Abstract

We present an efficient approach to determine the axial temperature profile in tube furnaces by analyzing the diameter profile after stretching from a silica glass rod. Given the temporal load for stretching, the effective axial temperature profile can be deduced. This approach neglects diameter change due to surface tension by using rods. It also considers homogeneous cross sectional temperatures as effective axial temperatures. Since the effective temperature is derived from the rod’s viscous behavior, it yields optimized temperature profiles for heat treatment with glass rods, such as all-solid fiber drawing. This provides better temperature profiles than using thermal couples, because they are measured by the glass rods from the working material. Silica glass was used in this approach for its well-studied viscosity-temperature relationship. The derived method was carried out in an inductive graphite furnace under argon flow at 1400 °C and 1800 °C, respectively. The results will be validated by the mean of saphire fiber Bragg grating sensor measurements. Finally, the problems of the stretched axis and spatial resolution will be discussed.

© 2015 Optical Society of America

1. Introduction

Tube furnaces [1] are devices that provide a cylindrical heat cavity in which one or more heat elements can be equipped to provide multiple heat zones along its axis. It generally provides one dimension of freedom along its axis. This can be utilized vertically, such as in fiber drawing [2], or horizontally, such as in tube collapsing [3] and preform sintering [4] with rotation.

Although most tube furnaces are designed to provide a homogeneous temperature within the cylindrical cavity, the actual temperature profile is not in “rectangular” form, especially at a high temperature. The condition for heat transfer via radiation, conduction, and convection is complex and varies from position to position. For graphite tube furnaces over 2000 °C, it is also necessary to continuously purge the hot cylindrical cavity with argon (or any other inert gas) to remove residual air. Therefore, the actual temperature profile deviates from the design. For this reason, the characterization of the temperature profile of tube furnaces is of significant importance.

A point-by-point scan with a temperature sensor, such as a thermal couple, is not efficient due to the waiting time necessary for each temperature point to reach equilibrium. Secondly, electronic thermal sensors may not be resistant to the strong electromagnetic field in an inductive furnace. Thirdly, the thermal sensor material usually differs in its properties of heat conductivity, emissivity, and surface roughness from the desired working material. The scanned value may not reflect the real temperature profile encountered by the actual working material in the tube furnace.

Here we present a highly-efficient, non-scanning method of measuring the effective axial temperature profile for tube furnaces via viscous stretching of silica glass rod as shown in Fig. 1. Silica glass is widely used and investigated for different high temperature applications such as optical fiber drawing and glass tube collapsing. The temperature measurements with silica glass itself are not disturbed much by additional impacts from the material of the measurement equipment, e.g. using thermal couples. In fact at a sufficient high temperature, any rods that can be considered as Newtonian laminar flow [5] and has known viscosity-temperature relation, can also be used in this approach. The diameter profile analysis in this approach is limited to the small or moderate strain level, those cases with large strain level (e.g. neck down region during fiber drawing) or for hollow structures (circular and non-circular tubes), refer to references [3,15] concerning the effect of surface tension and the treatment of viscosity.

 figure: Fig. 1

Fig. 1 Schema of viscous stretching in temperature profiles with different length of gage section.

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For amorphous materials such as glass and polymers, the viscosity-temperature relationship corresponds, in principle, to the Arrhenius equation [6] due to the fact that viscous flow is a thermally-activated process. However, the activation energy is not normally constant across the entire temperature range; therefore, most types of glass deviate from the Arrhenius equation. The activation energy generally consists of two parts [7]: the low-temperature part, which stands for bond breaking, and the high-temperature part for overcoming friction during viscous motion. When these two activation energies deviate a little, glass is usually characterized as strong otherwise fragile. Thus, the temperature range can be divided into three parts. At a temperature far below the glass transition temperature (Tg), the viscosity-temperature profile can be approximated very well in the Arrhenius equation (with an activation energy for bond breaking). At a high temperature (i.e., far above Tg), where the activation energy for motion dominates, the viscosity-temperature profile can also fit well into Arrhenius form. The middle region, however, is not suitable to fit into Arrhenius form. Instead, the Vogel-Fulcher-Tammann equation is used to provide a good fit at the region near Tg, which is marked as the middle region. To fit all temperature ranges, a mathematical two-exponential equation with four parameters can be used.

2. Derivation of the Viscous Stretching Method

When a viscoelastic silica glass rod is stretched [8] along its axial direction at an elevated temperature, the specimen encounters successively elastic deformation, uniform plastic deformation, and non-uniform plastic deformation where localized necking occurs (Fig. 2). At a high temperature range the elastic contribution diminishes, the plastic deformation dominates the stretching process, and the contraction of the diameter only accounts for its viscous flow. Then, the tensile stress relates to the rate of strain with viscosity as the coefficient in the form of Eq. (1) [14]:

σ=3ηdεdt
where σ is the axial stress, η is dynamic viscosity, and dε/dt is the rate of strain. This equation assumes incompressible Newtonian flow of the glass at an elevated temperature, which is also the assumption throughout this paper. Equation (1) is the fundamental equation for the viscous stretching method, which is used to determine the temperature profile.

 figure: Fig. 2

Fig. 2 Stress-strain relationship in uniaxial tensile stretching for viscoelastic material.

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If a glass rod is stretched from one side at a certain (such as Gaussian) temperature distribution as shown in Fig. 3, the rate of axial strain is represented as a function of position and time dε(x,t)/dt, where x is an axial position with its origin at the center. The axial stress has a time-and-position-dependent form σ(x,t). When applying Eq. (1) to each cross section with a different diameter at axial position x, we obtain

 figure: Fig. 3

Fig. 3 Schema of glass rod stretched from one side at normal temperature distribution T(x).

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ε˙=σ(x,t)3η(T(x,t))

Equation (2) is the general form of the stress-strain equation for each axial section at the neck. It is known from this equation that the strain rate of each axial section is inversely proportional to the viscosity at said position. Thus, the shape of the neck reveals the ratio of the viscosities between each section. Based on the Newtonian flow condition, viscosity is only a function of temperature, which is assumed to be homogeneously distributed in each cross section but differently distributed in the axial positions. For incompressible flow, the stress and strain are defined as

σ(x,t)=FA(x,t)
ε˙(x,t)=1LdLdt=1A(x,t)dA(x,t)dt
where F is the load at the ends of the rod, A(x,t) is the cross-sectional area at position x in time t, L is the length of the section of interest of the rod. Equation (3) and Eq. (4) concern the variations of time and position in the cross-sectional area. When necking occurs, the concepts of length and elongation are no longer restrictedly defined. The strain defined by the cross-sectional areas is obtained. For small strains (<0.1) [9], Eq. (3) and Eq. (4) can be reduced to Eq. (5) and Eq. (6), respectively.
σ(x,t)=FA0
ε˙(x,t)=ΔLL0
where L0, A0 and ∆L are the initial length, initial cross-sectional area, and elongation, respectively. The definitions in Eq. (5) and Eq. (6) are referred to as engineering strain-stress [7] in some references. Bridgman [10] has analyzed the stress-strain at the neck as part of the tensile test of rod specimens. The term σ(x,t) stands for average axial stress for each cross section at the neck; it achieves its maximum at the smallest section of the neck. By obtaining the average stress at the smallest section of the neck as a reference, Bridgman has provided a uniaxial stress value which is equivalent to the value for the case in which necking is not introduced. This is an alternative method that renders σ(x,t) independent of position, which is expressed by σ(x,t) = σequivalent(x). However, for a more general approach, our method treats σ(x,t) as a function that also depends on its position. A necessary assumption for our method is that the acceleration of stretching is so slow that it is basically force balanced; thus, the load applied on each section is the same, which means that F(t) is used instead of F(x,t). Now, by inserting Eq. (3) and Eq. (4) into Eq. (2), we obtain
dA(x,t)dt=F(t)3η(T(x,t))
Subsequently, by tensile streching a perfect cylinder with strain from A0 to A(x,t) at time 0 to t and integrating Eq. (7), we get
A0A(x,t)=0tF(t)3η(T(x,t))
The integration on the right-hand side of Eq. (8) is complex. Therefore, further approximation should be made. If the temperture encoutered by each axial section within a period of time t is almost constant – in other words, if the temperture is only a function of position – then Eq. (9) applies
T(x,t)T(x)
Equation (9) is a good approximation if the stretched length is small compared to the gage section or if the temperture fluctuates so slightly in the axial direction that the corresponding variation in viscosity is negligible. With this approximation, viscosity is no longer time dependent and, therefore, Eq. (8) can be reduced to
A0A(x,t)=M3η(T(x))
Where M=0tF(t)dt is the integration of load force F over time, which can be obtained experimentally. For silica glass, the Arrhenius form of the viscosity-temperature relationship is used here at two different temperature ranges. Details are shown in the further sections in this paper. In general, viscosity is related to temperature in the Arrhenius form
η(T(x))=Cexp(B/RT(x))
where B and C are the activation energy and fitting parameters, respectively, and R is the gas constant. In the case of a more general viscosity-temperature relationship, the two-exponential equation for the viscosity-temperature relationship can be used as shown in Eq. (12):
η(T(x))=DT(x)(1+Eexp(L/RT(x)))(1+Fexp(H/RT(x)))
where D, E and F are fitting parameters, L and H are activation energies dominating the low and high temperature range, respectively. Since silica glass fits well into Arrhenius form in our case, Eq. (12) is not necessary. For convenience, inserting Eq. (11) into Eq. (10), we obtain
f(x)=A0A(x,t)=M3Cexp(B/RT(x))
where f(x) stands for the change in the cross-sectional area at position x. The temperature profile can then be expressed by
T(x)=BR(ln(M/3C)lnf(x))
Since B and C are fit parameters, what we need for the temperature profile T(x) is the load-time integration for M and the change in the exponential strain f(x) derived from the localized neck profile.

2.1 Exclusion of the contribution of elasticity in M

Since the above-mentioned derivation attributes the strain (the change in the cross-sectional area) to the viscous flow in the neck, it is necessary to exclude the elastic contribution in the integration of the load-time curve for the calculation of M in Eq. (14) when the glass rod exhibits viscoelastic behavior at lower temperature ranges. This tendency will be shown later in the results and discussion section. The exclusion of elastic strain can be carried out by setting the integration interval for M to [ty, t] in Eq. (14), where ty is the time it takes to reach the yield point (Fig. 2). The mathematical transformation from strain to time requires a record of the temporal velocity. Therefore, it would be convenient if it were possible to distinguish the elastic contribution directly in the load-time curve. This is possible if the load-time relationship can be mapped to linearly correspond to the engineering stress-strain relationship. In other words, if the engineering stress can be expressed by the linear product of the load and a constant, the engineering strain can be expressed by the product of time and another constant; the load-time curve is just a rescaled picture of the stress-strain curve with a similar “shape”. Thus, by taking the end of the linear part in the load-time curve as the beginning of the integration for M, we can exclude the contribution of elasticity conveniently. From Eq. (5), the load and engineering stress are related linearly by the constant 1/A0 while the engineering strain is

ε(t)=ΔL(t)L0=vxtL0
where vx is the axial stretching rate. Since L0 is constant, the engineering strain ε(t) is linearly related to time t only if vx is constant. For this reason, the rod should be stretched at a constant rate to make it possible to avoid the elastic strain by selecting the upper and lower limits in the integration of M in the load-time curve. When the temperature is high enough, the elastic contribution can be neglected. In this case, it would be convenient to use a constant load instead of a constant stretching rate since the integration of M equals the production of constant load F and elapsed time t. This means a force sensor can be saved for high temperature measurement by using a constant load.

2.2 Spatial resolution of the temperature profile

Since the silica rod is stretched during measurement, position in the rod is actually shifted after stretching from the initial position x as demonstrated in Fig. 4. If the furnace is fixed, there will be a relative displacement d = x´- x between the moving section and the position of the temperature profile in the furnace. According to Eq. (9), the temperature within said relative displacement d has to be constant. For this reason, d can be defined as spatial resolution for our temperature profile measurement. In the case of the fixed furnace, the interval of d is [0, ∆L], which is distributed from the onset of the neck at the fixed end to the stretching end on the other side. If the furnace moves in the same direction (i.e., toward the stretching end) at velocity vf, d’s interval will be shifted to [-vf t,L- vf t]. The absolute value of d achieves its minimum when vf t = ½ ∆L, which provides the best spatial resolution at the middle of [-½∆L, ½ ∆L]. This is anadvantage for those cases in which the middle zone of the furnace is of most interest. Now we need to know the relationship between x and x’. Consider stretching for a period of time t: the cross section located in position x is shifted to position x’ after stretching. If we set the origin at the left onset of the neck (as in Fig. 4), x is also the partial length. In the case of volume conservation before and after stretching, the partial volume Vp determined by partial length x (light gray region) should be equal to that determined by partial length x’ (dark gray region) after stretching. Thus, we get

Vp=xA0=0x´A(x,t)dx
where A0 and A(x,t) are the cross-sectional areas before and after stretching at time t. It should be noted here that the condition of the initial constant cross-sectional area A0 is of significant importance. Otherwise the problem is a complex one to be solved in analytical form. A perfect cylinder with little roughness compared to the diameter change is required for measurement. For the derivation of the final temperature profile in our method, the strain at position must be mapped back to x according to Eq. (16).

 figure: Fig. 4

Fig. 4 Schema of a cylindrical rod after stretching with localized necking has occurred. The axial position at x is shifted to x’ after stretching. The volume determined by position x before stretching (light gray) is the same as that determined by x’ after stretching (dark gray) for incompressible flow.

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3. Experimental setup

The setup for demonstration of the rod stretching method for temperature profile measurement is shown in Fig. 5. A glass lathe with immobile/mobile tailstocks for the fixing rod is used. A mobile inductive furnace (a maximum of 2500 °C) is used to provide a normal temperature distribution and track at half the speed of the stretching tailstock. An optical micrometer is fixed next to the furnace and measures the rod diameter using laser beam scans. The diameter profile is scanned at 0° and 90° before and after stretching at a precision of 0.5 µm, after which an average value can be used. The scanned diameter profile has an axial interval of 0.5 mm. The load applied to the rod at different times is monitored using a force sensor (a maximum of 100 N at a precision of 0.1 N), which is fixed to the mobile tailstock. We have chosen silica glass as the specimen with a dimension of 1 mm in diameter. The rod is fixed to the tailstocks through the mobile inductive furnace with a desired gage section length of 70 mm. As a reference, the temperature profile has been calibrated via scanning (20 mm/min) with sapphire fiber Bragg grating (FBG) sensors in a (ID = 1 mm/OD = 2 mm) silica tube [11]. The measurements of 1400 °C with a 1 mm rod and 1800 °C with a 9 mm rod have been used in this paper for demonstration. The working temperature is measured via pyrometer monitoring on the outer surface of the graphite heating element (ID = 45 mm, length = 70 mm).

 figure: Fig. 5

Fig. 5 Schema of the experimental setup for demonstration of the rod stretching method for temperature profile measurement.

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4. Results and discussions

4.1 Strain profile and surface tension

Figure 6 shows the diameter profiles obtained before and after stretching a 1 mm silicate glass rod at 1 mm/min for 2 min at 1400 °C. If we consider a diameter change of 25 µm across a localized neck of 80 mm (gage section): the axial principle curvature is so small that the corresponding stress that arises from the surface tension can be neglected. However, the stress caused by radial principle curvature, which is the driving force in tube collapsing, does not contribute to the change in the cross-sectional area. In general, if the changed diameter is much smaller than the gage section, the surface tension can be safely neglected in the smooth sections.

 figure: Fig. 6

Fig. 6 Diameter profile of a 1 mm silicate glass rod before and after stretching at 1400°C in a gage section of about 80 mm.

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This is the case of stretching with a small total strain where some of the surface roughness is comparable to the signal (diameter change). Its resulting uncertainty will be demonstrated later. According to Eq. (4), the total strain at position x after stretching in a period of time t can be calculated using Eq. (17)

ε(x,t)=lnD0D90D0(x,t)D90(x,t)
Where D0(x,t) and D90(x,t) are the diameters at position x after a period of stretching time t under orientation angle of 0° and 90°, respectively, D0 and D90 are constants representing the average diameter at two measuring angles which are not position dependent.

4.2 Integration of M

Figure 7 demonstrates the load-time curve when stretching at 1400 °C. To conveniently distinguish the elastic contribution in the load-time curve as discussed above, the stretching rate is kept constant (1 mm/min). The shaded area is obtained by the integration of M, which is 1355 Ns in this case. Figure 8 demonstrates the elastic contribution at different temperatures for silica glass and its tendency to become significant towards lower temperatures. For silica glass, a constant stretching rate should be used for those measurements under 1600 °C; otherwise, a constant load is beneficial because force sensors are not necessary in this case.

 figure: Fig. 7

Fig. 7 Load-time curve and its integration over the plastic strain area (shaded area) at 1400 °C at 1 mm/min. The bright area under the curve can be attributed to the elasitc contribution at the initial stage of stretching.

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 figure: Fig. 8

Fig. 8 Stress-strain data from the necking method with silica glass rod specimen of 9 mm in diameter at different temperatures.

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4.3 Temperature profile measurement at 1400 °C

Now we will demonstrate two measurements at 1400 °C and 1800 °C with small and large total strain respectively. According to Doremus’s suggestion6, using the Arrhenius form as in Eq. (11), the most reliable viscosity data parameters for silica are shown in Table 1.

Tables Icon

Table 1. Fitting parameters for viscosity-temperature relationship for silica glass in Arrhenius form.

The temperature profile T(x) can be deduced according to Eq. (14) by determining M and using the fit parameters B and C from literature. A measurement for our inductive furnace that operates at 1400 °C has been demonstrated with a total stretching of 2.5%. The detailed diameter profile can be seen in Fig. 6. The results are then compared to the measurement results from the sapphire FBG sensor as demonstrated in Fig. 9. Both measurements show about 1290 °C as the maximum temperature near the 0 mm position where a temperature gap of about 100 K has been located. Since the working temperature of 1400 °C is defined as the surface temperature of the graphite heat element (walls of the heat cavity), the real temperature inside the cavity may not reach the same temperature level due to the heat loss towards both open ends of the tube furnace. This result is reasonable and the correctness of Eq. (14) has been demonstrated.

 figure: Fig. 9

Fig. 9 Comparison of temperature profile measured via the viscous stretching method (circle) using a 1 mm silica glass rod and scanning with an FBG sensor (triangle), respectively.

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The total strain in this measurement is very small. Some surface defects are comparable to the diameter changes near the onsets of the neck. A small amount of total strain results in fine (small average interval for temperature) spatial resolution in overall axial measurement positions. However, we can see the random deviations located near ± 40 mm in Fig. 9 which originate from the small signal-to-noise ratio13 (SNR) near and beyond the onsets of the localized neck in Fig. 6. The central region near 0 mm has a larger SNR and, therefore, agrees well with the reference.

In addition, several 5 μm height peaks (dusts, bumps) were found in the initial diameter profiles at the positions −30 mm, 20 mm, and 40 mm with a corresponding diameter change of 5μm, 15 μm, and 3 μm, respectively, in Fig. 6. Due to their different SNRs at different positions, the deviation generated at 20 mm can be omitted while those at −30 mm and 40 mm are significantly affected. According to the observations above, the gage section for this inductive furnace is about 80 mm (from −40 mm to 40 mm).

The total stretched length is 2 mm at 1400 °C, which is about 2.5% of the width of the localized neck (gage section). Figure 10 shows the distribution of the spatial resolution in the measurement at 1400 °C according to Eq. (16) with both a fixed furnace and a moving furnace (half the stretching rate). The highest (smallest) spatial resolution has been shifted to the center with the moving furnace. In addition, the maximum spatial resolution has also been reduced. This is advantageous when the center of the furnace is of most interest. It provides a temperature profile with the worst spatial resolution of 1.0 mm at the onset of the neck.

 figure: Fig. 10

Fig. 10 Spatial resolution of viscous stretching measurement at 1400°C with fixed furnace setup and moving furnace setup (half the stretching rate).

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Although both methods yield similar results, the viscous stretching method is more efficient since it is a single measurement while the other point-by-point scanning measurements using temperature sensors such as FBG sensors or thermal couples are more time consuming. Assuming the time to achieving temperature equilibrium is 1 min, the measurement time for viscous stretching is then 1 min + 2 min (stretching) = 3 min. For FBG sensors or any other scanning method, the time needed to scan a length of 70 mm is 140 min for a resolution of 0.5 mm in the axial position.

4.4 High temperature profile measurement at 1800 °C

Figure 11 shows a high temperature profile measurement using a 9 mm rod at 1800 °C, in which a much larger total strain of 33% (29 mm out of a gage section of 80 mm) has been carried out. It is shown that the temperature profile is very smooth, even near ± 40 mm due to the large SNRs. However, the large total strain has also stretched the x axis for the temperature profile; therefore, it needs to be corrected according to Eq. (16). These temperature profiles have been demonstrated in Fig. 11, in which the corrected temperature profile has the similar width of 80 mm, as we found in the 1400 °C measurement. In addition, the temperature gap between the working temperature and the maximum temperature measured in Fig. 11 is about 30 °C, which is much smaller than the gap at 1400 °C. This smaller temperature gap can be attributed to the larger rod surface (9 mm) for radiation absorption. The corresponding spatial resolution for this measurement has been shown in Fig. 12. Large total strain causes poorer spatial resolution.

 figure: Fig. 11

Fig. 11 Temperature profile measured via the viscous stretching method using a 9 mm silica glass rod at 1800°C for the inductive tube furnace.

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 figure: Fig. 12

Fig. 12 Spatial resolution of viscous stretching at 1800°C with large total strain.

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Still, we are able to choose a reliable section for the temperature profile measurement in the x axis according to an acceptable spatial resolution desired. For example, if the acceptable spatial resolution is 5 mm, which means an average viscosity/temperature is being considered within this range, the reliable region will be [-9 mm, 8 mm]. This could be expanded to [-25 mm, 21 mm] if the acceptable spatial resolution is 10 mm.

5. Conclusion and outlook

We have demonstrated a simple and efficient method of temperature profile measurement for a tube furnace by stretching a silica glass rod for which the viscosity-temperature relationship is well known (using data from Urbain et al. and Hetherington et al.). The correctness of the method and its derivation have been demonstrated in the measurement of the temperature profile at 1400 °C compared to sapphire FBG sensor temperature scanning. Large total strain improves the SNRs but lowers the spatial resolution and stretches the positions. The correction method for position stretching has also been demonstrated at high temperature measurement at 1800 °C. By choosing suitable total strain, a section in position for temperature profile measurement according to the required spatial resolution has also been derived and demonstrated.

This method is, in fact, able to provide an effective axial temperature profile for cases with a different heat load such as, for example, silica rods with different diameters. Furthermore, if the temperature profile of a rod is known, the viscosity can be derived by this method in an “inverse” fashion.

References and links

1. Laboratory Tube furnaces, (Carbolite, 2015), http://www.carbolite.com/products/furnaces/laboratory-tube-furnaces/.

2. J. Kobelke, K. Schuster, D. Litzkendorf, A. Schwuchow, J. Kirchhof, V. Tombelaine, H. Bartelt, P. Leproux, V. Couderc, A. Labruyere, and R. Jamier, “Highly germanium and lanthanum modified silica based glasses in microstructured optical fibers for non-linear applications,” Opt. Mater. 32(9), 1002–1006 (2010). [CrossRef]  

3. J. Kirchhof and A. Funke, “Reactor problems in modified chemical vapour deposition (II),” Cryst. Res. Technol. 21(6), 763–770 (1986). [CrossRef]  

4. K. Schuster, S. Unger, C. Aichele, F. Lindner, S. Grimm, D. Litzkendorf, J. Kobelke, J. Bierlich, K. Wondraczek, and H. Bartelt, “Material and technology trends in fiber optics,” Adv. Opt. Technol. 3(4), 447–468 (2014).

5. Y. Nakayama and R. F. Boucher, Introduction to Fluid Mechanics (Elsevier, 2002), Chap. 4.

6. R. H. Doremus, “Viscosity of silica,” J. Appl. Phys. 92(12), 7619 (2002). [CrossRef]  

7. M. I. Ojovan, K. P. Travis, and R. J. Hand, “Thermodynamic parameters of bonds in glassy materials from viscosity–temperature relationships,” J. Phys. Condens. Matter 19(41), 415107 (2007). [CrossRef]  

8. J. R. Davis, Tensile Testing (ASM international, 2004).

9. Y. Ling, “Uniaxial true stress-strain after necking,” in AMP Journal of Technology 5. (AMP incorporated, 1996), pp. 37–48.

10. P. W. Bridgman, Studies in Large Plastic Flow and Fracture (McGraw-Hill, 1952).

11. T. Habisreuther, T. Elsmann, Z. Pan, A. Graf, H.-J. Pißler, M. Rothhardt, R. Willsch, H. Bartelt, and M. A. Schmidt, “Optical Sapphire Fiber Bragg Gratings As High Temperature Sensors,” Proc. SPIE 8794, 87940B (2013). [CrossRef]  

12. G. Urbain, Y. Bottinga, and P. Richet, “Viscosity of liquid silica, silicates and alumino-silicates,” Geochim. Cosmochim. Acta 46(6), 1061–1072 (1982). [CrossRef]  

13. D. H. Johnson, “Signal-to-noise ratio,” http://www.scholarpedia.org/article/Signal-to-noise_ratio. [CrossRef]  

14. E. L. Bourhis, Glass Mechanics and Technology (Wiley-VCH, 2007).

15. Y. M. Stokes, P. Buchak, D. G. Crowdy, and H. Ebendorff-Heidepriem, “Drawing of micro-structure fibres: circular and non-circular tubes,” J. Fluid Mech. 755, 176–203 (2014). [CrossRef]  

References

  • View by:

  1. Laboratory Tube furnaces, (Carbolite, 2015), http://www.carbolite.com/products/furnaces/laboratory-tube-furnaces/ .
  2. J. Kobelke, K. Schuster, D. Litzkendorf, A. Schwuchow, J. Kirchhof, V. Tombelaine, H. Bartelt, P. Leproux, V. Couderc, A. Labruyere, and R. Jamier, “Highly germanium and lanthanum modified silica based glasses in microstructured optical fibers for non-linear applications,” Opt. Mater. 32(9), 1002–1006 (2010).
    [Crossref]
  3. J. Kirchhof and A. Funke, “Reactor problems in modified chemical vapour deposition (II),” Cryst. Res. Technol. 21(6), 763–770 (1986).
    [Crossref]
  4. K. Schuster, S. Unger, C. Aichele, F. Lindner, S. Grimm, D. Litzkendorf, J. Kobelke, J. Bierlich, K. Wondraczek, and H. Bartelt, “Material and technology trends in fiber optics,” Adv. Opt. Technol. 3(4), 447–468 (2014).
  5. Y. Nakayama and R. F. Boucher, Introduction to Fluid Mechanics (Elsevier, 2002), Chap. 4.
  6. R. H. Doremus, “Viscosity of silica,” J. Appl. Phys. 92(12), 7619 (2002).
    [Crossref]
  7. M. I. Ojovan, K. P. Travis, and R. J. Hand, “Thermodynamic parameters of bonds in glassy materials from viscosity–temperature relationships,” J. Phys. Condens. Matter 19(41), 415107 (2007).
    [Crossref]
  8. J. R. Davis, Tensile Testing (ASM international, 2004).
  9. Y. Ling, “Uniaxial true stress-strain after necking,” in AMP Journal of Technology 5. (AMP incorporated, 1996), pp. 37–48.
  10. P. W. Bridgman, Studies in Large Plastic Flow and Fracture (McGraw-Hill, 1952).
  11. T. Habisreuther, T. Elsmann, Z. Pan, A. Graf, H.-J. Pißler, M. Rothhardt, R. Willsch, H. Bartelt, and M. A. Schmidt, “Optical Sapphire Fiber Bragg Gratings As High Temperature Sensors,” Proc. SPIE 8794, 87940B (2013).
    [Crossref]
  12. G. Urbain, Y. Bottinga, and P. Richet, “Viscosity of liquid silica, silicates and alumino-silicates,” Geochim. Cosmochim. Acta 46(6), 1061–1072 (1982).
    [Crossref]
  13. D. H. Johnson, “Signal-to-noise ratio,” http://www.scholarpedia.org/article/Signal-to-noise_ratio .
    [Crossref]
  14. E. L. Bourhis, Glass Mechanics and Technology (Wiley-VCH, 2007).
  15. Y. M. Stokes, P. Buchak, D. G. Crowdy, and H. Ebendorff-Heidepriem, “Drawing of micro-structure fibres: circular and non-circular tubes,” J. Fluid Mech. 755, 176–203 (2014).
    [Crossref]

2014 (2)

K. Schuster, S. Unger, C. Aichele, F. Lindner, S. Grimm, D. Litzkendorf, J. Kobelke, J. Bierlich, K. Wondraczek, and H. Bartelt, “Material and technology trends in fiber optics,” Adv. Opt. Technol. 3(4), 447–468 (2014).

Y. M. Stokes, P. Buchak, D. G. Crowdy, and H. Ebendorff-Heidepriem, “Drawing of micro-structure fibres: circular and non-circular tubes,” J. Fluid Mech. 755, 176–203 (2014).
[Crossref]

2013 (1)

T. Habisreuther, T. Elsmann, Z. Pan, A. Graf, H.-J. Pißler, M. Rothhardt, R. Willsch, H. Bartelt, and M. A. Schmidt, “Optical Sapphire Fiber Bragg Gratings As High Temperature Sensors,” Proc. SPIE 8794, 87940B (2013).
[Crossref]

2010 (1)

J. Kobelke, K. Schuster, D. Litzkendorf, A. Schwuchow, J. Kirchhof, V. Tombelaine, H. Bartelt, P. Leproux, V. Couderc, A. Labruyere, and R. Jamier, “Highly germanium and lanthanum modified silica based glasses in microstructured optical fibers for non-linear applications,” Opt. Mater. 32(9), 1002–1006 (2010).
[Crossref]

2007 (1)

M. I. Ojovan, K. P. Travis, and R. J. Hand, “Thermodynamic parameters of bonds in glassy materials from viscosity–temperature relationships,” J. Phys. Condens. Matter 19(41), 415107 (2007).
[Crossref]

2002 (1)

R. H. Doremus, “Viscosity of silica,” J. Appl. Phys. 92(12), 7619 (2002).
[Crossref]

1986 (1)

J. Kirchhof and A. Funke, “Reactor problems in modified chemical vapour deposition (II),” Cryst. Res. Technol. 21(6), 763–770 (1986).
[Crossref]

1982 (1)

G. Urbain, Y. Bottinga, and P. Richet, “Viscosity of liquid silica, silicates and alumino-silicates,” Geochim. Cosmochim. Acta 46(6), 1061–1072 (1982).
[Crossref]

Aichele, C.

K. Schuster, S. Unger, C. Aichele, F. Lindner, S. Grimm, D. Litzkendorf, J. Kobelke, J. Bierlich, K. Wondraczek, and H. Bartelt, “Material and technology trends in fiber optics,” Adv. Opt. Technol. 3(4), 447–468 (2014).

Bartelt, H.

K. Schuster, S. Unger, C. Aichele, F. Lindner, S. Grimm, D. Litzkendorf, J. Kobelke, J. Bierlich, K. Wondraczek, and H. Bartelt, “Material and technology trends in fiber optics,” Adv. Opt. Technol. 3(4), 447–468 (2014).

T. Habisreuther, T. Elsmann, Z. Pan, A. Graf, H.-J. Pißler, M. Rothhardt, R. Willsch, H. Bartelt, and M. A. Schmidt, “Optical Sapphire Fiber Bragg Gratings As High Temperature Sensors,” Proc. SPIE 8794, 87940B (2013).
[Crossref]

J. Kobelke, K. Schuster, D. Litzkendorf, A. Schwuchow, J. Kirchhof, V. Tombelaine, H. Bartelt, P. Leproux, V. Couderc, A. Labruyere, and R. Jamier, “Highly germanium and lanthanum modified silica based glasses in microstructured optical fibers for non-linear applications,” Opt. Mater. 32(9), 1002–1006 (2010).
[Crossref]

Bierlich, J.

K. Schuster, S. Unger, C. Aichele, F. Lindner, S. Grimm, D. Litzkendorf, J. Kobelke, J. Bierlich, K. Wondraczek, and H. Bartelt, “Material and technology trends in fiber optics,” Adv. Opt. Technol. 3(4), 447–468 (2014).

Bottinga, Y.

G. Urbain, Y. Bottinga, and P. Richet, “Viscosity of liquid silica, silicates and alumino-silicates,” Geochim. Cosmochim. Acta 46(6), 1061–1072 (1982).
[Crossref]

Buchak, P.

Y. M. Stokes, P. Buchak, D. G. Crowdy, and H. Ebendorff-Heidepriem, “Drawing of micro-structure fibres: circular and non-circular tubes,” J. Fluid Mech. 755, 176–203 (2014).
[Crossref]

Couderc, V.

J. Kobelke, K. Schuster, D. Litzkendorf, A. Schwuchow, J. Kirchhof, V. Tombelaine, H. Bartelt, P. Leproux, V. Couderc, A. Labruyere, and R. Jamier, “Highly germanium and lanthanum modified silica based glasses in microstructured optical fibers for non-linear applications,” Opt. Mater. 32(9), 1002–1006 (2010).
[Crossref]

Crowdy, D. G.

Y. M. Stokes, P. Buchak, D. G. Crowdy, and H. Ebendorff-Heidepriem, “Drawing of micro-structure fibres: circular and non-circular tubes,” J. Fluid Mech. 755, 176–203 (2014).
[Crossref]

Doremus, R. H.

R. H. Doremus, “Viscosity of silica,” J. Appl. Phys. 92(12), 7619 (2002).
[Crossref]

Ebendorff-Heidepriem, H.

Y. M. Stokes, P. Buchak, D. G. Crowdy, and H. Ebendorff-Heidepriem, “Drawing of micro-structure fibres: circular and non-circular tubes,” J. Fluid Mech. 755, 176–203 (2014).
[Crossref]

Elsmann, T.

T. Habisreuther, T. Elsmann, Z. Pan, A. Graf, H.-J. Pißler, M. Rothhardt, R. Willsch, H. Bartelt, and M. A. Schmidt, “Optical Sapphire Fiber Bragg Gratings As High Temperature Sensors,” Proc. SPIE 8794, 87940B (2013).
[Crossref]

Funke, A.

J. Kirchhof and A. Funke, “Reactor problems in modified chemical vapour deposition (II),” Cryst. Res. Technol. 21(6), 763–770 (1986).
[Crossref]

Graf, A.

T. Habisreuther, T. Elsmann, Z. Pan, A. Graf, H.-J. Pißler, M. Rothhardt, R. Willsch, H. Bartelt, and M. A. Schmidt, “Optical Sapphire Fiber Bragg Gratings As High Temperature Sensors,” Proc. SPIE 8794, 87940B (2013).
[Crossref]

Grimm, S.

K. Schuster, S. Unger, C. Aichele, F. Lindner, S. Grimm, D. Litzkendorf, J. Kobelke, J. Bierlich, K. Wondraczek, and H. Bartelt, “Material and technology trends in fiber optics,” Adv. Opt. Technol. 3(4), 447–468 (2014).

Habisreuther, T.

T. Habisreuther, T. Elsmann, Z. Pan, A. Graf, H.-J. Pißler, M. Rothhardt, R. Willsch, H. Bartelt, and M. A. Schmidt, “Optical Sapphire Fiber Bragg Gratings As High Temperature Sensors,” Proc. SPIE 8794, 87940B (2013).
[Crossref]

Hand, R. J.

M. I. Ojovan, K. P. Travis, and R. J. Hand, “Thermodynamic parameters of bonds in glassy materials from viscosity–temperature relationships,” J. Phys. Condens. Matter 19(41), 415107 (2007).
[Crossref]

Jamier, R.

J. Kobelke, K. Schuster, D. Litzkendorf, A. Schwuchow, J. Kirchhof, V. Tombelaine, H. Bartelt, P. Leproux, V. Couderc, A. Labruyere, and R. Jamier, “Highly germanium and lanthanum modified silica based glasses in microstructured optical fibers for non-linear applications,” Opt. Mater. 32(9), 1002–1006 (2010).
[Crossref]

Kirchhof, J.

J. Kobelke, K. Schuster, D. Litzkendorf, A. Schwuchow, J. Kirchhof, V. Tombelaine, H. Bartelt, P. Leproux, V. Couderc, A. Labruyere, and R. Jamier, “Highly germanium and lanthanum modified silica based glasses in microstructured optical fibers for non-linear applications,” Opt. Mater. 32(9), 1002–1006 (2010).
[Crossref]

J. Kirchhof and A. Funke, “Reactor problems in modified chemical vapour deposition (II),” Cryst. Res. Technol. 21(6), 763–770 (1986).
[Crossref]

Kobelke, J.

K. Schuster, S. Unger, C. Aichele, F. Lindner, S. Grimm, D. Litzkendorf, J. Kobelke, J. Bierlich, K. Wondraczek, and H. Bartelt, “Material and technology trends in fiber optics,” Adv. Opt. Technol. 3(4), 447–468 (2014).

J. Kobelke, K. Schuster, D. Litzkendorf, A. Schwuchow, J. Kirchhof, V. Tombelaine, H. Bartelt, P. Leproux, V. Couderc, A. Labruyere, and R. Jamier, “Highly germanium and lanthanum modified silica based glasses in microstructured optical fibers for non-linear applications,” Opt. Mater. 32(9), 1002–1006 (2010).
[Crossref]

Labruyere, A.

J. Kobelke, K. Schuster, D. Litzkendorf, A. Schwuchow, J. Kirchhof, V. Tombelaine, H. Bartelt, P. Leproux, V. Couderc, A. Labruyere, and R. Jamier, “Highly germanium and lanthanum modified silica based glasses in microstructured optical fibers for non-linear applications,” Opt. Mater. 32(9), 1002–1006 (2010).
[Crossref]

Leproux, P.

J. Kobelke, K. Schuster, D. Litzkendorf, A. Schwuchow, J. Kirchhof, V. Tombelaine, H. Bartelt, P. Leproux, V. Couderc, A. Labruyere, and R. Jamier, “Highly germanium and lanthanum modified silica based glasses in microstructured optical fibers for non-linear applications,” Opt. Mater. 32(9), 1002–1006 (2010).
[Crossref]

Lindner, F.

K. Schuster, S. Unger, C. Aichele, F. Lindner, S. Grimm, D. Litzkendorf, J. Kobelke, J. Bierlich, K. Wondraczek, and H. Bartelt, “Material and technology trends in fiber optics,” Adv. Opt. Technol. 3(4), 447–468 (2014).

Litzkendorf, D.

K. Schuster, S. Unger, C. Aichele, F. Lindner, S. Grimm, D. Litzkendorf, J. Kobelke, J. Bierlich, K. Wondraczek, and H. Bartelt, “Material and technology trends in fiber optics,” Adv. Opt. Technol. 3(4), 447–468 (2014).

J. Kobelke, K. Schuster, D. Litzkendorf, A. Schwuchow, J. Kirchhof, V. Tombelaine, H. Bartelt, P. Leproux, V. Couderc, A. Labruyere, and R. Jamier, “Highly germanium and lanthanum modified silica based glasses in microstructured optical fibers for non-linear applications,” Opt. Mater. 32(9), 1002–1006 (2010).
[Crossref]

Ojovan, M. I.

M. I. Ojovan, K. P. Travis, and R. J. Hand, “Thermodynamic parameters of bonds in glassy materials from viscosity–temperature relationships,” J. Phys. Condens. Matter 19(41), 415107 (2007).
[Crossref]

Pan, Z.

T. Habisreuther, T. Elsmann, Z. Pan, A. Graf, H.-J. Pißler, M. Rothhardt, R. Willsch, H. Bartelt, and M. A. Schmidt, “Optical Sapphire Fiber Bragg Gratings As High Temperature Sensors,” Proc. SPIE 8794, 87940B (2013).
[Crossref]

Pißler, H.-J.

T. Habisreuther, T. Elsmann, Z. Pan, A. Graf, H.-J. Pißler, M. Rothhardt, R. Willsch, H. Bartelt, and M. A. Schmidt, “Optical Sapphire Fiber Bragg Gratings As High Temperature Sensors,” Proc. SPIE 8794, 87940B (2013).
[Crossref]

Richet, P.

G. Urbain, Y. Bottinga, and P. Richet, “Viscosity of liquid silica, silicates and alumino-silicates,” Geochim. Cosmochim. Acta 46(6), 1061–1072 (1982).
[Crossref]

Rothhardt, M.

T. Habisreuther, T. Elsmann, Z. Pan, A. Graf, H.-J. Pißler, M. Rothhardt, R. Willsch, H. Bartelt, and M. A. Schmidt, “Optical Sapphire Fiber Bragg Gratings As High Temperature Sensors,” Proc. SPIE 8794, 87940B (2013).
[Crossref]

Schmidt, M. A.

T. Habisreuther, T. Elsmann, Z. Pan, A. Graf, H.-J. Pißler, M. Rothhardt, R. Willsch, H. Bartelt, and M. A. Schmidt, “Optical Sapphire Fiber Bragg Gratings As High Temperature Sensors,” Proc. SPIE 8794, 87940B (2013).
[Crossref]

Schuster, K.

K. Schuster, S. Unger, C. Aichele, F. Lindner, S. Grimm, D. Litzkendorf, J. Kobelke, J. Bierlich, K. Wondraczek, and H. Bartelt, “Material and technology trends in fiber optics,” Adv. Opt. Technol. 3(4), 447–468 (2014).

J. Kobelke, K. Schuster, D. Litzkendorf, A. Schwuchow, J. Kirchhof, V. Tombelaine, H. Bartelt, P. Leproux, V. Couderc, A. Labruyere, and R. Jamier, “Highly germanium and lanthanum modified silica based glasses in microstructured optical fibers for non-linear applications,” Opt. Mater. 32(9), 1002–1006 (2010).
[Crossref]

Schwuchow, A.

J. Kobelke, K. Schuster, D. Litzkendorf, A. Schwuchow, J. Kirchhof, V. Tombelaine, H. Bartelt, P. Leproux, V. Couderc, A. Labruyere, and R. Jamier, “Highly germanium and lanthanum modified silica based glasses in microstructured optical fibers for non-linear applications,” Opt. Mater. 32(9), 1002–1006 (2010).
[Crossref]

Stokes, Y. M.

Y. M. Stokes, P. Buchak, D. G. Crowdy, and H. Ebendorff-Heidepriem, “Drawing of micro-structure fibres: circular and non-circular tubes,” J. Fluid Mech. 755, 176–203 (2014).
[Crossref]

Tombelaine, V.

J. Kobelke, K. Schuster, D. Litzkendorf, A. Schwuchow, J. Kirchhof, V. Tombelaine, H. Bartelt, P. Leproux, V. Couderc, A. Labruyere, and R. Jamier, “Highly germanium and lanthanum modified silica based glasses in microstructured optical fibers for non-linear applications,” Opt. Mater. 32(9), 1002–1006 (2010).
[Crossref]

Travis, K. P.

M. I. Ojovan, K. P. Travis, and R. J. Hand, “Thermodynamic parameters of bonds in glassy materials from viscosity–temperature relationships,” J. Phys. Condens. Matter 19(41), 415107 (2007).
[Crossref]

Unger, S.

K. Schuster, S. Unger, C. Aichele, F. Lindner, S. Grimm, D. Litzkendorf, J. Kobelke, J. Bierlich, K. Wondraczek, and H. Bartelt, “Material and technology trends in fiber optics,” Adv. Opt. Technol. 3(4), 447–468 (2014).

Urbain, G.

G. Urbain, Y. Bottinga, and P. Richet, “Viscosity of liquid silica, silicates and alumino-silicates,” Geochim. Cosmochim. Acta 46(6), 1061–1072 (1982).
[Crossref]

Willsch, R.

T. Habisreuther, T. Elsmann, Z. Pan, A. Graf, H.-J. Pißler, M. Rothhardt, R. Willsch, H. Bartelt, and M. A. Schmidt, “Optical Sapphire Fiber Bragg Gratings As High Temperature Sensors,” Proc. SPIE 8794, 87940B (2013).
[Crossref]

Wondraczek, K.

K. Schuster, S. Unger, C. Aichele, F. Lindner, S. Grimm, D. Litzkendorf, J. Kobelke, J. Bierlich, K. Wondraczek, and H. Bartelt, “Material and technology trends in fiber optics,” Adv. Opt. Technol. 3(4), 447–468 (2014).

Adv. Opt. Technol. (1)

K. Schuster, S. Unger, C. Aichele, F. Lindner, S. Grimm, D. Litzkendorf, J. Kobelke, J. Bierlich, K. Wondraczek, and H. Bartelt, “Material and technology trends in fiber optics,” Adv. Opt. Technol. 3(4), 447–468 (2014).

Cryst. Res. Technol. (1)

J. Kirchhof and A. Funke, “Reactor problems in modified chemical vapour deposition (II),” Cryst. Res. Technol. 21(6), 763–770 (1986).
[Crossref]

Geochim. Cosmochim. Acta (1)

G. Urbain, Y. Bottinga, and P. Richet, “Viscosity of liquid silica, silicates and alumino-silicates,” Geochim. Cosmochim. Acta 46(6), 1061–1072 (1982).
[Crossref]

J. Appl. Phys. (1)

R. H. Doremus, “Viscosity of silica,” J. Appl. Phys. 92(12), 7619 (2002).
[Crossref]

J. Fluid Mech. (1)

Y. M. Stokes, P. Buchak, D. G. Crowdy, and H. Ebendorff-Heidepriem, “Drawing of micro-structure fibres: circular and non-circular tubes,” J. Fluid Mech. 755, 176–203 (2014).
[Crossref]

J. Phys. Condens. Matter (1)

M. I. Ojovan, K. P. Travis, and R. J. Hand, “Thermodynamic parameters of bonds in glassy materials from viscosity–temperature relationships,” J. Phys. Condens. Matter 19(41), 415107 (2007).
[Crossref]

Opt. Mater. (1)

J. Kobelke, K. Schuster, D. Litzkendorf, A. Schwuchow, J. Kirchhof, V. Tombelaine, H. Bartelt, P. Leproux, V. Couderc, A. Labruyere, and R. Jamier, “Highly germanium and lanthanum modified silica based glasses in microstructured optical fibers for non-linear applications,” Opt. Mater. 32(9), 1002–1006 (2010).
[Crossref]

Proc. SPIE (1)

T. Habisreuther, T. Elsmann, Z. Pan, A. Graf, H.-J. Pißler, M. Rothhardt, R. Willsch, H. Bartelt, and M. A. Schmidt, “Optical Sapphire Fiber Bragg Gratings As High Temperature Sensors,” Proc. SPIE 8794, 87940B (2013).
[Crossref]

Other (7)

D. H. Johnson, “Signal-to-noise ratio,” http://www.scholarpedia.org/article/Signal-to-noise_ratio .
[Crossref]

E. L. Bourhis, Glass Mechanics and Technology (Wiley-VCH, 2007).

Laboratory Tube furnaces, (Carbolite, 2015), http://www.carbolite.com/products/furnaces/laboratory-tube-furnaces/ .

Y. Nakayama and R. F. Boucher, Introduction to Fluid Mechanics (Elsevier, 2002), Chap. 4.

J. R. Davis, Tensile Testing (ASM international, 2004).

Y. Ling, “Uniaxial true stress-strain after necking,” in AMP Journal of Technology 5. (AMP incorporated, 1996), pp. 37–48.

P. W. Bridgman, Studies in Large Plastic Flow and Fracture (McGraw-Hill, 1952).

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Figures (12)

Fig. 1
Fig. 1 Schema of viscous stretching in temperature profiles with different length of gage section.
Fig. 2
Fig. 2 Stress-strain relationship in uniaxial tensile stretching for viscoelastic material.
Fig. 3
Fig. 3 Schema of glass rod stretched from one side at normal temperature distribution T(x).
Fig. 4
Fig. 4 Schema of a cylindrical rod after stretching with localized necking has occurred. The axial position at x is shifted to x’ after stretching. The volume determined by position x before stretching (light gray) is the same as that determined by x’ after stretching (dark gray) for incompressible flow.
Fig. 5
Fig. 5 Schema of the experimental setup for demonstration of the rod stretching method for temperature profile measurement.
Fig. 6
Fig. 6 Diameter profile of a 1 mm silicate glass rod before and after stretching at 1400°C in a gage section of about 80 mm.
Fig. 7
Fig. 7 Load-time curve and its integration over the plastic strain area (shaded area) at 1400 °C at 1 mm/min. The bright area under the curve can be attributed to the elasitc contribution at the initial stage of stretching.
Fig. 8
Fig. 8 Stress-strain data from the necking method with silica glass rod specimen of 9 mm in diameter at different temperatures.
Fig. 9
Fig. 9 Comparison of temperature profile measured via the viscous stretching method (circle) using a 1 mm silica glass rod and scanning with an FBG sensor (triangle), respectively.
Fig. 10
Fig. 10 Spatial resolution of viscous stretching measurement at 1400°C with fixed furnace setup and moving furnace setup (half the stretching rate).
Fig. 11
Fig. 11 Temperature profile measured via the viscous stretching method using a 9 mm silica glass rod at 1800°C for the inductive tube furnace.
Fig. 12
Fig. 12 Spatial resolution of viscous stretching at 1800°C with large total strain.

Tables (1)

Tables Icon

Table 1 Fitting parameters for viscosity-temperature relationship for silica glass in Arrhenius form.

Equations (17)

Equations on this page are rendered with MathJax. Learn more.

σ = 3 η d ε d t
ε ˙ = σ ( x , t ) 3 η ( T ( x , t ) )
σ ( x , t ) = F A ( x , t )
ε ˙ ( x , t ) = 1 L d L d t = 1 A ( x , t ) d A ( x , t ) d t
σ ( x , t ) = F A 0
ε ˙ ( x , t ) = Δ L L 0
d A ( x , t ) d t = F ( t ) 3 η ( T ( x , t ) )
A 0 A ( x , t ) = 0 t F ( t ) 3 η ( T ( x , t ) )
T ( x , t ) T ( x )
A 0 A ( x , t ) = M 3 η ( T ( x ) )
η ( T ( x ) ) = C exp ( B / R T ( x ) )
η ( T ( x ) ) = D T ( x ) ( 1 + E exp ( L / R T ( x ) ) ) ( 1 + F exp ( H / R T ( x ) ) )
f ( x ) = A 0 A ( x , t ) = M 3 C exp ( B / R T ( x ) )
T ( x ) = B R ( ln ( M / 3 C ) ln f ( x ) )
ε ( t ) = Δ L ( t ) L 0 = v x t L 0
V p = x A 0 = 0 x ´ A ( x , t ) d x
ε ( x , t ) = ln D 0 D 90 D 0 ( x , t ) D 90 ( x , t )

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