Simulation - Guru Mahesh

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International Journal of Mechanical and Production

Engineering Research and Development (IJMPERD )

ISSN 2249-6890

Vol. 2 Issue 2 June 2012 68-78

© TJPRC Pvt. Ltd.,

SIMULATION OF THE CRACK PROPAGATION USING FRACTURE

MECHANICS TECHNIQUES IN TITANIUM-BASED ALLOY SKIN

G. GURUMAHESH1 & Y. SURESH BABU2

1,2 Rajiv Gandhi Memorial College of Engineering And Technology, Nandyal, A.P, India.

ABSTRACT

The focus of this project is to investigate how a crack propagates and grows in a typical Ti-6Al-4V

material plate. By using the finite element method software (ANSYS13) were used to simulate crack

growth and to compute the stresses and the stress-intensity factor. A specific plate design was selected

and a corner crack was investigated. This configuration was used since the engineers often detect this

type of crack in plates. The Von Mises stress near the crack tip is compared against the yield strength of

the material. The Mode I stress-intensity factor is compared against the material’s fracture toughness.

The results show that the plate can tolerate small cracks in the structure. The fatigue strength of the

structure is recommended to be assessed in the future.

KEYWORDS: Fracture Mechanics, Alloy Skin, Titanium.

INTRODUCTION/LITERATURE REVIEW

Airliners progressively more demand for high performance and fuel-efficient aircraftsdue to the

increasing gasoline price. In order tomeet the market needs, original equipment manufacturers are

developing smaller and lighter aircraft engines. Industry analysts are expecting the engine

components in the next decades to be very space efficient. As a result, light but high strength

materials are very valuable and competitively sourced to reduce weight and cost in manufacturing

aircraft engine.

Aero engine designers design component in various shapes and sizes for mounting bleed air ducting,

starter air duct, fuel lines and hydraulic lines to the engine core. One can find more than one hundred

mounting points in an engine. Many of the ducting supports have the shapes of L, T and Z (multiple

bends) to accommodate multiple tubes in the tight space found in an aircraft engine. In general, the platethickness ranges from .125 to .500 inch because the plate not only must be thin to reduce weight and

cost, but also must be functional and serve its purposes in the extreme environment, e.g. Extremely high

temperature, combination of vibration load, bending moment and maneuver load. Any crack found

in a plate may cause the ducting to become unstable during a mission, and thus induce high cycle fatigue

load on the overall major structures and shorten the structures life. From the economic standpoint, it is a

cost saving strategy to replace brackets before they are completely damaged due to replacing broken

machine plate components mitigates the risk of damaging other major components, such as the ducting,

which are more costly to replace. Besides, replacing a bracket before it completely breaks can avoid

many engineering catastrophes and save many lives. On the other hand, knowing the fatigue life of a





G. Gurumahesh & Y. Suresh Babu 70

intensity factors and compare it against material toughness to determine the influence of the crack on the

plate.

METHODOLOGY

Engineers strive to optimize part geometry by designing the thinnest possible components

because this approach not only reduce engine weight but also reduce the risk of brittle structure often

found in bulk materials. Being able to determine the rate of crack growth, an engineer can schedule

inspection accordingly and repair or replace the part before failure happens. Being able to predict the

path of a crack helps a designer to incorporate adequate geometric tolerance in structural design to

increase the part life. The methodology used to investigate the mechanics of crack propagation consists

of the following steps:

• Model creation

• Elastic stress analysis of the UNcracked body

• Flaw implementation

• Crack propagation

• Elastic stress analysis of the cracked body

• Calculation of stress intensity factor

• Interpretation of results

MODEL AND MATERIAL PROPERTIES

Model is having with the dimensions of 100 mm in height, 200 mm in width,and 10 mm in

thickness. In addition, the boundary condition chosen by Thomas for his optimum plate research was

applied to the plate in this study. At the base of the plate, one end is clamped in all degree of freedom,

while the other end is a slider. The plate is pulled downward and top of the section.



Simulation of the Crack Propagation using Fracture Mechanics

Techniques in Titanium-Based Alloy Skin

71

Among all the titanium based alloys, according to Immarigeon et al [3], Ti-6Al-4V is by far the most

widely used, accounting for almost half of all titanium used in aircraft because the material can increase

the strength+to+weight ratio in structures and provide heat resistance with weight savings. However, the

significant weight savings permitted by these titanium application developments generate specificdrawbacks that need particular technological developments. Among the most important concerns are

brittle inclusions, which are difficult to detect by non-destructive testing, can initiate cracks and

produce early failure of the structures [2]. Materials imperfections due to manufacturing process, for

example, voids and impurities can develop flaws that may cause the material to become weak. For those

reasons, the material chosen in this study is Titanium+6Al+4V and the properties are summarized in

Table 1.

Table 1 : Material properties of Titanium-6-Al-4V

Material Properties Comments

Density 4.43gm/cc

Tensile strength, ultimate 1170 Mpa

Tensile strength, yield 1100Mpa

Elongation at break 10.0%

Modulus of elasticity 114 Gpa Average of tension and compression

Compressive yield strength 1070 Mpa

Notched tensile strength 1550 mpa Kt(stress concentration factor)=6.7

Poission’s ratio 0.33

Fatigue strength @no.of cycles 1.00e+7 Kt(stress concentration factor)=3.3

160 Mpa

@no.of cycles 1.00e+7 Un notched

700 Mpa

Fracture toughness 43.0Mpa-m1/2

Shear modulas 44.0 Gpa

Shear strength 760Mpa Ultimate shear strength

a) Infinite plate with centre through crack under tension




The concept of stress intensity factor plays a central role in fracture mechanics. We now refer to

Tada [19] to present some classical examples of cracked geometries - represented in Figure 3 - for which

the stress intensity factor has been computed or approximated explicitly. It is assumed that crack

propagation may not occur, i.e., the problem is static.

b) Semi-Infinite plate with a center through crack under tension

Figure 3

The stress intensity values for these geometries are as follows, where the letters a) - d) used to

identify the formulas are in correspondence with those of the pictures in figure 3

a) KI=σ√(∏a): ------------- (1)

b) b) KI=1.1215σ√(∏a) ------------ (2)

c) KI=σ√ (∏a) (1-0.025(a/b) 2+0.06(a/b) 4) √ (sec (∏a/2b)) ------------ (3)





73

d) KI=σ√ (∏a) √ ((2b/ ∏a) tan (∏a/2b)) 0.75+2.02(a/b) +

0.37(1-sin(∏a/2b)3)/cos(∏a/2b)) ----------- (4)

The previous examples involved geometries of infinite dimensions. Ansys [1] computed the stress

intensity factor for the finite geometry represented in Figure 4.

Fig. 4 : Finite plates with a centre through crack under tension

He considered a rectangular plate, of height 2h, width 2b, with a central through crack of length 2a,

which was loaded from its upper and lower edges by a uniform tensile stress (σ). For this particular

geometry, he estimated

KI= σ√(∏a)(1+0.043(a/b)+0.491(a/b)2+7.125(a/b)3-28.403(a/b)4+59.583(a/b)5-65.278(a/b)6+29.762(a/b)7

(5)

Using the numerical package Ansys 13, we also determined the value of the stress intensity factor KI

for the same geometry. This was computed using finite elements on a mesh with quadratic triangular

elements on the vicinity of the crack tip, and quadratic rectangular elements everywhere else. Quarter

point elements, formed by placing the mid-side node near the crack tip at the quarter point, were used to

account for the crack singularity. More details can be seen in [4]. In Table 2 we display some values of

KI /K0, whereK0 =σ√πa, up to two significant digits. It can be seen that our results, identified by

theoretical values by using of empirical formulas, are in line with those predicted by Ansys, and even

more so for smaller values of a/b. We further illustrate this analysis in Figure 5, for which more data

points were taken.




Table 2 : Stress intensity factors

a/b=0.1 a/b=0.2 a/b=0.4 a/b=0.6 a/b=0.8

Ansys 1.01 1.07 1.24 1.50 2.02

Theoretical 1.01 1.06 1.22 1.48 1.97

0

0.5

1

1.5

2

2.5

a/b=0.1 a/b=0.2 a/b=0.4 a/b=0.6 a/b=0.8

Ansys

Theoretical

Figure 5

As we had already mentioned, the stress intensity factor depends on the geometry of the plate we are

considering. In particular, it depends on the ratio h/b. On Table 3 we display the values of K I /K0,

determined again using Ansys, for different geometries.

Table 3 : Values of KI /KO for different geometries

a/b=0.1 a/b=0.2 a/b=0.4 a/b=0.6 a/b=0.8

h/b=0.25 1.17 1.57 2.65 4.03 7.44

h/b=0.5 1.04 1.17 1.63 2.42 3.77

h/b=1 1.01 1.06 1.22 1.48 1.99

h/b=2 1.01 1.02 1.11 1.30 1.81

h/b=α 1.01 1.02 1.11 1.30 1.81

We note that as the value of h/b increases, the values of KI /K0 tend to the values of the last line

(h/b=∞), which refers to values that we would expect for an infinite stripe with a centre through crack

under tension, as in Figure 3-c. We have computed these using the values of K I from the formula c). To

better illustrate this idea, we conclude with the graphical representations of the values of Table 3 on

Figure 6.





75

0

1

2

3

4

5

6

7

8

a/b=0.1 a/b=0.2 a/b=0.4 a/b=0.6 a/b=0.8

h/b=0.25

h/b=0.5

h/b=1

h/b=2

h/b=α

Figure 6

SIMULATION OF CRACK GROWTH

A static fracture analysis was performed, where the goal was merely to compute the stress intensity

factors. Now, we are ready for a quasi-static analysis, which means we now look into the actual

propagation of the crack. We observe that this is not a dynamic analysis in the sense that dynamic effects

such as wave propagation are not taken into account.

The thing we have to do is to follow the position of the crack tip, as the propagation occurs. We use

a step by step process. For that, we consider an increment a, which is the distance between two

consecutive points of the crack tip. Once a has been chosen, a possible algorithm will be as follows.

Given: a step size a, the initial crack tip coordinates x(l)

tip for l = 0.

1- Increment l;

2- Compute the values of the stress intensity factors, KI and KII;

3- Determine the angle of propagation θp(K)using equation

4- Update the crack tip coordinates.

5- Back to step 1.

We use Ansys and Mat lab to implement this algorithm with a= 0.025.We thus obtain a set of

consecutive crack tip coordinates, as displayed in Table 4. These constitute a discrete approximation for

the crack path.




Xtip Ytip

0 0.10 0.10

1 0.13 0.095

2 0.15 0.090

3 0.18 0.088

4 0.20 0.086

Xtip Ytip

5 0.23 0.085

6 0.25 0.084

7 0.28 0.084

8 0.30 0.084

9 0.33 0.084

Xtip Ytip

10 0.35 0.085

11 0.38 0.085

12 0.40 0.086

13 0.43 0.088

14 0.45 0.089

Fig. 7 : Crack Tip Region





77

Initial crack crack path n=6

Crack path n=15

Figure 8

Finally Figure 10 represents the deformed state of the plate for the initial crack, and again after 7 and

15 steps.

a) Initial crack b) crack path n=6 c) Crack path n=15

From these figures it seems that there is a tendency for the crack propagation to occur mainly in

modeI, during continued fracture. This agrees with the predictions of [6, 9].




REFERENCES

[1] M. H. Aliabadi and M. H. L´opez.Database of stress intensity factors.Computational Mechanics

Publications, 1996.

[2] S. N. Atluri. A path independent integrals in finite elasticity and inelas-ticity with body forces,

inertia and arbitrary crack-face conditions.Eng.Fract. Mech, 16:341-364, 1982.

[3] M. F. Ashby and D. R. Jones.Engineering materials 1, an introduction to their properties and

applications, Butterworth Heinemann, 1996.

[4] R. S. Barsoum. Triangular quarter-point elements as elastic and perfectly-plastic crack tip

elements.Int. J. for Num. Meth.in Eng.,11:85-98, 1977.

[5] D. Broek.Elementary engineering fracture mechanics. Kluwer Academic Publishers, Dordrecht,

1986.

[6] G. P. Cherepanov.Mechanics of Brittle Fracture.MacGraw-Hill, New York 1979.

[7] L. B. Freund.Dynamic Fracture Mechanics.Cambridge University Press, 1990.

[8] A. A. Griffith. The phenomena of rupture and flows in solids.Phil. Trans.Roy. Soc. London,

A221:163-197, 1921.

[9] D. Hegen.An Element-free Galerkin Method for Crack Propagation in Brittle Materials. PhD

thesis, Eindhoven University of Technology, 1997.

[10] C. E. Inglis. Stresses in a plate due to the presence of cracks and sharp corners.Proc.Inst. Naval

Architects, 60, 1913.

[11] G. R. Irwin. Analysis of stresses and strains near the end of a crack transversing a plate. Trans.

A.S.M.E., J. Applied Mechanics, 361-364,1957.

[12] G. R. Irwin. Fracture.Encyclopedia of Physics (Handbuch der Physic),Vol IV, Springer, Berlin,

1958.

[13] G. R. Irwin. Fracture.Encyclopedia of Physics (Handbuch der Physic)”,Vol VI, Fl¨ugge (Ed.),

Springer Verlag, Berlin 551-590, 1958.

Simulation - Guru Mahesh

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