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ASSESSMENT OF FATIGUE CRACK RATE UNDER  

CYCLIC LOADING 
 

Assistant Lecturer: Muhanad Hamed Mosa 

University of Al-Qadisiya, College of Engineering, Department of Mechanical Engineering 

Email: alafaq_eng@yahoo.com 

Received 7 April 2014         Accepted 21 May 2014      

 

ABSTRACT 

Fatigue crack growth (FCG) in structures subjected to variable amplitude (VA) loading is a 

complex phenomenon. Analysing of FCG rate is important for the reliable life of engineering 

structures. It is difficult to model all the parameters influence FCG correctly due to the random 

nature of the VA loading as well as the number and complexity of the mechanisms involved in 

the FCG problems. . It has been found from the literature review that no universal model has 
been developed to analyse the crack growth condition under VA loading. In addition, no general 

understanding has also been agreed among researchers for any available models. Therefore, the 

main objective of this work is to investigate the FCG under VA loading based on FCG models 

also to propose a suitable model for VA loading. The work describes some of the FCG models 

for predicting the fatigue lives and FCG rates. For the simulation part of this study, towards 

prediction of crack propagation under cyclic, variable and random loading were used. The results 

had been carried out based on the Austen, modified Forman and NASGRO models. There are 

many factors affecting the FCG, which shown with great influence such as; initial crack length, 

load sequence, aspect ratio and stress ratio. 

 

Keyword: Fatigue crack growth (FCG), variable amplitude (VA), finite element method (FEM) 

 

INTRODUCTION 

The phenomenon of fatigue has been discovered in the post-incident findings of the Versailles 

incident in 1842.  Since then, engineers and scientists have developed models to predict the 

fatigue life of components and to incorporate the fatigue analysis in the design.  When pressure 

vessels and piping are subjected to fluctuations in stress, they may lead to the development of 

fatigue cracks. Fatigue cracks extend slowly, and this is generally with a very small increment of 

crack growth occurring with each cycle, and with little or no evidence of plastic deformation. 

The cracks can continue to grow until they completely cause failure of the component, member 

or structure by fast fracture, plastic collapse or other mode, which prevent service duties being 

performed. The service load histories may completely in random pattern. It is either to be simple 
and repetitive or at the other extreme. In general, prediction models published in the literature 

employ basic material fatigue data as references. Such data can be fatigue limits, stress-life (S-N) 

data, fatigue diagrams, crack growth data, and fracture toughness for the final failure. These 

prediction models that are used for crack growth under variable amplitude (VA) loading vary 

from simple modifications on the constant amplitude (CA) baseline up to the complex models 

with detailed descriptions of relevant fracture mechanisms. For instance, some models calculate 

crack growth by averaging over the applied load spectrum, while many others tend to calculate it 

using the cycle-by-cycle analysis. Despite the on-going development of the prediction models 

mailto:alafaq_eng@yahoo.com


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towards a more accurate description of the phenomena, there seems to be no general agreement 

about which mathematical description is the most useful. In fact, the simplest prediction models 

are still being used by many engineers. The alternative to make prediction is to carry out 

experiments for specific fatigue questions when they arise. Unfortunately, testing is not always 

possible due to geometry complication, time, costs etc (Simonsen & Tornqvist 2004; Richard et 

al. 2008).  

This is even complicated by the fact that it is not at all easy to accomplish experimental fatigue 

conditions which will give a relevant answer to the question raised. Thus, the overall fatigue life 

of a component is generally defined as follows: “Total life = Initiation life + Propagation life”. 

This can be represented in a block diagram of Figure 1.1. 

The prediction of fatigue properties of structures and avoiding structural fatigue were recognised 

as engineering problems in the early decades of the 20th century. High stress concentrations 

were understood and taken as possibly harmful and should therefore be avoided. The 

significance of stress concentration factors had been identified before 1950 and designers also 

realised that the fatigue performance of a structure was dependent upon improved detail design 

(Schijve 2003). In 1950s, many investigators mentioned how early in the fatigue life they could 

observe microcracks. Since then, it has been clear that the fatigue life under cyclic loading 

consists of two phases, namely the crack initiation life, followed by a crack growth period until 

failure (Khan et al. 2004). 

 Predicting fatigue crack growth in metals under random loadings is a difficult task, particularly 

because of the load history effects, which are known for decades to stem from plastic 

deformation in the vicinity of the crack tip (Schijve 1999; Skorupa 1999; Hamam et al. 2007). In 

addition, history effects are closely related to the elastic–plastic behaviour of the material. 

Although many fatigue crack growth (FCG) models have been developed, none of them enjoys 

universal acceptance. Each model can only account for one or several phenomenological 

factor(s). 

From an engineering point of view, the experience of dealing with fatigue problems in the 

beginning of the previous century was still a matter of trial and error. In later decades, however, 

continued research (Schijve 2003; Huang et al. 2005) clearly indicated, that the number of 

variables that could affect the fatigue strength and the fatigue life of a structure is large. In 

principle, it is correct to consider fatigue as a phenomenon, which is characterised by microcrack 

initiation, crack growth as an invisible microcrack, and later as a visible macrocrack which 

finally leads to a complete failure. Unfortunately, this concept does not mean that the fatigue 

phenomenon occurs in the same way in all metallic materials (Schijve 2003).Though many 

damage models have been developed, none of them enjoys universal acceptance (Fatemi & Yang 

1998; Paris et al. 1999; Molent et al. 2008). Each damage model can only account for one or 

several phenomenological factors, such as load dependence, multiple damage stages, nonlinear 

damage evolution, load sequence and interaction effects, overload effects, small amplitude cycles 

below fatigue limit and mean stress. Due to the complexity of the problem, none of the existing 

predictive models can encompass all of these factors. Therefore, the applicability of each model 

varies from case to. 

Fracture mechanics seeks to establish the local stress and strain fields around a crack tip in terms 

of global parameters such as the loading and the geometry of the structure. The fracture 

mechanical approaches are usually divided into linear elastic solutions and non-linear methods.  

For linear elastic materials, Irwin (1957) suggested describing the stresses in the vicinity of the 

crack by stress intensity factors (SIFs). There are basically three different types of SIFs, with 

each describing one of the deformation modes illustrated in Figure 1.2.  The superposition of 



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these three modes forms the general case of cracking. Meanwhile, the deformation modes can be 

characterised as follows: 

 

(a) Mode I: The in-plane tensile mode where the crack surface is symmetrically opened, 

(b) Mode II: The sliding or shear mode which is present when the crack is exposed to skew-

symmetric in-plane loading, 

(c) Mode III: An anti-plane mode where the crack surface is twisted by forces perpendicular to 

the crack plane. 

 

 In the simulations of crack propagation in linear elastic fracture mechanics (LEFM), it is usually 

the combination of modes I and II which is of greatest interest. Irwin (1957) analysed this in-

plane mixed mode problem, and using the Westergaard (1939) stress functions, he found an 

analytical solution for the stress distribution in the vicinity of the crack tip. 

 

SIMULATION OF FATIGUE CRACK GROWTH 

Assessing the behaviour of fatigue cracks in any structure is a complex and uncertain process 

due to the large number of parameters that may influence the crack growth process. For instance, 

pipes or tubes contain defects from the manufacturing, installation and servicing processes. The 

defects can affect the safety of the structures, and even depress their service life that may lead to 

enormous economic costs and jeopardise the surrounding ecological environments. Numerous 

models have been developed over the past few decades and these have taken different 

approaches to modelling FCG; however, no universal or all-encompassing model exists. Most 

models are developed for a specific application and they incorporate the main parameters 

influencing FCG for the structure in question. A successful model is like a balance between 

being able to accurately predict the crack growth rate and its simplicity. Such a model is 

beneficial in planning inspection intervals, determining required maintenance and minimising the 

problem of crack propagation in any future designs. 

The fatigue life prediction has brought significant advantages to designers, not least of all the 

abilities to do up-front fatigue calculations long before a prototype available.  However, the 

combination of the two technologies also poses challenges. The engineering need for prediction 

models and some of the CA and VA loading FCG models were discussed. In this chapter, the 

methodology to select three of these models (the Austen, modified Forman and NASGRO 

models) and the fatigue life and FCG prediction as well as the factors affecting it. Then, the 

experiments setting are presented so as to enable the readers to fully understand how this work is 

accomplished. 

The start of a fatigue failure is a strictly local process and also depends on the dynamics of the 

system. It is important to highlight that the time history of stress or strain, at the exact location 

where a crack is going to start, is the critical factor whereas the general distribution of these 

parameters throughout the component is of secondary interest. Accurate measurement, data 

acquisition, analysis, and testing are among the key factors in the process of calculating structure 

performance. Most structures survive in a VA loading environment and a predominant failure 

mechanism under these conditions is fatigue. Exploiting fatigue knowledge and the use of 

computer-based analysis techniques, at an early stage in the design process, can dramatically 

reduce the developmental period. Therefore, the designer has the opportunity to estimate the 

effects of changing geometry, material and loading performance. 

Many shell structures such as pressure vessels or pipelines, contain defects from the 

manufacturing and servicing processes. These defects can affect the safety of the structure and 

even depress their life; may lead to enormous costs and jeopardize the surrounding ecological 



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environments. For fatigue life prediction (S-N), the finite element technique was used for 

modelling and simulating the case study of the shell as a cylinder with infinite length in three-

dimensional mesh, as shown in Figure 1.3.Selecting the right techniques of meshing are based on 

the geometry, model topology, analysis objectives and engineering judgment. Triparametric 

solids with the topological shape of a brick or wedge can be meshed either hex or wedge 

elements. Any other form of the triparametric solid can only be meshed with tetrahedral (TET) 

elements. The auto TetMesh approach is a highly automated technique for meshing solid regions 

of the geometry. It creates a mesh of tetrahedral elements for any closed solid including 

boundary representation (B-rep) solid. Tetmeshing produces high quality meshing for B-rep 

solids model imported from the computer aided design (CAD) systems. Since the tetrahedral is 

found to be the best meshing technique, the TET4 (4 nodes tetrahedral) version of the cylinder 

head was then used for the initial analysis. In addition, the TET4 compared to the TET10 (10 

nodes tetrahedral) mesh using the same global mesh length. The TET10 mesh is presumed to 

represent a more accurate solution since TET4 meshes are known to be dreadfully stiff. For this 

purpose, the finite element analysis technique, with tetrahedral elements (10 nodes) shown in 

Figure 1.4, was applied to model and simulate based on the MSC Nastran/Patran analysis codes 

(MSC 2005). In order to study the fatigue life prediction, a shell with T/D < 0.1 (where T is the 

thickness and D is the diameter) and L/D > 10 (where L is the length) was analysed. 

The cycle-by-cycle analysis can be performed with or without involving interaction effects, i.e. 

the effects of a load cycle on the crack growth in later cycles. Both the constant and random 

amplitude loadings are shown in Figure 1.5. For the VA loading, similar programmable VA load 

histories, as given by with different values, were used in the analysis with different load 

sequences and values ranging from high to low or from low to high, as shown in Figure 1.6. The 

presence of the interaction effects is always altering the crack growth rate, under the application 

of the VA loading. In order to correctly predict the crack growth under VA loading, it is 

necessary to involve the interaction effects, while developing the prediction models, as a part of 

cycle by cycle analysis using different models. The CA loading has been developed to correlate 

FCG rates for different values. 

The steel used in the fabrication of pressure vessels is usually of two kinds, viz. carbon steel and 

alloy steel. In the pressure vessel steel, carbon is of prime importance because of its 

strengthening effect. It also raises the transition temperature, lowers the maximum energy values 

and widens the temperature range between completely brittle behaviour.  Manganese (up to 

1.5%), on the other hand, improves low temperature properties (ASM 1993). Of all the different 

kinds of steel, those produced in greatest quantity fall within the low carbon classification. These 

steel types generally contain less than 0.25 wt. %C and are unresponsive to heat treatments 

intended to form martensite, as strengthening is accomplished by cold work. Meanwhile, 

microstructure consists of ferrite and pearlite constituents. As a consequence, these alloys are 

relatively soft and weak, but have outstanding ductility and toughness, in addition to the fact that 

they are machinable, weldable and most important of all, steel is the least expensive to produce. 

They typically have yield strength of 585 MPa, tensile strengths between 637 and 825 MPa, and 

a ductility of 20-25 %EL. In particular, ASTM A533 is one such kind of steel which has 

applications in pressure vessels (ASTM standard V1.04).In this analysis, the material of the shell 

ASTM A533 steel has been used. 

 

MODELLING AND SIMULATION 

It is important to note that the fatigue life prediction based on modelling is very time consuming. 

For example, an off-shore platform designed to withstand 108 cycles in a 30 year life would 

require a 30 year test at 0.2 Hz to validate the design (MSC 2005). Computer-based prediction 

methodologies can perform the same validation with an equal accuracy in a very small fraction 



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of the time (e.g. less than two hours) at a very small fraction of the cost. Such prediction 

software requires the fatigue crack growth to be accumulated on a cycle-by-cycle basis and it 

must also incorporate the complex effects of the stress history, crack closure, static fracture 

modes and notches and be able to predict the environmental influences on fatigue crack growth 

rates (corrosion fatigue). Stephens et al. (2001) have estimated that between 50% and 90% of 

these failure are due to fatigue.  As a result of the recent advances in the finite element method 

(FEM) and its application to fracture mechanics, various numerical methods exist to calculate the 

fracture parameters and to evaluate the FCG models. One of the advantages of the numerical 

method is that accurate results can be obtained for complex problems of any geometry and crack 

configuration in a fraction of the time and cost, as compared to the alternative of 

experimentation. 

 

Finite Element Approach 

The start of a fatigue failure is a strictly local process and it is also one that depends on the 

dynamics of the system. The time history of stress or strain, at the exact location where a crack is 

going to start, is a critical factor and a general distribution of these parameters throughout the 

component is of secondary interest. This is precisely why the finite element analysis (FEA) is 

important in this particular discipline. With FEA, an analyst can choose any location within a 

model and concentrate on it. In fact, using FEA can give a tighter control over the move from 

general geometry and loading to the local parameters, and allows dynamic factors to be dealt 

with more analytically. 

The concepts of fracture mechanics that were derived prior to 1960 are applicable only to 

materials that obey the Hook’s law. Although corrections for small-scale plasticity were 

proposed as early as 1948, these analysis are restricted to structures whose global behaviour is 

linear elastic. Since 1960, fracture mechanics theories have been developed to account for 

various types of nonlinear material behaviour (i.e. plasticity and viscoplasticity) as well as 

dynamic effects. All of these more recent results, however, are extensions of LEFM. Thus a solid 

background in the fundamentals of LEFM is essential to an understanding of more advanced 

concepts in fracture mechanics. The linear static analysis was performed using the finite element 

software to determine the stress and strain. The finite element technique was used for modelling 

and simulation of the case study of the shell as a cylinder with infinite length in three-dimension 

mesh shown in Figure 1.3. For this purpose, the F technique with tetrahedral elements (10 nodes) 

shown in Figure 1.4 was being used for the modelling and simulating based on MSC 

Nastran/Patran analysis codes. To study the fatigue life prediction a shell with T/D < 0.1 and L/D 

> 10 (where T is the thickness, D is the diameter and L is the length) was analysed.The results of 

the maximum principle stresses and strains are used for the subsequent fatigue life analysis, 

using the cyclic load history (discussed above) as an internal pressure to the shell. Figure 1.7 

shows the contour (image) of the stress distribution on the shell. 

 

Software Codes 

It is important to note that more than one software was used in this work for the modelling or the 

experiments. Firstly, the fatigue life predication (S-N) was performed based on the MSC 

Nastran/Patran software (MSC 2005). The crack growth software, developed by the nCode 

(2003), offers engineers the facilities to predict defect tolerant fatigue life based on all these 

aspects. These diagnostic techniques enable iteration and interpolation with different material, 

geometry and history inputs, as well as probably the most important variable, i.e. the initial crack 

size. As for the modelling part of this work, the Glyphwork software was used to predict FCG 

after building the block system of all parts shown in Figure 1.8. 

 



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RESULTS AND DISCUSSION 

In general, the plastic-zone size at crack front increases as it grows in a metallic material under 

cyclic loading. At low stress-intensity factor levels, the plane-strain conditions should prevail as 

the plastic-zone size becomes large compared to thickness. The FCG life under CA loading, 

using three different FCG models (the Austen, modified Forman and NASGRO models) is 

shown in Figure 1.9. Based on the simulation results, the Austen model was found to give the 

lowest value of life, i.e. 35% less than the NASGRO model. On the contrary, the modified 

Forman model gave a better life than that by 20%, while the NASGRO model revealed a higher 

life, i.e. 128 x 103 cycles to failure. The differences in fatigue lives predicted are mainly due to 

basic theoretical background for each model. The effects of using different fatigue crack growth 

models on the fatigue crack growth rate are shown in Figure 1.10. The differences are related to 

the results presented in the a-N curves.  

The modified Forman model is capable of describing region III of the fatigue rate curve as well 

as included the stress ratio effect, while the NASGRO model take into account the crack closure 

effect as well as the previous factors. The Austen model is including the crack closure effect and 

capable of representing data in region II same as the other two models. So the NASGRO model 

represents more factors than the other two models and gives better representation of the data. 

 

CONCLUSION 

The methodology of simulation and modelling as well as the cyclic experimental setting 

suggested clarifying the objectives of this work. The present research focused on studying FCG 

under variable and random loadings. The finite element modelling and analysis of pressure 

vessel or piping were utilised to predict stress-life. The factors affecting the S-N curve, such as 

mean stress, wall thickness, surface finish, type of elements and mesh size, were also presented. 

The TPB specimen of ASTM A533 steel material was used in the assessment study on FCG 

under different loading conditions, such as cyclic, variable and random loadings. Therefore, the 

emphasis in this study was more on the assessment of FCG under variable and random loadings, 

based on three different FCG models, namely, the Austen, modified Forman and NASGRO. It is 

important to note that different models give different fatigue lives. In order to validate these 

results, a set of experiments were performed under the same boundary conditions and they were 

then compared with the predicted values. More importantly, a good agreement was found 

between the findings in the present study and those in the previous research. 

 The regression technique was used and shown by the polynomial equations for each set of the 

experiments and the comparison revealed discrepancies as a result of different loading 

conditions.  Among the factors affecting the FCG rate were initial crack length, stress ratio, 

aspect ratio and load sequence. The results also showed that increasing the initial crack length, 

stress ratio and aspect ratio would tend to minimise the fatigue life. For the cases of the load 

sequence effect, the number of blocks and the stress ratio has significantly affected the crack 

growth rate. 

 

 

REFRENCES 

 

Fatemi, A. & Yang, L. 1998. Cumulative fatigue damage and life prediction theories: a survey of 

the state of the art for homogeneous materials. International Journal of Fatigue 20(1): 9-34. 

 

Hamam, R., Pommier, S. & Bumbieler, F. 2007. Variable amplitude fatigue crack growth, 

experimental results and modelling. International Journal of Fatigue 29: 1634-1646. 

 



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MSC. FATIGUE. 2005. User manual. Los Angles: MSC. Software Corporation. 

nCode. 2003.  User guide manual. ICE-FLOW cracks growth. V 2.0. 

 

Molent, L., McDonald, M., Barter, S. & Jones, R. 2008. Evaluation of spectrum fatigue crack 

growth using variable amplitude data. International Journal of Fatigue 30: 119-137. 

 

Paris, P.C., Tada, H. & Donald, J.K. 1999. Service load fatigue damage-a historical perspective. 

International Journal of Fatigue 21: S35-S46. 

 

Richard, H., Sander, M., Fulland, M. & Kullmer, G. 2008. Development of fatigue crack growth 

in real structures. Engineering Fracture Mechanics 75: 331-340. 

 

Simonsen, B.C. & Tornqvist, R. 2004. Experimental and numerical modeling of ductile crack 

propagation in large-scale shell structures. Marine Structures 17: 1-27. 

 

Schijve, J.1999. The significance of fractography for investigations of fatigue crack growth 

under variable amplitude fatigue. Fatigue Fracture Engineering Material Structures  22 :87-99. 

 

Schijve, J. 2003. Fatigue of structures and materials in the 20th century and the state of the art. 

International Journal of Fatigue 25: 679-702. 

 

Skorupa, M. 1999. Load interaction effects during fatigue crack growth under variable amplitude 

loading- a literature review Part II: qualitative interpretation. Fatigue Fracture Engineering 

Material Structures 22(10): 905-926. 

 

Skorupa, M. 2001. Load interaction effects during fatigue crack growth under variable amplitude 

loading- a literature review. Part 1: Empirical trends, part II: Qualitative interpretation. Fatigue 

Fracture Engineering Materials Structures 21: 987-1006. 

 

Stephens, R.I., Fatemi, A., Stephens, R.R. & Fuchs, H.O. 2001. Metal Fatigue in Engineerring. 

2nd Edition. New York: J. Wiley & Sons. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 



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Figure 1.2: Three models of cracking Source: Anderson 2005 

Figure 1.3: A finite element mesh of a shell structure 



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Figure 1.5: Display of constant and random loading 

histories 

Figure 1.4: Integration grid points of the TET10 nodes 

 

Case 2 Case 1 

Figure 1.6: Display of the load histories with different two sequences  



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Figure 1.7: Contour displays of stresses on shell structure 

Figure 1.8: The FCG worksheet based on the Glyphwork 

softwareSource: nCode 2003 

 



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Figure 1.9: Fatigue crack growth using different models 

Figure 1.10: Fatigue crack growth rates for various TPB specimen geometries based on 

different FCG models 

∆K   (MPa. m
1/2

) 
0.001

0.01

0.1

1

d
a
/d

N
  

(m
m

/c
y

c
le

)

Austen model

mod. Forman model

NASGRO model