
Research Article


Simulating Fatigue Propagation Life of Martensitic Steel 

O.O. Ajayi
and
J. Ikotun



ABSTRACT

The linear elastic fracture mechanics equation together
with that for stress intensity factor range Î”K, was used to develop
a fatigue propagation life model, after substituting parameters of material
constants. The model was then employed in creating simulation software
which can be used at any time to generate data, make design consideration
and predict response to variable loading. This became useful in predicting
the life of metal from the point of crack initiation; investigate behaviour
to changes in crack sizes and also determine adequate damage tolerance
for the metal.







INTRODUCTION
Fracture is the fragmentation of a solid into two or more parts under
the action of stress (Dieter, 1988). Its process can be considered to
be made up of crack initiation and growth and can be classified into two
broad headings namely ductile and brittle fracture. While ductile fracture
is characterized by appreciable amount of plastic deformation prior to
and during the propagation of crack, with an amount of gross deformation
present at the crack surface, brittle fracture is characterized by a rapid
rate of crack propagation, without gross deformation but little micro
deformation (Dieter, 1988; Shaffer et al., 1999). It occurs in
engineering structures and can be costly in terms of human and/ or property
damage (Benham et al., 1996; Odukwe et al., 2007).
Many researchers have reviewed some structural damages, beginning from
the late 1800s (Dieter, 1988; Benham et al., 1996; Odukwe et
al., 2007), when members of the British Iron and Steel Institute reported
the mysterious cracking of steel in a brittle manner. Investigation shows
that the most failures were initiated in welds that are defective. Later
discoveries also reveal that crack initiation and growth in structures
resulted in sudden failures (Anderson, 1995). However, in spite of design
improvements, increased use of crack arresters, use of quality materials
and so on, brittle fractures still occurs in ships within the 1950s. This
occurrence continued up to 1988 when the fuselage of a 19 year old Boeing
737 airline ripped open in flight (Chau and Chan, 2002). The cause was
found to be due to fatigue crack propagation from a rivet hole in the
fuselage of the jet. In real sense, most cracks nucleate in areas of increased
stress concentration and propagate unnoticed within the microstructure
of the material (Courtney, 2000; Dieter, 2000). Fracture mechanics have
shown that because of the interrelations among materials, design, fabrication
and loading, fatigue failure cannot be eliminated merely by using materials
with improved notch toughness (Tomkins et al., 1981; Anderson,
1995; Miller and Akid, 1996). The failures still occurs today. Moreover,
materials subjected to fluctuating stresses can fail at stresses much
lower than their yield stress (Odukwe et al., 2007).
Reliability studies had shown that structural members contain preexisting
cracks which may have resulted during fabrication (Tomkins et al.,
1981; Dieter, 1988, 2000; Odukwe et al., 2007). These cracks are
assumed to be known and therefore, efforts are geared towards determining
the remaining number of cycle of loading to failure (Dieter, 2000). The
number of these cycles from the point of crack initiation to failure is
the fatigue propagation life of the material, which depends on when and
how a crack initiates and the rate at which it grows through the microstructure
(Tomkins et al., 1981). Most fatigue failure have been discovered
to be caused by cracks, which initiates unnoticed and grows at a fast
rate through the structural member, it is easily the most common type
of failure of engineering structures in service and difficult to foresee,
because conditions causing it are frequently not easily recognizable (Sobczyk,
1987; Odukwe et al., 2007).
More over, advances in fatigue design had shown that relationship exist
between the rate of crack growth da/dN and the stress intensity factor
range Î”K, obtained by plotting a loglog graph of da/dN on the ordinate
against Î”K on the abscissa. The resulting plot is a sigmoid curve
which can be divided into regions IIII (Dieter, 1988, 2000; Courtney,
2000; Ashby and Jones, 2005). Region I is called the nonpropagating crack
region, II is the linear elastic fracture mechanics range and III is the
accelerated crack growth region that is followed by failure respectively.
The region II is represented by the power law as (Benham et al.,
1996; Courtney, 2000):
where, A, M are material constants determined by experimental procedures.
Further study on Eq. 1 was carried out by Odukwe et
al. (2007), which analyzed experimental conclusion on high yield strength
(Ïƒ_{y} = 100 Ksi (689 MN m^{2}), critical stress
intensity factor (K_{c}) = 150 Ksi [in]^{1/2 }(165 MN
mm^{2})) Martensitic steel in air environment. It was discovered
that Î”K explains about 99.85% of the changes in da/dN and that as
a result of experimental analysis A = 7.244x10^{9} and M = 2.22
for Martensitic steel. When crack growth and stress intensity factor are
correlated, it is possible to use data generated under constant amplitude
loading to determine the fatigue life of a component (Dieter, 2000; Odukwe
et al., 2007). The knowledge of when a material will fail and the
required number of cycles to failure will aid in knowing the rate of changeability
and maintenance of such material, therefore, researchers must look into
the problem of structural failures with a view to bring ideas and develop
new packages, which can be used to generate data and evolve new design
considerations, so that the problem of catastrophic failure will be a
forgone issue. This can be achieved by developing interactive software
which can be used on the field and at any time to determine design consideration
and loading parameters. This research focused on achieving this, by using
Eq. 1 together with the stress intensity factor range
to develop a simple model and therefore carryout the simulation process
for Martensitic steel. It is also sort to extend such simulation to include
other metals.
MATERIALS AND METHODS
The linear elastic fracture mechanics equation, represented by Eq.
1 was employed with the relation for Î”K presented as Eq.
2. Constant amplitude stresses, Ïƒ of 15 Ksi (103.43 MPa), 20
Ksi (137.90 MPa) and 25 Ksi (172.38 MPa), together with values of material
constants A and M developed by Odukwe et al. (2007) were used to
develop a possible fatigue life model of Martensitic steel containing
edge and surface cracks as follows:
Where:
C 
= 
Constant for edge and surface cracks = 1.12 (Odukwe
et al., 2007) 
a 
= 
Average crack size 
Putting (2) into (1) gives:
Integrating from N_{o} to N_{f} gives:
Substituting values of A, M and C given earlier gives:
The Eq. 5 above is therefore the constant amplitude
fatigue life model of Martensitic steel containing edge and surface crack.
This is employed to develop a simulation that can be used at any time
to generate data for design consideration, change and modify design parameter,
predict and determine safe operating conditions or damage tolerance for
the metal. The simulation is Fig. 1.
where, A = Are you working on Martensitic steel? And B = Do you have
more data?
RESULTS AND DISCUSSION
Simulation Result
The result of the number of cycles corresponding to the crack sizes
from 0.2 inches (5.08 mm) and above is shown in Table 1.
This is obtained from the simulation software after the desired crack
sizes are inputted.
Table 1 show that a preexisting crack size of 0.2
inches (5.08 mm) was assumed present. Crack sizes were presuggested and
the corresponding numbers of crack propagation cycles were then determined
by employing the software developed from the simulation (Fig.
1). The plot of Fig. 2 reveals that the constant
amplitude stress is inversely related to the number of cycles but directly
to the crack size. This is expected because, as the stress builds up,
the energy release rate required to propagate the crack on a precracked
surface increases and thus, crack increment occurs when the increase in
the energy release rate is at least equal to the energy required to create
a new crack surface (Dieter, 1988; Anderson, 1995; Miller and Akid, 1996),
thereby, lowering the number of crack propagation cycles required to create
the new crack length. Thus, inferring that, as the constant amplitude
stress increases, the required surface energy of the cracked surface increases,
the material resistance to crack extension reduces and thus, crack growth
takes place with reduced number of cycles of loading (Pascual and Meeker,
1999). The findings of many researchers (Benham et al., 1996; Felbeck
and Atkins, 1996; Shaffer et al., 1999; Courtney, 2000; Dieter,
2000; Shigley et al., 2004; Smith and Hashemi, 2004) further support
this. More so, at a constant amplitude stress value, the number of cycles
increases with increasing crack lengths. This is the basis for the linear
elastic fracture mechanics which relates the rate of crack growth to the
average crack length at constant amplitude stress (Benham et al.,
1996; Courtney, 2000). In addition, Fig. 2 reveals the
following: that, at constant crack size, the number of cycles decreases
with increasing stress values and also that, at constant number of cycles,
crack sizes increases with increasing stresses. These information can
be used adequately in mechanical reliability design of machine parts,
where the designer may wish to know the behaviour of such part under different
loading conditions, when it is desired that crack size or number of cycles
are to be kept constant. This would aid in gaining quick access to valid
information and also determine safe operating conditions. Moreover, the
use of the simulation software, if employed in design, will create flexibility,
as design parameters can be changed at ease and corresponding predictive
response can easily be determined without recourse to rigorous and expensive
experimentation.
Table 1: 
The number of cycles of loading corresponding to desired
crack sizes, obtained from simulation software 


Fig. 1: 
Simulation flowchart representing fatigue propagation
life of Martensitic Steel, extended to include other metals 

Fig. 2: 
Plot of desired crack sizes against corresponding number
of crack propagation cycles, determined from the simulation software 
CONCLUSION
The simulation of the fatigue propagation life of Martensitic steel has
been done. A model developed from the linear elastic fracture mechanics
equation together with the relation for the stress intensity factor range
was employed in the modeling and simulation work. The resulting simulation
software was then used to investigate the behaviour of the metal to different
constant amplitude loading stresses and found to be adequate, as result
obtained agreed with previous findings. This became useful in predicting
the life of the metal from the point of crack initiation and to investigate
its behaviour to changes in crack sizes and also determine adequate damage
tolerance for the metal. Thus, removing the cumbersome computational and
repetitive processes involved in data generation and adds pleasantness
and speed to design.

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