IN
DEGREE PROJECT MECHANICAL ENGINEERING, SECOND CYCLE, 30 CREDITS
,
STOCKHOLM SWEDEN 2017
Estimation of Fatigue Life for
Welded Gears Using LEFM
RICHARD TYNANDER
Estimation of Fatigue Life for Welded Gears
Using LEFM
Richard Tynander
Degree project in Solid Mechanics Second level, 30.0 HEC Stockholm, Sweden 2017
Abstract
The aim of this thesis is to estimate the high cycle fatigue life of welds using finite element methods and liner elastic fracture mechanics (LEFM). Develop a new method of calculating the fatigue life that can replace the old method in use today at GKN Driveline. This is a continuation of an earlier master thesis1. When gears have been subjected to high cycle fatigue testes, cracks have been observed propagating though the welds and resulted in component failure. The current method of estimating the welds fatigue life has a tendency to overestimate the life and thus a new method with better accuracy is sought after.
The crack propagation is simulated by starting with an initial crack in the weld which is then incrementally increased in a pre-determined direction. The energy release rate for each node at the crack tip is calculated using virtual crack closure technique (VCCT). Due to multiaxial loads on the weld, an effective stress intensity factor is used together with a Paris Law type equation to estimate the propagation speed. The total amount of cycles it takes the crack to propagate between each of the increments is then estimated.
The results show a good correlation to experimental data and takes approximately 40% longer time compared to the old method. The new method has a high dependency on good material parameters and thus important that these are chosen accurately, which can be especially hard for welds. The results can also be linearized which means that the life can be estimated for different load levels from one single simulation.
1
Uppskattning av utmattningslivslängden för
svetsade kugghjul med hjälp av LEFM
Richard Tynander
Examensarbete i Hållfasthetslära Avancerad nivå, 30 hp Stockholm, Sverige 2017
Sammanfattning
Målet med detta arbete har varit att uppskatta utmattningslivslängden för svetsar genom finita elementmetoder och linjär elastisk brottmekanik (LEFM). Skapa en ny metod för att uppskatta livslängden som ska ersätta den nuvarande metoden som just nu används hos GKN Driveline. Detta arbete bygger vidare på ett tidigare examensarbete2 som gjorde hos GKN. I högcykel utmattningsprov har det observerats att sprickor bildas i svetsen som sedan växer tills det ger upphov till ett haveri. Den nuvarande metoden för att uppskatta svetsarnas livslängd har en tendens att överskatta livslängden, vilket har gjort att GKN söker efter en ny metod för att räkna livslängden på deras svetsar.
Sprickpropageringen simuleras genom att börja med en startspricka i svetsen som sedan stegvis ökas i storlekt i en förutbestämd riktning. Energi frigörelsen för varje nod längs sprickspetsen beräknas med hjälp av virtuell sprickförslutningsteknik (VCCT). Svetsarna utsätts för fleraxliga laster vilket gör det nödvändigt att använda en effektiv spänningsintensitetsfaktor tillsammans med en Paris Lag liknande ekvation. Med hjälp av detta kan sedan sprickhastigheten mellan de olika stegen beräknas.
Resultaten visar god korrelation mot gjorda test samt att den nya metoden tar cirka 40% längre tid än den nuvarande metoden. Den nya metoden är också väldigt beroende på att materialparametrarna, vilket kan vara svåra att erhålla för svetsat gods. Resultaten kan linjäriseras, vilket betyder att olika last nivåer kan undersökas från en simulering.
2
Foreword
Table of content
1 Introduction ... 1 1.1 Nomenclature ... 4 2 Theory ... 5 2.1 Manufacturing ... 5 2.2 Fatigue ... 52.3 Effective Notch Method ... 6
2.4 LEFM ... 7 2.5 Paris law ... 9 2.6 J-integral ... 11 2.7 VCCT ... 12 3 Modeling ... 15 3.1 General ... 15 3.2 Boundary Condition ... 16 3.3 Different Approaches ... 19 3.3.1 2D automatic remeshing ... 19
3.3.2 Automatic crack propagation with remeshing ... 20
3.3.3 Element separation ... 21
3.3.4 ΔK-method ... 23
4 Experiments ... 26
5 Material and Constants ... 28
6 ΔK-Method ... 29
6.1 Method ... 29
6.2 Validation ... 31
7 Results ... 32
7.1 Validation of LEFM ... 32
7.2 Linear approximation of ΔKeff ... 32
7.3 Life time approximation ... 33
7.4 Estimation of time ... 35
8 Discussion ... 36
9 Summary ... 38
1
1 Introduction
This thesis was written at GKN Driveline in Köping, Sweden. GKN is a global company with a focus in four major areas, automotive (Driveline), aerospace, land systems and powder metallurgy. The factory plant in Köping has a lot of experience in developing and producing different units in the driveline of passenger cars.
At the Köping factory GKN Driveline manufactures components for AWD (All Wheel Drive), it is basically a four wheel drive but the torque can be adjusted on specific wheels which will in turn give better performance for the car. GKN is selling automotive components for almost all major car manufactures, such as Volvo, Ford and Volkswagen. The two big drive components in the AWD are PTU (Power Transfer Unit) and the RDU (Rear Drive Unit), see figure (1).
Figure 1: Consept of the drive line.
The PTU is connected to the gearbox and distributes the torque to the front wheels and to the propeshaft, which in turn is connected to the RDU, see figure (1). The RDU then distributes the power to the rear drive shafts.
2
Figure 2: The housing of the PTU is open to show interactions of the gears.
3
Crack propagation in structures is a very complex area, because there are a lot of different factors that contribute to the crack propagation. At the moment when GKN Driveline is calculating if the welds will meet the requirements they use the effective notch method (ENM). This method is well-documented and GKN Driveline has a clear standard and procedure on how this should be conducted. The result from ENM does not have the best correlation against HCF tests so GKN are currently looking at new ways of determining the fatigue life of the welds to lower the safety factor and get more design freedom.
The aim of the thesis work was to try and establish a new method for simulating the crack propagation in welds, in order to get a better estimation of the welds fatigue life. To reach the goal, four different ways of simulating the crack propagation was investigated:
2D automatic remeshing
3D automatic remeshing
Element separation
ΔK-method
4
1.1 Nomenclature
Crack length
Constant in Paris Law
Damage from Palmgren-Miner Damage calculation Young’s modulus (plane stress)
Young’s modulus (plane strain) Fat-value for the stress
Energy release rate
Energy release rate for mode I, II & III Vickers pyramid number (hardness)
J-integral
Cyclic hardening coefficient
Stress Intensity factor for mode I, II & III Critical stress intensity factor
Critical stress intensity factor for mode I Exponent in Paris Law
Exponent in effective notch calculation
Number of surviving cycles at specific Fat-value Cyclic hardening exponent
Amount of cycles from Paris Law integration Amount of cycles for incremental crack increase
Expected amount of cycles to failure
Total amount of cycles to failure Safety factor constant
Applied pinion torque Potential energy J-integral path
Stress Intensity ranges for mode I, II & III Effective stress intensity factor range
Threshold value for crack propagation Maximum principal stress range
Friction coefficient Poisson's ratio
Cyclic Yield strength (0,2%)
Yield stress for uniaxial cyclic fatigue loading
5
2 Theory
2.1 Manufacturing
The crown wheel and the tube shaft are both created out of steel, the gear (crown wheel) is then heat treated to achieve desired material properties. The ring gear is press fitted to the tubular shaft and then welded with a laser weld, how the weld is done depends on the specific PTU model. The welding is done with a laser beam that heats the material of the ring gear and the tube shaft so they are melted together. The laser has a fixed position so the component is rotated around and in the process adds no material, it just fuses the metals together. After the welding process the welds are tested with ultrasound, the PTU is sunken down in a medium of water and then tested. The test is done to confirm that the weld has been done in a correct way. The ultrasound test has the ability to detect cracks bigger than . Before the start of each new production one welded component is first cut and inspected to see how weld performed, if the weld meets all requirements the production starts.
2.2 Fatigue
6
Figure 4: The Whöler curve shows how many cycles a component can endure at each stress level until failure occures. On the horisontal axsis is the amount of cycles and the vertical is the applied stress, [1].
Creating a Whöler curve is straight forward, the component or material is loaded with a constant amplitude load until failure occurs. The amount of cycles it could endure before failure and applied load is then plotted in a graph. Then the same procedure is repeated for a wide variety of loads, when enough data points are obtained a line can be drawn between them to create the Whöler curve, see figure (4). The Whöler curve usually takes a lot of time and effort to create, so ways of calculating the curve is sought after.
2.3 Effective Notch Method
The effective notch method as mentioned earlier is the current method in use at GKN Driveline. This method is a common way to determine the fatigue life of welds in a structure and this is done by modeling a notch at the position of the weld. The notch size and dimensions in determined from a standard [2] and then calculated using finite element. The theory is that the small notch (i.e. key hole) will account for the troublesome defects that exist in a weld. After the notch has been inserted, the model is meshed and the maximum and minimum principal stresses are evaluated around the notch. With help of the maximum and minimum stress around the notch, an average for the maximum principle stress range ( ) is calculated. The cycles to failure can be calculated using following equation.
(
)
7
Where are material/component dependent parameters which are chosen according to [2]. If and , it implies that the structure will survive 2 million cycles at [ ], these values depends on the weld and material of the component. The values are determined empirically from fatigue tests that have previously been done by [2]. is a constant that is derived from statistical evaluations from earlier data extracted by [2]. The values of varies between , depending on what probability of failure the calculations are supposed to evaluate, more values can be seen in [15, Table 1]. The component is then loaded at an arbitrary load level because the stresses is assumed to have a linear correlation against applied load. Then the stresses for the component can be approximated for every load in a load collective, then a Palmgren-Miner damage calculation [3] can be calculated to estimate the damage for that load collective,
∑
The total damage is and is the value calculated from equation at that particular load level and is the amount of revolutions it supposed to endure according to the load collective.
2.4 LEFM
Linear Elastic Fracture Mechanics (LEFM) is a way to calculate the crack growth in a structure with a linear approximation but it is more complex and computational demanding process than the effective notch. In elastic materials the stresses at a crack tip should be infinite, but this is off course not the case, then all structures with cracks fracture at an instant. But if the material is elastic-plastic like a metal [21], the area in front of the crack will experience plasticity, which will give rise to lower stresses. In figure (5) the characteristic stress distribution is shown and the plastic zone.
8
The stresses in the vicinity of the crack are assumed to depend on three major factors: stress, crack length and geometry. The LEFM way of dealing with small scale yielding is to calculate the stress intensity factor (SIF), usually denoted as .
Irwin [4] showed that a crack can be loaded in mainly three different ways, these are called mode I, II & III.
Figure 6: Illustration of the different stress intensity modes, [5].
Mode I are usually the worst way a crack can be loaded, this is loading in pure tensile opening and the stress intensity factor for this case is called . Modus and are not assumed to have the same effect on the cracks as mode , also fatigue cracks tend to grow in a direction of mode . Modus is sliding in the crack plane and modus is sliding perpendicular to the crack plane, see figure (6).
Here is an example of the stress intensity factors for a 2D case [6],
√
√
√
9
Figure 7: Crack in which equation is intended for.
If the stress intensity factor for a crack reaches a certain value the Fracture Toughness ( ) the crack will grow in an unstable way and give complete failure. The Fracture Toughness is an experimentally determined material parameter. For simple cases with uniaxial loading there are handbook solutions for calculating the SIF, but for more complex situations the use of finite element method is needed. The use of LEFM is valid when the so called small scale yielding is small compared to the characteristic dimensions of the crack body, and as a rule of thumb the ASTM E399 [7] condition can be used,
( )
Where is the smallest characterstic length of the body and is the yield stress for the material. If this condition is fulfilled it means that the plastic zone is roughly 20 times smaller than the smallest characteristic length.
2.5 Paris law
The crack propagation can be split up into three main stages, stage 1 is the crack initiation. In stage 1 the crack is initiated at the microscopic level and starts to grow to a macro crack. In stage 2 the crack has a stable propagation and stage 3 is the final rapture. It is in stage 3 the crack growth is unstable and failure will occur.
10
Paris law is a way of calculating the crack propagation in stage 2, it was first described by Paris in 1960’s [8]. The Paris power law calculates the length of crack propagation against the cycle increment, with the help of the stress intensity factor. The Paris law was first only valid for crack propagation in mode and it assumes a linear propagation. The basic form of Paris law is,
Here is the stress intensity range in the fatigue cycle. and are material parameters and they describe the properties of the crack propagation and these needs to be experimentally determined. is the crack increment and is the cycle increment. By doing an integration of equation the number of fatigue cycles for a crack that propagates from to can be determined as,
∫
Here and are the different crack lengths. is assumed to be which means that the stress intensity factor is above the threshold values for crack propagation and below the critical stress intensity factor . The stress intensity range is the range where the crack is loaded, and is calculated as,
For the threshold value is the value of the SIF when the crack propagation can be expected to happen.
11
Figure 9: Illustartion of the small plastic zoone.
In order to ensure this, a requirement that closely resembles that of the ASTM requirement is used [9],
(
)
In equation (2.10) is the smallest characteristic length (i.e. crack length, thickness or ligament) and is the yield stress for cyclic loading.
There are also modified growth laws, one was suggested by Forman [10]: 2.6 J-integral
12 Where is the energy release rate and is the crack and is the potential energy of the system, is the thickness and depends if it is an edge crack or inner crack.
The J-integral was introduced by Rice [13] and is a method of calculating the amount of energy that is traveling to a crack. This integral is path independent as long as it begins and ends at the two sides of the crack. For a 2D case, see figure (10), the integral is then written as,
∫ ( )
where is the path around the crack, and is the deformation work by volume unit,
∫
Figure 10: An arbitrary crack with the J-integral path drawn [6].
For a linear elastic material the J-integral is equal to the energy release rate.
2.7 VCCT
One option to obtain the energy release rate is to use the Virtual Crack Closure Technic (VCCT) [14], this option a general way for obtaining the energy release rates. Energy release rate is often depicted as and it is calculated from the opening displacement and forces that acts on the nodes for the different modes. For the case of a 3D solids in figure (11) the energy release rate can be calculated for the different modes as [14],
13
Where are forces that acts on node and ( ) are the displacements at the different nodes all in the coordinate system displayed in figure (11). is the crack surface area , all forces and displacements are obtained from the finite elements analysis.
Figure 11: Display of how the energy release rates is calculated using VCCT, according to [14], the crack surface is indicated with a slight grey and brown coolor.
From equation (2.17-2.19) the total energy release rate is then the sum of each mode,
(2.20)
The stress intensity factors can be obtained from this relationship,
(2.21)
(2.22)
15
3 Modeling
3.1 General
Here are the different methods of modelling the crack propagation described and evaluated if they are feasible ways of calculating. Four different methods are evaluated and they are, 2D approximation, automatic crack propagation with remeshing, element separation and manual crack propagation ( ).
The PTU that’s investigated here is a PTU in the development and prototype stage. The loads that are acting on the gears are coming from the pinion and theses are varied according to how the car is driven. The weld between crown wheel and tube shaft, see figure (12), is the objective that this thesis.
Figure 12: Picture showing how the crown wheel, tube shaft and pinion is connected, and the welding area.
16
Figure 13: Cross section of an approved weld, the initial gap shown in the picture is due to the welding process.
3.2 Boundary Condition
17
Figure 14: Showing where the boundary conditions are applied for translation and rotation. The light blue elements are elements that have nodes linked to the RBE2’s.
18
To simulate a full revolution, four loads are applied in sequence and the position of the gear mesh point is rotated 90 degrees, see figure (16). The applied gear load for the first force is shown in Table (1), and the other loads have same amplitude but just rotated from that.
Figure 16: Indications of where the forces are applied and in what direction.
Table 1: Applied gear forces at varying pinion torque in x, y and z, the subscript 1 indicates that this is in figure
(16) and the other 3 will same amplitudes but rotated.
[ ] [ ] [ ]
[ ]
[ ]
[ ]
19
3.3 Different Approaches
3.3.1 2D automatic remeshing
The 2D-model was created from a cross section of the PTU. This cross section was then meshed in three different bodies, the crown gear and the tubular shaft with a more coarse mesh and then the weld part with a fine mesh. Quadratic first order elements was used and the load was applied with the use of a RBE2 (Ridged Body Element) on the crowns outer nodes. The model was then bounded in all degrees of freedom at the place of the bearings. The welded part has a glued contact to the crown gear and the tube. The tube and crown gear have a touching connection with friction in between. All connections are established with (node to segment) according to [5].
Figure 17: Corss section of the crown wheel and the tube shaft with boundary condition and RBE2.
20
The crack is initiated in the notch part and is then propagating in a 45° angle outwards towards the boundary, see figure (18).
This is a somewhat feasible way of calculating the crack propagation but due to the difficulties of transferring the multi axial loads from the 3D case to a 2D plane, this was decided not be a good way of solving the crack propagation. Due to the differences would differ too much from the real 3D case. Therefore this method is not further investigated.
3.3.2 Automatic crack propagation with remeshing
The same gear as Figure (12) is meshed using Altair Hypermesh 14. The meshed is split into four different segments crown gear, weld part and the tubular shaft. The welded part is then split up into two separate parts, one with a very fine mesh where the crack will propagate and one with a more coarse mesh. This is due to ease the computational time. The mesh is first order hex-elements except for the fine mesh of the weld which is first order tetra elements.
Figure 19: Display of the finite element model used in calculations, fine and coarse mesh is indicated by arrows.
The element used for the crown gear, tubular shaft and coarse weld is first order hexahedral elements and first order tetrahedral elements for the finely meshed weld part.
21
Figure 20: Arbitrary crack inserted to show how it looks, note that the mesh here is not the mesh used in the VCCT calculation, the mesh will be updated before crack calculations is done, see figure (28).
The cracks are inserted at the root of the weld due to in house experiments at GKN has shown that the root is the initiation place for the cracks, see figure (3). The loads are then applied in sequence with the use of rigid body elements as described earlier. Then a press fit between crown gear and the tube shaft is simulated with the use of an interference fit at the touching elements with a closure of 5µm. The shaft is locked in translation at the bearing locations and in rotation at the splines. The crack properties and remeshing properties is then added in. The contacts between all bodies are glued except for the contact crown gear tube shaft which have touching contact to allow for press fit and friction.
The automatic remeshing process will create a very fine mesh around the crack tip to accurately calculate the energy release rate and after a full fatigue cycle, grow the crack and then remesh it.
Due to the models complexity the automatic remeshing was not able to automatically remesh the body and another way of calculating this was done, see section 3.3.4, K method.
3.3.3 Element separation
22
Figure 21: Display of finite element model used in analysis.
Figure 22: The differense between the sizes of the coarse and fine mesh can be seen.
23
Figure 23: The contact between the two welded surfaces, the crack initiation place is indicated.
The crack is then chosen to propagate forward from that initial state.
This was a very promising way of calculating the crack propagation but due to high complexities in the geometry, the software could not handle it. Similar problems have occurred in [15] that some design did not work properly. After contact with the MSC support service, this problem should be solved in the next update of the program (2017). So this method is put on ice for eventual future work.
3.3.4 ΔK-method
24
The cracks are modeled in Altair HyperMesh and they start as a semi ellipse and then with an incrementally increasing radius of each time. The initial crack has a crack depth of and a width of , it can be seen in figure (24). The cracks are inserted with a tilt of roughly 45° degrees, same tilt as the observed cracks from testing. The cracks are inserted at the weld root in the finely meshed part of the weld. Each new crack has a uniformly increasing area and some of the inserted cracks can be seen below in figure (24-27).
Figure 24: This is the initial crack with a depth of 1 mm. Figure 25: Increment 2, crack depth 1,6 mm.
Figure 26: Increment 9, crack depth 3,4 mm. Figure 27: Increment 15, crack depth 5,2 mm.
25
26
4 Experiments
GKN Driveline is testing the components with both high cycle fatigue and low cycle fatigue before they are put into production. The PTU that’s investigated in this report have been tested until failure at three different pinion torques; and . The tests are conducted with a constant speed and pinion torque until breakage occurred.
Failure in the weld occurred times at torque and 1 time at torque but at other components failed before the weld, so no experimental data at that load level exists. The failure results can be seen in table 2.
Table 2: Results from HFC tests, note that the number of endured cycles are pinion revolutions.
Pinion Torque Number of cycles* Failed component
650 3 450 000 Weld
650 3 250 000 Weld
650 6 255 000 Weld
650 5 310 000 Weld
825 520 000 Weld
* Pinion revolutions, gear ratio between pinion and ring gear is 2.58.
One fractured weld was investigated from the fatigue test and the welded cross section was cut open and polished and photographed, the crack path can be seen in figure
Figure 29: Picture from fractured cross section for fatigue test with a pinion torque of
27
(A) (B)
28
5 Material and Constants
The material of the crown wheel and tube shaft is both steel with a Young’s modulus of and a Poisson’s ratio of . The crown wheel has been heat treated to achieve a harder surface. The chemical composition of the ring gear is displayed below.
The ultimate tensile strength of the weld area is unknown, but an approximation can be done according to [16]
The material parameter is unknown but it can be estimated from the ultimate tensile strength. By inserting the from equation (5.1) into FEMFAT (software) and assuming ‘general structure carbon steel’ behavior. FEMFAT then estimates the cyclic strength coefficient and cyclic hardening exponent to be:
[ ]
[ ]
The cyclic yield strength is not known for the material, but can be estimated from [17] to be
Where and comes from equation (5.2). The value for cyclic yield strength then approximated to be, [ ].
The Paris law parameters are chosen according to [2] recommendations for welds. These are & , for [ √ ]and [ ]. Also recommended in [2] are when there are no reliable data for residual stresses, the threshold value should be chosen to be [ √ ]. Values for the critical stress intensity factor is approximated with regards to similarly materials, to be
[ √ ].
Chemical composition
Component C SI MN P S CR NI MO CU AL
29
6 ΔK-Method
6.1 Method
To ease the computational time only a small section of the weld was investigated with a crack, see figure (19). So the assumption is when this crack has grown through to the outer side, the whole weld has failed. It is assumed that other cracks will grow in the same speed and direction and when one crack reaches the boundary the other has done it as well.
Because the inserted cracks have a uniformly increasing shape, i.e. the shape of the crack is increasing in a uniform way. The average of the stress intensity factors is calculated for each mode along the crack tip. So, for each crack increment there are 4 different values of the three modes. In order to get the stress range , all modes are added together to identify which load is produced the largest sum. When the load that produces the maximum and minimum sum is identified, the ranges of can be calculated. I.e. the maximum is and minimum is . The subscript of the forces tells where the force is applied, see figure (16). The stress range is then calculated according to,
In equations , force 3 is the force that gives the maximum and gives the lowest and thus can be evaluated.
The crack closure effect is not taken into account here, due to this would make the stress intensity range too complex to calculated, so this phenomena is not assumed to have a large effect. In order to be able to estimate this, specific experimentally test would have to be conducted [18].
By having self-contact with friction for the crack surface, the effects of will be taken into account, when the SIF for the other modes are calculated. When the crack is loaded with compression, i.e. , will be reduced due to the friction at the surface of the crack. The friction coefficient is set to be 0,1 because it is friction between steel and steel.
When have been calculated the effective stress intensity factor according to [19],
√
30
where is Poisson’s ratio. The is calculated in order to use a Paris Law type of equation to estimate how many cycles it took between the incremental increase of the cracks. The crack is assumed to propagate linearly between the small increments according to LEFM.
In order to be able to compare the results against a load collective, the needs to be estimated for every crack at different torque levels. The analysis can’t be run at different torque levels due to computational time, so a linear approximation is used to estimate . First one full analysis is done at one torque levels and the is calculated for every crack increment. Then a linear approximation is done for each crack to extract the change of against torque. The assumption for the linear approximation is that the stress intensity factor has a linear change against torque for a fixed crack length.
In order to calculate the amount of cycles it took the crack to propagate one increment, a slightly modified way of calculating between the cracks is used [20],
In equation means that it is the for the first crack ( ) and is for crack , except for when the crack is initiated, then . Then equation (6.5) is inserted into equation (2.7) to estimate the number of cycles it took for the crack to propagate between & ,
∫
( ) Then equation (6.6) is done for all of the incremental growth steps, from to . The total amount of cycles it toke the crack to propagate is the sum of to
, as
Equation is an estimation of the amount of revolutions (cycles) it takes for the crack to propagate from initial crack size to the outer boundary. Equation (6.6 - 6.7) is then done for all torque levels in the load collective. After all calculations are done for the different torque levels, a Palmgren-Miner damage calculation [3] is done to be able to estimate the damage from each torque level on the total life,
∑
31
Where is the amount of revolutions it is supposed to endure at a particular torque level according to the load collective. is the estimation of revolution it survives according equation at that torque level. Then this is calculated for all torque levels from to with a incremental increase of the torque.
6.2 Validation
The use of LEFM is that the plastic zone size is small compared to the rest of the characteristic dimensions. There are no clearly stated rules for when LEFM can be used and especially not when there are multiaxial loads, but for simple uniaxial loads the ASTM criterions can be used, shown in equation . Assuming that in equation will be change to and the ASTM condition then becomes,
(
32
7 Results
7.1 Validation of LEFM
The use of LEFM for this problem is here validated according to equation and parameters according to section . For an initial crack of and a maximum √ which would be the worst case, equation is still satisfied, . The crack that gives the maximum is a crack with the depth of and gives √ and this still satisfies equation , . Equation is in a sense increasing the , due to every torque level from to is investigated and thus a great variety of will be encountered. So to satisfy this condition only √ will be regarded in the life estimation, and initial cracks with a √ will be regarded to have infinite life.
7.2 Linear approximation of ΔKeff
In this report the is calculated for all crack increment for a torque level of . From these calculated values a linear approximation of was done for torque levels from to with an increase of . In the table below all functions for is displayed.
Table 3, Table displaying the linear approximation of [ √ ] variation against applied pinion torque for
every crack from to . In function the symbolizes applied pinion torque in [ ].
33
To check if a linear approximation were valid, the was calculated for two cracks sizes and at more than two torque levels to see if the behavior was linear, see figures (31,32) below.
Figure 31, estimation of with 2 calculated values for crack with .
Figure 32, estimated with 3 calculated values.
From these values the seems to have a linear variance against the applied pinion torque.
7.3 Life time approximation
When the life is estimated the amount of cycles at a particular torque level is first calculated, this is then compared to a load collective. The load collective is a requirement from the customers of GKN and it tells how many revolutions the product should endure at each specific load. The load collective should resemble the products
34
whole life in the hands of the final customer. To compare the calculations of the life to the load collective a damage calculation needs to be done, this is done with Palmgren-Miner damage calculation according to equation .
To get a good visualization of the results the load collective is plotted against the calculated life, then the experimentally HCF test is plotted for 50% failure. These values is also compared to the effective notch method, earlier describes in section 2.3. The for the initial crack was just above the new threshold value at pinion torque which is indicated with a red solid line in figure (33). Everything below the red line should be regarded to have infinite life for the ΔK –method.
Figure 33, the green line is the load collective, purple line is the Effective notch method for 50% probability of failure and blue line is the ΔK method. The two black dots are the experimentally evaluated values for 50%
35
Comparison for the total damage from Palmgren-Miner calculation when the expected life is infinite for pinion torque below in the ΔK-method:
Table 4: Calculated damage for both the effective notch-method and the ΔK-method against the load collective according to equation (6.8), the load collective can be seen in figure (33) as the green line.
Method Total damage [-]
Effective Notch-Method
ΔK-method
7.4 Estimation of time
To get an estimation of how long time it takes to estimate the weld life for the new ΔK-method compared to the old effective notch ΔK-method, a rough estimation of the time from the beginning of the project until the last results are calculated and presented. The estimated time includes creating the model and doing all calculations and everything in between.
Table 5: Estimation of total time for a weld investigation, comparison for new method and old, displayed time in hours [h].
Method Estimated Time [h]
Effective Notch-Method
36
8 Discussion
Of the four methods tried only one was sufficient to solve the task. The 2D case was dismissed due to the multiaxial loads, and that it was meaningless to continue with it because the results would not reflect the real case. The two other methods, 3.3.2 & 3.3.3 where too complex for the software. After having discussed the problems with MSC support, the problems that were encountered will be fixed in the next version of MSC Marc.
The goal for this thesis was to find a way to estimate the fatigue life of the welds with the use of LEFM, which has been done with ΔK-method. But to arrive to this results there are off course simplifications and assumptions.
The smallest crack that can be detectible is around deep. So the initial crack size was modeled to have a depth of with a semi elliptical shape. The angle of the crack is assumed to be constant throughout the propagation. Assumption of a constant angle of the crack comes from when the fatigue tested gears had been cut open and investigated, the crack path seems to have a constant incline of , see figure (29). The decision to load the crack four times was picked from trial and error, with the goal to have as few loads as possible, but at the same time good data. One could have six, eight or even more loads, but the increases in computational time compared to the increase of accuracy in results were observed to be small, so a decision to use four loads was made.
As mentioned in section 2.4, cracks tend to grow in mode 1 but in some cases when the material has been weakened they can grow in areas dominated by other modes. The crack propagation in this thesis appeared to be dominated mainly by mode 2 & 3. So the crack was assumed to grow in theses modes due to results from the investigations of the fractured welds, which indicated that the cracks had this behavior. The crack was set to propagate in a small section of the welded area, this was necessary in order to get a decent computational time, tries with a full uniform crack was done but with no success, due to issues in the software and to a rapid increasing in computational time. The software issues are supposed to be fixed in a newer release of the program (2017 and newer).
The assumption that the crack closure effect doesn’t have a large effect on the result is not certain, but the closure effect should make the crack to propagate slower, thus make the results from the ΔK-method more conservative.
37
equation to estimate the crack propagation speed is not an obviously decision when the loads are multi axial, but it has been observed that it can be done. This thesis uses [19] by Keisuke Tanaka, as an effective stress intensity factor and it can be seen in equation (6.5). As mentioned by Tanaka, equation (6.5) has shown good correlation to tests as long as the crack is extending in the direction of the existing crack.
The small differences that occurred in figure (31), e.g. that it is not really linear is probably due contact issues. The mesh has the same size for the crack but some variation can occur when the glued contact is established between the crown wheel or tube shaft to the weld, which is believed to be the cause of the slightly varying results. The size difference between the crack propagation body and the other parts is large as can be seen in figure (19). The size difference can affect the contact search, so small mesh differences are preferred.
The incremental increase of for each crack was picked but one could have a smaller increment to reach a better result, but smaller increments will rapidly increase the computational time. Advice if this method should be used in the future is to not have a larger increment then , but smaller can of course be used if the user have large computational power or time.
The -method is highly dependent on the Paris Law parameters ( ) in equation ( ), and it is of high importance that these accurately describes the material. Currently these parameters are taken from [2] but my advice is that they should be experimentally determined if this method should be used. It is those parameters that will describe the weld and all its different properties. The residual stresses around the weld are only taken into account by according to [2], but one could try to use some modified Paris Law growth function for this in the future.
In figure ( ) a close resemblance between the -method and the ENM can be seen. This is thought to be due to both are assumed to have a linear variance towards the applied torque and both has the same exponent in their calculations. The ENM has in equation ( ), which is the same exponent as the Paris Law type equation in equation ( ), . This resemblance gives the -method credibility when it comes to determining the fatigue life.
The effects of negative mode 1 SIF (i.e. < 0) is assumed to be taken into account for by having a self-contact option with friction in the crack surfaces. So the effect of ( ) on the other modes ( ) is accounted for by the help of the friction at the crack surfaces.
The simplification to only use a small section where the crack is propagating could be misleading and maybe a full size crack should be used to estimate the crack propagation, due to the changes in the overall stiffness. This was not feasible to do in this thesis but for further investigations this would be interesting to compare against current method.
38
new product is investigated using this ΔK-method it can be difficult to figure out how the crack will propagate.
This thesis is solving the end goal with an estimation of the fatigue life with the use of LEFM. The estimated time for the new method is slightly more than the old ENM. The estimated time for the new method can be improved further with for example automated scripts for the calculations and crack modeling.
The future of ΔK-method is now to validate it with other welds and not only the PTU that was investigated here, to see if the method is robust enough to constantly give a good prediction of the fatigue life.
9 Summary
Different methods for calculating the crack propagation in welds was investigated and one method was determined to do the task.
New method was created for calculating the crack propagation welds with the use of VCCT.
The new method is currently depending on the material parameter to accurately describing the welded area crack propagation characteristics.
39
10 References
[1] Sundström, B., ed. (2010) Handbok och formelsamling i hållfasthetslära. Department of Solid Mechanics, KTH, Stockholm, Sweden, pp. 285.
[2] Hobbacher, A., (2008) Recommendation for Fatigue Design of Welded Joints and Components. International Institute of Welding, doc. XIII-2151r4-07/XV-1254r4-07, Paris, France.
[3] Miner, M.A., Cumulative Damage in Fatigue, J. Appl. Mech., 67 A, pp. 159-164, (1945).
[4] Irwin, G.R., Analysis of Stresses and Strains Near the End of a Crack Traversing a Plate, J. Appl. Mech., Vol. 24, pp. 361-364, (1957).
[5] Marc® 2016, Volume A: Theory and User Information, pp. 149.
[6] Sundström B., ed. (2010) Handbok och formelsamling i hållfasthetslära. Department of Solid Mechanics, KTH, Stockholm, Sweden. pp. 237. [7] ASTM E-399, Test method for plane-strain fracture toughness of metallic
materials. Annual Book of ASTM Standards, Vol. 03. 01, American Society for Testing and Materials, West Conshohocken, Pa.
[8] Paris P. C., M. P. Gomez, and W. P. Anderson, A rational analytic theory of fatigue, The Trend in Engineering, Vol. 13, pp. 9-14, 1961.
[9] Sundström B., ed. (2010) Handbok och formelsamling i hållfasthetslära. Department of Solid Mechanics, KTH, Stockholm, Sweden, pp. 251-252. [10] Forman, R. G., Kearney, V. E., Engle, R. M., Numerical Analysis of Crack
Propagation in Cyclic Loaded Structure, J. Basic Engineering, Trans. ASME, Series D, Vol. 89, pp. 459-463, 1967.
[11] Griffith, A.A., The Phenomena of Rapture and Flow in Solids, Philosophical Transactions of Royal Society, Series A, Vol. 221, pp. 163-198, (1920).
[12] Irwin, G.R., Handbook of physics, Springer Verlag, Berlin, Germany, Vol. 6, 1958, pp. 551-590.
[13] Rice, J. R., A Path Independent Integral and the Approximate Analysis of Strain, J. Appl. Mech., Vol. 35, pp. 379-386, 1968.
[14] Krueger. R., Virtual Crack Closure Technique: History, Approach and Applications, Appl. Mech. Rev., Vol 57:2, pp. 109-143, 2004.
[15] Fredriksson, E., Accuracy Study in Predicting Fatigue Life for a Welding Joint, Master Thesis, KTH, Stockholm, Sweden, 2015.
[16] Hertter, T., Calculative Proof of Strength of the Fatigue Capacity for
40
[17] Li, J., Sun, Q., Zhang, Z.P., Li, Y.J., Qiao, Y.J., Theoretical estimation to the cyclic yield strength and fatigue limit for alloy steels, Mechanics Research
Communications, Vol. 36, Issue 3, 2009, pp. 316-321.
[18] Nilsson, F., Fracture Mechanics from Theory to Applications, Department of Solid Mechanics, KTH, 2001.
[19] Tanaka, K. Fatigue Crack Propagation From a Crack Inclined to the Cyclic Tensile Axis, Engineering Fracture Mechanics, Vol. 6, pp. 493-507, 1974.
[20] Marc® 2016, Volume A: Theory and User Information, pp. 161-162.
