Proceedings of the 5th International Conference on Integrity-Reliability-Failure, Porto/Portugal 24-28 July 2016 Editors J.F. Silva Gomes and S.A. Meguid Publ. INEGI/FEUP (2016)

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PAPER REF: 6233

MODELLING FRAMEWORK FOR 3D FATIGUE CRACK

PROPAGATION IN WELDS OF OFFSHORE STEEL STRUCTURES

Jie Zhang(*), Stijn Hertelé, Nahuel Micone, Wim De Waele

Soete Laboratory, Dept. Electrical Energy, Systems and Automation, Ghent University, Belgium (*)Email: jie.zhang@ugent.be

ABSTRACT

Fatigue is a common failure mechanism of welds in offshore structures. Their lifetime prediction is hampered by a large number of influence factors related to weld (e.g., residual stresses) and environment (e.g., variable amplitude loading, hydrogen embrittlement). These challenges (among others) are tackled in the Flemish research programme MaDurOS. This paper describes an extended finite element method (XFEM) based framework, developed within the scope of this programme. It allows to include weld and environment related aspects into the numerical lifetime prediction of welded structures. This paper focuses on the three-dimensional nature of crack propagation predictions. A numerical optimization and benchmark validation are discussed to illustrate the possibilities of the framework. In future work, the developed software will be coupled with residual stress in the welds and hydrogen embrittlement and be validated with experimental input/output, allowing to evaluate the ability of numerical tools for weld lifetime predictions in offshore fatigue conditions.

Keywords: 3D fatigue crack propagation, extended finite element method, weld, offshore steel

INTRODUCTION

Fatigue is one of the most common failure modes for structures subjected to complex service loading. For offshore structures, such loading is caused by welding residual stresses, variable wind and sea wave loads. Meanwhile, offshore structures suffer from the corrosive reaction of salty sea water which deteriorates the material strength and interacts with the fatigue mechanism. The so-called MaDurOS programme, executed by a consortium of Flemish academic and industrial partners, focuses on the material durability of offshore structures. One aspect is to identify the combined damage mechanisms of both fatigue and corrosion, and to generate improved numerical and experimental predictive models, therefore, to analyze corrosion fatigue crack initiation and propagation.

Existing flaws, which are inevitably introduced by manufacturing processes and in-service damage, can be conservatively considered as an initial crack. It is of significance to predict their propagation and, thus, judge on their criticality and impact on residual life of the cracked structure. The limitation of conventional finite element methods to do so is the challenge of singularity topology of crack domain. To accurately model and predict the crack growth phenomenon, finite element analysis algorithms such as remeshing methods (Maligno, Rajaratnam et al. 2010), element deletion (Lee, Choi et al. 2009) and boundary element method (Mi and Aliabadi 1994) have been developed. These, however, are computationally challenging and time-consuming in complex 3D geometries (Shi, Chopp et al. 2010). Compared herewith, the extended finite element method (XFEM) (Belytschko T 1999) shows

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a significant advantage by removing the strong mesh dependency of conventional FEM approaches. To this end, discontinuous enrichment functions are added for cut-through elements and asymptotic functions are defined for crack tip elements (Moës, Sukumar et al. 2000). In respect of the development of the theoretical background of XFEM, researchers intensively investigated enrichment functions, mixing elements, integration schemes, conditions of contact surface of a crack, enrichment domains, error estimations and rates of convergence. On the other hand, in the respect of XFEM application and its prevalence in the research of cracks, several commercial softwares packages have developed or added XFEM calculation, such as ABAQUS, ALTAIR RADIOSS, ASTER, ANSYS, CAST3M, VAST, MORFEO, DYNAFLOW, GETFEM++ and others.

In this paper, a universal 3D fatigue crack propagation tool is proposed to model fatigue crack growth using the XFEM solver of the commercial software ABAQUS® (version 6.14). A cubic spline description for crack geometry is used to characterize the evolving crack front, whose rate and three-dimensional shape of growth is controlled by a crack propagation law. The crack front mapping approach has been implemented in an iterative framework that allows to include (and update as the crack grows) distributions of residual stress and hydrogen embrittlement created by the corrosive nature of the offshore environment (part of ongoing research). It is a promising tool to carry out fatigue crack propagation analysis and life time prediction of offshore welded structures.

MODELLING FRAMEWORK

The iterative framework for fatigue crack propagation modelling is programmed using Python including pre-processing and post-processing modules. Python is an object-oriented and open source programming language which allows for communication with the API of ABAQUS®, including definition of simulation input and extraction of output. The framework consists of three modules (pre-processing module, post-processing module and analysis module) which are developed to control all the commands during the simulation process in respect of different functionalities. The modules are introduced briefly in the following, and Fig demonstrates the flow chart for crack propagation simulation framework.

In the pre-processing module, when initiating the iterative framework, a geometric model of the structure with material properties, boundary conditions and external loads are imported. At the same time, the residual stress field of welds or hydrogen content field can be imported as initial service condition. Then, a crack will automatically be inserted according to the initially proposed or iteratively updated crack profile. Next, the calculation job based on XFEM is solved. As the job has completed, the post-processing module reads the calculation output files and preliminarily decides whether the value of effective stress intensity factor is between the threshold value for crack growth and the fracture toughness of the material. If it is less than the threshold, the crack will not grow anymore and the iteration stops, returning an infinite fatigue life. Else if it exceeds the fracture toughness, the crack will grow tremendously fast, corresponding with failure. In such case, the iterative framework is also stopped, returning the fatigue life. In between these two extreme scenarios, the post-processing module performs further calculations based on the crack growth law to update the three-dimensional crack profile, and then puts an updated crack front into a new simulation job. The tool is iteratively executed by the analysis module which coordinates the first two modules, until it reaches either of the extreme scenarios.

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Fig. 1 - Iterative framework for crack propagation simulation with influence of residual stress and hydrogen embrittlement

METHODOLOGY

Crack Insertion

A crack profile is built in its local coordinate system and it is necessary to transform its orientation to the global one before being inserted into the main structure body. Euler angles of crack orientation are defined as show in Figure 2, and the transformation matrix ���, �, �� is given in equation (1):

cos cos sin sin cos sin cos cos sin cos sin sin

( , , ) cos sin sin cos cos sin sin cos cos cos sin

sin sin cos sin cos

R

ψ ϕ ψ ϕ θ ψ θ ψ ϕ θ ϕ θψ θ ϕ ψ ϕ ψ ϕ θ ψ ϕ ψ θ ϕ θ

ψ θ ψ θ θ

− + = − − − + −

(1)

These Euler angles have pre-defined values, defining the crack orientation.

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Fig. 2 - Euler angles definition representing rotations between original(xyz) system and

rotated(x`y`z`) system.

Crack Propagation

As the calculation job is completed, parameters of fracture mechanics, such as stress intensity factors in different modes (KI, KII, KIII), J-integral and T-stress, can be requested as output. Here, only stress intensity factors are used for multiaxial stress state analysis. The following algorithm for crack front profile update procedure is given below.

Step 1:

Coordinates of integration points of XFEM along the crack front and K values (KI, KII, KIII) in these points are read from the job output file. However, the distribution of these K values potentially exhibits an unwantedly irregular shape such as the zig-zag pattern, or exhibits erroneous values at the end points of an open crack. These anomalies are caused by the approximate nature of the numerical computing method. In this paper, to filter the zig-zag, each K value on the point along the crack front is first replaced by the average value of itself and the average value of two neighbors. Values on two ends points are replaced by linear extrapolation. This methodology is adopted from (Dhondt 1998).

For multiaxial loading, the equivalent stress intensity factor� is computed as:

2 2 2(1 )eq I II IIIK K K Kν= + + − (2)

All K_eq data are used for a least squares fit to a 2nd degree, which is a user-defined parameter, polynomial curve. K_eq values are replaced by interpolation on the fitting curve shown in Figure 3, the horizontal axis of which is the through-thickness coordinate of the CT specimen used in validation section. Following this procedure, the K distribution along an arbitrary crack front is determined. Upon application of a crack growth law (e.g Paris law(Paris and Erdogan 1963), Walker(Walker 1970) law or Elber law(Wolf 1970)), crack growth increments at each point along the crack front can be determined as explained below.

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Fig. 3 - Stress intensity factor distribution along a through-thickness crack in a CT

specimen Step 2:

There are two strategies for crack growth increment calculation. One consists of fixing a number of cycles ∆N and then calculating the crack growth ∆a along the crack front corresponding to these cycles. The drawback of this strategy is that at the beginning of the simulation, the crack propagation increment is probably extremely small and potentially even smaller than the numerical inaccuracies involved (Dhondt 1998). Conversely, towards the end of simulation, crack growth speed exponentially increases, possibly leading to instability of the simulation. The second strategy, referred to as “crack front mapping approach” (Shi, Chopp et al. 2010), consists of fixing the maximum crack growth increment for each step and, based on the crack growth law, calculating the corresponding crack growth at all points along the crack front. This iteration is likely to correspond with a variable number of multiple cycles as the crack grows throughout he iterative process.

In this paper, the second strategy is preferred. Hereby, equations (3-5) (Donahue, Clark et al. 1972, Kurihara, Katoh et al. 1986) are used, thus taking into account effects of crack closure and loading ratio (R-ratio):

( )meff

daC K

dN= ∆ (3)

where da/dN is crack growth rate, C and m are material parameters according to crack growth law, and Δ is the range of effective stress intensity factor within a load cycle:

1(1 )eq

eff op

KK K

R γ−

∆∆ = − ∆

− (4)

where �� is the range of equivalent stress intensity factor determined by equation (2), R is loading ratio, � relates to material properties, �� takes crack closure effect into account. Concretely, �� is taken equal to crack growth threshold ��.

maxmax

mi

effi

eff

Ka a

K

∆∆ = ∆ ∆

(5)

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where Δ�� is ith crack growth increment, Δ ��� is the maximum value along the crack front,

Δ���� is the maximum crack increment along the crack front, occurring at the point corresponding to Δ

���. ���� is a user-defined algorithm parameter which allows for a desired trade-off between accuracy and calculation time consumption.

The advantage of the crack front mapping approach is that the simulation converges more straightforwardly and overall computation time can be easier estimated by comparing the predefined maximum crack growth increment with the size of the crack ligament.

Step 3:

Figure 4 illustrates the crack propagation direction. In this study, out-plane growth direction for each point is calculated based on the maximum tangential stress criterion, using either the

condition ����

��� 0 or � � � 0, resulting in equation (5):

2 4 2 21

2 2

3 8sgn( )cos

9II I I II

II

I II

K K K KK

K Kθ −

+ + = − +

(5)

Other direction criteria will be implemented into the framework in the future to compare them with each other.

The following procedure is used to obtained the in-plane growth direction for each point !�" .

Firstly, the two closest crack tip points from the last step ( !#"$% and !&

"$%) are identified. The normal direction of the plan formed by these three points is the normal direction '() of in-plane crack growth. The direction corresponding to the line that connects its two neighbor points is the tangential direction *) of the crack front. Then, the in-plane growth direction +) is determined by the cross-multiplication of the normal direction and the tangential direction. Equations (6-8) show the algorithm to obtain these directions. Note that these vectors are not unit direction vectors, but they are sufficient to obtain the angles between global and local coordinate systems.

1 1 1j j j j

A i A Bn P P P P− − −= ×

uuuuuur uuuuuuuurr (6)

1 1j j

i it P P− +=uuuuuurr

(7)

q n t= ×r r r

(8)

Fig. 4 - Overview of 3D arbitrary crack growth

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Coordinate transformations between the global cordinate system (x,y,z) and the new local coordinate system (q,n,t) can be calculated by the following equation (9):

cos( , ) cos( , ) cos( , )

cos( , ) cos( , ) cos( , )

cos( , ) cos( , ) cos( , )

q x

n y

t z

x q y q z qu u

u x n y n z n u

u ux t y t z t

=

r r ur r r r

r r ur r r r

r r ur r r r (9)

So far new crack front control points U have been obtained, and then the shape of the crack front ,�-� is a cubic spline fitting curve to these discrete points. In addition, the first and second derivatives of the spline are continuous.

Step 4:

Assuming constant amplitude loading for this study, there are no retardation and acceleration effects of crack growth. It is numerically reasonable to presume that in each crack length increment there is a linear relation between equivalent stress intensity factor and crack length, as shown in Figure 5 and in equation (10-11). Number of loading cycles within each crack increment can be calculated by integrating the crack growth law, as shown in equation (12).

effK aα β∆ = + (10)

1( ) j j

eff eff effda d K K Kβ += ∆ = ∆ − ∆ (11)

( )

11 1( )

(1 )

mj

eff j

jmj

eff

KdaN

C mK β

−+ +∆

= =−∆

∫ (12)

Fig. 5 - Schematic of stress intensity factor and crack length growth

The total crack propagation life can finally be calculated by summing up all amounts of cycles corresponding to the crack increments.

VALIDATION EXAMPLE

The validation example is based on a compact tension specimen with an additional hole and uses the same loading and material properties and compares the numerical and experimental results in Miranda`s work (Miranda, Meggiolaro et al. 2003). The modified CT specimen

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geometry and mesh grids are depicted in Figure 6. The additional hole asymmetrically redistributes the stress field so that the crack trajectory is deviated towards the hole. The position of the hole is determined by A=8.3mm B=8.1mm. Young`s modulus of the material E=205GPa and Poisson ratio ν=0.3. The crack propagation law is characterized by m=2.1 and C=4.5e-10, and the fatigue crack growth threshold 〖∆K〗_th=11.6MPa√m. Meshing design is demonstrated in Figure 6, finer mesh grid with 0.4mm length in the vicinity of missing hole and center of the specimen is built. Although XFEM is meshing insensitive, fine and regular meshing strategy can produce more integration points along the crack front, which improves the accuracy of description of crack front.

Fig. 6 - Geometry of modified CT specimen and meshing strategy

In Figure 7 it is obviously shown that the crack growth trajectory simulated with the tool developed in this paper is the same as the experimental data reported. Figure 8 shows 3D view. The 3D crack profile illustrates that the surface of crack consists of several parts, and within each part the surface is slanted in stead of th flat. At the different points of these parts, the crack surface demonstrates various crack orientations.

Fig. 7 - Left: experimentally observed fatigue crack path from Miranda (Miranda, Meggiolaro et al. 2003),

Right: crack path predicted using the iterative tool developed in this study.

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Fig. 8 - Perspective view of an arbitrary three-dimensional crack profile during propagation

In this paper, the sensitivity of the pre-defined maximum crack increment is investigated. The final crack profiles corresponding to different maximum increments (0.9mm, 1.3mm and 1.8mm respectively are plotted relatively to compare with data reported in Figure 9. Figure 10 illustrates an example of crack propagation life prediction vs crack length in constant loading condition. Furthermore, three different mesh sizes (0.4mm,0.57mm,0.8mm) in the vicinity of the missed hole are implemented to investigate the sensitivity of the framework mesh sizes as shown in Figure 11. The results confirm the meshing independence of XFEM used in the framework.

Fig. 9 - Crack profiles for different maximum increment values

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Fig. 10 - Predicted fatigue crack propagation life vs crack length

Fig. 11 - Crack profiles for different mesh sizes at the center of the specimen

CONCLUSION

An iterative tool for arbitrary 3D weld crack propagation due to fatigue loading has been developed. It utilizes a crack growth mapping concept based on static XFEM modelling incorporated in ABAQUS®. This paper explains the algorithm used and shows a successful comparison with reported test data. The sensitivity study of crack increment indicates that smaller increments result in the same crack trajectory but consume more calculation time. Meanwhile, the mesh density sensitivity study shows that mesh size does not influence the crack path during the growth.

0

5000

10000

15000

20000

0 5 10 15 20

n/cycle

a/mm

crack length vs number of cycles

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In future work, residual stress distributions and hydrogen induced material degradation will be implemented in this framework and other advanced features such as variable amplitude loading conditions, additional crack propagation direction criteria and crack growth laws will be explored. Robustness and universality of the program will be further improved, and numerical and experimental validations will be performed.

ACKNOWLEDGMENTS

The authors gratefully acknowledge the financial support of DeMoPreCI-MDT project from SIM Flanders (Strategic Initiative Materials) and Vlaio (Flemish Agency for innovation and Entrepreneurship).

REFERENCES

[1]-Belytschko T, B. T. (1999). "Elastic crack growth in finite elements with minimal remeshing." International Journal for Numerical Methods in Engineering 45(5): 601-620.

[2]-Dhondt, G. (1998). "Automatic 3-D mode I crack propagation calculations with finite elements." International Journal for Numerical Methods in Engineering 41(4): 739-757.

[3]-Donahue, R. J., H. M. Clark, P. Atanmo, R. Kumble and A. J. McEvily (1972). "Crack opening displacement and the rate of fatigue crack growth." International Journal of Fracture Mechanics 8(2): 209-219.

[4]-Kurihara, M., A. Katoh and M. Kawahara (1986). "Analysis on fatigue crack growth rates under a wide range of stress ratios." Journal of pressure vessel technology 108(2): 209-213.

[5]-Lee, H., J. Choi, K. Jung and Y. Im (2009). "Application of element deletion method for numerical analyses of cracking." Journal of Achievements in Materials and Manufacturing Engineering 35(2): 154-161.

[6]-Maligno, A. R., S. Rajaratnam, S. B. Leen and E. J. Williams (2010). "A three-dimensional (3D) numerical study of fatigue crack growth using remeshing techniques." Engineering Fracture Mechanics 77(1): 94-111.

[7]-Mi, Y. and M. H. Aliabadi (1994). "Three-dimensional crack growth simulation using BEM." Computers & Structures 52(5): 871-878.

[8]-Miranda, A. C. O., M. A. Meggiolaro, J. T. P. Castro, L. F. Martha and T. N. Bittencourt (2003). "Fatigue life and crack path predictions in generic 2D structural components." Engineering Fracture Mechanics 70(10): 1259-1279.

[9]-Moës, N., N. Sukumar, B. Moran and T. Belytschko (2000). "An extended finite element method (X-FEM) for two-and three-dimensional crack modeling." ECCOMAS, Barcelona.

[10]-Paris, P. and F. Erdogan (1963). "A critical analysis of crack propagation laws." Journal of Fluids Engineering 85(4): 528-533.

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[11]-Shi, J., D. Chopp, J. Lua, N. Sukumar and T. Belytschko (2010). "Abaqus implementation of extended finite element method using a level set representation for three-dimensional fatigue crack growth and life predictions." Engineering Fracture Mechanics 77(14): 2840-2863.

[12]-Walker, K. (1970). "The effect of stress ratio during crack propagation and fatigue for 2024-T3 and 7075-T6 aluminum." Effects of environment and complex load history on fatigue life 462: 1-14.

[13]-Wolf, E. (1970). "Fatigue crack closure under cyclic tension." Engineering Fracture Mechanics 2(1): 37-45.