The theory of critical distances: a review of its applications in fatigue

This paper attempts to review the most interesting findings in the use of the theory of critical distances (TCD) to predict fatigue strength of notched mechanical components. Initially, the most modern formalisations of the TCD are considered, showing their peculiarities and differences. An ad-hoc section is then focused on the multiaxial high-cycle fatigue problem, considering all the open questions arising in the presence of complex stress fields damaging the fatigue process zone in the vicinity of the stress concentrator apex. Subsequently, the physical idea on the structural volume concept is briefly investigated showing some peculiar results generated in the high-cycle fatigue regime under both uniaxial and biaxial fatigue loading. Finally, our idea to extend the use of the TCD down to the low-medium cycle fatigue regime is briefly explained.Working in collaboration with Prof. David Taylor, we have spent the last five years investigating this theory both to better understand its physical meaning and to systematically check its accuracy in predicting notch fatigue strength under different loading conditions. After so much work done in this area we feel so confident to proudly and loudly say that the TCD is a powerful engineering tool suitable for assessing real mechanical components in situations of practical interest. Finally, it can be highlighted also that the best TCD formalisations were seen to be those based on the use of linear-elastic stresses. This suggests that such a theory can successfully be used to post-process simple linear-elastic finite element (FE) models reducing time and costs of the design process.     

A novel formulation of the theory of critical distances to estimate lifetime of notched components in the medium-cycle fatigue regime

In the present paper, the theory of critical distances (TCD) is reformulated in order to make it suitable for predicting fatigue lifetime of notched components in the medium-cycle fatigue regime. This extension of the TCD takes as its starting point the idea that the material characteristic length, L, changes as the number of cycles to failure, N_f, changes. In order to define the L versus N_f relationship two different strategies were investigated. Initially, we attempted to determine it by using the L values calculated considering material properties defined at the two extremes, namely static failure and the fatigue limit. This strategy, though correct from a philosophical point of view, contained some problems in its practical application. We subsequently attempted to determine the L versus N_f relationship by means of two calibration fatigue curves; (one generated by testing plain specimens and the second one generated by testing notched specimens). This second strategy was found to be much more simple to apply to practical problems, resulting in estimations characterized by a higher accuracy. The reliability of the devised method was systematically checked by using experimental results generated by testing notched specimens of low-carbon steel containing different geometrical features and tested using various loading types, stress ratios and specimen thicknesses. The accuracy of the method was further verified by using several data sets taken from the literature. Our method was seen to be successful giving predictions falling always within the scatter band of the data from the parent material. These results are very interesting, especially considering that the TCD is very easy to use because it requires only a linear-elastic stress analysis.     

On the prediction of high-cycle fretting fatigue strength: Theory of critical distances vs. hot-spot approach

This paper summarises the results we obtained when applying both the theory of critical distances (TCD) and the hot-spot approach to predict high-cycle fretting fatigue strength. In particular, the accuracy of such approaches was checked considering some experimental results taken from the literature and ad hoc generated to explicitly investigate the size effect in fretting fatigue. According to the well-known experimental outcome that initiation and initial growth of fretting fatigue cracks is mainly mixed-mode dominated, both the TCD and the hot-spot approach were used in conjunction with two different multiaxial fatigue criteria: the so-called modified Wöhler curve method (MWCM), that is, a conventional critical plane approach, and the well-known mesoscopic criterion due to Dang Van. Considering cylindrical-on-flat contacts tested under partial slip conditions, it was seen that the TCD is successful in predicting the size effect in fretting fatigue, resulting in more accurate predictions than those obtained by applying the classical hot-spot approach. Moreover, the present study revealed that the overall best accuracy is obtained when applying the TCD along with the MWCM. This result is very promising, especially in light of the fact that such a design methodology can be employed by simply post processing linear-elastic FE results, making it suitable for being used to assess real mechanical assemblies without the need for carrying out time-consuming elasto-plastic analyses.     

Mean stress and plasticity effect prediction on notch fatigue and crack growth threshold,combining the theory of critical distances and multiaxial fatigue criteria

In the present work, we propose a robust calibration of some bi‐parametric multiaxial fatigue criteria applied in conjunction with the theory of critical distances (TCD). This is based on least‐square fitting fatigue data generated using plain and sharp‐notched specimens tested at two different load ratios and allows for the estimation of the critical distance according to the point and line method formulation of TCD. It is shown that this combination permits to incorporate the mean stress effect into the fatigue strength calculation, which is not accounted for in the classical formulation of TCD based on the range of the maximum principal stress. It is also shown that for those materials exhibiting a low fatigue‐strength‐to‐yield‐stress ratio σ_fl,R = ?1/σ_YS, such as 7075‐T6 (σ_fl,R = ?1/σ_YS = 0.30), satisfactorily accurate predictions are obtained assuming a linear‐elastic stress distribution, even at the tip of sharp notches and cracks. Conversely, for any materials characterized by higher values of this ratio, as quenched and tempered 42CrMo4 (σ_fl,R = ?1/σ_YS = 0.54), it is recommended to consider the stabilized elastic‐plastic stress/strain distribution, also for plain and blunt‐notched samples and even in the high cycle fatigue regime still with the application of the TCD.     

The theory of critical distances and fatigue from notches in aluminium 6061

Fatigue failure of metal components containing notches, cracks and other defects has been an active research topic for many years because of its important practical and theoretical implications. Recently, Taylor and his colleagues have re‐visited this topic and proposed the theory of critical distance (TCD), which summarizes the early work by Neuber, Peterson and others in a unifying theory and predicts fatigue fracture with the use of a critical distance, L_o. In this paper, an experimental and numerical study of the fatigue of notched and un‐notched 6061 aluminium alloys is used to verify the TCD and some of the limitations of the TCD are discussed on this basis.     

Assessment of notched structural steel components using failure assessment diagrams and the theory of critical distances

When the structural integrity of notched components is analysed, it is generally assumed that notches behave as cracks, something which generally provides overconservative results. The proposal of this paper consists, on the one hand, in the application of the theory of critical distances for the estimation of the notch fracture toughness and, therefore, for the conversion of the notched situation into an equivalent cracked situation in which the material develops a higher fracture resistance. On the other hand, once the notch fracture toughness has been defined, the assessment is performed using the failure assessment diagram methodology, and assuming that the notch effect on the limit load is negligible. The methodology has been applied to 336 CT notched fracture specimens made of two different structural steels, covering temperatures from the corresponding lower shelf up to the upper shelf, providing satisfactory results and a noticeable reduction in the overconservatism derived from the analyses in which the notch effect is not considered.     

Fatigue crack initiation prediction of cope hole details in orthotropic steel deck using the theory of critical distances

Orthotropic steel decks are vulnerable to fatigue cracking in welded connections and complex geometrical details. A total of three fatigue tests were conducted on segments of orthotropic steel deck to evaluate the fatigue performance of trough‐to‐crossbeam connections with various cut‐out configurations. In the tests, the specimens were subjected to cyclic three‐point bending load and the fatigue cracks were more likely to initiate from the cope holes in the crossbeam web rather than the trough‐to‐crossbeam fillet welds. Three‐dimensional finite element models (FEM) of the specimens were built and validated by the measured deflections and stresses. Using the validated FEM, the characteristic stresses based on the theory of critical distances (TCD) were calculated for the stress concentrations along the cope holes. The fatigue crack initiation life, predicted by the TCD‐based stress combined with the plain material S–N curve, agreed reasonably with the fatigue test results. The TCD method could further form a basis of fatigue crack propagation analysis using the fracture mechanics approaches.     

A fretting damage correction factor applicable to the McDiarmid criterion of plain high-cycle fatigue

This paper presents an assessment of the performance of a set of multi-axial high-cycle fatigue criteria on the basis of a series of fretting fatigue experiments. We carried out tests on a creep-resistant chromium steel material used for steam-turbine blades. The first type of experiment employed the classical cylinder-on-flat geometry with flat dog-bone specimens. The second set of experiments adopted dovetail geometry. Various loads were applied in order to capture a wide range of contact slip amplitudes. A set of eight plain multi-axial fatigue criteria was applied to the numerically simulated stress response in the contacts during a single load cycle. Methods, which originated in the so-called theory of critical distances, were used for correcting the results in order to take the stress gradient effect into account. A simple factor based on slip amplitudes is introduced in order to consider the surface damage and is calibrated for the McDiarmid method. This criterion provided the best estimates of the most probable cracking sites.     

On the application of multiaxial high-cycle fatigue criteria using the theory of critical distances

This paper proposes that the application of multiaxial fatigue criteria in terms of the theory of critical distances requires the use of a distance which may be different from the widely adopted value given by half of the El Haddad’s intrinsic crack length. Three criteria (Modified Wöhler Curve Method, Crossland and Dang Van) are evaluated at the appropriate critical distance and compared with experimental data obtained from specimens containing small and/or sharp notches under proportional loading. The Modified Wöhler Curve Method provided the best estimates. It is also shown that this theory can be extended to the prediction of the loading ratio, R, effect on the threshold stress intensity factor range for propagation of long cracks.     

Prediction methods of fatigue critical point for notched components under multiaxial fatigue loading

Two methods based on local stress responses are proposed to locate fatigue critical point of metallic notched components under non‐proportional loading. The points on the notch edge maintain a state of uniaxial stress even when the far‐field fatigue loading is multiaxial. The point bearing the maximum stress amplitude is recognized as fatigue critical point under the condition of non‐mean stress; otherwise, the Goodman's empirical formula is adopted to amend mean stress effect prior to the determination of fatigue critical point. Furthermore, the uniaxial stress state can be treated as a special multiaxial stress state. The Susmel's fatigue damage parameter is employed to evaluate the fatigue damage of these points on the notch edge. Multiaxial fatigue tests on thin‐walled round tube notched specimens made of GH4169 nickel‐base alloy and 2297 aluminium‐lithium alloy are carried out to verify the two methods. The prediction results show that both the stress amplitude method and the Susmel's parameter method can accurately locate the fatigue critical point of metallic notched components under multiaxial fatigue loading.     

Multiaxial fatigue limits and material sensitivity to non-zero mean stresses normal to the critical planes

This paper is concerned with an attempt to reformulate the so-called Modified Wöhler Curve Method (MWCM) in order to more efficiently account for the detrimental effect of non-zero mean stresses perpendicular to the critical planes. In more detail, by taking as a starting point the well-established experimental evidence that engineering materials exhibit different sensitivities to superimposed tensile static stresses, an effective value of the normal mean stress relative to the critical plane was attempted to be calculated by introducing a suitable correction factor. Such a mean stress sensitivity index was assumed to be a material constant, i.e. a material parameter to be determined by running appropriate experiments. The accuracy of the novel reformulation of the MWCM proposed here was systematically checked by using several experimental data taken from the literature. In particular, in order to better explore the main features of the improved MWCM, its accuracy in estimating multiaxial high-cycle fatigue damage was evaluated by considering fatigue results generated not only under non-zero mean stresses but also under non-proportional loading. Such a validation exercise allowed us to prove that the systematic use of the mean stress sensitivity index resulted in estimates falling within an error interval equal to about ±10%, and this held true independently of considered material and complexity of the investigated loading path. Finally, such a novel reformulation of the MWCM was also applied along with the Theory of Critical Distances (TCD) to predict the high-cycle fatigue strength of notched samples tested under in-phase bending and torsion with superimposed tensile and torsional static stresses: again our method was seen to be highly accurate, correctly predicting high-cycle multiaxial fatigue damage also in the presence of stress concentration phenomena.     

The critical distance theory for fatigue analysis of notched aluminium specimens subjected to repeated bending

This paper provides new insights in the use of the critical distance method for fatigue analysis of notched aluminium components subjected to constant amplitude bending loading. A straightforward test setup was developed to load test samples with different stress concentrations in repeated bending at high test frequency. The mean values of the local endurable stress amplitudes are determined with the staircase method and the Dixon and Mood theory using a minimum amount of test samples. The critical distance is determined using these fatigue limits and the corresponding stress gradients determined by means of finite element analysis. The results indicate a unique critical distance of 0.22 mm for fatigue crack initiation. Consequently, the critical distance theory can be successfully applied for fatigue analysis of notched specimens or engineering components of aluminium EN AW 7075 T7351 with geometrical features of various size and shape subjected to fluctuating loading in bending.     

Sensitivity of direct metal laser sintering Maraging steel fatigue strength to build orientation and allowance for machining

This work deals with the effect of build orientation and of allowance for machining on DMLS‐produced Maraging Steel MS1. The experimental results, arranged by tools of Design of Experiment, have been statistically processed and compared. The outcomes were that, probably due to effect of the thermal treatment, machining, and material properties, the aforementioned factors do not have a significant impact on the fatigue response. This made it possible to work out a global curve that accounts for all the results, consisting in a high amount of data points. This can be regarded as one of the most complete and reliable fatigue models in the current literature. Fractographic and micrographic studies have been performed as well, to individuate the crack initiation points, usually located at subsurface porosities, and to investigate the location of internal inclusions and the actual martensitic microstructure along the stacking direction and on the build plane.     

An improved critical plane and cycle counting method to assess damage under variable amplitude multiaxial fatigue loading

The plane with the maximum variance of the resolved shear stress is taken as the critical plane. Two algorithms are used along with the maximum variance method (MVM) to determine the orientation of the critical plane. The maximum variance of the normal stress on the potential critical planes is calculated to determine the one experiencing the maximum extent of fatigue damage. A new multiaxial cycle counting method is proposed to count cycles on the critical plane. The modified Wöhler curve method is used to assess fatigue damage. About 200 experimental results were collected from the technical literature to validate the approaches being proposed. The results show that the improved design technique being proposed is successful in assessing fatigue damage under variable amplitude multiaxial cyclic loading.     

Application of additive manufacturing techniques in sports footwear

Additive manufacturing (AM) enables faster prototype development for design visualisation, performance studies and personalisation in the sports footwear industry. Among the available AM techniques, stereolithography (SLA), PolyJet (PJ), selective laser sintering (SLS) and three-dimensional printing (3DP) have been used for sports footwear prototyping. A five-point scoring system was used to rate the performance of AM techniques in four important characteristics namely accuracy, surface finish, range of materials supported and building time. Key elements of AM-based footwear personalisation and customisation methodology were also discussed.     

The role of manufacturing defects in munition component failures

The U.S. Army Research Laboratory, Weapons and Materials Research Directorate performs numerous failure analysis investigations on munition-related components. Many of the failures are attributable to defects that can be traced back to the manufacturing process. Typical manufacturing defects encountered include those associated with the material, forging, casting, welding, and heat treatment processes. Dimensional anomalies have also been noted. Munition component failures are very costly and may seriously affect the safety and readiness of the armed forces. Additionally, repeated failure may lead to the grounding (removal from service) of a system, depending on the severity of the problem. Specific examples of component defects/failures discussed in this report include bomb fin retaining bands, general purpose bomb suspension lugs, missile launcher attachment bolts, cluster bomb tailcones, general purpose bomb fins, and Gatling gun breech bolt assemblies. This paper focuses on the importance of proper manufacturing techniques to the munitions industry and, by inference, to numerous other industries.     

Effect of hollow pores on the detection characteristics of trans-stilbene crystals grown by vertical Bridgman technique

Bulk crystals of trans-stilbene were grown using Vertical Bridgman Technique with progressive modification of growth parameters. The grown crystals were studied for their performance in time resolution setup with two anti collinear gamma rays from ²²Na sources. The crystals were subjected to optical microscopic and Scanning Electron Microscopic (SEM) studies. The microscopic defects (hollow pores) in the grown crystals and their effect on the detection characteristics are reported.     

Additive Manufacturing of Metal Structures at the Micrometer Scale

Currently, the focus of additive manufacturing (AM) is shifting from simple prototyping to actual production. One driving factor of this process is the ability of AM to build geometries that are not accessible by subtractive fabrication techniques. While these techniques often call for a geometry that is easiest to manufacture, AM enables the geometry required for best performance to be built by freeing the design process from restrictions imposed by traditional machining. At the micrometer scale, the design limitations of standard fabrication techniques are even more severe. Microscale AM thus holds great potential, as confirmed by the rapid success of commercial micro‐stereolithography tools as an enabling technology for a broad range of scientific applications. For metals, however, there is still no established AM solution at small scales. To tackle the limited resolution of standard metal AM methods (a few tens of micrometers at best), various new techniques aimed at the micrometer scale and below are presently under development. Here, we review these recent efforts. Specifically, we feature the techniques of direct ink writing, electrohydrodynamic printing, laser‐assisted electrophoretic deposition, laser‐induced forward transfer, local electroplating methods, laser‐induced photoreduction and focused electron or ion beam induced deposition. Although these methods have proven to facilitate the AM of metals with feature sizes in the range of 0.1–10 µm, they are still in a prototype stage and their potential is not fully explored yet. For instance, comprehensive studies of material availability and material properties are often lacking, yet compulsory for actual applications. We address these items while critically discussing and comparing the potential of current microscale metal AM techniques.