Modelling of Fatigue Crack Propagation in Part-Through Cracked Pipes Using Gamma Function

Mechanics, Materials Science & Engineering, September 2016

ISSN 2412-5954

Modelling of Fatigue Crack Propagation in Part-Through Cracked Pipes Using Gamma Function Pawan Kumar 1,a, Vaneshwar Kumar Sahu2, P.K.Ray2, B.B.Verma2 1

Institute For Frontier Materials, Deakin University, Australia

2

National Institute of Technology, Rourkela, India

a

pkumar@deakin.edu.au DOI 10.13140/RG.2.2.16973.03043

Keywords: fatigue crack propagation, part-through cracked pipes, gamma model.

ABSTRACT. In the present investigation a gamma model has been formulated to estimate the fatigue crack growth in part-through cracked pipe specimens. The main feature of the model is that the gamma function is correlated with various physical variables like crack driving parameters and materials properties in non-dimensional form so that the proposed model can be used for different loading conditions. The validation of model has been done with experimental data in order to compare its accuracy in predicting fatigue crack growth.

Introduction. The fatigue crack growth Nomenclature behaviour of surface crack in a pipe in a semi circumferential crack length; radial direction is one of the most serious problems associated with piping systems ai initial crack length (mm); as it is responsible for detection of leak aj final crack length (mm);; before break. There are different piping integrity systems used in aircrafts, A, B, C, D curve fitting constants offshore oil drilling, and coolant pipes in da/dN crack growth rate; high pressure nuclear reactors which encounter fluctuating loading condition. K stress intensity factor (MPa); Due to this kind of loading condition a new KC fracture toughness (MPa m); surface crack can generate or an existing K stress intensity factor range (MPa m); crack can propagate. This leads to damage in structure and integrity. Many M specific growth rate; researchers have studied fatigue crack growth problems in pipes. Different mij specific growth rate in interval( j-i); techniques like numerical analysis, finite N number of cycles; element method, boundary integral have P been used to address fatigue crack growth N j predicted fatigue life using exponential model; in pipes. Jhonson et al. [1] used boundary R load ratio; integral method for tension loading and providing a numerical solution to the Ri Internal radius of specimen; problem. Daond and Cartwright [2] t Thickness of specimen; applied strain energy release rate method w width of specimen in uniform tension as well as pure bending condition in pipes. Delate et al. [3] reported numerical results for exterior cracks of semi-elliptical shape using line spring model.

In present research a modified gamma function is used to model fatigue crack growth in part- through cracked pipe in radial direction.

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Experimental Procedure. In the present investigation TP316L grade of stainless pipe was used. The chemical composition and mechanical properties of the material is presented in Tables 1 and 2 respectively. A part-through notch of angle 45o was machined by wire EDM and shown in Fig.1.

Fatigue crack growth test was conducted using a servo-hydraulic dynamic testing machine Instron 8800 on part-through cracked pipe specimens. The tests were conducted in air and at room temperature under constant amplitude 4-point bend loading condition. The schematic loading diagram is shown in Fig. 2. Seven specimens were tested in order to formulate the model and the 8th specimen was tested for validation of the model. Fig. 3 shows the fractured surface of the break opened specimen after fatigue test.

Table 1. Chemical Composition of TP316L stainless steel.

Element

C

Mn

Si

Weight (%)

0.03

2.00

0.75

Cr

Ni

16-18 10-14

P

S

Mo

N

Fe

0.045

0.030

2-3

0.10

balance

Table 2. Mechanical Properties ofTP316L stainless steel.

Modulus of elasticity (E), GPa 220

Yield strength ( MPa 0.3

366

Fig. 1. Cross sectional view of specimen (All dimensions are in mm).

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ys ),

Ultimate tensile strength ( ut), MPa 611

Mechanics, Materials Science & Engineering, September 2016

ISSN 2412-5954

Fig. 2. Four-point bend schematic loading diagram (All dimensions are in mm).

Fig. 3. Fracture surface of specimen.

Formulation of Model. Fatigue crack propagation, a natural physical process of material damage, is characterised by rate of increase of crack length (a) with number of cycles (N). It requires a discrete set of crack length vs. number of cycle data generated experimentally. The experimental a-N data is shown in Fig. 5.

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Crack length, a (mm)

5.8 4.8 3.8

2.8 1.8 0.8 0.00E+00

2.00E+04

4.00E+04

6.00E+04

No. of cycles, N

Fig. 5. Experimental a-N curve.

Gamma function is defined as the generalization of the factorial function to non-integral values, introduced by the Swiss mathematician Leonhard Euler in the 18th century. For a positive whole number n, the factorial (written as n!) is defined by n n n. To extend the factorial to any real number n > 0 (whether or not n is a whole number), the gamma function is defined as [5-7]:

(1)

In our present investigation a modified gamma model has been proposed to predict crack growth in part-through cracked pipe. Here term t is replaced by number of cycles N. The term z is chosen in such a way that it becomes non-dimensional and represents the parameters that affect crack growth.

The integral is chosen so that it is non-dimensional and represents crack growth at the end of a fixed number of loading cycles. Generally fatigue crack growth depends on the initial crack length, material properties and specimen geometry, loading conditions etc. The non-dimensional parameter is chosen in such a way so that it includes all these variables and properties. The expression for predicting the final crack length at the end of N cycle is given by:

,

(2)

where z has been replaced by [m a/w)]; w is the specimen thickness; m is defined as a non-dimensional parameter whose value remains approximately constant for a given cycle interval.

The value of m includes all the properties which affect crack growth.

Fatigue crack growth behaviour depends upon initial crack length and load history. Therefore, while using gamma model each previous crack length is taken as initial crack length for the present step and non-dimensional parameter m is calculated for each step in incremental manner. The experimental a-N data have been used to determine the parameter m for each step using MATLAB programming. At first, initial value of m is assumed for a given interval of cycle. The iteration is continued till the value of m for which LHS and RHS of the equation (2) becomes nearly equal (within +0.01) is taken as the value of m for given cyclic interval.

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The RHS of equation (2) was solved with the help of MATLAB programming.

The expression of m is given by: ,

(3)

where A, B, C, and D are curve fitting constants. The non-dimensional number m is correlated with another parameter l which takes into account two crack driving forces K and Kmax as well as material parameters Kc, E and YS.

.

(4)

The stress intensity factor K is calculated by equation (5) [8]:

,

where

(5)

is the bending stress; is the axis-symmetrical stress which is zero in present case.

The predicted crack length was calculated as:

,

(6)

where m has been given by equation (3) after putting average value of curve fitting constants (for seven specimens) and validate it by using eq. (6) for 8th specimen.

Discussion. The predicted a-N curve obtained from proposed gamma model was compared with experimental test data (Fig. 6). The experimental a-N data of specimen no. 1, 2, 3, 4, 5, 6, and 7 were used for formulation of model, and its validation has been checked by 8th specimen. The experimental test data was compared with predicted a-N curve obtained from proposed gamma model (Fig. 6). The average values of curve fitting constants (for seven different specimens) for gamma model have been given in Table 3. The predicted and experimental da/dN- K curves for the 8th specimen are presented (Fig. 7) for comparison. It can be seen that results from predicted gamma model are in good agreement with experimental data.

Table 3 Values of coefficients for gamma model.

A 23.33 E+07

B

C

35.12E+06

-17.65E+05

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D 29.63E+03

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crack length, a (mm)

5.5 5

a ( experimental) a ( predicted)

4.5 4 3.5 3 5.00E+04

7.00E+04

9.00E+04

1.10E+05

No. of cycle, N

Fig. 6. a-N curve (gamma model).

5.5E-05

da/dN (mm/cycle)

5E-05

da/dN experimental da/dN predicted

4.5E-05 4E-05 3.5E-05 3E-05 17

18

19

20

21

K (Mpa*m^1/2)

Fig. 7. da/dN- K curve (gamma model).

The performance of gamma model is evaluated by comparing the predicted results with experimental data for part-through cracked pipes under constant amplitude loading condition. There are three criteria that have been used for comparison of predicted results and experimental data, which are: 1. Percent deviation of predicted result from the experimental data as:

2. Prediction ratio which is defined as ratio between experimental data to predicted result as:

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3. Error bands i.e. the scatter of the predicted life on either side of experimental line within certain error limits.

The percentage deviations and the prediction ratio of exponential model are presented in Table 4 and Table 5.

Table 4. Model Performances (for crack length).

Test specimen

% Dev

Prediction ratio

TP316L stainless steel

1.74

0.99

Table 5. Model performances (for number of cycle).

Test specimen TP316L stainless steel

% Dev model) 1.02

Prediction ratio 1.01

The error band scattered is plotted in order to evaluate performance of gamma model as shown in Fig. (8-9). It has been observed that the error band scatter of gamma model lies in the range of +0.03% to -0.03% for experimental number of cycles and +0.03 to -0.03% of experimental crack length.

1.20E+05

Prpredicted N to a given crack length

1.10E+05 1.00E+05

N (Predicted)

9.00E+04 8.00E+04 7.00E+04 6.00E+04 60000

80000

100000

Experimental N to a given crack length

Fig. 8. Error band scatter for number of cycle (gamma model).

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120000

Mechanics, Materials Science & Engineering, September 2016

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Crack length, a (mm) Predicted

5.5 Predicted 5

4.5

4

3.5

3 3

3.5

4

4.5

5

5.5

6

crack length, a (mm) Experimental

Fig. 9. Error band scatter for crack length (gamma model).

Summary 1. The fatigue crack propagation in part-through cracked pipes can be determined effectively using gamma model of the form 2. Specific growth rate ((m) is expressed as where l is correlated with various crack driving forces and material properties is an important parameter for gamma model.

3. Using Gamma model it is possible to predict the crack extension corresponding to a given number of cycles or to predict the number of cycles required for a given crack extension.

References [1] Johnson, RN. "Fracture of a cracked solid circular cylinder (Investigation of cracks in radial plane of solid right circular cylinder)[Ph. D. Thesis]." (1972).

[2] Daoud, O. E. K., and D. J. Cartwright. "Strain energy release rates for a straight-fronted edge crack in a circular bar subject to bending." Engineering Fracture Mechanics,19.4 (1984): 701-707.

[3] Delale, F., and F. Erdogan. "Application of the line-spring model to a cylindrical shell containing a circumferential or axial part-through crack." Journal of Applied Mechanics 49.1 (1982): 97-102.

[4] Andrews, George E., Richard Askey, and Ranjan Roy. "Special Functions, volume 71 of Encyclopedia of Mathematics and its Applications." (1999).

[5] Rice, J. A. (1995). Mathematical Statistics and Data Analysis (Second Edition). p. 52 53 [6] Abramowitz, Milton, and Irene A. Stegun, eds. Handbook of mathematical functions: with formulas, graphs, and mathematical tables. No. 55. Courier Corporation, 1964.

[7] Journal of Fatigue., Elsevier, vol. 31, pp. 418-424, 2009, doi: 10.1016/j.ijfatigue.2008.07.015

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[8] Al Laham, S., and Structural Integrity Branch. Stress intensity factor and limit load handbook.

British Energy Generation Limited, 1998.

Cite the paper Pawan Kumar, Vaneshwar Kumar Sahu, P.K.Ray & B.B.Verma (2016). Modelling of Fatigue Crack Propagation in Part-Through Cracked Pipes Using Gamma Function. Mechanics, Materials Science & Engineering Vol.6, doi: 10.13140/RG.2.2.16973.03043

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