Fracture Mechanics in Quartz Lamps

Transcription

1 Fracture Mechanics in Quartz Lamps M.M. Joosten MT06.16 Coaches: J.H.A. Selen, P.H.M. Timmermans W.A.M. Brekelmans, M.G.D. Geers June 28, 2006

2 Abstract There are many consumer products in the market that use lamps. Some applications require lamps that emit relatively much light from a small point. Ultra High Performance (UHP) lamps meet these requirements and are therefore widely used in products like beamers and special type of TVs. Philips Lighting BV produces UHP lamps in Turnhout, Belgium. Different types of burners are produced for several applications. After production of the lamps tests are done to check the light quality. During these tests it sometimes happens that lamps fail at the first time burning. Causes for these early failures can be initial cracks after production or defects due to crystallization of the material. In both cases an initial crack exists in the quartz before the lamp starts burning. The goal of this project is to find out whether it is possible and useful to implement J Integral calculations in the current model and to examine the influence of the size and the position of the crack on the value of the J Integral. Various cracks are modeled, at different locations of the bulb and with different sizes. For these cracks simulations are done with both elastic as viscoelastic material behavior. In calculations with elastic material behavior the J Integral gives a value for the energy release rate, or the energy that is available for the growth of a crack. In calculations with viscoelastic material behavior this is not true anymore. This means that simulations with viscoelastic material behavior can only be used for qualitative comparison. After the simulations the resulting values of the J Integral can be used to determine the stress intensity factor K I. This stress intensity factor can be compared with the fracture toughness of the material. However, the fracture toughness of quartz is temperature dependent and unknown for higher temperatures. This makes it not possible to compare the stress intensity factors from the simulations with the fracture toughness of the material. This all leads to the conclusion that it is possible to implement fracture mechanics in the current model and that it gives more information than the model without fracture mechanics, but that there are too much insecurities regarding the interpretation of the J Integral calculations with viscoelastic material behavior and comparing the results with a fracture toughness, that at this point it can not be used as an instrument to evaluate the behavior of initial cracks under certain circumstances.

3 Contents 1 Introduction 3 2 UHP Lamps Gas discharge lamps High pressure mercury lamps Production of UHP lamps The gas discharge bulb The current lead The assembly of the current lead The filling The sealing The assembly of the lamp Fracture Mechanics Introduction Non-thermal stress conditions Theory Numerical implementation Thermal stress conditions Numerical implementation Material Properties Thermal properties Mechanical properties Viscoelastic material behavior Theory Structural relaxation J Integral

4 5 The Modeling of the Lamp Introduction Thermal boundary conditions Initial conditions Mechanical boundary conditions Simulations Interpretation of J Integrals The position of the crack Elastic material behavior Viscoelastic material behavior The size of the crack Elastic material behavior Viscoelastic material behavior Discussion Conclusions 33 8 Recommendations 34 9 References 35 A Heat Distribution 36 B Crack Mesh Procedure 37 2

5 Chapter 1 Introduction There are many consumer products in the market that use lamps. Some applications require lamps that emit relatively much light from a small point. Ultra High Performance (UHP) lamps meet these requirements and are therefore widely used in products like beamers and special type of TVs. Philips Lighting BV produces UHP lamps in Turnhout, Belgium. Different types of burners are produced for several applications. After production of the lamps tests are done to check the light quality. During these tests it sometimes happens that lamps fail at the first time burning. Causes for these early failures can be initial cracks after production or defects due to crystallization of the material. In both cases an initial crack exists in the quartz before the lamp starts burning. In fracture mechanics there are ways to determine whether or not an existing crack will grow and eventually cause an explosion at the given thermal and mechanical loads. One way is to calculate the stress intensity factor and compare this factor, which is dependent on the amount of stress and the geometry of the crack, with the critical fracture toughness. The critical fracture toughness is a material parameter that can be temperature dependent. The stress intensity factor can be calculated using the J Integral along a crack front. For this purpose a finite element model is needed which includes the initial crack(s). A finite element model to simulate the behavior of the UHP lamp is already available. This model is used to compare stresses and temperatures in the lamp at different geometries and material properties. In the course of time this model is optimized and results are compared with results of experiments. The goal of this project is to find out whether it is possible and useful to implement J Integral calculations in the current model and to examine the influence of the size and the position of the crack on the value of the J Integral. The next chapter is about the working principle of the UHP lamp and the way it is produced. In the third chapter the principles of fracture mechanics are treated, followed by a chapter about the material behavior of the used materials. The total model is described in the fifth chapter. Chapter six deals with the simulations and the results of the simulations. Finally there are the conclusions and some recommendations. 3

6 Chapter 2 UHP Lamps 2.1 Gas discharge lamps An UHP (Ultra High Performance) lamp is a high pressure gas discharge lamp. Compared to low pressure gas discharge lamps, high pressure gas discharge lamps are usually more compact, have a higher internal pressure, higher power and deliver more light. The burner of a high pressure gas discharge lamp is a tube of quartz with sealed-in electrodes, filled with a metal vapor. The electrons emitted by the electrodes are accelerated by the electric field and can have collisions with the gas atoms and molecules. These collisions lead to heat generation and excitation or ionization of the gas atoms. Excited atoms can generate electromagnetic radiation. Ionization increases the electron concentration in the gas. Figure 2.1: UHP lamp An electric discharge is only possible if the number of charged particles remains at a sufficiently high and constant level so that an electric current can flow through the vapor or gas. Since most vapors and gases are good insulators nothing is likely to happen when a voltage is applied to the two electrodes of a discharge tube. That is why in every discharge lamp a 4

7 Figure 2.2: Burner of an UHP lamp noble gas or a mixture of noble gases is added to the metal vapor. When a low electrical current between the two main electrodes has become established, the temperature will rise because of excitation and ionization processes in the bulb. Due to the temperature rise the metal filling can vaporize. With the vapor pressure also the number of excited metal atoms increases. Discharge of the metal vapor takes over the discharge of the noble gas more and more. Finally, an equilibrium is reached. The composition of the electromagnetic radiation depends on the discharge tube filling. A part of the electromagnetic radiation might be visible. The luminous flux and the spectral distribution of the emitted radiation are highly influenced by the metal vapor pressure in the discharge tube. The pressure depends on the temperature of the coldest spot in the tube in case of a saturated burner. For unsaturated burners the amount of metal and the temperature distribution determine the pressure. The available power determines the lamp temperature. The lamp voltage of a gas discharge is determined by the nature and pressure of the filling and by the electrode spacing. Because the lamp temperature influences the vapor pressure, discharge lamps are so constructed that the influence of changes in ambient temperature on the discharge is reduced to a minimum, for instance through the use of an outer bulb. 2.2 High pressure mercury lamps The basic principle of the high pressure mercury lamp is the radiation of a discharge in mercury vapor at a relatively high pressure. To get a good luminous efficacy a pressure of 20 MPa has to be achieved. Any metal vapor pressure depends on the temperature of the metal and the amount of mercury. The high vapor pressure and the high wall temperature require a special highly resistant material. Quartz with its high softening point and a very good transmission for wavelengths between 185 and 4000 nm, proved a good material, although it has some disadvantages. First of all the high processing temperature of 1600 to 1700 C and besides that the low coefficient of expansion of quartz glass, which makes it impossible to make a metal to quartz seal in the same way as a metal to glass seal. This problem can be solved with the use of a thin molybdenum foil as the current lead. The electrode is made of tungsten, a metal with a high boiling point, so the electrode material will not evaporate quickly at the high temperatures prevalent during operation. Blackening of the discharge tube will be limited. 5

8 2.3 Production of UHP lamps The production of the UHP burner takes place in 6 steps The gas discharge bulb In the first step a gas discharge bulb is made in the middle of a quartz tube. To produce this bulb a quartz tube of about 20 centimeters is picked up and rotated. Then it is shortly heated by a flame, to remove stresses in the quartz. Continuing rotating, the quartz is heated in the middle of the tube and during this heating the ends of the tube are moved to each other which results in an expansion of the tube where it is heated. At this point a die is placed around the tube to produce the right form of the bulb The current lead In this step the different parts of the current lead are mounted to each other. There are three parts: the supply wire, the molybdenum feed-trough and the tungsten rod. The process starts with the supply wire that is turned into a spring, what is necessary later in the production when the total current lead is put into the quartz tube. The spring is mounted onto the molybdenum feed-through using a resistance welding process. This process is also used for mounting the tungsten rod onto the molybdenum feed-through. The most important issue in this step of the process is the aligning of the three parts, because that is necessary to achieve a light source that is as small as possible The assembly of the current lead The next process step is to put the complete current leads into the quartz tube. Because of the spring the current lead is not moving when it is put into the tube. When both current leads are put into the tube, they have to be moved to the middle of it, with both rods at a certain distance from each other. This is done by two pins that are pushed into both sides of the tube. With the use of x-rays the end of the rods can be detected and so the rods can be positioned at the right place The filling One of the ends of the tube is closed by a laser beam that melts the quartz. After that the air is pumped out of the tube. When the tube is completely vacuum the tube is filled with mercury and argon (the starting gas). When the tube is filled the other end of the tube is closed The sealing The mercury has to be sublimated to make sure that it stays at the bottom side of the tube. Then the upper part of the tube is sealed by a laser beam. First close to the bulb, and then with the beam moving upwards. After cooling of the tube, it is turned upside down. At this 6

9 moment it is more difficult to keep the mercury sublimated, so the bulb has to be cooled (at -200 C) with liquid nitrogen, while the other side of the tube is sealed (at 1800 C). After this step the ends are cut off by a leaser beam and the burner is ready for assembly into the lamp The assembly of the lamp There are a lot of different applications of the UHP burner, which requires a lot of different assemblies. The assembly process starts with a burner that is put in a reflector. The burner is ignited and heated until it reaches the maximum gas discharge. At that moment the burner is put by hand into the focal point of the reflector. The position of that focal point is found by measuring the light intensity of the burner. Because the burner has a very small light source, a small movement to or from the focal point of the reflector makes a significant difference in the light intensity. When the burner is positioned at the right place, it is fixed with a ceramic glue. 7

10 Chapter 3 Fracture Mechanics 3.1 Introduction The magnitude of an elastic crack tip stress field can be described by the stress intensity factor K. A crack in a solid can be loaded in three different modes, as illustrated in figure 3.1. Normal stresses lead to the opening mode or mode I loading. The displacement of the crack surfaces are perpendicular to the plane of the crack. In-plane shear results in mode II or the sliding mode, the displacements of the crack surfaces are in the plane of the crack and perpendicular to the leading edge of the crack. The tearing mode or mode III is caused by out-of-plane shear. Crack surface displacements are in the plane of the crack and parallel to the leading edge of the crack. The superposition of the three modes describes the general case of loading. mode I mode II mode III Figure 3.1: The three modes of loading The stress intensity factors are determined using J Integrals calculated by MSC.Marc. The way this is done is first explained for non-thermal stress conditions. Then for thermal stress conditions, and finally for the case where the material behaves visco-elastic. 8

11 3.2 Non-thermal stress conditions Theory The J Integral is a line integral that, in cases of elastically behaving material, gives a value for the energy release rate. Because of some important properties, the J Integral is suitable for using it in a crack-growth criterion. To show this, first a homogeneous body of linear elastic material is considered, free of body forces and subjected with all stresses σ ij depending only on two Cartesian coordinates x 1 (= x) and x 2 (= y) (figure 3.2). X 2 A 0 Γ X 1 Figure 3.2: Homogeneous body In this case the definition of the J Integral is [Rice1968]: J = Γ ( ) u i W n 1 T i ds (3.1) x 1 where W is the strain energy, Γ is a closed path in the x 1 x 2 plane, and s is the distance along the path. Also T i = σ ij n j (with n i the component of the unit normal in x i direction on Γ) and u i are components of the traction and displacement vectors, respectively. The stresses σ ij are related to the strains ɛ ij with E (Young s modulus) and ν (Poisson s ratio), σ ij = which, using W = 1 2 ɛ ijσ ij, leads to: νe (1 + ν) (1 2ν) ɛ kkδ ij + E (1 + ν) ɛ ij (3.2) W (ɛ ij ) = 1 νe 2 (1 + ν) (1 2ν) (ɛ kk) 2 E + 2 (1 + ν) ɛ ijɛ ij (3.3) To prove that the value for the J Integral is equal to zero over a closed path, the line integral can be transformed into a surface integral, using the Green-Gauss theorem. J = A 0 [ W x 1 ( ) u ] i σ ij da (3.4) x j x 1 9

12 where A 0 is the area enclosed by the path Γ. Here it is assumed that no singularities are enclosed by the path Γ. Because the strain energy W has been written as a function of strain components only (see equation (3.3)), W x 1 can (in case of no body forces), using equilibrium, be written as: W = W [ ( ) ɛ ij ɛ ] ij ui = σ ij = σ ij = ( ) u i σ ij x 1 ɛ ij x 1 x 1 x j x 1 x j x 1 (3.5) Substituting the above equation into equation (3.4) results in the integrand of that equation being equal to zero and therefore J = 0. Thus the value of the J-line integral over a closed path, enclosing an area free of singularities, is equal to zero. Now it has been proven that the J Integral over a closed path is equal to zero, the integral can be calculated over any path surrounding a crack tip to determine the stress intensity factors. x 2 n Ω Γ + Γ Γ B n x 1 Γ A Figure 3.3: Path independence Consider the closed path (Γ A + Γ + + Γ B + Γ ) shown in figure 3.3. It follows that where [ ] + [ ] + [ ] + [ ] = 0 (3.6) Γ A Γ B Γ + Γ ( [ ] = W n 1 T i u i x 1 ) ds (3.7) If path Γ B is circular with radius r 1 around the crack tip and if r 1 approaches 0, then the stresses and displacements in the region through which the path Γ B passes, can be described using the crack tip equations depending on the mode I and mode II stress intensity factors 10

13 [Schreurs1996]. Here only mode I stress intensity factors are considered, which for plane strain conditions results in an evaluation of the contribution J ΓB along Γ B and using equation (3.6) (assuming that tractions on Γ + and Γ are negligible) eventually leads to: (1 ν 2 ) KI 2 = J ΓA (3.8) E Because of path independence of the J-line Integral, K I can be evaluated using equation (3.8) with J ΓA evaluated over any path Γ A which surrounds the crack tip Numerical implementation The J Integral evaluation in MSC.Marc is based upon the domain integration method. Due to difficulties in considering the integration path Γ A, a direct evaluation of J ΓA is not very practical in a finite element analysis. Therefore a derivation of the domain integral expression for the energy release rate is determined, using equation (3.1) as a starting point. To determine the stress intensity factor K I, J ΓA needs to be calculated. To convert the line integral into a surface integral a function q(x, y) is added to the integral. This can only be done if q(x, y) = 1 on Γ A and q(x, y) = 0 on Γ B : J ΓA = [ ] ds = [ ] q(x, y)ds + [ ] q(x, y)ds + [ ] q(x, y)ds + [ ] q(x, y)ds Γ A Γ A Γ B Γ + Γ with = [ ] q(x, y)ds = ( [ ] = A [ W q x 1 σ ij u i x 1 q x j W n 1 T i u i x 1 ] da (3.9) ) ds (3.10) Equivalent to the path independence of equation (3.1), equation (3.9) is domain-independent so that any domain can be chosen for the purpose of evaluating J. This makes it possible to choose the domain in accordance with the mesh design. In a three dimensional case there is a crack front instead of a crack tip. In this case it is not possible to calculate one single value for the J Integral. Therefore the J Integral is calculated at several points along the three dimensional crack front. This results in multiple values for the J Integral for one crack. 3.3 Thermal stress conditions In the case of no thermal stress conditions, the strain energy density W was only depending on the strain ɛ ij (see equation 3.3). When thermal stresses are involved, W is also depending on the temperature T (T is temperature rise with respect to a reference temperature). Therefore the J Integral over a closed path, as defined in equation (3.1), is not equal to zero. If the J 11

14 Integral is combined with an area integral inside the closed path, crack tip stress intensity factors can be calculated for any path surrounding the crack tip. This is shown in this section. The definition of the J Integral is still as defined by equations (3.1), with in case of thermal stress conditions: W = W (ɛ ij, T ) = 1 2 ɛ ijσ ij (3.11) The elastic relationship between stress and strain components is now σ ij = λɛ ii δ ij + 2µɛ ij Eα 1 2ν T δ ij (3.12) where T is the temperature and α is the coefficient of thermal expansion. When, analogous to the non-thermal case, W x 1 is derived, an extra term has to be added, due to the thermal strain: W = W ɛ ij + W T (3.13) x 1 ɛ ij x 1 T x 1 Using equations (3.2) and (3.12), equation (3.13) can be given the more specific form W x 1 = σ ij ɛ ij x 1 + Eα [ 1 1 2ν 2 T (T ɛ ii ) ɛ ii x 1 x 1 ] (3.14) which finally leads to J = Eα [ 1 1 2ν Γ 2 T (T ɛ ii ) ɛ ii x 1 x 1 ] da (3.15) Since the value for this J Integral is not equal to zero in this case, there is also an extra term needed in the calculation for the stress intensity factor. ( 1 ν 2 ) E K 2 I = J ΓA Numerical implementation Eα [ 1 1 2ν Γ 2 T (T ɛ ii ) ɛ ii x 1 x 1 ] da (3.16) In MSC.Marc the calculation of J ΓA in equation (3.16) takes place as explained in section When thermal stresses are involved, also the extra term is calculated. The result of the J Integral calculations in MSC.Marc contains both terms and includes therefore the entire right side of equation (3.16). 12

15 Chapter 4 Material Properties In this chapter the behavior and properties of the different materials are discussed, first the thermal properties and then the mechanical behavior. Because of the symmetry of the burner, only a quarter of the burner is modeled. The materials used in the model are quartz, tungsten, molybdenum and mercury (for the gap between the the molybdenum lead and the quartz tube). At modeling the geometry of the model some assumptions are made to avoid numerical problems. To compensate for these geometry assumptions some material properties are adapted. The first geometry part where this occurs is the tungsten coil. In reality the coil is a wire with length l and diameter d. But because l >> d a large amount of elements would be needed for accurate numerical computations. To avoid this the coil is modeled as a cone around a tungsten rod. This leads to a correction in the thermal conductivity and the emissivity, as described later in this chapter. The other geometry part is the molybdenum foil. This foil is in reality very thin in an attempt to reduce the difference in expansion caused by a temperature rise between the molybdenum foil and the quartz tube. Again for numerical reasons, the thin foil is modeled as a cilinder. This is corrected by a different thermal conductivity. 4.1 Thermal properties Table 4.1 shows thermal material properties of all used materials [Peters1995]. The specific heat of quartz and both the thermal conductivity and the emissivity of quartz and tungsten are temperature dependent. These properties are shown in the indicated figures (figure 4.1, 4.2, 4.3 and 4.4). The behavior of the thermal conductivity of tungsten, as can be seen in figure 4.3, simulates the melting process of the material. The transition from solid tungsten to liquid tungsten causes the rise in thermal conductivity. 13

16 Material Part Specific Thermal Emissivity Density heat conductivity J/gK W/mmK - g/mm 3 quartz Bulb figure 4.1 figure 4.2 figure tungsten 1 Rod figure 4.3 figure tungsten 2 Coil figure 4.3 figure mercury Capillary molybdenum 1 Foil molybdenum 2 Lead thermal conductivity of molybdenum 1 in length direction is W/mmK Table 4.1: Thermal properties 1.4 Specific heat of quartz quartz 6 x 10 3 Thermal conductivity of quartz quartz specific heat [J/gK] conductivity [W/mmK] Temperature [K] Temperature [K] Figure 4.1: Specific heat (J/gK) Figure 4.2: Conductivity (W/mmK) Thermal conductivity of tungsten tungsten 1 tungsten Emissivity of quartz and tungsten quartz tungsten 1 tungsten conductivity [W/mmK] emissivity [ ] Temperature [K] Temperature [K] Figure 4.3: Conductivity (W/mmK) Figure 4.4: Emissivity ( ) 4.2 Mechanical properties In the mechanical analysis only the quartz bulb and the tungsten electrode rod are taken into account. Simulations are done with both elastic and viscoelastic material behavior. During 14

17 Material Part E-modulus Poisson s ratio Thermal expansion coefficient N/mm 2-1/K quartz Bulb tungsten 1 Rod Table 4.2: Mechanical properties the simulation mechanical stresses appear due to gas pressure, temperature gradients and thermal expansion mismatch between quartz and tungsten rod. The mechanical properties of the materials when both the quartz and the tungsten are behaving elastically are shown in table 4.2 [Peters1995]. Because the material properties are much more complicated when the material behavior is viscoelastic, this is described in a separate section. 4.3 Viscoelastic material behavior Theory The viscoelastic material behavior is described by a multi-mode Maxwell model, a parallel connection of an amount of Maxwell models that all have their own relaxation time τ i and contribution to the stiffness E i (see figure 4.5). E 1 E 2 E 3 E i E η 1 η 2 η 3 η i Figure 4.5: Generalized Maxwell model In the case of complete relaxation, E = 0. Maxwell model is described by: In this case the behavior of the generalized σ (t) = ε 0 E i exp ( t/τ i ) (4.1) i 15

18 with: τ i = η i /E i (4.2) The stress response of the material after a step ε 0 in the strain at t = 0 is shown in figure 4.6. After a sudden increase of the stress, stress relaxation occurs which results in a smooth decrease of the stress to a stationary value σ. When σ = 0, there is complete relaxation. ε σ σ 0 ε 0 σ t 0 t t 0 t Figure 4.6: Viscoelastic stress response At higher temperatures the stress decreases faster than at lower temperatures. This is shown in figure 4.7. log E T Figure 4.7: Temperature dependent E modulus log t It can be observed that for increasing temperatures the shape of the curve stays the same, it only moves to the left. For decreasing temperatures the curve moves to the right. So viscoelastic behavior at one temperature can be related to another temperature by a change in time-scale only. To use the fact that each individually measured curve at a different temperature has the same shape, one master curve can be determined. By doing experiments at different temperatures in the same range of time, this master curve can be generated. This is shown in figure 4.8. In this figure can be seen the measurements at the different 16

19 log E T 7 T 6 T 5 T 4 T 3 T 2 T 1 log t Figure 4.8: Generating master curve, T 1 < T 2 < T 3 < T 4 < T 5 < T 6 < T 7 temperatures in the same range of time. The results of the measurements can be shifted over the x-axis which results in one single master curve. The temperature that belongs to the part of the curve that is not shifted is used as the reference temperature of this master curve. For all other temperatures a shift function is necessary to calculate the right viscoelastic behavior. There are several functions that can be used as a shift function, one of them is the Arrhenius function: ln a T = A R ( 1 T 1 T0 ) (4.3) where A is the activation energy, R is the universal gas constant and T is the temperature (in degrees Kelvin). R had the value of kjmol 1 K 1. The quantity a T is defined as: a T = τ(t ) τ(t 0 ) (4.4) Using this shift function, where τ is the time belonging to the measurements at a certain temperature, the viscoelastic material behavior at a specific temperature can be determined Structural relaxation It is possible that after production of the lamp, the glass is not in its volumetric equilibrium, due to fast cooling down from a temperature higher than the glass transition temperature (T g ) to room temperature. At turning on the lamp, a temperature rise in the glass can cause structural relaxation, which means that the material wants to approach its volumetric equilibrium, as a result of an altered thermal expansion coefficient. In the case of quartz glass there is not much structural relaxation, because of the small thermal expansion coefficient. Therefore the structural relaxation is neglected in this model. 17

20 4.3.3 J Integral Calculation of the J Integral gives a value for the energy that is available in a system for the growth of a crack. When it concerns linear material behavior, all this energy can be used for the growth of the crack, but in the case of viscoelastic material a part of this total energy is dissipated. To find out how much energy is available for the growth of the crack an extra term in the formulation of the J Integral is needed, as shown by Guttierez [Guttierez2002]. However, this formulation is not used by MSC.Marc. This means that the value of the J Integral in viscoelastic simulations does no longer describe only the energy that is available for the growth of a crack. Therefore the results of viscoelastic simulations are mainly used for comparing the different simulations and not to compare them with a fracture toughness. 18

21 Chapter 5 The Modeling of the Lamp 5.1 Introduction An important part in the analysis of the mechanical behavior of the lamp is the heat exchange between the different materials in the lamp and between the lamp and the environment. Heat is generated by the current flow through the electrodes. This heat is conducted through all the components of the lamp to the outer surface, where heat losses appear caused by convection and radiation effects. 5.2 Thermal boundary conditions δ 1 δ 2 δ 3 δ 4 δ 5 Mercury Tungsten (rod) Tungsten (coil) Molybdenum (foil) Molybdenum (lead) Quartz Figure 5.1: Cross section of the lamp There are several thermal boundary conditions applying on the lamp. The boundary of the lamp is therefore divided in several parts, as can be seen in figure 5.1. Heat is generated by the current flow through the electrodes. The amount of heat that is 19

22 generated is dependent on the lamp power and the operating voltage, which depends on the plasma pressure and the distance between the electrodes. where: q t is the total heat flux caused by current flow [W] V tip is the operating voltage [V] I lamp is the electric current [A] A tip is the area of the electrode tip [m 2 ] Q t = V tip I lamp (5.1) Because of this heat generation both the electrodes and the mercury plasma are heated. The total amount of heat can be divided in three parts, namely the heat that directly warms the electrodes (q d ) and heat that warms the mercury plasma and so indirectly the inner wall of the bulb (q w ) and the outside of the electrodes (q e ). This means that Q t = q d A d + q w A w + q e A e (5.2) where: q t is the total heat flux caused by current flow [W/(m 2 )] q d is the heat flux at the electrode tip [W/(m 2 )] q w is the heat flux at the inner wall of the bulb [W/(m 2 )] q e is the heat flux at the outside of the electrodes [W/(m 2 )] A d is the area of the electrode tip [m 2 ] A w is the area of the inner wall of the bulb [m 2 ] A e is the area of the outside of the electrodes [m 2 ] Heat losses appear by convection and radiation to the environment. Free convection at the outside of the lamp can be calculated as follows: q c = h c (T w T ) (5.3) where: q c is the heat flux caused by convection [W/(m 2 )] h c is the average convection heat transfer coefficient [W/(m 2 K)] T w is the temperature of the wall [K] is the temperature of the environment [K] T Radiation effects are described by: ( ) q r = σε Tw 4 T 4 (5.4) where σ is called the Stefan-Boltzmann constant and has the value of σ = W/ ( m 2 K 4) and: q r is the heat flux caused by radiation [W/(m 2 )] ε is the emissivity of the material (ε = 1 for an ideal radiator) [-] 20

23 The applied boundary conditions for each boundary are: boundary conditions on δ 1 q = q w (5.5) boundary conditions on δ 2 ( ) q = q e σε Tw 4 T 4 (5.6) boundary conditions on δ 3 ( ) q = q d σε Tw 4 T 4 (5.7) boundary conditions on δ 4 and δ 5 ( ) q = h c (T w T ) σε Tw 4 T 4 (5.8) 5.3 Initial conditions Both the temperature of the environment and the temperature of the lamp at t = 0 are equal to 300 K: T = 300 K T (t = 0) = 300 K 5.4 Mechanical boundary conditions The most important mechanical load is a result of the internal pressure in the bulb when the lamp is burning. This mechanical load is applied on δ 1 in figure 5.1. Other mechanical boundary conditions consist of fixing the geometry at the symmetry planes. 21

24 Chapter 6 Simulations The model is used as a starting point for different kind of simulations. The goal of the simulations is to find out whether the position and the size of a possible crack are important for the stress intensity factor at the crack. Before the results of these simulations can be evaluated, it has to be determined which values are suitable for comparison. All simulations have the same loading history. The internal pressure is shown in figure 6.1. The lamp is turned on at t = 0 s and turned off at t = 300 s (later in this chapter when viscoelastic material behavior is used, the lamp is turned of at t = 600 s) pressure [N/mm 2 ] time [s] Figure 6.1: Internal pressure during simulations When the lamp is burning an initial crack is opening due to thermal and mechanical loading. This can be seen in figure 6.2. However, in this figure the displacements are multiplied by In reality the crack opening during the simulations is so small that it is not visible. 6.1 Interpretation of J Integrals Every simulation results in 7 tables with for every increment 13 J Integral values along the crack front. There are 7 tables because there are 7 paths selected around the crack tip. Since 22

25 Figure 6.2: Geometry of an opened crack the calculation of the J Integral should be path independent, the values in these 7 tables should be equal. Due to a coarser mesh away from the crack front, the values in the tables with the results for these paths are not the same anymore and thus not reliable. That is why the results of the 3 rd path are chosen, for evaluating the results of the simulations Figure 6.3: Position of the crack in the bulb Figure 6.4: Crack front (cross section) The differences in the values along the crack front are dependent on the shape of the crack. Because the shape of the crack is constant for all simulations, first an analysis of the variations in the values along the crack front is done. The nodes along the crack front that are used in the J Integral calculations are numbered 1 to 13. Node 1 is located at the outer side of the lamp and node 13 is located at the symmetry plane of the bulb. A possible position of the crack area in the bulb is shown in figure 6.3 and a cross section of the crack geometry in figure 6.4. To examine the variations along the crack front the values of the J Integral of one of the 23

26 simulations are plotted for all nodes as a function of the time (see figure 6.5). J Integral 16 x J Integral along a crack front node 1 node 2 node 3 node 4 node 5 node 6 node 7 node 8 node 9 node 10 node 11 node 12 node time [s] Figure 6.5: J Integral along a crack front It can be observed that the value of the J Integral in the first few nodes (from the outside of the bulb) is higher than in the other nodes. This means that if an initial crack in the bulb has a geometry like the crack in figure 6.4, the crack will start to grow in the direction of those nodes. Figure 6.3 indicates that a crack that grows in this direction proceeds along the surface of the bulb and not towards the middle of it. Optimizing the geometry of the crack would result in a more elliptic form of the crack. Since the highest value, and thus the most critical value, of the J Integral is found at node 1, the values of the J Integral around this node will be compared to each other, while varying the position of the crack and the size of the crack. 6.2 The position of the crack To examine the influence of the position of the crack, simulations are done with both elastic and viscoelastic material behavior. First the results of the simulations with elastic material behavior are presented, after that the results of the simulations with viscoelastic material behavior. 24

27 6.2.1 Elastic material behavior First the J Integral values along the bulb are compared as a function of the time, for cracks at 20 different positions along the bulb. The procedure used for meshing the cracks in shown in Appendix B. The first position is at the top of the bulb with the subsequent ones shifting to the neck with small steps. This is visualized in figure 6.6. In the heat flux at the inner side of the wall, also gravity is taken into account (see for the heat distribution appendix A). For this reason one side of the bulb reaches higher temperatures than the other side. The cracks in these simulations are situated at the side where the lamp reaches the highest temperatures. Figure 6.6: Cracks along the bulb First the results of the simulations using elastic material behavior are shown in figure 6.7. What can be observed in this figure is the value of the J Integral for the different cracks as a function of the time. In the first 60 seconds the value of the J Integral increases, corresponding with the pressure. When the lamp is turned off at t = 300 s, the value of the J Integral decreases immediately. When comparing the values of the J Integral for the different cracks, the first thing that can be concluded is that there are big J Integral variations between the different cracks and that the values are more or less constant in the first 300 seconds (when the lamp is burning) and in the last 300 seconds (when the lamp is not burning). When the lamp is burning crack 11 is the most critical, followed by crack 10, crack 9 and crack 12. When the lamp is not burning the differences are much smaller, but crack 20 has the most critical value of the J Integral. To compare the J Integral values of the different cracks they are drawn as a function of the 25

28 x 10 4 J Integral [N/mm] 2 1 crack 1 crack 2 crack 3 crack 4 crack 5 crack 6 crack 7 crack 8 crack 9 crack 10 crack 11 crack 12 crack 13 crack 14 crack 15 crack 16 crack 17 crack 18 crack 19 crack time [s] Figure 6.7: J Integral for various crack locations using elastic material behavior x x J Integral [N/mm] 2 1 J Integral [N/mm] arc length [mm] arc length [mm] Figure 6.8: J Integral along the bulb at t=60 s (lamp burning) Figure 6.9: J Integral along the bulb at t=600 s (lamp off) arc length when the lamp is burning (for t = 60 s), and when the lamp is not burning (for t = 600 s). This is shown in the figures 6.8 and 6.9. In these figures the value of the J Integral is plotted as a function of the arc length. The arc length is measured from the top of the bulb in the length direction along the 20 cracks (see figure 6.6). When the lamp is not burning (figure 6.9) the value of the J Integral increases along the bulb. However, the maximum value of the J Integral (at crack 20) is still very small (mind the scaling of the vertical axis) compared to the values of the J Integral when the lamp is burning (figure 6.8). 26

29 In this case there is a clear maximum visible, around crack 10 and 11 (arc length about 3.5 mm). In the direction of crack 20 (arc length 4.7 mm) the value of the J Integral decreases quite fast. In the other direction, towards the top of the bulb (arc length 0.3 mm), the value of the J Integral decreases less, but cracks at those positions are obviously less critical than crack 10 and Viscoelastic material behavior J Integral [N/mm] 1.2 x crack 1 crack 2 crack 3 crack 4 crack 5 crack 6 crack 7 crack 8 crack 9 crack 10 crack 11 crack 12 crack 13 crack 14 crack 15 crack 16 crack 17 crack 18 crack 19 crack time [s] Figure 6.10: J Integral for various crack using viscoelastic material behavior The same simulations are also carried out with viscoelastic material behavior. The only difference is that instead of a simulation time of 600 seconds for the simulations with elastic material behavior, the simulations with viscoelastic material behavior have a simulation time of 1200 seconds, because of the time dependency of the viscoelastic material behavior. The lamp is turned on at t = 0 s and turned off at t = 600 s. Again first the J Integral values along the bulb are compared as a function of the time (figure 6.10). For these simulations the same trends are visible as for the simulations with elastic material behavior. After t = 60 s, the stress increases or decreases as a result of the viscoelastic material behavior. This can be explained by the temperature distribution in the bulb. The temperatures at the inside of the bulb are the highest, so there the material is most viscous. The behavior of the material at the inside of the bulb results in stress relaxation in some parts at the outside of the bulb, and a stress increase at other places. In general the viscoelastic material behavior does not have much influence on the trends, but an important difference is the maximum value of the 27

30 J Integral. The maximum value is two times as high for the simulations with elastic material behavior as a result of higher stresses. Another difference is the position of the crack with the maximum value, which is not crack 11 (as it was for the simulations with elastic material behavior), but crack 10. Again the values of the J Integral are plotted as a function of the arc length in the figures 6.11 and 6.12 to be compared to the figures 6.8 and 6.9. The results for viscoelastic behavior confirm the trends as observed for the simulations with elastic material behavior. 1.2 x x J Integral [N/mm] J Integral [N/mm] arc length [mm] arc length [mm] Figure 6.11: J Integral along the bulb at t=600 s (lamp burning) Figure 6.12: J Integral along the bulb at t=1200 s (lamp off) 6.3 The size of the crack The analysis of the influence of the crack size is also performed by simulations with elastic and viscoelastic material behavior. Again, first the results of the simulations with elastic material behavior are presented and in the following section the results of the simulations with viscoelastic material behavior Elastic material behavior To examine the influence of the size of the crack on the value of the J Integral, cracks with a different size are modeled. The geometry of the crack is taken constant, only the crack depth changes. When the value of the J Integral is not increasing for a deeper crack, then the crack might stabilize at a certain size. The simulations with three different cracks and elastic material behavior are shown in figure In this figure crack 1 is the deepest crack, crack 3 the smallest. So if the crack size increases, also the value of the J Integral increases. This means that in this range of crack depths there will be no stabilization of the crack. To show this in another way the value of the J Integral is plotted as a function of the crack depth, see figure In this figure two lines are visible. The line indicated with node 1 is fitted through the values of the J Integral around node 1 as plotted in figure It can be observed that for a deeper crack, the value of the J Integral increases more. This means that 28

31 x 10 4 crack 1 crack 2 crack 3 J Integral [N/mm] time [s] Figure 6.13: J Integral elastic material behavior different crack sizes 2.2 x node 1 node 1 node 13 node J Integral [N/mm] crack depth [mm] Figure 6.14: J Integral elastic material behavior different crack sizes a crack in this range of sizes in this direction will be unstable. To determine whether the same is true for crack growth in the direction towards the center of the lamp the values of the J Integral around node 13 are plotted in the same figure. These values are lower, as already established earlier this chapter, but the same trend is visible. So also for crack growth in the direction of the center of the lamp, cracks with a size in this range are unstable. 29

32 6.3.2 Viscoelastic material behavior In figure 6.15 the results of the simulations for viscoelastic material behavior are shown. There is hardly any difference visible in the trend compared to the simulations using elastic material behavior, however the values of the J Integral are much lower. Nevertheless, again the values of the J Integral are plotted as a function of the crack depth. This is visible in figure This figure confirms the conclusion that is drawn after analyzing the results of the simulations for elastic material behavior. So also for viscoelastic material behavior it can be concluded that when a crack within this crack size range will start growing, the crack growth will be critical, both in the direction over the surface of the bulb as in the direction towards the center of the bulb. 9 x crack 1 crack 2 crack 3 7 J Integral [N/mm] time [s] Figure 6.15: J Integral viscoelastic material behavior different crack sizes 6.4 Discussion The results of the simulations lead to some conclusions. First of all the fact that the cracks as implemented in the current model are more likely to grow over the surface of the bulb instead of in the direction of the center of the bulb. A second result shows the weakest spot on the bulb under the used circumstances. At this spot (arc length about 3.5 mm) the value of the J Integral is clearly higher than at other spots. Since the main goal of this research is to evaluate the importance of using fracture mechanics in the current model, the results of the simulations are compared to the stress distribution in the lamp. When the results of the J Integral simulations show the same critical places as would be expected by examining the stress distribution, it could be concluded that implementing J Integral calculations is not giving extra information. In figures 6.17 and 6.18 both the J Integral values along the arc length (as defined in section 6.2.1) as the stress distribution along the arc length are shown. Both the values of the J Integral as the stress distribution show at t = 60 s a maximum around an arc length of 4 mm. However, close to the top of 30

33 10 x node 1 node 1 node 13 node 13 8 J Integral [N/mm] crack depth [mm] Figure 6.16: J Integral viscoelastic material behavior different crack sizes the bulb (arc length 0) the stress decreases much more than the value of the J Integral. This can be caused by the influence of higher temperatures or by the geometry of the bulb. At t = 600 s both the value of the J Integral and the stress show the same trend, until the neck of the burner is reached. Figure 6.17: J Integral compared to stress distribution at t=60 s Figure 6.18: J Integral compared to stress distribution at t=600 s The next conclusion is that an initial crack with a size between 0.1 and 0.3 mm is unstable. This means that when the value of the J Integral of such a crack exceeds the critical value of the J Integral, the crack will keep growing, and will not reach a stable state. A final point of discussion is the comparison of the values of the J Integral with a critical value. This is possible when the value of the J Integral is used to determine the stress intensity factor K I, which can be compared with the fracture toughness of the material. However, the fracture toughness of quartz is temperature dependent, at higher temperatures the fracture 31

34 toughness will increase due to the viscoelastic material behavior of quartz. The values of the fracture toughness of quartz at higher temperatures are not known. When equation (3.8) is used to determine the maximum stress intensity factors during the simulations, this leads to K I = 0.12 MPa m in the case of elastic material behavior and K I = 0.09 MPa m in the case of viscoelastic material behavior. The fracture toughness of quartz at room temperature is around 0.7 MPa m. Since the temperatures during the simulations are much higher than room temperature, the fracture toughness will also be higher, and cracks of this size under the given circumstances will not be critical. 32

35 Chapter 7 Conclusions The main goal of the project was to determine whether initial cracks in UHP lamps can be examined using fracture mechanics. Fracture mechanics theories can indicate when an existing crack is critical under certain mechanical and thermal loads. It can be concluded that the implementation of J Integral calculations is possible in the current model. A 3-dimensional crack can be meshed using the procedure described in Appendix B. Simulations lead to values of the J Integral over the crack front. In the case of viscoelastic material behavior the results of the simulations are not very accurate because for such material behavior an extra term is required in the calculation of the J Integral. However, this extra term is not implemented in MSC.Marc which results in a value for the J Integral that cannot be used as the amount of energy that is available for crack growth. Still, simulations with viscoelastic material behavior are done, to compare them with each other and to examine the resulting trends. The simulations done with both elastic and viscoelastic simulations lead to several conclusions. First of all there is the fact that the cracks as implemented in the current model are more likely to grow over the surface of the bulb instead of in the direction of the center of the bulb. A second result shows the weakest spot on the bulb under the used circumstances. At this spot (arc length about 3.5 mm) the value of the J Integral is clearly higher than at other spots. The final result from the simulations is that an initial crack with a size between 0.1 and 0.3 mm is unstable. This means that when the value of the J Integral of such a crack exceeds the critical value of the J Integral, the crack will keep growing, and will not reach a stable state. Finally there is the comparison of the values of the J Integral with a critical value. This is possible when the value of the J Integral is used to determine the stress intensity factor K I, which can be compared with the fracture toughness of the material. However, the fracture toughness of quartz is temperature dependent, at higher temperatures the fracture toughness will increase due to the viscoelastic material behavior of quartz. The values of the fracture toughness of quartz at higher temperatures are not known. This all leads to the conclusion that it is possible to implement fracture mechanics in the current model and that it gives more information than the model without fracture mechanics, but that there are too much insecurities regarding the interpretation of the J Integral calculations with viscoelastic material behavior and comparing the results with a fracture toughness, that at this point it can not be used as an instrument to evaluate the behavior of initial cracks under certain circumstances. 33