A critical assessment of methods for the determination of the shear stress amplitude in multiaxial fatigue criteria belonging to critical plane class

Multiaxial high cycle fatigue criteria based on the critical plane approach necessitate unambiguous definitions of the amplitude and mean value of the shear stress (τ_a and τ_m) acting on the material planes. Four of the existing definitions relate the values of τ_a and τ_m to a geometrical element of the curve described by the tip of the shear stress vector (curve Ψ), respectively, the radius of the Minimum Circumscribed Circle, the Longest Chord, the Longest Projection, the diagonal of the Maximum Rectangular Hull (MRH).In this paper a critical assessment of the above definitions is proposed, focusing on that based on the concept of MRH, which is the most recently developed. The main issues of the comparison are the uniqueness of the solution in the determination of τ_a and τ_m, the ability to differentiate proportional and non-proportional stresses, the differences of the values of τ_a obtained by each of the 4 methods for differently shaped curves Ψ.     

Efficiency of algorithms for shear stress amplitude calculation in critical plane class fatigue criteria

Fatigue criteria that belong to the critical plane class necessitate unambiguous definitions of the amplitude and mean value of the shear stress acting on a material plane. This is achieved through the construction of the minimum circle circumscribing the path described by the tip of the shear stress vector on each plane. By definition, the centre and the radius of this circle provide the mean shear stress and the shear stress amplitude, respectively. The search of the minimum enclosing circle is an optimisation problem for which efficient numerical solution schemes are required. Several algorithms exist for similar situations; however these are not necessarily related to the fatigue strength of metals. In this paper some algorithms are studied to assess their computational efficiency within the engineering framework of the application of fatigue criteria of the critical plane type.     

Determination of the critical plane by a weight-function method based on the maximum shear stress plane under multiaxial high-cycle loading

A weight function method for the determination of the critical plane is here proposed for the case of specimens under combined bending and torsion in the high cycle fatigue regime. The critical plane is assumed to be coincident with the mean maximum absolute shear stress plane, which is calculated by averaging the instantaneous angle between the specimen axis and the normal to the maximum absolute shear stress plane. Two kinds of weight functions are proposed to determine such a plane. The proposed method to determine the critical plane is verified by employing fatigue data available in the literature in terms of experimental fracture planes, and the multiaxial fatigue life is also predicted by a reformulation of the criterion proposed by Carpinteri et al. to verify the determined critical plane. The results show that the proposed method can be applied to determine the critical plane under both constant and variable amplitude loading.     

Estimating the orientation of Stage I crack paths through the direction of maximum variance of the resolved shear stress

Our formalisation of the Shear Stress-Maximum Variance Method takes as a starting point the hypothesis that, in ductile materials subjected to fatigue loading, the crack initiation planes, i.e. the so-called Stage I planes, are those containing the direction experiencing the maximum variance of the resolved shear stress. From a computational point of view, the most remarkable implication of the above assumption is that, as soon as the variance and covariance terms characterising the considered load history are known, the effective time needed to estimate the orientation of the critical plane does not depend on the length of the load history itself. Further, such a computational efficiency is seen to be associated with an high-level of accuracy in estimating fatigue lifetime of both plain and notched engineering components, this holding true under constant as well as under variable amplitude uniaxial/multiaxial fatigue loading. In this scenario, by assuming that the orientation of Stage I planes can directly be determined through the orientation of Stage II crack paths, the present paper investigates whether, independently from the degree of multiaxiality and non-proportionality of the applied loading history, the direction of maximum variance of the resolved shear stress is also capable of accurately estimating the orientation of Stage I crack paths.     

Multiaxial fatigue damage parameter and life prediction for medium-carbon steel based on the critical plane approach

The tension–torsion fatigue characteristics were investigated under proportional and non-proportional loading in this paper. The fatigue cracks on the surface of multiaxial fatigue specimens were observed and analyzed by a scan electron microscope. On the basis of the investigation on the Kindil–Brown–Miller and Fatemi–Socie’s critical plane approaches, a shear strain based multiaxial fatigue damage parameter was proposed by von Mises criterion based on combining the maximum shear strain and the normal strain excursion between adjacent turning points of the maximum shear strain on the critical plane. The proposed multiaxial fatigue damage parameter does not include the weight constants. According to the proposed multiaxial fatigue damage parameter, the multiaxial fatigue life prediction model was established with the Coffin–Manson equation, which is used to predict the multiaxial fatigue life of medium-carbon steel. The results showed that the proposed multiaxial fatigue damage parameter could be used under either multiaxial proportional or non-proportional loading.     

The equivalent stress on the critical plane determined by the maximum covariance of normal and shear stresses

The paper presents a new form of the stress criterion of multiaxial random fatigue. The criterion has been defined as a sum of normal and shear stresses with weight coefficients on the critical plane defined as the plane where the maximum covariance between shear and normal stresses occurs. Two forms of the criterion have been proposed.     

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.     

Proportional/nonproportional constant/variable amplitude multiaxial notch fatigue: cyclic plasticity,non‐zero mean stresses,and critical distance/plane

This paper deals with the formulation and experimental validation of a novel fatigue lifetime estimation technique suitable for assessing the extent of damage in notched metallic materials subjected to in‐service proportional/nonproportional constant/variable amplitude multiaxial load histories. The methodology being formulated makes use of the Modified Manson‐Coffin Curve Method, the Shear Strain–Maximum Variance Method, and the elasto‐plastic Theory of Critical Distances, with the latter theory being applied in the form of the Point Method. The accuracy and reliability of our novel fatigue lifetime estimation technique were checked against a large number of experimental results we generated by testing, under proportional/nonproportional constant/variable amplitude axial‐torsional loading, V‐notched cylindrical specimens made of unalloyed medium‐carbon steel En8 (080M40). Specific experimental trials were run to investigate also the effect of non‐zero mean stresses as well as of different frequencies between the axial and torsional stress/strain components. This systematic validation exercise allowed us to demonstrate that our novel multiaxial fatigue assessment methodology is remarkably accurate, with the estimates falling within an error factor of 2. By modelling the cyclic elasto‐plastic behaviour of metals explicitly, the design methodology being formulated and validated in the present paper offers a complete solution to the problem of estimating multiaxial fatigue lifetime of notched metallic materials, with this holding true independently of sharpness of the stress/strain raiser and complexity of the load history.     

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.     

A novel engineering method based on the critical plane concept to estimate the lifetime of weldments subjected to variable amplitude multiaxial fatigue loading

This paper summarizes an attempt at proposing a new engineering method suitable for estimating the fatigue lifetime of steel‐ and aluminium‐welded connections subjected to variable amplitude multiaxial fatigue loading. In particular, the proposed approach is based on the use of the so‐called Modified Wöhler Curve Method (MWCM), i.e. a bi‐parametrical critical plane approach, whose accuracy has been checked so far solely in addressing the constant amplitude multiaxial fatigue problem. In order to extend the use of our criterion to variable amplitude situations, the critical plane is suggested here as being determined by taking full advantage of the maximum variance concept, that is, such a plane is assumed to be the one containing the direction along which the variance of the resolved shear stress reaches its maximum value. The main advantage of such a strategy is that the cycle counting can directly be performed by considering the shear stress resolved along the maximum variance direction: by so doing, the problem is greatly simplified, allowing those well‐established cycle counting methods specifically devised to address the uniaxial variable amplitude problem to be extended to those situations involving multiaxial fatigue loading. The validity of the proposed methodology was checked by using two different datasets taken from the literature and generated by testing both steel and aluminium tube‐to‐plate welded connections subjected to in‐phase and 90° out‐of‐phase variable amplitude bending and torsion. This new fatigue life assessment technique was seen to be highly accurate allowing the estimates to fall within the calibration scatter bands not only when the constants in the governing equations were calculated by using the experimental uniaxial and torsional fully reversed fatigue curves, but also when they were determined by using the reference curves supplied, for the investigated geometry, by the available standard codes. These results seem to strongly support the idea that, thanks to its peculiar features, our method can be considered as an effective engineering approach capable of performing multiaxial fatigue assessment under variable amplitude loading which fully complies with the recommendations of the available standard codes.     

Notch and mean stress effect in fatigue as phenomena of elasto-plastic inherent multiaxiality

The present paper summarises an attempt of estimating fatigue lifetime of notched metallic materials by directly accounting for the degree of multiaxiality of the local elasto-plastic stress/strain-fields acting on the fatigue process zone. In more detail, the proposed approach takes as its starting point the assumption that Stage I is the most important stage to be modelled to accurately estimate fatigue damage, and this holds true independently of the sharpness of the assessed geometrical feature. According to this initial idea, and by taking full advantage of the so-called Modified Manson-Coffin Curve Method (MMCCM), the hypothesis is then formed that the crack initiation plane is always coincident with that material plane experiencing the maximum shear strain amplitude. Subsequently, to devise an efficient design method capable of taking into account the detrimental effect of stress/strain gradients arising also from severe stress/strain concentration phenomena, the MMCCM is suggested here as being applied in terms of the Theory of Critical Distances (TCD), the latter being used in the form of the Point Method (PM). Further, in light of the well-known fact that the value of the mean stress/strain components in the vicinity of the stress/strain raisers’ apices can be different from the corresponding nominal values due to the actual elasto-plastic behaviour of the material being assessed, it is shown, through the MMCCM itself, that also the mean stress effect can directly and accurately be treated as a problem of inherent multiaxiality. Finally, as a preliminary validation, the accuracy and reliability of the proposed approach is checked through several experimental results taken from the literature and generated by testing, under uniaxial fatigue loading, samples containing a variety of geometrical features, the effect of different nominal load ratios being investigated as well.     

Criteria of multiaxial random fatigue based on stress,strain and energy parameters of damage in the critical plane

In this paper generalized criteria of multiaxial random fatigue based on stress, strain and strain energy density parameters in the critical plane have been discussed. The proposed criteria reduce multiaxial state of stress to the equivalent uniaxial tension–compression or alternating bending. Relations between the coefficients occurring in the considered criteria have been derived. Thus, it is possible to take into account fatigue properties of materials under simple loading states during determination of the multiaxial fatigue life. Presented models have successfully correlated fatigue lives of cast iron GGG40 and steel 18G2A specimens under constant amplitude in‐phase and out‐of‐phase loadings including different frequencies.     

On the interaction of normal and shear stresses in multiaxial fatigue damage

One of the important issues in assessing multiaxial fatigue damage is interactions between different components of stress such as normal and shear stresses. The present study investigated this interaction effect on the fatigue behavior of materials with shear failure mode when subjected to multiaxial loading conditions. A method is introduced to model this interaction based on the idea that two types of influence are caused by the normal stress acting on the critical plane orientation. These two types of influence are affecting roughness induced closure, as well as fluctuating normal stress which affects the growth of small cracks in mode II. Shear‐based critical plane damage models which use normal stress as a secondary input, such as FS damage model, could then use the summation of these terms. In order to investigate the effect of the method, constant amplitude load paths with different levels of interaction between the normal and shear stresses, as well as variable amplitude tests with histories both taken from service loading conditions and generated using random numbers were designed for an experimental program. The proposed method was observed to result in improved fatigue life estimations where significant interactions between normal and shear stresses exist.     

High cycle fatigue analysis in presence of residual stresses by using a continuum damage mechanics model

This study attempts to predict the high cycle fatigue life of steel butt welds by numerical method. At first, FE simulation of plate butt welding is carried out to obtain the weld-induced residual stresses employing sequentially coupled three-dimensional (3-D) thermo-mechanical FE formulation. Then, a nonlinear damage cumulative model for multiaxial high cycle fatigue based on continuum damage mechanics (CDM), which can incorporate the effect of welding residual stresses, is derived using FE technique. The high cycle fatigue damage model is applied to the butt welds subjected to cyclic fatigue loading to calculate the fatigue life considering the residual stresses, and the computed total fatigue life which takes into account the fatigue crack initiation and the propagation is compared with the test result. In addition, the fatigue life prediction of the welds without considering the residual stresses is implemented to investigate the influence of welding residual stresses on the fatigue performance. The FE results show that the high cycle fatigue damage model proposed in this work can predict the fatigue life of steel butt welds with high accuracy, and welding residual stresses should be taken into account in assessing the fatigue life of the welds.     

Use of an energy‐based/critical plane model to assess fatigue life under low‐cycle multiaxial cycles

For engineering components subjected to multiaxial loading, fatigue life prediction is crucial for guaranteeing their structural security and economic feasibility. In this respect, energy‐based models, integrating the stress and strain components, are widely used because of their availability in fatigue prediction. Through employing the plastic strain energy concept and critical plane approach, a new energy‐based model is proposed in this paper to evaluate the low‐cycle fatigue life, in which the critical plane is defined as the maximum damage plane. In the proposed model, a newly defined NP factor κ^*  is used to quantify the nonproportional (NP) effect so that the damage parameter can be conveniently calculated. Moreover, a simple estimation method of weight coefficient is developed, which can reflect different contributions of shear and normal plastic strain energy on total fatigue damage. Experimental data of 10 kinds of materials are employed to assess the effectiveness of this model as well as three other energy‐based models.     

Numerical algorithms for cyclic phase transformation hysteresis in a shape memory plate subject to axisymmetric plane stress

We present numerical algorithms for calculating stress fields in an annulus composed of a shape memory material under conditions of quasi‐static edge loading at constant temperature. The algorithms track the material microstructure in terms of the volume fraction of austenite (A) and martensite (M), the latter of which provides a transformation strain. The dependence on load path imparts significant hysteresis in the stress induced transformation between A and M. A previous study that was restricted to proportional loading in the direction of forward transformation (J. Appl. Mech. 2005; 72 :44–53) is here generalized to consider arbitrary loadings. The shooting algorithm that was robust for the previously considered proportional loadings is found to be subject to numerical instability for the most general transformation possibilities considered here. This motivates the development of an alternative iterated mapping algorithm that is found to generate a robust semi‐analytical finite difference procedure. The algorithm efficiently determines the operative transformation type, as is illustrated in cases where forward and reverse loading are occurring simultaneously at different plate locations. At those locations where phase transformation is inactive, the algorithm continues to account for martensite reorientation that alters the local transformation strain. Copyright © 2006 John Wiley & Sons, Ltd.     

The development of a naturalistic data collection system to perform critical incident analysis: an investigation of safety and fatigue issues in long-haul trucking

Traditionally, both epidemiological and empirical methods have been used to assess driving safety. This paper describes an alternative, hybrid, naturalistic approach to data collection that shares advantages with each traditional approach. Though this naturalistic approach draws on elements of several safety techniques that have been developed in the past, including the Hazard Analysis Technique, instrumented vehicle studies, and fleet studies of driving safety interventions, it has a number of unique elements. Sophisticated instrumented vehicles collected over 400,000 km of commercial vehicle data to address the long-haul trucking application described in this paper. The development of this data collection and analysis method and data collection instrumentation has resulted in a set of valuable tools to advance the current state-of-the-practice in driving safety assessment. An application of this unique approach to a study of long-haul truck driver performance, behavior, and fatigue is described herein.     

Analysis of the influence of the level of residual stresses on the ultimate stresses in a cycle for welded structures according to the results of testing of small-size specimens

We propose a procedure for the determination of the diagrams of ultimate stresses in a cycle for welded joints with preliminarily induced steady-state residual stresses according to the results of testing of small-size specimens without residual stresses. Translated from Problemy Prochnosti, No. 3, pp. 107–115, May–June, 2009.     

Evaluation of the ultimate stresses in a cycle for welded structures with high residual stresses based on test results for small specimens without residual stresses

We propose an equation for the evaluation of the fatigue limits of welded joints of low-carbon and low-alloy steels with high residual stresses based on the results of testing of small specimens without residual stresses. The independence of the amplitude of stresses on the mean stress in a cycle is proved for welded joints with high residual stresses. __________ Translated from Problemy Prochnosti, No. 2, pp. 66–81, March–April, 2008.     

The impact response of a half plane crack in a transversely isotropic solid due to 3-D mixed mode loading

Three-dimensional analysis of a half plane crack in a transversely isotropic solid is performed. The crack is subjected to two opposed pairs of shear line loads on its faces. Transform methods are used to reduce the boundary value problem to a set of coupled integral equations that can be solved by the Wiener-Hopf technique. The Cagniard-de Hoop method is employed to invert the transforms. Exact expressions are derived for the mode II and III stress intensity factors as functions of time and position along the crack edge. Some features of the solutions are discussed through numerical results.