The porcelain component of a porcelain-fused-to-metal restoration L is strengthened by residual (tempering) stresses which are induced by cooling procedures followed in dental laboratories. The thermophysical properties of materials and cooling rate are the main factors which determine the residual stress. In this paper, the temperatures in the midplane of body-porcelain disks were measured from a heat-soak temperature (1000'C) to room temperature during two different cooling procedures: slow cooling in air and forced-air cooling. Experimental results approximated exponential cooling wherein the cooling rates could be represented by a linear equation of temperature. Residual stresses, as affected by the tempering method and thickness of a porcelain disk, were calculated by computer simulation for regions away from the edges. The following temperature-dependent factors were incorporated into the simulation: elastic modulus, viscosity, and coefficient of thermal expansion. The cooling rate dependencies of the glass transition temperature and the temperature distribution during cooling were also included. The cooling rates used in this simulation were derived from the tempering data. The agreement between development of transient and residual stresses-calculated by computer simulation for various cooling methods, and the tendency toward failures of porcelain disks subjected to the tempering processeswas examined. Simulated residual stresses were also in good agreement with those measured by the indentation fracture method of Marshall and Lawn (1977) and Anusavice et al. (1989). J Dent Res 71(9):1623-1627, September, 1992

Introduction. Elastic bending stresses developed between porcelain and metal from thermal contraction mismatch during cooling from the fired state can be evaluated by use of the Timoshenko (1925) bimetallic strip equation: To

where r is the bending stress; k is a function of dimensions and elastic properties of each material; T is an upper, stress-free temperature; To is the temperature oi interest; and a is the difference in coefficients ofthermal contraction of the alloy and the dental porcelain from T to T for the given materials. Eq. (1) is not affected by the cooling rate, since it assumes only an elastic, uniform-temperature state. This is contrary to the fact that residual stress in a porcelain-fused-to-metal (PFM) strip is affected by the rate of cooling. Evidence for this is shown by the different bending curvatures found by Bertolotti (1980). Fairhurst et al. (1981) considered T1 in Eq. (1) as equal to the deformation point, Td, of a bending porcelain beam during heating and measured Received for publication March 4, 1991 Accepted for publication April 7, 1992 This work was partially supported by a Grant-in-Aid for Scientific Research (C) from the Japanese Ministry of Education, Science and Culture and by Interagency Agreement Y01-DE-30001 from the National Institute of Dental Research, NIH, Bethesda, MD.

the effect of the heating rate on Td. The effects of cooling rates on residual stress in PFM strips were then considered as related to T1, determined from the heating. According to dental laboratory practice, PFM restorations are fired several hundred degrees above the porcelain's glass transition temperature, removed from the furnace, and allowed to cool in air at ambient temperature. Fairhurst et al. (1989) reported that the rate of cooling through the glass transition temperature range, in dental laboratory practice (ambient-air cooling), approached about 10'C/s. The cooling rate, however, is not a constant during convective cooling. Asaoka and Tesk (1990, 1991) pointed out that viscoelastic deformation and the coefficients of thermal expansion of the porcelain and metal, in the porcelain glass-transition range, play an important role in compatibility ofPFM strips. They used computer simulation to clarify the mechanisms of stress development in dental porcelain and PFM strips during cooling. The durability of porcelain and the success of tempering are thought to depend strongly on the levels of stress which can be generated in the porcelain during thermal tempering. Before a study of the most suitable thermal tempering techniques for PFM restorations can be undertaken, the tempering of porcelain alone as it is affected by its thermo-physical properties andthickness should beinvestigated. A previous paper by Asaoka and Tesk (1989) calculated tempering stresses as developed by constant cooling rates. DeHoff and Anusavice (1989) calculated transient and residual stresses of a porcelain slab subjected to convective cooling. They selected three values for the convective heat-transfer coefficients that were used for their calculations oftemperature distributions in a semi-infinite plate. However, the heat-transfer coefficients used were arbitrarily selected and were not associated with measurement of specific cooling conditions. HoJjatie et al. (1989) calculated the heat-transfer coefficient for temperings under the assumption of temperatureindependent thermo-physical properties of porcelain, and used those results to calculate the transient stresses by an elastic finiteelement method. They used temperature gradients calculated with a constant heat-transfer coefficient during convective cooling. The present study calculates transient and residual stresses from tempering, by use of a viscoelastic model. An experimentally determined cooling profile (found to be approximately Newtonian) was used for calculation ofthe needed temperature distributions, which were applicable for specified cooling conditions, without the use of arbitrary heat transfer coefficients. The objective ofthis study was to examine the correlations between transient and residual stresses, calculated by computer simulation for two tempering methods, and the resistance to the failures of porcelain disks duringthe tempering treatment. The results may be used to infer modifications which could be made to tempering treatments and will be used for further studies in the future. In general, porcelain behaves as a viscoelastic material in the glass transition range where thermo-physical properties-such as the coefficient of thermal expansion, glass transition temperature, and viscosity-have strongtemperature dependency and determine the residual stress. In this study, the cooling rates of porcelain specimens as affected by tempering methods were measured by thermocouples embedded at the porcelain midplane. The results were used to develop relationships between cooling rates and residual stresses in the porcelain, as calculated by consideration of 1623

1624

J Dent Res September 1992

ASAOKA et al.

0.10 1000

0.08

900 800 Ls

G1)

700 Cl,

600

0.06

4-'

E

500

0.04

a)

400

0.02

300 0

30

60

90

0 2 4 6 8 10 12 14 16

120

Thickness (mm)

Time (s) Fig. 1-Logarithm of temperature vs. time for porcelain specimens, 10 mm in diameter and 10 mm thick (solid line) and 1.5 mm thick (dashed line), removed from a furnace at 10000C and cooled in ambient air. The slope m is a constant from Eq. (3) in the text.

Fig. 2-Relationship between the value of m and thickness of a porcelain disk, 10 mm in diameter, with ambient-air cooling.

to one of two methods: (1) suspended in air at ambient temperathe temperature-dependent properties according to the method of ture, or (2) with forced convection by horizontal flow ofcompressed Asaoka and Tesk (1989). air (0.6 MPa) parallel to the suspended porcelain surface from a distance of 30 cm to the porcelain (diameter of the nozzle used for blasting, 7 mm). Materials and methods. Temperatures in the porcelain disks were measured at oneExperimental methods. -Porcelain powder was formed into disks second intervals during cooling. A data-recorder (DL-9100, TEAC, by means of a steel mold (internal diameter of 10 mm) and a press. Tokyo, Japan) was connected with a computer (PC-9800, NEC, Chromel-alumel(CA)thermocoupleswithO.13-mm-diameterstrands Tokyo, Japan) by an RS-232C buss to save the accumulated data. Analytical method.-When the temperature within the porcewere embedded in the porcelain compact. The porcelain disks were fired according to manufacturer's instructions. After one lain is reasonably uniform duringheating and cooling, the temperafiring procedure, the disks were re-heated on thin platinum foil, ture may be determined by what is known as a "lumped system 20 gm thick, to 1000TC and held there for five min. Disks were analysis" (LSA) (Ozisik, 1987). Under this condition, when the removed from the furnace by use of the foil and cooled according temperature, T, is changed by an incremental dT during a time

TABLE THERMO-MECHANICAL PROPERTIES OF THE PORCELAIN USED IN THE SIMULATION

Property Elastic modulus, E Glass transition onset temperature, Tg(onset) Thermal diffusivity, K

Viscosity, Xr Coefficient of thermal expansion, a

Value E =70.2 GPa at 300C; 66.7 GPa at 4750C

Reference (1)

Tg(onset) = 3450C determined at cooling rate of 0.08'C/s

(2)

K = 0.66 mm2/s

(3)

= 6.0 x 10- exp(30800/T) pa s Oa = (8.5 + 0.01 T) x 10-6/OC a = t(a' + 0.07(T - Tg)} x 106/OC a = 35 x106/0C

(4) (2)

q

for T T g(onset) for TLjquid T T g(onset) for T = T Luid (1) KIse et al. (1985), (2) Measured data, (3) Piddock et al. (1989), and (4) Asaoka et al. (1990).

0

100

1000 900 800 700

Midplane 50 co

a) :3 CD

a)

0-

1625

TEMPERING STRESS IN PORCELAIN

Vol. 71 No. 9

CL) Cfo a)

500

Nl.

--Oo

-~~~~~~~~~~~~~~~~~l,'

-50

4-J

CU)

E(3)

r

0

600

--_____

__--~~

400

Surface

-100

300

-1 i fl

nv

I,

0

30

15

60

45

Time (s) Fig. 3-Logarithm of temperature vs. time for porcelain cylinders, 10 mm in diameter and 2 mm thick, which were removed from a furnace at 1000W and cooled by forced air. due dt, interval, to the loss of heat

across the

boundary,

the

200

0

400 600 Temperature (C)

800

Fig. 4-Results of simulation of transient-stress profiles in the surface and at the midplane of a 10-mm-diameter porcelain disk, 2 mm thick. Solid lines represent ambient-air cooling, and dashed lines represent forced-air cooling. Here, compressive stresses are negative.

change~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

interval, dt, due to the loss of heat across the boundary, the change ofthe porcelain for VC decrements. The thermo-physical properties in temperature is given by

C dT = -U (T- T) S dt

(2)

Here, C is the heat capacity and is equal to VpC , where: V is the volume; p is the density; and C is the specific heat of the porcelain; U is the coefficient of heat convection, assumed to be constant; T is the temperature of the porcelain; T is the ambient temperature; and S is the surface area of the porcelain. Eq. (3) follows from Eq. (2), and the porcelain surface cools according to an exponential curve. T = Ta + A, exp(-mt)

(3)

with Ai=T,-TaO where T. and T are the heat-soak temperature and the ambient air temperature, respectively; t is the elapsed time after removal of the porcelain from a furnace; and m is given by m = SU/ (VpCP)

of the porcelain used are as shown in the Table. For the present study, the cooling process was modified from the previous constant cooling rate to a temperature-dependent cooling rate as shown by Eq. (5), i.e., the surface of the porcelain was cooled according to Eq. (3). The LSA is considered to provide a reasonable approximation for describing the cooling behavior when the Biot number, Bi, < 0.1 (Ozisik, 1987). With slow ambient air cooling, Bi = 0.05 for the porcelain specimen 10 mm in diameter and 10 mm thick, and Bi = 0.02 for the specimen of 2-mm thickness. With forced-air cooling, Bi = 0.08 for a 2-mm-thick specimen. Under these conditions, Eqs. (1) to (5) can be considered to apply. However, temperature gradients can exist. They are just relatively small in comparison with the temperature. For a surface at a temperature, T = A[1 - exp(-fPt)], for t > 0, the temperature distribution in a slab which has a zero initial temperature has an exact solution (Carslaw and Jaeger, 1959). With the surface temperature, T, modified to T - T = A exp(- Pt), the temperature distribution can be computed from the following Eq.:

(4) T - Ta= A2exp (-pt) COS x (1/K)1/2 cos 1 (13/K)"12

The cooling rate, q, of the porcelain surface is calculated from Eq.(3): q = aT/dt = - mA1 exp(-mt) = -m(T- Ta)

(5)

+

16pA2I2

n=O

cos (2n +1)irx (6) ( 1)ne-K(2n+1rlV24l2 21 (2n +1) [4 l2-KUr2(2n+ 1)2]

Or

and m is experimentally determined from the cooling profile by the use of Eq. (3). Residual stresses in tempered porcelain specimens had previously been estimated by computer simulation according to a method ofcalculation as reported in detail by Asaoka and Tesk (1989, 1990). Creep relaxation was assumed to follow a Voigt model. The incremental time-step was determined from the surface temperature

where K is the thermal diffusivity; 1 is a half-thickness of the slab; and x is the distance from the midplane of the slab, - I