** **

STEPAN MAJOR,

MICHAL RUZICKA,

Department of Technical Education

University Hradec Králové

Rokitanského 62, Hradec Králové

CZECH REPUBLIC

s.major@seznam.cz

**Abstract: - **
These article is devout to the fatigue crack growth and it’s
simulation. The special problem of planar stress is discussed. The
effectiveness of three-dimensional and two-dimensional crack is
discussed. Both models are compared. In the article is shown that the
simulation based only on the stress intensity factor K is much less
accurate than the model based on combination of stress intensity factor*K* and stress triaxiality *T*. The stress triaxiality *T* can describe local influence of constrains and so improve
crack model.

**Key-Words: -**
Fatigue crack, stress intensity factor *K*, triaxiality * T*, constrain, plane stress, plane deformation.

**1 Introduction**

This article is devout to the problem of simulation of fatigue crack in the specimen made from aluminum alloys.

In the case of engineering structure, its failure is often caused by materials fatigue. The material fatigue is cumulative damage of material under cyclic loading [1,2,3]. The fatigue fracture typically initiates in the region of maximal stress, i.e. in the zone of stress concentrations [3]. Corners and wholes in structure are acting as such stress raisers. These crack initiation sites are subjected to the same loading spectrum as other parts of structure.

In the work are discussed options for simulations and prediction of fatigue
crack [4,5,6]. The model of fatigue crack was improved by stress
triaxiality *T *[7,8,9]. The stress triaxiality *T *was used
to describe crack growth, primarily to determine crack velocity.

**2 Experimental Procedure**

Each theoretical model must be verified by comparison with experimental data. An experiment is needed in which the crack will propagate from the corner of the square hole in the flat plate. The geometry of specimen is shown in Fig. 1. The specimen is subjected to cyclical loading (pull – push) in the direction parallel to the plate. The fatigue crack propagates from the corners of central hole. Results of theoretical simulations are compared with experimental results obtained from fractographycal analysis of fracture surface [10,11,12,13]. The fractographycal analysis allows to track changes in the geometry of crack forehead. The experimental sample was prepared from aluminium alloy 7075.

Fig. 1 Sample geometry

**3 Computional Model Based On Stress Intensity Factor**

**
3.1 Relationship between crack velocity v and stress intensity
factor ΔK
**

In an effort to obtain functional model of planar cracks several approaches
have been proposed. Most of them are based on first amplitude of
stress-intensity factor *ΔK*, which is understood as a variable
along the forehead the crack. In engineering practice different types of
computer programs are used, these programs take into account loading and
material characteristics. Control algorithms of these programs are based on
concept of interacting plastic zones. Also, algorithms based on analytic
assessment of crack closure and effective amplitude of *ΔK _{eff}* are used. The latter is used only rarely. Simple
phenomenological hypothesis was postulated. This hypothesis says:

The velocity *v* of crack growth (at a certain point on the
forehead, the velocity is perpendicular to the forehead of the crack) is
controlled by local value of relative stress intensity factor *K' _{r}*. The relative stress intensity factor is specific
for given loading mode. The value of relative stress intensity factor

However, this hypothesis has obvious shortcomings. Firstly, the fatigue
experiments used to produce function
are relatively complex. For each loading mode (or it's combination), it is
necessary realize separate set of fatigue experiments. Also, detailed
fractographyc analysis of crack surface is necessary. The assessment of *K'*_{ }is also considerably complex because radius of
crack's forehead changes over time.

**3.2 **
**
Results of simulation based on stress intensity factor
conception
**

Quantitative analysis of the material on the fatigue crack forehead requires complicated calculations. These calculations require 3D-model of specimen with very dense mesh (high number of cells). Moreover, the deformations on the cracks forehead are elasto-plastic, i.e. finite element model used in calculation is non-linear. Whole problem is complicated by requirement to simulate the process in a longer time.

However, basic trends can be obtained relatively easily by 2D-simulation. These 2D-simulation are based by assumption of small deformations and plane stress or plane strain.

The results of such simulations are shown in Fig. 2 to Fig. 5. All graphs
are related to the point of maximal loading. Horizontal axes display the *x *coordinate in the spreading direction. Vertical lines indicate
the current positions of crack forehead. The shift of crack forehead *u _{y}* (Fig. 2) is engraved on the vertical axis

Fig. 2 Results of simulation: shift of crack forehead *u _{y.}*

**
**

Fig. 3 Results of simulation: plastic deformation

In the case of general 3D-object, the crack constrain along the curved forehead is variable. The crack constrain is function of series of variables such as plastic zone shape, plastic zone thickness, specimens shape or forehead profile. With increasing penetration depths of crack, the effect of crack constrain is growing.

Fig. 4 Results of simulation: shift of crack forehead *u _{y.}*

*
_{
}
*

Fig. 5 Results of simulation: shift of crack forehead *u _{y.}*

** **

**
4 Triaxiality and K-factor As Interacting Controling
Parameters
**

Previous results confirm fact that the driving effect of *K* factor
must be taken in the account during the simulation of motion of the fatigue
cracks forehead. Another fact which must be taken into account during
simulations is local effect of constraint variations. As a parameter of
local constraint, the stress triaxiality *T* was used. The stress
triaxiality is defined as ratio between medium stress

(1)

and Von Mises effective stress , is defined as sum of multiplied differences between principal elements of stress tensor:

(2)

where

(3) .

If the triaxility *T* is growing, the effect of constrain is growing
too. As a result of these facts the controlling function
mast be written as

(4)

This new model needs to be tested. Testing will be based on assumptions: (1) the state of stress is changing along the whole length of cracks forehead; (2) stress is changing with growing distance from cracks forehead. The model assumed cyclic pull-push loading.

The state of stress triaxility will be determined by 3D static calculation
of triaxility *T* in the distance 45μm from cracks forehead. The
state of deformation is elasto-plastic during whole computing cycle.
Computing cycles are repeated roughly until the moment when one third of
maximum stress is reached.

The value of velocity *v, K'* factor and triaxiality *T* were
calculated along two fractographycaly identified foreheads. These two
foreheads correspond to two symmetrical cracks in the specimen with central
opening. This hole determines central plane of symmetry and plane of crack.
Both of cracks are initiated from central hole.

The foreheads velocity *v* is growing according to increase of both
quantities *K' *and also triaxiality *T*. This way you can
get a substitution for function
Eq.1 as count of two independent functions. These two functions are *K'* factor and triaxiality *T*. These functions can be
written as:

** **

(5)

This modified function can be used for simulation of two different models. These two models represents fatigue crack in the corner of the structure with distinct geometry.

Both models are subjected to the same loading with the same time course. In
bout cases the crack initiated in the corner of the specimen. The results
of simulations are shown in the Fig. 6. Full lines in the Fig. 6 correspond
to the smoothed foreheads of fatigue crack. For each of these cracks
(respectively for each stopping of forehead) the method of border elements
was used. In these manner *K' *was* *correlated. For every
third forehead the triaxiality *T* was determined by finite element
method and using the results obtained by border element analysis.

Then the time course of perpendicular velocity *v _{P}* can
be determined. For each new location of crack the procedure is repeated. In
the Fig. 6 are rectangles with a number of cycles

Both pairs of simulations and tests lead to a good match of the results.
Theoretically obtained positions of foreheads correspond to the points on
the fracture surface. This match of the theoretical curves and points will
stand out in comparison with results obtained by calculation based only on *K'* (look at the Fig. 5 and compare results in the case (a) with
results in the case (b)). This agreement is the consequence of reduction of
driving effect of *K*-factor at the first stage of crack growth. The *K*-factor will fall only lightly after the introduction of
triaxiality, but the effect on the shape and velocity of forehead is
considerable.

Utilization of triaxiality in the fatigue crack growth simulations leads also to the good agreement between theoretical a real velocity of forehead. Only in the case of first model the theoretical velocity is considerable higher than in the reality (compare Fig. 5 (a) and (b)).

Causes of this distortion are not conclusively clearly. The fractographical
analysis of real specimen shows that in the reality the fatigue crack
initiate in multiple places. For example in the studied specimen the crack
initiate on three places. These three particular cracks grow on the three
different planes. These three particular planes are very slightly diverted.
These three cracks merge together after changing the crack propagation
mechanism. The development of the fracture could thus be hampered in the
initial stage of the process. The model did not count on the existence of
three initiations. Distortion could also be caused due to the fact that the
bulk of the growth of the through-crack was controlled by *v*, *K'*, *T*. These three values are obtained for crack growing
from the corner of flat specimen. All these values were determined for
symmetrically deployed pairs of cracks. If the symmetry of the cracks were
not used, the function would be dependent on specimens shape.

Fig. 6 Fractographycal analysis and simulations. (a) model base on K-factor
conception; (b) model based on combination of *K* and *T*.
Black dots at are experimental points obtained by fractography.

**5 Conclusion**

Simulation of planar fatigue crack growth with curved forehead in 3D-body
should be based on assumptions: the calculations must take not only stress
intensity factor *K* in mind, but also constrain at the crack tip
and its course. The value of crack constrains affects value of cyclic
stress in the critical region before crack. The local value of constrains
can be characterized by stress triaxiality. The stress triaxiality is
determined by ratio between medium value of normal stress and Von Misess
effective stress. The theory was verified in practice. This theory says
that the local value of forehead velocity is dependent on the local value
of stress intensity factor for normal stress normalized to stress 1 MPa and
local value of triaxiality. The triaxiality is estimated by calculations in
elasto-plastic region. Parameters used in this calculation are obtained
from fatigue test on the specimen with simple geometry. In this case
numerical simulations lead to the satisfactory results.

**Acknowledgment: **
This project was prepared by finantial support
*
“Metoda konečných prvků a automatizace experimentu ve výuce mechaniky a
technických laboratoří na katedře technických předmětů”
*
18/I zč SV 2119 - pracoviště 1440.

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