A correction in the algorithm of fatigue life calculation based on the critical plane approach

doi:10.1016/j.ijfatigue.2015.10.011

International Journal of Fatigue

Volume 83, Part 2, February 2016, Pages 174-183

https://doi.org/10.1016/j.ijfatigue.2015.10.011 Get rights and content

Highlights

•
A correction in the algorithm for fatigue life calculation is presented.
•
Constants applied in multiaxial fatigue criteria become function of fatigue life.
•
Improvement of convergence in calculated and experimental fatigue lives is shown.

Abstract

The paper presents the algorithm for calculating the fatigue life taking into account the variability of coefficients occurring in the multiaxial fatigue criterion depending on the number of cycles to failure. The algorithm has been analysed under uniaxial cyclic loads and a combination of bending and torsion for four structural materials. Significant increase of convergence of calculated and experimental fatigue life using the new algorithm as compared to the classical approach for five selected multiaxial fatigue criteria based on a critical plane has been demonstrated.

Introduction

Fatigue defined as the degradation of mechanical properties of the material under loadings which vary in time is one of the main causes of the limited life time of machines and structures. This results in an increase in operating costs and is the reason for ongoing research on the complex phenomenon of fatigue failure. One research areas is the multiaxial fatigue criteria which aim at evaluating the fatigue degradation of the material at any load run. This evaluation is usually carried out by comparing the reduced complex state of stress to a scalar value equivalent to a uniaxial stress (the so-called fatigue limit). The proposed function for reducing the complex stress to a uniaxial state is an essential part of the multiaxial fatigue criterion. Among the many proposed functions, one can distinguish a group characterized by the assumption that the components of the stress associated with the plane of a certain orientation are responsible for the initiation of fatigue cracks. The orientation of the plane should coincide with the plane of the fatigue crack. This proposal, known as the critical plane concept, has gained great interest in the academia [1], [2], [3], [4], [5], [6], [7], [8]. Despite the considerable amount of literature and research on this concept, there is no proposal of a criterion accepted by the wider group of researchers and applying to different materials and loads.

The reduction functions proposed in the criteria are also used to calculate the fatigue life N_cal by comparing the equivalent value of stress σ_eq to stress σ(N_f) from the fatigue characteristics (e.g. Wöhler or Basquin), assuming that N_cal = N_f. Fatigue characteristics obtained during cyclic torsion σ(N_f) = τ_f(N_f), tension–compression, and bending σ(N_f) = σ_f(N_f) are the most commonly used. Correctly proposed reducing function applied to any case of a uniaxial load, for example torsion, tension–compression, or bending stresses, but with the same fatigue life, brings these stresses to the equivalent state, thus $σ_{eq} (N,Torsion) = σ_{eq} (N,Bending) = σ (N = N_{f}) .$

Reducing functions based on the critical plane are usually linear or non-linear function of material parameters and shear τ_ns, normal σ_n (in the critical plane), or hydrostatic σ_h stresses (invariant of the stress). Material parameters are determined in such a way that the stress reduction satisfies Eq. (1). Typically, the fatigue criteria in their original form are proposed to assess the limit state, that is, for the so-called fatigue limit. Simplifying the problem by adopting the theoretical fatigue limit (for steel) σ_af corresponding to the number of cycles N = 2·10⁶, that is, σ(N_f = 2·10⁶) = σ_af, Eq. (1) is reduced to $σ_{eq} (Torsion) = σ_{eq} (Bending) = σ_{af} .$

Therefore, the material parameters are relations of fatigue limits from uniaxial stress states. Applying the proposed reduction function to the fatigue life other than the limiting one requires looking for material parameters which satisfy Eq. (1). Unfortunately, the fatigue criteria, or rather the reducing functions used to calculate the so-called reduced fatigue life (N < 2·10⁶ for steel) are usually assumed with coefficients which are the functions of fatigue limits [9], [10], [11], [12], [13]. This approach is valid only for a certain class of materials, for which $\frac{σ_{f} (N_{f})}{τ_{f} (N_{f})} = const,$ that is for materials with parallel fatigue characteristics. This fact was noted in several papers including [14], [15], [16].

The aim of this paper is to propose an algorithm for determining the fatigue life by using generally known fatigue criteria, taking into account the correct determination of material parameters which are a function of the number of cycles to failure. Validation of the proposed algorithm is performed using stress based criteria applicable in the high cyclic fatigue regime. However, the main idea of correction could be implemented also in strain or energy based criteria.

Section snippets

Stanfield (1935), Stullen-Cummings (1954), Findley (1959) criterion: C1

Stanfield [17] was the first to propose the calculation of the critical shearing stress value τ_c (fatigue strength, limiting value for failure) for a multiaxial stress state based on a linear combination of the shear τ_ns and normal σ_n stresses in the plane of the material at a certain orientation $τ_{c} = \max_{n} {τ_{ns, a} + k σ_{n, a}},$ where k is a material constant. According to Stanfield, τ_c is calculated in a plane (with normal n) on which a linear combination of (4) is at the maximum. Stanfield has not

Generalization of the analysed criteria for a limited number of cycles to failure

The application of fatigue criteria for estimating the fatigue life is based on calculations of the equivalent stress value σ_eq and comparing it to the accepted material fatigue characteristics f(N_f). This study analyses time-varying loads, but with component variables of the stress which are proportionally relative to each other without the mean value. Accordingly, the equivalent stress can be represented as function of stress amplitudes. The generalized equation which enables to calculate the

Corrected algorithm for calculating the fatigue life under proportional loads

The algorithm for calculating the number of cycles to failure N_cal for proportional loads, which takes into account the variability of the coefficients a and b (according to Table 1) depending on the number of cycles N is shown in Fig. 1a. In order to exhibit the change in algorithm the classical version is presented in Fig. 1b.

In the first step, the stress amplitudes σ_ij_,_a are loaded for the global coordinate system O_xyz and material constants defining two fatigue characteristics: σ_f(N_f) and τ_f

Experimental data

The proposed algorithm has been verified based on experimental data taken from literature, which relate to four materials, i.e. S355J2WP steel [26] (according to PN: 10HNAP), S355J2G3 steel [26], [27] (according to PN: 18G2A), 30CrNiMo8 steel [28], [29] and 2017A-T4 aluminium alloy [30], [31], [32]. All experimental data apply to specimens subjected to: (i) uniaxial cyclic plane bending, (ii) uniaxial cyclic torsion, (iii) proportional cyclic bending and torsion. Only waveforms with a zero mean

Parameters for criteria evaluation

The evaluation of the effectiveness of criteria for multiaxial fatigue of materials for a limited number of cycles to failure typically involves the comparison of the calculated strength N_cal with the experimental one N_exp on a log–log diagram [36], [37], [38], [39], [40], [41] with the additionally calculated parameters of the scatter of results [9], [33], [41]. In order to evaluate the performance of the proposed algorithm, the original function Pr(T) has been proposed along with the

Evaluation of the convergence of the proposed algorithm

The convergence of the proposed algorithm (Fig. 1) has been examined by analysing the waveforms of the objective function (19) by iteratively changing the number of cycles N for uniaxial loads and load combinations. The calculation results shown in Fig. 2 demonstrate the existence of a local minimum of the function Er(N) for all analysed reducing functions (criteria) and all types of applied loads. The occurrence of a local minimum with the value of Er = 0 guarantees the convergence of the

Results and discussion

The number of cycles to failure N_cal has been calculated using two algorithms which, for the clarity of presentation, are designated as: NA – new algorithm proposed in this paper, which takes into account the variability of coefficients a(N), b(N) (Table 1) depending on the number of cycles N; CA – the classical algorithm, wherein the a and b coefficients are constants corresponding to the theoretical fatigue limit, i.e. for N = 2·10⁶ cycles for steel and N = 10⁷ cycles for aluminium alloy. Fig. 3,

Conclusions

We can draw the following conclusions based on the foregoing analyses:

1.
The proposed algorithm for calculating the fatigue life, which takes into account the variability of the coefficients existing in fatigue criteria depending on the number of cycles is convergent in the analysed areas of proportional cyclic loads.
2.
A good consistency of experimental and calculated fatigue life has been obtained using the new algorithm for three analysed materials, i.e. S355J2WP and S355J2G3 steel, and 2017A-T4

References (41)

J. Papuga
A survey on evaluating the fatigue limit under multiaxial loading
Int J Fatigue
(2011)
I.V. Papadopoulos et al.
A comparative study of multiaxial high-cycle fatigue criteria for metals
Int J Fatigue
(1997)
A. Carpinteri et al.
Multiaxial fatigue assessment using a simplified critical plane-based criterion
Multiaxial Fatigue Models
(2011)
J. Li et al.
Multiaxial fatigue life prediction for various metallic materials based on the critical plane approach
Int J Fatigue
(2011)
A. Karolczuk et al.
Selection of the critical plane orientation in two-parameter multiaxial fatigue failure criterion under combined bending and torsion
Eng Fract Mech
(2008)
J. Park et al.
Evaluation of an energy-based approach and a critical plane approach for predicting constant amplitude multiaxial fatigue life
Int J Fatigue
(2000)
K. Slámečka et al.
Fatigue life of cast Inconel 713LC with/without protective diffusion coating under bending, torsion and their combination
Eng Fract Mech
(2013)
A. Karolczuk et al.
Non-local stress gradient approach for multiaxial fatigue of defective material
Comput Mater Sci
(2008)
A. Carpinteri et al.
Multiaxial high-cycle fatigue criterion for hard metals
Int J Fatigue
(2001)
J. Papuga et al.
Two new multiaxial criteria for high cycle fatigue computation
Int J Fatigue
(2008)

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Citation Excerpt :
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A correction in the algorithm of fatigue life calculation based on the critical plane approach

Highlights

Abstract

Introduction

Section snippets

Stanfield (1935), Stullen-Cummings (1954), Findley (1959) criterion: C1

Generalization of the analysed criteria for a limited number of cycles to failure

Corrected algorithm for calculating the fatigue life under proportional loads

Experimental data

Parameters for criteria evaluation

Evaluation of the convergence of the proposed algorithm

Results and discussion

Conclusions

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