Department of Civil and Industrial Engineering
Master’s Degree in Mechanical Engineering
Master Thesis
Analysis of the interaction and propagation
of multiple cracks in weldments
In collaboration with
Supervisors
Prof. Francesco Frendo M. Sc. Julian Bernhard
Candidate Andrea Chiocca
Abstract
This thesis explores the recent field of fracture mechanics about the study of interaction and coa-lescence of multiple cracks. The work will be focused on propagation of multiple cracks in a butt welded joint specimen intended for heavy industrial application.
The thesis is based on previous research carried forward by Fraunhofer LBF group for Numerical Method and Component Design and all the groups belonging to the IBESS project. The study takes place on short cracks regime in which the use of elastic material hypothesis through the stress intensity factor loses its validity. Instead, an elastic plastic material behaviour is necessary with the introduction of the J-integral parameter, calculated through the use of FEM simulations of interacting and coalescing cracks’ models.
This leads to the definition of two factors necessary for a more accurate study of the propagation of fatigue cracks, the interaction and coalescence factors. A research is conducted to compare the crack growth between experimental results and the IBESS computational algorithm. Also, an improvement of the coalescence factor is required, to solve a singularity problem in the function. The work showed that the simulations with heat affected zone material behaviour leads to results closer to reality respect the base material behaviour. Furthermore, the new implemented coales-cence formula solved the problem of singularity at the beginning of coalescoales-cence, maintaining the behaviour of the function close to the experimental results.
Riassunto analitico
Il seguente lavoro è basato sulla teoria della meccanica della frattura e si focalizza sullo studio dell’interazione e coalescenza tra cricche multiple. Il tema della tesi è legato allo studio della propagazione di cricche multiple in giunti saldati per applicazioni industriali pesanti.
La tesi è basata su risultati derivanti da il gruppo di simulazione numerica del Fraunhofer LBF e da tutti i gruppi appartenenti al progetto IBESS. La ricerca riguarda lo studio di cricche corte, in cui l’uso dell’ipotesi di materiale elastico lineare, con l’utilizzo del fattore di intensificazione delle tensioni, perde di validità.
Viene invece richiesto l’utilizzo dell’ipotesi di materiale plastico, con l’introduzione del parametro J-integral, il quale viene calcolato direttamente tramite simulazioni agli elementi finiti.
Questo porta alla definizione di due fattori (interazione e coalescenza), i quali sono necessari per uno studio più accurato della propagazione di cricche a fatica. Uno studio è condotto per confrontare i risultati tra dati sperimentali e quelli derivanti dallo script IBESS. Inoltre, è richiesto un aggiornamento della funzione per il calcolo del fattore di coalescenza, poiché è necessario risolvere un problema legato ad una singolarità nella formula.
I risultati mostrano come l’utilizzo dell’ipotesi di materiale termicamente alterato porti a errori minori rispetto all’utilizzo del materiale base. Inoltre, la nuova formula di coalescenza risolve il problema della singolarità nella funzione mantenendo un comportamento molto simile ai risultati sperimentali.
Contents
List of Figures vii
List of Tables xii
List of Symbols xiv
1 Introduction 1
1.1 Structure of the work . . . 2
2 Theoretical principles 3 2.1 State-of-the-art on fatigue life prediction methods . . . 3
2.1.1 Stress-based approach . . . 3
2.1.2 Nominal stress approach . . . 4
2.1.3 Hot-Spot stress approach . . . 4
2.1.4 Notch stress approach . . . 4
2.2 Fracture Mechanics . . . 5
2.2.1 Crack shape parameters . . . 5
2.2.2 Linear Elastic Fracture Mechanics . . . 6
2.2.3 Elastic Plastic Fracture Mechanics . . . 9
2.2.4 Fatigue crack growth . . . 11
2.2.5 Definition of short and long cracks . . . 14
2.2.6 Interaction . . . 15
2.2.7 Colescence . . . 17
2.3 Growth of multiple cracks . . . 18
2.4 IBESS procedure . . . 19
2.4.1 Materials and geometry of the specimen . . . 20
2.4.2 Weight functions . . . 23
2.4.3 Plasticity effect . . . 23
2.4.4 Multiple cracks propagation . . . 25
2.4.5 Variability of weld toe geometry . . . 26
2.5 J-integral valutation with Abaqus®FEM code . . . 26
3.3 Conclusion . . . 34
4 IBESS script modification 35 4.1 Extension of the script . . . 37
4.1.1 Symmetric stress profile in specimen’s thickness . . . 37
4.1.2 Add a filter to choose the right intersection point between ellipses . . . . 37
4.1.3 Creation of a distribution without standard deviation . . . 38
4.1.4 One crack characterization for small interaction factor values . . . 38
4.1.5 Additional functions to calculate the stress intensity factor for high aspect ratio . . . 39
4.1.6 Interaction, coalescence and one crack characterization . . . 43
4.1.7 Input with an indefinite number of cracks . . . 43
4.1.8 Step-size optimization for the number of cycles . . . 43
4.2 Script validation . . . 44
4.2.1 Reference specimens for the thesis . . . 45
4.2.2 Single crack models . . . 47
4.2.3 Multiple cracks models . . . 48
4.3 Calculation of sminfunction . . . 52
4.3.1 Results . . . 53
5 Coalescence study 55 5.1 Symmetric model . . . 57
5.1.1 Build up of the model . . . 58
5.1.2 Mesh . . . 59
5.1.3 Results . . . 60
5.1.4 Conclusion . . . 61
5.2 Complete model . . . 61
5.2.1 Coalescence formula fitting . . . 62
5.3 Conclusion . . . 65
5.4 Future developments . . . 65
6 Summary 66 A Ramberg-Osgood’s equations and their relation 68 B Tables with numerical results for script validation 70 B.1 Single crack . . . 71
B.2 Multiple cracks . . . 72
B.3 Stress intensity factor functions . . . 78
B.4 Single crack propagation . . . 79
B.5 Multiple cracks propagation . . . 82
D Stress intensity factor function validation results 90 D.1 Spline and polynomial fitting . . . 90
E Symmetric coalescence cracks 94
E.1 Previous colascence models . . . 94 E.2 Partition plane inclination . . . 96 E.3 Crack’s center distance . . . 97
List of Figures
2.1 S-N curve for a general steel material, Stress over Log(Number of cycles) . . . . 3
2.2 a) Real crack b) Elliptical crack approximation . . . 5
2.3 Elliptical crack growth . . . 5
2.4 Diagram of different zones sorrounding the crack tip . . . 6
2.5 General stress field behavior depending on the stress intensity factor (K) . . . 6
2.6 Loading modes for a crack . . . 7
2.7 Reference system for Westergaard stress solution . . . 7
2.8 a) Non-linear elastic behavior b) Plastic behavior . . . 9
2.9 J-integral Γ contour around crack tip in 2D . . . 10
2.10 Crack Tip Opening Displacement . . . 11
2.11 Crack closure effects on specimen a) Definition of parameters b) Plasticity-induced mechanism c) Roughness-induced mechanism d) Oxide-induce mechanism, picture from [11] . . . 12
2.12 Paris law equation, d Nda over ∆K . . . 12
2.13 Masing behaviour . . . 13
2.14 a) Crack length scales according to [19] b) small crack propagation stages according to [12]. Figure from [22] . . . 14
2.15 Geometry parameters of two interacting craks . . . 15
2.16 Normalised SIF Vs. Angle of ellipse for interacting multiple semi-elliptical cracks from [7] . . . 16
2.17 Effect of cracks interspacing and aspect ratio on Interaction Factor from [7] . . . 16
2.18 Geometry parameters of two coalescing craks . . . 17
2.19 Variation of SIF distribution Vs. Angle of ellipse for coalescing semi-elliptical cracks from [7] . . . 18
2.20 Effect of cracks interspacing and aspect ratio on Coalescence Factor from [7] . . 18
2.21 Butt-welded joint, toe geometry 1, from Schork et al. [17] . . . 20
2.22 HAZ hardness for different weld region, from Kucharczyk et al. [10] . . . 21
2.23 Initiation and growth of the crack in the HAZ near the weld toe, from Schork et al. [17] . . . 22
2.24 Global and local weld geometry . . . 22
2.25 Plastic correction function . . . 24
2.28 Crack characterization in IBESS procedure . . . 25
2.29 Element C3D20, normal and collapse behaviour . . . 26
2.30 Nodes position for an elastic-plastic material behaviour . . . 27
2.31 Nodes position for an elastic material behaviour . . . 27
2.32 Integration volume for the calculation of J, inner and outer tube form the close volume . . . 27
3.1 a) FEM calculation of J-integral b) Find the convergence value of J-integral c) Fitting of IAF and COF values from FEM simulations d) Implementation of the function in the script . . . 28
3.2 IAF and COF factors over s for a fixed cracks geometry . . . 30
3.3 Comparison between Ramberg-Osgood and Masing curve . . . 32
3.4 Crack numeration for graph 3.5 and graph 3.6 . . . 32
3.5 Relative percentage error for IAF . . . 33
3.6 Relative percentage error for IAF . . . 33
3.7 Relative percentage error for COF in different crack’s points . . . 33
4.1 Crack growth script flowchart with the principal modifications: white boxes = past changes made by research group, green boxes = changes made in this thesis, grey boxes= original IBESS script . . . 36
4.2 Stress concentration factor behaviour over the thickness of the specimen for a butt weld joint . . . 37
4.3 Real solution for two intersecting ellipses . . . 38
4.4 Wang Lambert low aspect ratio with Glinka solution compared with Wang Lambert low aspect ratio with integral solution for a polynomial fitting, flank angle = 10◦ and weld toe radius = 0.05mm. Crack depth a is measured in mm . . . 40
4.5 Wang Lambert low aspect ratio with Glinka solution compared with Wang Lambert low aspect ratio with integral solution for a spline fitting, flank angle = 10◦ and weld toe radius = 0.05mm. Crack depth a is measured in mm . . . 41
4.6 Error between Wang Lambert Glinka for high aspect ratio and Newman Raju for a linear stress profile. Crack depth a is measured in mm . . . 42
4.7 Error between Goyal Glinka and Newman Raju for a linear stress profile. Crack depth a is measured in mm . . . 42
4.8 Intersections between more than two ellipses . . . 43
4.9 Wrong intersection between cracks due to a too high step-size . . . 43
4.10 Variable step-size over crack’s behaviour . . . 44
4.11 Left figure shows heat-tinting on the surface, right figure shows beach marks, the figure below shows thermographic images . . . 45
4.12 Positioning of the artificial cracks near the welding on the left, microscopic image of the artificial crack on the right . . . 45
4.16 Comparison between experimental and simulate crack growth of S-5-75-NG1-440, S335NL Base Material, weld toe radius=1.85mm ± 0.25mm, flank angle=27.1◦± 2◦ 48 4.17 Comparison between experimental and simulate crack growth of S-5-75-NG1-440,
S335NL HAZ Material, weld toe radius=1.85mm ± 0.25mm, flank angle=27.1◦± 2◦ 48 4.18 Comparison between experimental and simulate crack growth of S-5-65-NG1-388,
S335NL HAZ Material, weld toe radius=5.3mm ± 2.5mm, flank angle=12.23◦± 2◦,
left couple of cracks . . . 49
4.19 Crack length over number of cycles for the left couple of cracks of the specimen S5-65-388, results obtained using IBESS simulation,heat tinting technique and thermography. TG = Termographic-camera, UB = Upper-bound, LB = Lower-bound,HT = Heat-tinting . . . . 49
4.20 Comparison between experimental and simulate crack growth of S-5-65-NG1-388, S335NL HAZ Material, weld toe radius=5.3mm ± 2.5mm, flank angle=12.23◦± 2◦, middle couple of cracks . . . 50
4.21 Crack length over number of cycles for the middle couple of cracks of the specimen S5-65-388, results obtained using IBESS simulation, heat tinting technique and thermography. TG = Termographic-camera, UB = Upper-bound, LB = Lower-bound,HT = Heat-tinting . . . . 50
4.22 Comparison between experimental and simulate crack growth of S-5-65-NG1-388, S335NL HAZ Material, weld toe radius=5.3mm ± 2.5mm, flank angle=12.23◦± 2◦, right couple of cracks . . . 51
4.23 Crack length over number of cycles for the right couple of cracks of the specimen S5-65-388, results obtained using IBESS simulation, heat tinting technique and thermography. TG = Termographic-camera, UB = Upper-bound, LB = Lower-bound,HT = Heat-tinting . . . . 51
4.24 Symmetrical behavior of smindistance . . . 52
4.25 Double logaritmic graph of input values over function evaluated values . . . 54
4.26 Graph of residual error over the weight function . . . 54
5.1 a) Right shape b) Self-intersection of integration region around the crack front . . 55
5.2 Fillet radius with different possibilities of ak o al ai nt ratio . . . 56
5.3 Geometrical limitation on outer circle radius . . . 56
5.4 Different crack modelling with and without symmetry . . . 57
5.5 Partition of the integration volume around the crack and its mesh . . . 58
5.6 a) Crack seam b) Crack front on FEM model . . . 58
5.7 Boundary conditions and cinematic coupling of the force . . . 59
5.8 Overview of the mesh of the specimen . . . 60
5.9 Error relative percentage between J-intergral evaluation for symmetrical and com-plete FEM simulation . . . 61
5.10 Negative displacement of the crack’s nodes on symmetric FEM model . . . 61
5.11 New COFB after the implementation of the new models and the forced behavior in s= 0 . . . 63
5.13 Residual Error Plot . . . 64 5.14 Crack length over number of cycles for the right couple of cracks of the specimen
S5-65-388, results obtained using IBESS simulation fo the old and the update coalescence factor, heat tinting technique and thermography. TG = Termographic-camera,UB = Upper-bound, LB = Lower-bound, HT = Heat-tinting . . . . 64 5.15 Combining the ellipse parameters by using the area between fillet radius and ellipse
arcs . . . 65 A.1 Translation of the referece system from Ramberg-Osgood to Masing material
behaviour . . . 69 B.1 Comparison between WLG low-aspect-ratio with Glinka numerical integration and
WLG high-aspect-ratio with Glinka numerical integration . . . 78 B.2 Comparison between Goyal-Glinka with Glinka numerical integration and WLG
high-aspect-ratio with Glinka numerical integration . . . 78 B.3 Comparison between Goyal-Glinka with Glinka numerical integration and WLG
low-aspect-ratio with Glinka numerical integration . . . 79 B.4 Comparison between experimental and simulate crack growth of S-5-65-NG1-388,
S335NL Base Material, weld toe radius=5.3mm ± 2.5mm, flank angle=12.23◦± 2◦, Central crack . . . 79 B.5 Comparison between experimental and simulate crack growth of S-5-65-NG1-388,
S335NL HAZ Material, weld toe radius=5.3mm ± 2.5mm, flank angle=12.23◦± 2◦, Central crack . . . 80 B.6 Comparison between experimental and simulate crack growth of S-5-65-NG1-388,
S335NL Base Material, weld toe radius=5.3mm ± 2.5mm, flank angle=12.23◦± 2◦, Right crack . . . 80 B.7 Comparison between experimental and simulate crack growth of S-5-65-NG1-388,
S335NL HAZ Material, weld toe radius=5.3mm ± 2.5mm, flank angle=12.23◦± 2◦, Right Crack . . . 81 B.8 Comparison between experimental and simulate crack growth of
S-5-74-NG1-363, S335NL HAZ Material, weld toe radius=3.175mm ± 0.225mm, flank an-gle=10.84◦± 0.76◦, left couple of cracks . . . 82 B.9 Crack length over number of cycles for the left couple of cracks of the specimen
S5-74-363, results obtained using IBESS simulation, heat tinting technique and thermography. TG = Termographic-camera, UB = Upper-bound, LB = Lower-bound,HT = Heat-tinting . . . . 82 B.10 Comparison between experimental and simulate crack growth of
S-5-74-NG1-363, S335NL HAZ Material, weld toe radius=3.175mm ± 0.225mm, flank an-gle=10.84◦± 0.76◦, middle couple of cracks . . . 83 B.11 Crack length over number of cycles for the middle couple of cracks of the specimen
S5-74-363, results obtained using IBESS simulation, heat tinting technique and TG = Termographic-camera, UB = Upper-bound, LB =
Lower-B.12 Comparison between experimental and simulate crack growth of S-5-74-NG1-363, S335NL HAZ Material, weld toe radius=3.175mm ± 0.225mm, flank an-gle=10.84◦± 0.76◦, right couple of cracks . . . . 84 B.13 Crack length over number of cycles for the right couple of cracks of the specimen
S5-74-363, results obtained using IBESS simulation, heat tinting technique and thermography. TG = Termographic-camera, UB = Upper-bound, LB = Lower-bound,HT = Heat-tinting . . . . 84 D.1 Wang Lambert low aspect ratio with Glinka solution compared with Wang Lambert
low aspect ratio with integral solution for a polynomial fitting, flank angle = 10◦ and weld toe radius = 5mm . . . 90 D.2 Wang Lambert low aspect ratio with Glinka solution compared with Wang Lambert
low aspect ratio with integral solution for a spline fitting, flank angle = 10◦ and weld toe radius = 5mm . . . 91 D.3 Wang Lambert low aspect ratio with Glinka solution compared with Wang Lambert
low aspect ratio with integral solution for a polynomial fitting, flank angle = 45◦ and weld toe radius = 0.05mm . . . 91 D.4 Wang Lambert low aspect ratio with Glinka solution compared with Wang Lambert
low aspect ratio with integral solution for a spline fitting, flank angle = 45◦ and weld toe radius = 0.05mm . . . 92 D.5 Wang Lambert low aspect ratio with Glinka solution compared with Wang Lambert
low aspect ratio with integral solution for a polynomial fitting, flank angle = 45◦ and weld toe radius = 5mm . . . 92 D.6 Wang Lambert low aspect ratio with Glinka solution compared with Wang Lambert
low aspect ratio with integral solution for a spline fitting, flank angle = 45◦ and weld toe radius = 5mm . . . 93 E.1 Plane partition parameters . . . 96 E.2 Parameters of symmetric cracks . . . 97
List of Tables
2.1 Mechanical property for S355NL (plates with 8mm-100mm thickness) . . . 21 2.2 Material properties, comparison between base material and heat affected zone
material from Kucharczyk et al. [10]. Refer to figure 2.22 to understand the location of the two areas in the specimen . . . 21 4.1 Maximum relative percentage error between SIF evaluation functions for the A
and C positions . . . 43 5.1 Geometric limitations for the construction of the integration volume, the values in
the table represent a range of s for a given OC R= 0.01 mm . . . 57 5.2 New coalescence models implemented . . . 62 B.1 Table of results for S-5-75-NG1-440, S335NL Base Material, simulated with Wang
Lambert Glinka function, weld toe radius=1.85mm ± 0.25mm, flank angle=27.1◦± 2◦ 71 B.2 Table of results for S-5-75-NG1-440, S335NL HAZ Material, simulated with Wang
Lambert Glinka function, weld toe radius=1.85mm ± 0.25mm, flank angle=27.1◦± 2◦ 71 B.3 Table of results for S-5-65-NG1-388, S335NL Base Material, simulated with Wang
Lambert Glinka function, weld toe radius=5.3mm±2.5mm, flank angle=12.23◦±2◦, Right Crack . . . 71 B.4 Table of results for S-5-65-NG1-388, S335NL HAZ Material, simulated with Wang
Lambert Glinka function, weld toe radius=5.3mm±2.5mm, flank angle=12.23◦±2◦, Right Crack . . . 72 B.5 Table of results for S-5-65-NG1-388, S335NL Base Material, simulated with Wang
Lambert Glinka function, weld toe radius=5.3mm±2.5mm, flank angle=12.23◦±2◦, Central crack . . . 72 B.6 Table of results for S-5-65-NG1-388, S335NL HAZ Material, simulated with Wang
Lambert Glinka function, weld toe radius=5.3mm±2.5mm, flank angle=12.23◦±2◦, Central crack . . . 72 B.7 Table of results for S-5-65-NG1-388, S335NL HAZ Material, simulated with Wang
Lambert Glinka function, weld toe radius=5.3mm±2.5mm, flank angle=12.23◦±2◦, Central crack . . . 73 B.8 Table of results for S-5-65-NG1-388, S335NL HAZ Material, simulated with Wang
B.9 Table of results for S-5-74-NG1-363, S335NL HAZ Material, simulated with Wang Lambert Glinka function, weld toe radius=3.175mm ± 0.225mm, flank an-gle=10.84◦± 0.76◦, Left crack . . . . 75 B.10 Table of results for S-5-74-NG1-363, S335NL HAZ Material, simulated with
Wang Lambert Glinka function, weld toe radius=3.175mm ± 0.225mm, flank an-gle=10.84◦± 0.76◦, Central crack . . . . 76 B.11 Table of results for S-5-74-NG1-363, S335NL HAZ Material, simulated with
Wang Lambert Glinka function, weld toe radius=3.175mm ± 0.225mm, flank an-gle=10.84◦± 0.76◦, Right crack . . . . 77 E.1 Overview of coalescence models built up by the research group . . . 95
List of Symbols
Symbol Description
a crack depth (mm)
aint coalescence intersection (mm) akoal coalescence depth (mm)
c crack length (mm)
da
d N crack growth rate (mm/cycle) DN number of cycles step-size used in the script
E elastic module (M Pa)
G energy release rate (J/m2)
h weld reinforcement (mm)
J J-integral (J/m2)
K0 material constant for Ramberg-Osgood power law (M Pa) KI, KI I, KI I I stress intensity factor for fracture modes I-I I-I I I (M Pa
√ m) KC fracture toughness (M Pa
√ m) Kop crack opening stress intensity factor (M Pa
√ m) Kt stress concentration factor
L weld width (mm)
m(x, a) weight function
m crack’s center x-position (mm)
N number of cycles
Nf number of cycles for fracture
n0 hardening exponent
R stress ratio
Rm tensile strength (M Pa)
r fillet radius between coalescing cracks (mm) s crack’s tip distance (mm)
smin minimum crack’s tip distance to have no interaction (mm) t thickness of the specimen (mm)
U crack closure effects parameter ULC crack closure effects parameter for long crack
W half-plate width (mm)
α flank angle (deg)
ε0 yeld strain
εel elastic strain
εpl plastic strain
θ angle of crack’s edge position (deg)
ν poisson ratio
ρ weld toe radius (mm)
σij stress field around the crack tip (M Pa)
σY yeld stress (M Pa)
σr reversed stress in the S-N curve (M Pa) ∆J J-integral range for fatigue load condition (J/m2) ∆K stress intensity factor range for fatigue load condition (M Pa√m) ∆Ke f f effective stress intensity factor range (M Pa√m)
∆Kp stress intensity factor range corrected for plasticity (M Pa √
m) ∆Kth threshold value of stress intensity factor range (M Pa
√ m)
∆σ stress range (M Pa)
∆ε strain range
Abbreviation Description
CF D Cumulative Fatigue Damage
COF Coalescence Factor
CTOD Crack Tip Opening Displacement E PF M Elastic Plastic Fracture Mechanics
FCP Fatigue Crack Propagation
F E M Finite Element Method
GU I Graphical User Interface
H AZ Heat Affected Zone
HT Heat Tinting
I AF Interaction Factor
I BE SS Integral fracture mechanics determination of the fatigue strength of welds
L B Lower Bound
LE F M Linear Elastic Fracture Mechanics
OC R Outer Circle Radius
SI F Stress Intesity Factor
T G Termographic Camera
Introduction
In the following work, the general engineering approach based on fracture mechanics is used, one of the most important aspects of fracture mechanics is to improve understanding in the role of flaws in structures and how they might affect the performance. This can be translated in the development of damage tolerant design approach, that requires a comprehension of crack growth mechanisms and material behaviour in order to understand how a crack grows in service and when it becomes critical. The classic mechanics, in fact, does not include the presence of a defect in the material and if the permissible stress satisfies the comparison with the equivalent stress the part is safe. In a damage tolerant study it is supposed a initial damage on the part, and from this develop the stress field in the proximity of the crack, how the stress change and what are the stress values that permit a crack propagation.
Recent works have tried to extend the fracture mechanics concept at the study of interaction and coalescence between cracks (Bayley [1]; Patel [7]), but in many cases, a cracks symmetric approach is used, this does not allow to investigate the mutual effect between different geometries and, therefore, obtain different cracks growth rate. Another issue is the considered material behaviour, most of the researchers are focused on elastic material models, where the main parameter is the stress intensity factor, this approach leads to good results when the defects studied are long cracks. Nevertheless, for a small cracks regime, an elastic-plastic material behaviour must be used, with the introduction of the J-integral parameter.
It is quite common to find multiple cracks in engineering structures, so although the standard procedures can be used when the cracks are isolated, these are not right when the cracks are close to each other. During interaction and coalescence the crack growth rate change depending on the modification of the crack driving force, for this reason, there may be a variation in the lifetime of a specimen compare with the standard models.
There are many studies whose purpose is to calculate the fatigue life of a mechanical component, one of those is the German research project IBESS. The project is embodied in a computational algorithm but, at the moment, it does not include the interaction (IAF) and coalescence (COF) modelling, instead, when the cracks touch each other only a single crack characterization is used. Starting from the work already developed within the project an improvement in the structure of the script is required to implement the latest developments. Also, a more in-depth study is also
1.1 Structure of the work
In Chapter 2 after a general introduction about the state of the art for the fatigue approaches of lifetime calculation, the focus will be on the basic theory of fracture mechanics, it follows an overview about the theoretical principles develop inside the IBESS project and the minimum theory necessary to better understand this thesis.
The first part of the work is developed in Chapter 3, here a validation of FEM results for interacting and coalescing cracks is done based on Konterman theory, while in Chapter 4 are described the modifications performed in the IBESS script necessary to implement the new theory for crack propagation, interaction and coalescence.
In Chapter 5 the work will be focused on crack coalescence, the existing models will be updated through extrapolation of FEM results for different crack geometries and subsequent fitting of the data.
Theoretical principles
2.1 State-of-the-art on fatigue life prediction methods
In this part of the work a basic overview of the main fatigue life prediction methods will be given, however, the following thesis is based on fatigue crack propagation (FCP), so a broader description will be made on the fracture mechanics concepts, look at section 2.2.
Below is a description of the main methods belonging to the cumulative fatigue damage (CFD) approach.
2.1.1 Stress-based approach
The stress-based approach (S-N curve approach) is one of the first methods developed but still one of the most used, it is based on experimental data and takes the form of a curve shown in the figure 2.1. The curve is fitted using experimental points that derive from the coupon testing, this test is performed applying sinusoidal stress through a testing machine which also counts the number of cycles until the break of the specimen or the runout of the machine time.
The calculation of those points is highly influenced by many factors like mean stress, loading frequency, corrosion, residual stresses, and notches presence, for this reason from the same sample different results can be expected, this probabilistic nature of fatigue can be seen in figure 2.1, as matter of fact there are different cycles to failure for the same stress amplitude and this behavior increase with the decreasing in the stress amplitude.
In general, the plot for the first part of S-N curve (fatigue life versus stress amplitude) for a metal is given by the Basquin equation
σr = A(N)b (2.1)
where A and b are two material constants, σr is the reversed stress and N is the number of stress reversal after which fatigue failure will occur.
The stress for the fatigue analysis is calculated for a linear elastic material behavior in the potential trigger point of the fatigue fracture, if stress concentration is present, it must be taken into account through stress concentration factor (Kt) that multiples the nominal stress (σnom), as shown in the equation 2.2.
σmax = Ktσnom (2.2)
In a component, there are two types of stress concentration the first is due to structural geometry change or discontinuity and the second is due to welding. Taking into account the stress concen-tration, the stress-based approach can be further divided into nominal stress approach, the hot-spot stress approach and the notch stress approach.
2.1.2 Nominal stress approach
The nominal stress method is a non-local fatigue assessment method according to which the fatigue life of a joint can be calculated studying the global geometry of the joint and the nominal stress history in a specific location.
The global geometry is described by a design category and one or more detail category numbers while the stress history is studied by rainflow or reservoir method.
2.1.3 Hot-Spot stress approach
Similar to the nominal stress approach even the hot-spot approach is based on design category and detail category numbers, the main difference lies in the more competent measure of geometrical stresses, respect the nominal stresses calculation for the nominal stress method, for this reason, fewer detail categories are required for the hot-spot approach in order to provide comparable versatility. The geometrical stress is the maximum structural stress calculated at the base of the welding seam, it considers the effects due to global geometry and not to the micro-local geometry of the welding.
2.1.4 Notch stress approach
The notch stress method take into consideration the stress rising from the notch effects of the weld, unlike the hot-spot approach now the micro-geometry is considered, the true notch is substitute with a fictitious one of radius r and the stress is defined as the local maximum stress in linear-elastic weld models.
2.2 Fracture Mechanics
This section gives an overview of fracture mechanics for an elastic and plastic behaviour of the material. However the principal focusing of the work will be on the elastic-plastic fracture mechanics and how multiple cracks interact with each other, so more attention will be given to that part.
However, before introducing the other topics an brief introduction to the crack geometry is required
2.2.1 Crack shape parameters
A
C
RC
La
c
a
c
a)
b)
Figure 2.2: a) Real crack b) Elliptical crack approximation
The crack dimensioning in this work is based on the assumption of an elliptical crack shape, this is one of the many ways used in the approximation of the real crack structure, but usually, the simplified geometry depends on the type of problem studied. The approximation with an elliptical geometry is very useful, for the reason that the crack front could be described by only three points as shown in figure 2.2. The crack growth is simplified, all the necessary parameters are calculated for the A, CR and CL points and the geometry of the new ellipse is updated, see figure 2.3.
Δc
R1Δc
R2Δc
L1Δc
L2Δa
1Δa
2 Figure 2.3: Elliptical crack growth2.2.2 Linear Elastic Fracture Mechanics
K-dominance-zone elastic deformation
Plastic zone inelastic deformation
Mechanism-based strain gradient ( high-order stress dominates ) Classical plasticity
( HRR field )
Mechanism-based strain gradient ( stress dominates )
Figure 2.4: Diagram of different zones sorrounding the crack tip
The linear elastic fracture mechanics (LEFM) concepts are valid if the plastic zone, shown in figure 2.4, is much smaller than the singularity zones (K-dominance zone in figure 2.4). The objective of LEFM is to predict the critical load that will cause the growth of a crack in a brittle material.
It is possible the determination of the analytical expression of the field tension around the crack tip, assuming isotropic linear elastic behavior of material.
This is the "local" solution find out by Westergaard, with the hypothesis of a polar coordinate system (r, θ) center in the crack tip, figure 2.7:
σi j(r, θ) = K √ 2πr fij(θ) + O[r] (2.3)
where r is the crack’s tip radius, K the stress intensity factor and fij(θ) is a adimentional function in the variable θ depending on the fracture modes, for more information about the fij(θ) function see [6].
σ
r θ = 0
theory a specimen with a very small radius of the crack should break immediately even with small loads because the stress concentration factor reach infinite value.
In spite the stress and strain singularity can be introduced the stress intensity factor ( K or SI F ) that gives the amplitude of the singularity, it is not a characteristic of the material but depend on the length of the crack and nominal stress, so the K is the parameter that shows how the stress field around the tip changes.
The K value 1 is correlated with the loading modes of the crack, as shown in figure 2.6
Mode I
Mode II
Mode III
Figure 2.6: Loading modes for a crack
y
x r
θ
z
Figure 2.7: Reference system for Westergaard stress solution
LEFM considers three distinct fracture modes that encompass all possible ways of crack de-formation.
• Mode I : The opening mode, the forces are perpendicular to the crack, pulling the crack open.
• Mode I I : The in-plane shear mode, the forces are parallel to the crack, this creates a shear crack where the crack slides along itself.
• Mode I I I: The out-of-plane shear mode, the forces are transversal to the crack, the material separate moving out of its original plane.
The use of one mode or another depends on the type of load applied, below the formal definition of the K factors with reference to the coordinate system in figure 2.7
KI= lim r →0 √ 2πrσyy(r, 0) (2.4) KII= lim r →0 √ 2πrσyx(r, 0) (2.5) KIII= lim r →0 √ 2πrσyz(r, 0) (2.6)
The validity of LEFM could be estimated by the comparison of a characteristic dimension x of the specimens (remaining ligament size, thickness, crack length) with the plastic zone size, an example is given by the ASTM Standard [16] for the validity of LEFM for the mode I:
x > 2.5 KIC
σY 2
(2.7)
where KIC is the critical stress intensity factor value and σYis the yeld stress.
Standard tests are done to find the critical value of the SIF, KIC, this value depends only on the material and is called fracture toughness. The failure is defined every time the K exceeds the KC, this leads to the following safety factor for failure by fracture as
Sa f et yFactor = KC
K (2.8)
everytime the safety factor becomes less than one the failure occurs, the reader can refer to Juvinall et al. [6] for more informations.
By considering the fact that crack propagation always involves energy dissipation (like surface energy, plastic dissipation, etc.) an alternative criterion at the SIF could be postulated in terms of energy release rate, however, the whole theory behind it is omitted because it will not be used later in the work, for more information refer to [6].
The Energy Release Rate (G) is a global parameter contrary to the K that is a local crack-tip parameter, it represents the energy dissipated during fracture per unit of newly created fracture depth for two-dimensional problem ( or fracture surface area for three-dimensional problem ).
J = G = −δEp
δa (2.9)
where Epis the potential energy, a is the crack depth and J is the J-integral widely explained in the next section 2.2.3.
In this case the stress for failure could be calculated as r
For isotropic linear materials G is correlated with K in the mode I: Plane strain G= 1 − ν 2 E K 2 (2.11) Plane stress G= −K 2 E (2.12)
2.2.3 Elastic Plastic Fracture Mechanics
When the plastic zone is not small compared to the specimen ( reference in figure 2.4 ) the fundamental parameter K based on LEFM loses its meaning, different criteria should be introduced, like J-integral and CTOD explained later, to determine the expression for the stress components inside the plastic region.
However, the theory used in this work is not based on elastic-plastic material behaviour but rather the deformation theory of plasticity is considered, this allows fully plastic analysis of ductile metals, usually under small-displacement conditions. The model is based on a curve like equation 2.13, where the stress is defined by the total mechanical strain with no history dependence, it is, in fact, a nonlinear elastic model, as shown in the figure 2.8 a), where the unloading part follows the same path of the loading part.
ε
σ
a)
b)
σ
Loading Loading Unloading Unloadingε
Figure 2.8: a) Non-linear elastic behavior b) Plastic behavior
A material model suitable for metals is the Ramberg-Osgood model, a material power-law hardening form ε = εel+ εpl = σ E + σ K0 n01 (2.13)
where the first termσE is equal to the elastic part of the strain,
σ K0
n01
take into accounts the plastic part and K0and n0are material parameters.
consider-being the parameter used in the following thesis:
• J-integral, is a line integral (path-independent) around the crack tip
x2 x1
n
Γ
Figure 2.9: J-integral Γ contour around crack tip in 2D
J = ∫ Γ W dx2− ti δui δx1 ds (2.14)
where W is the density of strain energy
W= ∫ εi j
0
σi jdεi j (2.15)
Γis an arbitrary path clockwise around the tip of the crack
tiare the components of the vectors of traction
uithe components of the displacement vectors
dsan incremental length along the path Γ
σi j and εi j are the stress and strain tensors
The J-integral represents the rate of change of net potential energy with respect the ad-vance of the crack for a non-linear elastic solid.
This parameter characterizes the near crack tip situation because its path independence, that is obtained when contours are taken around the crack tip as shown in figure 2.9 and when the following conditions are reached:
– time independent processes – small strains
– homogeneous hyper-elastic material
– plane stress and displacement fields, i.e. no dependence on out-of-plane coordinate – straight and stress-free crack borders parallel to x
The J-integral is used to calculate the intensity of the stress field singularity near the crack tip even when the shape of the stress field is unknown. Budiansky and Rice (1973) showed that J-integral is correlated with the energy release rate like is shown is equation 2.9. Similar to LEFM could be found the stress field around the crack tip, below the HRR fields (Hutchinson, Rice and Rosengren)
σi j(r, θ) = σY
J
ασYε0Inr n1+1
σtilde,i j(θ) (2.16)
where the parameters σY, ε0, α, n characterize the material behavior from a Ramberg-Osgood power law equation like
ε ε0 = σ σY + α σ σY n (2.17) In this thesis the J-Integral approach will be used to calculate the crack driving force numerically as it is well established in the FEM Abaqus® code, it is robust and does not require intensive mesh refinement in the vicinity of the crack tip, more information can be found in section 2.5.
• Crack Tip Opening Displacement ( CTOD ) is the displacement at the original crack tip and the 90° intercept, look at figure 2.10.
Elastic-Plastic CTOD= d(ε0, n) J σ2 Y Linear-Elastic CTOD= K 2 σYE
The CTOD approach requires detailed mesh refinement in the vicinity of the crack tip and a definition of CTOD must be established because no unique definition of CTOD exists, neither for the numerical nor for the experimental determination. It is common use this method in experimental tests.
Actual crack tip CTOD
45°
Figure 2.10: Crack Tip Opening Displacement
2.2.4 Fatigue crack growth
Figure 2.11: Crack closure effects on specimen a) Definition of parameters b) Plasticity-induced mechanism c) Roughness-induced mechanism d) Oxide-induce mechanism, picture from [11]
for elastic-material behavior ( e.g. infinite plate with uniform uniaxial stress ∆KI = KI,max − KI,min = (σmax−σmin)
√ πa) and ∆J = ∫ Γ ∆W dx2− ∆ti δ∆ui δx1 ds (2.19)
for plastic-material behavior. Compared to the static load condition, where the crack propagates almost instantly once KCis reached, here the growth of the crack has a complete different behaviour, in fact the crack propagate before reaching ∆KC.
An easy way to describe the crack growth is through the Paris and Erdogan equation da
dN = C∆K
m (2.20)
whered Nda is the crack growth rate, C and m are constants that depend on the material, environment and stress ratio, ∆K is the stress intensity factor range.
Classically is used to plot the equation in a double logarithmic graph, d Nda over ∆K
Phase
I
L og(da/dN) Log( K)K
thK
ICC
1
m
Phase
II
Phase
III
slow crack growth high speed of propagation stable propagationFigure 2.12: Paris law equation, d Nda over ∆K
is difficult to predict since it depends on microstructure and flow properties of the material. The second one where the Paris law is valid, the crack growth rate is insensitive to the microstructure. The third one where the crack growth increase until the fail of the material, this area is rather sensitive to the microstructure and flow properties of the material.
Another important phenomenon is the crack closure on the propagation of the crack, this because the crack remain close even though an external tensile force is acting on the material.
The crack closure effect is induced principally by plasticity, corrosion, and roughness of surfaces, as can be seen in figure 2.11. This effect stop the crack growth rate and increase the life of material. If only plasticity is considered, this induced closure as the compressive residual stresses on the crack front remain after the unloading of material, in the next cycle the crack does not open if the applied load does not exceed a level to overcome the residual compressive stress.
The crack closure is included taking ∆Ke f f instead of ∆K
∆Ke f f = Kmax− Kop (2.21)
∆Ke f f ≤ ∆K (2.22)
where Kopis the K value in which the crack opens, see figure 2.11.
All this theory apply only when the plastic zone is sufficiently small around the crack tip, otherwise the J-Integral is needed. In the same way of ∆K, the ∆J path independent, for materials that shows a particular behavior (e.g. Masing behavior), is used to account the crack closure effect, and though ∆J is defined as shown in equation 2.19, the meaning of ∆Je f f is the same as ∆Ke f f but for a plastic behaviour of the material.
The Masing model equation is shown below
∆ε = ∆σ E + 2 ∆σ 2K0 n01 (2.23)
is widely used to describe hysteresis loops of metal materials under cyclic loading, the model assumes a symmetrical behavior of the material during the loading and unloading phase, figure 2.13.
Δε σ
Δσ
ε
2.2.5 Definition of short and long cracks
Figure 2.14: a) Crack length scales according to [19] b) small crack propagation stages according to [12]. Figure from [22]
In the crack initiation can be detected as microstructurally short crack the cracks with size in the order of the microstructure of the material, the plastic zone of these cracks is enclosed in few grains, and the growth or stop of these cracks depends by the local microstructure, look at figure 2.14. The application of classic fracture mechanics is not possible in that case.
A microstructurally short crack which is not arrested will become a mechanically short crack, the parameter K has no meaning in this area, is necessary the use of EPFM, because the size of the flaw is in the order of the plastic zone, as shown in figure 2.4.
For long crack the plastic zone is small compared to the crack length 2c and the crack depth a, is still possible the use of LEFM.
2.2.6 Interaction θ1 a1 c1 c2 a2 s C1L C1RC2L C2R A2 A1
Figure 2.15: Geometry parameters of two interacting craks
The interaction is the phase in which the growth in the length direction of the cracks increase as the two inner cracks approach each other, look at 2.15. During the interaction phase the propagation ratio change along the crack tip and it is influenced mainly by the geometrical parameters of the two cracks, as reported in [7]. If the cracks are separated by a distance greater than the length of the crack, then the effect of interaction could be neglected, so as to have a small error, as reported in [4] and figure 2.16 on the following page (it could be noticed that for high s/c values the interaction effect is negligible), but if the distances its less then we could take notice of the interaction. The interaction factor (IAF) is defined in literature as
I AF= SI Fdouble−crack SI Fsingle−crack
(2.24)
where SI Fdouble−crack represent the stress intensity factor along the edge of the crack with the presence of another crack in the neighbourhood and SI Fsingle−crack have the same definition but only for a single crack model.
A fitteded parametric formula by Patel [7] for the I AF in C position is shows below:
I AF = a0+ a1ln s 2c + ha2+ a3ln s 2c i a c + ha4+ a5ln s 2c i a c 2 + +ha6+ a7ln s 2c i a t + ha8+ a9ln s 2c i a c a t (2.25)
The equation is fitted by multivariable least square fit from FEM results.
As shown in figure 2.17 and equation 2.25 the most sensitive parameters are s/c and a/c, I AF increase with the increase in both parameters, while the angle θ of the crack tip and the loading mode (tension, bending) give marginally differences, so they could be neglected in the formula. The aiadimesional factors ( with i = 1, 2, ...9) give only the weight of the geometrical parameters, so a more detailed description can be found in [7].
Figure 2.16: Normalised SIF Vs. Angle of ellipse for interacting multiple semi-elliptical cracks from [7]
2.2.7 Colescence θ1 a1 c1 c2 a2 akoal aint r Coalescence plane A1 A 2 B C1L C2R E D
Figure 2.18: Geometry parameters of two coalescing craks
After that the two cracks touch each other the coalescence phase begins, and the SIF of the crack tips increase dramatically as shown in figure 2.19. The definition of the interaction factor for coalescence, or coalescence factor (COF) is same as described in the previous section
COF = SI Fdouble−crack
SI Fsingle−crack (2.26)
As for the IAF also for the COF could be found values in literature, here an example of COF for the coalescence plane position by [7]:
COF = ft sg h b0+ b1 a c + b2 a 2s + b3 a 2s a c + b4 a 2s a c 2 + +b5 a 2s 2a c + b6 a 2s 2a c 2i (2.27)
In the same way of IAF the important parameters are a, c and s for the other ones an explenation could be found directly in [7].
The coalescence factor is a strong function of aspect ratio (a/c) and thickeness ratio (a/s), and increase with increase in both parameters but the SIF does not show any clear trend respect to interspacing ratio (s/c) (hatch in figure 2.19). In the region adjacent to the coalescence plane, since coalescence depth (akoalfigure 2.18) increase it can be observed a rapid decrease in SIF due to the local curvature of the crack front [1]
Figure 2.19: Variation of SIF distribution Vs. Angle of ellipse for coalescing semi-elliptical cracks from [7]
The variation of COF with interspacing for various aspect ratios is shown in figure 2.20.
Figure 2.20: Effect of cracks interspacing and aspect ratio on Coalescence Factor from [7]
2.3 Growth of multiple cracks
The multiple cracks growth is quite different from the single crack behaviour when two or more cracks are closed to each other their growth is influenced by adjacent cracks. The multiple cracks propagation could be divided in four phases referring to [7]:
1. Isolate cracks, each cracks is studied with the single crack theory, because there is no interaction as consequence of their distance, s value in figure 2.15.
s decrease
2. Interaction, the growth of the cracks is mutually influenced because their distance.
3. Coalescence, the cracks are intersected and a new coalescing crack is formed by the merge of the previous ones.
s is negative
4. New single crack, model-assumption for this thesis of characterization of coalescencing cracks in single crack once akoal = 0.8amax.
s has no more
meaning
2.4 IBESS procedure
In this section, the minimum theory behind the IBESS procedure will be explained, so that the reader can understand the following work, being this procedure the main focus of this thesis.
The IBESS project "Integrale Bruchmechanische Ermittlung der Schwingfestigkeit von Schweißverbindungen" which is German for "Integral fracture mechanics determination of the fatigue strength of welds",
is divided into 8 sub-projects carry on by different German institutes, purpose of this project is the fracture mechanics based simulation of SN-curves of welded joints.
This approach tries to exceed the limitations of the previous studies such as
• a) used of LEFM even for small cracks where the plasticity-induced effects must be taken into account
• b) phenomenon of multiple crack initiation and interaction is not took into consideration • c) not take into account the variability of the weld toe geometry
Figure 2.21: Butt-welded joint, toe geometry 1, from Schork et al. [17]
During several tests conducted within the project IBESS, using non-destructive techniques such as beach-marking and heat-tinting it has been revealed that long part of fatigue life is spent in the short crack propagation regime.
The focus of this thesis will be addressed the propagation, interaction and coalescence of two cracks in a butt-welded joint for a short crack propagation regime. For the configuration that is studied, both initiation and propagation did occur in the vicinity of the weld toe, at the transition between the weld metal and HAZ [10].
The method integrate used fracture mechanics approaches for long fatigue crack propagation with some new elements such as:
• analytical determination of cyclic elastic-plastic crack driving force in short crack • crack closure phenomenon in short crack
• determination of initial crack size by crack arrest considerations • specification of the fatigue limit of a component based on crack arrest • stochastical treatment of the varying weld geometry along the weld toe • study of propagation and coalescence of multiple cracks
2.4.1 Materials and geometry of the specimen
In this section is given a brief description about the materials and the geometry of the specimen used in following study.
Material
S355NL is a normalised weldable fine grain structural steel to EN10025-3. S355NL steel has a yield strength of 355 MPa at 16 mm at room temperature falling to 275 MPa at 250 mm. Structural Steels S355NL is used in many applications, combining good welding properties with guaranteed strengths. S355NL steel is used above all for manufacturing highly stressed welded structures in the construction of heavy machinery, bridges and steel structures.
Grade Min. Yield (MPa) Tensile (MPa) Elongation (%) Impact Energy S355NL 235MPa-275MPa 470MPa-630MPa 22% −50◦ 27J
The minimum impact energy is longitudinal energy
Table 2.1: Mechanical property for S355NL (plates with 8mm-100mm thickness) .
Since crack propagation occurs most of the time near the weld toe, most of life of the crack is spent inside the HAZ zone, as clearly can be seen in figure 2.23, for this reason, the HAZ material properties are fundamental for the following work, look at figure 2.22. During all the research were used the material properties studied in the IBESS project by Kucharczyk et al. [10]. Below is a summary and comparison with the basic material:
Rm(M Pa) K0(M Pa) n0 Fatigue Limit (MPa)
Base material S355NL 555 1203 0.203 250
Coarse Grain-CP1 870 1527 0.170 421
Table 2.2: Material properties, comparison between base material and heat affected zone material from Kucharczyk et al. [10]. Refer to figure 2.22 to understand the location of the two areas in the specimen
Figure 2.23: Initiation and growth of the crack in the HAZ near the weld toe, from Schork et al. [17]
Specimen
For the following work, a precise specimen geometry will be used, as can be seen in figure 2.24. This thesis will be focused on the butt weld joint with the S355NL material, for this reason only the geometrical properties of the latter will be shown, figure 2.24.
Plate
thickness
T
Weld width
L
Weld
reinforcement
h
x
y
z
Plate width
2*W
Flank angle
α
Weld toe
radius
ρ
z
y
2.4.2 Weight functions
The calculation of the SIF in the IBESS procedure takes place through the use of weight functions. This method is based on the premise that the stress and displacement fields for an uncracked and cracked body are linked under a certain loading system.
The SIF is calculated as
K= ∫ a
0
σ(x)m(x, a)dx (2.28)
where σ(x) is the stress distribution for the uncracked specimen and m(x, a) is the weight function, it does not depend on the special stress distribution, but only on the geometry of the component. Once the weight function is known, for a particular geometry, the SIF can be calculated for any given stress field through the equation 2.28.
The general aspect of m(x, a) is for the A point
mA(x, a) = 2 p 2π(a − x) h 1+ M1A 1 − x a 12 + M2A 1 − x a + M3A 1 − x a 32i (2.29)
for the C point
mC(x, a) = √2 πx h 1+ M1C x a 12 + M2C x a + M3C x a 32i (2.30)
In this thesis two main weight functions will be used (refer to Wang et. al [21] and Glinka et. al [5]), all the additional informations are shown in the appendix C.
2.4.3 Plasticity effect
The IBESS procedure used a NASGRO semplified equation for crack propagation ( detailed infor-mation in [14] ) da dN = C[U(a)∆Kp] mh1 − ∆Kth(a) ∆Kp ip (2.31)
This equation respect to the Paris law describe in a better way the phase I in the graph 2.12, through the parameter p.
∆Kth (the threshold value of ∆K) is develop in a different way respect the long crack, infact due to the crack closure effects the ∆Kth is function of the crack length in a small crack regime, see figure 2.14, ∆Kthit could be defined as:
∆Kth =
A(∆a)b+ ∆Kth,e f f ∆a< ∆aLC
∆Kth, LC ∆a> ∆aLC (2.32)
The equation 2.31 consider short and long fatigue cracks, in the first is used an elastic-plastic crack driving forcetaken into account plasticity by defining a plasticity-corrected stress intensity factor ( ∆Kp) instead of ∆J. The ∆Kpand ∆J could be correlated, first of all calculating the ∆K by the LEFM theory, then using a plastic correction function ( f (∆Lr)) the ∆J is calculated and in the end the ∆Kpis obtained:
where E0= E plane str ess E 1−ν2 plane str ain (2.34) and ∆J = ∆K 2 E0 [ f (∆Lr)] −2 (2.35)
where f (∆Lr) is the plastic correction function, see [11] for specific informations, the general behaviour of this function is showed in figure 2.25.
Load
J,
K
Elastic-plastic
Elastic
f(ΔLr)
Figure 2.25: Plastic correction function
The crack closure effects is considered by
U = ∆Ke f f ∆K =
(Kmax− Kop) (Kmax− Kmin)
(2.36)
where U = 1 when no crack closure effects exist and gradually decrease reaching a crack indipen-dent constant value U= ULC < 1, see figure 2.26.
Figure 2.27: Partition of weld toe into equidistant stripes, picture by [11]
2.4.4 Multiple cracks propagation
An update that will hopefully lead to better results in life prediction is to take care about the interaction and coalescence behaviour of the cracks, in the IBESS project until now this effect is neglected, the cracks are studied as single cracks until they touch each other after that they are characterized as one single crack with the length sum of the two previous cracks length and as depth the maximum value between a1and a2, refer to figure 2.28.
anew= max(a1, a2) (2.37)
cnew= c1+ c2 (2.38)
a1 anew a2
c1 c2
cnew
Figure 2.28: Crack characterization in IBESS procedure
Purpose of this work is the improvement of this simplified behaviour, quoting [11] page 22: "...the coalescence of two or more cracks is activated when the surface points of these cracks touch one another. Then the cracks are transformed in one crack. Note, however, that this criterion does not correspond to the reality because of two aspects. First of all, adjacent cracks interact before coming into contact, because of which the crack driving force at the surface points is increased and the crack propagation at the
2.4.5 Variability of weld toe geometry
Another important issue is the modelling of the weld toe geometry, and consequently, the variable stress field on the weld toe, in fact, the crack initiation sites are correlated with the variation of the local geometry and this problem has a relevant influence on the expected lifetime.
In the IBESS the weld toe is partitioned in equidistant stripes and all the geometrical parameters of the weld (weld toe radius, flank angle secondary notch depths) are determined for every stripe by semi-random sampling from their statistical distribution, see figure 2.27. The through thickness stress distribution used in this work was determined by a parametric equation named BAMXT develop in [11].
2.5 J-integral valutation with Abaqus
®FEM code
In this section only the three-dimensional crack modeling will be described being the one used in the following work.
As already described in the sections above the stress (strain) field has a singularity on the crack front both for elastic and plastic behavior of the material. A distinction can be made between the elastic behaviour with a singularity proportional to√1
r, and the plastic behaviour with a singularity proportional tor1. In order to be able to reproduce this behaviour, collapsing elements are provided in Abaqus®. The elements named C3D20, figure 2.29, is a quadratic brick element with 3x3x3 integration points. If it is used as crack tip element, the collapsed model is adopted, see figure 2.29.
2 3 4 5 6 8 9 7 12 13 18 19 20 10 15 16 17 1 14 11 4 2 3 6 7 12 18 19 20 10 15 16 14 5 17 1 9 13 8 11
Crack front Collapse nodes
Figure 2.29: Element C3D20, normal and collapse behaviour
The nodes position in the collapse element depends on the type of material to be simulated, for elastic-plastic the singularity 1r is obtained with the mid edge node in half length position, as shown in figure 2.30, the nodes on the collapse crack line can be separeted during the simulation.
H H/2
Figure 2.30: Nodes position for an elastic-plastic material behaviour
If an elastic behavior of the material is required the singularity √1
r is obtained by positioning the nodes in one quarter of the edge length like showed in figure 2.31, in this case the nodes are treating like initial nodes and can not be separeted.
H 3*H/4
Figure 2.31: Nodes position for an elastic material behaviour
The J-integral in Abaqus is evaluated via a domain integral, the calculation is done over a volume within contours that incorporates the crack. The elements C3D20, explained above, is arranged around the crack to form a close volume, figure 2.32.
Inner tube Outer tube
Figure 2.32: Integration volume for the calculation of J, inner and outer tube form the close volume
Validation of previous results
3.1 Previous results
This work is based on previous researchs on propagation, interaction and coalescence of small cracks. This section summarizes the results developed up to now by the reserch group, and it refers to Bernhard [2], refer to it for more complete informations.
b) J-integral convergence value
contours J
a) J-integral integration region
c) Fitting IAF and COF
IAF = f (a1, a2, c1, c2, ρ, α ) COF = f (a1, a2, c1, c2, ρ, α ) 1 s COF IAF ai ci α ρ d) IBESS script
Input Crack growth function Output IBESS script
IAF COF
Abaqus
Matlab
Eureqa
Figure 3.1: a) FEM calculation of J-integral b) Find the convergence value of J-integral c) Fitting of IAF and COF values from FEM simulations d) Implementation of the function in the script
The study on propagation is based on experimental tests and FEM simulations. A formula was searched to link the geometrical parameters of the crack (a, c) and the specimen (α, ρ) at the IAF and COF, same definition of section 2.2. First of all with the use of Abaqus®, the J-integral on the crack front was calculated through integration on different circular contours around the crack tip, for this reason a integration region must be defined, as shown in figure 3.1 a). Then the J-integral convergence value over the contours is chosen, figure 3.1 b). After that with the equation 2.24-2.26 the IAF and COF were calculated and the results were fitted in Eur eqa® to find an equation function of all the parameters already explained above, figure 3.1 c). Finally the
The study focused only on the point C for the interaction, figure 2.15, because for the point A the IAF reaches appreciable values only for unusual shapes of the cracks (ac > 1.5). While for the coalescence the points A − B are considered , figure 2.18.
The equations are showed below: IAF for C position
I AFC = 1 + 0.419a1 c1 a2 c2 a2 c2 2s c1+c2+ a1 a2 2s c1+c2+ α 180◦ a2 c2 2 + a1 c1 2s c1+c2 a1 a2 2 +a2 c2 2 2s c1+c2 2 (3.1)
The principle differences by the Patel equation 2.25, is that in this case the simulations take care also for non-symmetrical cracks geometry, this could be seen from the fact that the parameters for both cracks are present in the equation (a1, c1, a2, c2). The parameter ρ is not present because gives marginally differences.
In the same way the coalescence is studied, the number of simulations is however reduced respect the interaction case, this for the limitations that this type of FEM simulations have, mainly due to impossibility to create an integration-region for the J-integral in Abaqus®for every type of crack’s geometry, see section 2.5 for more details.
Also for the COF an equation is developed by the data fitting of FEM results: For the B position
COFB = 2.53 + 1 s 2c1+2c2 0.0271 a2 c2 2 + 0.106a2 c2 a1 c1 2 + −0.233 a 2 c2 a 1 c1 − 0.663 a 1 c1 − 0.87 a 2 c2 a 1 c1 (3.2)
and for the A position
COFA= 0.922 + s 2c1+ 2c2 0.669a2 c2 − 1.69a1 c1 + 0.0494 a1 a2+ +3.1a1 t 2 + 0.493a1 c1 a2 a1 − 0.311a1 c1 (3.3)
Aim of this work will be also the expanse of the previous coalescence formulas, the old simulations will be integrated with new ones and a better data fitting will be created. The region near s= 0 at the moment is symplified by a constant orizontal curve at the value I AF(0), this because the COFB equation have a singularity for s → 0−, this region is hard to reach by FEM simulation for the limitations explained in section 2.5, (see figure 3.2).
-0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 s distance (mm) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Factor IAFC COFB COFA
FEM result IAFC
FEM result COFB
FEM result COFA
Extrapolate results a1=0.5 mm c1=0.5 mm a 2=0.4 mm c2=0.8 mm = 15 ° t = 10 mm
Figure 3.2: IAF and COF factors over s for a fixed cracks geometry
3.2 Validation of previous results
The use of IAF and COF occurs through their multiplication with ∆Kp, in this way the factors go directly to modify the cracks propagation law.
In this chapter the right position of the IAF and COF factors in the crack growth formula is validated, at the moment the used equation is the 3.4
da dN = C[U(a)X∆Kp] mh1 −∆Kth(a) ∆Kp ip (3.4)
where X represent the IAF and COF inside the equation.
The right position of the interaction and coalescence factors has to be verify because from the Abaqus®simulations the J-integral was based only on σ-ε and not on ∆σ-∆ε, for this reason the location of the factors inside the equation was not certain. In other words the insecurity in their positioning is due to the static load simulations run in Abaqus®, only the parameters for the load cycle are considered (Ramberg-Osgood equation). This leads to the calculation of the J instead of ∆J, for this reason being the X based only on J the equation 3.4 cannot be used before a validation. The two possible ways are showed in equation 3.5 and equation3.6.
X(Kp,max− Kp,min) (3.5)
or
A verification was done through the Konterman theory [9], he based his research on J-integral simulations with Masing behaviour of material, Konterman assumes that the difference between the J-integral based on Ramberg-Osgood material behaviour (σ-ε) and the J-integral based on Masing behaviour of material (∆σ-∆ε) has a factor of four
∆JM asing = 4JR−O (3.7)
but from the definition of IAF and COF this factor is deleted
I AFM asing = 4JDoubleCr acks
4JSingleCr ack = I AFR−O
(3.8)
COFM asing =
4JDoubleCr acks
4JSingleCr ack = COF
R−O (3.9)
and the solutions should be the same compared to the previous FEM simulations with the Masing’s hypothesis, this gives credit at the equation 3.5.
To follow the method by Konterman, new simulations has been done changing the material parameters follow the Masing’s hypotesis for plastic material behaviour, according to which the σ − ε variation during load reversal is described by a curve that correspond to the doubled initial cyclic loading curve. Graphically, the Masing behaviour is exhibit when the tip of hysteresis loop coincide at peak compressive stress, as already shown in figure 2.13. However is important to notice that Masing’s hypothesis is just an approximation to the cyclically stabilized stress–strain response of some metals.
The Masing’s equation is a Ramberg-Osgood like equation, the only differences are a scale factor of two and a range quantities consideration.
∆ε = ∆σ E + 2 ∆σ 2K0 n01 (3.10)
The material use for the FEM simulations it is a structural steel S355NL, for detailed information see section 2.4.1.
The parameters used in the Ramberg-Osgood equation
ε = σ E + σ K0 1 n0 (3.11) are • n0= 0.1961 • K0= 1115
they can be translated in the Ramberg-Osgood parameters for the equation used by Abaqus®
ε = σ E + α E |σ| σY n−1 σ (3.12)
• α= 0.7404
in the same way Masing’s parameters can be obtained • n= n10 = 5.099 • Choosing Rp0.1%σy = K00.001n 0 = 287.7 MPa • α= 2EσYn−1 2K0n = 0.04321
The complete equations that linked all the parameters are shown in appendix A.
Masing
Ramberg-Osgood
Strain (-)
Stress (M
Pa)
0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 0 100 1000 1500 2000 2500Figure 3.3: Comparison between Ramberg-Osgood and Masing curve
In Abaqus®new J-integral results are aquired changing the material parameters with the values described above and also the load is doubled as described in the paper by Konterman.
For the interacting cracks simulation 10 models are choosen out of the 250 created, the reason behind such a limited number lies in the long time necessary for each simulation, however the 10 models have been chosen in such a way to cover all the limit geometry cases for the cracks. For the coalescence only four models are chosen, this because less coalescence FEM simulations were available, but in the same way as for the interaction the geometrical limit cases are covered. All the results show a factor of four in the J-integral calculation, while a null error is found for the coefficients of interaction and coalescence as expected.
The difference between the IAF calculated for Ramberg-Osgood and Masing is shown in figure 3.5 and 3.6, the errors are so small that they can be compared with the numerical errors of the program, every bar of the graphs shows the maximum relative percentage error (maximum respect to the edge position of the crack) for both the interacting cracks ( C0left and C1right as shows figure 3.4 ).
