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,20163Eng. &Tech.Journal, Vol.34,Part (A), No.
513
Strain Behavior at Crack Tip in Thin Plate Using Numerical
and Experimental Work
Dr. Fathi A.Alshamma
Mechanical Engineering Department, University of Baghdad / Baghdad
Email: fathi_alshamma@yahoo.com
Bassam Ali Ahmed
Electromechanical Engineering Department, University of Technology/ Baghdad
Email: bsmty77@yahoo.com
Received on:7/10/2015 & Accepted on:20/1/2016
ABSTRACT
In this work, strains were studied and analyzed in a thin flat plate with a surface
crack at the center, subjected to cycling of low velocity impact loading for two types of
aluminum plates (2024, 6061). Experimental and numerical methods were
implemented to achieve this research. Two cases of boundary conditions were used in
this study; clamped-clamped with simply supported at the other edges, and clamped-
clamped with free at the other edges. Numerical analysis using program (ANSYS11-
APDL) based on finite element method used to analyze the strains with respect to time
at crack tip. In the experimental work, a rig was designed and manufactured for cyclic
impact loading on the cracked specimens. The grid points was screened in front of the
crack tip to measure the elastic-plastic displacements in the x and y directions by grid
method, from which the strains are calculated. The results show that the strains increase
with increasing the crack length. It was found that the cumulative number of cycles
leads to increase in the strain values.
Keywords: strain, crack tip, analysis, crack growth, crack plate.
)2024,6061 . (
. .
(ANSYS11-APDL)
.
.
.
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Eng. &Tech.Journal, Vol.34,Part (A), No.3,2016 Strain Behavior at Crack Tip in Thin Plate Using
Numerical and Experimental Work
514
INTRODUCTION
n real life, cracks may occur in some parts of structure that may lead to failure like
accidental cracking of welded connection, explosion of pressure vessel, buildings
and sudden failure of jet aircraft. Therefore, strain analysis and study the cracks
propagation within structures is very important to improve the design against fracture,
Wanhill, et al., 1989. In this paper, strain of surface cracked thin plate under cyclic
impact loading was analyzed. Cycling load involved in many structures such as
automobiles (piston inside cylinder), wing of aircraft, bridges, and machines structures.
In general, three different fluctuating stress-time modes are possible. One is represented
schematically by regular and sinusoidal time dependence. Where in the amplitude is
symmetrical about mean zero stress level, alternating from a maximum tensile stress to
a minimum compressive stress of equal magnitude, this is a reversed stress cycle.
Another type, termed repeated stress cycle illustrated the maximum and minimum are
asymmetrical relative to the zero stress level. Finally the stress may vary randomly in
amplitude and frequency. Gears are subjected to reversed stress cycles, while the
connected rod in a petrol engines and the wing of an aircraft are subjected to repeated
stress cycles, Stephens, et al., 2001.
There are many researchers were studied the strain at crack tip with different
fluctuating stress-time modes. Toribio, and Kharin, 2009, studied the plane-strain
crack subjected to mode I cyclic loading under small scale yielding. Abd- ALRsoul, et
al., 2011, studied the fatigue short and long cracks behaviour in 2024 T4 aluminum
alloy under rotating bending loading. In the short cracks region, cracks grow initially at
a fast rate but deceleration occurs quickly and, depending on the stress level, they either
arrest or are temporarily halted at a critical crack length. Saleh, et al., 2012, in this
paper, the buckling behavior for edge cracked plates under compression loading is
studied considering the influence of the crack parameters (i.e. size, location and
orientation), plate aspect ratio and plate boundary conditions. Sahoo, et al., 2007,
analyzed the effects of plasticity on the stress and deformation field near the crack tip,
while Boljanovic, 2012, proposed a computational model for estimating the crack
growth behavior and fatigue life of a plate with a semi-elliptical surface crack.
The aim of this work is to build up a model to describe the strain behavior at crack tip
in thin plate under cyclic loading. A rig system will be designed and manufacturing for
this purpose. ANSYS11-APDL package will be employed to build up the model and
analysis the strains.
Numerical Analysis
Numerical analysis of structures subjected to various kinds of actions is an
important issue for structural safety. A numerical method can be used to obtain an
approximate solution; approximate numerical procedures have become very accurate
and reliable for the solution with the advent of high speed digital computers, Kareem,
1998. Solving fracture mechanics problems involves performing a linear elastic or
elastic-plastic state analysis and then using specialized post processing commands or
macros to calculate the desired fracture parameters. The following topics describe the
two main aspects of procedure:
1. Modeling the crack region.
2. Selecting of element and meshing.
3. Calculating fracture parameters.
In this paper, the ANSYS software APDL is used for solving fracture problem.
Selecting of element as shown in fig.1 and meshing in fig.2. The element Solid185 in
I
Eng. &Tech.Journal, Vol.34,Part (A), No.3,2016 Strain Behavior at Crack Tip in Thin Plate Using
Numerical and Experimental Work
515
fig.5 is used for 3D modeling of solid structures. It is defined by eight nodes having
three degrees of freedom at each node, translations in the nodal x, y and z directions.
The element has plasticity, hyper elasticity, stress stiffening, creep, large deflection,
and large strain capabilities. It has mixed formulation capability for simulating
deformations of nearly incompressible elastoplastic materials, and fully incompressible
hyper elastic materials. The boundary conditions are clamped-clamped with simply
supported at the other edges plate and clamped-clamped with free at the other edges
plate.
The Crack Region Modeling
Stress and deformation fields around the crack tip generally have high gradients.
The precise nature of these fields depends on the material, geometry and other factors.
To capture the rapidly varying stress and deformation fields, use a refined mesh in the
region around the crack tip. For linear elastic problem the displacement near the crack
tip (or crack front) vary as √ , where (r) is the distance from the crack tip. The
stresses and strains are singular at the crack tip, varying as 1/√. To produce this
singularity in stresses and strains, the crack tip mesh should have certain
characteristics, the crack faces should be coincident and the elements around the crack
tip (or crack front) should be quadratic, with the mid side nodes placed at the quarter
points. Such elements are called singular elements.
Loads and Boundary Conditions
The cycling impact loading is applied on the center of the cracked thin flat plate;
the boundary conditions are clamped-clamped (ux=uy=uz=0) with simply supported at
the other edges plate (ux=uz=0) and clamped-clamped with free at the other edges
plate. Fig.6
Experimental Work
A rig system was designed and built up to achieve this work. The main purpose of
rig system design to get a cycling impact loading to strike vertically at center of plate’s
surface and measurement the induced deformations and calculated the strains. It
consists of electric motor, control number of cycle’s equipment, gearbox, one step of
pulleys and impactor arm. The specifications of electric motor were, power (100watt),
voltage (220 volt), frequency (50Hz), and rotation velocity (780rpm) was reduced by
gearbox that have a reduction ratio (1:40), the step of pulleys having (53mm) diameter
of pinion pulley and (64mm) diameter of wheel pulley so as a reduction ratio (1:1.2), so
that have a velocity for pinion pulley (19.5rpm) and suitable velocity for the wheel
pulley (16.25rpm). A control number of cycle’s equipment determinates the number of
cycles needed to strike a plate’s surface, for this work, the number of cycles was (1000
cycle/sec). The impactor mass (1.5kg), which have a hemispherical end (R=1.5mm),
moves vertically to strike the plate’s surface. The distance between end of impactor and
the surface of plate was constant (80mm). The samples were hold from four sides
(clamped-clamped with simply supported at the other edges) and once again through
(clamped-clamped with free at the other edges). Fig.7 is shows the experimental rig.
Two types of metals were used in this work, aluminum (2024) plate as shown in fig.3
and aluminum (6061) plate in fig.4. Plate dimensions were (200x150 mm&
150x150mm); plate's thickness is constant (6mm). Specifications of metals have shown
in tables 1 and 2 [9]. A grid has been printed in the front of the crack tip with square
Eng. &Tech.Journal, Vol.34,Part (A), No.3,2016 Strain Behavior at Crack Tip in Thin Plate Using
Numerical and Experimental Work
516
grid for measuring the displacements of each point after deformation by cycling impact
load.
Procedure of Work
Grid method is one of the methods of strain analysis, which is whole field in
nature. In order to determine displacements and strain components at given points of
arbitrarily shaped surfaces a grid can be engraved on the surface to be studied. This
grid acts as a reference element and the changes that the grid experiences from the
unreformed to the deformed conditions can be utilized to determine either
displacements or strains. Two difficulties are encountered which limit the use of grids
for measuring deformations; firstly, the strains to be measured are usually very small,
and in most cases the displacement readings are difficult to make with sufficient
accuracy. This is particularly true in strain analysis. However, this method is very much
suitable for the study of deformation in materials .Secondly, when the photographs of
the grid network are magnified by microscope, the images of the grid lines are usually
poorly defined introducing appreciable errors into the displacement readings. This
method has the advantages that a photographic record of deformations covers the entire
field of the specimen. This record can be obtained for either static, dynamic elastic or
plastic deformations. The strain was measured directly. The distance between the grid
lines on the model was measured by a microscope by keeping the magnification of
microscope same before and after loading. The specimen was impacted vertically
through a number of cycles by the impactor on the center of the sample. The number of
the cycles was controlled by controlling equipment. The grid method was used to
calculate the displacement in X-axis (u) and in Y–axis (v). The dimensions of grid were
(30 mm×30 mm) and the length of square is (1mm) .The grid was photographic before
and after the cycle of the sample and the measurements of the displacements was taken
by microscope as shown in Fig.8 for all the samples. Then the strains at the surface
crack tip were calculated in the plate.
Boundary Conditions Change under Cyclic Impact Load
Two types of boundary conditions were used in this work. The first, clamped-
clamped with simply supported at the other edges plate (CSCS). The second of
boundary condition, clamped-clamped with free at the other edges plate (CFCF).
Results and Discussion
The results showed substantial convergence in the numerical analysis with
experimental work and illustrated the effect of cyclic impact load on the strains at
surface crack tip due to number of cycles. Tables 3, 4 and 5 show a comparison
between numerical and experimental values with error percentage.
Numerical Analysis (ANSYS Program)
The effect of the crack lengths on the strains
The extension in length (deflection) is direct proportional with applied stresses that
means the increasing in crack length which leads to increase the values in strain as
shown in Fig.11 to 18. The instantaneous length of crack for aluminum 6061 is greater
than aluminum 2024.Because the aluminum 6061 is more ductile than the aluminum
2024 (young modulus for aluminum 2024 greater than aluminum 6061). Increasing in
the cumulative number of cycles leads to increase in the strains with nonlinear behavior
so that the increasing in crack length also will be nonlinear .The yield region will be
Eng. &Tech.Journal, Vol.34,Part (A), No.3,2016 Strain Behavior at Crack Tip in Thin Plate Using
Numerical and Experimental Work
517
not appoint, so that there is some limiting values that strain hardening will affect the
results therefor the rate of increasing curve slope will be low till nearly 600 cycles then
after that, the rate of curve slope will have a high increasing. Materials especially
metals tend to exhibit a yield stress, above which they deform plastically. This means
that there is always a region around the tip of a crack in a metal, where plastic
deformation occurs, and this plastic region is known as the crack tip plastic zone. The
plastic zone size varies with the number of cycles and it increases with increase the
number of cycles, because the increase in the number of cycles means increasing in the
applied stresses that is leading to increase in the plastic zone size.
The effect of the boundary conditions on the strains
Fig. 19 to 22 show, the results of the strain values at clamped-clamped with free at
the other edges boundary condition will be higher from the clamped –clamped with
simply supported at the other edges boundary condition by maximum discrepancy
percentage (17%) for Al-6061 and (18.9%) for Al-2024 between numerical analysis
and experimental work, this is because the value of the deflection at clamped-clamped
with free at the other edges will be higher from clamped-clamped with simply
supported at the other edges, that is leading to the stress and strain values become
higher.
Experimental Work
The effect of the crack lengths on the strains
The effect of the experimental combined load (cycling impact load) on the
strains at surface crack tip due to number of cycles with crack lengths (Lc) = (7mm,
10mm) and constant depth of crack (2mm) for aspect ratio (AR) = (1.33) of aluminum
plates. Maximum discrepancy percentage of strains is (12.5%) for aluminum 6061 and
(13%) for aluminum 2024 between experimental work and numerical analysis.
The effect of the boundary conditions on the strains
The effect of the experimental combined load (cyclic impact load) on the strains at
surface crack tip due to number of cycles with boundary conditions, clamped-clamped
with free at the other edges and clamped-clamped with simply supported at the other
edges, with crack length (12mm) for aspect ratio (1). When the results of experimental
work were compared with the results of the numerical analysis found the maximum
discrepancy in strain (18.9%). In experimental work it was found that values of
displacements were higher at crack-tip and reduced when leaving the crack tip. Thus,
the strains have greatest value at crack tip. To determine the time of entry of the
specimen in the plastic zone in experimental work was very difficult. But in the
ANSYS program was identified as the time when the specimen in the region of the
plastic zone
CONCLUSIONS
1. Plastic zone has a significant effect on crack growth velocity under cycling impact
loading for 2024 and 6061 aluminum plates used in this work. There are some specific
values of strains at which strain hardening effect on the results. Therefore the rate of
curve slope will be lower until about 600 cycles. After that, the rate of curve slope will
increase again.
Eng. &Tech.Journal, Vol.34,Part (A), No.3,2016 Strain Behavior at Crack Tip in Thin Plate Using
Numerical and Experimental Work
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2. The number of cycles has a significant effect on crack growth velocity specially at
400 cycles were the rate of increasing of crack growth velocity will be very high which
reflects the effect of plastic zone.
3. The effect of ductility of material on the strains under cycling impact loading
becomes more pronounced in aluminum 6061rather than in aluminum 2024.
4. With cumulative number of cycles under the cycling impact loading on the plate,
using clamped-clamped with simply supported at the other edges boundary condition
was better than clamped-clamped with free at the other edges.
Table(1). Mechanical Properties of aluminum 2024. [9]
Young modulus
(E)Gpa
Yield Tensile
Strength () Mpa
Ultimate tensile
strength (.) Mpa
Poisson's
ratio()
Density
()Kg/m3
73 325 470 0.33 2780
Table (2). Mechanical Properties of aluminum 6061. [9]
Young modulus
(E) Gpa
Yield Tensile
Strength () Mpa
Ultimate tensile
strength (.) Mpa
Poisson’s
ratio ()
Density
()Kg/m3
69 275 310 0.33 2700
In following tables, shows comparing between numerical and experimental values with
error percentage:
Table (3). Numerical and experimental strain values of (Al-6061)
Aspect ratio Crack length
(mm) Numerical Experimenta
l
Error
percentage
1 7 0.0011 0.001 9.1%
10 0.0021 0.0023 8.7%
1.33 7 0.0024 0.0021 12.5%
10 0.0044 0.0039 11.36%
Table(4). Numerical and experimental strain values of (Al-2024)
Aspect
ratio
Crack length
(mm) Numerical Experimental Error
percentage
1 7 0.0012 0.0011 9.1%
10 0.0017 0.0019 10.5%
1.33 7 0.0026 0.0023 13%
10 0.0031 0.0033 6%
Table (5). Numerical and experimental strain values of Al-6061 & Al-2024 for
boundary conditions
Aluminum Aspect
ratio
Crack
length
(mm)
Boundary
conditions
Numeric
al
Experiment
al
Error
percentage
6061
1 12
CFCF 0.0039 0.0047 17%
CSCS 0.0044 0.005 12%
2024 CFCF 0.003 0.0037 18.9%
CSCS 0.0031 0.0038 18.4%
Eng. &Tech.Journal, Vol.34,Part (A), No.3,2016 Strain Behavior at Crack Tip in Thin Plate Using
Numerical and Experimental Work
519
Figure (1). Solid185 element geometry.
Figure (2). Mesh200 Element Geometry.
Figure (3). A printed grid on Figure (4). A printed grid on
the plate (Al-2024) the plate (Al-6061)
Figure(5). 3D Model with solid185.
Eng. &Tech.Journal, Vol.34,Part (A), No.3,2016 Strain Behavior at Crack Tip in Thin Plate Using
Numerical and Experimental Work
520
Figure (6). Loading and B.C.
Figure (7). Rig system.
Figure (8). Crack with grid.
Eng. &Tech.Journal, Vol.34,Part (A), No.3,2016 Strain Behavior at Crack Tip in Thin Plate Using
Numerical and Experimental Work
521
Figure (9). CSCS-boundary condition
Figure (10). CFCF-boundary condition
Figure (11). x numerical and experimental with number of cycles
(AR=1.33, Lc=7mm & CSCS for Al-6061)
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0 200 400 600 800 1000 1200
x
Numberofcycles
ANSYS
EXP.
Eng. &Tech.Journal, Vol.34,Part (A), No.3,2016 Strain Behavior at Crack Tip in Thin Plate Using
Numerical and Experimental Work
522
Figure (12). y numerical and experimental with number of cycles
(AR=1.33, Lc=7mm & CSCS for Al-6061)
Figure (13). x numerical and experimental with number of cycles (AR=1.33,
Lc=10mm & CSCS for Al-6061)
0
0.001
0.002
0.003
0.004
0.005
0 200 400 600 800 1000 1200
y
Numberofcycles
ANSYS
EXP.
0
0.001
0.002
0.003
0.004
0.005
0.006
0 200 400 600 800 1000 1200
x
Numberofcycles
ANSYS
EXP.
Eng. &Tech.Journal, Vol.34,Part (A), No.3,2016 Strain Behavior at Crack Tip in Thin Plate Using
Numerical and Experimental Work
523
Figure (14). y numerical and experimental with number of cycles
(AR=1.33, Lc=10mm & CSCS for Al-6061)
Figure (15). x numerical and experimental with number of cycles
(AR=1.33, Lc=7mm & CSCS for Al-2024)
Figure (16). y numerical and experimental with number of cycles
(AR=1.33, Lc=7mm & CSCS for Al-2024)
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0 500 1000 1500
y
Numberofcycles
ANSYS
EXP.
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0 200 400 600 800 1000 1200
x
Numberofcycles
ANSYS
EXP.
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0.004
0.0045
0 500 1000 1500
y
Numberofcycles
ANSYS
EXP.
Eng. &Tech.Journal, Vol.34,Part (A), No.3,2016 Strain Behavior at Crack Tip in Thin Plate Using
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524
Figure (17). x numerical and experimental with number of cycles
(AR=1.33, Lc=10mm & CSCS for Al-2024)
Figure (18). y numerical and experimental with number of cycles
(AR=1.33, Lc=10mm & CSCS for Al-2024)
Figure (19). x. numerical with number of cycles
(B.C: AR=1, Lc=12mm for Al-6061)
0
0.001
0.002
0.003
0.004
0.005
0 200 400 600 800 1000 1200
x
Numberofcycles
ANSYS
EXP.
0
0.001
0.002
0.003
0.004
0.005
0.006
0 200 400 600 800 1000 1200
y
Numberofcycles
ANSYS
EXP.
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0 500 1000 1500
x
Numberofcycles
Clamped‐Free
Clamped‐Simply
supported
Eng. &Tech.Journal, Vol.34,Part (A), No.3,2016 Strain Behavior at Crack Tip in Thin Plate Using
Numerical and Experimental Work
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Figure (20). y numerical with number of cycles
(B.C: AR=1, Lc=12mm for Al-6061)
Figure(21). x numerical with number of cycles
(B.C: AR=1, Lc=12mm for Al- 2024)
Figure(22). y numerical with number of cycles
(B.C: AR=1, Lc=12mm for Al-2024)
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0 200 400 600 800 1000 1200
y
Numberofcycles
Clamped‐Free
Clamped‐Simply
supported
0
0.001
0.002
0.003
0.004
0.005
0.006
0 200 400 600 800 1000 1200
x
Numberofcycles
Clamped‐Free
Clamped_Simply
supported
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0 200 400 600 800 1000 1200
y
Numberofcycles
Clamped‐Free
Clamped‐Simply
supported
Eng. &Tech.Journal, Vol.34,Part (A), No.3,2016 Strain Behavior at Crack Tip in Thin Plate Using
Numerical and Experimental Work
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REFRENCES
[1]. Ewalds, H. L. and Wanhill, R. J. H., 1989, "Fracture Mechanics",
London, Edward Arnold.
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[4]. Abd- ALRsoul H., Basim H., and Khairallah S., 2011, "Crack Growth Model For
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[5] Saleh N. H. and Kuess S. K., 2012, "Studying a Buckling Behavior for Edge
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