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L. C. H. Ricardo et alii, Frattura ed Integrità Strutturale, 35 (2016) 456-471; DOI: 10.3221/IGF-ESIS.35.52                                                        
 

456 
 

 
 
 
  
Crack simulation models in variable amplitude loading - a review 
 
 
Luiz Carlos H. Ricardo 
Materials Technology Department, IPEN, University of São Paulo, Brazil, Instituto de Pesquisas Energéticas e Nucleares  
Av. Lineu Prestes 2242 - Cidade Universitária - São Paulo - SP BRASIL- CEP: 05508-000. 
lricardo@ipen.br 
 
Carlos Alexandre J. Miranda 
Nuclear Engineering Department, IPEN, University of Sao Paulo, Brazil, Instituto de Pesquisas Energéticas e Nucleares  
Av. Lineu Prestes 2242 - Cidade Universitária - São Paulo - SP BRASIL- CEP: 05508-000  
 

 
ABSTRACT. This work presents a review of crack propagation simulation models considering plane stress and 
plane strain conditions. It is presented also a chronological different methodologies used to perform the crack 
advance by finite element method. Some procedures used to edit variable spectrum loading and the effects 
during crack propagation processes, like retardation, in the fatigue life of the structures are discussed. Based on 
this work there is no consensus in the scientific community to determine the best way to simulate crack 
propagation under variable spectrum loading due the combination of metallurgic and mechanical factors 
regarding, for example, how to select and edit the representative spectrum loading to be used in the crack 
propagation simulation.  
  
KEYWORDS. Fatigue; Crack propagation simulation; Finite element method; Retardation. 
 
 
INTRODUCTION  
 

he most common technique for predicting the fatigue life of automotive, aircraft and wind turbine structures is 
Miner’s rule [1]. Despite the known deviations, inaccuracies and proven conservatism of Miner’s cumulative 
damage law, it is even nowadays being used in the design of many advanced structures. Fracture mechanics 

techniques for fatigue life predictions remain as a back up in design procedures. The most important and difficult problem 
in using fracture mechanics concepts in design seems to be the use of crack growth data to predict fatigue life. The 
experimentally obtained data is used to derive a relationship between stress intensity range (K) and crack growth per 
cycle (da/dN). In cases of fatigue loaded parts containing a flaw under constant stress amplitude fatigue, the crack growth 
can be calculated by simple integration of the relation between da/dN and K. However, for complex spectrum loadings, 
simple addition of the crack growth occurring in each portion of the loading sequence produces results that, very often, 
are more erroneous than the results obtained using Miner’s rule with an S-N curve. Retardation tends to cause 
conservative results using Miner’s rule when the fatigue life is dominated by the crack growth. However, the opposite 
effect generally occurs when the life is dominated by the initiation and growth of small cracks. In these cases, large cyclic 
strains, which might occur locally at stress raisers due to overload, may pre-damage the material and lower its resistance to 
fatigue.  
The experimentally derived crack growth equations are independent of the loading sequence and depend only on the 
stress intensity range and the number of cycles for that portion of the loading sequence. The central problem in the 
successful utilization of fracture mechanic techniques applied to the fatigue spectrum is to obtain a clear understanding of 

T 



 

                                                      L. C. H. Ricardo et alii, Frattura ed Integrità Strutturale, 35 (2016) 456-471; DOI: 10.3221/IGF-ESIS.35.52 
 

457 
 

the influence of loading sequences on fatigue crack growth [2]. Investigations covering the effects of particular interest, 
after high overload, in the study of crack growth under variable-amplitude loading in the growth rate region, called crack 
growth retardation, seem to have little interest nowadays.  
Stouffer & Williams [3] and other researchers show a number of attempts to model this phenomenon through 
manipulation of the constants and stress intensity factors in the Paris-Erdogan equation however little appears to have 
been done in the effort to develop a completely rational analysis of the problem. Probably, the only one reason that the 
existing models of retarded crack growth are not satisfactory is that these models are deterministic whereas the fatigue 
crack growth phenomenon shows strong random features. In addition, most of the reported theoretical descriptions of 
the retardation are based on data fitting techniques, which tend to hide the behavior of the phenomenon. If the retarding 
effect of a peak overload on the crack growth is neglected, the prediction of the material lifetime is usually very 
conservative [4]. Accurate predictions of the fatigue life will hardly become possible before the physics of the peak 
overload mechanisms is better clarified. According to the existing findings, the retardation is a physically very complicated 
phenomenon which is affected by a wide range of variables associated with loading, metallurgical properties, environment, 
etc., and it is difficult to separate the contribution of each of these variables [5].  
 
 
CRACK PROPAGATION CONCEPTS  
 

rwin [6,7] defines in his work a release energy rate G, which is a measure of the available energy, dП-potential of 
energy and A-crack area, to provoke crack propagation as shown in Eq. (1). The term rate as employed is not related 
to a derivate in relation to the time but is referred to a change in the potential energy rate in the crack area.  Later, 

this quantity has been called K, and is used to characterize the stress state ("stress intensity") near a crack tip caused by a 
remote load or residual stress in isotropic and elastic bodies. The stress field in the crack tip is given by Eq. (2),  
 

 
d

G
dA


            (1) 

  
 1/2 1/22 3( 2 ) ( ) ( ) ( ) ......ij ij ij ijK r f A g A h r 

             (2) 
 
where K is the stress intensity factor; r and  are the distance from the crack tip and the angle between the crack tip and 
the plane of the crack, respectively; Ai is a constant of the material; fij (), gij () and hij() are functions of ..After 
years, the stress-intensity factors for a large number of crack configurations have been generated; and these have been 
collated into several handbooks (see, for example, Refs [8,9]). The use of K is meaningful only when small-scale yielding 
conditions exist. Plasticity and nonlinear effects will be covered in the next section. Because fatigue-crack initiation is, in 
general, a surface phenomenon, the stress-intensity factors for a surface- or corner-crack in a plate or at a hole, such as 
those developed by Raju and Newman [10,11], are solutions that are needed to analyze small-crack growth. Some of these 
solutions are used later to predict fatigue-crack growth and fatigue lives for notched specimens made of a variety of 
materials [12]. 
Frost and Dugdale [13] have evidenced that the size of the plastic zone increases in the same ratio that of the crack length. 
One can notice that the results of the equation depend linearly on the crack length a; however, Frost and Dugdale [13] 
also argued by dimensional analysis that the incremental propagation in the crack length da, for an incremental number of 
cycles dN, should be directly proportional to the crack length a. Thus,  
 

 
da

Ba
dN

            (3) 

 

where B is a function of the applied stresses.  
Paris & Erdogan [14] conducted a revision on the crack propagation approach from Head [15] and others and discussed 
the similarity of these theories and the differences of results between them, isolated and in group tests. Paris suggested 
that, for a cyclical load variation, the stress field in the crack tip for a cycle can be characterized by a variation of the stress 
intensity factor, 
 

 max minK K K             (4) 

I 



 

L. C. H. Ricardo et alii, Frattura ed Integrità Strutturale, 35 (2016) 456-471; DOI: 10.3221/IGF-ESIS.35.52                                                        
 

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where Kmax and Kmin  are the maximum and the minimum stress intensity factors, respectively. In the crack propagation 
curve, the linear part represents the Paris - Erdogan law, when plotting the values of K vs da/dN in logarithmic scale.  
Fatigue crack initiation and growth under cyclic loading conditions is controlled by the plastic zones that result from the 
applied stresses and exist in the vicinity (ahead) of a propagating crack and in its wake or flanks of the adjoining surfaces. 
For example, the fatigue characteristics of a cracked specimen or component under a single overload or variable amplitude 
loading situations are significantly influenced by these plastic zones. In modelling the fatigue crack growth rate this is 
accounted by the incorporation of accumulative damage cycle after cycle and should include plasticity effects. During the 
crack propagation the plastic zone should grown and the plastic wake will have compressive plastic zones that can help to 
keep the crack close.  
Prediction of the fatigue behaviour of structural components subjected to overloads and variable amplitude loading 
requires an estimation of the plastically affected regions ahead of the crack-tip. One of the most widely used plasticity 
models in fatigue is the Dugdale’s yield strip model [16] In this model the plastically affected zone (ry or rp) is assumed to 
be small as shown Fig. 1. 
 

 
Figure 1: Elastic and Elastic-Plastic Zone Sizes.  

 
Hairman & Provan [17] discuss the problems pertaining to fatigue loading of engineering structures under single overload 
and variable amplitude loading involving the estimation of plasticity affected zones ahead of the crack tip. The models of 
Irwin [6,7] and Dugdale [16]  give an idea of the size of the plastic zone but not of its shape. The size, in general, is 
estimated as a circle of certain diameter (ry or rp) obtained on the basis of reasoning given in the above models for crack-
tip-plasticity. In these models the effect of the shape of the plasticity affected zones is not taken into account.  
To obtain a better idea of the plastic zone shape, the components of stress in the radial and circumferential directions of a 
mode-1 type of loading were derived using an eigenfunction expansion method developed by Williams [18] and with a 
modification to take into consideration crack-tip blunting. The resulting equations are: 

 

 

 

 

 

1 3
5 cos cos , ,

4 2 22

1 3
3 cos cos , ,

4 2 22

1 3
sin sin , ,

4 2 22

rr

r

r

K
T f r

r

K
f r

r

K
f r

r





 
 




 
  




 
 



            
                                                

     (5) 

 
The first terms in Eq. (5) represent the singular terms as r  0 and are, therefore, dominant near the crack-tip. The 
second term in Eq. (5) arises from a consideration of higher power terms. This term is known as the T-stress, it is not 
singular as r  0 but it can affect the elastic-plastic crack-tip stress state. The third terms arise as a contribution from 
crack-tip blunting and are not given in Williams [18]. The contribution of crack-tip blunting has been discussed in Rolfe – 



 

                                                      L. C. H. Ricardo et alii, Frattura ed Integrità Strutturale, 35 (2016) 456-471; DOI: 10.3221/IGF-ESIS.35.52 
 

459 
 

Barsom [19] and the contribution of this term is p=K/()  for a sharp elliptic or hyperbolic notch with a crack-tip 
radius, .  
The above equations can now be used to obtain the principal stresses after the simplifying assumptions of negligible 
contributions of Trr and f(,r,) are assumed. Hence, the principal stresses, as derived from Eq. (5), become: 
 

 

 

1

2

3
21

cos 1 sin
2 22

1
cos 1 sin

4 2 22

0

K

r

K

r

 



 





  

       
               
 
  

          (6) 

  
This, in conjunction with the von Mises and Tresca yield criteria, gives the expressions for the plastic zone shape as 
follows: 
 
von Mises: 

 

 
2

2 2
2

2
2

2

3
sin ( ) (1 2 ) 1 cos( )

4 2
( )

3
1 sin ( ) cos( )

4 2

ys

p

ys

K

r
K

  



 



     
 

 
      

      (7) 

 
Tresca: 

 

2
2

2

222

2

22
2

2

cos sin
2 2 2

( )

cos 1 2 sin
2 2 2

cos 1 sin
2 2 2

ys

p

ys

ys

K

r

K

K

 




 




 




    
    

   



 


               

          

    

       (8) 

 
 
 

 min max
( )

1

m

c

C Kda

dN K K K K



     

 

max

( )m

c

C Kda

dN K K





 

1
max( ) ( )

m mda C K K
dN

   

 

Table 1: Empirical crack growth equations for constant amplitude loading [14]. 

 

Plane Stress 

Plane Strain 

Plane Stress 

Plane Strain 

Plane Stress 

Plane Strain 



 

L. C. H. Ricardo et alii, Frattura ed Integrità Strutturale, 35 (2016) 456-471; DOI: 10.3221/IGF-ESIS.35.52                                                        
 

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In the original Paris crack propagation equation [14] the driving parameters are C, K and m. In Tab. 1 it is possible to 
see some other crack propagation equations for constant amplitude loading, which are modifications of the Paris 
equation, relating the mentioned parameters. 
Murthy et al. [20] discuss crack growth models for variable amplitude loading and the mechanisms and contribution to 
overload retardation. There are many authors which have been developing fatigue crack growth models for variable 
amplitude loading. Tab. 2 presents some authors and the application of their models. 
 

Yield Zone Concept Crack Closure Concept 

Wheeler [21]  Elber [28] 

Willenborg, Engle, Wood [22]  Bell and Creager (Generalized Closure) [29] 

Porter [23]  Newman (Finite Element Method) [30]  

Gray (Generalized Wheeler) [24]  Dill and Staff (Contact Stress ) [31] 

Gallagher and Hughes [25]  Kanninen, Fedderson, Atkinson [32]  

Johnson [26]  Budiansky and Hutchinson [33] 

Chang et al. [27]  de Koning [34]  
 

Table 2: Fatigue crack growth models [20].  
 

 
 

RETARDATION PHENOMENON 
 

orbly & Packman [35] present some aspects of the retardation phenomenon some of which are presented below. 
1. Retardation increases with higher values of peak loading peak for constant values of lower stress levels [36,37]. 
2. The number of cycles at the lower stress level required to return to the non-retarded crack growth rate is a 

function of Kpeak, Klower, Rpeak,, Rlower and number of peak cycles [38]. 
3. If the ratio of the peak stress to lower stress intensity factors is greater than l.5 complete retardation at the lower stress 

intensity range is observed. Tests were not continued long enough to see if the crack ever propagated again [38].  
4. With a constant ratio of peak to lower stress intensity the number of cycles to return to non-retarded growth rates 

increases with increasing peak stress intensity [37,38].  
5. Given a ratio of peak stress to lower stress, the number of cycles required to return to non-retarded growth rates 

decreases with increased time at zero load before cycling at the lower level [38].  
6. Increased percentage delay effects of peak loading given a percent overload are greater at higher baseline stress 

intensity factors [39].  
7. Delay is a minimum if compression is applied immediately after tensile overload [40]. 
8. Negative peak loads cause no substantial influence of crack growth rates at lower stress levels if the values of R > 0 for 

the lower stress [41].  
9. Negative peak loads cause up to 50 per cent increase in fatigue crack propagation with   R = - 1 [40]. 
10. Importance of residual compressive stresses around the tip of crack [42]. 
11. Low-high sequences cause an initial acceleration of the crack propagation at the higher stress level which rapidly 

stabilizes [43].  
 

 
SMALL SCALE YIELD MODELS 
 

hile the basic layout of the small scale yield model has been established by Dill & Saff [44], only improvements 
introduced later by Newman [45] made this approach applicable to general variable amplitude loading. The 
small scale yield model employs the Dugdale [16] theory of crack tip plasticity modified to leave a wedge of 

plastically stretched material on the fatigue crack surfaces. The fatigue crack growth is simulated by severing the strip 
material over a distance corresponding to the fatigue crack growth increment as shown Fig. 2. In order to satisfy the 
compatibility between the elastic plate and the plastically deformed strip material, a traction must be applied on the 

C 

W 



 

                                                      L. C. H. Ricardo et alii, Frattura ed Integrità Strutturale, 35 (2016) 456-471; DOI: 10.3221/IGF-ESIS.35.52 
 

461 
 

fictitious crack surfaces in the plastic zone (a  x < aafict), as in the original Dugdale model, and also over some distance in 
the crack wake (aopen  x < a), where the plastic elongations of the strip L(x) exceed the fictitious crack opening 
displacements V(x).  
The compressive stress applied in the crack wake to insure L(x)=V(x) are referred to as the contact stresses. The fatigue 
crack growth is simulated using the strip material as shown schematically in Fig. 2.  
 

 
Figure 2: Schematic Small Scale Yield Model.  

 
Ricardo et al. [46] discuss the importance in the determination of materials properties like crack opening and closing stress 
intensity factor. The development of crack closure mechanisms, such plasticity, roughness, oxide, corrosion, and fretting 
product debris, and the use of the effective stress intensity factor range, has provided an engineering tool to predict small 
and large crack growth rate behavior under service loading conditions.  
The major links between fatigue and fracture mechanics were done by Christensen [47] and Elber [48]. The crack closure 
concept put crack propagation theories on a firm foundation and allowed the development of practical life prediction for 
variable and constant amplitude loading, by such as experienced by modern day commercial aircrafts. Numerical analysis 
using finite elements has played a major role in the stress analysis crack problems. Swedlow [49] was one of the first to use 
finite element method to study the elastic-plastic stress field around a crack.  
The application of linear elastic fracture mechanics, i.e. the stress intensity factor range, K, to the “small or short” crack 
growth have been studied for long time to explain the effects of nonlinear crack tip parameters. The key issue for these 
nonlinear crack tip parameters is crack closure. Analytical models were developed to predict crack growth and crack 
closure processes like Dugdale [16], or strip yield, using the plasticity induced approach in the models considering 
normally plane stress or strain effects. Schijve [50], discussing the relation between short and long cracks presented also 
the significance of crack closure and growth on fatigue cracks under services load histories. The ultimate goal of 
prediction models is to arrive at quantitative results of fatigue crack growth in terms of millimeters per year or some other 
service period. Such predictions are required for safety and economy reasons, for example, for aircraft and automotive 
parts.  
Sometimes the service load time history (variable amplitude loading) is much similar to constant amplitude loading, 
including mean load effects. In both cases quantitative knowledge of crack opening stress level Sop is essential for crack 
growth predictions, because Keff is supposed to be the appropriate field parameter for correlating crack growth rates 
under different cyclic loading conditions. The correlation of crack growth data starts from the similitude approach, based 
on the Keff, which predicts that same Keff cycles will produce the same crack growth increments. The application of 
Keff to variable amplitude loading require prediction of the variation of Sop, during variable amplitude load history, which 
for the more advanced prediction models implies a cycle by cycle prediction. The Fig. 3 shows the different K values. 
 



 

L. C. H. Ricardo et alii, Frattura ed Integrità Strutturale, 35 (2016) 456-471; DOI: 10.3221/IGF-ESIS.35.52                                                        
 

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Figure 3: Definitions of K Values, Schijve [50]. 

 
The application of Keff is considerably complicated by two problems: (1) small cracks and (2) threshold K values 
(Kth). Small cracks can be significant because in many cases a relatively large part of the fatigue life is spent in the small 
crack length regime.  
The threshold problem is particularly relevant for fatigue under variable amplitude spectrum, if the spectrum includes 
many “small” cycles. It is important to know whether the small cycles do exceed a threshold K value, and to which 
extension it will occur. The application of similitude concept in structures can help so much, but the results correlation is 
not satisfactory and the arguments normally are: 

 The similarity can be violated because the crack growth mechanism are no longer similar 
 The crack can be too small for adopting K as a unique field parameter 
 Keff and others conditions being nominally similar, it is possible that other crack tip aspects also affect crack 
growth, such as crack tip blunting and strain hardening, Schijve [50]. 

Newman and Armen [51-53] and Ohji et al. [54] were the first to conduct the two dimensional analysis of the crack 
growth process. Their results under plane stress conditions were in quantitative agreement with experimental results by 
Elber [28], and showed that crack opening stresses were a function of R ratio (Smin/Smax) and the stress level (Smax/0), 
where 0 is the flow stress i.e: the average between ys and u. 
Blom and Holm [55] and Fleck and Newman [56-57] studied crack growth and closure under plane-strain conditions and 
found that cracks did close but the cracks opening levels were much lower than those under plane stress conditions 
considering same loading condition. Sehitoglu et al. [58] found later that the residual plastic deformations that cause 
closure came from the crack. McClung [59-61] performed extensive finite element crack closure calculations on small 
cracks at holes, and various fatigue crack growth models. Newman [62] found that Smax/0 could correlate the crack 
opening stresses for different flow stresses (0). This average value was used as stress level in the plastic zone for the 
middle crack tension specimen, McClung [61] found that K analogy, using Kmax/K0 could correlate the crack opening 
stresses for different crack configurations for small scale yielding conditions where K0=o(a) . (K-analogy assumes that 
the stress-intensity factor controls the development of closure and crack-opening stresses and that, by matching the K 
solution among different cracked specimens, an estimate can be made for the crack opening stresses). 
Matos & Nowell [63] present a literature review of the phenomenon of plasticity-induced fatigue crack closure under 
plane strain conditions and mention that there are controversial topics concerning the mechanics of crack propagation. In 
general there is no consensus in the scientific community. Fleck [64] used finite elements to simulate plasticity induced 
crack closure under plane strain conditions and predicted that the nature of the closure process changes from continuous 
to discontinuous after a sufficient increment of crack growth. He suggested that closure involves only a few elements 
relatively distant from the current crack tip and the closure levels decay steadily as the crack grows beyond its initial 
length. In the limit, the closure would not occur at all. Tab. 3 presents an adapted chronologic review crack advance 
scheme from Matos & Nowell [63]. 
 



 

                                                      L. C. H. Ricardo et alii, Frattura ed Integrità Strutturale, 35 (2016) 456-471; DOI: 10.3221/IGF-ESIS.35.52 
 

463 
 

Year Author 
Node Release 

Scheme 
Constraint 

 
Target 

Element 
Type 

1985 Blom and Holm [55] Maximum load PStress; 
PStrain 

COP and CCL Triangle linear 

1986 Fleck [64]  Maximum load PStress; 
PStrain 

COP Triangle linear 

1989 McClung and 
Sehitoglu [65] 

Maximum load PStress; 
PStrain 

COP Quadrilateral 
linear 

1989 McClung et al. [66] Maximum load PStress; 
PStrain 

COP Quadrilateral 
linear 

1991 Sun and Sehitoglu [67] Maximum load PStress; 
PStrain 

COP Quadrilateral 
linear 

1992 Sehitoglu and Sun [68] Maximum load; 
Minimum load 

PStress; 
PStrain 

COP Quadrilateral 
linear 

1996 Wu and Ellyin [69] Maximum load PStress COP and CCL Quadrilateral 
linear 

1999 Ellyin and Wu [70] Maximum load PStress COP and CCL Quadrilateral 
linear 

2000 Wei and James [71]  Maximum load PStress; 
PStrain 

COP and CCL Triangle linear 

2002 Ricardo et al. [72] Minimum Load PStress COP and CCL Triangle 
quadratic 

2002 Pommier [73]  Minimum Load PStrain COP and CCL Quadrilateral 
linear 

2003 Ricardo [74] Minimum Load PStress  CCL Triangle 
quadratic 

2003 Solanki et al. [75]  Maximum load PStress; 
PStrain 

COP and CCL 
by COEL 

Quadrilateral 
linear 

2004 Solanki  et al. [76]  Maximum load PStress; 
PStrain 

COP and CCL 
by COEL 

Quadrilateral 
linear 

2004 Zhao et al. [77]  Maximum load PStrain COP and CCL 
by CME 

Quadrilateral 
linear 

2005 Gonzalez-Herrera and 
Zapatero [78]  

Maximum load PStress; 
PStrain 

COP and CCL 
by DME 

Quadrilateral 
linear 

2007 Matos & Nowell [79] Minimum load PStress COP and CCL 
by COEL 

Quadrilateral 
linear 

PStress- plane stress; PStrain- plane strain; COP- crack opening; CCL- crack closing;  
COEL- crack opening and closing by contact element;  
CME- crack opening and closing by compliance method;  
DME- crack opening and closure by displacement method 

 

Table 3: Chronological crack advance scheme. 
 
In Singh et al. [80] the authors provide a review of some crack propagation issues. The paper cover the transients and 
single overload effects as well as the plasticity induced crack closure. In this topic Singh et al [80] presented a discussion 
regarding how the researchers normally work in crack propagation simulation considering overload-induced crack closure.     
Lei [81] determine the crack closure by finite element method in a compact specimen. In the work Lei [81] use ABAQUS 
[82] to perform the crack propagation simulation using the crack face method was good agreement with experimental 
data. 
Ricardo et al. [72] present an example of small scale yielding under constant amplitude loading. A compact tension 
specimen was modeled using a commercial finite element code Ansys version 6.0 [83]. A half of the specimen was 
modeled and symmetry conditions were applied. Fig. 4 shows the compact tension specimen from ASTM 647-E95a [84]. 
A value of 19 MPam was applied as an equivalent force using the expression (9) in the model. Fig. 5 shows the model 
used in this work and Fig. 6 shows an example of post-processing of the small scale yielding stress intensity factor.  



 

L. C. H. Ricardo et alii, Frattura ed Integrità Strutturale, 35 (2016) 456-471; DOI: 10.3221/IGF-ESIS.35.52                                                        
 

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 max
KBW

P
a

f
w


 
 
 

           (9) 

where, K is the stress intensity factor; Pmax is the maximum applied load; B is the specimen thickness; a is the crack length; 
W is the specimen width; a/w is the crack length to width relation for the specimen and f(a/W) is the characteristic 
function of the specimen that can be found in ASTM 647-E95a [84]. 
 

 
 

 
 

Figure 4: Compact Tension (CT) Specimen. Figure 5: FEM Model of CT specimen. 
 

 
 

Figure 6: Post-Processing of Small Scale Yield Model. 
 
 
 

GENERATION OF VARIABLE AMPLITUDE LOADINGS 
 

achniewicz [85-86] presents methodologies for fatigue crack growth models considering metallic materials. In 
the part I Machniewicz [85] present a review of crack growth predictions models and the deterministic models 
like AFGROW [87] and Willenborg et al. [22] models. Crack closures models are presented with their 

characteristics to apply under constant and variable amplitude loading. Machniewicz in part II [86] is presented the 
constraint factors normally used in plane stress constraint. FASTRAN [88] and NASGRO [89] are the most codes used in 
plane stress constrain to determine plastic strip stresses and strain. 
Heuler & Klätschke [90] discuss the procedure and how the generation of standards loadings can support the 
development of structures and components considering crack growth phenomenon under variable amplitude loading. It is 
well-known that data and models that characterize the fatigue behavior of materials and structures under baseline constant 
amplitude loading may not be appropriate or sufficient to adequately assess their fatigue performance under irregular 

M 



 

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465 
 

variable amplitude loading. Basic research is conducted under use of simplified load sequences such as single overload or 
underload or block loading with alternate mean loads.  
Phenomena like crack growth retardation or acceleration are described making reference to base-line constant amplitude 
data. It is generally agreed, however, that real life load spectra also need to be applied in order to get a realistic picture of 
the relevance and significance of the mechanisms involved. Standardized load sequences or Load–time Histories (SLH’s) 
presently available provide an appropriate selection of load sequences to be used in the development of components, but 
they can also advantageously be used for other tasks. In this section it will be presented an overview on and a 
summarizing description of standardized load–time histories. With the need for optimum light-weight design, originally 
the aircraft industry was the main driver for these efforts. Two of the most well known SLH’s are the TWIST [91] and 
FALSTAFF [92] sequences for transport and fighter aircraft, respectively, which have been and are still being applied for 
numerous studies on materials, joints and other structural elements.  
For automotive applications, the CARLOS [93] series of SLH’s have been presented including the very recent load 
sequence for car trailer couplings, CARLOS-TC. In the US, activities were mainly centered on the derivation of test load 
sequences to be used for evaluation and development of fatigue life prediction methodology. Bodies like the SAE Fatigue 
and Evaluation Committee took a pragmatic approach by selecting test load sequences from existing strain measurements, 
which were felt to be typical for the ground vehicle industry. Altamura & Straub [94] presents a work where discuss 
different ways to work with variable amplitude loading and the strategies to conduct fatigue analysis in structures. It is 
shown the methodology for discretization of random loads in blocks to be used in the development of components. And, 
also, it is presented the procedure to evaluate crack growth under constant and variable amplitude loading. Probabilistic 
fatigue crack growth is discussed as well the mathematics models available to use like Monte Carlos simulation. It is 
generally agreed that the structural load variations should be characterized in the time domain since in most cases the 
range (or amplitude) of a load, stress or strain cycle and its respective max or mean value can be considered as fatigue-
relevant. Furthermore, the sequence or mixing of load cycles of different ranges and mean values must not be neglected. 
Analyses in the frequency domain give insight into the frequency content of a load signal which is particularly useful for 
flexible structures, but do not deliver the above-mentioned values. Many structural loading environments can be described 
as sequences of different modes [95] which may be a particular flight, driving a car on certain road types, a sea state of a 
given severity, etc. These modes of operation contain load cycles of different, but typical magnitudes and frequencies. 
Often distinct patterns of grouped load cycles can be distinguished, they are called a loading event or element, such as 
braking or cornering of a car, different flight phases or maneuvers of an aircraft.  
Zheng [96] provides a criterion for omitting small loads. In past, the underload (or subload) was defined as the nominal 
stress amplitude lower than or equal to the endurance limit, and the underload effect on fatigue life was investigated 
experimentally by using smooth specimens. Test results showed that underload cycles applied to smooth specimens 
increased the fatigue life or the endurance limit of low-carbon steel [96] and cast iron [97], which was called “coaxing”. 
However, past research on the underload effect was not associated with the omission of small load cycles in life prediction 
[98,99]. The omission of small load cycles is necessary and important in compilation of the load spectrum [100,101], once 
the accumulated damage will not affect the prediction of the fatigue life and the assessment of the fatigue reliability of 
structures [102,103], and it is most cost effective in fatigue tests of components and structures under long-term variable-
amplitude or random loading histories [104]. Up to date, some empirical criteria have been proposed and used [105,106]. 
However, how to omit the small loads in life prediction by using the local strain approach was not clearly set forth [106]. 
In the discussion of the importance of crack growth under variable amplitude loading, Youb & Song [106], using results 
obtained from single edge crack bending (SEB), mentioned that Schijve [101] was one of the first works covering this 
topic. Kikukawa et al. [107] have extensively measured crack opening behavior under various random loadings and 
reported that crack opening point is controlled by the maximum range-pair load cycle (which we call hereafter “the largest 
load cycle”) in a random load history and is identical to the crack opening result of constant amplitude loading 
corresponding to the largest load cycle. Based on this crack opening behavior, they proposed a simple prediction 
procedure for crack growth under random loading.        
The phenomenon of plasticity-induced fatigue crack closure under plane strain conditions is one of the most controversial 
topics concerning the mechanics of crack propagation. No general consensus exists among the scientific community 
concerning the physical mechanism for crack closure under plane strain conditions. One of the problems is on how to 
prepare the mesh and the procedure used in crack propagation. With three-dimensional models it becomes necessary to 
use normal contact approach to node release; in plane stress, spring is normally used to help the crack propagation, using 
contact resources for crack propagation and considering material nonlinear analysis it will result in a big result file and will 
spend a considerable time processing to end the simulation. According to Fleck [108] the source of discontinuous closure 
appears to be a residual wedge of material on the crack flanks, located just ahead of the initial position of the crack tip. 



 

L. C. H. Ricardo et alii, Frattura ed Integrità Strutturale, 35 (2016) 456-471; DOI: 10.3221/IGF-ESIS.35.52                                                        
 

466 
 

More recently Wei and James [109] reported that after growing a virtual plane strain fatigue crack for a few cycles, there is 
no contact in the region immediately behind the crack tip and the contact pressure along the crack faces is discontinuous. 
Zao et al. [110] modelled a CT specimen under plane stress and plane strain. They did not observe plasticity-induced crack 
closure under plane strain during steady state crack growth under cyclic tension, although they found significant levels of 
closure under plane stress.  
Solanki et al. [75] present a review of crack propagation in plane stress and plane strain conditions. A M(T) specimen was 
modeled with an externally induced T -stress to observe the subsequent change in closure levels under plane-strain. A T-
stress was induced by applying tractions parallel to the crack in addition to the conventional tractions perpendicular to the 
crack. Fig. 7 shows the variation in the crack tip plastic zone size accordingly with mesh. Fig. 8 shows the difference of 
result in node release at minimum and maximum load compared by Solanki et al. [76]. 
 

 

 
 

 

Figure 7: Variation in Crack Tip Plastic Zone Size with Mesh 
[75]. 

Figure 8: Comparison of Crack opening values based on crack 
advance scheme [75]. 

 

 

  

Figure 9: Middle-Crack Tension Specimen Subjected to uniform 
stress [112]. 

Figure 10: Crack Propagation Model Quarter of Middle Tension 
[113]. 

 

 
Chermahini [111] present some crack propagation analyses using 3D model and plane strain model to determine the crack 
opening level. On the specimen surface and in the mid-plane the crack-opening stress levels tend to be two-dimensional 
solutions for plane stress and plane strain conditions, respectively. Fig. 9 shows the geometry used by Chermahini et al. 
[112].  



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467 

In Fig. 10, it is possible to see the finite element model used for crack propagation elaborated by Wu & Ellyin [113]. The 
model was prepared using layers of elements, considering the size of the smaller elements in the reverse plastic zone 
computed by Irwin equation and then increasing the size of the hexahedron elements until arriving the region where the 
results will not affect the stress level in the crack propagation area. Spring elements were used for node release, cycle after 
cycle, as in Newman [45].  
Wu and Ellyin [113] had used a truss element together with pairs of contact elements and the element death option for 
crack propagation simulation. This technique used in plane stress and plane strain models is usual in commercial finite 
element codes. The element death option was incorporated to remove truss elements.  With their approach, a node can be 
released any time during a load cycle irrespective of the magnitude of the deformation caused by the release of the node. 
Consequently, fewer problems with convergence were encountered and also several nodes could be released 
simultaneously if desired.  

CONCLUSIONS 
 
he paper provides a review of some crack retardation models under variable amplitude loadings. It was discussed, 
also, the small scale yield model using finite element method. The Miner’s rule crack initiation approach can be 
conservative in some applications, in special if the structures should develop cracks under variable amplitude 

loading. It is presented the standards loadings histories normally used in automotive and aeronautics structures. Several 
crack advance schemes are presented and it is possible to observe that there is no agreement in the science community 
about the best strategy to edit experimental signals to be applied in numerical models aiming to obtain good correlation 
between numerical and experimental data. The crack propagation simulation under constant amplitude loading in plane 
stress has good agreement with experimental data. Plane strain need complex models with large number of nodes and it is 
necessary to define and work with contact between the crack surfaces and, therefore, perform nonlinear analysis to 
identify when the crack open or close.  
Regarding variable amplitude loading until the moment the authors do not identify a consistent methodology and 
procedure for crack propagation simulation. The problem should be related with the random fatigue phenomenon and to 
determine when the crack opens or closes, either using experimental or numerical data, is a challenge to be achieved. The 
computers are improving their processing and storage capacity with possibility to increase the size of models and 
decreasing the element size becoming more realistic the crack propagation simulation. In the near future it will be 
necessary to perform more and more tests to validate the numerical models hoping that the correlation between numerical 
and experimental results becomes better and better. 

REFERENCES 

[1] Miner, M. A., Cumulative damage in fatigue, Journal of Applied Mechanics, ASME, USA, 12 (1945) A159-A164.
[2] Schijve, J., Fatigue crack propagation in light alloy sheet material and structures, NLR, Report MP195, Amsterdam 

(1960).
[3] Stouffer, D. C. , Williams, J. F., A method for fatigue crack growth with a variable stress intensity factor, Eng. 

Fracture Mechanics, 11, (1979), 525-536.  DOI: 10.1016/0013-7944(79)90076-6.
[4] Ditlevsen, O., Sobczyk, K., Random fatigue crack growth with retardation, Eng. Fracture Mech., 24(6) (1986) 861-

878. DOI:10.1016/0013-7944(86)90271-7
[5] Wei, R. P., Shih, T. T., Delay in fatigue crack growth, Int. Journal Fracture, 10 (1974) 77-85.

DOI: 10.1007/bf00955082
[6] Irwin, G. R., Journal of Applied Mechanics, 79 (1953) 361.
[7] Irwin, G. R., Journal of Basic Engineering, Trans., ASME, series D, 82(2) (1960) 417. DOI: 10.1115/1.3662608
[8] Tada, H., Paris, P.C, Irwin, G.R., The stress analysis of cracks handbook, Bethlehem, PA, Del Research Corporation, 

USA,(1985).
[9] Murakami, Y., editor. Stress intensity factors handbook. New York: Pergamon Press, USA (1987).
[10]  Raju I.S., Newman JC Jr. Stress-intensity factors for a wide range of semi-elliptical surface cracks in finite-thickness 

plates, Eng. Fracture Mechanics, 11 (1979) 817-829. DOI: 10.1016/0013-7944(79)90139-5. 

T 



L. C. H. Ricardo et alii, Frattura ed Integrità Strutturale, 35 (2016) 456-471; DOI: 10.3221/IGF-ESIS.35.52 

468 

[11] Newman, J.C. Jr., Raju, I., Stress-intensity factor equations for cracks in three-dimensional finite bodies subjected to
tension and bending loads, Computational Methods in the Mech. of Fracture. In: Atluri SN, editor. Amsterdam:
Elsevier, (1986) 311-334. DOI: 10.1002/zamm.19870671132.

[12] Newman, J. C, The merging of fatigue and fracture mechanics concepts: a historical perspective, Progress in
Aerospace Sciences, 34 (1998) 347-390.  DOI:10.1016/s0376-0421(98)00006-2.

[13] Frost, N. E., Dugdale, D. S., Journal of Mechanics and Physics of Solids, 6(2) (1958) 92-110.
DOI:10.1016/0022-5096(58)90018-8.

[14] Paris, P. C., Erdogan, F., Journal of Basic Engineering, 85 (1963) 528.
[15] Head, A. K., The Philosophical Magazine, 44(7) (1953) 253.
[16] Dugdale, D. S., Yielding of steel sheets containing slits, J. Mech. Phys. Solids, 8(2) (1960) 100-104.

DOI:10.1016/0022-5096(60)90013-2.
[17] Hairman, G. A., Provan, J. W., Fatigue crack tip plasticity revisited, The Issue of Shape Addressed, Theoretical and

Appl. Frac. Mech. Journal, 26 (1997) 63-79.  DOI: 10.1016/s0167-8442(96)00036-5.
[18] Williams, M. L., On the stress distribution at base stationary crack, Journal of Applied Mechanics, 24 (1957) 111-114.
[19] Rolfe, S. T., Barsom, J. M., Fracture and fatigue control in structures – applications of fracture mechanics, Prentice -

Hall, New Jersey, (1977). DOI: 10.1520/MNL41-3RD-EB.
[20] Murthy, A. R. C.; Palani, G. S., Iyer, N. R., State of art review on fatigue crack growth analysis under variable

amplitude loading, IEI Journal, (2004) 118-129.
[21] Wheeler, O. E., Spectrum loading and crack growth, Transactions of the ASME Series D’, Journal of Basic

Engineering, 94 (1972) 181-186.
[22] Willenborg, J. D.; Engle, R. M., Wood, H. A., A crack growth retardation model using an effective stress concept,

AFFDL, TM-71-FBR, Air Force Flight Dynamics Laboratory, Wright Patterson Air force Base, OH, (1971).
[23] Porter, T. R., Method of analysis and prediction of variable amplitude fatigue crack growth, Eng. Fracture Mechanics

4(4) (1972) 717-736.  DOI:10.1016/0013-7944(72)90011-2.
[24] Gray, T. D., Gallagher, J. P., Predicting fatigue crack retardation following a single overload using a modified Wheeler

model, ASTM STP 590, (1976). DOI: 10.1520/STP590-EB.
[25] Gallagher, J. P., Hughes, T. F., Influence of the yield strength on overload fatigue crack growth behavior of 4340

steel, AFFDL – TR-74-27, Air Force Flight Dynamics Laboratory, Wright Patterson Air force Base, OH, (1974).
[26] Johnson, W. S., Multi-parameter yield zone model for predicting spectrum crack growth, ASTM STP 748, (1981)  85-

102. DOI: 10.1520/E0748-02R08 .
[27] Chang, J. B., Hiyama, R. M., Szamossi, M., Improved methods for predicting spectrum loadings effects, AFWAL-TR-

81-3092, Air Force Flight Dynamics Laboratory, Wright Patterson Air force Base, OH, (1984).
[28] Elber, W., The significance of fatigue crack closure, ASTM STP 486, (1971) 230- 242. DOI: 10.1520/STP486-EB.
[29] Bell, P. D., Creager, M., Crack growth analyses for arbitrary spectrum loading, AFFDL-TR-74-129, Air Force Flight

Dynamics Laboratory, Wright Patterson Air force Base, OH, (1974).
[30] Newman, J. C., A finite element analysis fatigue crack closure, NASA TM X 72005,  NASA, Hampton, VA, (1975).
[31] Dill, H. D., Saff, C. R., Spectrum crack growth prediction method based on crack surface displacement and contact

analyses, Fatigue crack growth under spectrum loads, ASTM STP 595, (1976) 306–319. DOI: 10.1520/STP595-EB.
[32] Kanninnen, M. F., Atkinson, C., Feddersen, C. E., A fatigue crack growth analysis method based on a single

representation of crack tip plasticity, ASTM STP 637, (1977) 122-140. DOI: 10.1520/STP637-EB.
[33] Budianski, B., Hucthinson, J. W., Analysis of closure in fatigue crack growth, Journal of Applied Mechanics, 45

(1978) 267-276.  DOI 10.1115/1.34.24286.
[34] de Koning, A. U., A Simple crack closure model for predictions of fatigue crack growth rates under variable

amplitude loading, ASTM STP 743, (1981) 63-85. DOI: 10.1520/STP743-EB.
[35] Corbly, D. M., Packman, P. F., On The influence of single and multiple peak overloads on fatigue crack propagation

in 7075-T6511 aluminum, Eng. Fracture Mech., 5 (1973) 479-497. DOI:10.1016/0013-7944(73)90034-9.
[36] Schijve, J., Brock, D., de Rigle, P., NLR, Report M2094, Amsterdam, (1962).
[37] Hardrarth, H. F., McEvily, A . T., In: Proc. Crack propagation symposium, Cranfield, 1 (1961).
[38] Jonds, D., Wei, R. P., An exploratory study of delay effects in fatigue crack growth, Int. J. Fracture Mechanics, 7

(1971).
[39] von Ewu, E., Hertzberg, R., Roberts, R., Delay defects in fatigue crack propagation, nat. symposium F.M., (1971) 7.
[40] Hudson, C. M., Effect of stress ratio on fatigue-crack growth in 7075-T6 and 2024-T3 alumina-alloy specimens.

TNL-5390, NASA, (1969).
[41] Crooker, T. W., Effect of T. C., Cycling on fatigue grade growth in high strength alloys, NRL Report 7220, (1971).



 

                                                      L. C. H. Ricardo et alii, Frattura ed Integrità Strutturale, 35 (2016) 456-471; DOI: 10.3221/IGF-ESIS.35.52 
 

469 
 

[42] Hudson, C.M., Hardrath, H. F., Effects of changing stress amplitude on the rate of fatigue-crack propagation in two 
aluminum alloys, NASA TN D-960, September, (1961). 

[43] Mcmillan, J. C., Pelloux, R. M., ASTM STP 415, (1966) 505. DOI: 10.1520/STP415-EB 
[44] Dill, H. D., Saff, C. R., Spectrum crack growth prediction method based on crack surface displacement and contact 

analyses, Fatigue crack Growth under Spectrum Loads, ASTM STP 595, (1976) 306–319. DOI: 10.1520/STP595-EB. 
[45] Newman Jr, J. C., A Crack closure model for predicting fatigue crack growth under aircraft spectrum loading. In: 

Chang JB, Hudson CM, editors. Methods and models for predicting fatigue crack growth under random loading, 
ASTM STP 748, (1981) 53–84. DOI: 10.1520/STP748-EB 

[46] Ricardo, L. C. H., Pimenta, P. M., Spinelli, D., Crack closure simulation by finite element. In, Blom, A.F. ,Fatigue 
2002, 4/5 (2002) 2293 – 2300. 

[47] Christensen, R. H., Fatigue Crack growth affected by metal fragments wedged between opening closing crack surface, 
Appl. Mater. Res., 2 (1963) 207-210.  

[48] Elber W., Equivalent constant amplitude concept for fatigue crack growth under spectrum loading, ASTM STP 595, 
(1976) 236-250. DOI: 10.1520/STP595-EB 

[49] Swedlow, J. L., Effects and plastic flow in cracked plates, PhD Thesis, California Institute of Technology, Pasadena, 
USA, (1965).  

[50] Schijve, J., Observations on the prediction of fatigue crack growth propagation under variable amplitude loading, 
ASTM STP 595, (1976) 3-23. DOI: 10.1520/STP595-EB. 

[51] Newman J. C. Jr., Finite element analysis of fatigue crack propagation, including the effect C-(T) of crack closure, 
PhD Thesis, Virginia Polytechnic Institute, (1974).  

[52] Newman, J. C. Jr.; Armen, H. Jr. Elastic-plastic analysis of fatigue crack under cyclic loading, AIAA Journal, 13 
(1975) 1017-1023. DOI: 10.2514/3.60499. 

[53] Newman, J. C. Jr., A finite element analysis of fatigue crack closure, ASTM 490, (1976) 281-301.  
DOI: 10.1520/STP490-EB. 

[54] Ohji, K., Ogura, K. Ohkubo, Y., Cyclic analysis of a propagation crack and its correlation with fatigue crack growth; 
Eng. Fracture Mechanics, 7 (1975) 457-464. DOI: 10.1016/0013-7944(75)90046-6. 

[55] Blom, A. F., Holm. D. K., An experimental and numerical study of crack closure; Eng. Fracture Mechanics, (1985) 
907-1011. DOI: 10.1016/0013-7944(85)90039-6. 

[56] Fleck, N. A., Finite element analysis of plasticity induced crack closure under plane strain conditions, Eng. Fracture 
Mechanics, 25 (1986) 441-449. DOI:10.1016/0013-7944(86)90258-4. 

[57] Fleck, N. A., Newman, J. C. Jr, Analysis of crack closure under plane strain conditions , ASTM  STP 982, (1988) 319-
341. DOI: 10.1520/STP982-EB. 

[58] Sehitoglu, H. Gall, K.,  Garcia, A. M., Recent advances in fatigue crack growth modeling, Int. Journal of Fracture, 
80(2) (1996) 165-192. DOI: 10.1007/BF00012668. 

[59] McGlung, R. C., Sehitoglu, H., Finite element analysis of fatigue crack closure – numerical results, Eng. Fracture 
Mechanics, 33 (1989) 253–272. DOI: 10.1016/0013-7944(89)90028-3. 

[60] McGlung, R. C., Finite element modeling of fatigue crack growth – theoretical concepts and numerical analysis of 
fatigue, In: Blom A, Beevers, C. Editors, West Midlands, Emas, (1992) 153-172. 

[61] McGlung, R. C., Finite Element Analysis of Specimen Geometry Effects on Fatigue Crack Closure, Fatigue Fract. 
Eng. Mater. Structures, 17 (1994) 861-872. DOI: 10.1111/j.1460-2695.1994.tb00816.x. 

[62] Newman, J. C. Jr., A Finite element analysis of fatigue crack closure, ASTM 490, (1972) 281-301. 
DOI: 10.1520/STP490-EB. 

[63] Matos, P. F. P., Nowell, D., Numerical simulation of plasticity induced fatigue crack closure with emphasis on the 
crack growth scheme: 2D and 3D analyses, Eng. Fracture Mechanics, 75 (2008) 2087-2114.           
DOI: 10.1016/j.engfracmech.2007.10.017. 

[64] Fleck, N. A., Finite element analysis of plasticity-induced crack closure under plane strain conditions, Eng. Fracture 
Mechanics, 25(4) (1986) 441–449. DOI: 10.1016/0013-7944(86)90258-4. 

[65] McClung, R. C., Sehitoglu, H., On the finite element analysis of fatigue crack closure 1. basic modeling issues. Eng. 
Fracture. Mechanics, 33 (2) (1989) 237–252.  DOI: 10.1016/0013-7944(89)90027-1.  

[66] McClung, R. C., Sehitoglu H., On the finite element analysis of fatigue crack closure 2. numerical results. Eng. 
Fracture. Mechanics, 33(2) (1989) 253–272.  DOI: 10.1016/0013-7944(89)90028-3. 

[67] Sehitoglu, H., S Wei, Modelling of plane strain fatigue crack closure. J. Eng. Mater. Technol, 113 (1991) 31–40.  
DOI :10.1115/1.2903380. 



 

L. C. H. Ricardo et alii, Frattura ed Integrità Strutturale, 35 (2016) 456-471; DOI: 10.3221/IGF-ESIS.35.52                                                        
 

470 
 

[68] Sun, W., Sehitoglu, H., Residual stress fields during fatigue crack growth, Fatig. Fract. Eng. Mater. Struct., 15(2) 
(1992) 115–128. DOI: 10.1111/j.1460-2695.1992.tb00042.x. 

[69] Wu, J., Ellyin, F., A study of fatigue crack closure by elastic-plastic finite element analysis for constant-amplitude 
loading. Int. Journal of Fracture, 82 (1996) 43–65. DOI:10.1007/bf00017863. 

[70] Ellyin, F., Wu, J., A numerical investigation on the effect of an overload on fatigue crack opening and closure 
behavior, Fatig. Fract. Eng. Mater. Struct., 22 (1999) 835–847. DOI: 10.1046/j.1460-2695.1999.00223.x. 

[71] Wei, L. W., James, M. N., A study of fatigue crack closure in polycarbonate CT specimen., Eng. Fracture Mechanics, 
66 (2000) 223–242.  DOI: 10.1016/s0013-7944(00)00014-x. 

[72] Ricardo, L. C. H., Pimenta, P. M., Spinelli D.,  Andrade, A. H. P., Crack closure simulation by finite element method; 
In: Blom, A. F (Ed.), Fatigue 2002, Emas Publishing, Stockholm, Sweden, 4 (2002) 2863-2869. 

[73] Pommier S., Plane strain crack closure cyclic hardening, Eng. Fracture Mechanics, 69(3) (2002) 25–44. 
DOI:10.1016/s0013-7944(01)00061-3. 

[74] Ricardo, L. C. H., Modeling fatigue crack opening and closing phenomenon by finite element method, PhD Thesis, 
Department of Structures and Foundations, University of Sao Paulo (In portuguese), (2003). 

[75] Solanki, K., Daniewicz, S. R., Newman, Jr J. C., Finite element modelling of plasticity-induced crack closure with 
emphasis on geometry and mesh refinement effects. Eng. Fracture Mechanics, 70 (2003) 1475–89.  
DOI:10.1016/s0013-7944(02)00168-6.        

[76] Solanki, K., Daniewicz, S. R.,  Newman, Jr J. C., A new methodology for computing crack opening values from finite 
element analyses, Eng. Fracture Mechanics, 71 (2004) 1165–1175.  DOI:10.1016/S0013-7944(03)00113-9. 

[77] Zhao, L. G., Tong, J., Byrne, J., The evolution of the stress–strain fields near a fatigue crack tip and plasticity-induced 
crack closure revisited, Fatig. Fract. Eng. Mater. Struct., 27(1) (2004) 19–29. DOI: 10.1111/j.1460-2695.2004.00716.x. 

[78] Gonzalez-Herrera, A., Zapatero, J., Influence of minimum element size to determine crack closure stress by finite 
element method, Eng. Fracture Mechanics, 72 (2005) 337–355.  DOI:10.1016/j.engfracmech.2004.04.002. 

[79] Matos, P.F.P.,  Nowell D., On the accurate assessment of crack opening and closing stresses in plasticity-induced 
crack closure problems, Eng. Fracture Mechanics, 74(10) (2007) 1579–1601. 
DOI: 10.1016/j.engfracmech.2006.09.007. 

[80] Singh, K.D., Parry, M.R., Sinclair, I., Variable amplitude fatigue crack growth behavior – a short overview, Journal of 
Mechanical Science and Technology, 25(3) (2011) 663-673. DOI: 10.1007/s12206-011-132-6. 

[81] Lei, Y., Finite element crack closure analysis of a compact tension specimen, International Journal of Fatigue, 30 
(2008) 21–31. DOI:10.1016/j.ijfatigue.2007.02.012. 

[82] ABAQUS, V6.3, Hibbitt, Karlsson & Sorensen, Inc., Providence, RI, (2002). 
[83] Ansys Inc, Ansys Version 6.0, USA, (2002).  
[84] ASTM, Standard test method for measurement of fatigue crack growth rates, E647–95a, (1995).  

DOI: 10.1520/E0647-15. 
[85] Machniewciz, T., Fatigue crack growth prediction models for metallic materials, part I: overview of prediction 

concepts, Fatigue Fract Engng Mater Struct, 36 (2012) 293–307. DOI: 10.1111/j.1460-2695.2012.01721.x. 
[86] Machniewciz, T., Fatigue crack growth prediction models for metallic materials, Part II: Part II: Strip yield model – 

choices and decisions, Fatigue Fract Engng Mater Struct, 36 (2012) 361–373. DOI: 10.1111/ffe.12009. 
[87] Harter, J. A. AFGROW users guide and technical manual. Air Force Research Laboratory, Report No. AFRL-VA-

WP-TR-2006-XXXX, (2006). 
[88] Newman, J. C. FASTRAN II – A fatigue crack growth structural analysis program. NASA Technical Memorandum  

No. 104159. NASA Langley Research Center, Hampton, (1992). 
[89] Ten Hoeve, H. J., de Koning, A. U., Reference manual of the strip yield module in the NASGRO or ESACRACK 

software for the prediction of retarded crack growth and residual strength in metal materials, Report No. NLR TR 
97012 L, NLR, Amsterdam, The Netherlands, (1997). 

[90] Heuler, P., Klätschke, H., Generation and use of standardized load spectra and load- times histories, International 
Journal of Fatigue, 27 (2005) 974-990. DOI: 10.16/j.ifatigue.2004.09.012. 

[91] Schütz, D., Lowack, H., de Jonge J. B.,  Schijve, J., A standardized load sequence for flight simulation tests on 
transport aircraft wing structures, LBF Report FB 106, NRL- Report TR 73, (1973).  

[92] Aircher, W., Branger, J., van DijK, G. M., Ertelt J., Hück, M.,  de Jonge J. B., Description of a fighter aircraft loading 
for standard for fatigue evaluation FALSTAFF, Common Report of FCW Emmen, LBF, NRL, IABG, (1976).  

[93] Schütz, D., Klätschke, H., Steinhilber, H., Heuler, P.,  Schütz, W., Standardized load Sequences for car wheel 
suspension components, car loading standard- CARLOS, Fraunhofer Institute für Betriebsfestigkeit (LBF), 
Darmstadt, Industrieanlagen – Betriebsgesellschaft MBH (IABG), Ottobrunn, LBF – Report No FB- 191, (1999). 



 

                                                      L. C. H. Ricardo et alii, Frattura ed Integrità Strutturale, 35 (2016) 456-471; DOI: 10.3221/IGF-ESIS.35.52 
 

471 
 

[94] Alessandra A., Straub, D., Reliability assessment of high cycle fatigue under variable amplitude loading: Review and 
solutions, Eng. Fracture Mechanics, 121–122 (2014) 40–66.  DOI:10.1016/j.engfracmech.2014.02.023. 

[95] Ten Have, A. A., European approaches in standard spectrum development, In: Potter J. M., Watanable R. T., editors, 
Development of Fatigue Loading Spectra, ASTM-STP 1006, ASTM, (1989) 35-75. DOI: 10.1520/STP1006-EB. 

[96] Zheng, X., On some basic problems of fatigue research in engineering, International Journal of Fatigue, 23 (2001) 
751-766. DOI:10.1016/S0142-1123(01)00040-8. 

[97] Frost, N. E., Marsh, K. J., Pook, L. In: Metal fatigue, Oxford: Clarendon Press, (1974) 20-22. 
[98] Heuler, P., Seeger, T., A criterion for omission of variable amplitude loading histories, International Journal of 

Fatigue, 8(4) (1986) 225-230. DOI:10.1016/0142-1123(86)90025-3. 
[99] ECCS Recommendations for fatigue design of steel structures, Institute of Metal Construction (IOCM) of Swiss 

Federal Institute of Technology, Lausanne, Switzerland, ECCS, (1985).  
[100] Shi, Y. J., Yan, Y. M., Li, Z. R. Shi, Z. J.,  Hou, W. W., Assessment of remaining life of steel beams of Chang- 

Tai-Guam bridge on the line from Beijing to Guangzhou, Technical Report, Beijing, Institute of Railway Engineering, 
Chinese Academy of Railway Science, (1990). 

[101] Schijve, J., Observations on the prediction of fatigue crack growth propagation under variable amplitude loading, 
ASTM STP 595, (1976) 3-23. DOI: 10.1520/STP595-EB. 

[102] Barsom, J. M., Fatigue crack growth under variable amplitude loading in various bridge steels, ASTM STP 595, 
(1976) 217-235. DOI: 10.1520/STP595-EB. 

[103] Hudson, C. M., A root mean square approach for predicting fatigue crack growth under random loading, ASTM 
STP 748, (1981) 41-52. DOI: 10.1520/STP748-EB. 

[104] Johnson, W. S., Multi-parameter yield zone model for predicting spectrum crack growth, ASTM STP 748, (1981)  
85-102. DOI: 10.1520/STP748-EB. 

[105] Rudd, J. L., Engle Jr, R. M., Crack growth behavior of center cracked panels under random spectrum loading, 
ASTM STP 748, (1981) 1033-1114. DOI: 10.1520/STP748-EB. 

[106] Youb, Y.,  Song, J. H., Fatigue crack closure and growth behavior under random loading, Eng. Fracture 
Mechanics, 49(1) (1994) 105-120. DOI:10.1016/0013-7944(94)90115-5. 

[107] Kikukawa, M., Jono, M., Kondo, Y., Mikami, S., Fatigue crack closure and estimation method of crack 
propagation rate under stationary varying loading conditions including random loading (1st report, Effects of mean 
load and study on wave counting method), Trans. Jpn Soc. Mech. Engrs, 48 (1982)1496-1504.  

[108] Fleck, N. A., Finite-element analysis of plasticity induced crack closure under plane strain conditions. Eng. 
Fracture Mechanics., 25 (1986) 441-449.  DOI:10.1016/0013-7944(86)90258-4. 

[109] Wei, L. W., James, M. N., A study of fatigue crack closure in polycarbonate ct specimens. Eng. Fracture 
Mechanics, 66 (2000) 223–42.   DOI:10.1016/S0013-7944(00)00014-X. 

[110] Zhao, L. G., Tong, J., Byrne, J., The evolution of the stress–strain fields near a fatigue crack tip and plasticity-
induced crack closure revisited. Fatig. Fract. Eng. Mater. Struct., 27(1) (2004) 19–29. 

[111] Chermahini, R. G., Three-dimensional elastic- plastic -finite-element analysis of fatigue crack growth and closure. 
PhD Thesis, Old Dominion University, Norfolk, VA, (1986). 

[112] Chermahini, R. G., Shivakumar, K. N., Newman, J. C. Jr., Three-dimensional finite-element simulation of fatigue 
crack growth and closure. ASTM STP 982, (1988) 398-413. DOI: 10.1520/STP982-EB. 

[113] Wu J, Ellyin F., A study of fatigue crack closure by elastic–plastic finite element analysis for constant-amplitude 
loading. Int. Journal of Fracture, 82 (1996) 43–65. DOI: 10.1007/bf00017863. 

 
 
 
 
 
 

 

















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    /NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken die zijn geoptimaliseerd voor prepress-afdrukken van hoge kwaliteit. De gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe Reader 5.0 en hoger.)
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    /ENU (Use these settings to create Adobe PDF documents best suited for high-quality prepress printing.  Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.)
  >>
  /Namespace [
    (Adobe)
    (Common)
    (1.0)
  ]
  /OtherNamespaces [
    <<
      /AsReaderSpreads false
      /CropImagesToFrames true
      /ErrorControl /WarnAndContinue
      /FlattenerIgnoreSpreadOverrides false
      /IncludeGuidesGrids false
      /IncludeNonPrinting false
      /IncludeSlug false
      /Namespace [
        (Adobe)
        (InDesign)
        (4.0)
      ]
      /OmitPlacedBitmaps false
      /OmitPlacedEPS false
      /OmitPlacedPDF false
      /SimulateOverprint /Legacy
    >>
    <<
      /AddBleedMarks false
      /AddColorBars false
      /AddCropMarks false
      /AddPageInfo false
      /AddRegMarks false
      /ConvertColors /ConvertToCMYK
      /DestinationProfileName ()
      /DestinationProfileSelector /DocumentCMYK
      /Downsample16BitImages true
      /FlattenerPreset <<
        /PresetSelector /MediumResolution
      >>
      /FormElements false
      /GenerateStructure false
      /IncludeBookmarks false
      /IncludeHyperlinks false
      /IncludeInteractive false
      /IncludeLayers false
      /IncludeProfiles false
      /MultimediaHandling /UseObjectSettings
      /Namespace [
        (Adobe)
        (CreativeSuite)
        (2.0)
      ]
      /PDFXOutputIntentProfileSelector /DocumentCMYK
      /PreserveEditing true
      /UntaggedCMYKHandling /LeaveUntagged
      /UntaggedRGBHandling /UseDocumentProfile
      /UseDocumentBleed false
    >>
  ]
>> setdistillerparams
<<
  /HWResolution [2400 2400]
  /PageSize [612.000 792.000]
>> setpagedevice