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.     

Influence of loading path on fatigue crack growth under multiaxial loading condition

In this study, the specimens made of carbon steel S45 with an initial surface straight edge notch were subjected to combined cyclic axial‐torsion loading at room temperature. The fatigue life, surface crack extension direction and crack length were experimentally investigated. The effects of loading path, stress amplitude ratio and phase angle on the crack growth behaviour were also discussed. The results showed that, under the combination of cyclic axial and torsion loading, the tension stress amplitude had more effect on the initial crack growth path than the latter. The shear stress amplitude contributed mainly to the latter crack extension. The crack extension path was mainly determined by the stress amplitudes and the ratio of the normal stress to the shear stress, and almost independent of the mean stresses. The increase of the tension stress amplitude and shear stress amplitude would both accelerate the crack growth rate.     

Long life fatigue under multiaxial loading

A life prediction model in the field of high-cycle (i.e. long-life) fatigue is presented in this paper. The proposed model applies in the case of constant amplitude multiaxial proportional and non-proportional loading. The problems of the fatigue limit criterion and of the fatigue life prediction are both addressed and comparisons with experimental data are shown. Some limited discussion of the stress gradient effect is also offered. Although the particular model developed here is better suited for ferritic steels, it is explained in the paper that the methodology used to obtain this model can be adequately adapted to derive mathematically consistent models for other classes of metallic materials.     

Crack initiation and growth path under multiaxial fatigue loading in structural steels

In real engineering components and structures many accidental failures occur due to unexpected or additional loadings, such as additional bending or torsion. There are many factors influencing the fatigue crack paths, such as the material type (microstructure), structural geometry and loading path. It is widely believed that fatigue crack nucleation and early crack growth are caused by cyclic plasticity. This paper studies the effects of multiaxial loading paths on the cyclic deformation behaviour, crack initiation and crack path. Three types of structural steels are studied: Ck45, medium carbon steel, 42CrMo4, low alloy steel and the AISI 303 stainless steel. Four biaxial loading paths were applied in the tests to observe the effects of multiaxial loading paths on the additional hardening, fatigue crack initiation and crack propagation orientation. Fractographic analyses of the plane orientations of crack initiation and propagation were carried out by optical microscope and SEM approaches. It is shown that these materials have different crack orientations under the same loading path, due to their different cyclic plasticity behaviour and different sensitivity to non-proportional loading. Theoretical predictions of the damage plane were conducted using the critical plane approaches, either based on stress analysis or strain analysis (Findley, Smith–Watson–Topper, Fatemi–Socie, Wang–Brown–Miller, etc). Comparisons of the predicted crack orientation based on the critical plane approaches with the experimental observations for the wide range of loading paths and the three structural materials are shown and discussed. Results show the applicability of the critical plane approaches to predict the fatigue life and crack initial orientation in structural steels.     

Energy dissipation under multiaxial thermomechanical fatigue loading

The paper presents an approach to the energy dissipation calculation under arbitrary multiaxial thermomechanical fatigue (TMF) loading. In such an approach the total area of plastic hysteresis loops is taken as a measure of dissipated energy. The calculation is based on the concept of the developed temperature dependent Prandtl type operator. Energy dissipation is associated to irreversible dislocation movements represented by slider shifts of three independent operators. The dissipated energy is then obtained continuously at any time by collecting dissipated energy increments of each operator. It is shown that the multiaxial operator approach gives us the same total energy of plastic deformation as compared to the classical integration approach. Furthermore, the presented approach enables to automatically split the obtained dissipated energy between the “true” dissipated energy and the elastically “stored” energy. In order to satisfy the request for a minimum number of dedicated material tests, the approach assumes fixed principal directions. Therefore, the proportional as well as non-proportional loading conditions are addressed in the same manner, which is currently the main deficiency of the approach.     

Estimating fatigue damage under variable amplitude multiaxial fatigue loading

The present paper is concerned with the use of the modified Wöhler curve method (MWCM) to estimate both lifetime and high‐cycle fatigue strength of plain engineering materials subjected to complex load histories resulting, at critical locations, in variable amplitude (VA) multiaxial stress states. In more detail, when employed to address the constant amplitude (CA) problem, the MWCM postulates that fatigue damage reaches its maximum value on that material plane (i.e. the so‐called critical plane) experiencing the maximum shear stress amplitude, fatigue strength depending on the ratio between the normal and shear stress components relative to the critical plane itself. To extend the use of the above criterion to those situations involving VA loadings, the MWCM is suggested here as being applied by defining the critical plane through that direction experiencing the maximum variance of the resolved shear stress. Such a direction is used also to perform the cycle counting: because the resolved shear stress is a monodimensional quantity, stress cycles are directly counted by the classical rain‐flow method. The degree of multiaxiality and non‐proportionality of the time‐variable stress state at the assumed critical sites instead is suggested as being measured through a suitable stress ratio which accounts for the mean value and the variance of the stress perpendicular to the critical plane as well as for the variance of the shear stress resolved along the direction experiencing the maximum variance of the resolved shear stress. Accuracy and reliability of the proposed approach was checked by using several experimental results taken from the literature. The performed validation exercise seems to strongly support the idea that the approach formalized in the present paper is a powerful engineering tool suitable for estimating fatigue damage under VA multiaxial fatigue loading, and this holds true not only in the medium‐cycle, but also in the high‐cycle fatigue regime.     

Short surface fatigue cracks growth under constant and variable amplitude loading

This paper deals with the fatigue strength of S355NL steel, of a common use within the shipbuilding industry, under uniaxial constant and variable loading. Indeed, ship structures are subjected to variable loading due to various sea states, wind and waves. As a consequence, a better knowledge of fatigue behavior under real loading conditions is needed. This study aims at analyzing the influence of loading conditions (load ratio and variable amplitude loading) on the short crack behavior and last, with a proposed model to assess the fatigue crack life. The tools used to prepare inspections in critical areas only take into account the long crack behavior. The results from the proposed model were compared to the assessments these tools are providing with.     

Modelling of fatigue lifetime under non-proportional multiaxial alternating loading

This paper presents two approaches to lifetime prediction under non‐proportional multiaxial alternating loading; a phenomenological approach using the Manson–Coffin relation and a microstructural approach. Both models have in common the use of a new multiaxiality factor. The data sets for the adaptation and validation of both models are taken from the authors' own experiments. In these tests, both the load paths and the phase shift are varied. The biaxial test apparatus allows for an application of fixed principal stress or strain directions even under non‐proportional loading. A fairly good agreement with our multiaxial lifetime results is obtained with both models. For an advanced assessment of the quality of the approaches used, the models are compared with several other well‐known models from the literature. An additive yardstick is used for the comparison of the different models.     

Microcrack propagation and fatigue lifetime under non-proportional multiaxial cyclic loading

Non-proportional multiaxial fatigue tests of tubular specimens were performed under purely alternating strain-controlled loading. Different loading paths with different phase shifts were applied. With increasing phase shift at the same equivalent load, the lifetime was found to increase. For lifetime prediction a model based on the Manson–Coffin law was developed. By including the hydrostatic loading part, it was possible to compare the results of the multiaxial fatigue tests with uniaxially ascertained results. To obtain more information about the microcrack behaviour under multiaxial non-proportional loading, sonic emission studies and fractographic analyses were performed. The results suggest a discontinuous microcrack propagation. Motivated by the good agreement between these observations and some microcrack propagation models known from literature, a simplified model was proposed for micro and short crack propagation. This model which is based on the J-integral range ΔJ yields a quite good agreement between the experimentally observed and the calculated lifetimes.     

Local and structural multiaxial stress states in weldedjoints under fatigue loading

A cylindrical stiffener on a plate and a tube through a hollowed plate, both circularly welded, were tested under uniaxial fatigue loading. Their fatigue cracking behaviour was seen to be really complex due to the fact that crack initiation sites changed their position as the number of cycles to failure increased. To investigate this anomalous behaviour, an accurate numerical investigation was carried out to study the distribution of both local and structural linear elastic stresses along the weld toe circumferences. The numerical results proved that the weld beads were subjected to complex stress states, even though the applied nominal load was uniaxial. By the light of this evidence, the fatigue behaviour of the investigated welded joints was then re-interpreted from a multiaxial fatigue point of view by applying the Modified Wöhler Curve Method in terms of hot-spot stresses. The proposed approach was seen to be successful allowing us to estimate both crack initiation sites and fatigue lifetime with a high precision level. This fact is very interesting because it strongly supports the idea that our method can be used to assess real welded components subjected to multiaxial fatigue loading by simply post-processing linear-elastic finite-element results.     

Fatigue crack growth of filled rubber under constant and variable amplitude loading conditions

Service conditions experienced by rubber components often involve cyclic loads which are more complex than a constant amplitude loading history. Consequently, a model is needed for relating the results of constant amplitude characterization of fatigue behaviour to the effects of variable amplitude loading signals. The issue is explored here via fatigue crack growth experiments on pure shear specimens conducted in order to evaluate the applicability of a linear crack growth model equivalent to Miner's linear damage rule. This model equates the crack growth rate for a variable amplitude signal to the sum of the constant amplitude crack growth rates associated with each individual cycle. The variable amplitude signals were selected to show the effects of R-ratio (ratio of minimum to maximum energy release rate), load level, load sequence, and dwell periods on crack growth rates. In order to distinguish the effects of strain crystallization on crack growth behaviour, two filled rubber compounds were included: one that strain crystallizes, natural rubber, and one that does not, styrene-butadiene rubber. The linear crack growth model was found to be applicable in most cases, but a dwell effect was observed that is not accounted for by the model.     

Description of fatigue damage in carbon black filled natural rubber

The present paper describes macroscopic fatigue damage in carbon black‐filled natural rubber (CB‐NR) under uniaxial loading conditions. Uniaxial tension‐compression, fully relaxing uniaxial tension and non‐relaxing uniaxial tension loading conditions were applied until sample failure. Results, summarized in a Haigh‐like diagram, show that only one type of fatigue damage is observed for uniaxial tension‐compression and fully relaxing uniaxial tension loading conditions, and that several different types of fatigue damage take place in non‐relaxing uniaxial tension loading conditions. The different damage types observed under non‐relaxing uniaxial tension, loading conditions are closely related to the improvement of rubber fatigue life. Therefore, as fatigue life improvement is classically supposed to be due to strain‐induced crystallization (SIC), a similar conclusion can be drawn for the occurrence of different types of fatigue damage.     

The growth of fatigue cracks at high temperatures under predominantly elastic loading

The extent of applicability of linear elastic fracture mechanics (LEFM) to high temperature fatigue-crack growth in $\text{1}\text{2}$ and 1% Cr-Mo-V turbine-casing steels has been established using a range of specimen geometry and thickness. Crack growth rates exceeded those at room temperature and generally increased with declining frequency. The relation for crack growth was: dL/dN = CΔK², where L was cracuength, N cycle number, C a frequency-dependent term and ΔK the stress intensity amplitude. There was little effect of increasing the maximum stress intensity of the cycle (or mean stress level) provided widespread creep deformation was avoided. A limiting crack tip displacement (C.O.D.) for LEFM analysis was established for the cast Cr-Mo-V steels examined.The relative contributions of creep and oxidation to fatigue-crack growth were examined. Oxidation was found to predominate at high frequencies and low amplitudes, whilst creep effects were significant at low (10^?3 Hz) frequencies.     

Spectral method of fatigue life calculation under random multiaxial loading

We develop a new spectral method for the evaluation of fatigue life under random multiaxial loading. This method is a generalization of the known formulas of Miles, Kowalewski, Raikher, and Bolotin based on the power spectral density function of stresses under uniaxial random loads. The power spectral density function of the equivalent stress determined according to a linear criterion of multiaxial random fatigue is introduced in the indicated formulas. It is shown that, in reducing the multiaxial state of loading to the uniaxial state according to linear criteria, the frequency bands of the components of the stress state are transformed into the frequency band of the equivalent stress without increasing its width. This favorable result cannot be obtained if the equivalent stress is calculated according to nonlinear multiaxial fatigue criteria. Technical University of Opole, Opole, Poland, Published in Fizyko-Khimichna Mekhanika Materialiv, Vol. 32, No. 3, pp. 86–96, May–June, 1996.     

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 stress effects on fatigue behavior of filled natural rubber

This work explores multiaxial stress effects on fatigue crack nucleation and growth in filled natural rubber based on experiments using short thin-walled cylindrical specimens subjected to axial and twist displacements. Cyclic stress–strain response exhibits significant initial softening relative to the monotonic response, followed by a more gradual additional softening. Irreversible breakage of various types of bonds is believed to cause the initial softening, while the presence of fillers and their influence on network chain breakage is believed to cause the additional softening. Crack nucleation planes and crack growth paths were monitored and cracks were observed to form on specific plane(s). Correlations of crack nucleation lives and growth rates with respect to the peak maximum principal strain were obtained and are discussed.     

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.     

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.