47
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
Fracture Mechanics in Quartz Lamps
M.M. JoostenMT06.16
Coaches:J.H.A. Selen, P.H.M. TimmermansW.A.M. Brekelmans, M.G.D. Geers
June 28, 2006
Abstract
There are many consumer products in the market that use lamps. Some applications requirelamps 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 andspecial type of TVs.
Philips Lighting BV produces UHP lamps in Turnhout, Belgium. Different types of burnersare produced for several applications. After production of the lamps tests are done to checkthe light quality. During these tests it sometimes happens that lamps fail at the first timeburning. Causes for these early failures can be initial cracks after production or defects dueto crystallization of the material. In both cases an initial crack exists in the quartz before thelamp starts burning. The goal of this project is to find out whether it is possible and usefulto implement J Integral calculations in the current model and to examine the influence of thesize 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. Forthese cracks simulations are done with both elastic as viscoelastic material behavior. Incalculations with elastic material behavior the J Integral gives a value for the energy releaserate, or the energy that is available for the growth of a crack. In calculations with viscoelasticmaterial behavior this is not true anymore. This means that simulations with viscoelasticmaterial behavior can only be used for qualitative comparison.
After the simulations the resulting values of the J Integral can be used to determine the stressintensity factor KI . This stress intensity factor can be compared with the fracture toughnessof the material. However, the fracture toughness of quartz is temperature dependent andunknown for higher temperatures. This makes it not possible to compare the stress intensityfactors 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 thecurrent model and that it gives more information than the model without fracture mechan-ics, but that there are too much insecurities regarding the interpretation of the J Integralcalculations with viscoelastic material behavior and comparing the results with a fracturetoughness, that at this point it can not be used as an instrument to evaluate the behavior ofinitial cracks under certain circumstances.
Contents
1 Introduction 3
2 UHP Lamps 4
2.1 Gas discharge lamps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 High pressure mercury lamps . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Production of UHP lamps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3.1 The gas discharge bulb . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3.2 The current lead . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3.3 The assembly of the current lead . . . . . . . . . . . . . . . . . . . . . 6
2.3.4 The filling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3.5 The sealing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3.6 The assembly of the lamp . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 Fracture Mechanics 8
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2 Non-thermal stress conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2.2 Numerical implementation . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.3 Thermal stress conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.3.1 Numerical implementation . . . . . . . . . . . . . . . . . . . . . . . . . 12
4 Material Properties 13
4.1 Thermal properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.2 Mechanical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.3 Viscoelastic material behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.3.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.3.2 Structural relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.3.3 J Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1
5 The Modeling of the Lamp 19
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5.2 Thermal boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5.3 Initial conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5.4 Mechanical boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . 21
6 Simulations 22
6.1 Interpretation of J Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
6.2 The position of the crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
6.2.1 Elastic material behavior . . . . . . . . . . . . . . . . . . . . . . . . . 25
6.2.2 Viscoelastic material behavior . . . . . . . . . . . . . . . . . . . . . . . 27
6.3 The size of the crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6.3.1 Elastic material behavior . . . . . . . . . . . . . . . . . . . . . . . . . 28
6.3.2 Viscoelastic material behavior . . . . . . . . . . . . . . . . . . . . . . . 30
6.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
7 Conclusions 33
8 Recommendations 34
9 References 35
A Heat Distribution 36
B Crack Mesh Procedure 37
2
Chapter 1
Introduction
There are many consumer products in the market that use lamps. Some applications requirelamps 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 andspecial type of TVs.
Philips Lighting BV produces UHP lamps in Turnhout, Belgium. Different types of burnersare produced for several applications. After production of the lamps tests are done to checkthe light quality. During these tests it sometimes happens that lamps fail at the first timeburning. Causes for these early failures can be initial cracks after production or defects dueto crystallization of the material. In both cases an initial crack exists in the quartz beforethe lamp starts burning.
In fracture mechanics there are ways to determine whether or not an existing crack will growand eventually cause an explosion at the given thermal and mechanical loads. One way isto calculate the stress intensity factor and compare this factor, which is dependent on theamount of stress and the geometry of the crack, with the critical fracture toughness. Thecritical fracture toughness is a material parameter that can be temperature dependent. Thestress intensity factor can be calculated using the J Integral along a crack front. For thispurpose 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. Thismodel is used to compare stresses and temperatures in the lamp at different geometries andmaterial properties. In the course of time this model is optimized and results are comparedwith results of experiments. The goal of this project is to find out whether it is possible anduseful to implement J Integral calculations in the current model and to examine the influenceof 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 chapterabout the material behavior of the used materials. The total model is described in the fifthchapter. Chapter six deals with the simulations and the results of the simulations. Finallythere are the conclusions and some recommendations.
3
Chapter 2
UHP Lamps
2.1 Gas discharge lamps
An UHP (Ultra High Performance) lamp is a high pressure gas discharge lamp. Compared tolow pressure gas discharge lamps, high pressure gas discharge lamps are usually more com-pact, have a higher internal pressure, higher power and deliver more light. The burner of ahigh pressure gas discharge lamp is a tube of quartz with sealed-in electrodes, filled with ametal vapor. The electrons emitted by the electrodes are accelerated by the electric field andcan have collisions with the gas atoms and molecules. These collisions lead to heat generationand excitation or ionization of the gas atoms. Excited atoms can generate electromagneticradiation. 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 sufficientlyhigh and constant level so that an electric current can flow through the vapor or gas. Sincemost vapors and gases are good insulators nothing is likely to happen when a voltage isapplied to the two electrodes of a discharge tube. That is why in every discharge lamp a
4
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, thetemperature will rise because of excitation and ionization processes in the bulb. Due to thetemperature rise the metal filling can vaporize. With the vapor pressure also the number ofexcited metal atoms increases. Discharge of the metal vapor takes over the discharge of thenoble gas more and more. Finally, an equilibrium is reached.
The composition of the electromagnetic radiation depends on the discharge tube filling. Apart of the electromagnetic radiation might be visible. The luminous flux and the spectraldistribution of the emitted radiation are highly influenced by the metal vapor pressure in thedischarge tube. The pressure depends on the temperature of the coldest spot in the tube incase of a saturated burner. For unsaturated burners the amount of metal and the temperaturedistribution 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 fillingand 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 onthe 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 inmercury vapor at a relatively high pressure. To get a good luminous efficacy a pressure of 20MPa has to be achieved. Any metal vapor pressure depends on the temperature of the metaland the amount of mercury.
The high vapor pressure and the high wall temperature require a special highly resistantmaterial. Quartz with its high softening point and a very good transmission for wavelengthsbetween 185 and 4000 nm, proved a good material, although it has some disadvantages. Firstof all the high processing temperature of 1600 to 1700 C and besides that the low coefficientof expansion of quartz glass, which makes it impossible to make a metal to quartz seal inthe same way as a metal to glass seal. This problem can be solved with the use of a thinmolybdenum foil as the current lead.
The electrode is made of tungsten, a metal with a high boiling point, so the electrode materialwill not evaporate quickly at the high temperatures prevalent during operation. Blackeningof the discharge tube will be limited.
5
2.3 Production of UHP lamps
The production of the UHP burner takes place in 6 steps.
2.3.1 The gas discharge bulb
In the first step a gas discharge bulb is made in the middle of a quartz tube. To producethis bulb a quartz tube of about 20 centimeters is picked up and rotated. Then it is shortlyheated by a flame, to remove stresses in the quartz. Continuing rotating, the quartz is heatedin the middle of the tube and during this heating the ends of the tube are moved to eachother which results in an expansion of the tube where it is heated. At this point a die isplaced around the tube to produce the right form of the bulb.
2.3.2 The current lead
In this step the different parts of the current lead are mounted to each other. There are threeparts: the supply wire, the molybdenum feed-trough and the tungsten rod. The process startswith the supply wire that is turned into a spring, what is necessary later in the productionwhen the total current lead is put into the quartz tube. The spring is mounted onto themolybdenum feed-through using a resistance welding process. This process is also used formounting the tungsten rod onto the molybdenum feed-through. The most important issue inthis step of the process is the aligning of the three parts, because that is necessary to achievea light source that is as small as possible.
2.3.3 The assembly of the current lead
The next process step is to put the complete current leads into the quartz tube. Because ofthe spring the current lead is not moving when it is put into the tube. When both currentleads are put into the tube, they have to be moved to the middle of it, with both rods at acertain distance from each other. This is done by two pins that are pushed into both sides ofthe tube. With the use of x-rays the end of the rods can be detected and so the rods can bepositioned at the right place.
2.3.4 The filling
One of the ends of the tube is closed by a laser beam that melts the quartz. After that theair is pumped out of the tube. When the tube is completely vacuum the tube is filled withmercury and argon (the starting gas). When the tube is filled the other end of the tube isclosed.
2.3.5 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 thenwith the beam moving upwards. After cooling of the tube, it is turned upside down. At this
6
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). Afterthis step the ends are cut off by a leaser beam and the burner is ready for assembly into thelamp.
2.3.6 The assembly of the lamp
There are a lot of different applications of the UHP burner, which requires a lot of differentassemblies. The assembly process starts with a burner that is put in a reflector. The burner isignited and heated until it reaches the maximum gas discharge. At that moment the burner isput by hand into the focal point of the reflector. The position of that focal point is found bymeasuring 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 inthe light intensity. When the burner is positioned at the right place, it is fixed with a ceramicglue.
7
Chapter 3
Fracture Mechanics
3.1 Introduction
The magnitude of an elastic crack tip stress field can be described by the stress intensityfactor 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 cracksurfaces are perpendicular to the plane of the crack. In-plane shear results in mode II orthe ’sliding mode’, the displacements of the crack surfaces are in the plane of the crack andperpendicular to the leading edge of the crack. The ’tearing mode’ or mode III is caused byout-of-plane shear. Crack surface displacements are in the plane of the crack and parallel tothe leading edge of the crack. The superposition of the three modes describes the generalcase 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. Theway this is done is first explained for non-thermal stress conditions. Then for thermal stressconditions, and finally for the case where the material behaves visco-elastic.
8
3.2 Non-thermal stress conditions
3.2.1 Theory
The J Integral is a line integral that, in cases of elastically behaving material, gives a value forthe energy release rate. Because of some important properties, the J Integral is suitable forusing it in a crack-growth criterion. To show this, first a homogeneous body of linear elasticmaterial is considered, free of body forces and subjected with all stresses σij depending onlyon two Cartesian coordinates x1 (= x) and x2 (= y) (figure 3.2).
X2
X1
Γ
A0
Figure 3.2: Homogeneous body
In this case the definition of the J Integral is [Rice1968]:
J =∫
Γ
(Wn1 − Ti
∂ui
∂x1
)ds (3.1)
where W is the strain energy, Γ is a closed path in the x1x2−plane, and s is the distancealong the path. Also Ti = σijnj (with ni the component of the unit normal in xi direction onΓ) and ui are components of the traction and displacement vectors, respectively.
The stresses σij are related to the strains εij with E (Young’s modulus) and ν (Poisson’sratio),
σij =νE
(1 + ν) (1− 2ν)εkkδij +
E
(1 + ν)εij (3.2)
which, using W = 12εijσij , leads to:
W (εij) =12
νE
(1 + ν) (1− 2ν)(εkk)
2 +E
2 (1 + ν)εijεij (3.3)
To prove that the value for the J Integral is equal to zero over a closed path, the line integralcan be transformed into a surface integral, using the Green-Gauss theorem.
J =∫
A0
[∂W
∂x1− ∂
∂xj
(σij
∂ui
∂x1
)]dA (3.4)
9
where A0 is the area enclosed by the path Γ. Here it is assumed that no singularities areenclosed by the path Γ. Because the strain energy W has been written as a function of straincomponents only (see equation (3.3)), ∂W
∂x1can (in case of no body forces), using equilibrium,
be written as:
∂W
∂x1=
∂W
∂εij
∂εij
∂x1= σij
∂εij
∂x1= σij
[∂
∂xj
(∂ui
∂x1
)]=
∂
∂xj
(σij
∂ui
∂x1
)(3.5)
Substituting the above equation into equation (3.4) results in the integrand of that equationbeing equal to zero and therefore J = 0. Thus the value of the J-line integral over a closedpath, 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 canbe calculated over any path surrounding a crack tip to determine the stress intensity factors.
x2
x1
Ω
Γ+
Γ−ΓB
ΓA
~n
~n
Figure 3.3: Path independence
Consider the closed path (ΓA + Γ+ + ΓB + Γ−) shown in figure 3.3.
It follows that
∫
ΓA
[ ] +∫
ΓB
[ ] +∫
Γ+[ ] +
∫
Γ−[ ] = 0 (3.6)
where
[ ] =(
Wn1 − Ti∂ui
∂x1
)ds (3.7)
If path ΓB is circular with radius r1 around the crack tip and if r1 approaches 0, then thestresses and displacements in the region through which the path ΓB passes, can be describedusing the crack tip equations depending on the mode I and mode II stress intensity factors
10
[Schreurs1996]. Here only mode I stress intensity factors are considered, which for plane strainconditions results in an evaluation of the contribution JΓB
along ΓB and using equation (3.6)(assuming that tractions on Γ+ and Γ− are negligible) eventually leads to:
(1− ν2)E
K2I = JΓA
(3.8)
Because of path independence of the J-line Integral, KI can be evaluated using equation (3.8)with JΓA
evaluated over any path ΓA which surrounds the crack tip.
3.2.2 Numerical implementation
The J Integral evaluation in MSC.Marc is based upon the domain integration method. Dueto difficulties in considering the integration path ΓA, a direct evaluation of JΓA
is not verypractical in a finite element analysis. Therefore a derivation of the domain integral expressionfor the energy release rate is determined, using equation (3.1) as a starting point.
To determine the stress intensity factor KI , JΓAneeds 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 bedone if q(x, y) = 1 on ΓA and q(x, y) = 0 on ΓB:
JΓA=
∫
ΓA
[ ] ds =∫
ΓA
[ ] q(x, y)ds +∫
ΓB
[ ] q(x, y)ds +∫
Γ+[ ] q(x, y)ds +
∫
Γ−[ ] q(x, y)ds
=∮
[ ] q(x, y)ds =∫
A
[W
∂q
∂x1− σij
∂ui
∂x1
∂q
∂xj
]dA (3.9)
with
[ ] =(
Wn1 − Ti∂ui
∂x1
)ds (3.10)
Equivalent to the path independence of equation (3.1), equation (3.9) is domain-independentso that any domain can be chosen for the purpose of evaluating J . This makes it possible tochoose 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 notpossible to calculate one single value for the J Integral. Therefore the J Integral is calculatedat several points along the three dimensional crack front. This results in multiple values forthe 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 onthe strain εij (see equation 3.3). When thermal stresses are involved, W is also depending onthe temperature T (T is temperature rise with respect to a reference temperature). Thereforethe J Integral over a closed path, as defined in equation (3.1), is not equal to zero. If the J
11
Integral is combined with an area integral inside the closed path, crack tip stress intensityfactors 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 thermalstress conditions:
W = W (εij , T ) =12εijσij (3.11)
The elastic relationship between stress and strain components is now
σij = λεiiδij + 2µεij − Eα
1− 2νTδij (3.12)
where T is the temperature and α is the coefficient of thermal expansion.
When, analogous to the non-thermal case, ∂W∂x1
is derived, an extra term has to be added, dueto the thermal strain:
∂W
∂x1=
∂W
∂εij
∂εij
∂x1+
∂W
∂T
∂T
∂x1(3.13)
Using equations (3.2) and (3.12), equation (3.13) can be given the more specific form
∂W
∂x1= σij
∂εij
∂x1+
Eα
1− 2ν
[12
∂
∂x1(Tεii)− εii
∂T
∂x1
](3.14)
which finally leads to
J =Eα
1− 2ν
∫
Γ
[12
∂
∂x1(Tεii)− εii
∂T
∂x1
]dA (3.15)
Since the value for this J Integral is not equal to zero in this case, there is also an extra termneeded in the calculation for the stress intensity factor.
(1− ν2
)
EK2
I = JΓA− Eα
1− 2ν
∫
Γ
[12
∂
∂x1(Tεii)− εii
∂T
∂x1
]dA (3.16)
3.3.1 Numerical implementation
In MSC.Marc the calculation of JΓAin equation (3.16) takes place as explained in section
3.2.2. When thermal stresses are involved, also the extra term is calculated. The result of theJ Integral calculations in MSC.Marc contains both terms and includes therefore the entireright side of equation (3.16).
12
Chapter 4
Material Properties
In this chapter the behavior and properties of the different materials are discussed, firstthe thermal properties and then the mechanical behavior. Because of the symmetry of theburner, 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 thequartz tube).
At modeling the geometry of the model some assumptions are made to avoid numerical prob-lems. 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 withlength l and diameter d. But because l >> d a large amount of elements would be neededfor accurate numerical computations. To avoid this the coil is modeled as a cone around atungsten rod. This leads to a correction in the thermal conductivity and the emissivity, asdescribed later in this chapter.
The other geometry part is the molybdenum foil. This foil is in reality very thin in an attemptto reduce the difference in expansion caused by a temperature rise between the molybdenumfoil 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 specificheat of quartz and both the thermal conductivity and the emissivity of quartz and tungstenare 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, simulatesthe melting process of the material. The transition from solid tungsten to liquid tungstencauses the rise in thermal conductivity.
13
Material Part Specific Thermal Emissivity Densityheat conductivityJ/gK W/mmK - g/mm3
quartz Bulb figure 4.1 figure 4.2 figure 4.4 2.206 · 10−3
tungsten 1 Rod 0.160 figure 4.3 figure 4.4 19.2 · 10−3
tungsten 2 Coil 0.160 figure 4.3 figure 4.4 19.2 · 10−3
mercury Capillary 0.105 1.5 · 10−3 - 5.0 · 10−4
molybdenum 1 Foil 0.290 0.120 1 0.12 10.2 · 10−3
molybdenum 2 Lead 0.290 0.120 0.12 10.2 · 10−3
1 thermal conductivity of molybdenum 1 in length direction is 0.0286 W/mmK
Table 4.1: Thermal properties
0 500 1000 1500 2000 2500 3000
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
Temperature [K]
spec
ific
heat
[J/g
K]
Specific heat of quartz
quartz
Figure 4.1: Specific heat (J/gK)
0 500 1000 1500 2000 2500 3000 3500 40001.5
2
2.5
3
3.5
4
4.5
5
5.5
6x 10
−3 Thermal conductivity of quartz
Temperature [K]
cond
uctiv
ity [W
/mm
K]
quartz
Figure 4.2: Conductivity (W/mmK)
0 500 1000 1500 2000 2500 3000 3500 40000.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2Thermal conductivity of tungsten
Temperature [K]
cond
uctiv
ity [W
/mm
K]
tungsten 1tungsten 2
Figure 4.3: Conductivity (W/mmK)
0 500 1000 1500 2000 2500 3000 3500 40000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Emissivity of quartz and tungsten
Temperature [K]
emis
sivi
ty [−
]
quartztungsten 1tungsten 2
Figure 4.4: Emissivity (−)
4.2 Mechanical properties
In the mechanical analysis only the quartz bulb and the tungsten electrode rod are taken intoaccount. Simulations are done with both elastic and viscoelastic material behavior. During
14
Material Part E-modulus Poisson’s ratio Thermal expansion coefficientN/mm2 - 1/K
quartz Bulb 72 · 103 0.18 5.0 · 10−7
tungsten 1 Rod 350 · 103 0.28 5.3 · 10−6
Table 4.2: Mechanical properties
the simulation mechanical stresses appear due to gas pressure, temperature gradients andthermal expansion mismatch between quartz and tungsten rod. The mechanical propertiesof the materials when both the quartz and the tungsten are behaving elastically are shownin table 4.2 [Peters1995].
Because the material properties are much more complicated when the material behavior isviscoelastic, this is described in a separate section.
4.3 Viscoelastic material behavior
4.3.1 Theory
The viscoelastic material behavior is described by a multi-mode Maxwell model, a parallelconnection of an amount of Maxwell models that all have their own relaxation time τi andcontribution to the stiffness Ei (see figure 4.5).
E1 E2 E3 Ei
E∞
η1 η2 η3 ηi
Figure 4.5: Generalized Maxwell model
In the case of complete relaxation, E∞ = 0. In this case the behavior of the generalizedMaxwell model is described by:
σ (t) = ε0
∑
i
Eiexp (−t/τi) (4.1)
15
with:
τi = ηi/Ei (4.2)
The stress response of the material after a step ε0 in the strain at t = 0 is shown in figure4.6. After a sudden increase of the stress, stress relaxation occurs which results in a smoothdecrease of the stress to a stationary value σ∞. When σ∞ = 0, there is complete relaxation.
ε
ε0
t0 t
σσ0
σ∞
t0 t
Figure 4.6: Viscoelastic stress response
At higher temperatures the stress decreases faster than at lower temperatures. This is shownin figure 4.7.
log E
log t
↑ T
Figure 4.7: Temperature dependent E modulus
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. Soviscoelastic behavior at one temperature can be related to another temperature by a changein time-scale only. To use the fact that each individually measured curve at a differenttemperature has the same shape, one master curve can be determined. By doing experimentsat 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
T7
log E
T4
T5
T6
T3T2T1
log t
Figure 4.8: Generating master curve, T1 < T2 < T3 < T4 < T5 < T6 < T7
temperatures in the same range of time. The results of the measurements can be shifted overthe 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 thereference temperature of this master curve. For all other temperatures a shift function isnecessary to calculate the right viscoelastic behavior. There are several functions that can beused as a shift function, one of them is the Arrhenius function:
ln aT =A
R
(1T− 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 8.314 · 10−3 kJmol−1K−1. The quantity aT is definedas:
aT =τ(T )τ(T0)
(4.4)
Using this shift function, where τ is the time belonging to the measurements at a certaintemperature, the viscoelastic material behavior at a specific temperature can be determined.
4.3.2 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(Tg) to room temperature. At turning on the lamp, a temperature rise in the glass cancause structural relaxation, which means that the material wants to approach its volumetricequilibrium, as a result of an altered thermal expansion coefficient. In the case of quartz glassthere is not much structural relaxation, because of the small thermal expansion coefficient.Therefore the structural relaxation is neglected in this model.
17
4.3.3 J Integral
Calculation of the J Integral gives a value for the energy that is available in a system for thegrowth of a crack. When it concerns linear material behavior, all this energy can be used forthe growth of the crack, but in the case of viscoelastic material a part of this total energyis dissipated. To find out how much energy is available for the growth of the crack an extraterm 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 JIntegral in viscoelastic simulations does no longer describe only the energy that is availablefor the growth of a crack. Therefore the results of viscoelastic simulations are mainly usedfor comparing the different simulations and not to compare them with a fracture toughness.
18
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 exchangebetween the different materials in the lamp and between the lamp and the environment. Heatis generated by the current flow through the electrodes. This heat is conducted through all thecomponents of the lamp to the outer surface, where heat losses appear caused by convectionand radiation effects.
5.2 Thermal boundary conditions
Mercury
Tungsten (rod)
Tungsten (coil) Quartz
Molybdenum (lead)
Molybdenum (foil)
δ4
δ5
δ2
δ1
δ3
Figure 5.1: Cross section of the lamp
There are several thermal boundary conditions applying on the lamp. The boundary of thelamp 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
generated is dependent on the lamp power and the operating voltage, which depends on theplasma pressure and the distance between the electrodes.
Qt = VtipIlamp (5.1)
where:qt is the total heat flux caused by current flow [W]Vtip is the operating voltage [V]Ilamp is the electric current [A]Atip is the area of the electrode tip [m2]
Because of this heat generation both the electrodes and the mercury plasma are heated. Thetotal amount of heat can be divided in three parts, namely the heat that directly warms theelectrodes (qd) and heat that warms the mercury plasma and so indirectly the inner wall ofthe bulb (qw) and the outside of the electrodes (qe). This means that
Qt = qdAd + qwAw + qeAe (5.2)
where:qt is the total heat flux caused by current flow [W/(m2)]qd is the heat flux at the electrode tip [W/(m2)]qw is the heat flux at the inner wall of the bulb [W/(m2)]qe is the heat flux at the outside of the electrodes [W/(m2)]Ad is the area of the electrode tip [m2]Aw is the area of the inner wall of the bulb [m2]Ae is the area of the outside of the electrodes [m2]
Heat losses appear by convection and radiation to the environment. Free convection at theoutside of the lamp can be calculated as follows:
qc = hc (Tw − T∞) (5.3)
where:qc is the heat flux caused by convection [W/(m2)]hc is the average convection heat transfer coefficient [W/(m2K)]Tw is the temperature of the wall [K]T∞ is the temperature of the environment [K]
Radiation effects are described by:
qr = σε(T 4
w − T 4∞
)(5.4)
where σ is called the Stefan-Boltzmann constant and has the value of σ = 5.67·10−8 W/(m2K4
)
and:qr is the heat flux caused by radiation [W/(m2)]ε is the emissivity of the material (ε = 1 for an ideal radiator) [-]
20
The applied boundary conditions for each boundary are:
boundary conditions on δ1
q = qw (5.5)
boundary conditions on δ2
q = qe − σε(T 4
w − T 4∞
)(5.6)
boundary conditions on δ3
q = qd − σε(T 4
w − T 4∞
)(5.7)
boundary conditions on δ4 and δ5
q = −hc (Tw − T∞)− σε(T 4
w − T 4∞
)(5.8)
5.3 Initial conditions
Both the temperature of the environment and the temperature of the lamp at t = 0 are equalto 300 K:
T∞ = 300 KT (t = 0) = 300 K
5.4 Mechanical boundary conditions
The most important mechanical load is a result of the internal pressure in the bulb whenthe lamp is burning. This mechanical load is applied on δ1 in figure 5.1. Other mechanicalboundary conditions consist of fixing the geometry at the symmetry planes.
21
Chapter 6
Simulations
The model is used as a starting point for different kind of simulations. The goal of thesimulations is to find out whether the position and the size of a possible crack are importantfor the stress intensity factor at the crack. Before the results of these simulations can beevaluated, 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 whenviscoelastic material behavior is used, the lamp is turned of at t = 600 s).
0 100 200 300 400 500 6000
20
40
60
80
100
120
140
160
180
time [s]
pres
sure
[N/m
m2 ]
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 by1000. 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 thecrack front. There are 7 tables because there are 7 paths selected around the crack tip. Since
22
Figure 6.2: Geometry of an opened crack
the calculation of the J Integral should be path independent, the values in these 7 tablesshould be equal. Due to a coarser mesh away from the crack front, the values in the tableswith the results for these paths are not the same anymore and thus not reliable. That is whythe results of the 3rd path are chosen, for evaluating the results of the simulations.
Figure 6.3: Position of the crack in thebulb
1
13
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 variationsin the values along the crack front is done.
The nodes along the crack front that are used in the J Integral calculations are numbered 1to 13. Node 1 is located at the outer side of the lamp and node 13 is located at the symmetryplane of the bulb. A possible position of the crack area in the bulb is shown in figure 6.3 anda 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
simulations are plotted for all nodes as a function of the time (see figure 6.5).
0 100 200 300 400 500 600−2
0
2
4
6
8
10
12
14
16x 10
−5 J Integral along a crack front
time [s]
J In
tegr
al
node 1node 2node 3node 4node 5node 6node 7node 8node 9node 10node 11node 12node 13
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 ofthe bulb) is higher than in the other nodes. This means that if an initial crack in the bulbhas a geometry like the crack in figure 6.4, the crack will start to grow in the direction ofthose nodes. Figure 6.3 indicates that a crack that grows in this direction proceeds along thesurface of the bulb and not towards the middle of it. Optimizing the geometry of the crackwould 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 varyingthe 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 elasticand viscoelastic material behavior. First the results of the simulations with elastic materialbehavior are presented, after that the results of the simulations with viscoelastic materialbehavior.
24
6.2.1 Elastic material behavior
First the J Integral values along the bulb are compared as a function of the time, for cracksat 20 different positions along the bulb. The procedure used for meshing the cracks in shownin Appendix B. The first position is at the top of the bulb with the subsequent ones shiftingto the neck with small steps. This is visualized in figure 6.6. In the heat flux at the inner sideof the wall, also gravity is taken into account (see for the heat distribution appendix A). Forthis reason one side of the bulb reaches higher temperatures than the other side. The cracksin 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 cracksas 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 theJ Integral decreases immediately.
When comparing the values of the J Integral for the different cracks, the first thing that canbe concluded is that there are big J Integral variations between the different cracks and thatthe values are more or less constant in the first 300 seconds (when the lamp is burning) andin the last 300 seconds (when the lamp is not burning). When the lamp is burning crack 11 isthe most critical, followed by crack 10, crack 9 and crack 12. When the lamp is not burningthe 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
0 100 200 300 400 500 6000
1
2
x 10−4
time [s]
J In
tegr
al [N
/mm
]
crack 1crack 2crack 3crack 4crack 5crack 6crack 7crack 8crack 9crack 10crack 11crack 12crack 13crack 14crack 15crack 16crack 17crack 18crack 19crack 20
Figure 6.7: J Integral for various crack locations using elastic material behavior
0 1 2 3 4 50
1
2
x 10−4
arc length [mm]
J In
tegr
al [N
/mm
]
Figure 6.8: J Integral along the bulb att=60 s (lamp burning)
0 1 2 3 4 50
1
2
3
4
5
6
7
8x 10
−6
arc length [mm]
J In
tegr
al [N
/mm
]
Figure 6.9: J Integral along the bulb att=600 s (lamp off)
arc length when the lamp is burning (for t = 60 s), and when the lamp is not burning (fort = 600 s). This is shown in the figures 6.8 and 6.9. In these figures the value of the JIntegral is plotted as a function of the arc length. The arc length is measured from the topof the bulb in the length direction along the 20 cracks (see figure 6.6). When the lamp isnot burning (figure 6.9) the value of the J Integral increases along the bulb. However, themaximum value of the J Integral (at crack 20) is still very small (mind the scaling of thevertical axis) compared to the values of the J Integral when the lamp is burning (figure 6.8).
26
In this case there is a clear maximum visible, around crack 10 and 11 (arc length about 3.5mm). In the direction of crack 20 (arc length 4.7 mm) the value of the J Integral decreasesquite fast. In the other direction, towards the top of the bulb (arc length 0.3 mm), the valueof the J Integral decreases less, but cracks at those positions are obviously less critical thancrack 10 and 11.
6.2.2 Viscoelastic material behavior
0 200 400 600 800 1000 12000
0.2
0.4
0.6
0.8
1
1.2x 10
−4
time [s]
J In
tegr
al [N
/mm
]
crack 1crack 2crack 3crack 4crack 5crack 6crack 7crack 8crack 9crack 10crack 11crack 12crack 13crack 14crack 15crack 16crack 17crack 18crack 19crack 20
Figure 6.10: J Integral for various crack using viscoelastic material behavior
The same simulations are also carried out with viscoelastic material behavior. The onlydifference is that instead of a simulation time of 600 seconds for the simulations with elasticmaterial behavior, the simulations with viscoelastic material behavior have a simulation timeof 1200 seconds, because of the time dependency of the viscoelastic material behavior. Thelamp is turned on at t = 0 s and turned off at t = 600 s. Again first the J Integral valuesalong the bulb are compared as a function of the time (figure 6.10). For these simulationsthe same trends are visible as for the simulations with elastic material behavior. After t = 60s, the stress increases or decreases as a result of the viscoelastic material behavior. This canbe explained by the temperature distribution in the bulb. The temperatures at the inside ofthe bulb are the highest, so there the material is most viscous. The behavior of the materialat 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 nothave much influence on the trends, but an important difference is the maximum value of the
27
J Integral. The maximum value is two times as high for the simulations with elastic materialbehavior as a result of higher stresses. Another difference is the position of the crack withthe maximum value, which is not crack 11 (as it was for the simulations with elastic materialbehavior), but crack 10. Again the values of the J Integral are plotted as a function of the arclength in the figures 6.11 and 6.12 to be compared to the figures 6.8 and 6.9. The results forviscoelastic behavior confirm the trends as observed for the simulations with elastic materialbehavior.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.2
0.4
0.6
0.8
1
1.2x 10
−4
arc length [mm]
J In
tegr
al [N
/mm
]
Figure 6.11: J Integral along the bulb att=600 s (lamp burning)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
1
2
3
4
5
6
7
8x 10
−6
arc length [mm]
J In
tegr
al [N
/mm
]
Figure 6.12: J Integral along the bulb att=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 elasticand viscoelastic material behavior. Again, first the results of the simulations with elasticmaterial behavior are presented and in the following section the results of the simulationswith viscoelastic material behavior.
6.3.1 Elastic material behavior
To examine the influence of the size of the crack on the value of the J Integral, cracks witha different size are modeled. The geometry of the crack is taken constant, only the crackdepth changes. When the value of the J Integral is not increasing for a deeper crack, thenthe crack might stabilize at a certain size. The simulations with three different cracks andelastic material behavior are shown in figure 6.13. 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. Toshow this in another way the value of the J Integral is plotted as a function of the crackdepth, see figure 6.14. In this figure two lines are visible. The line indicated with node 1 isfitted through the values of the J Integral around node 1 as plotted in figure 6.13. It can beobserved that for a deeper crack, the value of the J Integral increases more. This means that
28
0 100 200 300 400 500 6000
1
x 10−4
time [s]
J In
tegr
al [N
/mm
]
crack 1crack 2crack 3
Figure 6.13: J Integral elastic material behavior different crack sizes
0.1 0.15 0.2 0.25 0.3 0.350.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2x 10
−4
crack depth [mm]
J In
tegr
al [N
/mm
]
node 1node 1node 13node 13
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 thesame is true for crack growth in the direction towards the center of the lamp the values of theJ Integral around node 13 are plotted in the same figure. These values are lower, as alreadyestablished earlier this chapter, but the same trend is visible. So also for crack growth in thedirection of the center of the lamp, cracks with a size in this range are unstable.
29
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 elasticmaterial behavior, however the values of the J Integral are much lower. Nevertheless, againthe values of the J Integral are plotted as a function of the crack depth. This is visible infigure 6.16. This figure confirms the conclusion that is drawn after analyzing the results of thesimulations for elastic material behavior. So also for viscoelastic material behavior it can beconcluded that when a crack within this crack size range will start growing, the crack growthwill be critical, both in the direction over the surface of the bulb as in the direction towardsthe center of the bulb.
0 200 400 600 800 1000 12000
1
2
3
4
5
6
7
8
9x 10
−5
time [s]
J In
tegr
al [N
/mm
]
crack 1crack 2crack 3
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 asimplemented in the current model are more likely to grow over the surface of the bulb insteadof in the direction of the center of the bulb.
A second result shows the weakest spot on the bulb under the used circumstances. At thisspot (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 mechanicsin the current model, the results of the simulations are compared to the stress distribution inthe lamp. When the results of the J Integral simulations show the same critical places as wouldbe expected by examining the stress distribution, it could be concluded that implementingJ Integral calculations is not giving extra information. In figures 6.17 and 6.18 both the JIntegral values along the arc length (as defined in section 6.2.1) as the stress distributionalong the arc length are shown. Both the values of the J Integral as the stress distributionshow at t = 60 s a maximum around an arc length of 4 mm. However, close to the top of
30
0.1 0.15 0.2 0.25 0.3 0.353
4
5
6
7
8
9
10x 10
−5
crack depth [mm]
J In
tegr
al [N
/mm
]
node 1node 1node 13node 13
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 theneck of the burner is reached.
Figure 6.17: J Integral compared to stressdistribution at t=60 s
Figure 6.18: J Integral compared to stressdistribution 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 ofthe 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 criticalvalue. This is possible when the value of the J Integral is used to determine the stress intensityfactor KI , which can be compared with the fracture toughness of the material. However, thefracture toughness of quartz is temperature dependent, at higher temperatures the fracture
31
toughness will increase due to the viscoelastic material behavior of quartz. The values of thefracture toughness of quartz at higher temperatures are not known.
When equation (3.8) is used to determine the maximum stress intensity factors during thesimulations, this leads to KI = 0.12 MPa
√m in the case of elastic material behavior and
KI = 0.09 MPa√
m in the case of viscoelastic material behavior. The fracture toughnessof 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 behigher, and cracks of this size under the given circumstances will not be critical.
32
Chapter 7
Conclusions
The main goal of the project was to determine whether initial cracks in UHP lamps can beexamined using fracture mechanics. Fracture mechanics theories can indicate when an existingcrack is critical under certain mechanical and thermal loads. It can be concluded that theimplementation of J Integral calculations is possible in the current model. A 3-dimensionalcrack can be meshed using the procedure described in Appendix B. Simulations lead to valuesof the J Integral over the crack front. In the case of viscoelastic material behavior the resultsof the simulations are not very accurate because for such material behavior an extra term isrequired in the calculation of the J Integral. However, this extra term is not implemented inMSC.Marc which results in a value for the J Integral that cannot be used as the amount ofenergy that is available for crack growth. Still, simulations with viscoelastic material behaviorare 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 morelikely 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 thisspot (arc length about 3.5 mm) the value of the J Integral is clearly higher than at otherspots. The final result from the simulations is that an initial crack with a size between 0.1and 0.3 mm is unstable. This means that when the value of the J Integral of such a crackexceeds the critical value of the J Integral, the crack will keep growing, and will not reach astable state.
Finally there is the comparison of the values of the J Integral with a critical value. This ispossible when the value of the J Integral is used to determine the stress intensity factor KI ,which can be compared with the fracture toughness of the material. However, the fracturetoughness of quartz is temperature dependent, at higher temperatures the fracture toughnesswill increase due to the viscoelastic material behavior of quartz. The values of the fracturetoughness of quartz at higher temperatures are not known.
This all leads to the conclusion that it is possible to implement fracture mechanics in thecurrent model and that it gives more information than the model without fracture mechan-ics, but that there are too much insecurities regarding the interpretation of the J Integralcalculations with viscoelastic material behavior and comparing the results with a fracturetoughness, that at this point it can not be used as an instrument to evaluate the behavior ofinitial cracks under certain circumstances.
33
Chapter 8
Recommendations
When the decision is made to continue using fracture mechanics in the current models somerecommendations can be made. To simplify the procedure of meshing the cracks the originalmodel should be remeshed. This could make it easier to implement all kinds of cracks indifferent lamp models.
Since in the simulations some variations were found in the value of the J Integral along thecrack front, some investigation could be done to the geometry of the crack. At this point onlycircular cracks are modeled. This could be changed into all kinds of elliptic geometries.
When it is desirable to compare the results of the simulations with the fracture toughness,the possibilities to determine this temperature dependent parameter using experiments couldbe examined. This will require experiments at high temperatures and is therefore not trivial.
As described in the introduction, another reason for early failure can be initial cracks at theinside of the bulb. These cracks are a result of the production process, because of thermalexpansion mismatch between quartz and tungsten. It could be possible to do simulations withcracks at the inside of the bulb. However, when this is done it is important to investigatethe mechanical interaction between the quartz, the tungsten and a possible gap filled withmercury between them.
34
Chapter 9
References
Gutierrez-Lemini, D. (2002). The initiation J-integral for linear viscoelastic solids with con-stant Poisson’s ratio. International Journal of Fracture 113, 27-37
Peters, L. (1995). D.T. UHP 150 W 1.3; test frame modelling, Ansys input of UHP Lampmodel, Philips CDL
Rice, J.R. (1968). A Path Independent Integral and the Approximate Analysis of StrainConcentration by Notches and Cracks. Journal of Applied Mechanics June 1968, 379-386
Schreurs, P.J.G. (1996). Breukmechanica, Syllabus bij de cursus ”Breukmechanica”, Tech-nische Universiteit Eindhoven
Shih, C.F., Moran, B., Nakamura, T. (1986). Energy release rate along a three-dimensionalcrack front in a thermally stressed body. International Journal of Fracture 30, 79-102
Timmermans, P.H.M., Voncken, R.M.J. and Sluis, O. van der (2004). Numerical simula-tion of crack propagation in brittle materials. Philips Electronics Nederland B.V. Eindhoven,CTB591-04-1528
Wilson, W.K., Yu, I.-W. (1979). The use of the J-integral in thermal stress crack problems.International Journal of Fracture 15, 377-387
35
Appendix A
Heat Distribution
Subroutine used by MSC.Marc to describe the heat distribution at the inner side of the bulb.SUBROUTINE FLUX(F,TS,N,TIME)IMPLICIT REAL *8 (A-H, O-Z)DIMENSION TS(6), N(7)i = timecon = -0.08nn = 2.5x = TS(4)y = TS(5)z = TS(6)rad = sqrt(x**2+z**2)phi = acos(x/rad)axi = yphi1 = cos(phi/2)phin = ((phi1)**nn)Qwin = 3.8050fac = 0.6381598A = 16.787theta = atan(rad/axi)if (theta.ge.1.04719755) thenthet2 = 1elsethet2 = (theta/1.04719755)endifif (i.le.300.d0) thenqi = ((con*cos(phi)+phin)*thet2)F = qi * facelseF = 0endifRETURNEND
36
Appendix B
Crack Mesh Procedure
Subroutine used in MSC.Marc to mesh a crack.
First one element of the original mesh is taken out of the model and isolated. The 8 cornersand 6 planes of this element are numbered in a strict order. Afterwards the coordinates ofpoints 1, 2, 4 and 5 are filled in the following file:
|***********************************************|Crack geometry||Dimensions: [mm], [kg], [s]|***********************************************|
|VUL VOLGENDE PARAMETERS IN; RESP. DE COORDINATEN VAN POINT 1, 2, 4 & 5
define x1 3.800164786618define y1 2.357976318776define z1 0define x2 3.645215272481define y2 2.588158313239define z2 0define x4 3.245743217215define y4 2.273151179188define z4 0define x5 3.670677311549define y5 2.357976318776define z5 -9.835550213045e-1
system resetrenumber allconvert points |convert point to nodes1 5 8 2 #add elements
37
1 3 4 2
sub divisions4 4 1subdivide elementsall existing
remove elementsall existingrenumber all
add elements7 12 13 88 13 14 2
select filter noneselect method singleselect elements1 #store elements Deel1all selectedselect clearselect elements2 #store elements Deel2all selectedselect clear
sub divisions8 4 1subdivide elements1 #subdivide elements2 #
select setsDeel1visible selectedremove elements21 22 25 26 29 30 33 34 #add elements54 53 64 6453 52 64 6452 57 64 6457 62 64 6462 67 64 6467 72 64 64
38
72 73 64 6473 13 64 64remove unused nodesselect setsDeel1select elements67 68 69 70 71 72 73 74 #store elements Deel1all selectedselect clearsub divisions1 4 1subdivide elements67 68 69 70 71 72 73 74 #change elements quadraticall visiblesweep nodesall visible
|GA NAAR MOVE|GA NAAR CENTROID|KLIK IN HET MODEL DE CRACKTIPNODE AAN|GA VERDER MET C2.PROC
After this part of the procedure, the crack tip has to be defined with a mouse click. Thesecond part of the routine:
*set move scale factors0.5 0.5 0.5set move translations0 0 0move nodes266 272 285 298 311 324 337 350 363 #select nodes266 123 264 122 262 121 260 54 235 49 226 44 217 39 208 34 #store nodes nodesDeel1all selectedselect clear
select setsDeel2visible selectedremove elements51 52 55 56 59 60 63 64 #add elements115 116 105 105116 117 105 105
39
117 112 105 105112 107 105 105107 102 105 105102 97 105 10597 96 105 10596 95 105 105remove unused nodesselect setsDeel2select elements163 164 165 166 167 168 169 170 #store elements Deel2all selectedselect clearsub divisions1 4 1subdivide elements163 164 165 166 167 168 169 170 #change elements quadraticall visiblesweep nodesall visibleset move scale factors0.5 0.5 0.5move nodes607 594 581 568 555 542 529 516 510 #select nodes607 442 606 441 605 440 604 604 95 472 90 463 85 454 80 445 75 #store nodes nodesDeel2all selectedselect clearinvisible selected
system alignx1 y1 z1x2 y2 z2x4 y4 z4
select setsDeel1set expand point0 0 0set expand rotations7 0 0set expand translations0 0 0set expand repetitions
40
12expand elementsall selectedremove unused nodes
select setsDeel2expand elementsall selectedrenumber allremove unused nodes
select clearselect setsnodesDeel1nodesDeel2sweep nodesall unselectedselect clearselect filter surfaceselect method planeselect faces859:2select nodes facesall selected
move to geom mode directedset move to geom directionx2-x1 y2-y1 z2-z1
select filter surfaceselect method planeselect faces859:2select nodes facesall selected
detach nodes45 #detach elements709 710 711 712 713 714 715 716 717 718 719 720 #set move point0 0 0set move scale factors1 1 1set move translations
41
-0.01 0 0move elementsall existing
move nodes to surface6all selected
select clearselect faces199:0|*face quads|all selected|*select elements class|quad8|*set expand rotations|0 0 0|*set expand translations|-0.015 0 0|*set expand repetitions|1 |*expand elements|all selected
move to geom mode directedset move to geom directionx1-x2 y1-y2 z1-z2
|*select clear|*select faces|1506:5select nodes facesall selected
move nodes to surface4all selected
elements wireframeedges fullregenerateselect clearselect filter noneselect method single
|RELAX DE MIDSIDENODES VAN DE GEEXPANDEERDE DELEN
42
When after this part of the procedure the midside nodes of the expanded elements are relaxed,the final part of the routine is as follows:
*elements solidregenerate
move to geom mode directedset move to geom directionx4-x1 y4-y1 z4-z1
system alignx1 y1 z1x4 y4 z4x2 y2 z2
select clearselect filter surfaceselect method planeselect faces960:1959:1958:1957:1956:1955:1face quadsall selectedselect elements classquad8set expand rotations0 0 0set expand translations0.2 0 0set expand repetitions1expand elementsall selected
select clearselect filter surfaceselect method planeselect faces1428:51427:51426:51425:51424:5
43
1423:5select nodes facesall selectedmove nodes to surface5all selected
move to geom mode directedset move to geom directionx5-x1 y5-y1 z5-z1
system alignx1 y1 z1x5 y5 z5x2 y2 z2
select clearselect filter surfaceselect method planeselect faces1423:2954:1953:1952:1951:1950:1937:1face quadsall selectedselect elements classquad8set expand rotations0 0 0set expand translations0.5 0 0set expand repetitions1expand elementsall selected
select clearselect filter surfaceselect method planeselect faces1550:51532:51531:5
44
1530:51529:51528:51521:5select nodes facesall selectedmove nodes to surface2all selected
select clearselect setsnodesDeel1nodesDeel2sweep nodesall unselectedremove unused nodesremove unused points
|RELAX OVERIGE MIDSIDE NODES|CHECK INSIDE OUT|NAKIJKEN OF DE NODES DE OPPERVLAKKEN RAKEN (VOORAL BIJ VLAK 1)
45
