Acknowledgements. Daniel Arvidsson, Stockholm, July 2018

Transkript

1 Additive weld manufacturing and material properties effect on structural margins (Additiv svetstillverkning och materialegenskapers effekt påstrukturella marginaler) DANIEL ARVIDSSON Master Thesis carried out at GKN Aerospace Sweden, Trollhättan, Sweden, for the Department of Aeronautical and Vehicle Engineering Divison, Lightweight Materials KTH Royal Institute of Technology Stockholm, Sweden [2018]

2 iii Abstract This thesis investigates how structural margins are affected by including ansiotropic material as well as weld material in the FE analyses. Traditionally all parts are modeled with isotropic base material. Analyses are made on a part of the nozzle which includes both a butt weld and metal deposition and which is an interface to another part causing loads that has to be sustained by the weld and the MD. As a small part of this thesis was also a fatigue study made to a spot weld test specimen. In order to strengthen the nozzle to prevent structural damage, an outer layer is added to the already existing metal cone by material deposition, MD, or additive manufacturing. During the manufacturing process the material will indicate some degree of anisotropic properties. The key purpose of this thesis was to analyze how this anisotropic behaviour might affect the structural stiffener connected to this anisotropic material when exposed to a load at the end of the stiffener. Further analysis due to fatigue was also done to parts of the structure. The procedure was done by building a model and setting up the different anisotropic properties with help of a finite element program, Ansys. The material properties regarding the anisotropy of the material was changed and compared in order to see how it affected stresses and strains in the anisotropic material and it s surrounding materials. Further analysis was made to the properties of the weld such as the yield limit. The result would indicate that for loadings that did not generate plastic deformations, hence elastic deformations, there were no significant difference for the different trial values of the yield ratios. However, the differences became parent when studying large plastic deformations. Variation of the Young s modulus would show some differences in the monitored properties for both elastic and plastic deformations. Studies of degrading the welds yield limit would show no diffrences when elastic deformations were present, but would have a big impact when large plastic deformations were present. The J-values variations for the spotweld would indicate huge differences depending on the yield limits for the spotweld and base material.

3 iv Sammanfattning Detta examensarbete undersöker hur strukturella marginaler påverkas av att inkludera anisotropiska egenskaper för delar av en struktur samt att inkludera en svets som har andra material egenskaper såsom sträckgräns jämfört med anliggande material. Tidigare beräkningar är gjorda med antagandet att ingående material delar i strukturen är isotropiska. Analyserna omfattar styvhets bracketter som svetsas mot den anisotropiska delen av raket munstycket som är utsatta för laster som uppstår genom anslutning via andra delar påmunstycket. En mindre analys gjordes också av en test detalj där två metall plattor svetsas ihop med hjälp av en punktsvets som förekommer påmunstycket. Analyser med hänsyn till uttmattning har också varit en del i detta arbete. För att styva upp raket munstyckets övre del så adderas ett yttre hölje genom additativ tillverkning. Denna tillverknings metod ger upphov till att det pålagda höljet uppvisar anisotropiska egenskaper. Inledningsvis så skapades en finit element model i programmet Ansys där det anisotropiska materialet definierades samt material egenskaper för svetsen. Beräkningar gjordes sedan utifrån att en last pålades i utkanten av styvhets bracketten och laster och töjningar noterades för varje beräknings cykel för dom olika delarna av strukturen. Resultaten visade att för laster som inte genererade några plastiska töjningar, dvs elastiska töjningar, att det inte var några skillnader när olika värden påsträckgränserna för den anisotropiska delen av strukturen ändrades. Däremot var ändringarna mer påtagliga när töjningarna var plastiska, dvs för större pålagda laster som genererade stora kvarvarande töjningar i strukturen. Variationer av styvheterna i den anisotropiska delen i strukturen visade skillnader för både elastiska och plastiska töjningar. Analyser där svetsens sträckgräns var mindre än närliggande material visade inga direkta skillnader för elastiska töjningar men visade stora skillnader för plastiska töjningar. J-integral värdena i punktsvetsen som funktion av pålagd last visade sig vara beroende av sträckcgränsen för punktsvetsen samt för bas materialet.

4 v Acknowledgements I would like to thank my supervisor at GKN Aerospace Sweden, Phd. Robert Tano for all his help throughout the creation of this thesis and my KTH supervisor, Associate Professor Zuheir Barsoom. Daniel Arvidsson, Stockholm, July 2018

5 Contents Contents List of Figures List of Tables List of Acronyms vi viii ix xi 1 Introduction Background Purpose and Objective Approach Limitations Theoretical background Anisotropic material Plasticity Von mises yield criterion Hill s yield criterion Hardening model Crack initiation and stability Fracture toughness J-integral Welding mechanics Butt welds Spot welds Fatigue design Linear Elastic Fracture Mechanics (LEFM) Model description General Models Main model vi

6 CONTENTS vii Spot weld test specimen Material data Results Stiffener with butt weld Base material for all parts Variation of Young s-modulus Variation of Hill ratios Lowered yield strength of weld material Coffin-Manson Stress Intensity Factors, SIF Crack propagation, Paris law Spot weld test specimen Discussion and conclusions Stiffener with butt weld Base material for all parts vs weld Variation of Young s-modulus Variation of Hill ratios Spot weld test specimen Future work Bibliography 35

7 List of Figures 1.1 Sketch of a rocket engine nozzle, including a cross section of the reinforcement jacket seen, from above Von mises yield surface as a cylinder Isotropic and kinematic hardening Paris law graph Main model of the stiffener Mesh convergence graph Spot weld model The stress-strain curve for the base material FE-models results for the reference calculations Stresses and strains when E z = 147 GP a and F AX = Stresses and strains when r zz = 0.8 and F AX = Stresses and strains when c 0.8 = 0.8 and F AX = Coffin-manson graph of base material Graph of J-values and applied load Main model of the spot weld test specimen viii

8 List of Tables 3.1 Load factors used in the studies Design parameters for the base material Variables table for the reference calculation Results with constant Young s-modulus, Hill s ratios and F AX = Results with constant Young s-modulus, Hill s ratios and F AX = Iteration table with varying Young s-modulus and constant Hill s ratios Results with varying Young s-modulus and constant Hill s ratios, F AX = Results with varying Young s-modulus and constant Hill s ratios, F AX = Iteration table with varying Hill s ratios and constant Young s-modulus Results with varying Hill s ratios, constant Young s-modulus and F AX = Iteration table with constant Young s-modulus, Hill s ratios, F AX = 1.3/7.3 and c 2 = Results table with constant Young s-modulus, Hill s ratios, F AX = 1.3 and c 2 = Result table with constant Young s-modulus, Hill s ratios, F AX = 7.3 and c 2 = Variables for the Coffin-Manson graph Iteration table with varying Young s-modulus and constant Hill s ratios Case table with K I values Case table with da/dn values ix

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10 List of Acronyms MD FEM LEFM Metal deposition Finite element method Linear elastic fracture mechanics xi

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12 Chapter 1 Introduction 1.1 Background Between 2003 and 2018, 99+ successful launches has taken place with the Ariane 5 rocket, 80+ with vulcain 2.0 engine. The payload of those launches has exclusively been satellites that have had different functions such as telecommunications, GSP etc. The Ariane 5 rocket is equipped with the Vulcain 2 engine that can generate 1390kN of thrust in vacuum. The engine is equipped with Vulcain 2.0 exhaust nozzle, which is one of the engine s key components.this component is developed and delivered by GKN Aerospace Sweden. However, studies in this thesis is regarding Vulcain 2.1 engines that are equipped with SWAN exhaust nozzles that are under development. It s design can be described as super-alloy sheets with cooling channels inside, see figure 1.1. At the top, a reinforced jacket is applied, which function is to strengthen the nozzle during compressive axial loads. This jacket is manufactured with a method called metal deposition, MD [3]. Because of this manufacturing method, the material will become anisotropic to some degree. Attached to the exhaust nozzle are stiffeners that are connected to other equipment on the nozzle and therefore loads will be applied to the stiffeners. One of those stiffeners have been investigated in this thesis. The stiffeners is joined by butt welding to the MD part and to the stiffener itself, see figure 3.1. A small part of this thesis has also been to investigate a test specimen joined by spot welding and can be seen in figure 3.3. There have been other studies made at GKN Aerospace Sweden, which purpose has been to investigate the degree of anisotropy created when using MD for the upper jacket. The thesis "Anisotropy and its influence on a MD structure" was made at the company 2009 [3] which has worked as a foundation for this thesis. In that thesis, hypothesis were made in regard to anisotropy of the jacket, some of them were directly structural ones. 1

13 2 CHAPTER 1. INTRODUCTION Figure 1.1: Sketch of a rocket engine nozzle, including a cross section of the reinforcement jacket seen, from above. The most important one for this thesis were stated as: "An anisotropic material in the nozzle reinforcement jacket will significantly increase the level of elastic deformation". No evidence of this was supported, however, some facts made it plausiable, hence this hypothesis couldn t be rejected. The overall conclusions from that thesis were that the properties in the tangential direction would effect the elastic strains and structural margins the most. Simplifications were also made with regards to the anisotropy such as possions ratios and that the anisotropy in tension and compression were the same. There were also studies for buckling of the nozzle and what effects the anisotropic properties could have on the results. In this thesis, there was an underlying investigation made with our results regarding the buckling and it was concluded in the SWAN project that it had small and negligable effects on the results. No details on the MD manufacturing method is covered in this thesis. 1.2 Purpose and Objective The main task of this thesis is to investigate how the structural margins are affected by including anistropic material properties for part of the structure and material properties for the weld used to join the anisotropic part and the stiffener. - How are the margins for welds and the surrounding materials affected by including the actual weld in the FEM analysis? - How are the margins for additive material and surrounding materials affected by modeling the AM part with anisotropic material? 1.3 Approach The commercial FEM program Ansys has been used through out this thesis together with Matlab for creation of graphs and other calculations.

14 1.4. LIMITATIONS Limitations Only parts of the exhaust nozzle is analyzed as a part of this thesis. The temperature is restricted to 294K (20.85 C) for all calculations. All material data has been assumed and is not valid for any typical nozzle material.

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16 Chapter 2 Theoretical background 2.1 Anisotropic material Anisotropic materials are materials that shows different material properties in all its directions, compared to most materials that are isotropic and shows no difference in its directions. Material properties such as tensile, shear yield limits and possion s ratios are gathered thru extensive material testing and experiments. The generalized Hooke s law for 3D anisotropic materials can be given in the following matrix form [2] [5]: σ = Cɛ (2.1) where C is a matrix containing material constants, which are normally obtained through experiments. The consitutive equation can be written as: σ xx C 11 C 12 C 13 C 14 C 15 C 16 ɛ xx σ yy C 22 C 23 C 24 C 25 C 26 ɛ yy σ zz σ yz = C 33 C 34 C 35 C 36 ɛ zz Sym. C 44 C 55 C 56 ɛ yz σ xz C 55 C 56 ɛ xz σ xy C 66 ɛ xy Since C ij = C ji, there are 21 independent material constants C ij, which is true for a fully anisotropic material. Materials with three orthogonal planes of symmetry for its material properties, are called orthotropic materials. This material is fully characterized by 9 independent coefficients: 5

17 6 CHAPTER 2. THEORETICAL BACKGROUND C 11 C 12 C C 22 C C = C C Sym. C 55 0 C 66 For the orthotropic material, the poisson s ratios are given as: υ ij E ii = υ ji E jj (2.2) where i,j=1,2,3 or x,y,z. For a orthotropic material it then becomes: υ 12 E 11 = υ 21 E 22 (2.3) υ 31 E 33 = υ 13 E 11 (2.4) hence, three of these are independent. υ 32 E 33 = υ 23 E 22 (2.5) For isotropic materials, C can be reduced to: C 11 C 12 C C 11 C C C = (C Sym. 11 C 12) (C 11 C 12) 2 0 where, (C 11 C 12) 2 C 11 = C 12 = E(1 υ) (1 2υ)(1 + ɛ) Eυ (1 2υ)(1 + ɛ) (2.6) (2.7)

18 2.2. PLASTICITY 7 G = C 11 C 12 2 (2.8) where E, υ and G are Young s modulus, Poisson s ratio and the shear modulus of the material. 2.2 Plasticity The software Ansys, uses a variety of material models that can be adopted to mimic different material properties and effects of a structure. A material model, or constitutive model is a mathematical representation of the expected behavior of a given material in response to an applied load. In this thesis a combination of linear elasticity, plasticity with kinematic hardnening has been used. For the material showing isotropic behaviour, the von-mises theorem has been used and for the material showing anisotropic behaviour, the Hill s theroem has been used. Below, a brief description is given Von mises yield criterion The von-mises yield criterion predicts that yielding will occur whenever the distortion energy in a unit volume equals the distortion energy in the same volume when uni-axially stressed to the yield strength, see equation σ e = 2 [(σ 1 σ 2 ) 2 + (σ 2 σ 3 ) 2 + (σ 3 σ 1 ) 2 ] (2.9) When the von-mises stress exceeds the uni-axial material yield strength, hence the stress state is outside of the cylinder, see figure 2.1, and yielding will occur Hill s yield criterion The Hill yield criterion is an anisotropic criterion that depends on the orientation of the stress relative to the axis of anisotropy[1]. When a coordinate system is aligned with the anisotropy coordinate system, the Hill yield criterion can be expressed as f(σ, σ y ) = F (σ 22 σ 33 ) 2 + G(σ 33 σ 11 ) 2 + H(σ 11 σ 22 ) 2 + 2Lσ Mσ Nσ 2 12 σ 2 y = 0 (2.10) The coefficients in the criterion are functions of the ratio of the scalar yield stress

19 8 CHAPTER 2. THEORETICAL BACKGROUND Figure 2.1: Von mises yield surface as a cylinder parameter and the yield stress in each of the stress components: F = 1 2 ( 1 R R R11 2 ) (2.11) G = 1 2 ( 1 R R R22 2 ) (2.12) H = 1 2 ( 1 R R R33 2 ) (2.13) L = 3 2 ( 1 R23 2 ) (2.14) M = 3 2 ( 1 R13 2 ) (2.15) N = 3 2 ( 1 R12 2 ) (2.16) where the directional yield stress ratios are the user-input parameters to Ansys and

20 2.2. PLASTICITY 9 are related to the isotropic yield stress parameter by: r xx = R 11 = σy 11 σ y (2.17) r yy = R 22 = σy 22 σ y (2.18) r zz = R 33 = σy 33 σ y (2.19) r xy = R 12 = 3 σy 12 σ y (2.20) r yz = R 23 = 3 σy 23 σ y (2.21) r xz = R 13 = 3 σy 13 σ y (2.22) where σ y i is the yield stress in the direction indicated by the value of subscript i. The stress directions are in the anisotropy coordinate system which is aligned with the element coordinate system Hardening model When a material is loaded above the yield criterion, it starts to deform plastically. This in turn, creates an increase in the yield criterion upon further loading. The two common types of hardening rules are the isotropic and kinematic hardening [1]. The hardening rules describes how the yield surface changes as a result of plastic deformation. The two types can be seen in figure 2.2. In this thesis we have used a multi-linear kinematic hardening stress-strain curve which implies that the curve is a piece-wise linear approximation in the plastic region, see figure 3.4.

21 10 CHAPTER 2. THEORETICAL BACKGROUND Figure 2.2: Isotropic and kinematic hardening 2.3 Crack initiation and stability Crack initiation is a process where cracks normally forms at the surface of a material. The primary reason for the formation of cracks on any surface is fatigue, but also due to monotonic loads, if high enough. Fatigue leads to progressive and localized structural damage when any material experiences cyclic loading. Due to cyclic loading, the material experiences continuous and repeated loads or forces at various points on the material. When such loads are high enough, they lead to crack initiation, growth of cracks and ultimately a fracture. The crack development can be divided into three phases: Initiation stage: Development of micro crack due to cyclic loading. For welds, one always assume existing cracks so the inition phase is ignored. Propagation stage: The crack is advancing with each cycle. Final breakage: The structure can t longer take the load and fails. In order to analyze the initiation stage, the Coffin-Manson model was used. It s definied as: ɛ 2 = σ f E (2N f ) b + ɛ f (2N f ) c (2.23) where ɛ is the strain range, E elastic modulus, σ f, ɛ f, b and c are material constants and Nf is the number of cycles it takes in order to from a crack. The above function can then be plotted in order to be used to estimate the required cycles for a particular stress or strain range that the structure is exposed to.

22 2.3. CRACK INITIATION AND STABILITY Fracture toughness If a material is assumed to be linear elastic, the stress at the crack tip will in theory grow to infinity, due to the assumption of linear elastic behaviour, but in reality a small area around the crack tip will act non-linear so that the stress will not go to infinity. The parameter that controls what happens in this area is called the stress intensity factor, SIF and can be seen as the variable that defines the severity of the stresses at the tip of the crack and is defined mathematicaly as: K I = σ aπf (2.24) where f is a dimensionless parameter that depends on the geometry,crack shape and size, specimen shape and size and loading method. The variable a is half of the crack length and σ is the stress at a distance from the crack. Experiments has shown that cracks that reaches a critical value, K IC, starts to grow. This value is called the fracture toughness value and characterises the resistance of a material to fracture in the presence of a crack. An important problem with LEFM is how to determine the stress-intensity factors and there are several possibilities available. One is to use tables from handbooks [10]. In this thesis, a surface crack has been assumed to exist in the weld, see appendix C, case J-integral When the combination of geometry and loading is such that the plastic zone is not confined, the LEFM is not sufficient. Hence, for larger plastic strains, one must use another method. According to the pioneering work by Griffith [8], the energy relations at a moving crack tip have been considered of importance. The idea is that a crack tip consumes a certain amount of energy per unit of newly created crack surface and that this energy has to be taken from the surrounding body. The energy transfer to a crack tip can be described as: where, for plane strain and for plane stress: G = K2 I E (2.25) E = E 1 υ 2 (2.26) E = E (2.27) Another way to describe the energy flow to the crack tip from the surrounding body

23 12 CHAPTER 2. THEORETICAL BACKGROUND is by an path independent integral called the J-integral. Physically, the integral can be interpreted as the energy flow at the crack tip and is described as: J = (A u i d y σ ij n j x d s) (2.28) Γ where Γ is the curve surrounding the crack tip, A is the deformation work density, σ ij is the stress acting on the crack, u i is the displacement vector and d s is an element of the arc length along Γ. 2.4 Welding mechanics Butt welds Butt welding is a commonly used technique in welding that can either be automated or done by hand on steel pieces. It is used to attaching two pieces of metal together by adding a third metal that may or may not have the same material as the metal it is applied to Spot welds Resistance spot welding is a process in which contacting metal surface points are joined by the heat obtained from resistance to electric current Fatigue design Welds like any other material will contain flaws to a certain degree. This will decrease the strength of the weld. When speaking long term effects one has to consider fatigue. When a structure is exposed to cyclic loading, a crack inside the weld will start to form and grow with every loading cycle. The end result is also of utter importance for a welds lifetime. Design error, root gaps, cracks, incomplete weld penetration, bonding errors and internal effects are all factors for the affecting time. When a structure is exposed to over 1000 loading cycles, a fatigue study is recommended [7]. There are different methods for this: Nominal method, hot spot method, effective notch stress method and linear elastic fracture mechanics. The mention methods are ranked in degree of accuracy and cost Linear Elastic Fracture Mechanics (LEFM) The LEFM method is used to predict the behaviour of cracks in solids subjected to fatigue loading [7]. It is as known today the best method to establish the crack growth. The benefits is that one can establish inspection intervals and the fatigue life. The method assumes an initial crack length or a postulated crack already is

24 2.4. WELDING MECHANICS 13 present in the weld. The fatigue life is determined by the number of cycles the structure can be subjected to before a crack of a critical length is formed. The rate at which a crack is formed is called Paris law and is the linear relationship between the crack propagation rate and the stress intensity range: da dn = C( K I) n (2.29) where K I K I K IC and C, n are material parameters. The K th is the threshold value. The stress range are controlling the crack propagation according to: K I = K I,max K I,min (2.30) The curve can be plotted in a log-log diagram, see figure 2.3: For region I, K need to reach a certain value, K th in order to start the crack propagation. This value depends on material and the stress intensity factor ratio [11]. In region III, Paris law in no longer valid due to unstable crack propagations and propagates quickly for every load cycle, N.

25 14 CHAPTER 2. THEORETICAL BACKGROUND Figure 2.3: Paris law graph.

26 Chapter 3 Model description 3.1 General In this thesis, the work has been focused mainly on two different models. The main model consists of a stiffener from the SWAN nozzle and consists of the inner sandwich wall of material Hayes 230 and the outer anisotropic MD-material IN625, where a butt weld is used to join the stiffener to the MD-material. The stiffener is connected to other equipment on the nozzle that generates loads at the end of the stiffener. The second model consists of two plates joined with a spot weld. Different analyses was made on the different models to help GKN Aerospace Sweden with how to model welds and MD in FE-analyses. 3.2 Models Main model Below follows a description of the different Ansys models used in this thesis. The main model will be described more in detail since it was created as a part of this thesis. The other models will be described only briefly Design The main model was made in Ansys APDL 18.1, see figure 3.1. The inner sandwich wall is simplified to be a solid. The actual nozzle is not fully axisymmetric in this region but a simplification has been made w.r.t computational time.the model is a rotated volume, but only a part of the volume is analyzed in order to save computation time. The FE-model was modeled with 20 node elements, referred in Ansys as SOLID186. Area A1 is the stiffener itself, A2 the weld and the rest are the outer and inner mantels of the nozzle.the model is built up so that it is possible to easily change 15

27 16 CHAPTER 3. MODEL DESCRIPTION Figure 3.1: Main model of the stiffener. material properties for all the areas. The areas regarding the outer core have showed anisotropic properties, the others have displayed isotropic properties. In Chapter 4 it is described in detail which parameters been controlled and analyzed Anisotropy in Ansys To define an anisotropic material in Ansys, it needs the elastic modulus in all the directions, shear modules and also the shear ratios or the stiffness matrix. In order to setup the anisotropic material correctly in Ansys, one has to check the actual elements planned to be used in the structural analyses, which can be found in Ansys theory manual [1]. In this case it was solid186 elements. By default, this elements coordinate system, that defines the material properties, are aligned with the global coordinate systems [6]. The anisotropic part in the main model is not aligned with the cylindrical coordinate system which meant that the the element coordinate system had to be rotated for the MD-material Plasticity As mentioned briefly in the theory chapters, a multi-linear hardening model has been used in the analyses. In order to define this in Ansys, the temperature were defined to 294K, and the points on the stress-strain curve for the plastic zone. Since the MD-material are defined as an anisotropic material, the Hill yield criterion needed to be defined since Ansys by definition uses the von-mises criterion. In order to define the Hill criterion in Ansys, the ratios discussed in the theory chapters are needed as inputs to Hill s yield surface. In [3] the ratios were chosen to be in the range which corresponded to a ±25 % change in the yield

28 3.2. MODELS 17 limit in tension/compression and shear. In this thesis the range was , hence corresponding to a change of ±20 %, and the equations 2.20, 2.21 and 2.22 were estimated as: r xy = (r xx + r yy ) 2 (3.1) r yz = (r yy + r zz ) 2 (3.2) r xz = (r xx + r zz ) 2 (3.3) Loads A load is applied at the end of the stiffener and a load factor, F AX has been used in order to reach certain desirable conditions in the structure, see table 3.1. The force is applied at the nodes on the outer surface with help of surf154 elements. Table 3.1: Load factors used in the studies. Load factor, F AX Value Stress range in MD,weld and stiffener, MPa 1.3 Plastic deformations, 2-4% Boundary conditions The boundary conditions used in the model are purely mechanical and there is no coupling between other physics than solid mechanical ones. The lower boundary are set to be locked in z-direction and the upper ones are coupled and hence allowed to move in the z-direction(axial direction). The boundaries in tangential direction(ydirection) are locked, i.e axisymmetric Mesh The mesh convergence study is done by study and looking at how the von-mises stresses were changing with number of elements in the weld and the MD-material. The von-mises stress as a function of number of elements is shown in figure 3.2. The stress values stabilized for approximate elements.

29 18 CHAPTER 3. MODEL DESCRIPTION Figure 3.2: Mesh convergence graph Spot weld test specimen The spot weld test model is shown in figure 3.3. It consists of two metal plates joined together with a spot weld. The main purpose was to investigate how the J-integral value is changing with respect to weld material and the applied load. See Chapter 4 for details and results. (a) Model with spot weld, 1. (b) Model with spot weld, 2. Figure 3.3: Spot weld model. 3.3 Material data In this thesis, two input material files have been used through out all the assignments and tasks. The first one is describing psycial material properties such as E-modulus, shear modulus and poisson s ratios. The second file describes the parameters used for plasticity such as what type of hardening model and the stressstrain curve for the base material in elastic and plastic region. When simulating a weld material, a degrading or a upgrading have been established by changing the yield limit parameter c 2 between values of in order to investigate the effect

30 3.3. MATERIAL DATA 19 in the response in the structure. In table 3.2 the physical properties are given and in figure 3.4 the stress-strain curve is shown. Table 3.2: Design parameters for the base material. Design Parameter Value Elastic modulus in x-direction, E x 210 GPa Elastic modulus in y-direction, E y 210 GPa Elastic modulus in z-direction, E z 210 GPa Shear modulus in xy-plane, G xy GPa Shear modulus in yz-plane, G yz GPa Shear modulus in xz-plane, G xz GPa Poissons ratio, υ x = υ y = υ z 0.30 MD-angle, φ 10 Figure 3.4: The stress-strain curve for the base material. For the MD material that shows anisotropic behaviour, the shear modulus were calculated according to Hooke s linear elastic laws: G xy = E x 2(υ + 1) (3.4)

31 20 CHAPTER 3. MODEL DESCRIPTION G yz = E y 2(υ + 1) (3.5) G xz = E z 2(υ + 1) (3.6)

32 Chapter 4 Results 4.1 Stiffener with butt weld In order to see the connections between the material properties that are varying depending on the orientation of the MD material, the welds yield limit, many analysis were done. The Young s-modulus, Hill ratios were varied and comparison was made to a lowering of the welds yield strength. The von-mises stresses and the plastic deformations in the MD-material, weld and stiffener were values of most interest Base material for all parts Keeping all the Young s-modulus, Hill s ratios at a constant value and a load factor of F AX = 1.3 that will generate elastic deformations, see table 4.1. Table 4.1: Variables table for the reference calculation. E x (GP a) E y (GP a) E z (GP a) r xx r yy r zz r xy r yz r xz F AX The results are reported in table 4.2: Table 4.2: Results with constant Young s-modulus, Hill s ratios and F AX = 1.3. Parameter Von mises stress in the weld, σ weld Von mises stress in MD, σ md Von mises stress in stiffener, σ stiff Value 109 MPa 123 MPa 106 MPa 21

33 22 CHAPTER 4. RESULTS Further, to achieve large plastic deformations, an increase in the load factor F AX = 7.3 was used. The result are displayed in table 4.3: Table 4.3: Results with constant Young s-modulus, Hill s ratios and F AX = 7.3. Parameter Value F AX Von mises stress in the weld, σ weld 421 MPa 7.3 Von mises stress in MD, σ md 426 MPa 7.3 Von mises stress in stiffener, σ stiff 409 MPa 7.3 Von mises plastic strains, ɛ p.weld 2.87% 7.3 Von mises plastic strains, ɛ p.md 3.06% 7.3 Von mises plastic strains, ɛ p.stiff 2.45% 7.3 (a) Stresses for the reference calculation. (b) Plastic strains for the reference calculation. Figure 4.1: FE-models results for the reference calculations. In the following sections there is an extensive comparison made between elastic and plastic deformations and stresses created when changing values of the material properties Variation of Young s-modulus In order to study the changes in the structure due to changes in the MD material, different values of Young s-modulus was studied, see table 4.4.

34 4.1. STIFFENER WITH BUTT WELD 23 Table 4.4: Iteration table with varying Young s-modulus and constant Hill s ratios. E x (GP a) E y (GP a) E z (GP a) r all F AX 1 147/ / / / / /7.3 The result for elastic deformations when using a load factor of F AX = 1.3 can be seen below, see table 4.5: Table 4.5: Results with varying Young s-modulus and constant Hill s ratios, F AX = 1.3. σ weld σ iso,el σ MD σ iso,el σ stiff σ iso,el / / / / / / / / /1.05 The result for large plastic deformations when using a load factor of F AX = 7.3 can be seen below, see table 4.6: Table 4.6: Results with varying Young s-modulus and constant Hill s ratios, F AX = 7.3. σ weld σ iso,pl σ MD σ iso,pl σ stiff σ iso,pl ɛ weld ɛ iso,pl ɛ MD ɛ iso,pl ɛ stiff ɛ iso,pl / / / / / / / / / / / / / / / / / /0.86 The result with largest plastic strains was obtained when having low stiffness in the z-direction, see figure 4.2.

35 24 CHAPTER 4. RESULTS (a) Stresses for E z = 147 GP a. (b) Plastic strains for E z = 147 GP a. Figure 4.2: Stresses and strains when E z = 147 GP a and F AX = Variation of Hill ratios In table 4.7, the iterations calculated when varying the Hill ratios can be seen below. Table 4.7: Iteration table with varying Hill s ratios and constant Young s-modulus. r xx r yy r zz r xy r yz r xz E all F AX 1 0.8/ / / / / / / / / / / / / / / / / / Results regarding elastic deformations did not show any changes when alternating the ratios, as expected, since the loadings never exceds the yield limits, so no such table are presented. The result for large plastic deformations using a load factor of F AX = 7.3 can be seen below, see table 4.8. Table 4.8: Results with varying Hill s ratios, constant Young s-modulus and F AX = 7.3. σ weld σ iso,pl σ MD σ iso,pl σ stiff σ iso,pl ɛ weld ɛ iso,pl ɛ MD ɛ iso,pl ɛ stiff ɛ iso,pl / / / / / / / / / / / / / / / / / /0.89 The highest plastic strains was obtained when the ratio was low in z-direction,

36 4.1. STIFFENER WITH BUTT WELD 25 see figure 4.3. (a) Stresses for r zz = 0.8 GP a. (b) Plastic strains for r zz = 0.8. Figure 4.3: Stresses and strains when r zz = 0.8 and F AX = Lowered yield strength of weld material Analyses was further conducted keeping the Young s-modulus and Hill s ratios constant and a load factor F AX = 1.3 that would generate elastic deformations when the welds yield limit was lowered. To achieve lower yield strength for the weld, the yield strength parameter c 2, was set to c 2 = 0.8. See table 4.9. Table 4.9: Iteration table with constant Young s-modulus, Hill s ratios, F AX = 1.3/7.3 and c 2 = 0.8. E x E y E z r all F AX C / The results can be seen in table 4.10:

37 26 CHAPTER 4. RESULTS Table 4.10: Results table with constant Young s-modulus, Hill s ratios, F AX = 1.3 and c 2 = 0.8. σ weld σ iso,pl σ MD σ iso,pl σ stiff σ iso,pl For comparison, a analysis was made with a loading factor set to F AX = 7.3 that would generate large plastic deformations, see table Table 4.11: Result table with constant Young s-modulus, Hill s ratios, F AX = 7.3 and c 2 = 0.8. σ weld σ iso,pl σ MD σ iso,pl σ stiff σ iso,pl ɛ weld ɛ iso,pl ɛ MD ɛ iso,pl ɛ stiff ɛ iso,pl In figure 4.4 below, the result when the yield limit is lowered for the weld can be seen. (a) Stresses for c 2 = 0.8. (b) Plastic strains for c 2 = 0.8. Figure 4.4: Stresses and strains when c 0.8 = 0.8 and F AX = Coffin-Manson From equation 2.23, a graph was created in Matlab using the following values, see table 4.12: The graph was used to estimate the number of cycles to from a crack in the MD part of the stiffener, since cracks are already formed in the welds due to the welding process, see fig 4.5:

38 4.1. STIFFENER WITH BUTT WELD 27 Table 4.12: Variables for the Coffin-Manson graph. Parameter Value σ f 700 MPa ɛ f 0.03 b -0.5 c E 200 GPa Figure 4.5: Coffin-manson graph of base material. The objective was to establish a relation between the stiffness in the structure by changing the Young s-modulus in the different directions and use the created strains due to the applied loads in order to see how the number of cycles would change. In order to prevent plastic strains and measure the cycles for elastic strains, a loading factor F AX = 2.6 was used. The result can be displayed in table 4.13: Table 4.13: Iteration table with varying Young s-modulus and constant Hill s ratios. E x (GP a) E y (GP a) E z (GP a) r all F AX N f 1 147/ / / / / /

39 28 CHAPTER 4. RESULTS Stress Intensity Factors, SIF Due to formed cracks during the welding process, a study was made regarding the stress intensity factors for the weld itself. By using handbook solutions [10], appendix C, 20.7 was used to calculate SIF values. Assuming K IC = 50 MP a m, a/c = 1 and a = 0.76 mm. The study was restricted to the elastic cases for the variations of Young s-modulus. See table 4.14 below: Table 4.14: Case table with K I values. Case E x E y E z σ x K I K I K IC K I K th Crack propagation, Paris law For the stress range we have assumed the applied load being applied in both directions, hence the stress range becomes: K I = K I,max ( K I,max ) = 2K I,max (4.1) Comparing the crack propagation rates depending on what Young s-modulus used in the analysis, a table can be established and then be compared under which circumstances the crack propagation will have it s highest value. Using values from [11], and excluding effects due to residual stresses, the following values are obtained, see table 4.15: 4.2 Spot weld test specimen For the spot weld test specimen, the key parts of interest was to create data that showed a connection between the J-integral and the applied load in the radial di-

40 4.2. SPOT WELD TEST SPECIMEN 29 Table 4.15: Case table with da/dn values. da Case E x E y E z σ x K I K I dn rection. When compared the J-integral with the applied load, five curves were created. The different curves applies to different parameters for the material in the base material and the material properties such as yield limit, for the spot weld. The result can be seen below in figure 4.6: Figure 4.6: Graph of J-values and applied load.

41 30 CHAPTER 4. RESULTS (a) Spot weld model of the stiffener, 1. (b) Spot weld model of the stiffener, 2. Figure 4.7: Main model of the spot weld test specimen.

42 Chapter 5 Discussion and conclusions 5.1 Stiffener with butt weld Base material for all parts vs weld Comparing the result tables, 4.2 and 4.10, it can be noted that the stresses remains the same. So using a load factor that are generating elastic deformations, there is no changes in the observed parts of the structure due to a lowering of the yield strength of 20 %(c 2 = 0.8), for the weld. However, using a load factor that are generating plastic deformations, it can be noted that it makes a significant difference, see tables 4.3 and In figure 4.1, the result of a load factor that generates large plastic deformations with the weld material having the same yield limit as the other surrounding materials is shown, figure 4.4 displays the result when the weld has a lower yield limit for the same applied load factor. It can be noted that the stress field is changing and stop being consistent through the weld compared with the stress field in the MD-material Variation of Young s-modulus When studying the result for elastic strains, table 4.5 can be studied. From this table it can be seen and be concluded that there is an increase of about 9 % for a lowering in stiffness in x-direction, E x = 147 GPa. In the MD-material, the increase is to about 8 % when the stiffness are lowered in z-direction, E z = 147 GPa. For plastic deformations, studying table 4.6, the stresses almost remains the same, also for the MD-material, however, there are notable diffrences in plastic strains compared to the isotropic case, displayed in table 4.6. Further considering plastic deformations, the variation of the stiffness in different directions would overall show that a lower stiffness in the x-direction, E x = 147 GPa, give the lowest plastic strains, see table 4.6. The lowest plastic strains was also shown when having higher stiffness in the z-direction, E z = 273 GPa, see table 4.6. The largest plastic strains 31

43 32 CHAPTER 5. DISCUSSION AND CONCLUSIONS was obtained when setting the stiffness low in z-direction,e z = 147 GPa, see figure 4.2. When studying the number of cycles to develop a crack in the MD-material, we can observe the result according to table 4.13 and see that the case when the stiffnes is lower in the z-direction, we reach the minimum value of cycles to form a crack. Further studying table 4.13, when using higher values of the stiffness, it also reaches the lowest values for the stiffness in z-direction, so clearly, this is the crucial one to observe. For the stress intensity factors, it can be seen that for all the studied cases, all K I values are higher than the thresshold value, K th, which means the the existing crack grows for each load cycle. It can also be observed that the results are far away from the K IC value which means that no unstable crack growth or rupture will occur for the applied load. The crack propogation rates can be seen in table It can be observed in this table that the propogation speed is the highest for when having a low stiffness in the z-direction Variation of Hill ratios As mentioned in Chapter 4, no changes were noted for the elastic case when alternated the values of the ratios as to be expected since not plastic deformations occur. For the variations of the ratios regarding plastic deformations, the lowest plastic strains were obtained when r yy = 0.8 and r zz = 1.2. Table 4.8 shows the results from those analyses. Studying table 4.8, note that there are significant increases in plastic strains. Largest changes in plastic strains can be noted in the weld and MD-material. The highest plastic strains was obtained when the ratio was set low in z-direction, r zz = 0.8, hence, having 20% lower yield limit in that direction, see figure Spot weld test specimen From figure 4.6, note a few interesting phenomena. In the beginning a linear relationship between the energy release rate J and the applied load P can be seen. Passing 10 kn, note that the curves takes a more non-linear behaviour. The first curve to take this approach is when the yield limit for the weld is lower than the base material in the two joined plates. The last curve is when both the weld and the base material has the same yield limits, but increased by 20 % (c 2 = 1.2).Passing the 15 kn mark things starts to get more interesting. We can see a 100 % increase in the energy release value J between the curves with higher yield limit for both the base and the weld, hence c 2,bas = 1.2 and c 2,weld = 1.2 and the curve with a lowered yield limit for only the weld, hence c 2,bas = 1.0 and c 2,weld = 0.8.

44 5.3. FUTURE WORK Future work During launch of the rocket and its journey through the atmosphere, there is variations in the loading that the stiffeners are exposed to. This loading cases would be needed in order to do a thorough study of the weld to see at what level of stresses it can take to sustain a lifetime that are desirable for the structure. The weld used would also need to be x-rayed in order to establish the initial crack size in order to use LEFM method described in Chapter 2. Regarding the MD material, extensive experiments would give the values for the yield ratios and the poission s ratios that would give a more accurate result of the calculations since it can be observed that the structural margins is affected by the variations of the yield limts ratios in Hill s yield criterion.

45

46 Bibliography [1] Theory References for Ansys APDL Ansys Inc. [2] Sundström, B. Handbok och Formelsamling i Hållfasthetslära, KTH, Stockholm, [3] Briland, Anders. Anisotropy and its Influence on a Metal Deposition Structure, Msc.thesis, Chalmers University of Technology, [4] Knuth: Computers and Typesetting, uno/abcde.html [5] G. R. Liu, S.S. Quek The finite element method, A practical course, 2003, ISBN: , pp [6] Erdogan Madenci, Ibrahim Guven The Finite Element Method and Applications in Engineering Using Ansys, ISBN: , pp [7] J. Samuelsson1, Z. Barsoum2, B. Jonsson Advanced Design of Welded Structures, Introduction to fatigue assessment methods. [8] Nilsson, Fred Fracture Mechanics-from Theory to Applications ISBN: [9] Olsson, Claes Konstruktionshandbok för svetsade produkter ISBN: [10] Nilsson, Fred Fracture Mechanics-from Theory to Applications ISBN: ,, Appendix C, pp

47 36 BIBLIOGRAPHY [11] Hobbacher, A Recommendations for Fatigue Design of Welded Joints and Components ISBN: , Table 3.15