a s ) a init b 1 − ( a ∗ − a s ) ∆ K 2 th , lc . w , e ff Y

(2)

As the problem of finding σ w , e ff ( n ) is recursive in nature, no closed form solution exists. However, the numerical implementation is rather simple. The constant amplitude fatigue limit used for the base loading, σ w , CA = 147MPa, is obtained when the number of loading spikes is zero. The parameter values used are Y = 1, a init = 250 µ m, ∆ K th , lc = 15MPa √ m, ∆ K th , e ff = 2 . 5MPa √ m, and ∆ σ th , 0 = 2 × 1 . 6 HV = 640MPa.

4. Results and discussion

The analysis results concerning the e ff ect of over- and underloads on fatigue limit are provided next, along with relevant discussion. Fig. 5 presents the reduction of σ w , e ff in terms of the ratio to the constant amplitude fatigue limit, σ w , CA , as a function of the number of loading spikes, n . The reduction of the e ff ective fatigue limit is substantial in all cases, and is primarily a consequence of crack growth. It can be seen that σ w , e ff /σ w , CA saturates with a large enough number of underloads in the loading history. Eq. (3) provides the limit value of σ w , e ff , governed by the intrinsic threshold ∆ K th , e ff and initial crack size a init ; once the stress amplitude is lowered enough for the initial ∆ K to equal ∆ K th , e ff , no crack propagation occurs. It is worth noting that Eq. (3) corresponds to the intrinsic threshold line in the Kitagawa–Takahashi-diagram.

∆ K th , e ff 2 Y √ π a init

lim n →∞

σ w , e ff ( n ) =

(3)

A considerable di ff erence in the rate of saturation of σ w , e ff exists depending on the loading spike, or R-curve shape. The significant delay of the R-curve saturation corresponding to the large overload (Fig. 3(b)) translates into an increased sensitivity of σ w , e ff to these loading spikes; fatigue limit is reduced 30 % after only one large overload. This result is qualitatively in alignment with the experimental data from Pompetzki et al. (1990), who considered relatively large over- and underloads (near yield strength) and found overloads to be more e ff ective at lowering the fatigue limit. For most other loading spikes, a pure resetting of the R-curve (unaltered R-curve shape), corresponding to the underload case herein, can be assumed for a simple and conservative analysis. However, the question of how large of