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Complex thin-walled structures are commonly employed in turbine components to improve cooling efficiency. However, thin-walled blades are prone to thermal fatigue failure due to the high thermal stress induced by frequent and abrupt temperature fluctuations during duty cycles. The thermal fatigue behavior of Ni3Al-based single crystal alloy with thin-walled structure during 25 °C ↔ 1000 °C/1100 °C was investigated by combining thermal fatigue tests and finite element simulation. The thermal fatigue demonstrated a significant thickness debit effect, with thermal fatigue property dropping as peak temperature and wall thickness increased. Wall thickness had a more pronounced effect than peak temperature. Visible slip traces were found on the wall thickness surface, and their density was positively correlated with wall thickness. Further investigation revealed that thermal fatigue is primarily driven by the octahedral slip system ({111}), with significant contributions from (111)[101‾] and (1‾1‾1)[101]. Simulations of thermal stress distribution and J-integral at the crack tip during crack initiation and propagation stages indicated that wall thickness influences thermal fatigue properties by affecting thermal stress concentration at the notch and crack tip. Higher peak temperatures increased thermal stress and reduced yield strength, thus diminishing thermal fatigue performance. A thermal fatigue crack propagation prediction model was constructed incorporating Paris' law and the J-integral. The model revealed an unfavorable correlation between material Cj/ nj and both peak temperature/wall thickness, indicating that both factors adversely affect thermal fatigue resistance.

This work proposes a finite element (FE)‐based numerical model that uses the moving mesh technique for simulating fatigue crack propagation phenomena inside material components subjected to cyclic loads. Precisely, the computational mesh is adjusted during the numerical simulation according to conditions dictated by the criteria of fracture mechanics. Unlike standard FE procedures, which perform remeshing for each increment of the crack length, the proposed model updates the computational mesh only when FEs are distorted excessively because of the mesh nodes' motion, thus reducing computational burdens considerably. The validity of the proposed method is assessed through comparisons with existing experimental and numerical results. Highlights: An FE‐based model is developed for the fatigue analysis of material components.A moving mesh strategy is used to reproduce the fatigue crack growth.The fatigue life is estimated using an incremental crack extension approach.The Paris' law calculates the number of cycles at each crack length increment. [ABSTRACT FROM AUTHOR]

This work presents an elevated temperature fatigue study on additively manufactured (AM) commercially pure (CP) alpha Ti alloy with dispersed TiB particles. This study consists of both experimental and analytical investigations. Experimental data presented in this work were collected using a rotating beam fatigue (RBF) tester and a scanning electron microscope (SEM). The experimental findings presented in this work show that the AM Ti–TiB alloy lost its fatigue strength considerably at 350 °C compared to its reported room temperature fatigue strength. This loss in fatigue life is attributed to the numerous surface crack propagations. Moreover, a novel modification to the Paris’ law equation is introduced in this study. This modified equation is designed to predict fatigue life under elevated temperatures, especially in scenarios involving extensive major crack propagation. Notably, this modified equation improved the fatigue failure prediction accuracy, measured in terms of R2 value from 0.34 to 0.93.

The assessment of the fatigue crack growth rate behavior of welded structural components made of ultra-high strength steels (UHHS) is very important to gain a comprehensive understanding of these materials under high-cycle fatigue loadings and to enhance their applications in the construction industry. The literature on the fatigue crack growth rates of welded ultra-high strength steel plates with nominal yield strengths higher than 690 MPa is very scarce and only very limited research has been done thus far to provide the Paris' Law fatigue material constants for welded UHSS components in terms of the base metal (BM), the weld metal (WM), and the heat-affected zone (HAZ) regions, which are essential for fatigue life prediction. Hence, the fatigue crack growth rates of butt-welded UHSS plates with three grades (S700, S960, and S1100) are experimentally investigated in this study using Paris' Law for these three different regions of the welded components. Gas metal arc welding (GMAW) and laser welding (LW) methods are adopted to analyze the influence of welding methods on each steel grade with respect to their static and fatigue strength, microstructural changes, and hardness. In the fatigue crack growth rate test, it was observed that the S700 base material shows the lowest fatigue crack propagation resistance amongst these three grades. In welded components, laser welding outperforms gas welding in terms of the fatigue crack growth resistance. Comparisons of fatigue crack growth behaviors are made among all three tested UHSSs as well as with those from the literature. [ABSTRACT FROM AUTHOR]

The most important characteristics for service safety of welded joints are those describing crack initiation and growth caused by variable loading. Crack initiation and growth caused by variable loading is the subject of numerous investigations. This paper shows the determination of parameters of the fatigue crack for constituents of welded joints produced of high strength low alloyed steel. The crack growth law of Paris establishes the relation between the applied variable load quantity or the corresponding stress intensity factor range and crack growth per cycle. Results have shown that the position of the notch and crack initiation affect the values of the stress intensity range of fatigue threshold DKth and parameters in the Paris' equation.

Estimating the remaining useful life (RUL) of bearings from healthy to faulty is important for predictive maintenance. The bearing fault severity can be estimated based on the energy or root mean square (RMS) of vibration signals, and a stopping criterion can be set based on a threshold given by an ISO standard. However, the vibration RMS is often not monotonically increasing with damage, which renders a challenge for predicting the RUL. This study proposes a novel method for splitting the vibration signal into multiple frequency bands before RMS calculations to generate multiple health indicators. Monotonic health indicators are identified using the Spearman coefficient, and the RUL is afterward estimated for each indicator using a suitable model and parameter update scheme. Historical failure data is not required to set any parameters. The proposed method is tested with the Paris' law, where parameters are updated by particle filters. Experimental results from two test rigs validate the performance of the proposed method.

The fatigue crack propagation of the composite rock-concrete interface under constant-amplitude flexural fatigue loading below the initial cracking load was investigated in the present study. Fatigue pre-crack initiation and propagation tests were first conducted on three types of rock-concrete interfaces under three-point bend loading. It was shown that for the rock-concrete interface, the damage evolution throughout the fatigue loading process can be divided into deceleration and acceleration stages according to the rate curves of flexural stiffness, deformation and crack propagation. The crack propagation rate exhibited two distinct regions when plotted on a log scale, namely the small-crack growth region and the long-crack growth region. The small-crack growth region for the studied rock-concrete interfaces was not particularly pronounced. The long-crack growth region was the Paris' law region and included both the acceleration and part of the deceleration stages. Based on the experimental results, the fatigue crack propagation rate in the Paris' law region was then described using the empirical Paris' law. The exponent m 1 in the Paris' law was found to be approximately constant, with a value of m 1 ≈ 21 for the rock-concrete interface. The pre-factor C 1 was found to be dependent on both the load ratio R and the type of the rock-concrete interface (represented by an elastic mismatch parameter α). By introducing R and α into the empirical Paris' law, an adjusted Paris' law was proposed. The adjusted Paris' law can describe the fatigue crack propagation rate in a unified way for the same rock-concrete interface of different R and different interfaces in the scope of this research. Based on the adjusted Paris' law, the fatigue cycles during the crack propagation and fatigue lifetimes of all tested rock-concrete interface specimens were reasonably predicted. For practical applications, an approach was suggested involving a combination of the adjusted Paris' law and a material coefficient β (approximately β ≈ 0.143 in this study) to estimate the fatigue lifetimes of structures or specimens with existing pre-cracks at the rock-concrete interface. • Fatigue crack evolution can be divided into deceleration and acceleration stages. • Long-crack growth region includes both the acceleration and part of deceleration stages. • The exponent in the Paris' law equals to 21 for the rock-concrete interface. • An adjusted Paris' law is proposed to predict fatigue cycles in crack growth. [ABSTRACT FROM AUTHOR]

Fatigue crack propagation is an undesirable phenomenon that may lead to catastrophic failures in many components and structures, therefore it is important to understand its underlying mechanics. To that effect, systematic parametric studies of fatigue crack propagation laws are interesting to determine how fatigue life varies with the constants that define the mechanical behavior of a given material in a fatigue situation, such as the Paris’ law constants, fracture toughness (Kc) or the stress range ??. The parametric studies performed in the present work assess the influence of several parameters, assuming that failure occurs when K>Kc, but also when all the material ahead of the crack is yielding. It was found that m and C, the Paris’ law parameters, are the most influential parameters in terms of fatigue life. The present study should help future designers when choosing materials for components or structures subjected to cyclic loads.

• A mechanistic model for quantitative analysis and prediction of the fatigue crack growth rate of paving materials is proposed. • The derived J-integral serves as the driving force for crack growth and it is a function of the horizontal tensile stress, initial resilient modulus, damage density, specimen size, and surface energy of the paving materials. • The fatigue life of cement-treated aggregates is primarily influenced by the crack initiation stage, while for asphalt mixtures, the crack growth stage plays a more significant role. • The Paris' law coefficients exhibit minimal variation across different stress levels and loading frequencies, and can be utilized to evaluate the fatigue crack resistance of paving materials. Fatigue crack growth rate (FCGR) is a critical indicator that reflects the rate at which material deterioration occurs under cyclic loads. However, there is currently a lack of a mechanistic quantitative analysis and prediction of the FCGR of paving materials under external indirect tensile (IDT) cyclic loads. To address this issue, this study aims to develop a mechanistic model for quantitative analysis and prediction of the FCGR in paving materials subjected to external IDT cyclic loads based on the J-integral based Paris' law. Two types of paving materials, namely cement-treated aggregates and asphalt mixtures, were selected to demonstrate the results and model of the FCGR. The results indicate that the derived J-integral serves as the driving force for crack growth and is influenced by horizontal tensile stress, initial resilient modulus, damage density, specimen size, and surface energy of the paving materials. The fatigue life of cement-treated aggregates is primarily influenced by the crack initiation stage, while for asphalt mixtures, the crack growth stage plays a more significant role. The application of Paris' law is found to be effective in predicting the FCGR at stable crack growth stage. The coefficients of Paris' law exhibit minimal variation across different stress levels and loading frequencies, and can be utilized to evaluate the fatigue crack resistance of different paving materials. [ABSTRACT FROM AUTHOR]

Curling stresses in rigid pavements are the result of temperature gradient through the thickness of the concrete slab. These stresses have a periodic nature, and the induced cyclic flexural curling stresses lead to the cracking of the concrete slab. In a previous research by the authors, a phase change material (PCM)-rich concrete overlay was proposed as an effective method to control this failure mechanism. A PCM-rich concrete overlay is a concrete surface layer that contains a relatively high percentage of PCM. In the previous study, a phenomenological approach was employed, and it was demonstrated that the incorporation of PCM improves the curling fatigue performance of the concrete slab. Using fracture mechanics, this paper builds upon the findings of the previous study, and it serves as a mechanistic verification of the conclusions of the previous work. A modified Paris' Law was adopted to calculate the cumulative fatigue-fracture damage of the concrete slab due to cyclic thermally-induced curling stresses. The findings of this research demonstrated that the PCM-modified concrete slab undergoes a curling-induced fracture process that is less severe than the slab without the PCM. The impact of this reduction was quantified in terms of an improved fatigue-fracture performance of the pavement. [ABSTRACT FROM AUTHOR]

Paris' law is a commonly used formula in the assessment of fatigue life of structures. In the present study, we sought to improve the accuracy of this law in marine engineering using a probability model to experimentally assess fatigue life. First, a fatigue crack growth test was performed using compact tension specimens made of marine high-strength steel to obtain a reasonable range of material coefficients C and m in Paris' law. The material coefficient m was then determined as a constant, C followed a log-normal distribution, and Paris' law with different probabilities dependent on C was established. Finally, the probability model based on Paris' law was used to calculate fatigue life of a welded component with a surface crack caused by combined compressive and bending stresses. The results were then compared with the data from a fatigue test that used a crack front marking technique to depict the fractography. The findings of this study show a linear relationship between log C and m that caused a mutual constraint of the material coefficients. In summary, the structure fatigue life calculated by the probability mode based on Paris' law is reliable, thus it can greatly improve safety and accuracy of fatigue life assessment in marine engineering. [ABSTRACT FROM AUTHOR]

In order to alleviate the fatigue cracking problem of asphalt pavements. The development of large-stone asphalt mixtures (LSAM) is a worthwhile research direction. In this paper, different loading frequencies, test temperatures and stress ratios are designed to investigate the influence of semicircular bending (SCB) test conditions for LSAM fatigue properties. Comparative tests were also conducted with AC-13 fine-grained asphalt mixtures. The fatigue property of LSAM was analyzed by applying Paris' law, measuring fatigue crack length and using fatigue rupture energy theory. The test results indicated that fatigue life increased approximately 12 times with increasing loading frequency, 5 times with lower test temperatures, and 4 times with reduced stress ratio. Respectively, the magnitude of the single fatigue rupture energy and the total fatigue rupture energy also increased. Compared to AC-13 fine-grained asphalt mixtures, LSAM has a lower rate of rise in the displacement curve and smaller fracture displacement. After logarithmic transformation, the fatigue rupture energy and fatigue life of LSAM show an obvious linear relationship, specifically W f = 0.000806 N f 1.12975 . In summary, based on the SCB test conditions, LSAM exhibited superior fatigue performance compared to AC-13 fine-grained asphalt mixture. Additionally, LSAM demonstrated better resistance to fatigue loading-induced deformation. ● Investigation of fatigue properties of asphalt using semicircular bending test. ● Determined semicircular bending test parameters applicable to fatigue testing. ● Mathematical expression of fatigue rupture energy and fatigue life established. [ABSTRACT FROM AUTHOR]

Understanding the fatigue response of adhesive joints under cyclic shear stress is important to avoid fatigue failure of the bonded structures. Recently, a cohesive zone modelling (CZM) technique was developed where the cohesive properties of the elements were degraded by a proposed empirical relation. However, based on this degradation approach, the fatigue life of the joints should be known before running the analysis. The aim of the current study is to improve the numerical method in which the fatigue life of the joints will be automatically estimated during the analysis. To achieve this, the concepts of fracture mechanics are considered by using the Paris' law combined with the degradation model. A user material subroutine is developed to conduct the numerical calculations based on the concepts of CZM. End notched flexure (ENF) tests were carried out to evaluate the numerical data. Based on the results, it was found that the modified approach, in which the total fatigue life is obtained automatically, can be employed for fatigue life assessment of adhesive joints subjected to cyclic shear stresses. [ABSTRACT FROM AUTHOR]

Carbon fiber (CF) reinforced polyether ether ketone (PEEK) laminates are applied to the members of transportation structures exposed to severe conditions, because carbon fibers and PEEK have excellent heat-resistant and chemical stability. Therefore, it is important to ensure the long-term reliability of CF/PEEK laminates and evaluate the propagation of transverse cracks, which are generally the initial damage incurred by multidirectional laminates under fatigue loading. This study evaluated the detailed mechanism of transverse crack propagation toward the thickness direction in CF/PEEK quasi-isotropic laminates under fatigue loading. The transverse crack growth was observed using the replica method, and the energy release rate associated with the transverse crack propagation toward the thickness direction was calculated using a three-dimensional virtual crack closure-integral method. Thus, it was found that the transverse crack propagated in a stable manner until it passed through the 90° layer toward the thickness direction. Additionally, it was found that the crack growth rate increased in the range wherein the crack length was short, and maintained an approximately constant growth rate in the middle stage of propagation. According to Paris’ law, when the crack depth was relatively shallow, the experimental results are approximately consistent with the analytical evaluation results for the energy release rate associated with the transverse crack propagation toward the thickness direction. Finally, it is also concluded that the high ductility of the PEEK resin and good fiber/matrix interfacial property of the CF/PEEK laminates are important influencing factors for the stable and characteristic crack propagation observed in the experiment.

The present contribution investigates the crack‐size effects on Paris' law in accordance with dimensional analysis and intermediate asymptotics theory, which makes it possible to obtain a generalised equation able to provide an interpretation to the various empirical power‐laws available in the Literature. Subsequently, within the framework of fractal geometry, scaling laws are determined for the coordinates of the limit‐points of Paris' curve so that a theoretical explanation is provided to the so‐called short cracks problem. Eventually, the proposed models are compared with experimental data available in the literature which seem to confirm the advantage of applying a fractal model to the fatigue problem. [ABSTRACT FROM AUTHOR]

The characterization of fatigue crack propagation behavior is crucial for performance and reliability evaluation of aerospace materials. In this study, high-temperature (maximum: 650°C) fatigue crack propagation experiments of Ni-based superalloy GH4169 were conducted. The bi-prism-based single-lens 3D digital image correlation (BSL 3D DIC) technique was used to in-situ measure the displacement and strain fields near fatigue crack tip. Based on the deformation information, the mode I stress intensity factor range ΔK and the crack opening displacement (COD) were determined for characterizing the crack closure effect. As the major fatigue crack growth model, the parameters of modified Paris' law were obtained based on the effective stress intensity factor range ΔKeff and the fatigue crack propagation rate (FCPR). Additionally, two kinds of J integrals, JP (path integral method) and JK (stress intensity factor K method), were used to evaluate the small-scale yielding approximation. [ABSTRACT FROM AUTHOR]

• The crack growth performance of cement-treated construction and demolition waste was characterized by Paris law. • A new model for predicting the fatigue damage growth of cement-treated construction and demolition waste was proposed. • The fatigue damage characteristic of cement-treated construction and demolition waste as a base layer was analyzed by FEM. • The stress-dependent characteristic of cement cement-treated construction and demolition waste was evaluated. Fatigue damages of cement-treated construction and demolition waste (CDW) aggregate base materials were predicted using a nonlinear elastic-damage model coupled with a finite element (FE) program. CDW with two different gradations (BC1 and BC2) and one control cement-treated base material using quarried aggregate (N1) were tested. Three-point bending strength and fatigue tests were conducted to obtain the fracture property of the three base materials and inputs of the proposed nonlinear elastic-damage model. Pavement structures with the three base layers were modelled by FE simulations, to evaluate the mechanical and cracking performance under three different vehicle load levels of 700 kPa, 840 kPa and 1050 kPa, respectively. The result of experiments shows that the BC1 with a skeleton-dense gradation has the highest fracture strength, which was followed by the N1, and is much higher than the BC2. In the FE simulations, the proposed nonlinear elastic-damage model can well reflect the damage evolution and modulus distribution of cement-treated base layers under vehicle loading. With the accumulation of vehicle loading cycles, the damage of base layers increases, and pavement structure with the BC2 base layer shows the fastest damage evolution. Besides, the stress-dependent modulus shows a non-uniform distribution in the base layers, presenting very high values at the surface and bottom of the base layer due to the high compressive stress and tensile stresses. [ABSTRACT FROM AUTHOR]

With an increasing demand for adhesives, the durability of joints has become highly important. The fatigue resistance of adhesives has been investigated mainly for epoxies, but in recent years many other resins have been adopted for structural adhesives. Therefore, understanding the fatigue characteristics of these resins is also important. In this study, the cyclic fatigue behavior of a two-part acrylic-based adhesive used for structural bonding was investigated using a fracture-mechanics approach. Fatigue tests for mode I loading were conducted under displacement control using double cantilever beam specimens with varying bond-line thicknesses. When the fatigue crack growth rate per cycle, da/dN, reached 10−5 mm/cycle, the fatigue toughness reduced to 1/10 of the critical fracture energy. In addition, significant changes in the characteristics of fatigue crack growth were observed varying the bond-line thickness and loading conditions. However, the predominance of the adhesive thickness on the fatigue crack growth resistance was confirmed regardless of the initial loading conditions. The thicker the adhesive bond line, the greater the fatigue toughness.

A numerical lattice-based approach equipped with a brittle erosion algorithm and material forces in the context of LEFM and configurational mechanics is introduced to analyze contact fatigue life and crack propagation of brittle solids under low amplitude cyclic loading using the classical Paris's law. Material force vectors at lattice nodes were employed in identifying potential crack path in high-cycle fatigue applications. A 2D plane strain pad-substrate system with contact formulations under constant compression and cyclic sinusoidal shear with constant amplitude was considered to investigate fatigue crack's growth rate and its shape in the substrate, made of hydrided brittle Zircaloy-4 used in nuclear industry. The capability of the lattice in predicting the crack extension path using the material force vectors at the crack tip is verified by comparing with analytical solutions for a centered crack domain under pure shear and direct tension loading, and also for an initial small surface crack for the contact fatigue pad-substrate system. Obtaining the total fatigue life by assuming a failure value for the crack length and confirming the Paris' law crack growth, the lattice results demonstrate that the fatigue crack path in the substrate is a curved trajectory which gradually reduces as the fatigue crack tip grows deeper into the substrate. Having simple constitutive formulation and straightforward erosion algorithm, this approach together with configurational mechanics implications can be employed in analyzing fretting fatigue problems. [ABSTRACT FROM AUTHOR]

The fatigue crack growth rate of 0.3C–CrMoV (ESR) steel has been investigated on parent metal and weld under sinusoidal and block loading conditions to simulate the proof pressure testing and service conditions of rocket motor case. Parent metal exhibited higher resistance to fatigue crack propagation at lower stress intensities during sinusoidal loading, and as the stress intensity increased, the crack propagation rates in weld and parent were almost identical. Block loading introduced a plastic zone ahead of the crack tip, retarding the crack propagation rates. During block loading with continuous increase in stress intensity levels, crack growth rates were slow compared to the specimens tested at different ranges of initial stress intensity levels. The Paris' law constants C and m were evaluated. [ABSTRACT FROM AUTHOR]

Highlights • A fatigue crack length estimation technique is proposed for continuous monitoring. • The Paris' law parameters are estimated from an S-N curve through optimization. • The Paris' law parameters obtained from the S-N curve are validated experimentally. • The proposed technique estimated the crack lengths with errors less than 2 mm. Abstract This paper proposes an estimation technique of fatigue crack length for an aluminum 6061-T6 plate structure with a notch using a stress-life (S-N) curve. First, an S-N curve is obtained from multiple specimen tests by applying constant-amplitude cyclic loading with different stress levels. Next, the parameters of the Paris' law are estimated from the S-N curve by solving an optimization problem. Then, based on the Paris' law, the current crack length is continuously estimated by counting the elapsed fatigue loading cycles. The novelty of this study lies on the followings: (1) estimation of fatigue crack length using only the S-N curve for online fatigue crack monitoring, (2) estimation and experimental validation of the Paris' law parameters estimated from the S-N curve, and (3) experimental validation of the crack length estimation technique for notched aluminum 6061-T6 plates with various thicknesses. [ABSTRACT FROM AUTHOR]

Highlights • Employs a new higher order smoothing scheme that resolves the lack of accuracy observed when constant smoothing is employed. • Requires fewer integration points when compared to the conventional XFEM. • The proposed formulation is applied to simulate fatigue cracks in homogeneous isotropic materials. • Proposed method involves rational integration procedure employing Green's theorem. Abstract In this paper, the recently proposed linear smoothed extended finite element method (LSmXFEM) is employed to simulate the fatigue crack growth. Unlike the conventional extended finite element method, the LSmXFEM does not require special numerical integration technique to integrate the terms in the stiffness matrix. The stress intensity factors (SIFs) are evaluated by using the domain form of the interaction integral technique. The fatigue crack growth rate is evaluated using the generalized Paris' law in conjunction with the maximum hoop stress criterion. The robustness of the method is demonstrated with a few examples for which the results are available in the literature. Then, the fatigue crack growth from the numerical simulation is compared with the experimental investigations performed on CR5 grade cold formed steel. It is seen that the fatigue life and the crack path obtained from the proposed method is in close agreement with the experimental observation. [ABSTRACT FROM AUTHOR]

This study analyzes crack growth behavior of woven carbon fiber reinforced plastics under tension–tension fatigue varying the stress ratio (R). The composite studied here is 3K‐70‐P carbon fiber plain weave with EPOLAM 2015® epoxy resin, manufactured by vacuum bagging procedure. The modified París' law for composites is used to estimate the delamination crack growth (da/dN) in terms of interlaminar fracture toughness ratio (Gmax/GIC). Delamination's dependence on R factor is discussed, as crack growth rate increases when stress ratio becomes higher. Also, fractographic examination is provided, as fatigue striation density is inversely dependent on fatigue stress ratio. [ABSTRACT FROM AUTHOR]

As often done in design practice, the Wöhler curve of a specimen, in the absence of more direct information, can be crudely retrieved by interpolating with a power-law curve between static strength at a given conventional low number of cycles N0 (of the order of 10-103), and the fatigue limit at a “infinite life”, also conventional, typically N∞=2·106 or N∞=107 cycles. These assumptions introduce some uncertainty, but otherwise both the static regime and the infinite life are relatively well known. Specifically, by elaborating on recent unified treatments of notch and crack effects on infinite life, and using similar concepts to the static failure cases, an interpolation procedure is suggested for the finite life region. Considering two ratios, i.e. toughness to fatigue threshold FK=KIc/DKth, and static strength to endurance limit, FR =sR /Ds0, qualitative trends are obtained for the finite life region. Paris’ and Wöhler’s coefficients fundamentally depend on these two ratios, which can be also defined “sensitivities” of materials to fatigue when cracked and uncracked, respectively: higher sensitivity means stringent need for design for fatigue. A generalized Wöhler coefficient, k’, is found as a function of the intrinsic Wöhler coefficient k of the material and the size of the crack or notch. We find that for a notched structure, k