1. INTRODUCTION
Failure of the
components is the results of two reasons one is fatigue loading and other one
is working environment like temperature that is the most common factor for
environment affected failure (VIKRAM; RAGHUVIR, 2015). In real life mostly
complex loading environment in which the components work but at the time of
analysis whether it can be experimental, theoretical or numerical we consider
the ideal loading environment to achieve the solutions easily. Fatigue is very
common reason of crack initiation, propagation to critical size (VIKRAM;
RAGHUVIR; KUMAR, 2014; BROEK, 1982) at which
sudden fracture occurs.
Crack extension
takes place due to high stress at the crack tip and due to plastics deformation
at the crack tip during cyclic loading; many efforts have been made to relate
the stress intensity range ratio with stress intensity factor “K” at the crack
tip. A well-established formula was given which is shown in “equation (1)” by
Paris and Erdogan (NICCOLLS, 1976).
∆K)n (1)
Where “C” and “n”
are material property coefficients. It is realized that for different values of
stress ratios, R, for a material a large deviation was obtained in data from
the curve fitted by “equation (1)”. The range of cyclic stress intensity
factors for describing fatigue crack growth rate (FCGR) is based on the
assumption that the crack tip starts opening as soon as load is completely
relaxed.
Elber (ASTM, 1967;
PARIS; ERDOGAN, 1963; WALKER, 1970; BROEK, 1982) studied that cyclic plasticity gives
rise to the formation of residual plastic deformation in the wake of the crack
tip causing the fatigue load crack to close and presented this as crack closure
phenomenon and advised that the fatigue crack growth can take place only during
the portion of the loading cycle in which the crack is fully open. Based on
this suggestion, effective stress range is defined as:
rseff= sm - so (or scl ) (2)
The ratio of rseff to the total stress range (rs) is known as the stress
intensity range ratio, U, and is shown by
U= =
(3)
Elber (VIKRAM;
KUMAR, 2013) further advised that the crack growth can be written in the
following form:
∆K)n (4)
In the crack
propagation equation” replaced by“
, the factors
which influence U are stress intensity range (rs), material properties (sy ,sf) , crack length (a) and stress ratio R.
Elber (VIKRAM; KUMAR, 2013), showed that U depends only on stress ratio R. So
many laws are available which give crack growth rate as a function of rK and material properties. Many other
researchers (ASTM, 1967; PARIS; ERDOGAN, 1963; FOREMAN;
MEARNEY; ENGLE, 1967; VIKRAM; KUMAR, 2013; VIKRAM; RAGHUVIR; KUMAR, 2014; VIKRAM; RAGHUVIR, 2015; KUMAR;
GARG, 1988; PEARSON, 1972; OSGOOD, 1982; ZHENG, 1983) had
also given their contribution to formulate the crack growth and crack closure.
In the present study, effort has been made on side edge notched specimen (SEN)
to show the effect of strain hardening on crack closure and form an empirical
relationship for aluminum alloys 3003, 5052, 6061, 6063, 6351.
2. MATERIALS
AND SPECIMEN GEOMETRY ANALYZED
2.1.
Material properties:
Five Aluminum
Alloy have been used to prepare specimens are 3003 Al, 5052 Al, 6061-T6 Al,
6063-T6 Al, 6351 Al that’s chemical and mechanical properties are given in
Table no.1 and 2 respectively.
Table 1:
Chemical composition
Table 2: Physical
properties
2.2.
Specimen Geometry:
Side edge notched
Specimen has been modeled with the dimensions as below
Length
(H)- 180 mm
Width
(W) - 50 mm
Thickness
(t) – 3 mm
Initially a notch of
6 mm had been made at en edge for crack propagation under the load applications
on the specimen during the fatigue test. The geometry is shown in Figure 1.
Figure 1:
Specimen Geometry
3. METHODOLOGY
The methodology
adopted for this study has certain specific steps which start from experiments
for fatigue testing of the specimen given in Figure 1 on MTS machine and result
data collected for the validation with finite element method and tabulated all
result parameters together to perform regression analysis to determine the
dependency of strain hardening on fatigue crack closure. All steps are shown in
Figure 2.
Figure 2:
Flow Diagram of Methodology
4. FINITE
ELEMENT ANALYSIS OF CRACK
4.1.
3D modeling using Catia V5
R19:
3D
modeling of specimen had been done on CATIA V5 R19 as shown in Fig: 01 the
dimensions of the specimen were based on ASTM standard E399 for fatigue testing
and then it has been imported to Abaqus® 6.10 as a deformable solid
part and analyzed to determine various parameters during crack propagation.
4.2.
FEM Modeling
A
crack had been developed in Abaqus® 6.10 itself as a shell
deformable part. After modeling both the instances were called in assemble
module to insert the crack in the specimen. C3D8R elements were used to mesh
the specimen but not the crack. Crack remains unmeshed throughout the analysis.
Because the whole analysis were done for Mode I as Figure
2 so that one side of the specimen were kept fixed and other end was loaded.
XFEM module was used to study the onset and propagation of cracking in
quasi-static problems. XFEM allows us to study crack growth along an arbitrary,
solution-dependent path without needing to remesh our model. We can choose to
study a crack that grows arbitrarily through our model or a stationary crack.
We defined an XFEM crack in the Interaction module. We specified
the initial location of the crack. Alternatively, we allowed Abaqus®
to determine the location of the crack during the analysis based on the value
of the maximum principal stress or strain calculated in the crack domain.
4.3.
Initial Conditions
Initial values of stresses, temperatures, field variables,
solution-dependent state variables, etc. specified as
follows.
4.4.
Boundary Conditions
Specimen has been
kept in mode I fracture mode that is called as crack opening mode as shown in Figure
3 in this mode tensile forces are exerted on the top and bottom face of the
specimen in this case displacement will be normal to the crack surface.
Figure 3:
Mode I Fracture Modes
Boundary conditions applied to the displacement and rotation
degrees of freedom for the SEN Specimen. One side kept fixed (use Encastre
Boundary condition) and on other side stress applied. During the analysis,
boundary conditions had an amplitude definition that is cyclic over the step.
Following loading conditions were
considered:
- Case1:Pmin=0kN;Pmax=6.2kN;R=0;
OLR=1.0
- Case2:Pmin=0kN;Pmax=6.2kN;R=0;
OLR=1.5
- Case3:Pmin=0kN;Pmax=6.2kN;R=0;
OLR=1.8
- Case4:Pmin=0kN;Pmax=6.2kN;R=0;
OLR=2.0
4.6.
Fields Output
Fields output variables ‘PHILSM’, ‘PSILSM’ and STATUSXFEM under the
Failure/Fracture and Status category respectively are selected to calculate
crack length with no of load cycle.
4.7.
Result visualization
|
|
|
Figure 4:
Crack Propagation
5. REGRESSION
ANALYSIS:
After
FEM analysis, Linear Regression analysis was done on SPSS® 16. From
the output we have drawn the graphs between U Vs. n fitted the trend line and
got coefficients value for trend line equation for each material. After getting
equation for each material we formed a generalized equation that suits the
result of all five aluminum alloys and with the help of this we can predict the
approximation for crack closure of other Aluminum alloys too. The scheme of the
curves is shown below.
Coefficients (For 3003 Al)
|
|
Unstandardized Coefficients |
Standardized Coefficients |
T |
Sig. |
|
|
|
B |
Std. Error |
Beta |
||
|
n |
0.01 |
0.003 |
0.074 |
.350 |
0.000 |
|
(Constant) |
0.773 |
0.009 |
|
86.9 |
0.000 |
|
|
Unstandardized Coefficients |
Standardized Coefficients |
T |
Sig. |
|
|
|
B |
Std. Error |
Beta |
||
|
N |
0.803 |
0.004 |
0.015 |
0.092 |
0.000 |
|
(Constant) |
0.824 |
0.001 |
|
70.223 |
0.092 |
Coefficients (For 6061 Al)
|
Coefficients (For 5052 Al) |
|||||
|
|
Unstandardized Coefficients |
Standardized Coefficients |
T |
Sig. |
|
|
|
B |
Std. Error |
Beta |
||
|
N |
0.207 |
0.015 |
0.001 |
0.003 |
0.000 |
|
(Constant) |
0.798 |
0.048 |
|
10.753 |
0.000 |
Coefficients (For 6063 Al)
|
|
Unstandardized
Coefficients |
Standardized Coefficients |
T |
Sig. |
|
|
|
B |
Std. Error |
Beta |
||
|
N |
0.960 |
0.001 |
0.096 |
0.612 |
0.000 |
|
(Constant) |
0.176 |
0.002 |
|
380.804 |
0.000 |
Coefficients
(For 6351 Al)
|
|
Unstandardized
Coefficients |
Standardized
Coefficients |
T |
Sig. |
|
|
|
B |
Std.
Error |
Beta |
||
|
N |
0.021 |
0.000 |
0.121 |
0.313 |
0.000 |
|
(Constant) |
0.777 |
0.001 |
|
53.4 |
0.000 |
Table 3: List
of equations obtained after regression analysis
|
Material |
Equations after
Regression Analysis |
|
3003
Al |
U=0.160*n+0.973 |
|
6061
T6 Al |
U=0.207*n+0.824 |
|
5052
Al |
U=0.397*n+0.198 |
|
6063
T6 Al |
U=0.415*n+0.136 |
|
6351
Al |
U=0.220*n+0.777 |
6. GENERALIZED
RESULTS
With the help of
these equations we can form a generalized equation
i.e. U=0.22*n+0.77
6.1.
Validation of the Generalized
Equation:
Table 4: Variation
Check for n=3.3 (Value of n is taken as arbitrary)
|
Material |
U (by generalized
Equation) For n=3.3 |
U(by individual
equation) For n=3.3 |
Variation (%) |
|
3003
Al |
1.503 |
1.501 |
1.397 |
|
5052
Al |
1.503 |
1.507 |
1.803 |
|
6061
T6 Al |
1.503 |
1.508 |
1.870 |
|
6063
T6 Al |
1.503 |
1.505 |
1.703 |
|
6351
Al |
1.503 |
1.503 |
1.530 |
7. MODIFIED
PARIS LAW:
Putting the above
relationship between U and n in the Paris law modified Paris Relationship was
formed which gives approximate 2% variation while calculating no. of cycles to
failure which is very well suitable for aluminum alloy
da/dN =
8. CONCLUSION
A plane stress analysis using XFEM and thereafter regression analysis at
different stress range ratio were performed on side edge notched specimen and
effect of strain hardening on crack closure were noticed that the value of
effective stress intensity range ratio (U) increases with the increasing strain
hardening exponent at the crack tip. A generalized relationship was formed for
evaluation of U accordingly a modified Paris relationship was obtained.
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