Biaxial fatigue for proportional and non‐proportional loading paths

In order to study the use of a local approach to predict crack‐initiation life on notches in mechanical components under multiaxial fatigue conditions, the study of the local cyclic elasto‐plastic behaviour and the selection of an appropriate multiaxial fatigue model are essential steps in fatigue‐life prediction. The evolution of stress–strain fields from the initial state to the stabilized state depends on the material type, loading amplitude and loading paths. A series of biaxial tension–compression tests with static or cyclic torsion were carried out on a biaxial servo‐hydraulic testing machine. Specimens were made of an alloy steel 42CrMo4 quenched and tempered. The shear stress relaxations of the cyclic tension–compression with a steady torsion angle were observed for various loading levels. Finite element analyses were used to simulate the cyclic behaviour and good agreement was found. Based on the local stabilized cyclic elastic–plastic stress–strain responses, the strain‐based multiaxial fatigue damage parameters were applied and correlated with the experimentally obtained lives. As a comparison, a stress‐invariant‐based approach with the minimum circumscribed ellipse (MCE) approach for evaluating the effective shear stress amplitude was also applied for fatigue life prediction. The comparison showed that both the equivalent strain range and the stress‐invariant parameter with non‐proportional factors correlated well with the experimental results obtained in this study.     

A modification of Morrow and Smith–Watson–Topper mean stress correction models

A modification of the Morrow and the Smith, Watson and Topper (SWT) mean stress correction models is proposed to account for the mean stress effect on fatigue life. The capability and accuracy of the proposed model are compared to those of the original Morrow and the SWT model using published mean stress fatigue test data. The proposed mean stress correction model was found to be superior to both the SWT and the Morrow model in the case of the Incoloy 901 superalloy and the ASTM A723 steel. On the other hand both the proposed and the original SWT model provided equally good correlation with experimental data in the case of 7075‐T561 aluminium alloy and 1045 HRC 55 steel. The Morrow model was found to give the least accurate predictions for all four materials analysed.     

Fatigue life prediction of vulcanized natural rubber under proportional and non-proportional loading

To investigate the multiaxial fatigue properties of vulcanized natural rubber (NR), a series of tests including both proportional and non-proportional loading paths on small specimens were performed. The existing fatigue life prediction approaches are evaluated with life data obtained in the tests. It is shown that the equivalent strain approach presents a good prediction of the fatigue life although it has a certain shortcoming. Compared with the strain energy density (SED) model, the cracking energy density (CED) model represents the portion of SED that is available to be released by virtue of crack growth on a given material plane, so it gives better results in the life prediction. Some of the approaches based on critical plane which are widely used for metal fatigue are also tested in this paper, and the results show that the Chen-Xu-Huang (CXH) model gives a better prediction, compared with the Smith-Watson-Topper (SWT) and Wang–Brown (WB) model. A modified Fatemi–Socie's model has also been introduced, and the results show that the modified model can be used to predict the fatigue life of rubber material well.     

Low‐cycle fatigue life prediction of various metallic materials under multiaxial loading

This paper proposed a simple life prediction model for assessing fatigue lives of metallic materials subjected to multiaxial low‐cycle fatigue (LCF) loading. This proposed model consists of the maximum shear strain range, the normal strain range and the maximum normal stress on the maximum shear strain range plane. Additional cyclic hardening developed during non‐proportional loading is included in the normal stress and strain terms. A computer‐based procedure for multiaxial fatigue life prediction incorporating critical plane damage parameters is presented as well. The accuracy and reliability of the proposed model are systematically checked by using about 300 test data through testing nine kinds of material under both zero and non‐zero mean stress multiaxial loading paths.     

A critical plane-strain energy density criterion for multiaxial low-cycle fatigue life under non-proportional loading

A series of multiaxial low-cycle fatigue experiments was performed on 45 steel under non-proportional loading. The present evaluations of multiaxial low-cycle fatigue life were systematically analysed. A combined energy density and critical plane concept is proposed that considers different failure mechanisms for a shear-type failure and a tensile-type failure, and from which different damage parameters for the critical plane-strain energy density are proposed. For tensile-type failures in material 45 steel and shear-type failures in material 42CrMo steel, the new damage parameters permit a good prediction for multiaxial low-cycle fatigue failure under non-proportional loading. The currently used critical plane models are a special and simple form of the new model.     

Multiaxial fatigue life prediction method based on path‐dependent cycle counting under tension/torsion random loading

A path‐dependent cycle counting method is proposed by applying the distance formula between two points on the tension‐shear equivalent strain plane for the identified half‐cycles first. The Shang–Wang multiaxial fatigue damage model for an identified half‐cycle and Miner's linear accumulation damage rule are used to calculate cumulative fatigue damage. Therefore, a multiaxial fatigue life prediction procedure is presented to predict conveniently fatigue life under a given tension and torsion random loading time history. The proposed method is evaluated by experimental data from tests on cylindrical thin‐walled tubes specimens of En15R steel subjected to combined tension/torsion random loading, and the prediction results of the proposed method are compared with those of the Wang–Brown method. The results showed that both methods provided satisfactory prediction.     

A computerized procedure for long-life fatigue assessment under complex multiaxial loading

A computerized procedure is presented and evaluated for application examples of long-life fatigue analyses of metallic materials under complex multiaxial loading. The method is based on the stress invariants and uses the minimum circumscribed ellipse approach for evaluating the effective shear stress amplitude under complex multiaxial loading. The applicability of the procedure for handling non-proportional loading is examined through typical examples such as combined normal/shear stresses and combined bi-axial normal stresses with complex stress time histories. The effects of phase shift angles, frequency ratios and waveforms on fatigue endurance were re-analysed and compared with available experimental results from the literature. The comparison shows that the presented procedure based on stress invariants is a potential conservative engineering approach, very suitable for fast fatigue evaluation in the integrated computer aided fatigue design.     

An improved multiaxial high‐cycle fatigue criterion based on critical plane approach

Several groups of fatigue damage parameters are discussed and then an improved multiaxial high‐cycle fatigue criterion based on critical plane defined by the plane of maximum shear stress range is presented in this paper. A compromising solution to consider the mean normal stress acting on the critical plane is also proposed. The new fatigue criterion extends the range of metallic materials which is valid for the ratio 1.25 < f_?1/t_?1 < 2. The predictions based on the presented model show a good agreement with test data.     

Prediction of rubber fatigue life under multiaxial loading

The process of fatigue failure of materials is generally described by two phases: crack initiation and crack propagation. This study concerns the crack initiation in rubbers submitted to a cyclic loading. A parameter based on the strain energy density (SED) and predicting the onset of primary crack and its probable orientation has been identified for such materials according to the investigations of Mars and Fatemi. More precisely, this criterion has been analytically developed in the cases of simple tension, biaxial tension and simple shear loadings by assuming large strains. The results denote the possibility to predict the orientation plane in which the primary crack would be expected to occur in a material. Then, it was implemented in a finite‐elements (FE) program in order to be applied to structures under any kind of the strain states. A good agreement was obtained between FE and analytical results for the usual strain states. Finally, to evaluate lifetime up to crack nucleation, we have achieved a set of experimental fatigue tests using uniaxial tension (UT) and pure shear (PS) test specimens containing a hole in order to localize the crack initiation. The obtained results proved the efficiency of the criterion to describe the fatigue life of rubbers under multiaxial loading.     

Multiaxial fatigue of V‐notched steel specimens: a non‐conventional application of the local energy method

The paper deals with the multi‐axial fatigue strength of notched specimens made of 39NiCrMo3 hardened and tempered steel. Circumferentially V‐notched specimens were subjected to combined tension and torsion loading, both in‐phase and out‐of‐phase, under two nominal load ratios, R=?1 and R= 0, also taking into account the influence of the biaxiality ratio, λ=τ_a/σ_a. The notch geometry of all axi‐symmetric specimens was a notch tip radius of 0.1 mm, a notch depth of 4 mm, an included V‐notch angle of 90° and a net section diameter of 12 mm. The results from multi‐axial tests are discussed together with those obtained under pure tension and pure torsion loading on plain and notched specimens. Furthermore the fracture surfaces are examined and the size of non‐propagating cracks measured from some run‐out specimens at 5 million cycles. Finally, all results are presented in terms of the local strain energy density averaged in a given control volume close to the V‐notch tip. The control volume is found to be dependent on the loading mode.     

A new damage parameter for fatigue life predictions under local strain analysis

A new damage parameter is proposed for fatigue life prediction using a local stress-strain approach. This parameter has a physical energy basis, and makes it possible to obtain the same accuracy as, and better life assessments than, the well-known Smith et al parameter using significantly less calculation time for load history treatment. Comparisons of life predictions obtained using the proposed parameter with experimental results and other predictions are presented.     

Modelling the dependency of the Smith–Watson–Topper parameter on the cycles‐to‐failure using serial hybrid neural networks

Calculating the fatigue damage with a strain‐based approach requires an ?–N durability curve that links the strain amplitude to the corresponding number of cycles‐to‐failure. This ?–N curve is usually modelled by the Coffin–Manson relationship. If a loading mean‐level also needs to be considered, the original Coffin–Manson relationship is modified using a Smith–Watson–Topper parameter. In this article a methodology for modelling the dependence of the Smith–Watson–Topper parameter on the number of cycles‐to‐failure is presented. The core of the presented methodology represents a multilayer perceptron neural network combined with the Smith–Watson–Topper analytical model. The article presents the theoretical background of the methodology, which is applied for the case of the experimental fatigue data. The results show that it is possible to model ?–N curves for different influential parameters, such as the specimen's diameter and the testing temperature. The results further show that it is possible to predict ?–N curves even for those combinations of the influential parameters for which no experimental data about the material endurance is available. This fact makes the presented model very suitable for the application in an R&D process when a durability of a product should be estimated on the basis of a very limited set of experimental data about the material endurance characteristics.     

A high-cycle fatigue life model for variable amplitude multiaxial loading

A high‐cycle fatigue life model for structures subjected to variable amplitude multiaxial loading is presented in this paper. It treats any kind of repeated blocks of variable amplitude multiaxial loading without using a cycle counting method. This model based on a mesoscopic approach is characterized by the following features: (i) the choice of a damage factor related to the accumulated mesoscopic plastic strain per stabilised cycle; (ii) the use of a mesoscopic mechanical behaviour taking into account the fatigue mechanisms such as plasticity and void growth. This behaviour is a von Mises elastoplastic model with linear kinematic hardening and hydrostatic stress dependent yield stress. The fatigue life model has six parameters identified with one SN curve and two fatigue limits. In‐phase and out‐of‐phase experimental tests from the literature are simulated. The predicted fatigue lives are compared to experimental ones.     

Cumulative fatigue damage and life prediction of elastomeric components

Elastomeric components are widely used in many applications due to their good damping and energy absorption characteristics. The type of loading normally encountered by these components in service is variable amplitude cyclic loading. Therefore, fatigue failure is a major consideration in their design. In this work capabilities of Rainflow cycle counting procedure, maximum principal strain as a damage criterion, and Miner's linear cumulative damage rule are evaluated with both specimen and component tests. An automotive cradle mount is used as an illustrative component. Comparison of predicted and experimental fatigue lives in both specimen and cradle mount variable amplitude load tests indicate satisfactory predictions in both cases.     

A curvilinear integral method for multiaxial fatigue life computing under non-proportional, arbitrary or random stressing

Some popular concepts for reducing three variable stress components σ_x(t), σ_y(t), τ_xy(t) to one equivalent amplitude spectrum, and the use of the linear damage accumulation hypothesis, have been evaluated as not fully correct when these components vary non-proportionally and arbitrarily. A different approach is suggested: computing damage accumulation by means of an integral directly on the non-radial arbitrary path, called the ‘trajectory’, described in the σ_x−σ_y plane when τ_xy(t) = 0, in the σ_x−τ_xy plane when σ_y(t) = 0, or in a special coordinate space where this trajectory is invariant of stress directions x, y. If the trajectory is random, it may be replaced by a statistical two-dimensional density of distribution. The integrand, called the R-function, is derived from various S−N fatigue curves under different determined loadings. Thus the traditional S−N function is replaced by the R-function for direct damage summation with differential analysis, which allows the loading to be arbitrary (non-cyclic, multiaxial and non-proportional). The method works by means of computer programs and is applicable to real structures.     

A fatigue criterion for general multiaxial loading

An incremental fatigue damage model is proposed. The model incorporates the critical plane concept in multiaxial fatigue, plastic strain energy and material memory in cyclic plasticity. With an incremental form the model does not require a cycle counting method for variable amplitude loading. The model is designed to consider mean stress and loading sequence effects. Features of the new model are discussed and the determination of material constants is detailed. Verification of the model is achieved by comparing the predictions obtained by using the new model and experimental data of four materials under different loading conditions.     

Fatigue damage of medium carbon steel under sequential application of axial and torsional loading

Fatigue tests were conducted on S45C steel under fully reversed strain control conditions with axial/torsional ( at ) and torsional/axial ( ta ) loading sequences. The linear damage value (n₁/N₁+n₂/N₂) was found to depend on the sequence of loading mode ( at or ta ), sequence of strain amplitude (low/high or high/low) and life fraction spent in the first loading. In general, at loading yields larger damage values than ta loading and the low–high sequence of equivalent strain leads to larger damage values than the high–low sequence. The material exhibits cyclic softening under axial cyclic strain. Cyclic hardening occurs in the torsion part of ta loading, which elevates the axial stress in the subsequent loading, causing more damage than in pure axial fatigue at the same strain amplitude. Fatigue life is predicted based on the linear damage rule, the double linear damage rule, the damage curve approach and the plastic work model of Morrow. Results show that overly conservative lives are obtained by these models for at loading while overestimation of life is more likely for ta loading. A modified damage curve method is proposed to account for the load sequence effect, for which predicted lives are found to lie in the factor‐2 scatter band from experimental lives.     

Three-dimensional calculations of high cycle fatigue life under out-of-phase multiaxial loading

In this study, we investigate the prediction of fatigue life at a high number of cycles (>5 × 10⁴ cycles) for three-dimensional structures. An approach has been developed that includes the results of fatigue tests in a program using the finite element method. Numerical fatigue life calculations using three fatigue criteria were conducted to predict S – N curves. To complete the study and validate this approach, tests were carried out on FGS 700/2 cast iron with different geometrical structures and different fatigue loadings.