DRAFT
Crack propagation analysis around the holes in the plates under the effect
of external stresses using the finite element model
CHORFI Sofiane and NECIB Brahim
Mechanical Engineering Department, Laboratory of Mechanics,
University of Frères Mentouri Constantine
Campus Chab Ersas 25000 Constantine, Algeria
Email : [email protected] and [email protected]
Abstract: The presence of holes in the isotropic, anisotropic, discrete or continuum plates are of great
importance in the application of modern technological fields of mechanics, aeronautics, civil and other
engineering technical fields. They are used for the assembly of different structures by riveting, screws or
bolts. However during their lifestyle or their state of drilling, cracks may appear around these holes which
may cause destruction or the total damage of the structures or the mechanical systems where they are
incorporated. In fact, this phenomenon is reflected at the macroscopic scale, by the creation of a surface of
discontinuity called the coves or cracks which leads to the weakening of the structure during its operation
due to the levels of stress at the level of the elasto-plastic areas. As a result, the analysis of the
propagation of cracks around the holes in the plates under the effect of different external loads plays
an important role to prevent the total destruction of these structures and the increase in their life time.
Our work consists on a study of the crack propagation around the holes in two-dimensional
isotropic continuous plates under the effect of external loads. Different modes of cracking with
different orientation angles of cracking along the plane of a plate with a central hole were analyzed.
The theoretical results based on the theory of the Griffith strain energy and compared to numerical
models using the quadrangular finite element methods and experimental results [9-10]. It was found
that the obtained results concerning the spread of cracks around the holes using the finite element
method are in good agreement with those obtained based on the Griffith theory and the experiments.
Key words: holed plate, crack modes, crack length, intensity factors, spread angle, stress, strain, loads.
Nomenclature
Kt stress concentration factor,
max. local stress
max local strain
N nominal strain
N nominal stress
Kc tenacity
KI stress concentration factor (mode I)
KII stress concentration factor (mode II)
a long of the crack
w wide of the plate
a crack length
w plate wide
K eq equivalent stress intensity factor
c hole radius
F1 experimental following factor [9-10]
Introduction
The presence of holes in the plates, notches and other connecting means leads to a
weakening of the structure due to the presence of local stresses or stress concentrations at
these holes. It is therefore advised to avoid as much as possible, drilling machining defects or
functional portions of such kind that increase the likelihood of the presence of cracks [1-2].
However, where the presence of these factors is unavoidable, it is necessary to know the stress
concentration factor associated with each geometry to size these structures, to avoid their
disaster and increase their screw length; something that has been considered in our problem.
In fact, holed plates have many practical applications especially in mechanical, aerospace,
biomechanics and other structures assembled by bolts or riveted, etc.
Our work is to analyze the propagation of the crack in the vicinity of a hole of a twodimensional plate under the effect of external stimuli in tension along the axis (y). Cracks also
considered in this case are initiated itself horizontally in the axis of the hole or at the angle
45°. Using the mesh described above and having regard to the symmetry of the plate, only one
quarter of the plate is considered. The following stress concentration factors for various
management modes and for different crack orientation angles are determined using the
theoretical model and the finite element model and the results are compared with
experimental results given by [8].
In this case a two-dimensional isotropic plate pierced by a circular hole and elliptical was
considered separately in order to determine the stress and strain around the holes.
Local stresses and deformation can be obtained from the stress concentration factor [3] Kt
given by:


Kt  max  max
(1)


N
N
Where, max and max are the local maximum stress and strain; while N and N are the
nominal stress and strain respectively. On the other hand, Griffith developed that the presence
of these defects in a material could amplify the local constraint. The latter can reach a value
equal to the tensile strength without applied external stress is high. This principle can also be
applied to any geometrical discontinuities in a material as an internal or external crack [4].
Generally there are three types of crack patterns [5] and are shown schematically as follows:
Figure 1: Three stress modes of a crack. a) I mode, b) Mode II, c) Mode III
The approach based on linear fracture mechanics is three variables: the applied stress, the
toughness Kc which replaces the limited elasticity and an additional variable is the size of the
defect. However, there are two alternative approaches to fracture mechanics: one using the
concept of critical stress intensity (toughness of the material) and the other energy criteria.
These two approaches are equivalent under certain conditions [5].The stress intensity factors
for the sample or specimen CTS Richard [6] obtained solutions of KI and KII to a central
crack, planar and normal to the side faces. The stress intensity factors at angles different
loading and initial crack orientation are given by the following expressions [7]:
KI  
K II  
cos 
a
a 

1 

W


a


0,26  2,65

W  a 
2
a
a




1  0,55
  0,08

W  a 
W  a 
(2)
sin 
a
a 

1 

W 

a


 0,23  1,40

W  a 
2
a
a




1  0,67
  2,08

W  a 
W  a 
(3)
The intensity of strain K factor is calculated by finite elements; while the intensity of the
equivalent stress ΔKeq factor is the combination of KI stress intensity factor and KII, and
which is given by [8]:

K eq  K I4  8K II4

O , 25
(4)
Consequently and for cracks emanating from an elliptical hole as shown in Figure 4, the
intensity of KI stress factors is given by the following formula [9-10]:
K I  F1   (a  c)
c
(5)
a
Figure 2: Elliptical whole crack length c
2. Effect of the mesh on the maximum stress applied to the plate:
The creation of the finite element mesh is the most important step in this analysis. It is
manually performed by selecting the elements and discrete lines, then a free automatic mesh
is applied to the entire model and a spider web configuration can be obtained [11]. A refined
and well constructed mesh consists of quadrangular elements near the crack tip to visualize
the spread of the latter. A search algorithm using the results of the connectivity matrix in the
regular integration areas, concentric is used to evaluate the integral J [12-13].
For the geometry of our plate, we consider that the exact or theoretical stress concentration
factor is given by: Kt = 2.17 [13]. Accordingly, the maximum stress can be obtained taking
into account the surface of the transverse portion (St = 0.002 mm2) and for an excitation
pressure (P = 1.0 Pa) and is: σmax = 4.34 Pa.
Furthermore, the values of maximum stress are numerically calculated using two types of
finite element meshes. The first less refined mesh (Figure.3a) gives a value of 4.59 Pa,
resulting in an error of 5.8%; while for a more refined mesh (Figure.3b), the maximum stress
obtained is 4.38 Pa, an error of 1%. In conclusion, over the mesh is refined over the results of
the maximum stress converges to the exact values.
Meshed areas
aerea
a) Normal mesh
(b) Refined Mesh
Figure 3: stress σmax for different meshes
3. Effect of the geometry of the hole and external loads on the stress concentration factor
The geometry of the hole (a/ b) and the types of external loading (F) have a great influence
on the stress of the plate concentration factor. Indeed, when “a” is greater than b hole
becomes harmful and can be regarded as a crack (Figure 4). In our case, the effect of the
position of the crack is considered from zero to 45 ° (Figure 5 and 6); then the orientation of
the external load will be considered (Figure 7).
A holed aluminum plate is considered. Its thickness, width and length are respectively:
3mm, 350mm and 1000 mm; and the whole diameter is 100 mm. Due the symmetry of the
plate, only a quarter of the plate is considered in order to analysis the crack propagation
around the hole using the finite element model. We note that this symmetry analysis is often
used for large dimension problems in order the execution time using the FME model; also our
problem has been validated using a solved problem.
The center of the hole is considered the origin of XY coordinates. Numerical methods is
used to determine the maximum horizontal stress in the patch of the plate then compared to
the calculated maximum value which is obtained using tabulated values for stress
concentration factors. Interactive commands will be used to formulate and solve the problem.
3.1. Effect of the geometry of the hole
In applied tensile force along the y axis and varied by the ratio a / b of the hole in the plate
and we observe the influence of this change on the results are shown in Figures 5 and 6.
F
a/b=1
a/b=2
a/b=4
Figure 4: stress versus ratio a / b
Figure 5: constraint of Variation with
horizontal patch
Figure 6: constraints based on the ratio a / b
3.2. Effect of the position of the crack hole
a) Crack initiated at zero degrees:
Application the same tensile load along the y axis is considered and a hole has circle form
of a/b=1. It is assumed also that the crack initiated in the point A of the edge of the hole has
an angle of 0° relatively to the overall view original XY (Figure 7). Then the analysis of the
stress concentration factor variation (K) with the crack length and different orientation angles
(see Figure 9 and 10).
Figure 7: a plate-quarter charge and a crack in the point A
a.1) Change of crack length
Figure 8: Stress distribution along the horizontal patch
Figure 9: Stress versus crack length variation
Figure 10: stress intensity factor variation
based on the variation of propagation angle
a.2) Change of the propagation angle
Angle=60°
Angle= 0°
Angle= 30°
Figure 11: Variation of the firing angle
Angle= 90°
Angle= 30°
Angle=60°
Angle= 90°
Angle= 0°
Figure 12: Variation of stress as a function of the firing angle.
Figure 13: Variation of stress as a function of
the propagation angle.
Figure14: Variation of the stress intensity
factor as a function of the angle of
propagation.
b) Crack initiated at 45 °:
Applying the same tensile load along the y axis is taken and a hole circle of a / b = 1, it is
assumed that the crack initiated in the point B on the edge of the hole has an angle of 45 °
relative to the overall origin XY view (Figure.15), and analyzes the variation of K (stress
concentration factor) according to the crack length and the different angles of orientation.
Figure 15: a plate-quarter charge and a crack in the point B
- Change of the angle propagation
Angle= 0°
Angle=30°
Angle=60°
Figure 16: Variation of stress as a function of the firing angle
Angle= 90°
Figure 17: Change in constraints angle
function
Figure18: stress concentration factor of
variation in angle function
3.3. Effect of the inclination of the loading
By applying a tensile load oriented at a different angle with respect to the origin oxy, it
takes a hole circle a / b = 1, it is assumed that the initiated crack it in the point A of the hole
edge has an angle of 0 ° with respect to the overall XY origin (Figure.19), and analyzes the
variation of K (stress concentration factor) according to the crack length and different angle of
orientation, thus can decompose load following the two direction (see table below).
F
F
F
F
F
F
Figure19: half a plate loaded with tensile strength of different inclined angle pre crack on the axis
Figure 20: Q variation as a function of the
crack length for different loading angle.
Figure 21:% error different numerical
approach based on the crack length.
4. Discussion of results:
In the case of the effect of geometry, while away from the edge pierces one notices that
there is a rapid decrease in stress until they became constant (Figure 5), increased report / b
spirited similar evolution of numerical and analytical constraints. By transforming Cartesian
global coordinate (OXY) in the local coordinate while following the split point and the
direction of propagation of this crack to find the stresses and stress concentration factor in
every step of propagation.
Regarding to the effect of crack initiation, there is only one mode is the mode I (pure
mode); the angle 0°, then there is a growth constraint following the evolution of the crack
length (Figure.9) (Figure.10): IDEM.
When there is a change of crack propagation angle we notice the existence of a mixed
mode (Mode I + II), or the change in stress is inversely proportional to the increase in the
angle of the spread of 0° to 90° (Figure13).
The evolution of the stress intensity factor for mode I KI at 0° is maximum and then
decreases gradually to reach its minimum value at the angle 90°, whereas for the mode II at 0°
the intensity factor stress is zero and then increases until it reaches its maximum value at 90°
or it coincides with the minimum value of I mode (Figure 14).
For a crack initiated at 45° generates two mode (Mode I and II) or call the mixed mode,
and do vary the angle of propagation is observed that the mode I at the angle 0° KI is
maximum value and then decreases gradually until the angle 90°, and the mode II at 0° KII
factor is the minimum value and then increases, and it coincides with the mode I to an angle
45° then it continues to increase up to an angle 90° or becomes greater than KII KI (Figure
18).
Then Keq  for the effect of the inclination of the load is observed that the is inversely
proportional with the direction of the propagation angle, and% error results enlargement
presenting the evolution of the crack length for different approaches (Figure.20-21).
When a crack is urged to an inclined tensile load (c-a-d)ie d. Mixed mode is observed that
there is a bifurcation that it characterized by a change in the direction of propagation. For
tensile loads inclined 30°, 45° and 60°, cracks develop in the direction of -48°, 40° and 32°
respectively in the numerical results and -46°, 41° and 36° respectively in the experimental
results. So a difference of 2° are observed, 1° and 4° between the numerical results and test
results [14,15].
Conclusion
Modeling the initiation of a crack is almost impossible (80-99%), but the spread is feasible,
for it makes the creation of the crack initiation. According to the numerical study on the
growth of fatigue cracks and behavior of the crack in a perforated plate under the effect of a
single charge or inclined traction, the following conclusions can be obtained:
Interest refining the mesh size and the choice of the element and the selected zone (crack
propagation region).
 The influence of the reduction of the radius pierces on increasing the stress
concentration. when the radius tends towards zero the hole behaves as a crack.
 The importance of knowledge of the starting point of the crack propagation angle and
makes us change the existing mode if it is pure or mixed with the same tensile
loading.
 The proportionality between the crack propagation and stress intensity factor.
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