Visco-elastic deformation of dental porcelain and porcelain-metal compatibility K. Asaoka* J.A. Tesk**
*Department of Dental Engineering Tokushima University School of Dentistry 3 Kuramoto-cho Tokushima, 770, Japan **Dental and Medical Materials National Institute of Standards and Technology Gaithersburg, MD 20899 Received January 2, 1990 Accepted July 28, 1990 Dent Mater 7:30-35, January, 1991
Abstracl-A computer simulation using a visco-elastic stress analysis was conducted to clarify the effect of the heating rate on deformation temperature of dental porcelain during firing. In this simulation, the following temperaturedependent factors were incorporated: elastic modulus, viscosity, and coefficient of thermal expansion. The cooling/heating rate dependencies of both the glasstransition temperature and the temperature distribution in the slab were also included. Thermal expansion curves of porcelain with an applied load at various heating rates were computed. Effects of the applied stress and the heating rate on the deformation temperature of porcelain were revealed. The results suggest that the temperature where the incompatibility stress develops in the porcelain-fused-tometal strips during cooling can be estimated closely from the deformation point of the heating curve of the porcelain with an applied stress of about 1.2 - 3.1 MPa. A method for measuring temperature dependence of viscosity as represented by an Arrhenius equation is proposed.
irst-order transformations in single-phase systems are characterized by a discontinuous change in volume at constant temperature and pressure. In multiphase systems such as alloys, the transition may extend over an appreciable temperature range and, under non-equilibrium conditions, generally begins at higher temperatures on heating than on cooling, with hysteresis as the result. Second-order transformations are generally characterized by the disappearance of ordered structures, with the result that a discontinuous change occurs in the coefficient of thermal expansion at constant temperature and pressure. A glassy material has a second-order transformation in the temperature range of the glass transition and has a higher temperature dependence of the thermal expansion coefficient in the transition range than that of the glassy state, as shown in Fig. 1. Dental porcelains behave like glasses (because they contain large amounts of continuous glassy structure), and at high temperatures, the coefficient of thermal expansion for the liquid porcelain is difficult to measure. Because of their low viscosity at high temperatures, stresses relax, and the high-temperature thermal expansion need not be considered in the discussion of transient and residual stresses. In this paper, the transient and residual stresses developed in porcelain slabs during cooling from the liquid state were simulated with consideration of the cooling rate dependence of the glass-transition temperature, the temperature distribution in the slab, and the temperature dependencies of the elastic modulus, viscosity, and coefficient of thermal expansion. The effects of the cooling rate, thickness of the slab, heat-soak temperature, and coefficient of thermal expansion on the transient and residual stress were calculated by simulations considering the
F
30 ASAOKA & TESK/VISCO-ELASTIC DEFORMATION OF PORCELAIN
second-order transition of the porcelain according to a method employed by Asaoka and Tesk (1987, 1989). DeHoff and Anusavice (1989) also calculated the tempering stresses in feldspathic porcelain in a somewhat related manner. The computer simulation technique was also applied for the heating process of the porcelain. The effect of the viscosity on thermal expansion behavior during heating was analyzed from the simulated results. A relationship between the temperature at which stress just begins to develop in a PFM strip during cooling was examined relative to the temperature at which the porcelain under load in a dilatometer changes from expansion deformation to contraction during heating. Calculations of the residual stress in porcelain-fused-to-metal (PFM) strips (Fairhurst et aL, 1981) have been based on the Timoshenko bimetallic strip equation (Timoshenko, 1925) as follows: ¢r = k
I?
Act d T
(1)
0
where To is the temperature of interest; T1, the temperature at which stress just begins to develop in the porcelain during cooling; and Act, the difference between the coefficient of thermal expansion for the porcelain and that of the alloy. A deeper understanding of the residual stress in PFM strips can be acquired by examination of the method used for determination of the temperature T1. Twiggs et al. (1986) used the three-point bending beam viscometer to measure the deformation temperature, Td, for commercial porcelains at various heating rates*, and found that the depend+Paper relatedto dental porcelainby Fairhurstet al. (1981) used T~ as having the same meaning as Td in this paper. But the glass-transition temperature, Tg, is related to the transition range from solid-like glass to liquid and is not an exact point.
ence of Td on the heating or cooling rate, q, follows the equation of Moynihan et al. (1974): d(Inlql) d(1/Td)
- Q R
(2)
where Q is the activation enthalpy of shear viscosity and R is the gas law constant. The relation between the temperature T~ in Eq. (1) and a deformation temperature, T~, which is measured from the expansion curve of the porcelain during heating, is an important point and has not been previously analyzed by a model with relevance to dentistry. MATERIALS AND METHODS
A computer simulation was conducted for a model of a push-rod dilatometer of the kind which has been used for determining the expansion of porcelain during heating. The analysis is based on the assumption that the one-dimensional visco-elastic behavior of porcelain can be approximated by the two elements of a spring and a dashpot model. For the simulation, a porcelain beam was modeled as if composed of 50 thin, equal, layers. Temperature distributions which developed in the beam during cooling/heating were calculated by the equation of Williamson and Adams (1919), where the surfaces of the slab were presumed to cool/heat at a constant rate. The thermal diffusivity used for the porcelain was chosen as 0.5 mm2/s, the same as in a previous report (Asaoka and Tesk, 1989). Each layer has a unique elastic modulus, thermal expansion coefficient, and viscosity, as determined by its temperature. The Tg in each layer was determined by its cooling/ heating rate by use of the relation in Eq. (2) and a Tg from a reference heating rate. Internal stresses of the beam were calculated by equilibrium of internal moments, strain continuity at the interfaces, and with internal forces set equal to the applied load, as reported in detail by Asaoka and Tesk (1989). The net strain in any layer is equal to the sum of the strains due to thermal expansion/ contraction, normal internal stress, bending strain (almost zero at any temperature because of the sym-
TABLE 1 THERMO-MECHANICAL PROPERTIES OF THE PORCELAIN USED IN THE SIMULATION 70.2 GPa at 30°C 66.7 GPa at 651°C
Elastic modulus (a) Thermal expansion coefficient (b) TTo Liquid Transition temperature (c) Viscosity (c) References: (a) K~ise et aL (1985). (b) Asaoka, experimental results. (c) Bertolotti and Shelby (1979).
c( = (1.3 + 0.020-T) × 10-6/°C c~ = {(x' + 0.28 (T - To)} × 10-6/°C Here, c~' is the coefficient at TQ c~ = 35 × 10-6/°C 460°C at 0.17°C/s -q = 3.6 × 10 -9 exp(404OO/r) Pa • s
metrical temperature distribution), and viscous relaxation. Transient stresses were calculated by an incremental time-step method either during cooling f r o m T~ ( h e a t - s o a k temperature) to TR (30°C) or heating from TR to Tb (load-bearing temperature), with temperature stages I°C apart. Stresses were computed for every stage. The constants in the temperature dependence of the coefficient of thermal expansion for the porcelain were computed from data on heating curves from 40 to 400°C after previous slow-cooling of porcelain which had been fired three times. The modulus of elasticity for the porcelain was taken from the data by K~ise et al. (1985). The viscosity of the porcelain was from the data of Bertolotti and Shelby (1979). Creep relaxation was assumed to follow either a Voigt model for cooling or a Maxwell model (for example, see McClintock and Argon, 1966) for heating. The incremental time step, At, was determined for the cooling/heating rate at the surface for 1°C temperature decrements/increments. The creep relaxation for the cooling process was then calculated according to the following equations: e/eo = 1 -
exp(-E.At/'q)
~c = ~o/n
(5)
where ¢ro is the applied stress, and is the viscosity as in Eq. (4). Table 1 shows the thermo-mechanical properties of the porcelain used in this simulation. RESULTS
Effect of Heating Rate and Applied Stress on Deformation of Porcelain Using the Maxwell Model.- The factors which determine the amount of the viscous flow (creep strain) of porcelain at each constant-temperature step are the applied stress, porcelain viscosity, and holding time. The strain rate during the heating of the porcelain is determined by the stress applied by the push rod and the viscosity, as shown in Eq. (5). The strain rates, ~¢, at fixed temperature, plotted against the applied stresses, ao, are straight lines, as shown in Fig. 2. The results of the simulation agree with the experimental data by Oda et al. (1984) obtained with applied A
(3) E
= no e x p ( Q / R T )
(4)
where eo is the normal strain due to internal stress from the previous time step; E and ~, elastic modulus and viscosity, respectively; no, constant; Q and R, activation enthalpy for shear viscosity and gas law constant; and T, absolute temperature. For the heating process, the creep strain rate, ~, was assumed to follow (McClintock and Argon, 1966):
> l//
=o
j
i.,--rangeTemperature, T °C
Fig. 1. Variation of specific volume of well-annealed glass being cooled at a specified rate. Dental Materials/January 1991 31
1.6 1.4
Deformation temperature, Td ~C
P=IO MPa q = 1 ~3/s
100
700
600
50(
.UJ
10
°C
'
~
10
600'(:;
03
,
1
5
0 •~ e¢Io o. x
,
10
50 100
"'
Stress, a MPa Fig. 2. Strain rate ~due to viscous flow of porcelain under applied stress and fixed temperature. Data points are from the experimental results for the same material by Oda ef al. (1984).
stresses higher than 10 MPa. These results indicate that the Maxwell model is appropriate for simulating the viscous flow of porcelain during heating. Fig. 3 shows an example of the simulated expansion curve for porcelain. Here, the conditions are a heating rate of l°C/s and an applied stress of 10 MPa. The solid line shows the simulated result of thermal expansion with creep relaxation, and the dotted line is the thermal expansion without creep. The glass-transition temperature, T., is defmed as the point of the inflection of the slope. The temperature where the increase in length of the specimen due to thermal expansion is balanced by the negative deformation due to the applied stress is defined here as the deformation point, T~. The loadbearing temperature, Tb, is the point at which the net negative deformation of the specimen above T~ is 2%. It is possible to calculate Tb from the following equation: ~Tb
a(T)dT
(6)
+ titR
UOdT = - 0.02 q~o exp( Q/RT)
R
where the first term represents the amount the specimen would expand in the absence of both externally applied forces and surface tension, and the second term represents the creep strain rate due to a compressive stress (defined as negative) at any temperature, T, integrated over the temperature range with a heating
32
=T
0.8
/
/q=l C/s
,
0.6 0.9
,(i,,!/, tO
1.1
1.2
13
Temperature, IO00/T fK Fig. 4. Reciprocal of the deformation temperature, T~, of the porcelain as a function of applied stress. Here, cooling rates are 0.1, 1.0, and IO.O°C/s, respectively. Thickness of the slab is 2 ram.
0.4 0.2 0.0
q=10~;/s///
b
1
1.0
|
100
300
500
700
I
900
Temperature, T °C Fig. 3. Result of the simulated expansion curve for a 2-mm-thick porcelain slab. Here, the heating rate is 1°C/s, and the applied stress is 10 MPa. The solid line shows the simulated result, and the dashed line is the thermal expansion without viscous flow. Ta, T~, and T~ represent glass-transition temperature, deformation temperature, and load-bearing temperature, respectively.
rate of q. The sum of these two must, by definition, be -0.02. The second term has no exact integral, but by making the substitution u = Q/RT and using the relation (Coats and Redfern, 1984),
: e-uu-bdu
-- u l - b e
-u
( - 1)~(b)~ n=O
U n+ l
(7)
it can be calculated numerically. Table 2 shows measured values of Tb by Oda et al. (1984), Tb calculated from Eq. (6), and T b determined from simulations. Here, the heating rate is 0.5 °C/s. The simulations were conducted two ways using the Maxwell model and one of two conditions: (1) The slab maintained a stress distribution when the porcelain was at high temperature and had low viscosity, or (2) uniform stress when ~/ E of the porcelain became less than AT/q (the relaxation time of the porcelain). The two simulations and calculated results are in good agreement with the experimental data. Experimental data with stresses of 2 and 10 MPa are close to the range of the
ASAOKA & TESK/VISCO-ELASTICDEFORMATIONOF PORCELAIN
results from the simulations made with the two different assumptions. This is discussed i,. the next paragraph. For various heating, at~, the results of the simulation ~_ ,)w that 1/Td and the negative logarithm of the applied stress, uo, are proportional, as shown in Fig. 4. But T~ becomes imprecisely determined when the applied stress is under 1 MPa. This is because internal stresses generated during heating are of this magnitude, and the deformation rate at any point is e = (oro _+ UT)/'O,depending on the sign of aT, where uo is the applied stress and (rT is the t h e r m a l stress. In o r d e r for the overall deformation of the specimen to be controlled by the applied stress, the lowest value of applied stress must be greater than aT. From an elastic stress calculation based on the temperature distribution in a slab of 2-mm thickness, the surface of the specimen has about 1 MPa compressive stress during heating at a constant rate of l°C/s. When the heating rate is 10°C/s, the surface of the same slab has a calculated compressive stress of about 10 MPa. Therefore, the stress which is caused by the temperature distribution in the slab is significantly higher than the applied stress under ordinary conditions for measurement of thermal expansion by use of a push-rod dflatometer. Under these conditions, the relationship between Td and the externally applied stress loses its significance, and simulations and calculations are imprecise.
Transient Stress Profiles During Coolingand Re.heating.- It may be of interest,
TABLE2 MEASUREDAND CALCULATEDLOAD-BEARINGTEMPERATURE,Tb°C
for dental laboratory practice, to note the transient stress profiles developed during rapid cooling and after re-heating of the porcelain, as revealed by computer simulations. Stresses for the cooling process were calculated by use of the program as reported by Asaoka and Tesk (1989). In that study, the transient stress developed in porcelain during rapid cooling and the residual stress at room temperature resulting from the rapid cooling were computed. Fig. 5(a) shows the transient stresses at the surface and the center of the porcelain with 1-mm thickness during cooling from a heat-soak temperature of 800°C. The cooling rate is 30°C/s, which is comparable with that experienced in commercial dental laboratories. The parabolic residual stress distribution induced in a slab cooled at a constant rate is shown in Fig. 5(b). The calculated compressive residual stress near the surface was strongly dependent on the glasstransition temperature, viscosity, coefficient of thermal expansion at T > Tg, slab thickness, cooling rate, and heat-soak temperature. The present study shows the transient stresses developed during re-heating at 0.5°C/ s in a slab which has residual stresses from being cooled (Fig. 5c). The residual stresses remained nearly unchanged until Tg was reached. Above Tg, the porcelain dropped sharply to a stress-free state over a limited temperature range of about 100°C. The porcelain temperature reached Td after passing the stress-free temperature. These simulations demonstrate the effect of viscosity on the contraction/ expansion behavior of porcelain while being cooled and re-heated.
Results Applied Experiment Calculated by Simulated Stress (MPa) 1st firing 3rd firing Eq. (6) Results 2 659 699 709 660-711" 10 660 669 672 660-674* 50 638 638 639 640 100 630 638 625 624 *Lower and upper values are simulated results under assumptions (1) and (2), respectively, in the text.
DISCUSSION The exact temperature where the stress develops while PFM strips are being cooled is an important variable in the thermo-mechanical compatibility of PFM strips. The stress in a PFM strip and porcelain viscosity are functions of the temperature and the cooling rate. During cooling, the visco-elastic behavior of the porcelain can be represented by a Voigt model. If the stress and the viscosity are assumed to be independent of the cooling rate for a small temperature
step, the delayed elasticity can be represented by: (T
ev = ~ [1 - exp(-E'At/'q)]
(8)
where ev is creep strain; (r, stress at T; E, elastic modulus; ~1, viscosity; and At, time. The net strain mismatch between the alloy and the porcelain is calculated as the sum of the thermal contraction mismatch, et, between the alloy and the porcelain for a temperature step, AT, and the viscous strain, %, for the porcelain. Here, et is represented by" et = Act. AT
(9)
where Act is the difference between the coefficient of thermal expansion of the alloy and the porcelain; and AT, the temperature step in time, At. The cooling rate, q, is: When T1 is the effective temperature at which the porcelain displays delayed elasticity, dt
~o/~ = ct" q
(10)
q = AT/At
fo E
and from Eqs. (10), (12), and (14), we can obtain relative relations among AT, At, a, and T1. The calculated creep relaxation ratios for the porcelain are shown in Fig. 6. The Tl's for slow cooling (I°C/ s) and rapid cooling (50°C/s) are indicated in the Fig. The porcelain is almost stress-free for temperatures above T , because viscous flow relaxes the stress from the thermal contraction mismatch, Act, between the alloy and the porcelain. For heating of the porcelain slab only, the Maxwell model provides good agreement with experimental results, as shown in Fig. 2 and Table 2. The deformation temperature, Td, is the point where the strain rate from thermal expansion during heating equals that due to compressive viscous deformation,
1
(11)
where v is a relaxation time for which 63% of the strain is relaxed over the temperature interval. From this, E . I c = q'~lo
(12)
f?
Cro/'q • d t =
fo
%'I~_
T,
exp(Q/RT)
(13)
At that temperature, the viscous strain of the porcelain in Eq. (8) is equal to 63% of the mismatch of the thermal strain for a temperature step as shown in Eq. (9). At T , AT =
(7"
E " Act
(14)
(16)
ct-AT
(17)
00" q where, Ih =
dt
ct " qdt
then,
[~rd
Td+AT
r T1-AT
(15)
where ct is coefficient of thermal expansion of porcelain at Td. From Eq. (15),
where, Ic =
at Td
dt
exp(Q/RT)
If T1 is assumed to be equal to T~ for the same q, [ Ih ] = I Ic ], a relation between a and ct can be derived with Eqs. (12) and (14): c~
%/¢r = ~
(18)
For ordinary commercial porcelains and alloys for metal-ceramics, the instantaneous ct at the deformation
Dental Materials/January 1991
33
15 10 o. ¢t3
b
,,;
q~
point of the porcelain is about 30 40 x 10-6PC and instantaneous Aa = ( 3 0 - 4 0 × 10 .6 ) P C - ( 1 5 - 17 × 10-6)/°C = 1 3 - 25 × 10-6/°C, respectively (Asaoka and Tesk, 1988), .~ leading to approximate values of
Center
5
% / a = 1.2 - 3.1
0
-10 -15
I
I
100
I
I
I
I
300 500
I
I
700 900
Temperature, T °C
10 I I
=E t:> ¢/) ¢D L__
-10
-200 0
0.5
1.0
Thickness mm
15 10 =.t
=Z b
¢/3
5
Center//T~~ l
(20)
. ~
at Td
(21)
Fig. 7 shows the thermal expansion coefficient for the porcelain plotted
-5
Surface~.~ Td
Fig. 5. (a) Simulated result of the transient stresses
-10 I
I
I
I
I
I
I
J
100 300 500 700 900
Temperature, T °C 34
• a • ~ 6)
In accordance with the above discussion, the activation enthalpy for viscosity, Q, and the viscosity constant, no, can be calculated from determination of Td from expansion curves of the porcelain with loads greater than 1.2 - 3.1 MPa under a constant heating rate. Table 3 shows Td calculated from Eq. (20) and from the results of the simulations. All of the temperatures calculated here are in good agreement. When the applied stress is under 1 MPa, T~ occurs at temperatures higher than 630°C. The temperature where ~/E becomes less than 1.0 s is over 640°C. The exact value of Td is difficult to measure because of the low viscosity of the porcelain, as discussed for Tb in Table 2. Eq. (20) shows that T~ increases by only 4°C when the coefficient of thermal expansion at Td is increased from 30 x 10 -6 to 40 x 10-6/°C. The heating rate also affects T~; the increase in Td when the heating rate is increased from l°C/s to 50°C/s is about 50°C. If Eq. (15) is re-written: ct = % / q
0
-15
If T1 is the temperature at which a stress of i MPa develops in the porcelain during the cooling of PFM strips, Td should be determined for the same porcelain with a load on the push-rod of the dilatometer which produces 1.2 - 3.1 MPa during heating. F r o m Eq. (15), T~ can be obtained: T,~ = ( Q / R ) / l n ( a o / q
0
(I3
80 70 60
40
/
50
30
-5
a.
(19)
100
near the surface and at the center of a porcelain slab, 1 mm thick. Here, the heat-soak temperature is 800°C, and the cooling rate is 30°C/s. (b) Residual stress distribuUon at room temperature in the slab after being cooled. (c) Simulated calculation for reheating of the porcelain slab with the residual stresses as shown in (b). Transient stresses are shown for the center and near the surface.
ASAOKA & TESK/VISCO-ELASTIC DEFORMATION OF PORCELAIN
20 10
•
o
T~ .-"
11
450 500 550 600 650 700 750 800
Temperature, T
Fig. 6. Percent creep relaxation occurring within one temperature step at any temperature, T, depending on the cooling rate for the porcelain.
against T~. When Figs. 6 and 7 are compared, T1 and T d can be seen to be almost equal for 1 and 50°C/s, with a ~o of 2.0 MPa.
CONCLUSIONS The thermal expansion and contraction of porcelain while being heated and cooled were simulated by the computer with a one-dimensional visco-elastic model. The macroscopic expansion curves of porcelain with applied stresses at various heating rates and the transient stresses developed in it were computed. The simulated results clarify the effects of the applied stress and the heating rate on the deformation temperature of the porcelain. The results suggest that the deformation temperature for a heating curve of the porcelain under an applied stress of about 1.2 - 3.1 MPa corresponds to the temperature where an incompatibility stress develops in PFM strips while being cooled. A method for determining the temperature dependence of viscosity (as described by the Arrhenius equation) was proposed t h r o u g h m e a s u r e m e n t s of changes in deformation temperatures with changes in either heating rate or applied stress.
ACKNOWLEDGMENTS This work was partially supported by Interagency Agreement Y01-DE30001 from the National Institute of Dental Research, NIH, Bethesda, MD, and by a Grant-in-Aid for Scientific Research (C) from the Japanese Ministry of Education, Science and Culture.
lO -3
TABLE 3 ~/q' MPa/C's20 ,,"
oo
o iiI
i/
10 -s
=
I-
"'
I//o,2,,,'/o.o,
0
--¢1=
ii *'1
10-"
SIMULATED AND CALCULATEDDEFORMATIONTEMPERATURE,T~ °C
/ //! ,,
i1/I
,///
,1'
Td Td 10-7 ,' /, / ,/ , , , 450 500 550 600 650 700 750 800 Temperature, T ~C
Fig. 7. The thermal expansion coefficient of liquid porcelain as a function of the deformation temperature, T~. Calculated results are shown for the ratios of the applied stress to the heating rate, ~o/q, are shown for o'o/q equal to 20, 2, 0.2, and 0.04 MPa/°C.s, respectively.
REFERENCES
ASAOKA, K. and TESZ, J.A. (1987): Residual Stress in Dental Porcelain Related to Cooling Rate, J Dent Res 66: 270, Abstr. No. 1306. ASAOKA, K. and TESK, J.A. (1988): Transient and Residual Stress in Dental Porcelain-Fused-to-Metal Restorations as Affected by the Thermal Expansion Coefficients of the Alloy, Trans 3rd World Biomater Cong 11: 518.
Applied Stress (MPa) 2 10 50 100
Results Calculated by Eq. (20) 627 595 566 554
ASAOKA, K. and TESK, J.A. (1989): Transient and Residual Stresses in Dental Porcelain, Dent Mater J 8: 925. BERTOLOTTI, R.L. and SHELBY, J.E. (1979): Viscosity of Dental Porcelain as a Function of Temperature, J Dent Res 58: 2001-2004. COATS, A.W. and REDFERN,J.P. (1984): Kinetic Parameters from Thermogravimetric Data, Nature 201: 68-69. DEHOFF, P.H. and ANUSAVICE, K.J. (1989): Tempering Stresses in Feldspathic Porcelain, J Dent Res 68: 134138. FAIRHURST, K.J.; ANUSAVICE, K.J.; RINGLE, R.D.; and TWIGGS, S.W. (1981): Porcelain-Metal Thermal Compatibility, J Dent Res 60: 815-819. KASE, H.R.; TESK, JA.; and CASE, E.D. (1985): Elastic Constants of Two Dental Porcelains, J Mater Sci 20: 524531. MCCLINTOCK, F.A. and ARGON, A.S.
Simulated Results 628 595 568 555
(1966): Mechanical Behavior of Materials, Reading, MA: Addison-Wesley, p. 244. MOYNIHAN, C.T.; EASTEAL, A.J.; WILDER, J.; and TUCKER, J. (1974): Dependence of Glass Transition Ternperature on Heating and Cooling Rate, J Phys Chem 78: 2673-2677. ODA, Y.; KOBAYASHI,H.; and SUMIZ, T. (1984): Softening Temperature of Porcelain for Metal-ceramics, Jpn J Dent Mater 3:775--779 (in Japanese). TIMOSHENKO,S. (1925): Analysis of Bimetal Thermostats, J Opt Soc A m 11: 233-255. TWIGGS, S.W.; HASHINGER, D.T.; MORENA~ R.; and FAIRHURS~, C.W. (1986): Glass Transition Temperature of Dental Porcelain at High Heating Rates, J Biomed Mater Res 20: 293--300. WILLIAMSON, E.D. and ADAMS, L.H. (1919): Temperature Distribution in Solids during Heating or Cooling, Phys ReD 14: 99-114.
Dental Materials~January 1991 35
*Department of Dental Engineering Tokushima University School of Dentistry 3 Kuramoto-cho Tokushima, 770, Japan **Dental and Medical Materials National Institute of Standards and Technology Gaithersburg, MD 20899 Received January 2, 1990 Accepted July 28, 1990 Dent Mater 7:30-35, January, 1991
Abstracl-A computer simulation using a visco-elastic stress analysis was conducted to clarify the effect of the heating rate on deformation temperature of dental porcelain during firing. In this simulation, the following temperaturedependent factors were incorporated: elastic modulus, viscosity, and coefficient of thermal expansion. The cooling/heating rate dependencies of both the glasstransition temperature and the temperature distribution in the slab were also included. Thermal expansion curves of porcelain with an applied load at various heating rates were computed. Effects of the applied stress and the heating rate on the deformation temperature of porcelain were revealed. The results suggest that the temperature where the incompatibility stress develops in the porcelain-fused-tometal strips during cooling can be estimated closely from the deformation point of the heating curve of the porcelain with an applied stress of about 1.2 - 3.1 MPa. A method for measuring temperature dependence of viscosity as represented by an Arrhenius equation is proposed.
irst-order transformations in single-phase systems are characterized by a discontinuous change in volume at constant temperature and pressure. In multiphase systems such as alloys, the transition may extend over an appreciable temperature range and, under non-equilibrium conditions, generally begins at higher temperatures on heating than on cooling, with hysteresis as the result. Second-order transformations are generally characterized by the disappearance of ordered structures, with the result that a discontinuous change occurs in the coefficient of thermal expansion at constant temperature and pressure. A glassy material has a second-order transformation in the temperature range of the glass transition and has a higher temperature dependence of the thermal expansion coefficient in the transition range than that of the glassy state, as shown in Fig. 1. Dental porcelains behave like glasses (because they contain large amounts of continuous glassy structure), and at high temperatures, the coefficient of thermal expansion for the liquid porcelain is difficult to measure. Because of their low viscosity at high temperatures, stresses relax, and the high-temperature thermal expansion need not be considered in the discussion of transient and residual stresses. In this paper, the transient and residual stresses developed in porcelain slabs during cooling from the liquid state were simulated with consideration of the cooling rate dependence of the glass-transition temperature, the temperature distribution in the slab, and the temperature dependencies of the elastic modulus, viscosity, and coefficient of thermal expansion. The effects of the cooling rate, thickness of the slab, heat-soak temperature, and coefficient of thermal expansion on the transient and residual stress were calculated by simulations considering the
F
30 ASAOKA & TESK/VISCO-ELASTIC DEFORMATION OF PORCELAIN
second-order transition of the porcelain according to a method employed by Asaoka and Tesk (1987, 1989). DeHoff and Anusavice (1989) also calculated the tempering stresses in feldspathic porcelain in a somewhat related manner. The computer simulation technique was also applied for the heating process of the porcelain. The effect of the viscosity on thermal expansion behavior during heating was analyzed from the simulated results. A relationship between the temperature at which stress just begins to develop in a PFM strip during cooling was examined relative to the temperature at which the porcelain under load in a dilatometer changes from expansion deformation to contraction during heating. Calculations of the residual stress in porcelain-fused-to-metal (PFM) strips (Fairhurst et aL, 1981) have been based on the Timoshenko bimetallic strip equation (Timoshenko, 1925) as follows: ¢r = k
I?
Act d T
(1)
0
where To is the temperature of interest; T1, the temperature at which stress just begins to develop in the porcelain during cooling; and Act, the difference between the coefficient of thermal expansion for the porcelain and that of the alloy. A deeper understanding of the residual stress in PFM strips can be acquired by examination of the method used for determination of the temperature T1. Twiggs et al. (1986) used the three-point bending beam viscometer to measure the deformation temperature, Td, for commercial porcelains at various heating rates*, and found that the depend+Paper relatedto dental porcelainby Fairhurstet al. (1981) used T~ as having the same meaning as Td in this paper. But the glass-transition temperature, Tg, is related to the transition range from solid-like glass to liquid and is not an exact point.
ence of Td on the heating or cooling rate, q, follows the equation of Moynihan et al. (1974): d(Inlql) d(1/Td)
- Q R
(2)
where Q is the activation enthalpy of shear viscosity and R is the gas law constant. The relation between the temperature T~ in Eq. (1) and a deformation temperature, T~, which is measured from the expansion curve of the porcelain during heating, is an important point and has not been previously analyzed by a model with relevance to dentistry. MATERIALS AND METHODS
A computer simulation was conducted for a model of a push-rod dilatometer of the kind which has been used for determining the expansion of porcelain during heating. The analysis is based on the assumption that the one-dimensional visco-elastic behavior of porcelain can be approximated by the two elements of a spring and a dashpot model. For the simulation, a porcelain beam was modeled as if composed of 50 thin, equal, layers. Temperature distributions which developed in the beam during cooling/heating were calculated by the equation of Williamson and Adams (1919), where the surfaces of the slab were presumed to cool/heat at a constant rate. The thermal diffusivity used for the porcelain was chosen as 0.5 mm2/s, the same as in a previous report (Asaoka and Tesk, 1989). Each layer has a unique elastic modulus, thermal expansion coefficient, and viscosity, as determined by its temperature. The Tg in each layer was determined by its cooling/ heating rate by use of the relation in Eq. (2) and a Tg from a reference heating rate. Internal stresses of the beam were calculated by equilibrium of internal moments, strain continuity at the interfaces, and with internal forces set equal to the applied load, as reported in detail by Asaoka and Tesk (1989). The net strain in any layer is equal to the sum of the strains due to thermal expansion/ contraction, normal internal stress, bending strain (almost zero at any temperature because of the sym-
TABLE 1 THERMO-MECHANICAL PROPERTIES OF THE PORCELAIN USED IN THE SIMULATION 70.2 GPa at 30°C 66.7 GPa at 651°C
Elastic modulus (a) Thermal expansion coefficient (b) TTo Liquid Transition temperature (c) Viscosity (c) References: (a) K~ise et aL (1985). (b) Asaoka, experimental results. (c) Bertolotti and Shelby (1979).
c( = (1.3 + 0.020-T) × 10-6/°C c~ = {(x' + 0.28 (T - To)} × 10-6/°C Here, c~' is the coefficient at TQ c~ = 35 × 10-6/°C 460°C at 0.17°C/s -q = 3.6 × 10 -9 exp(404OO/r) Pa • s
metrical temperature distribution), and viscous relaxation. Transient stresses were calculated by an incremental time-step method either during cooling f r o m T~ ( h e a t - s o a k temperature) to TR (30°C) or heating from TR to Tb (load-bearing temperature), with temperature stages I°C apart. Stresses were computed for every stage. The constants in the temperature dependence of the coefficient of thermal expansion for the porcelain were computed from data on heating curves from 40 to 400°C after previous slow-cooling of porcelain which had been fired three times. The modulus of elasticity for the porcelain was taken from the data by K~ise et al. (1985). The viscosity of the porcelain was from the data of Bertolotti and Shelby (1979). Creep relaxation was assumed to follow either a Voigt model for cooling or a Maxwell model (for example, see McClintock and Argon, 1966) for heating. The incremental time step, At, was determined for the cooling/heating rate at the surface for 1°C temperature decrements/increments. The creep relaxation for the cooling process was then calculated according to the following equations: e/eo = 1 -
exp(-E.At/'q)
~c = ~o/n
(5)
where ¢ro is the applied stress, and is the viscosity as in Eq. (4). Table 1 shows the thermo-mechanical properties of the porcelain used in this simulation. RESULTS
Effect of Heating Rate and Applied Stress on Deformation of Porcelain Using the Maxwell Model.- The factors which determine the amount of the viscous flow (creep strain) of porcelain at each constant-temperature step are the applied stress, porcelain viscosity, and holding time. The strain rate during the heating of the porcelain is determined by the stress applied by the push rod and the viscosity, as shown in Eq. (5). The strain rates, ~¢, at fixed temperature, plotted against the applied stresses, ao, are straight lines, as shown in Fig. 2. The results of the simulation agree with the experimental data by Oda et al. (1984) obtained with applied A
(3) E
= no e x p ( Q / R T )
(4)
where eo is the normal strain due to internal stress from the previous time step; E and ~, elastic modulus and viscosity, respectively; no, constant; Q and R, activation enthalpy for shear viscosity and gas law constant; and T, absolute temperature. For the heating process, the creep strain rate, ~, was assumed to follow (McClintock and Argon, 1966):
> l//
=o
j
i.,--rangeTemperature, T °C
Fig. 1. Variation of specific volume of well-annealed glass being cooled at a specified rate. Dental Materials/January 1991 31
1.6 1.4
Deformation temperature, Td ~C
P=IO MPa q = 1 ~3/s
100
700
600
50(
.UJ
10
°C
'
~
10
600'(:;
03
,
1
5
0 •~ e¢Io o. x
,
10
50 100
"'
Stress, a MPa Fig. 2. Strain rate ~due to viscous flow of porcelain under applied stress and fixed temperature. Data points are from the experimental results for the same material by Oda ef al. (1984).
stresses higher than 10 MPa. These results indicate that the Maxwell model is appropriate for simulating the viscous flow of porcelain during heating. Fig. 3 shows an example of the simulated expansion curve for porcelain. Here, the conditions are a heating rate of l°C/s and an applied stress of 10 MPa. The solid line shows the simulated result of thermal expansion with creep relaxation, and the dotted line is the thermal expansion without creep. The glass-transition temperature, T., is defmed as the point of the inflection of the slope. The temperature where the increase in length of the specimen due to thermal expansion is balanced by the negative deformation due to the applied stress is defined here as the deformation point, T~. The loadbearing temperature, Tb, is the point at which the net negative deformation of the specimen above T~ is 2%. It is possible to calculate Tb from the following equation: ~Tb
a(T)dT
(6)
+ titR
UOdT = - 0.02 q~o exp( Q/RT)
R
where the first term represents the amount the specimen would expand in the absence of both externally applied forces and surface tension, and the second term represents the creep strain rate due to a compressive stress (defined as negative) at any temperature, T, integrated over the temperature range with a heating
32
=T
0.8
/
/q=l C/s
,
0.6 0.9
,(i,,!/, tO
1.1
1.2
13
Temperature, IO00/T fK Fig. 4. Reciprocal of the deformation temperature, T~, of the porcelain as a function of applied stress. Here, cooling rates are 0.1, 1.0, and IO.O°C/s, respectively. Thickness of the slab is 2 ram.
0.4 0.2 0.0
q=10~;/s///
b
1
1.0
|
100
300
500
700
I
900
Temperature, T °C Fig. 3. Result of the simulated expansion curve for a 2-mm-thick porcelain slab. Here, the heating rate is 1°C/s, and the applied stress is 10 MPa. The solid line shows the simulated result, and the dashed line is the thermal expansion without viscous flow. Ta, T~, and T~ represent glass-transition temperature, deformation temperature, and load-bearing temperature, respectively.
rate of q. The sum of these two must, by definition, be -0.02. The second term has no exact integral, but by making the substitution u = Q/RT and using the relation (Coats and Redfern, 1984),
: e-uu-bdu
-- u l - b e
-u
( - 1)~(b)~ n=O
U n+ l
(7)
it can be calculated numerically. Table 2 shows measured values of Tb by Oda et al. (1984), Tb calculated from Eq. (6), and T b determined from simulations. Here, the heating rate is 0.5 °C/s. The simulations were conducted two ways using the Maxwell model and one of two conditions: (1) The slab maintained a stress distribution when the porcelain was at high temperature and had low viscosity, or (2) uniform stress when ~/ E of the porcelain became less than AT/q (the relaxation time of the porcelain). The two simulations and calculated results are in good agreement with the experimental data. Experimental data with stresses of 2 and 10 MPa are close to the range of the
ASAOKA & TESK/VISCO-ELASTICDEFORMATIONOF PORCELAIN
results from the simulations made with the two different assumptions. This is discussed i,. the next paragraph. For various heating, at~, the results of the simulation ~_ ,)w that 1/Td and the negative logarithm of the applied stress, uo, are proportional, as shown in Fig. 4. But T~ becomes imprecisely determined when the applied stress is under 1 MPa. This is because internal stresses generated during heating are of this magnitude, and the deformation rate at any point is e = (oro _+ UT)/'O,depending on the sign of aT, where uo is the applied stress and (rT is the t h e r m a l stress. In o r d e r for the overall deformation of the specimen to be controlled by the applied stress, the lowest value of applied stress must be greater than aT. From an elastic stress calculation based on the temperature distribution in a slab of 2-mm thickness, the surface of the specimen has about 1 MPa compressive stress during heating at a constant rate of l°C/s. When the heating rate is 10°C/s, the surface of the same slab has a calculated compressive stress of about 10 MPa. Therefore, the stress which is caused by the temperature distribution in the slab is significantly higher than the applied stress under ordinary conditions for measurement of thermal expansion by use of a push-rod dflatometer. Under these conditions, the relationship between Td and the externally applied stress loses its significance, and simulations and calculations are imprecise.
Transient Stress Profiles During Coolingand Re.heating.- It may be of interest,
TABLE2 MEASUREDAND CALCULATEDLOAD-BEARINGTEMPERATURE,Tb°C
for dental laboratory practice, to note the transient stress profiles developed during rapid cooling and after re-heating of the porcelain, as revealed by computer simulations. Stresses for the cooling process were calculated by use of the program as reported by Asaoka and Tesk (1989). In that study, the transient stress developed in porcelain during rapid cooling and the residual stress at room temperature resulting from the rapid cooling were computed. Fig. 5(a) shows the transient stresses at the surface and the center of the porcelain with 1-mm thickness during cooling from a heat-soak temperature of 800°C. The cooling rate is 30°C/s, which is comparable with that experienced in commercial dental laboratories. The parabolic residual stress distribution induced in a slab cooled at a constant rate is shown in Fig. 5(b). The calculated compressive residual stress near the surface was strongly dependent on the glasstransition temperature, viscosity, coefficient of thermal expansion at T > Tg, slab thickness, cooling rate, and heat-soak temperature. The present study shows the transient stresses developed during re-heating at 0.5°C/ s in a slab which has residual stresses from being cooled (Fig. 5c). The residual stresses remained nearly unchanged until Tg was reached. Above Tg, the porcelain dropped sharply to a stress-free state over a limited temperature range of about 100°C. The porcelain temperature reached Td after passing the stress-free temperature. These simulations demonstrate the effect of viscosity on the contraction/ expansion behavior of porcelain while being cooled and re-heated.
Results Applied Experiment Calculated by Simulated Stress (MPa) 1st firing 3rd firing Eq. (6) Results 2 659 699 709 660-711" 10 660 669 672 660-674* 50 638 638 639 640 100 630 638 625 624 *Lower and upper values are simulated results under assumptions (1) and (2), respectively, in the text.
DISCUSSION The exact temperature where the stress develops while PFM strips are being cooled is an important variable in the thermo-mechanical compatibility of PFM strips. The stress in a PFM strip and porcelain viscosity are functions of the temperature and the cooling rate. During cooling, the visco-elastic behavior of the porcelain can be represented by a Voigt model. If the stress and the viscosity are assumed to be independent of the cooling rate for a small temperature
step, the delayed elasticity can be represented by: (T
ev = ~ [1 - exp(-E'At/'q)]
(8)
where ev is creep strain; (r, stress at T; E, elastic modulus; ~1, viscosity; and At, time. The net strain mismatch between the alloy and the porcelain is calculated as the sum of the thermal contraction mismatch, et, between the alloy and the porcelain for a temperature step, AT, and the viscous strain, %, for the porcelain. Here, et is represented by" et = Act. AT
(9)
where Act is the difference between the coefficient of thermal expansion of the alloy and the porcelain; and AT, the temperature step in time, At. The cooling rate, q, is: When T1 is the effective temperature at which the porcelain displays delayed elasticity, dt
~o/~ = ct" q
(10)
q = AT/At
fo E
and from Eqs. (10), (12), and (14), we can obtain relative relations among AT, At, a, and T1. The calculated creep relaxation ratios for the porcelain are shown in Fig. 6. The Tl's for slow cooling (I°C/ s) and rapid cooling (50°C/s) are indicated in the Fig. The porcelain is almost stress-free for temperatures above T , because viscous flow relaxes the stress from the thermal contraction mismatch, Act, between the alloy and the porcelain. For heating of the porcelain slab only, the Maxwell model provides good agreement with experimental results, as shown in Fig. 2 and Table 2. The deformation temperature, Td, is the point where the strain rate from thermal expansion during heating equals that due to compressive viscous deformation,
1
(11)
where v is a relaxation time for which 63% of the strain is relaxed over the temperature interval. From this, E . I c = q'~lo
(12)
f?
Cro/'q • d t =
fo
%'I~_
T,
exp(Q/RT)
(13)
At that temperature, the viscous strain of the porcelain in Eq. (8) is equal to 63% of the mismatch of the thermal strain for a temperature step as shown in Eq. (9). At T , AT =
(7"
E " Act
(14)
(16)
ct-AT
(17)
00" q where, Ih =
dt
ct " qdt
then,
[~rd
Td+AT
r T1-AT
(15)
where ct is coefficient of thermal expansion of porcelain at Td. From Eq. (15),
where, Ic =
at Td
dt
exp(Q/RT)
If T1 is assumed to be equal to T~ for the same q, [ Ih ] = I Ic ], a relation between a and ct can be derived with Eqs. (12) and (14): c~
%/¢r = ~
(18)
For ordinary commercial porcelains and alloys for metal-ceramics, the instantaneous ct at the deformation
Dental Materials/January 1991
33
15 10 o. ¢t3
b
,,;
q~
point of the porcelain is about 30 40 x 10-6PC and instantaneous Aa = ( 3 0 - 4 0 × 10 .6 ) P C - ( 1 5 - 17 × 10-6)/°C = 1 3 - 25 × 10-6/°C, respectively (Asaoka and Tesk, 1988), .~ leading to approximate values of
Center
5
% / a = 1.2 - 3.1
0
-10 -15
I
I
100
I
I
I
I
300 500
I
I
700 900
Temperature, T °C
10 I I
=E t:> ¢/) ¢D L__
-10
-200 0
0.5
1.0
Thickness mm
15 10 =.t
=Z b
¢/3
5
Center//T~~ l
(20)
. ~
at Td
(21)
Fig. 7 shows the thermal expansion coefficient for the porcelain plotted
-5
Surface~.~ Td
Fig. 5. (a) Simulated result of the transient stresses
-10 I
I
I
I
I
I
I
J
100 300 500 700 900
Temperature, T °C 34
• a • ~ 6)
In accordance with the above discussion, the activation enthalpy for viscosity, Q, and the viscosity constant, no, can be calculated from determination of Td from expansion curves of the porcelain with loads greater than 1.2 - 3.1 MPa under a constant heating rate. Table 3 shows Td calculated from Eq. (20) and from the results of the simulations. All of the temperatures calculated here are in good agreement. When the applied stress is under 1 MPa, T~ occurs at temperatures higher than 630°C. The temperature where ~/E becomes less than 1.0 s is over 640°C. The exact value of Td is difficult to measure because of the low viscosity of the porcelain, as discussed for Tb in Table 2. Eq. (20) shows that T~ increases by only 4°C when the coefficient of thermal expansion at Td is increased from 30 x 10 -6 to 40 x 10-6/°C. The heating rate also affects T~; the increase in Td when the heating rate is increased from l°C/s to 50°C/s is about 50°C. If Eq. (15) is re-written: ct = % / q
0
-15
If T1 is the temperature at which a stress of i MPa develops in the porcelain during the cooling of PFM strips, Td should be determined for the same porcelain with a load on the push-rod of the dilatometer which produces 1.2 - 3.1 MPa during heating. F r o m Eq. (15), T~ can be obtained: T,~ = ( Q / R ) / l n ( a o / q
0
(I3
80 70 60
40
/
50
30
-5
a.
(19)
100
near the surface and at the center of a porcelain slab, 1 mm thick. Here, the heat-soak temperature is 800°C, and the cooling rate is 30°C/s. (b) Residual stress distribuUon at room temperature in the slab after being cooled. (c) Simulated calculation for reheating of the porcelain slab with the residual stresses as shown in (b). Transient stresses are shown for the center and near the surface.
ASAOKA & TESK/VISCO-ELASTIC DEFORMATION OF PORCELAIN
20 10
•
o
T~ .-"
11
450 500 550 600 650 700 750 800
Temperature, T
Fig. 6. Percent creep relaxation occurring within one temperature step at any temperature, T, depending on the cooling rate for the porcelain.
against T~. When Figs. 6 and 7 are compared, T1 and T d can be seen to be almost equal for 1 and 50°C/s, with a ~o of 2.0 MPa.
CONCLUSIONS The thermal expansion and contraction of porcelain while being heated and cooled were simulated by the computer with a one-dimensional visco-elastic model. The macroscopic expansion curves of porcelain with applied stresses at various heating rates and the transient stresses developed in it were computed. The simulated results clarify the effects of the applied stress and the heating rate on the deformation temperature of the porcelain. The results suggest that the deformation temperature for a heating curve of the porcelain under an applied stress of about 1.2 - 3.1 MPa corresponds to the temperature where an incompatibility stress develops in PFM strips while being cooled. A method for determining the temperature dependence of viscosity (as described by the Arrhenius equation) was proposed t h r o u g h m e a s u r e m e n t s of changes in deformation temperatures with changes in either heating rate or applied stress.
ACKNOWLEDGMENTS This work was partially supported by Interagency Agreement Y01-DE30001 from the National Institute of Dental Research, NIH, Bethesda, MD, and by a Grant-in-Aid for Scientific Research (C) from the Japanese Ministry of Education, Science and Culture.
lO -3
TABLE 3 ~/q' MPa/C's20 ,,"
oo
o iiI
i/
10 -s
=
I-
"'
I//o,2,,,'/o.o,
0
--¢1=
ii *'1
10-"
SIMULATED AND CALCULATEDDEFORMATIONTEMPERATURE,T~ °C
/ //! ,,
i1/I
,///
,1'
Td Td 10-7 ,' /, / ,/ , , , 450 500 550 600 650 700 750 800 Temperature, T ~C
Fig. 7. The thermal expansion coefficient of liquid porcelain as a function of the deformation temperature, T~. Calculated results are shown for the ratios of the applied stress to the heating rate, ~o/q, are shown for o'o/q equal to 20, 2, 0.2, and 0.04 MPa/°C.s, respectively.
REFERENCES
ASAOKA, K. and TESZ, J.A. (1987): Residual Stress in Dental Porcelain Related to Cooling Rate, J Dent Res 66: 270, Abstr. No. 1306. ASAOKA, K. and TESK, J.A. (1988): Transient and Residual Stress in Dental Porcelain-Fused-to-Metal Restorations as Affected by the Thermal Expansion Coefficients of the Alloy, Trans 3rd World Biomater Cong 11: 518.
Applied Stress (MPa) 2 10 50 100
Results Calculated by Eq. (20) 627 595 566 554
ASAOKA, K. and TESK, J.A. (1989): Transient and Residual Stresses in Dental Porcelain, Dent Mater J 8: 925. BERTOLOTTI, R.L. and SHELBY, J.E. (1979): Viscosity of Dental Porcelain as a Function of Temperature, J Dent Res 58: 2001-2004. COATS, A.W. and REDFERN,J.P. (1984): Kinetic Parameters from Thermogravimetric Data, Nature 201: 68-69. DEHOFF, P.H. and ANUSAVICE, K.J. (1989): Tempering Stresses in Feldspathic Porcelain, J Dent Res 68: 134138. FAIRHURST, K.J.; ANUSAVICE, K.J.; RINGLE, R.D.; and TWIGGS, S.W. (1981): Porcelain-Metal Thermal Compatibility, J Dent Res 60: 815-819. KASE, H.R.; TESK, JA.; and CASE, E.D. (1985): Elastic Constants of Two Dental Porcelains, J Mater Sci 20: 524531. MCCLINTOCK, F.A. and ARGON, A.S.
Simulated Results 628 595 568 555
(1966): Mechanical Behavior of Materials, Reading, MA: Addison-Wesley, p. 244. MOYNIHAN, C.T.; EASTEAL, A.J.; WILDER, J.; and TUCKER, J. (1974): Dependence of Glass Transition Ternperature on Heating and Cooling Rate, J Phys Chem 78: 2673-2677. ODA, Y.; KOBAYASHI,H.; and SUMIZ, T. (1984): Softening Temperature of Porcelain for Metal-ceramics, Jpn J Dent Mater 3:775--779 (in Japanese). TIMOSHENKO,S. (1925): Analysis of Bimetal Thermostats, J Opt Soc A m 11: 233-255. TWIGGS, S.W.; HASHINGER, D.T.; MORENA~ R.; and FAIRHURS~, C.W. (1986): Glass Transition Temperature of Dental Porcelain at High Heating Rates, J Biomed Mater Res 20: 293--300. WILLIAMSON, E.D. and ADAMS, L.H. (1919): Temperature Distribution in Solids during Heating or Cooling, Phys ReD 14: 99-114.
Dental Materials~January 1991 35
