Evaluation of stress intensity factors for multiple surface cracks in bi-material tubes

Evaluation of stress intensity factors for multiple surface cracks in bi-material tubes

ARTICLE IN PRESS Engineering Analysis with Boundary Elements 33 (2009) 1339–1343 Contents lists available at ScienceDirect Engineering Analysis with...

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ARTICLE IN PRESS Engineering Analysis with Boundary Elements 33 (2009) 1339–1343

Contents lists available at ScienceDirect

Engineering Analysis with Boundary Elements journal homepage: www.elsevier.com/locate/enganabound

Evaluation of stress intensity factors for multiple surface cracks in bi-material tubes J. Purbolaksono , A.A. Ali, A. Khinani, A.Z. Rashid Department of Mechanical Engineering, Universiti Tenaga Nasional, Km 7, Jalan Kajang-Puchong, Kajang 43009, Selangor, Malaysia

a r t i c l e in fo

abstract

Article history: Received 2 February 2009 Accepted 19 May 2009 Available online 23 June 2009

This paper presents stress intensity factors (SIFs) of multiple semi-elliptical surface cracks in bi-material tubes subjected to internal pressure by boundary element method. In this case the water-tube boiler with oxide scale formed on the inner surface due to prolonged exposure at elevated temperature is considered as the bi-material tubes. Variations of modulus of elasticity and thickness for the oxide scale are used to evaluate their effects on the stress intensity factors. The increasing of thickness of the oxide scale causes decreasing values of the normalized stress intensity factor as the modulus of elasticity for the oxide scale is greater than that of the tube metal. Conversely, if the modulus of elasticity for the oxide scale is smaller, the increasing of thickness of the scale would also give increasing values of the normalized stress intensity factor. & 2009 Elsevier Ltd. All rights reserved.

Keywords: Boundary element modeling Multiple surface crack Tubes Bi-material Stress intensity factors

1. Introduction Boundary element method (BEM) has become a robust tool for various engineering analyses. Brebbia [1] described applications of the boundary element method in providing an excellent basis for the development of integrated design and analysis systems. The technique possesses the advantages of simple model generation, easy-to-understand requirements, high accuracy, relative insensitivity to mesh refinements and the ability to accurately model the most difficult stress concentration problems. A BEM system that combines these features with close integration with CAD systems, highly automated mesh generation, in-built error estimation and design optimization technology provides a powerful tool for use in engineering analysis and design. Brebbia [2] also narrated a chronicle of recent innovations in BEM. Nowadays, the BEM technique has been proven to successfully analyse various engineering problems. This work is concerned with the problems of tubing in power plants. For prolonged exposure of service under elevated temperatures oxide scales may be formed on the inner surface of boiler tube as reported by Port and Herro [3] and French [4]. Surface cracks may be developed either on the internal or external surface of tube. In order to evaluate creep fatigue in tube as described by Viswanathan [5] or fatigue crack growth in tube as stated in API 579 [6], stress intensity factor (SIF) solutions are required. No available analytical expression for evaluating stress intensity factors of surface crack in a tube made of bi-material has  Corresponding author. Tel.: +60 3 89212213; fax: +60 3 89212116.

E-mail address: [email protected] (J. Purbolaksono). 0955-7997/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.enganabound.2009.05.004

been reported. This work presents the ability of the BEM in evaluating complex problems of tubes consisting of multiple semi-elliptical surface cracks in the presence of the oxide scale on the inner surface subjected to internal pressure. Analyses conducted are assumed and preserved to be using the concept of linear elastics fracture mechanics. Three different values of modulus of elasticity and three different values of thickness for the oxide scale are used in order to study their effects on the stress intensity factors. All the modeling and analysis are carried out by using the boundary element software package of BEASY [7].

2. Crack modeling in BEASY BEASY uses dual boundary element method (DBEM) for threedimensional crack analysis. The DBEM incorporates two independent boundary integral: the displacement equation applied at the collocation point on one of the crack surface, and the traction equation on the opposite surface. Analysis of three-dimensional crack problems by using the DBEM has been successfully carried out by Mi and Aliabadi [8]. The modeling strategy of threedimensional crack problems may be summarized as follows [8]:

 Crack surfaces are modeled with discontinuous elements.  Surfaces intersecting a crack surface are modeled with edgediscontinuous quadrilateral or triangular elements.

 The displacement integral equation is applied for collocation 

on one of the crack surfaces, i.e. the upper surface G+. The traction integral equation is applied for collocation on the opposite crack surface, i.e. the lower surface G.

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 The displacement integral equation is applied for collocation on all other surfaces and the continuous quadrilateral elements are used. Treatments of singular and hyper-singular integrals which appear in the dual boundary element formulations have also been established, and the detailed expressions were presented in [8]. The stress intensity factors along the crack front are successfully evaluated using the described technique of the DBEM.

3. Simulation models Model for multiple semi-elliptical surface cracks on the outer surface of the tube containing oxide scale is shown in Fig. 1. The model needs to be divided into two zones, i.e. inner region for the first zone (oxide scale) and the outer region for the second zone (tube metal). Modeling procedures for a compound structure have to be made in order to evaluate the problem accordingly. All the surfaces of the first zone are defined to be in outward normal direction. Next, for the second zone, the surface for the interface between the inner and outer regions has to be defined to be in inward normal direction, whereas the rest of the surfaces of the second zone are defined in outward normal direction. Two-dimensional quadratic elements are used to discretize all the surfaces of the model. Internal pressure of 4 MPa used for all the models is applied on inner surface of the tube. A spring boundary condition for displacement constraints in x, y and z directions is applied on either top or bottom of the cross sectional surface. The stiffness value of the spring is taken as 2% of Young’s modulus for the tube metal as recommended by BEASY [7].

tube metal

R25.4

oxide scale

R21.9

surface crack

Displacement constraints are also applied in normal directions on both top and bottom surfaces. Surface crack including its meshing is generated by using the Crack Wizard of BEASY [7]. The J-integral method is chosen to evaluate the stress intensity factors. The modulus of elasticity for the outer region may be taken to be constant with Eouter ¼ 163,410 MPa. Three different values for modulus of elasticity for the inner region are used, i.e. 150,000, 163,410 and 175,000 MPa [9]. Variations of the model for different values of modulus of elasticity for the inner and outer regions are evaluated to show their effects on the stress intensity factor values. As the modulus of elasticity for the inner region is 163,410 MPa, the model is considered as a free scale tube. The tube used in this study has outer radius and inner radius of 25.4 and 21.9 mm, respectively.

4. Numerical results and discussion In order to show the accuracy of the results, comparison of the SIFs results obtained by using BEASY [7] and from Mettu et al. [10] for a longitudinal semi-elliptical crack on the inner surface of a cylinder is made. The results are plotted in Fig. 2. It can be seen that the results are shown to be in good agreement. The largest difference is found at midpoint of the crack front (901) and is within 5%. Again, in order to demonstrate the accuracy of the results obtained by using BEASY [7] for the analyses of surface cracks in the tube made of bi-material, the model with three collinear external surface crack (a ¼ 0.5 mm each) in a tube composed of two zones made of the same material is evaluated. The distance of mid-crack to another adjacent mid-crack is 10.714 mm. Both zones are defined to have Young’s modulus of 163,410 MPa. The thickness of the oxide scale is 1.8 mm. The crack front is relatively close to the zones, interface which might affect the interaction of the geometries. It can however be seen from Fig. 3, the normalized stress intensity factors of both models are shown to be in very good agreement. The biggest difference occurred at outer points (angle of 01 or 1801 of position along the crack front) of the outer cracks, but it is within 1%. Fig. 4 shows the mesh refinements generated by Crack Wizard of BEASY [7] and Von Mises stress contour around crack tip. It shows that the tool possesses the advantages in generating mesh refinements with high accuracy and the ability to accurately model the complexity of the problem with stress intensities.

150 2c a

Dimension in mm Fig. 1. Studied model of the multiple surface cracks on the outer surface of a water-tube boiler in the presence of the oxide scale.

Fig. 2. Comparison of the SIF values for longitudinal semi-elliptical crack on inner surface of tube obtained by using BEASY [7] and by Mettu et al. [9] (a/c ¼ 1; a ¼ 0.5 mm).

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The normalized stress intensity factors KI/K0 for crack depth of 0.5 mm for different values of modulus of elasticity and different thickness of the scale are plotted in Figs. 5–8. Mid-crack is defined as the position at the crack front with an angle of 901, while outer points of a crack will be at angles 01 and 1801. The value of K0 is defined as pffiffiffiffiffiffi K 0 ¼ p pa

(1)

where p is the applied internal pressure and a is the crack depth (see Fig. 1). It can be seen from Fig. 5, for internal surface cracks the increasing of thickness of the oxide scale will give decreasing values of the normalized stress intensity factor as the modulus of elasticity for the scale region is greater than that of the tube metal

Fig. 3. Comparison of the SIF values for three collinear external surface cracks of the free scale tube and the tube composed of two zones with the same material.

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region. However, all the normalized stress intensity factors are also shown to be larger than those of the free scale tube. On the other hand, Fig. 6 shows that, if the modulus of elasticity for the scale region is smaller than that of the tube metal region, the increasing of thickness for the scale will also give increasing values of the normalized SIF. It can also be seen that all the normalized stress intensity factors are smaller than those of the free scale tube. It can be deduced from Fig. 5 that the stress intensity factors for the inner surface cracks, will have higher values than those of the free scale tube if the modulus of elasticity for the scale (inner region) is larger than that of the tube metal (outer region). However, otherwise feature is shown in Fig. 6. For external surface cracks, Fig. 7 shows that the increasing of thickness of the oxide scale will also give decreasing values of the

Fig. 5. Comparison of the normalized SIF values obtained from BEASY for multiple surface cracks on the inner surface with different scale thickness on the inner part of the tube (Escale ¼ 175,000 MPa; Etube ¼ 163,410 MPa).

55.4293 50.4607 45.4920 40.5234 35.5547 30.5861 25.6174 20.6488 15.6801 10.7115 5.7429 0.7742

Fig. 4. Mesh refinements generated by Crack Wizard of BEASY [7] and Von Mises stress contour around crack tip.

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Fig. 6. Comparison of the normalized SIF values obtained from BEASY for multiple surface cracks on the inner surface with different scale thickness on the inner part of the tube (Escale ¼ 150,000 MPa; Etube ¼ 163,410 MPa).

Fig. 8. Comparison of the normalized SIF values obtained from BEASY for multiple surface cracks on the outer surface with different scale thickness on the inner part of the tube (Escale ¼ 150,000 MPa; Etube ¼ 163,410 MPa).

Table 1 The normalized stress intensity factors KI/K0 ( ¼ b) for single surface crack (crack depth of a ¼ 0.5 mm). Young’s modulus (MPa)

Fig. 7. Comparison of the normalized SIF values obtained from BEASY for multiple surface cracks on the outer surface with different scale thickness on the inner part of tube (Escale ¼ 175,000 MPa; Etube ¼ 163,410 MPa).

normalized stress intensity factor as the modulus of elasticity for the scale region is greater than that of the tube metal region. In this case all the normalized stress intensity factors are shown to be smaller than those of the free scale tube. Similar to Fig. 6, Fig. 8 shows that, if the modulus of elasticity for the scale region is smaller than that of the tube metal region, the increasing of thickness for the scale will also give increasing values of the normalized stress intensity factors. It can however be seen that all the normalized stress intensity factors are larger than those of the free scale tube. It can be deduced from Fig. 7 that the stress intensity factors for the outer surface cracks will have smaller values than those of the free scale tube if the modulus of elasticity for the scale (inner region) is larger than that of the tube metal (outer region). Fig. 8 indicates otherwise deduction. The normalized stress intensity factors for the corresponding single surface cracks on inner and outer surfaces are also presented in Table 1. The results are used to study the effects due to the interactions of multiple surface cracks. The largest normalized stress intensity factors of multiple surface cracks are found at outer points (angle 01 or 1801 of position along the crack front) of outer cracks. It can be compared that the normalized stress intensity factors of multiple cracks for both internal and

Scale

Tube metal

175,000

163,410

163,410 150,000

163,410 163,410

Scale thickness (mm)

0.6 1.2 1.8 0 0.6 1.2 1.8

Internal cracks

External cracks

b at 01

b at 901

b at 01

b at 901

5.5325 5.4820 5.2021 5.0095 4.6565 4.6499 4.8151

4.7630 4.6740 4.5298 4.3894 4.0540 4.1450 4.2020

4.3664 4.2688 4.1919 4.3719 4.5122 4.5512 4.5650

3.8580 3.8010 3.7810 3.9449 3.9870 4.0524 4.1157

external of the middle surface cracks tend to be slightly lower than those of the corresponding single cracks. This feature might be due to the values of the principal stresses around the middle surface crack becoming slightly less. The normalized stress intensity factors presented in Table 1 show similar behaviors corresponding to the ratio of Young’s modulus of the inner and outer regions of the tubes.

5. Conclusions Normalized stress intensity factors for multiple semi-elliptical surface cracks on the inner and outer surfaces of tubes containing oxide scale on inner part subjected to internal pressure were presented. The increasing of thickness of the scale region (inner part) caused decreasing values of the normalized stress intensity factor as the modulus of elasticity for the scale is greater than that of the tube metal as the outer part. Conversely, if the modulus of elasticity for the inner region is smaller, the increasing thickness of the inner part would also give increasing values of the normalized stress intensity factor. However, comparisons between the normalized stress intensity factors of internal and external surface cracks and those of the free scale tubes for the same crack depth may be summarized as follows:

 If Young’s modulus of the inner part is greater than that of the outer part, the normalized SIFs for the internal surface cracks

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were higher than those of free scale tubes, whereas the normalized SIFs for the external surface cracks were smaller than those of free scale tube. If Young’s modulus of the inner part is smaller than that of the outer part, the normalized SIFs for the internal surface cracks were smaller than those of free scale tubes, whereas the normalized SIFs for the external surface cracks were higher than those of free scale tube.

Acknowledgements The authors wish to thank the Ministry of Science Technology and Innovation, Malaysia, for financial supports through the research project of IRPA 09-99-03-0033 EA001 and Science fund 04-02-03-SF0003. The authors would also like to thank Universiti Tenaga Nasional and TNB Research Sdn Bhd for permission of utilizing all the facilities and resources during completion of this study.

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