Interaction between weld residual stresses and warm pre-stressing
ELAHEH ETEMADI
KTH ROYAL INSTITUTE OF TECHNOLOGY
Table of content
NOMENCLATURE... 2
1 INTRODUCTION ... 4
1.1 Cleavage fracture ... 5
1.1.1 Crack initiation ... 5
1.1.2 Microcrack initiation ... 5
1.1.3 Propagation of the microcrack ... 6
1.2 Statistical modeling, application to fracture toughness ... 7
1.3 WPS ... 10
1.3.1 WPS cycles ... 10
1.3.2 The effect of WPS on cleavage fracture ... 11
1.4 Welding residual stresses ... 12
1.4.1 Residual Stress Effects on Brittle Fracture in Welded Structures ... 13
2 NUMERICAL ANALYSIS ... 14
2.1 Geometry and meshing ... 14
2.2 A model to combine residual stress with WPS ... 15
2.3 Load cycles ... 17
2.4 Material ... 20
2.5 Post-processor ... 20
2.6 Calibration ... 21
3 RESULTS AND DISCUSSION ... 22
4 CONCLUSIONS ... 26
5 ACKNOWLEDGEMENT ... 26
6 REFERENCES ... 27
NOMENCLATURE WPS AR
Warm pre-stressing.
As received
LCF Load-Cool-Fracture.
LUCF RSLUCF Equal force
J-modified
Load-Unload-Cool-Fracture Residual stress- LUCF
Pre-load in WPS processes calculated in a way which the reaction forces are equal Pre-load in WPS processes calculated in a way which the J-integrals are equal
RPV Reactor pressure vessel.
𝑃𝑓 Probability of failure.
𝑑𝑉 Infinitesimal volume.
𝑉0 Arbitrary volume.
ℎ A function that describes the tendency of
an imperfection causing a brittle fracture.
ℎ1 The function describing the tendency of
an inclusion fracturing.
𝑐 Constant that represents the number of
initiation points.
𝜀𝑒𝑝 Equivalent plastic strain.
ℎ2 The function describing the probability of
a fractured inclusion creating a microcrack.
𝜎𝑡ℎ Threshold stress describing the minimum
stress required for crack growth.
𝜎𝐼, 𝜎𝐼𝐼, 𝜎𝐼𝐼𝐼 Non-local principal stresses
𝜎̅ Non-local effective normal stress in the
weakest link model
𝑉𝐿 Integration volume for the non-local
stress tensor 𝜎̅̅̅̅ 𝑖𝑗
𝐿 Radius of volume 𝑉𝐿.
𝐸 Young’s modulus.
𝜈 Poisson’s ratio.
𝛾𝑠 Surface energy.
𝑎 Macroscopic crack length.
𝐾𝐼 KIc
Stress intensity factor modulus one Materials fracture toughness
KIaf/f
The local value of fracture toughness (carbide/ferrit)
The local value of fracture toughness (ferrit/ferrit)
𝑊 Specimen width.
𝐵 Specimen thickness.
Δ𝑉 Element volume.
𝒙 Element centroid coordinates.
𝐽 The J-integral.
1 INTRODUCTION
Cleavage fracture is a catastrophic phenomenon as it is a sudden structural collapse occurring to structures made of insufficiently ductile steels working at lower temperatures[1]. In this study, the focus is on the material used in reactor pressure vessels (RPV) in nuclear power plants. The steel is exposed to radiation that by time causes embrittlement of the material. Safe operation of the power plant with today’s engineering standards is possible by taking advantage of warm pre-stressing (WPS) effects.
Welding is an inevitable part of manufacturing; therefore, it is important to investigate the influence of residual stresses (RS) due to welding on the load-bearing capacity of cracked components. Additionally, these stresses can also affect the benefits of WPS cycles which as mentioned earlier improves the structure behavior.
In this study, the effects of two mechanisms in WPS on a RS field will be studied and compared. As the focus of the study is on the brittle fracture, a probabilistic model made by Kroon and Faleskog[4], will be used. To get an overview of the thesis, a summary of each topic will follow.
1.1 Cleavage fracture
A brief explanation of a series of phenomena that must occur in order to cleavage fracture happen is described in this section.
1.1.1 Crack initiation
During manufacturing or through welding, there will be a considerable amount of inclusions in the material.
These inclusions which mostly are carbides will crack at some point under increasing applied load.
The local maximum principal stress 𝜎𝐼 applied to the particle is given by:
𝜎𝐼= Σ𝐼+ 𝑘(Σ𝑒𝑞− 𝜎0) (1)
Where
Σ
𝐼 is the remote principal stress,Σ
𝑒𝑞 is the remote von Mises effective stress,𝜎
0 is the initial yield strength of the matrix, and𝑘
is a factor that depends on the particle shape and loading orientations [3]. A brittle fracture happens when𝜎
𝐼= 𝜎
𝐼𝑐, where𝜎
𝐼𝑐 is the smallest of the critical stress for particle cracking and particle-matrix decohesion [3].
Through the initiation step if the inclusion separates from the matrix it will create a small spherical void, see Figure 1; which typically can be a factor in ductile fracture while the load applied increases. However, if the inclusion does not separate from the matrix, then increasing the load may conclude to propagate a microcrack in the matrix[1].
1.1.2 Microcrack initiation
Suppose the cracked particle does not separate from the matrix; initiated crack subsequently continues to grow until it will be arrested at the border of particle and matrix shown as carbide/ferrit (c/f) in Figure 1.
In order to pass this border, the crack must reach the local value of fracture toughness KIac/f (carbide/ferrit).
Next, the crack will tear out to the neighboring matrix. This will complete the initiation of a microcrack[3].
Figure 1:Initiation of a microcrack from a particle [3].
KIac/f
KIaf/f
1.1.3 Propagation of the microcrack
After initiation of the microcrack, the crack will grow through the matrix, until it reaches the second barrier;
the grain boundary. Crack can be arrested at grain boundaries and if the load increases, an arrested crack can contribute to ductile fracture. But for cleavage fracture to occur, the crack must propagate freely across the grain boundaries; and for that to happen crack must reach the local value of fracture toughness KIaf/f
(ferrit /ferrit) shown in Figure 1 [3].
As it is known; microcrack starts to grow in a plane requiring least energy, often mentioned as the primary cleavage plane (BCC, {100} in Ferittic steels) [3]. The orientation of the cleavage planes changes by crossing the grain boundaries. But assuming microcracks initiate in neighboring grain along the same plane (100); by increasing the stress, crack in grain 2 can join those cracks in grain 1, as a result, the energy exceeds the grain boundary energy, consequently, grain boundary breaks, so the crack can continue to propagate in grain 2. Figure 2 illustrates the linking process schematically.
Figure 2: The four steps for the crossing of a twist grain boundary by a cleavage crack [3]
In this step, the crack starts to grow, if the stress ahead of the microcrack is enough to overcome the strength of material then the crack will not stop, it will tear through the material. As a result, a phenomenon called cleavage fracture will occur. At low temperatures cleavage fracture can be controlled by the initiation of microcracks, however, by increasing the temperature it is controlled by the propagation of the microcracks cross the grain boundaries, in this condition the cleavage stress is not constant with temperature [3]. Figure 3 demonstrates the cleavage fracture in the form of a flow chart.
Figure 3:A chart of chain of the pre-fracture phenomenon
1.2 Statistical modeling, application to fracture toughness
Depending on how the fracture triggering particles are distributed in the structure; the material's ability to resist fracture, so-called fracture toughness, will be different. Additionally, the applied load can be increased only until an inclusion has enough energy to propagate a microcrack into the macroscopic crack.
As a result, the critical load level for cleavage fracture is reliant on the location of these inclusions.
There are many different statistical models to predict cleavage fracture which all treat cleavage as the weakest link phenomenon. The weakest link method divides the specimen into an infinite amount of minuscule volume elements. The method then indicates that the failure occurs when at least one critical element fails. It is also assumed that the probability of failure of each element is specific and independent of other elements. The probability of failure can finally be inferred from the Poisson distribution as:
𝐹 = 1 − exp(−𝜌𝑉) (2)
Where V is the volume of material, and 𝜌 is critical particle per unit volume. Probability later should be integrated over individual volume elements
𝐹 = 1 − exp (− ∫ 𝜌𝑑𝑉) (3)
Among different models presented for the cumulative probability of failure by cleavage fracture, a model is proposed in the work done by Kroon and Faleskog [4]. In this model, the failure probability is defined as:
𝑃𝑓 = 1 − exp (− ∫ ℎ(𝜀𝑒𝑝, 𝜎̅̅̅)1 𝑑𝑉 𝑉0
𝑉
) (4)
Inclusion fractures Inclusion does not
fracture
𝜎 ≥ 𝜎𝐼𝑐
Crack initiates
𝜎 < 𝜎𝐼𝐶
Inclusion might separate from the matrix creating a void
Microcrack arrests and creates a void
Microcrack propagates through the material and cleavage fracture occurs
Stress Applied to structure
Where h is a function that includes the effects of triggering particles and the critical load on the cleavage fracture. They describe function h as:
ℎ(𝜖𝑒𝑝, 𝜎̅̅̅) = ℎ1 1(𝜖𝑒𝑝)ℎ2(𝜎̅̅̅) 1 (5)
Where the first part, h1 describes the initiation of the microcracks. This phenomenon is strain driven. In this model, the number of triggering particles is presented as a function of linear dependence on 𝜖𝑒𝑝 in the material and gives a dependence on the effective plastic strain.
ℎ1(𝜖𝑒𝑝) = 𝑐𝜖𝑒𝑝 (6)
Thus, h1 is evaluated locally, where c is a material parameter.
To include the propagation of the macrocrack from a microcrack it is needed to evaluate the non-local stress on a sufficiently large volume, considered as the mean value of the local stress in a spherical material volume, VL, with its center at x (see Figure 5). Non-local stress is evaluated as:
𝜎̅(𝑥) = 1
𝑉𝐿∫ 𝜎 (𝑥 − 𝑥̂) 𝑑𝑉̂ (7)
Figure 4: 𝝈̅ is the mean value of 𝝈 over a spherical volume with radius L, and with its center at {x,y,z} [4]
The main difference between Kroon and Faleskog's model with the earlier models is that the current model has an advantage of which, in lower stresses, failure probability remains close to zero [4]. Other advantages of the model are that microcrack size distribution employed as exponential distribution in the model and the dependents of the number of microcracks on the strain level is been accounted for. The model is validated by experiments [5].
As mentioned in section 1.2, a microcrack must propagate across the grain boundaries to get cleavage fracture. Cleavage planes will significantly differ in orientation across a boundary. These changes in orientations will affect the toughness locally. As a result, a cleavage fracture model based on the maximum principal stress is not necessarily the most appropriated one [6]. The random function ℎ2(𝜎̅̅̅) is taken as: 1
ℎ2(𝜎̅) = 𝑒𝑥𝑝 (− (𝜂𝜎𝑡ℎ
𝜎̅ )
2
) − exp(−𝜂2) for 𝜎̅ > 𝜎𝑡ℎ (8)
Where 𝜎𝑡ℎ and 𝜂 are the material parameters and as 𝜂 is related to the microcrack distribution, it is taken to be unity. The model introduced by Kroon and Faleskog has been developed further by Boåsen et al. (2019).
Boåsen in this model takes these orientations, and the effect of orientation changes on the toughness into account. To include the orientations, Boåsen introduces a volume average tensor according to:
𝜎̅𝐼𝐽(𝑥) = 1
𝑉𝐿∫ 𝜎𝑖𝑗 (𝑋̂ − 𝑋𝐾 𝑘) 𝑑𝑉̂ (9)
Where X=(X1 , X2 , X3 ) are the coordinates of the center of volume VL. Boåsen, assumes normal stress acting on a plane defined by the polar angle ∅ and the angle 𝜃 from principal stress system as illustrated in Figure 5, is expressed as:
𝜎𝑛
̅̅̅(𝜙, 𝜃) = 𝜎̅ cos𝐼 2𝜙 + 𝜎̅̅̅̅cos𝐼𝐼 2𝜃sin2𝜙 + 𝜎̅̅̅̅̅sin𝐼𝐼𝐼 2𝜃sin2𝜙 (10)
Figure 5: A non-local stress tensor 𝝈̅ is integrated over a spherical volume with radius 𝑳 from which the principal stresses are calculated. [6]
The non-local stress can be recalculated as:
𝜎̅ =(𝑛 + 1)𝜎̅ + 𝜎𝐼 ̅̅̅ + 𝜎𝑖𝑖 ̅̅̅̅̅𝐼𝐼𝐼
𝑛 + 3 (11)
An illustration of the effect of parameter 𝑛 on how effective normal stress changes is shown in Figure 6.
Based on three orthogonal cleavage planes, the orientation of the most critical plane with reference to the Figure 5 can due to symmetry reasons be represented by 𝜙 = (0,𝜋4) and 𝜃 = (0,𝜋2). The average normal stress on the most critical cleavage plane and comparing it to stress measure in equation (11) results in a value of 𝑛 = 4.57. [6]
Figure 6:Variation of the effective normal stress with the parameter n, mean stress is obtained when n=0 and the maximum stress when n ∞ [6]
1.3 WPS
A minimum critical stress 𝜎𝐼𝑐 for a macrocrack to propagate across the grain boundaries, is sensitive to temperature adjustments. An increase in temperature increases the stress required to crack a particle.
Therefore, one can divide the fracture of ferritic steels to different temperature regions. Brittle, ductile and the transition region, which is respectively corresponding to low temperature and low fracture toughness;
high temperature and high fracture toughness and the region in between.
Warm pre-stressing is a phenomenon that enhances the load to fracture in materials that show lower shelf cleavage fracture. WPS is an advantage of an increase of the fracture load in the lower shelf cleavage fracture, which is a result of the prior tensile loading of a cracked object at the ductile region. See Figure 7.
Figure 7: Warm pre-stressing process [7]
1.3.1 WPS cycles
During a load, unload, cool and fracture (LUCF) cycle as shown in Figure 7, the specimen is pre-loaded, in the ductile region, to a stress intensity factor, which is higher than the virgin materials fracture toughness, KIc, at the brittle lower shelf. The specimen is then completely unloaded. During unloading, compressive residual stress will generate around the crack tip. The specimen is then cooled at a zero load to the brittle lower shelf, it is then reloaded until a fracture occurs. The load applied during reloading reduces the compressive RS around the crack tip before developing a crack tip opening.
As for the Load, cool, fracture (LCF) cycle shown in Figure 7, the specimen is cooled while the preload is kept constant, undesired thermal stress acting with constant load can make the process more difficult [7].
Studies suggest that the LCF cycle usually provides the most enhance in fracture toughness [8].
Other than the processes mentioned above, different load and temperature scenarios, such as partial unloading to lower fracture toughness before reloading to fracture (LPUCF); is used in which the fracture behavior can be limited to the mentioned two cycles [9]. However, the two mentioned mechanisms LUCF and LCF will be the focus of this study.
1.3.2 The effect of WPS on cleavage fracture
There have been many studies done to explain the effects of WPS on the improvement of material behavior [2], [7], [9], [10]. Four principal reasons to describe the enhancement in load to fracture due to WPS are explained briefly in this section[16].
• Deactivation of the inclusions: As pre-loading occurs in the ductile region, inclusions either break or separate from the matrix, those that separate from the matrix can only cause a ductile fracture.
However, broken particles will initiate microcracks in the matrix material. Since loading occurs in the ductile region, the initiated microcrack will later be arrested at the nearby grain boundaries.
Arrested microcracks would create only voids and therefore can only cause a ductile fracture.
Therefore inclusions that break or separate are deactivated.
• Blunting of the crack tip: Pre-loading is done in the ductile region which makes it possible to subject the specimen to a higher load without experiencing structural collapse. This increases the plastic area with high plastic strains near the crack tip. The plastic strains will blunt the sharp crack tip and the theoretical stress singularity will be lost. Stress field around the crack tip will reduce due to larger plastic region and therefore the area in which the cleavage fracture would happen will be smaller, and as a result, the probability of cleavage fracture reduces.
• Closing residual stress field: After loading and unloading performed in the ductile region, there will be a plastic zone around the crack tip. After the unloading, the material will attempt to return to its origin but the deformed area around crack tip will not return to its undeformed state.
Therefore, it will be compressed by the surrounding material. As a result, there will be a compressive residual stress field around the macroscopic crack tip, which remains unchanged during cooling. Later, while reloading in the lower shelf region, a higher load is required to compensate for the residual compressive stresses. That can be explained by the superposition of the stresses. By lowering the stress field, the region that contains high stress is reduced and thus the probability of fracture is reduced as well.
• Increase in yield strength due to decreased temperature: The yield strength of the material increases while decreasing the temperature. As a result, the specimen experiences more plastic strains at the crack tip while loading in the ductile region. Plastic strains cause a lower and smoother stress field.
Thus specimen loaded in the ductile region experiences a lower stress field than if it was directly loaded in the brittle region.
1.4 Welding residual stresses
Welding is a process used to improve the manufacturing efficiency, despite this advantage, it will produce a significant amount of residual stresses that can affect load carrying capacity and fatigue- crack growth rate of the structure. During welding, a variety of phenomena happens most importantly are steps that produce residual stresses (RS). The difference between thermal expansion and contraction in the weld metal and parent material causes longitudinal and transverse residual stresses causing compressive and tensile zones within the weld, Figure 8, illustrates the chain of steps. For a better understanding, a brief explanation of the creation of RS follows. Immediately after welding the weld and the heat-affected zone will be extremely hot; however, the temperature according to the welding process will differ. After the temperature decreases by time, the area that has been hot contracts; but the parent material that has been jointed to the weld material intends to remain as before since it did not undergo temperature alteration as the shrinking part. So, the parent material resists the contraction which creates a tensile stress field near the weld. These stresses remain after cooling and build a RS field in the area. The RSs remain even after post-weld heat treatments in which the welded part is heated to its lower critical transformation temperature for a given amount of time to reduce the stresses to an acceptable level but in reality, structures are highly constrained, therefore they endure high residual stresses up to material yield strength magnitude specially near the weld.
[11]
Figure 8: An illustration of the initiation of residual stresses after welding [12]
Figure 9: As the distance from the weld line center increases, the RSs decrease gradually until they change into compressive stress
1.4.1 Residual Stress Effects on Brittle Fracture in Welded Structures
RSs overtop the effects of stresses due to loading on the crack tip stresses. As a result, dependent on whether they cause crack opening or closure, one can say RSs can either hinder or boost the fracture initiation. RSs affect the crack growth which can increase the instability of the crack growth [13]. As mentioned earlier structure endures compression and tension RSs dependent on the location of the observation, one can divide the structure into two regions, A and B as shown in Figure (9). It is rare to observe cracks in region A due to compression stresses, but there will be axial cracks in region B. In this region due to effect of weld in tension, propagation of axial cracks is faster and therefore they can hinder to onset of circumferential cracks [14]. In this study a residual stress field is modeled that obtains residual tensile stress at the crack deepest point in order to investigate how it affects the advantageous of WPS cycles.
A B
Longitudinal residual stresses
Transversal residual stresses
2 NUMERICAL ANALYSIS 2.1 Geometry and meshing
The aim of this study is to model a RS field that can represent welding RSs, in this section the geometry used, and meshing method is explained. To model the specimen, a geometry shown in Figure 10 is used.
Knowing that the spread of the RS field in advance of the crack front depends mainly on the geometric parameters, placement of pre-load and crack depth is chosen in a way that presented geometry maximizes the zone with tensile residual stress [15]. All geometric parameters of the specimen are parametrized and depeneds on W, the width of the specimen.
Figure 10: Geometry of the test specimen [15]
According to the geometry shown in Figure 10, A finite element model was created using ABAQUS software, a quarter of the total geometry was modeled with appropriate symmetry boundary conditions. The model contains hexahedral 8-node elements using reduced integration, ABAQUS element type C3D8R.
The crack was modeled by removing the symmetry boundary condition along the crack surface. An example of mesh is shown in Figure 11, which consists of 51841 elements, with 20 elements in the thickness direction with a bias towards the free end of the model. Finite strains were not accounted for in the analysis.
2.2 A model to combine residual stress with WPS
In this study, RSs are simulated by applying in-plane compression of the 3D FE model. Subsequently, a crack is introduced by changing boundary conditions while unloading the specimen. An illustration of loading history is shown in Figure 12. Loading has been chosen in this way to illustrate the RS field of the welded structure. After unloading prior to the WPS process, a RS field as shown in Figure 13 has been introduced in the specimen.
Figure 12: Compression that obtains the residual stress field
𝑊 𝑎
𝑊/2
R
5𝑊 𝛿Figure 11:An example of the mesh used in quarter symmetric FE model of specimen
Figure 13: Residual stress distribution by different indentations
The in-plane compression of the specimen leads to a stress field with compressive stresses and tension stresses normal to the crack surface during the loading, in order to obtain plastic deformation, Load Line Displacement( 𝛿𝐿𝐿𝐷) should be large enough. After unloading, a residual stress field is formed due to the plastic deformation of the material. Figure 13 shows changes in the residual stress field dependent on the magnitude of pre-loading.
After the in-plane compression that creates a desired residual stress field have been applied, LCF and LCUF processes are simulated, as shown in Figure 14.
Figure 14:Application of the pre-loading and WPS
𝑊 𝑎
𝑊/2
4𝑊 𝛿𝐿𝐿𝐷 5𝑊
a)
R
2.3 Load cycles
Initially, the specimen experiences an in-plane compressive pre-load to introduce the desired residual stress field as shown in Figure 15. The desired stress field was introduced by a compressive force of 62829 [N]
to illustrate a welding RS field.
Figure 15: Imposed residual stresses as initial stresses prior to WPS
LCF and LUCF cycles are done with two different pre-load levels at a temperature of 20ºC, and before being reloaded at temperature -150 ºC.
An illustration of the LUCF and LCF cycles is shown at the Figure 16 an 17 respectively.
Figure 16:A schematic view of LUCF cycles
𝑇 ሾ°𝐶ሿ 𝐽
ሾ𝑘𝑁/𝑚ሿ
107.5
25.2
−150 20
Figure 17:A schematic view of LCF cycles
As the material response is linear, using the small-scale yielding assumption, the 𝐽 -integral can be related to the stress intensity factor as 𝐽 =𝐾𝐼2
𝐸′, in case of rate-independent quasi-static fracture analysis. The load levels 𝐽 = 25.2𝑘𝑁𝑚 and 𝐽 = 107.5𝑘𝑁𝑚 in figures above are calculated from the allowed 𝐾𝐼 for level A/B and level C/D load cases using the Swedish procedure of assessing defects in Reactor Pressure Vessel[16].
RS-Load-Cool-Fracture(RSLCF) and RS-Load-Unload-Cool-Fracture(RSLUCF) cycles are modeled using two different methods to calculate the proper pre-load from the load level A/B and C/D in the RPV at a temperature of 20ºC, and before being reloaded at temperature -150 ºC.
The first method is to pre-load the RSLU with an Load Line Displacement that obtains the same applied load while preloading the LU, see Figures 18 and 19, because the forces are equal, in this study this method is mentioned as ‘Equal P’ method.
Figure 18:Reaction force at indentation point of the FE model for LU with maximum load 41.4[kN] and RSLU with maximum reaction force of 41.4[kN], load level A/B
𝑇 ሾ°𝐶ሿ 𝐽
ሾ𝑘𝑁/𝑚ሿ
107.5
25.2
−150 20
Figure 19: Applied load at indentation point of the FE model for LU with maximum load 57.9[kN] and RSLU with maximum reaction force of 57.7[kN], load level C/D
The next method used is choosing the pre-load is in such a way that the contribution of residual stresses to the 𝐽 -integral is taken to account in order to obtain equal crack tip conditions. In this method, an 𝛿𝐿𝐿𝐷 is used that by which the value of the 𝐽 -integral obtained for the case with RSs, RSLCF, and RSLUCF, named 𝐽 -modified in Figure 20; will by equal to 𝐽 -Integral for the case without RS, LCF and LUCF, shown in Figure 20. In this study, the method is mentioned as the ‘Jmodified’ method.
Figure 21: J-integral considering the contributions of residual stress field.
C/D
A/B
Figure 20:
𝑱-integral considering the contributions of residual stress field.
2.4 Material
The different parameters for the material properties of 18MND5 steel used in ABAQUS are shown in Figure 22 [16].
Figure 22:An illustration of the material curves described in Tabel 2 in Appendix 2.5 Post-processor
A post-processing script developed by KIWA [16], based on the model presented by M.Kroon and Faleskog [4], is used to generate a cumulative distribution function using data generated after the FE simulations are done. A volume average of the maximum principal stress over a sphere-shaped area with volume 𝑉𝐿 with radius 𝐿, see Figure 5 [6]. Equivalent plastic strain and the coordinates were evaluated at the centroid of each element since the increased elemental density assumed to be small. This assumption decreases the computational time. Another reason why the undeformed value of 𝑉0 is chosen will be that in the calibration part only value that changes is 𝑐, therefore it is reasonable to choose 𝑉0, as an arbitrary value that will be used only to give the 𝑑𝑃𝑓 the right dimension.
Equation (7), is discretized as in equation (11)
𝜎̅𝑖𝑗=∑ 𝜎𝑁𝐽 𝑖𝑗ΔV
∑ ΔV𝑁𝐽 , { √(𝑥𝑖− 𝑥𝑗)2+ (𝑦𝑖− 𝑦𝑗)2+ (𝑧𝑖− 𝑧𝑗)2≤ 𝐿} (12)
Where 𝑥𝑗, 𝑦𝑗 and 𝑧𝑗 are the undeformed centroid coordinates for element 𝐽. 𝑁 is the number of elements within VL.
T= -150°C T= 20°C
2.6 Calibration
The material models used in the probabilistic model in this project were calibrated using the experimental results from an on-going project at Kiwa Inspecta. Data obtained during the calibration process which are later used in this study are shown in Table 1 . The purpose of calibration is to choose the best combination of 𝐿, 𝜎𝑡ℎ that yields the smallest deviation or error 𝑅𝐿𝑠. Where 𝐿 is the radius of the sphere-shaped area in Figure 5, 𝜎𝑡ℎ is mentioned in equation (8). These parameters are used as material parameters in the post- processor script.
To calibrate the model, data from two sets of experiments are used, experiments are a part of an ongoing project at Kiwa Inspecta. A CT specimen with 𝑎/𝑤 = 0.55 generates a High constraint experiment data and one set of Low constraint data obtained from a three-point specimen with 𝑎/𝑤 = 0.1. In this study, a Weibull distribution used to be fitted to the experimental data.
The value of 𝑐 represented in equation (6), was calculated using an iterative numerical method based on the golden ratio for several combinations of L and 𝜎𝑡ℎ . The reason is that the golden ratio finds the global minimum error. Because two sets of experiments are used there might be a local minimum because each set has its own optimal value of 𝑐, but the local minimum will be so close to the global minimum, therefore this method will be good enough.
Table 1: Material parameters used in obtaining different cumulative distribution functions in this study.
L 𝜎𝑡ℎ [Mpa] c Error 𝑅𝐿𝑠
0 1265 3513.37 0.23
0.05 1228 4839.93 0.03
0.1 1192 8221.95 0.25
0.15 1162 17556.5 0.71
0.2 1132 30525.2 1.05
According to results parameters obtaining the minimum error of 0.03, are used in postprocessing script to calculate the probabilities of fracture.
3 RESULTS AND DISCUSSION
As mentioned earlier, the fracture specimen with a specified mesh configuration in the crack region experienced a compressive loading in order to introduce RS that represent welding residual stresses.
RS is imposed as initial stresses, subsequently, the crack was introduced. After introducing the crack, a RS field exists at the crack tip. See Figure 23.
Figure 23:RSs after introducing the crack
A series of analyses were done for the As Received (AR), RSLCF, RSLUCF, LCF and LUCF cycles with the mentioned magnitude of preloading. The results of all analyses were used to calculate the cumulative probability of fracture using the non-local approach model. Probability distribution predictions are shown in Figures 24, 25, 26 and 27.
Figure 24:Failure probability predictions for AR, LCF and RSLCF cycles of the 3D FE model, pre-load level A/B, at -150°C
Figure 25:Failure probability predictions for AR, LCF and RSLCF cycles of the 3D FE model, pre-load level C/D, at -150°C
Figure 26:Failure probability predictions for AR, LUCF and RSLUCF cycles of the 3D FE model, preload level A/B, at -150°C
Figure 27:Failure probability predictions for AR, LUCF and RSLUCF cycles of the 3D FE model, preload level C/D, cracked at -150°C
The WPS procedure introduces a significant amount of compressive RS at crack tip as mentioned in section 1.3.2. Comparing the RS filed with RSLU concludes that with a combination of Load-Unload and RS, a compressive stress field will still exist. See Figures 28 and 29.
Figure 28:Opening RS ahead of the crack at 20°C, preload level A/B
Figure 29:Opening RS ahead of the crack at 20°C, preload level C/D
Opening Residual stresses, in front of the crack tip, for RSCF, AR, LUCF, and RSLUCF cycles; at a constant fracture load corresponding to 50% fracture probability of AR are plotted in Figures 30 and 31.
These figures give a better understanding of the influences of RS and WPS on stresses ahead of the crack tip.
Figure 30:Opening mode stresses on the front of crack at the deepest point for various analyses at -150°C and a fracture load corresponding to 27.78[kN], Level A/B
Figure 31: Opening mode stresses on the front of crack at the deepest point for various analyses at -150°C and a fracture load corresponding to 27.78[kN], Level C/D
From results in Figure 30 and 31, it can be concluded that the lowest and the highest level of opening stresses observed at LUCF and RSCF respectively. This explains why LUCF and RSCF have a higher and lower force at a specific probability of fracture, respectively. See Figures 26 and 27.
However, the ‘deactivation mechanism’ as a benefit of the WPS, is not taken to account when deciding the ℎ1.
4 CONCLUSIONS
A series of analyses are performed to study the effects of RS representing welding, WPS cycles and the interaction of them on cleavage fracture. The preformed analyses indicated that:
• RSs have a significant influence on the fracture probability, in cases with equal crack tip conditions with respect to equal J-integral for pre-load level A/B. For cases with RSs, the fracture load corresponding to 50% fracture probability are 22 and 32kN lower for LUCF and LCF respectively.
Fracture load corresponding to 50% fracture probability without RSs is 37 and 50kN respectively.
• RSs have no influence on increasing the fracture probability, when choosing the ‘Equal force’
method (equal external force) on pre-load level C/D.
• The “change of yield strength” mechanism is the most advantageous mechanism.
• The results clearly show the importance of correctly considering RSs when utilizing the WPS.
5 ACKNOWLEDGMENT
Professor J.Faleskog and T.Bolinder are acknowledged for their helpful insights through the master thesis. I would like to thank Kiwa Inspecta for supporting this study. Finally, A. Eriksson is
acknowledged for providing the post-processing script used in this study.
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7 APPENDIX
The different parameters for the material properties of 18MND5 steel, used in ABAQUS are given in Table 2 below [16]
Table 2: Material properties used in FEM analysis
E [GPa] 𝝂 Temperature
211.1 0.3 -150
203.3 0.3 20
𝝈 [MPa] 𝜖𝑃𝑙𝑎𝑠𝑡𝑖𝑐 Temperature
765 0.0000 -150
766 0.0015 -150
766 0.0065 -150
767 0.0172 -150
820 0.0361 -150
875 0.0559 -150
958 0.0955 -150
994 0.1203 -150
1019 0.1402 -150
517 0.0000 20
518 0.0025 20
519 0.0075 20
566 0.0172 20
633 0.0369 20
675 0.0567 20
703 0.0770 20
723 0.0965 20
736 0.1164 20
753 0.1463 20
