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.     

An implicit gradient application to fatigue of complex structures

This paper presents a procedure to evaluate the stress gradient effect on the fatigue strength of steel welded joints and notched components. An effective stress is calculated by solving a second-order differential equation over all the component (the implicit gradient approach) independently of its geometric shape. The solution is obtained by assuming the isotropic linear elastic constitutive law for the material and the maximum principal stress as equivalent stress. The fatigue behaviour of geometrically complex steel welded joints is analysed and compared with previous fatigue scatter bands obtained for two-dimensional joints. In complex details, the actual critical point is derived from the analysis and is not assumed a priori. Implicit gradient analysis is also used to investigate high-cycle fatigue behaviour in the case of notches.In addition, it is shown that critical distance approaches can be obtained from the non-local theory by proper choice of the weight function.     

An implicit gradient application to fatigue of sharp notches and weldments

This paper addresses the problem of stress singularities at the tip of sharp V-notches by means of a non-local implicit gradient approach. A non-local equivalent stress is defined as a weighted average of a local stress scalar quantity computed on the assumption of linear elastic material behaviour. In the case of a crack, we propose an analytical solution for the non-local equivalent stress at the crack tip when the local equivalent stress assumes the analytical form proposed by Irwin. For open notches, several numerical procedures are possible.For welded joints, we assume that the material obeys a linear elastic constitutive law. In this case, the non-local equivalent stress obtained from the implicit gradient approach is assumed as the effective stress for assessments of joint fatigue. Using the principal stress as local equivalent stress and a notch tip or weld toe radius equal to zero, we analyse many series of arc welded joints made of steel and subjected to either tensile or bending loading, and we propose a unifying fatigue scatter band. If the welded joints are subjected only to mode I loading, an analytical relationship between the relevant Notch Stress Intensity Factors (NSIF) of mode I and the effective stress is established; otherwise, the effective stress is evaluated by means of a simplified numerical analysis. For complex welded structures, however, a completely numerical solution is proposed; when different crack initiation sites are present (i.e. either weld toes or roots), the proposed approach correctly estimates the actual critical point.     

Short-crack-growth-based fatigue assessment of notched components under multiaxial variable amplitude loading

A short crack model originally proposed for multiaxial constant amplitude loading is extended and applied to multiaxial variable amplitude loading. Load sequences have a significant influence on variable amplitude life; they are taken into account using algorithms originally proposed only for uniaxial loading. The estimated fatigue lives of unnotched tubular specimens and notched shafts under different in- and out-of-phase multiaxial constant and variable amplitude load histories are compared with the experimental results. The comparison reveals that the proposed short crack approach enables sufficiently accurate estimation. Moreover, the estimated critical planes, i.e., the planes of maximum crack growth rate or minimum life, are in good agreement with the experimental observations.     

A multiaxial high-cycle fatigue life evaluation model for notched structural components

A model for multiaxial high-cycle fatigue life evaluation of notched structural components is proposed, which considers the impact of the stress field on fatigue life by utilizing the Theory of Critical Distances (TCD) and Finite Element Method (FEM). The maximum shear stress range plane is defined as the critical plane, and the damage parameters are the maximum effective shear stress amplitude and the maximum effective normal stress, which are obtained by averaging the stress in the hemisphere volume around the maximum stress point. To validate the accuracy of the model, multiaxial fatigue tests are carried out for both smooth and notched specimens of Aluminum–Silicon alloy. The results indicate that the evaluated life and experimental life have a good agreement.     

A fatigue damage accumulation model for reliability analysis of engine components under combined cycle loadings

Rotor components of an aircraft engine in service are usually subjected to combined high and low cycle fatigue (CCF) loadings. In this work, combining with the load spectrum of CCF, a modified damage accumulation model for CCF life prediction of turbine blades is first put forward to take into account the effects of load consequence and load interaction caused by high‐cycle fatigue (HCF) loads and low‐cycle fatigue (LCF) loads under CCF loading conditions. The predicted results demonstrate that the proposed model presents a higher prediction accuracy than Miner, Manson‐Halford model does. Moreover, to evaluate the fatigue reliability of rotor components, reliability model with the failure mode of CCF is proposed on the basis of the stress‐strength interference method when considering the strength degeneration, and its results show that the reliability model with CCF is more suitable for aero‐engine components than that with the failure mode of single fatigue.     

A damage gradient model for fatigue life prediction of notched metallic structures under multiaxial random vibrations

In the present paper, a damage gradient model combing the damage concept with the theory of critical distance (TCD) is established to estimate the fatigue lives of notched metallic structures under multiaxial random vibrations. Firstly, a kind of notched metallic structure is designed, and the biaxial random vibration fatigue tests of the notched metallic structures are carried out under different correlation coefficients and phase differences between two vibration axes. Then, the fatigue lives of the notched metallic structures are evaluated utilizing the proposed model with the numerical simulations. Finally, the proposed model is validated by the experiment results of the biaxial random vibration fatigue tests. The comparison results demonstrate that the proposed model can provide fatigue life estimation with high accuracy.     

A one-dimensional model for non-linear behaviour of piezoelectric composite materials

In this paper, a one-dimensional analytical model is proposed to investigate the non-linear behaviour for piezoelectric and piezoelectric fibre reinforced composite (PFRC) materials in the fibre direction. The required linear and non-linear constants for purely piezoelectric materials are obtained using the curve-fitting method and the measured S₃–E₃ non-linear loops of the corresponding piezoelectric materials, whereas those for PFRC materials are determined by employing the quadratic non-linear constitutive equations for a purely piezoelectric material, the iso-field assumptions and linear and non-linear constants of the composite constituents. A numerical study is conducted. The numerical results reveal a significant effect of stress T₃ on S₃–E₃ non-linear behaviour for both soft PZT–5H ceramics and PZN–4.5%PT crystals. It is also found that the piezoelectric fibre volume fraction V_f and strain S₃ can significantly affect the T₃–E₃ non-linear behaviour for both PZT–5H/piezo-polymer polyvinylidene fluoride (PVDF) and PZN–4.5%PT/PVDF PFRC materials. A good correlation is noted between the piezoelectric constants d₃₃ and e₃₃ predicted using the present method and those recommended by manufacturer for PZT–5H ceramics and PZN–4.5%PT crystals.     

A mean stress correction model for tensile and compressive mean stress fatigue loadings

A new mean stress fatigue model based on the distortional strain energy is proposed to account for the mean stress effects on fatigue life. The proposed model is compared with the Morrow and the Smith‐Watson‐Topper (SWT) mean stress correction models using a number of experimental data sets for one cast iron, two steels and two aluminium alloys under tensile and compressive mean stress loadings. It is found that both the proposed mean stress correction model and the SWT model yield similar results and provide very good correlation for positive mean stress data and moderate negative mean stress data. For high compressive mean stresses, the proposed model shows reasonably good correlations, while the SWT model fails to correlate the fatigue data. The Morrow model was found to give poor correlations for all fatigue data analysed by yielding conservative results for compressive mean stresses and non‐conservative results for tensile mean stresses.     

Requirements for stress gradient‐based fatigue assessment of notched structures according to theory of critical distance

Notches, local stress raisers within structural components, are one of the most important locations for fatigue crack initiation. It is well known that fatigue is governed by the effective stresses in the vicinity of notches. Within this study, differences in prediction accuracy between different types of theory of critical distance methods, that is, point and line methods, are systematically investigated in conjunction with a sensitivity study regarding mesh refinement, assumed strength hypothesis and material behaviour. For this purpose, a finite element analysis parameter study on notched structures is performed and recommendations for the application of stress gradient methods are presented. Difference in effective stress of up to 30%, and hence a significant difference in fatigue life (e.g., 185% for a slope of S‐N curve of k = 4), is found for typical notch shapes, for example, in welded joints.     

A unifying approach to estimate the high-cycle fatigue strength of notched components subjected to both uniaxial and multiaxial cyclic loadings

This paper proposes an engineering method suitable for predicting the fatigue limit of both plain and notched components subjected to uniaxial as well as to multiaxial fatigue loadings. Initially, some well‐known concepts formalized by considering the cracking behaviour of metallic material under uniaxial cyclic loads have been extended to multiaxial fatigue situations. This theoretical extension allowed us to form the hypothesis that fatigue limits can be estimated by considering the linear–elastic stress state calculated at the centre of the structural volume. This volume was assumed to be the zone where all the main physical processes take place in fatigue limit conditions. The size of the structural volume was demonstrated to be constant, that is, independent from the applied loading type, but different for different materials. Predictions have been made by Susmel and Lazzarin's multiaxial fatigue criterion, applied using the linear–elastic stress state determined at the centre of the structural volume. The accuracy of this method has been checked by using a number of data sets taken from the literature and generated by testing notch specimens both under uniaxial and multiaxial fatigue loadings. Our approach is demonstrated to be a powerful engineering tool for predicting the fatigue limit of notch components, independently of material, stress concentration feature and applied load type. In particular, it allowed us to perform predictions within an error interval of about ±25% in stress, even though some material mechanical properties were either estimated or taken from different sources.     

A strain-energy-density-based lifetime prediction model for notched specimens: Polycarbonate under thermal fatigue

Polycarbonate is more and more extensively used in engineering because of its good mechanical properties. Pieces of polycarbonate are used in environments with variable temperature, especially in electronic devices. Thermal stresses could become important and, for this reason, the effects of thermal stresses must be taken into account in designing these pieces. We propose a method for the prediction of the life of notched specimens based on the density of dissipated strain energy. The laws of behavior of polycarbonate at various temperatures are determined, and the fatigue tests performed on smooth specimens give the laws of thermal fatigue of the material. The fatigue tests on notched specimens and finite-element-method computations enable us to establish the relationship between the stress concentration factor, the density of strain energy dissipated at the notch roots, and the density of nominal strain energy. A life-prediction model is proposed and discussed. Laboratory of Mechanical Reliability, Metz University, Metz, France. Published in Problemy Prochnosti, No. 6, pp. 32–42, November–December, 1998.     

A high order implicit algorithm for solving instationary non-linear problems

 In this paper a high order implicit algorithm is developed for solving instationary non-linear problems. This generic numerical method combines four mathematical techniques: a time discretization, a homotopy transformation, a perturbation technique and a space discretization. The time integration is performed by classical implicit schemes (Euler implicit for problems with a first order time derivative and Newmark for second order). The time-discretization leads to non-linear equations. In this paper a new technique is proposed to solve iteratively the latter equations. The key points in this approach are, first a high order solver based on perturbation techniques, second the possibility of choosing the iteration operator, which limits the number of matrices to be triangulated. To illustrate the performance of the proposed algorithm two examples are considered: the Korteweg-de Vries equation (KdV) and the non-linear oscillations of a 2D elastic pendulum. Received 10 February 2001 / Accepted 19 December 2001     

A non-linear damage accumulation fatigue model for predicting strain life at variable amplitude loadings based on constant amplitude fatigue data

A new phenomenological technique for using constant amplitude loading data to predict fatigue life from a variable amplitude strain history is presented. A critical feature of this reversal-by-reversal model is that the damage accumulation is inherently non-linear. The damage for a reversal in the variable amplitude loading history is predicted by approximating that the accumulated damage comes from a constant amplitude loading that has the strain range of the particular variable amplitude reversal. A key feature of this approach is that overloads at the beginning of the strain history have a more substantial impact on the total lifetime than overloads applied toward the end of the cycle life. This technique effectively incorporates the strain history in the damage prediction and has the advantage over other methods in that there are no fitting parameters that require substantial experimental data. The model presented here is validated using experimental variable amplitude fatigue data for three different metals.     

The influence of high R ratio on notched fatigue behaviour of 1045 steel with three different heat treatments

Three 1045 steels with hardness levels of R_c=10, 37, and 50 were tested under cyclic axial load control conditions using mildly notched specimens, K_t=1.65, at high R ratios of 0.8 and 0.9. The notched ultimate tensile strengths, S_un, for the three steels were greater than the unnotched ultimate tensile strengths, S_u. This allowed values of nominal maximum stress, S_max, and mean stress, S_m, to exceed S_u in most tests. Thus, fatigue limits based on S_max were higher than S_u in 5 of the 6 test conditions. S–N_f curves were very flat in 5 of the 6 test conditions with appreciable scatter. With S_max and S_m>S_u in most tests, usual S–N_f fatigue life models involving S_u could not be used. Replacing S_u with S_un, allowed calculations, but these were completely inaccurate. Local strain-life, –N_f, models were also completely inaccurate for these high R ratios. R_c=10 specimens failed by cyclic creep/ratcheting from internal microvoid coelescence and not from fatigue. R_c=37 specimens failed by fatigue from surface thumbnail cracks, but were influenced by cyclic creep/ratcheting. R_c=50 specimens failed by brittle fracture from minute surface fatigue cracking without cyclic creep/ratcheting. In design situations at long life, usual S–N_f models with these materials and high R ratios would restrict S_max to levels well below the experimental fatigue limits resulting in very conservative results.     

A fatigue model for sensitive materials to non-proportional loadings

The present study proposes a novel fatigue model based on virtual strain energy. This model separates loading paths based on their non-proportionality where directly takes into account the loading in fatigue life prediction. The proposed fatigue model is expressed in two tension-based and shear-based equations for two tensile and shear cracking failure modes. The model was validated against several experimental datasets available in the literature. In addition, obtained results were compared to predicted lives through some well-known fatigue models comprising maximum shear strain, Smith–Watson–Topper, and Fatemi–Socie. The results were strongly correlated with the experimental data indicating accuracy of the model.     

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.     

Variables affecting the fatigue resistance of PVD-coated components

The effect of intrinsic properties of CrN coatings on fatigue behaviour was studied in this paper. The coating layer microhardness and the residual stresses characterising the surface film were measured and the obtained results were introduced in a numerical modelling predicting fatigue life procedure of coated components. The effect of a CrN monolayer film deposited on bulk samples, produced in 2205 duplex stainless steel, H11 tool steel or 6082 aluminium alloy was investigated. The fatigue limit of coated and uncoated samples was experimentally determined while the development of FEM models, confirmed by means of experimental tests, represents a powerful tool to predict fatigue life of coated components. The effects on the fatigue strength of coating and bulk material defects like droplets and non-metallic inclusions were considered along with the residual stress gradient characterising the coating and evaluated by means of X-ray measurements. The influence of the substrate material plastic deformation on the integrity of the coating was evaluated too.     

Homogenised non-linear soft inclusion for simulation of impact damage in composite structures

This paper presents the development of a homogenised non-linear soft inclusion which captures the geometric and material non-linearity of impact damage zone loaded in tension and compression. The homogenised non-linear soft inclusion can present a conservative worst case damage zone or use experimental data to mimic the behaviour of a particular damage zone in a simple and computationally efficient way that can be used as a structural design tool for composite structures subjected to impact. The development of the non-linear soft inclusion, implemented in an ABAQUS/Explicit VUMAT, is presented at element and coupon level. The non-linear soft inclusion is validated against experimental coupon data and produces a conservative worst case estimate in all cases investigated.     

Environment and time dependent effects on the fatigue response of an advanced nickel based superalloy

The fatigue behaviour of the nickel based superalloy RR1000 is characterised using double edge notch specimens incorporating shot peening. Evaluations were conducted at two test temperatures, 300 °C and 650 °C, employing baseline and dwell waveforms. The effects of air and vacuum environments plus prior exposure at 650 °C were also assessed. It is demonstrated that surface oxidation does not control performance at the test conditions of interest. Rather, the modification to stabilized peak and mean stresses resulting from either thermal relaxation of peened stresses or a time dependent shake down of stress under mechanical loading governs ultimate behaviour.