Numerical fatigue-analysis of a threaded connection model based on linear elastic fracture mechanics

Numerical fatigue-analysis of a threaded connection model based on linear elastic fracture mechanics Diploma Thesis Thomas Havelka | 1405002 Civil Engineering Institute: Stahlbau und Werkstoffmechanik Adviser: Prof. Dr.-Ing. M. Vormwald, M.Sc. D. Panic Submitted: 23.11.2016

Contents Contents ....................................................................................................................................... i List of Figures ............................................................................................................................ iii List of Tables ............................................................................................................................. iv List of Diagrams ......................................................................................................................... v List of Symbols and Abbreviations .......................................................................................... vii 1

2

3

4

Introduction ........................................................................................................................ 1 1.1

Motivation ................................................................................................................... 1

1.2

Problem Statement....................................................................................................... 2

Background ......................................................................................................................... 3 2.1

Fundamentals of Fatigue ............................................................................................. 3

2.2

Crack Development ..................................................................................................... 4

2.3

Stress Field around the Crack Tip ............................................................................... 5

2.4

Methods of the Identification of the SIFs (K-concept) ............................................... 6

2.5

Fatigue Crack Growth ................................................................................................. 8

2.6

Crack Closure Effect ................................................................................................. 10

2.7

Stress Concentration along a Screw Thread .............................................................. 12

Finite Element Modeling in Fracture Mechanics ............................................................. 14 3.1

Determination of the SIFs ......................................................................................... 14

3.2

Mixed-Mode Interaction and 2D Crack Growth Trajectory...................................... 17

3.3

Crack Propagation by quasi-static Methods .............................................................. 18

Abaqus Model of the Threaded Connection without Crack ............................................. 19 4.1

Geometry ................................................................................................................... 20

4.2

Boundaries ................................................................................................................. 24

4.3

Loads ......................................................................................................................... 26

4.4

Fracture Mechanics in Abaqus .................................................................................. 27

4.5

Crack Initiation .......................................................................................................... 29

List of Figures

i

5

6

7

8

9

FRANC2D Model of the Screw Thread ........................................................................... 33 5.1

Translation Script and Contact Problems .................................................................. 34

5.2

Geometry ................................................................................................................... 35

5.3

Boundaries ................................................................................................................. 36

5.4

Loads ......................................................................................................................... 37

5.5

Crack Initiation and Propagation ............................................................................... 41

5.6

Comparison of the three Methods of Crack Propagation .......................................... 43

Case Study 1: Screw Thread with 1 Crack ....................................................................... 44 6.1

Crack Growth ............................................................................................................ 45

6.2

Load Redistribution ................................................................................................... 46

6.3

Accuracy of the FE Result ......................................................................................... 48

6.4

Fracture Mechanical Results ..................................................................................... 50

Case Study 2: Screw Thread with 2 Cracks ..................................................................... 56 7.1

Fundamental Consideration ....................................................................................... 57

7.2

Crack Growth ............................................................................................................ 59

7.3

Load Redistribution ................................................................................................... 61

7.4

Accuracy of the FE Result ......................................................................................... 62

7.5

Fracture Mechanical Results ..................................................................................... 63

Case Study 3: Screw Thread with multiple Cracks .......................................................... 67 8.1

Crack Growth ............................................................................................................ 68

8.2

Load Redistribution ................................................................................................... 70

8.3

Accuracy of the FE Result ......................................................................................... 71

8.4

Fracture Mechanical Results ..................................................................................... 72

Conclusions and Perspective ............................................................................................ 76

Appendix ..................................................................................................................................... I Bibliography ............................................................................................................................. XI Declaration of Authorship ..................................................................................................... XIII

List of Figures

ii

List of Figures Figure 1: crack propagation (s. [2] p. 1) ..................................................................................... 4 Figure 2: K-dominant region (s. [3] Lecture 5 p. 101) ............................................................... 5 Figure 3: mode I-III (s. [4] p. 85) ............................................................................................... 6 Figure 4: stress field in vicinity of the crack tip (s. [1] p. 23) .................................................... 7 Figure 5: Paris' Law (s. [2] p. 3) ................................................................................................. 8 Figure 6: crack closure effect (s. [7] p. 123) ............................................................................ 10 Figure 7: crack closure effects .................................................................................................. 10 Figure 8: stress increase along a screwed joint (s. [9] p. 4)...................................................... 12 Figure 9: stress distribution within a screw thread (s. [9] p. 6) ................................................ 13 Figure 10: crack tip models with ¼-point-elements: quadr. and triang. (s. [7] p. 194) ............ 15 Figure 11: crack tip meshing (s. [3] Lecture 4 p. 74) ............................................................... 16 Figure 12: hoop stress around crack tip .................................................................................... 17 Figure 13: schematic flow diagram in FRANC2D code .......................................................... 18 Figure 14: geometry of threaded M16 connection (s. [12]) ..................................................... 20 Figure 15: symbols of the geometric parameters of the bolt and the nut (s. [9] p. 13) ............ 21 Figure 16: meshed assembly of threaded connection ............................................................... 22 Figure 17: boundaries (s. [9] p. 36) .......................................................................................... 24 Figure 18: boundaries of the bolt thread and the nut thread ..................................................... 25 Figure 19: normal cut surface and stress area (s. [9] p. 12)...................................................... 26 Figure 20: assign line................................................................................................................ 27 Figure 21: q-vector ................................................................................................................... 27 Figure 22: meshed crack tip region .......................................................................................... 28 Figure 23: deformed crack tip region ....................................................................................... 28 Figure 24: numbering of the pitches ......................................................................................... 29 Figure 25: maximal principal stress above the first loaded thread’s flank ............................... 30 Figure 26: crack initiation in respect of the inclination α......................................................... 32 Figure 27: model of threaded connection in FRANC2D.......................................................... 35

List of Figures

iii

Figure 28: boundary conditions ................................................................................................ 36 Figure 29: equivalent loads on thread's flanks ......................................................................... 37 Figure 30: elastic widening of the nut (s. [9] p. 40) ................................................................. 40 Figure 31: principal stress state in FRANC2D model .............................................................. 40 Figure 32: process of crack initiation and crack propagation ................................................... 41 Figure 33: initial crack .............................................................................................................. 42 Figure 34: error message .......................................................................................................... 43 Figure 35: crack path predicted by FRANC2D without abortion criterion Kc ......................... 45 Figure 36: critical SIF range (s. [6] p. 109) .............................................................................. 51 Figure 37: numerical integration (s. [6] p. 130) ....................................................................... 53 Figure 38: two cracks in screw (s. job definition p. 2) ............................................................. 56 Figure 39: crack paths predicted by FRANC2D without abortion criterion Kc....................... 59 Figure 40: multiple-cracked screw thread with m=2.0 (FRANC2D) ....................................... 68

List of Tables Table 1: tolerance zone for M16 for 6g and 6H [mm] out of DIN13 ....................................... 21 Table 2: averaged normal pressure magnitudes of all five thread flanks ................................. 39 Table 3: crack length of change between the FE programs ...................................................... 44 Table 4: crack length of change between the FE programs ...................................................... 67 Table 5: inclination of initial cracks ......................................................................................... 67 Table 6: comparison of the crack growths (s. chapter 7.5)....................................................... 74

List of Tables

iv

List of Diagrams Diagram 1: stress distribution along the screw axis ................................................................. 30 Diagram 2: principal stress along the thread root ..................................................................... 31 Diagram 3: J-Integral depending on the angle of inclination α ................................................ 32 Diagram 4: linearization of normal pressure for thread flank 1 ............................................... 38 Diagram 5: linearization of tangential pressure for thread flank 2 ........................................... 38 Diagram 6: normal contact pressure of all five loaded thread flanks ....................................... 39 Diagram 7: comparison between the interaction theories ........................................................ 43 Diagram 8: crack path predicted by Franc2D........................................................................... 45 Diagram 9: normed normal force on thread flanks (Case Study 1) .......................................... 46 Diagram 10: stress redistribution in thread roots (Case Study 1) ............................................. 47 Diagram 11: accuracy of the SIFs between ABAQUS and FRANC2D (Case Study 1) .......... 48 Diagram 12: deviation of the SIFs before and after software change ...................................... 49 Diagram 13: ΔK-a-curve (Case Study 1) .................................................................................. 52 Diagram 14: a-N-curve (Study Case 1) .................................................................................... 54 Diagram 15: Crack 2 path......................................................................................................... 60 Diagram 16: comparison of Crack 1 path between Case Study 1 and 2 .................................. 60 Diagram 17: normed normal force on thread flanks (Case Study 2) ........................................ 61 Diagram 18: stress redistribution in thread roots (Case Study 2) ............................................. 61 Diagram 19: accuracy of the SIFs Kv between ABAQUS and FRANC2D (Case Study 2)..... 62 Diagram 20: deviation of the SIF results before and after software change ............................ 62 Diagram 21:comparison of Kv between Case Study 1 and Case Study 2 ................................ 63 Diagram 22: ΔK-a-curve of Crack 1 (Case Study 2)................................................................ 64 Diagram 23: SIF range curve of Crack 2 (Case Study 2) ........................................................ 65 Diagram 24: a-N-curve (Case Study 2) .................................................................................... 66 Diagram 25: comparison of Crack 1 path of Case Study 1 and 3 ............................................ 68 Diagram 26: crack paths of Crack 2-5 ...................................................................................... 69 Diagram 27:normed normal force on thread flanks (Case Study 3) ......................................... 70

List of Diagrams

v

Diagram 28: deviation of the SIFs Kv between ABAQUS and FRANC2D (Case Study 3) .... 71 Diagram 29: jumps of the SIFs due to software change ........................................................... 71 Diagram 30: comparison of Kv between Case Study 1 and Case Study 3 ............................... 72 Diagram 31: ΔK-a-curve od Crack 1 (Case Study 3) ............................................................... 73 Diagram 32: SIF range curve of Crack 2-5 .............................................................................. 74 Diagram 33: a-N-curve (Case Study 3) .................................................................................... 75

List of Diagrams

vi

List of Symbols and Abbreviations a

crack depth/length [mm]

Δa

quasi-static crack propagation step [mm]

da/dN

crack growth rate [mm/cycle]

C

constant of Paris’ equation relating to MPa*m1/2

C*

constant of Paris’ equation relating to MPa*mm1/2 / N*mm-3/2

Di

diameters of the nut [mm] out of DIN 13

di

diameters of the screw [mm] out of DIN 13

FA

amplitude [kN]

FM

mid-load [kN]

g(a,w)

geometrical correction function [-]

Ki

Stress Intensity Factor (SIF) [N*mm-3/2]

ΔK

stress intensity factor range Kmax – Kmin [N*mm-3/2]

Kc

fracture strength [N*mm-3/2]

m

constant of Paris’ equation [-]

MCTS

Maximum Circumferential Tensile Stress

NI

amount of load cycles caused by crack initiation [-]

NP

amount of load cycles caused by crack propagation [-]

P

height of the pitch [mm]

r

radial distance to the crack tip [mm]

R

stress ratio [-]

R1

radius of the root of thread [mm]

SIF

Stress Intensity Factor

List of Symbols and Abbreviations

vii

1

Introduction

1.1 Motivation In mechanical engineering, most components are subjected to variable load cycles. In conjunction with geometry and material discontinuities, these mechanical components in general fail because of fatigue. By improvements of the methods determining the mechanical material behavior of engineering components subjected to high stress, an increasing material-saving design was made possible. Caused by this, material fatigue came to the fore. In the past high efforts were expended to understand the phenomenon of fatigue and to develop avoidance strategies. In many cases, the avoidance is nevertheless not possible and the development of techniques for predicting the life-time of components was unalterable. For this purpose, it was important to acquire knowledge about the initiation and the propagation behavior of cracks. The linear elastic fracture mechanics is one of the fracture mechanical concepts that are applicable to linear elastic materials. In this concept, the existing crack inside a structure is researched. The stress intensity factors Ki characterize the nearby stress field around the crack tip. The trajectory of the fatigue crack and the life-time cycle is mainly controlled by these factors. In the past a lot of investigations were concerned with the development of analytical, experimental and later numerical methods for determining the stress intensity factors. In recent years, the numerical methods like the finite element methods gained more and more importance caused by the increase of computer performance.

1. Introduction

1

1.2 Problem Statement The main purpose of this work is to investigate the fracture mechanical behavior of a prestressed threaded M16 connection subjected to a cyclic loading. The screw thread has caused by its design several stress-raising locations that tend to develop fatigue cracks. This work is focused on the thread roots of the screw that are subjected to the adjacent interface loads between bolt thread and nut thread. Beside the investigation of the propagation of one crack, the possibility of multiple cracks and their crack growth behavior inside the screw is considered.

The fatigue-analysis is conducted with the help of the finite element methods. The threaded connection is reduced to a two-dimensional axially symmetric model of the screw thread and the nut thread with linear elastic material. The thread flanks of both mechanical components are assembled with interface elements that control the contact and the frictional behavior among each other. For this purpose, two finite element programs are used: ABAQUS and FRANC2D. ABAQUS is used for the simulation of the interface between the thread flanks. FRANC2D is used for the prediction of the crack path trajectory and the determination of the stress intensity factors. In FRANC2D, just the screw thread is modeled and the normal and tangential interface loads between the thread flanks are replaced by linearized equivalent loads. The stress intensity factors are periodically verified with ABAQUS. The process of the crack initiation and the micro-crack development is not considered in this work. This procedure is replaced by existing initial cracks of 200µm length at the decisive locations. In the first case study (Case Study 1), the presence of one crack is investigated. The expected results of this investigation are fracture mechanical curves that illustrate the relationship between the number of load cycles and the crack length. For this purpose, the Paris’ Law is used. Additionally, the stress intensity factor range is related to the crack length. In a further case study (Case Study 2), the crack trajectories of co-existing cracks and the possible influence on the life endurance are investigated. Especially the trajectory of the crack that finally causes ultimate failure and its fatigue-life is regarded and compared to Case Study 1. The final investigation (Case Study 3) is poorly academic. All the thread roots contain initial cracks. The aim is investigating the influence of the co-existing cracks on the crack that causes ultimate fracture.

1. Introduction

2

2

Background

2.1 Fundamentals of Fatigue In material science, fatigue studies the formation of cracks and its propagation within a structure due to cyclic loading. The objective target is making an assessment about the fatigue life of mechanical components subjected to repeated stress. The presence of cracks in a component has a high influence on its life expectancy and depends on a few factors. The most important factors are described below (s. [1]): •

Geometry of the structure: The geometry of a structure and the applied loads create a stress field that favors the development of cracks at locations of high stress concentration, e. g. sharp corners. After the micromechanical development of a crack, the stress field notably determines its continuing path.

•

Geometry of the crack: The crack depth, the location and direction within the structure and the notch base radius r mainly affect the stress concentration in vicinity of the crack tip. By increasing the crack depth, the ratio of the hot spot stress and the nominal stress is getting higher, therefore the crack is growing faster. The hot spot stress is in inversion proportion to the notch base radius.

•

Material: the material type and the deformation behavior strongly influence the deformation in the crack tip and affect the speed of the crack growth. Of course, it is of importance in context of predicting the crack path direction, if the material is isotropic or anisotropic.

•

Stress amplitude: the influence of the stress amplitude is visible in Wöhler curves (S/N curves). The increase of the stress amplitude results in a decrease of the life expectation, because the respective cycles have a more damaging influence.

•

Mean stress: the minimum-to-maximum stress ratio R, respectively the mean stress, additionally affects the speed of the crack growth. It additionally influences the parameters of the Paris’ law (s. chapter 2.5).

There are a lot of other factors like temperature and corrosive environment that influence the proceeding of the damage of the material. Nevertheless, those factors are not taken into consideration in this work. 2. Background

3

2.2 Crack Development The crack growth occurs in two stages (s. [2]): the crack initiation and the crack propagation. This process is shown in Figure 1. The two stages are considered in separate analyses. The shear-driven micro-crack development occurs in the initiation process (stage 1). These microcracks are inserted by slip bands. The reason is the roughness of the surface caused by intrusions and extrusions. After a certain micro-crack propagation, this process turns into a tension driven macro-crack propagation and this crack propagates tension driven. In this work, just the propagation of the crack (stage 2) is considered.

Figure 1: crack propagation (s. [2] p. 1)

2. Background

4

2.3 Stress Field around the Crack Tip There are several methods dealing with fatigue problems. The accuracy of the results compared to experimental series primarily depends on the permitted method. The material of the structure and its deformation behavior in the fracture process zone identifies the method. If the structure consists of brittle material and has only a minor plastic deformation zone around the crack tip, the methods of linear elastic fracture mechanics (LEFM) are used. The nonlinear effects of the process zone are disregarded (s. [1]). In general, LEFM is used to handle brittle fracture, if the process zone is very small in comparison to the K-dominant region (s. Figure 2). The stress intensity factors (SIFs) Ki are the decisive value that determines the stability and trajectory of the crack growth.

Figure 2: K-dominant region (s. [3] Lecture 5 p. 101)

In contrast, ductile fracture problems are handled with the methods of elastoplastic fracture mechanics (EPFM). The methods that are mainly used are J-integral (a method based on energy methods) and CTOD (method “crack tip opening displacement” that is based on the displacement of the crack flanks). The job definition pretends the analysis and evaluation with parameters of linear elastic fracture mechanics. Therefore, the methods of LEFM is used below.

2. Background

5

2.4 Methods of the Identification of the SIFs (K-concept) The stress intensity factors (SIFs) of the K-concept follow from the linear-elastic stress components inside the stress field. For the K-concept, just the SIFs at the crack tip are of note. The following regards are based on [4]. Due to three stress components in the threedimensional stress distribution, there are three SIF-components that determine the trajectory of the crack propagation (s. Figure 3): • Mode I: crack opening • Mode II: in-plane shear • Mode III: out-of-plane shear In general, the crack growth is caused by a mixed-mode stress field inside a structure. If it is possible to reduce the structure to a two-dimensional model, there are just mode I and II considered caused by the two stress components. In case of plane strain or axially symmetric problem types, the third stress component out of plane does not cause an out-of-plane shear stress component and therefore not mode III.

Figure 3: mode I-III (s. [4] p. 85)

Mode I causes the crack opening and results from the tensile stress orthogonal to the crack flanks inside the plane. This mode has the decisive influence on the life expectancy and mainly controls the crack growth speed. Mode II results from the shear stress tangential to the crack flanks inside the plane and controls the crack trajectory. The ratio between KI (Mode I) and KII (Mode II) is affected by the location and the inclination of the crack in the structure. 2. Background

6

In literature, a great many of calculation formulas for the determination of SIFs is listed for special cases. For more accurate determinations, the shape of the structure must be considered more precisely.

Stress field in vicinity of the crack tip shown in Figure 4 (s. [5] p. 66):

Figure 4: stress field in vicinity of the crack tip (s. [1] p. 23)

Mode I: = !=

√

"

/ #

$−

Mode II: = !=

√

"

%

− #

/

/

/ /

%

Plane strain: $ =

− )*

+

= *,

+

=/

Plane stress: $ =

2. Background

− +

/

+

/

' +

' −

'$ + + '$ − +

−*.

/ /

/ /

/

/

& /

/

( & (

/ /

/

/

/

(

(

-

+*

7

2.5 Fatigue Crack Growth In order to make a prediction about the fatigue life of a component, a connection between the number of load cycles N and the crack length a must be established. The most common way is using the Paris’ Law that describes the connection between the crack growth rate da/dN and the SIF range ΔK between the threshold value ΔKth and the fracture toughness ΔKc. The Paris’ Law is usually shown in a double logarithmic reference frame and can be divided into three regions (s. Figure 5): 01 =3∗∆ 02 ∆

6

= ∆ ∗ √ ∗ 1 ∗ 7 1, 9

Figure 5: Paris' Law (s. [2] p. 3)

2. Background

8

Region I:

Region 1 is the near threshold region. With exceeding the threshold ΔKth, the crack propagation begins. The crack propagation stops below this threshold value. The threshold value is influenced by the mean stress. A higher stress ration R results in a lower critical SIF (s. [1] p. 87).

Region II:

Between the threshold ΔKth and the fracture toughness ΔKc, the stable macroscopic crack propagation continues. In the double logarithmic diagram, this section of the curve shows the linear relationship between the crack growth rate da/dN and the stress intensity factor range ΔK with gradient m. This gradient is merely dependent on the material and additionally includes the mean stress dependency. A higher stress ratio R results in a lower inclination (s. [6] table 5-4).

Region III:

By approaching the fracture toughness ΔKc, the stable crack growth turns into an instable crack growth and ends in ultimate fracture of the mechanical component. The fracture toughness is dependent on the thickness of the component, the quality of the material, the mean stress and the temperature. A higher stress ration R results in a lower critical SIF (s. [1] p. 87). The critical SIFs are in general listed for plane strain, because there is an influence of thickness.

There are some other relationships between the fatigue crack growth rate and the stress intensity factor range that respect the in reality rounded out intersection near the two threshold values ΔKth and ΔKc (s. Figure 5), but the graphic rendition of the Paris’ Law has enforced itself in science. It is quite easy to handle and merely requires the two material parameters m and C.

2. Background

9

2.6 Crack Closure Effect Crack closure is an effect, in which the crack flanks during the unloading in fact closes, however the crack tip is still tensile-loaded. The consequence is a lower SIF range than the load range applied on the structure and therefore this effect increases the life endurance of the structure (s. Figure 6). :

;

≤:

Figure 6: crack closure effect (s. [7] p. 123)

The reasons are illustrated in Figure 7: plastic deformations (a) and phase transformations (b) near the crack tip and the corrosion (c) and roughness (d) of the crack flanks. Additionally, effects like viscous fluids between the crack flanks (e) can play a role.

Figure 7: crack closure effects

2. Background

10

It is in the context of LEFM not possible to consider the decisive crack closure effects like plastic deformation. The LEFM acts on the assumption that after an unloading of the structure, the deformation state is the same like at the beginning of loading. The single possibilities to consider this effect are solutions out of handbooks with the aim to directly increase the stress ratio R and thereby the SIF range ΔK.

[6] p. 118 describes a method for the determination of KI,op by Newman: @=

,=>

,61

= AB C D, E/ + E ∗ D + E ∗ D + E ∗ D

E/ = /, G H − /, ) ∗ I + /, /H ∗ I E = /, ) H − /, /P ∗ I ∗ E = E =

./ >

− E/ − E − E

61

∗J

K ∗

61 L

MN

R1S; TU ;TT ≤ I ≤ . / >

; >R1S; TU 1VS

[8] p. 405 offers the solution by Schijve: W=

=>

61

2. Background

= /. HH + /.

∗ D + /.

∗D

12

3. Finite Element Modeling in Fracture Mechanics

14

3.1.2

Quarter Point Elements

Crack tip elements are special elements that are developed for a more precise determination of the singularity of the stress and strain field near the crack tip. The order of quarter point elements around the crack tip is often used because of its very simple application and its excellent results. The implementation in finite element software is quite easy, thus the use of those elements is the most common method nowadays (Kuna, 2008). Those special elements are both implemented in Franc2D and Abaqus, therefore they are used in the context of this work. The characteristic of quarter point elements is the modification of an isoparametric element with quadratic shape. The mid nodes of two neighbored element edges are moved to the quarter position towards the crack tip. This simple modification is possible for both quadrilateral and triangular isoparametric elements with quadratic trial function (s. Figure 10). The gain of this method is a shape of the obtained stress function in the vicinity of the crack

tip that is similar to the Y Z[,\ -singularity. The assessment of the stress intensity factors is

possible with simple formulas.

Figure 10: crack tip models with ¼-point-elements: quadr. and triang. (s. [7] p. 194)

3. Finite Element Modeling in Fracture Mechanics

15

Figure 11 illustrates the mesh of a crack tip with triangular quarter-point elements.

Figure 11: crack tip meshing (s. [3] Lecture 4 p. 74)

The following equations for the SIF extraction are from [3] Lecture 4 p. 74: =

]∗√

√^∗

=

Z

]∗√

√^∗

Z

∗ ')

∗ ')

_

+

_

+

;

−)

;

−)

0

−

0

−

`(

`(

In [7] p. 205, the following mesh density is recommended: 1. element size near crack tip: L = a/20 – a/10 2. number of circumferential elements: 24 - 32

3.1.3

Hybrid Crack Tip Elements

Hybrid elements are an alternative to general elements with pure displacement functions. The underlying idea of hybrid elements are stress functions in addition to displacement functions. The results are slightly better than the ones of the quarter tip elements (s. [7] p. 217). However, these elements are quite difficult to implement in the finite element software and are in commercial software like ABAQUS not implemented by default.

3. Finite Element Modeling in Fracture Mechanics

16

3.2 Mixed-Mode Interaction and 2D Crack Growth Trajectory An important aspect besides the choice of a convenient method for obtaining the SIFs is the choice of the right theory for predicting the crack growth trajectory. FRANC2D offers 3 wellknown theories for the quasi-static crack growth. For K2