Advances in Materials Science and Engineering

Volume 2016, Article ID 9573524, 26 pages

http://dx.doi.org/10.1155/2016/9573524

## A Review on Fatigue Life Prediction Methods for Metals

^{1}Mechanical and Industrial Engineering Department, College of Engineering, Qatar University, Doha 2713, Qatar^{2}Department of Mechanical Engineering, Hanyang University, 222 Wangsimni-ro, Seongdong-gu, Seoul 133-791, Republic of Korea^{3}Dipartimento di Ingegneria Industriale e Scienze Matematiche (DIISM), Università Politecnica delle Marche, 60131 Ancona, Italy

Received 19 June 2016; Accepted 17 August 2016

Academic Editor: Philip Eisenlohr

Copyright © 2016 E. Santecchia et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Metallic materials are extensively used in engineering structures and fatigue failure is one of the most common failure modes of metal structures. Fatigue phenomena occur when a material is subjected to fluctuating stresses and strains, which lead to failure due to damage accumulation. Different methods, including the Palmgren-Miner linear damage rule- (LDR-) based, multiaxial and variable amplitude loading, stochastic-based, energy-based, and continuum damage mechanics methods, forecast fatigue life. This paper reviews fatigue life prediction techniques for metallic materials. An ideal fatigue life prediction model should include the main features of those already established methods, and its implementation in simulation systems could help engineers and scientists in different applications. In conclusion, LDR-based, multiaxial and variable amplitude loading, stochastic-based, continuum damage mechanics, and energy-based methods are easy, realistic, microstructure dependent, well timed, and damage connected, respectively, for the ideal prediction model.

#### 1. Introduction

Avoiding or rather delaying the failure of any component subjected to cyclic loadings is a crucial issue that must be addressed during preliminary design. In order to have a full picture of the situation, further attention must be given also to processing parameters, given the strong influence that they have on the microstructure of the cast materials and, therefore, on their properties.

Fatigue damage is among the major issues in engineering, because it increases with the number of applied loading cycles in a cumulative manner, and can lead to fracture and failure of the considered part. Therefore, the prediction of fatigue life has an outstanding importance that must be considered during the design step of a mechanical component [1].

The fatigue life prediction methods can be divided into two main groups, according to the particular approach used. The first group is made up of models based on the prediction of crack nucleation, using a combination of damage evolution rule and criteria based on stress/strain of components. The key point of this approach is the lack of dependence from loading and specimen geometry, being the fatigue life determined only by a stress/strain criterion [2].

The approach of the second group is based instead on continuum damage mechanics (CDM), in which fatigue life is predicted computing a damage parameter cycle by cycle [3].

Generally, the life prediction of elements subjected to fatigue is based on the “safe-life” approach [4], coupled with the rules of linear cumulative damage (Palmgren [5] and Miner [2]). Indeed, the so-called Palmgren-Miner linear damage rule (LDR) is widely applied owing to its intrinsic simplicity, but it also has some major drawbacks that need to be considered [6]. Moreover, some metallic materials exhibit highly nonlinear fatigue damage evolution, which is load dependent and is totally neglected by the linear damage rule [7]. The major assumption of the Miner rule is to consider the fatigue limit as a material constant, while a number of studies showed its load amplitude-sequence dependence [8–10].

Various other theories and models have been developed in order to predict the fatigue life of loaded structures [11–24]. Among all the available techniques, periodic* in situ* measurements have been proposed, in order to calculate the macrocrack initiation probability [25].

The limitations of fracture mechanics motivated the development of local approaches based on continuum damage mechanics (CDM) for micromechanics models [26]. The advantages of CDM lie in the effects that the presence of microstructural defects (voids, discontinuities, and inhomogeneities) has on key quantities that can be observed and measured at the macroscopic level (i.e., Poisson’s ratio and stiffness). From a life prediction point of view, CDM is particularly useful in order to model the accumulation of damage in a material prior to the formation of a detectable defect (e.g., a crack) [27]. The CDM approach has been further developed by Lemaitre [28, 29]. Later on, the thermodynamics of irreversible process provided the necessary scientific basis to justify CDM as a theory [30] and, in the framework of internal variable theory of thermodynamics, Chandrakanth and Pandey [31] developed an isotropic ductile plastic damage model. De Jesus et al. [32] formulated a fatigue model involving a CDM approach based on an explicit definition of fatigue damage, while Xiao et al. [33] predicted high-cycle fatigue life implementing a thermodynamics-based CDM model.

Bhattacharya and Ellingwood [26] predicted the crack initiation life for strain-controlled fatigue loading, using a thermodynamics-based CDM model where the equations of damage growth were expressed in terms of the Helmholtz free energy.

Based on the characteristics of the fatigue damage, some nonlinear damage cumulative theories, continuum damage mechanics approaches, and energy-based damage methods have been proposed and developed [11–14, 21–23, 34, 35].

Given the strict connection between the hysteresis energy and the fatigue behavior of materials, expressed firstly by Inglis [36], energy methods were developed for fatigue life prediction using strain energy (plastic energy, elastic energy, or the summation of both) as the key damage parameter, accounting for load sequence and cumulative damage [18, 37–45].

Lately, using statistical methods, Makkonen [19] proposed a new way to build design curves, in order to study the crack initiation and to get a fatigue life estimation for any material.

A very interesting fatigue life prediction approach based on fracture mechanics methods has been proposed by Ghidini and Dalle Donne [46]. In this work they demonstrated that, using widespread aerospace fracture mechanics-based packages, it is possible to get a good prediction on the fatigue life of pristine, precorroded base, and friction stir welded specimens, even under variable amplitude loads and residual stresses conditions [46].

In the present review paper, various prediction methods developed so far are discussed. Particular emphasis will be given to the prediction of the crack initiation and growth stages, having a key role in the overall fatigue life prediction. The theories of damage accumulation and continuum damage mechanics are explained and the prediction methods based on these two approaches are discussed in detail.

#### 2. Prediction Methods

According to Makkonen [19], the total fatigue life of a component can be divided into three phases: (i) crack initiation, (ii) stable crack growth, and (iii) unstable crack growth. Crack initiation accounts for approximately 40–90% of the total fatigue life, being the phase with the longest time duration [19]. Crack initiation may stop at barriers (e.g., grain boundaries) for a long time; sometimes the cracks stop completely at this level and they never reach the critical size leading to the stable growth.

A power law formulated by Paris and Erdogan [47] is commonly used to model the stable fatigue crack growth:and the fatigue life is obtained from the following integration:where is the stress intensity factor range, while and are material-related constants. The integration limits and correspond to the initial and final fatigue crack lengths.

According to the elastic-plastic fracture mechanics (EPFM) [48–50], the crack propagation theory can be expressed aswhere is the integral range corresponding to (1), while and are constants.

A generalization of the Paris law has been recently proposed by Pugno et al. [51], where an instantaneous crack propagation rate is obtained by an interpolating procedure, which works on the integrated form of the crack propagation law (in terms of - curve), in the imposed condition of consistency with Wöhler’s law [52] for uncracked material [51, 53].

##### 2.1. Prediction of Fatigue Crack Initiation

Fatigue cracks have been a matter of research for a long time [54]. Hachim et al. [55] addressed the maintenance planning issue for a steel S355 structure, predicting the number of priming cycles of a fatigue crack. The probabilistic analysis of failure showed that the priming stage, or rather the crack initiation stage, accounts for more than 90% of piece life. Moreover, the results showed that the propagation phase could be neglected when a large number of testing cycles are performed [55].

Tanaka and Mura [56] were pioneers concerning the study of fatigue crack initiation in ductile materials, using the concept of slip plastic flow. The crack starts to form when the surface energy and the stored energy (given by the dislocations accumulations) become equal, thus turning the dislocation dipoles layers into a free surface [56, 57].

In an additional paper [58], the same authors modeled the fatigue crack initiation by classifying cracks at first as (i) crack initiation from inclusions (type A), (ii) inclusion cracking by impinging slip bands (type B), and (iii) slip band crack emanating from an uncracked inclusion (type C). The type A crack initiation from a completely debonded inclusion was treated like crack initiation from a void (notch). The initiation of type B cracks at the matrix-particle interface is due to the impingement of slip bands on the particles, but only on those having a smaller size compared to the slip band width. This effect inhibits the dislocations movements. The fatigue crack is generated when the dislocation dipoles get to a level of self-strain energy corresponding to a critical value. For crack initiation along a slip band, the dislocation dipole accumulation can be described as follows:where is the specific fracture energy per unit area along the slip band, is the friction stress of dislocation, is the shear modulus, is the shear stress range, and is the grain size [58]. Type C was approximated by the problem of dislocation pile-up under the stress distribution in a homogeneous, infinite plane. Type A mechanism was reported in high strength steels, while the other two were observed in high strength aluminum alloys. The quantitative relations derived by Tanaka and Mura [58] correlated the properties of matrices and inclusions, as well as the size of the latter, with the fatigue strength decrease at a given crack initiation life and with the reduction of the crack initiation life at a given constant range of the applied stress [58].

Dang-Van [59] also considered the local plastic flow as essential for the crack initiation, and he attempted to give a new approach in order to quantify the fatigue crack initiation [59].

Mura and Nakasone [60] expanded Dang-Van’s work to calculate the Gibbs free energy change for fatigue crack nucleation from piled up dislocation dipoles.

Assuming that only a fraction of all the dislocations in the slip band contributes to the crack initiation, Chan [61] proposed a further evolution of this theory.

Considering the criterion of minimum strain energy accumulation within slip bands, Venkataraman et al. [62–64] generalized the dislocation dipole model and developed a stress-initiation life relation predicting a grain-size dependence, which was in contrast with the Tanaka and Mura theory [56, 58]: where is the surface-energy term and is the slip-irreversibility factor (). This highlighted the need to incorporate key parameters like crack and microstructural sizes, to get more accurate microstructure-based fatigue crack initiation models [61].

Other microstructure-based fatigue crack growth models were developed and verified by Chan and coworkers [61, 65–67].

Concerning the metal fatigue, after the investigation of very-short cracks behavior, Miller and coworkers proposed the immediate crack initiation model [68–71]. The early two phases of the crack follow the elastoplastic fracture mechanics (EPFM) and were renamed as (i) microstructurally short crack (MSC) growth and (ii) physically small crack (PSC) growth. Figure 1 shows the modified Kitagawa-Takahashi diagram, highlighting the phase boundaries between MSC and PSC [69, 71].