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Transient and Residual Stress in a Porcelain-Metal Strip K. Asaoka and J.A. Tesk J DENT RES 1990 69: 463 DOI: 10.1177/00220345900690020901 The online version of this article can be found at: http://jdr.sagepub.com/content/69/2/463
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Transient and Residual Stress in a Porcelain-Metal Strip K. ASAOKA and J.A. TESK1 Department of Dental Engineering, School of Dentistry, Tokushima University, Tokushima, Japan; and 'Dental and Medical Materials, Polymers Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899
Porcelain-fused-to-metal (PFM) restorations may develop cracks during processing or in-mouth service if the relative physico-mechanical properties of the porcelain and metal are highly mismatched. Precise conditions when this might occur are not known. Many processing and property variations can affect the stresses developed throughout a porcelain-metal system. To understand this, we conducted a computer simulation of stress developed in a PFM beam. The simulation considers cooling from temperatures higher than the porcelain sagpoint. The following temperature-dependent factors were incorporated: the elastic modulus, shear viscosity (porcelain), and coefficients of thermal expansion. The cooling rate dependencies of the glass transition temperature, (Tg), and the temperature distribution during cooling were also included. The results suggest that transient tensile stress at the porcelain alloy interface may result in cracks in the porcelain during cooling. Occlusal forces may set up stresses to cause cracking at the surface of the porcelain if the compressive residual stress is not high enough. PFM restorations with an alloy of high thermal expansion coefficient require rapid cooling; on the contrary, PFM restorations with the alloys of lower coefficients require slow cooling. A high cooling rate can make up for thermal expansion mismatches between the alloy and the porcelain up to 2 X 10-6/C. Finally, the results indicated that curvature was not a sensitive indication of stress for a multimaterial beam when visco-elastic relaxation and high cooling rates are involved. For the case modeled here, curvature varied inversely with a 112 to 1/7th power of the stress.
(1982) pointed out that Tg as defined by Fairhurst et al. (1981) is discussed at length in standard texts, and according to that definition it is called the "porcelain-softening temperature". The most common definition of the softening temperature is called the "Littleton point": It is the temperature which produces a viscosity of 4.5 x 106 Pa s (Kingery et al., 1975). In general, Tg as defined by Fairhurst is actually called a "deformation temperature" or "sag point". DeHoff and Anusavice (1986) developed an analytical model that considered the START DATA INPUT
CALCULATE TEMPERATURE DISTRIBUTION F. IF PORCELAIN IS LIQUID, CALCULATE Tg
CALCULATE TEMPERATURE DEPENDENCE OF PHYSICO-MECHANICAL PROPERTIES
J Dent Res 69(2):463-469, February, 1990
Introduction. The alloys and ceramics used for the construction of porcelainfused-to-metal (PFM) restorations must have coefficients of thermal expansion that are suitably chosen if undesirable thermomechanical tensile stresses are to be avoided. Residual stress has been calculated by some as a function of the coefficient of thermal expansion mismatches for a porcelain-alloy bimaterial composite. These calculations were based on the Timoshenko (1925) bimetallic strip equation, as follows:
CALCULATE VISCOUS FLOW, THERMAL SHRINKAGE AND INTERNAL STRAIN BY NOMINAL AND BENDING STRESS LAST ELEMENT ? STRESS ANALYSIS
(T1
C=kk
A
dT
(1)
To
where k k(T) and is a function containing the moduli of elasticity of the two materials and geometrical parameters for the strip; Aco is the difference between the coefficients of thermal expansion; To is a lower temperature of interest, such as the ambient service temperature; and T1 is the temperature at which stress just begins to be developed in the porcelain during cooling. For example, Fairhurst et al. (1981) proposed that stress will be developed when Tg (the glass-transition temperature of the porcelain) is equivalent to T1 and measured the relation between Tg and the cooling rate. Bertolotti and Fukui Received for publication April 14, 1989 Accepted for publication November 16, 1989 This study was partially supported by NIDR Interagency Agreement Y01-DE-30001 and by a Grant-in-Aid for Scientific Research of the Japanese Ministry of Education (C-62570878).
RESULTS PRINT-OUT ADD INCREMENTAL TIME-STEP
LAST STEP ? FINAL RESULTS OUT-PUT DRAW FIGURES BY X-Y PLOTTER END Fig. 1-Procedure for calculation of the transient and residual stresses in the composite beam.
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464
ASAOKA & TESK
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ro I0
x *-0
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18 k 16 F
0 0
C._n 141 C CX Q
x w
Alloy
12
10 io-
0
0) CD Low
a)
200 400 600 800 Temperature, T 0C
Fig. 2-Typical average thermal-expansion coefficients, o, between 40'C and temperature, T, for the alloy and the porcelain (showing temperature dependencies).
CD
I0 50
A flow chart of the simulation procedure is shown in Fig. 1. Temperature distributions that developed in the beam during cooling were calculated with use of the equation of Williamson and Adams (1919), where the surfaces of the slab were presumed to cool at a constant rate. The temperature distribution in the alloy was chosen as uniform because the thermal diffusivities of dental alloys are over six times that of porcelain for nickel-based alloys and nearly 30 times higher for gold alloys. Compared with temperature differences found in the porcelain, this assumption produces temperature errors of 4% or less in the alloy; this is considered to be insignificant. The thermal diffusivity used for the porcelains was 0.5 mm2/s, the same as, in the previous reports by Asaoka and Tesk (1987,
x
40
:) a
0
0 *
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c 0 a
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x w a)
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200
400
600 Temperature, T 0C
800
Fig. 3-True thermal-expansion coefficients at temperature, T, for the materials as shown in Fig. 2.
same
effects of applied load and heating rate on Tg for dental porcelain. They showed that, depending on the measurement technique, porcelain has different Tg values. The residual stress developed in porcelain after cooling depends mainly on the differences in thermal contraction between the porcelains and alloy. Fairhurst et al. (1980) and Whitlock et al. (1980) measured the thermal expansion and/or contraction for several kinds of porcelains. The data were reported as the coefficients of expansion from either 40 or 500C up to 400 to 500TC. These average coefficients, especially for the porcelains, vary widely, depending on the upper temperature. Whitlock et al. (1980) also reported on thermal expansion for metals. Bertolotti and Shelby (1979) measured the viscosity of dental porcelain as a function of the temperature, because the viscosity of the porcelain controls the effective value of T1 in Eq. (1) and affects the stress-strain distribution at room temperature. Tesk et al. (1981), using data from creep experiments, estimated the activation energy and viscosity constants of several commercial dental porcelains. Young's moduli and shear moduli for several commercial dental porcelains and alloys used for PFM restorations were measured by KIse et al. (1985) and Kase and Tesk (1984). The development of residual stress in a PFM restoration can be influenced by the temperature dependence of the elastic constants of both materials. The aim of this investigation was to use computer simulation to clarify our understanding of the build-up of either transient or residual stress in PFM restorations during cooling from 8000C, which is well above T. Included in the simulation are: the cooling rate dependence of Tg, temperature-dependent viscosity, the actual thermal expansion coefficients at each temperature, temperature-dependent elastic moduli, thermal diffusivity of the porcelains, the temperature distribution developed in the porcelain during cooling, and dimensional factors.
Materials and methods. For the modeling simulation, a composite beam was chosen having 50 thin, equal layers of porcelain and one layer of alloy.
0
0-
Febmary 1990
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 rate by use of the equation proposed by Moynihan et al. (1974) and a Tg from a reference cooling rate. The activation enthalpy of shear viscosity was taken as equal to the activation energy in the Andrade equation for viscosity. Internal stresses and curvatures of the beam were calculated by equilibrium of internal moments and by strain continuity being required at the interfaces, with external forces set equal to zero, as reported in more detail by Asaoka and Tesk (1989). The net strain in any layer is
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0
x
465
TRANSIENT AND RESIDUAL STRESS IN PFM STRIP
VoL 69No. 2
50
1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00
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Fig. 4-Percent
creep
relaxation occurring within
one
-60
0100 300 500 700 900 Temperature,
'100 300 500 700 900
Temperature,
T 'C
T. 'C
Fig. 5-Transient stress developed in the body porcelain during cooling. Cooling rates are PC/s (left) and 50'C/s (right).
temperature stcp
at any temperature, T, depending on the cooling rate for body porcelain.
TABLE INPUT PARAMETERS FOR COMPUTER SIMULATION OF STRESSES IN PORCELAIN-METAL COMPOSITE BEAM Thickness Body 0.8 mm: Opaque 0.2 mm: Alloy 0.5 mm Heat-soak Temperature 800'C Cooling Rate 1PC/s and 50'C/s Viscosity Body n = 5.3 x 10-9 exp(38,200/T) Pa * s Opaque = 3.6 x 10-9 exp(40,400/T) Pa * s Reference Temperature for Tg Body 440TC at 0.17'C/s
50 40 0.
30
at 20
n n
n) a)
-
10
Opaque 460'C at 0.17'C/s Thermal Diffusivity for Porcelains K = 5 X 10-1 mm2/s
Thermal-expansion Coefficient a = (1.3 + 0.022 * T) x 10-6/OC T < Tg Body a = {a' + 0.28 (T - Tg)} x 10-6/OC T > Tg Here, (x' is the coefficient at Tg. a = 35 x 10-6/'C Liquid T < Tg Opaque ax = (1.3 + 0.020 * T) x 10-6/'C a = {a' + 0.28 (T - Tg)} x 10-6/OC T > Tg Here, ax' is the coefficient at Tg. oa = 35 x 10-66/C Liquid a = (7.71 + 0.012 * T) x 10-6/OC Alloy
equal to the sum of the strains due to thermal contraction, normal internal stress, bending, and viscous relaxation. Transient stresses were calculated by an incremental time-step method, as shown in Fig. 1, and with the temperature from Ts (heat-soak temperature) to To (30'C) divided into stages, 10C apart. Stresses were computed for every stage. The average coefficients of thermal expansion for the alloy and the porcelain are shown in Fig. 2. Here, the temperature is represented on the horizontal axis, and the reference temperature is 40'C. The coefficients for both of the materials are clearly dependent on the temperature. In general, the coefficient of thermal expansion for a material in a temperature range which has no phase transformation has been found to fit the following Eq. to a high degree:
C1 + C2T (2) where (x is the thermal coefficient of expansion; C1 and C2 are a =
-10
-20
-30
0 100 300 500 700 900 Temperature, T. 'C
0100 300 500 700 900 Temperature, T 'C Fig. 6-Transicnt stress developed in the opaque porcelain during cooling. Cooling rates are 1PC/s (left) and 50'C/s (right). constants, characteristic of each material; and T is the absolute temperature. The true coefficients of thermal contraction for
the porcelain below Tg and for the alloys could be represented by Eq. (2). The constants Cl and C2 for the porcelains were computed from the data on heating curves from 40 to 400'C, for porcelains that had been fired three times and cooled slowly (0.017'C/s). Fig. 3 shows the true coefficients for the same porcelain and the alloy as in Fig. 2. Measurement of the thermal expansion of porcelain at high temperatures is difficult because of deformation of specimens near the softening temperature (-500'C). A reasonable assumption is then needed for the porcelain in this temperature range. In this case, a constant expansion was assumed, as shown by the solid line for the porcelain. Unique values for Tg do not exist. Tg is defined as a single temperature; however, as seen from thermal expansion data (Figs. 2 and 3), it is a fictitious property. It is well-known that there is not a unique temperature at which the thermal expansion changes abruptly during the transition from the glass to the liquid state, or vice versa. This change occurs
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466
ASAOKA & TESK
Dent Res Februan, 1990
CO,
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Interface
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-150
0 100 300 500 700 900o -C Temperature, T
-1 6.0 ' 0100 300 500 700 900 0100 300 500 700 900 Temperature, T 'C Temperature, T 'C Fig. 9-Strain contraction of the body porcelain during cooling. Solid lines are thermal strain contraction only, depending on the thermal-expansion coefficients. Dashed lines are the net strains with strain relaxations for the body porcelain at the surface and at the opaque interface. Cooling rates are 1 C/s (left) and 50'C/s (right).
-200
0100 300 500 700 900
eC Temperature, T Fig. 7-Transient stress developed in the alloy during cooling. Cooling rates are 1'C/s (left) and 50'C/s (right).
TV
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-
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Alloy
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-8.0
.2
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C.)
CU
0 100 300 500 700 900 Temperature, T 'C
Temperature, T 'C Fig. 8-Curvature of the composite beam during cooling. Cooling rates are 1'C/s (left) and 50'C/s (right). over a temperature range with a continually changing value off the coefficient of thermal expansion. As will be shown, the consideration of this behavior has pronounced effects on the calculated values of transient and residual stress. The moduli of elasticity for the porcelains and alloys were taken from the data by Kise et al. (1985) and Kase and Tesk (1984). The following viscosities of body and opaque porcelain, as found by Tesk et al. (1981), were used: viz., -q 5.3 x 10-9 exp (38,200/T) Pa s, and by Bertolotti and Shelby (1979), viz., m = 3.6 x 10-9 exp (40,400/T) Pa s, respec=
tively.
Creep relaxation was assumed to follow a Voigt model. The incremental time-step, At, was determined from the cooling rate at the surface for 10C temperature increments. Fig. 4 shows creep relaxation for two cooling rates calculated according to the following Eq. E/E0 = 1 - exp (-G A t/lr ) (3)
where EO is the normal strain due to internal stress from the previous time-step; G and A, shear modulus and shear viscosity, respectively; At, time-step.
Results. Transient stresses developed in a three-layered composite beam during cooling from a heat-soak temperature to room
-10.0
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0o -12.0
/::/
Opaque Porcelain
Opaque -14.0
/
Porcelain
-16.0
-14.0/
-16.0
0 100 300 500 700 900
0 100 300 500 700 900
Temperature, T 'C
Temperature, T 'C Fig. 10-Strain contraction of the opaque porcelain during cooling. Solid lines are thermal strain contraction only. Dashed lines are the net strains with strain relaxations for the opaque porcelain at the body-porcelain interface and at the alloy interface. Cooling rates are 10C/s (left) and 50TC/s (right).
temperature, and the residual stresses were calculated. In these calculations, the cooling rate was taken as constant. Under this condition, the temperature distribution was not changed during cooling to room temperature. The properties of specific body and opaque porcelains and the nonprecious alloy (for porcelain veneers) that were used in the model are shown in the Table. Transient stresses in the body, opaque, and alloy are shown in Figs. 5 to 7, respectively. Here, the cooling rate was chosen for two cases, that is, 10C/s for slow cooling and 50'C/s for rapid cooling. Above the T the coefficient of thermal expansion of the porcelain was higher than that of the alloy, and below the Tg it was less. When the porcelain had low viscosity, strain relaxation of each layer compensated for the thermal contraction mismatch between the porcelain and the alloy. In this temperature range, the porcelains and alloy were almost stress-free. The temperature range of stress build-up (or creep relaxation) is shown in Fig. 4. The transient stress that developed depended highly on the thermal contraction mismatch.
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40-
30
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20
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1c
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467
TRANSIENT AND RESIDUAL STRESS IN PFM STRIP
Vol. 69 No. 2
0.5 1.0 1.5 Body Opaque Alloy Thickness, mm
-50 -I 0.0 0.5 1.0
3 2
-
0
1 1.5
Body Opaque Alloy Thickness, mm
Fig. 11-Residual stress distribution in composite beam after cooling rates of 10C/s (left) and 50'C/s (right).
100
0
0
I
1
I
I
.- l
5 10
Cooling Rate, q
50
-
Interface
(Tension)
f-I
50 100 Cs-1
Fig. 13-Curvature of the composite beam at the room temperature related to the cooling rate. tures within the porcelain when flaws are present, for example, due to incomplete fusing of the frit during firing. For the low temperature range, strain relaxation was negligible. Transient stress was then due to the thermal contraction
in
10
0-
X)5
Surface
- At
(Compression)
1,. .. 1 5 10 50100
Cooling Rate, q 'Cs-1 Fig. 12-Maximum residual stress in the body porcelain related to cooling rate.
The stress in the alloy was due mostly to the bending of the composite beam. During the cooling process, the progressively increasing stresses reached a peak at the temperature at which the coefficient of thermal expansion of the alloy met that of the porcelain, as shown in Fig. 3. A high tensile stress developed in the porcelain at the porcelain-alloy interface. It can be speculated that this transient tensile stress may result in frac-
mismatch of the materials and their elastic moduli. In this region, the coefficient of thermal expansion of the alloy was greater than that of the porcelain, and the transient stress in the composite beam decreased during cooling at temperatures below the glass-transition range. When an alloy with a low coefficient of thermal expansion was used, the surface of the porcelain was in tension at room temperature. The stress in the alloy was almost entirely caused by the bending of the composite beam. These stresses were well below the yield stresses of dental porcelain alloys by a factor of almost four in most instances. With rapid cooling, Tg shifted to a higher temperature, as did the temperature of interest for stress buildup, as shown in Fig. 4. The temperature gradient in the porcelain was also higher. Rapid cooling generated higher transient stresses, which led to higher residual stresses in the composite beam. Fig. 8 shows the thermal deflection behavior of the porcelainized strip during cooling. Insufficient stress relaxation of the porcelain produced initial bending of the composite beam; this was caused by the temperature gradient. After that, the thermal contraction mismatch of the materials also became an important factor in determining the curvature. The general behavior, as shown in Fig. 8, is in good agreement with the experimental results of Tuccillo and Nielsen (1972). Here, rapid cooling produced a large curvature, which resulted in surface compressive stress. Dimensional changes of the body and the opaque porcelain during cooling are shown in Figs. 9 and 10. Solid lines represent thermal contraction related to the coefficient of the thermal expansion. Dashed lines are the total contraction (thermal
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J Dent Res
ASAOKA & TESK
468
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February 1990
a
E 3.0
E
60
un
40
0-
20
0
q=500C/s
a -,
le
0 T" x
2.0
3.,
q=1C/s * 1.0o
0
-20
7
8
9
10
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13
-60
I
13
9
8 Ci
7
X10-60C-l
I
14
10
I
15
a
16
10-60C-I
Fig. 14-Maximum transient tensile stress at body-opaque interface (BO) and opaque-alloy interface (OA), and residual stress at the surface of the porcelain (S) related to the coefficient of thermal expansion of the alloy. C1 is the constant, as shown in Eq. (2), and (x is the average thermalexpansion coefficient, ranging from 40 to 500'C. Cooling rates are 1 C/s (solid lines) and 50'C/s (dashed lines).
contraction and creep relaxation) at the porcelain-surface and the opaque-porcelain interface. For rapid cooling from the heatsoak temperature to room temperature, there was a smaller thermal contraction than for slow cooling. This was because the Tg shifted to a higher temperature, and glassy porcelain (below Tg) had a lower coefficient of thermal expansion than that of liquid porcelain (above Tg). However, rapid cooling produced a larger net contraction value because of reduced creep relaxation in porcelain during cooling. Fig. 11 shows the residual stress at room temperature. The body porcelain contained stress over a wide range of 7 MPa (compression) to 24 MPa (tension) for a cooling rate of 10C/ s, and from -35 MPa to 42 MPa for cooling at 50'C/s. Tensile strength of the body porcelain was previously estimated by Asaoka (1986) at about 70 MPa. On this basis, the surface compressive residual stress produced a strengthening of about -
14
15 16 a x10 6oC-1
Fig. 15-Curvature of the composite beam at room temperature related to the coefficient of thermal expansion of the alloy. Cooling rates are 10C/ s (solid line) and 50TC/s (dashed line).
50% by rapid cooling. In the alloy interface region, the opaque porcelain had a lower residual tensile stress than the stress in the body. The body porcelain may be more prone than the opaque to cracking under the application of an external load. The factors considered in determining the transient and residual stress in the composite beam are the coefficients of thermal expansion for alloys and porcelains, elastic moduli, cooling rate, thermal diffusivities, viscosities of porcelains, and dimensions of the composite. From a practical sense, isolation of the effects of all of these factors is difficult. For example, the thermal expansion of porcelain, which is important in development of the internal stress, is itself a phenomenon related to many factors, i.e., thermal history, phase transition, and viscosity as affected by temperature and the internal-phase structures. It is a complex topic in itself and is beyond the scope of this paper.
Discussion. Precise effects of cooling rate on the residual stress in PFM restorations are an interesting problem for dental laboratories. Fig. 12 shows the relation between the residual stress in the body porcelain and cooling rates. The compressive residual stress at the surface of the porcelain and tensile residual stress at the opaque interface were raised with an increasing cooling rate, according to the following empirical Eq. (4):
oa
-
k(q/qo)n
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(4)
TRANSIENT AND RESIDUAL STRESS IN PFM STRIP
Vol. 69 No. 2
Here, Co is the residual stress; q, cooling rate in 'C/s; qO = l0C/s; and k and n are constants. For the input data used, the constants for the surface of the body porcelain and opaque interface were k = - 4.2 MPa, n = 0.56, and k = 21 MPa, n = 0.16, respectively. Fig. 13 shows the relation between the curvature of the composite beam and the cooling rate. The following empirical Eq. (5) was found to apply: i/p = D(q/qo)M (5) Here, 1/p is curvature; (q/qo) is as previously defined; and D and m are constants. The constants D and m were calculated as 1.9 x 10-3 /mm and 0.08, respectively. Combining Eqs. (4) and (5) shows: a= k(pD)-(n/m) (6) As can be seen from either Eq. (6) or Fig. 13, the residual stress is highly dependent on curvature; here, curvature is not a sensitive indication of differences in residual stress developed from different cooling rates (n/m 2 7). Too rapid cooling and high net contraction differences (thermal contraction plus strain relaxation) can set up tensile stresses high enough to cause cracking at the alloy interface. Clinically, occlusal forces may add stresses sufficient to cause cracking at the surface of the porcelain, if the residual compressive surface stress is not high enough. The two factors that have the greatest effect on the incompatibility of PFM restorations are: (1) the difference of the coefficients of thermal expansion for the alloy and porcelains, and (2) the cooling rate. The relationship of these factors was calculated; that is, the transient and residual stresses were simulated when alloys with different coefficients of thermal expansion were used. Fig.14 shows the maximum tensile transient stresses in the body and the opaque porcelains, and the residual compressive stress at the surface of the body porcelain. Here, the abscissa represents the average thermal expansion coefficient of the alloy, c, from 40 to 500°C or the intercept C1, in Eq. (2). The solid lines and the dashed lines are for cooling at 1°C/s and 50°C/s, respectively. The designations S, BO, and OA are for the surface of the porcelain (S), body-opaque interface (BO), and opaque-alloy interface (OA). A high transient tensile stress develops at the opaque-alloy interface with rapid cooling, or when an alloy with a low thermal expansion coefficient is used. In these cases, processing cracks may occur easily. The surface of the porcelain has a tensile stress when slow cooling and an alloy with a high thermal-expansion coefficient are used. It is important to note that a PFM restoration with an alloy of high thermal-expansion coefficient is highly desirable for use with rapid cooling. On the other hand, a PFM restoration with an alloy of low thermalexpansion coefficient is desirable for use with slow cooling. The cooling rate can make up for some for the thermal-expansion mismatch between the alloy and the porcelain, up to -2 x 10-6/OC. Whitlock et al. (1980) reported average thermal expansion coefficients between 14.1 and 15.7 x 10-6/0C over a range of 40 to 5000C during cooling. -
469
Fig. 15 shows the relation between the curvature of the composite beam and the coefficient of the thermal-expansion coefficient for the alloy. Results indicate less distortion of a PFM restoration after firing when an alloy with high thermal expansion or slow cooling is used. These simulation results suggest that cracks in PFM restorations can be prevented when the proper cooling rate is chosen for a given porcelain-alloy system. REFERENCES ASAOKA, K. (1986): Estimation on Residual Stress in Porcelain/ Alloy System by Thermal Shock Test, Dent Mater J 5:145-157. ASAOKA, K. and TESK, J.A. (1987): Residual Stress in Porcelain Related to Cooling Rate, J Dent Res 66:270, Abst. No. 1306. ASAOKA, K. and TESK, J.A. (1989): Transient and Residual Stresses in Dental Porcelains as Affected by Cooling Rates, Dent Mater J 8:9-25. BERTOLOTTI, R.L. and FUKUI, H. (1982): Measurement of Softening Temperatures in Dental Bake-on Porcelains, J Dent Res 61:1066-1069. BERTOLOTTI, R.L. and SHELBY, J.E (1979): Viscosity of Dental Porcelain as a Function of Temperature, J Dent Res 58:20012004. DEHOFF, P.H. and ANUSAVICE, K.J. (1986): An Analytical Model to Predict the Effects of Heating Rate and Applied Load on Glass Transition Temperatures of Dental Porcelain, J Dent Res 65:643647. FAIRHURST, C.W.; ANUSAVICE, K.J.; HASHINGER, D.T.; RINGLE, R.D.; and TWIGGS, S.W. (1980): Thermal Expansion of Dental Alloys and Porcelains, J Biomed Mater Res 14:435446. FAIRHURST, C.W.; HASHINGER, D.T.; and TWIGGS, S.W. (1981): Glass Transition Temperature of Dental Porcelain, J Dent Res 60:995-998. KASE, H. and TESK, J.A. (1984): Elastic Constants of Three Representative Nonprecious Dental Alloys, J Dent Res 63:258, Abst. No. 791. KASE, H.R.; TESK, J.A.; and CASE, E.D. (1985): Elastic Constants of Two Dental Porcelains, J Mater Sci 20:524-531. KINGERY, W.D.; BOWEN, H.K.; and UHLMANN, D.R. (1975): Introduction to Ceramics, New York: John Wiley and Sons, p. 759. MOYNIHAN, C.T.; EASTEAL, A.J.; WILDER, J.; and TUCKER, J. (1974): Dependence of the Glass Transition Temperature on Heating and Cooling Rate, J Phys Chem 78:2673-2677. TIMOSHENKO, S. (1925): Analysis of Bimetal Thermostats, J Opt Soc Am 11:233-255. TESK, J.A.; HINMAN, R.W.; WHITLOCK, R.P.; HOLMES, A.; and PARRY, E.E. (1981): Temperature Dependence of Shear Viscosity for Several Dental Porcelains, IADR Prog & Abst 59: No. 839. TUCCILLO, J.J. and NIELSEN, J.P. (1972): Shear Stress Measurements at a Dental Porcelain Gold Bond Interface, J Dent Res 51: 626-633. WHITLOCK, R.P.; TESK, J.A.; WIDERA, G.E.O.; HOLMES, A.; and PARRY, E.E. (1980): Consideration of Some Factors Influencing Compatibility of Dental Porcelains and Alloys, Proc Int Precious Metals Conf 4:273-282. WILLIAMSON, E.D. and ADAMS, L.H. (1919): Temperature Distribution in Solids During Heating or Cooling, Phys Rev 14:99114.
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Transient and Residual Stress in a Porcelain-Metal Strip K. Asaoka and J.A. Tesk J DENT RES 1990 69: 463 DOI: 10.1177/00220345900690020901 The online version of this article can be found at: http://jdr.sagepub.com/content/69/2/463
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Transient and Residual Stress in a Porcelain-Metal Strip K. ASAOKA and J.A. TESK1 Department of Dental Engineering, School of Dentistry, Tokushima University, Tokushima, Japan; and 'Dental and Medical Materials, Polymers Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899
Porcelain-fused-to-metal (PFM) restorations may develop cracks during processing or in-mouth service if the relative physico-mechanical properties of the porcelain and metal are highly mismatched. Precise conditions when this might occur are not known. Many processing and property variations can affect the stresses developed throughout a porcelain-metal system. To understand this, we conducted a computer simulation of stress developed in a PFM beam. The simulation considers cooling from temperatures higher than the porcelain sagpoint. The following temperature-dependent factors were incorporated: the elastic modulus, shear viscosity (porcelain), and coefficients of thermal expansion. The cooling rate dependencies of the glass transition temperature, (Tg), and the temperature distribution during cooling were also included. The results suggest that transient tensile stress at the porcelain alloy interface may result in cracks in the porcelain during cooling. Occlusal forces may set up stresses to cause cracking at the surface of the porcelain if the compressive residual stress is not high enough. PFM restorations with an alloy of high thermal expansion coefficient require rapid cooling; on the contrary, PFM restorations with the alloys of lower coefficients require slow cooling. A high cooling rate can make up for thermal expansion mismatches between the alloy and the porcelain up to 2 X 10-6/C. Finally, the results indicated that curvature was not a sensitive indication of stress for a multimaterial beam when visco-elastic relaxation and high cooling rates are involved. For the case modeled here, curvature varied inversely with a 112 to 1/7th power of the stress.
(1982) pointed out that Tg as defined by Fairhurst et al. (1981) is discussed at length in standard texts, and according to that definition it is called the "porcelain-softening temperature". The most common definition of the softening temperature is called the "Littleton point": It is the temperature which produces a viscosity of 4.5 x 106 Pa s (Kingery et al., 1975). In general, Tg as defined by Fairhurst is actually called a "deformation temperature" or "sag point". DeHoff and Anusavice (1986) developed an analytical model that considered the START DATA INPUT
CALCULATE TEMPERATURE DISTRIBUTION F. IF PORCELAIN IS LIQUID, CALCULATE Tg
CALCULATE TEMPERATURE DEPENDENCE OF PHYSICO-MECHANICAL PROPERTIES
J Dent Res 69(2):463-469, February, 1990
Introduction. The alloys and ceramics used for the construction of porcelainfused-to-metal (PFM) restorations must have coefficients of thermal expansion that are suitably chosen if undesirable thermomechanical tensile stresses are to be avoided. Residual stress has been calculated by some as a function of the coefficient of thermal expansion mismatches for a porcelain-alloy bimaterial composite. These calculations were based on the Timoshenko (1925) bimetallic strip equation, as follows:
CALCULATE VISCOUS FLOW, THERMAL SHRINKAGE AND INTERNAL STRAIN BY NOMINAL AND BENDING STRESS LAST ELEMENT ? STRESS ANALYSIS
(T1
C=kk
A
dT
(1)
To
where k k(T) and is a function containing the moduli of elasticity of the two materials and geometrical parameters for the strip; Aco is the difference between the coefficients of thermal expansion; To is a lower temperature of interest, such as the ambient service temperature; and T1 is the temperature at which stress just begins to be developed in the porcelain during cooling. For example, Fairhurst et al. (1981) proposed that stress will be developed when Tg (the glass-transition temperature of the porcelain) is equivalent to T1 and measured the relation between Tg and the cooling rate. Bertolotti and Fukui Received for publication April 14, 1989 Accepted for publication November 16, 1989 This study was partially supported by NIDR Interagency Agreement Y01-DE-30001 and by a Grant-in-Aid for Scientific Research of the Japanese Ministry of Education (C-62570878).
RESULTS PRINT-OUT ADD INCREMENTAL TIME-STEP
LAST STEP ? FINAL RESULTS OUT-PUT DRAW FIGURES BY X-Y PLOTTER END Fig. 1-Procedure for calculation of the transient and residual stresses in the composite beam.
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464
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ro I0
x *-0
a) a1)
18 k 16 F
0 0
C._n 141 C CX Q
x w
Alloy
12
10 io-
0
0) CD Low
a)
200 400 600 800 Temperature, T 0C
Fig. 2-Typical average thermal-expansion coefficients, o, between 40'C and temperature, T, for the alloy and the porcelain (showing temperature dependencies).
CD
I0 50
A flow chart of the simulation procedure is shown in Fig. 1. Temperature distributions that developed in the beam during cooling were calculated with use of the equation of Williamson and Adams (1919), where the surfaces of the slab were presumed to cool at a constant rate. The temperature distribution in the alloy was chosen as uniform because the thermal diffusivities of dental alloys are over six times that of porcelain for nickel-based alloys and nearly 30 times higher for gold alloys. Compared with temperature differences found in the porcelain, this assumption produces temperature errors of 4% or less in the alloy; this is considered to be insignificant. The thermal diffusivity used for the porcelains was 0.5 mm2/s, the same as, in the previous reports by Asaoka and Tesk (1987,
x
40
:) a
0
0 *
30
c 0 a
c
20
C
x w a)
I-
10
0
200
400
600 Temperature, T 0C
800
Fig. 3-True thermal-expansion coefficients at temperature, T, for the materials as shown in Fig. 2.
same
effects of applied load and heating rate on Tg for dental porcelain. They showed that, depending on the measurement technique, porcelain has different Tg values. The residual stress developed in porcelain after cooling depends mainly on the differences in thermal contraction between the porcelains and alloy. Fairhurst et al. (1980) and Whitlock et al. (1980) measured the thermal expansion and/or contraction for several kinds of porcelains. The data were reported as the coefficients of expansion from either 40 or 500C up to 400 to 500TC. These average coefficients, especially for the porcelains, vary widely, depending on the upper temperature. Whitlock et al. (1980) also reported on thermal expansion for metals. Bertolotti and Shelby (1979) measured the viscosity of dental porcelain as a function of the temperature, because the viscosity of the porcelain controls the effective value of T1 in Eq. (1) and affects the stress-strain distribution at room temperature. Tesk et al. (1981), using data from creep experiments, estimated the activation energy and viscosity constants of several commercial dental porcelains. Young's moduli and shear moduli for several commercial dental porcelains and alloys used for PFM restorations were measured by KIse et al. (1985) and Kase and Tesk (1984). The development of residual stress in a PFM restoration can be influenced by the temperature dependence of the elastic constants of both materials. The aim of this investigation was to use computer simulation to clarify our understanding of the build-up of either transient or residual stress in PFM restorations during cooling from 8000C, which is well above T. Included in the simulation are: the cooling rate dependence of Tg, temperature-dependent viscosity, the actual thermal expansion coefficients at each temperature, temperature-dependent elastic moduli, thermal diffusivity of the porcelains, the temperature distribution developed in the porcelain during cooling, and dimensional factors.
Materials and methods. For the modeling simulation, a composite beam was chosen having 50 thin, equal layers of porcelain and one layer of alloy.
0
0-
Febmary 1990
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 rate by use of the equation proposed by Moynihan et al. (1974) and a Tg from a reference cooling rate. The activation enthalpy of shear viscosity was taken as equal to the activation energy in the Andrade equation for viscosity. Internal stresses and curvatures of the beam were calculated by equilibrium of internal moments and by strain continuity being required at the interfaces, with external forces set equal to zero, as reported in more detail by Asaoka and Tesk (1989). The net strain in any layer is
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0
x
465
TRANSIENT AND RESIDUAL STRESS IN PFM STRIP
VoL 69No. 2
50
1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00
40-
30
60
20-
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40
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10-
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20
ui
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q=50 C/s
C/)
0-
en
-10
eO -20
-20
-40
-30 -
450 500 550 600 650 700 750 800 Temperature, T 'C
d
Fig. 4-Percent
creep
relaxation occurring within
one
-60
0100 300 500 700 900 Temperature,
'100 300 500 700 900
Temperature,
T 'C
T. 'C
Fig. 5-Transient stress developed in the body porcelain during cooling. Cooling rates are PC/s (left) and 50'C/s (right).
temperature stcp
at any temperature, T, depending on the cooling rate for body porcelain.
TABLE INPUT PARAMETERS FOR COMPUTER SIMULATION OF STRESSES IN PORCELAIN-METAL COMPOSITE BEAM Thickness Body 0.8 mm: Opaque 0.2 mm: Alloy 0.5 mm Heat-soak Temperature 800'C Cooling Rate 1PC/s and 50'C/s Viscosity Body n = 5.3 x 10-9 exp(38,200/T) Pa * s Opaque = 3.6 x 10-9 exp(40,400/T) Pa * s Reference Temperature for Tg Body 440TC at 0.17'C/s
50 40 0.
30
at 20
n n
n) a)
-
10
Opaque 460'C at 0.17'C/s Thermal Diffusivity for Porcelains K = 5 X 10-1 mm2/s
Thermal-expansion Coefficient a = (1.3 + 0.022 * T) x 10-6/OC T < Tg Body a = {a' + 0.28 (T - Tg)} x 10-6/OC T > Tg Here, (x' is the coefficient at Tg. a = 35 x 10-6/'C Liquid T < Tg Opaque ax = (1.3 + 0.020 * T) x 10-6/'C a = {a' + 0.28 (T - Tg)} x 10-6/OC T > Tg Here, ax' is the coefficient at Tg. oa = 35 x 10-66/C Liquid a = (7.71 + 0.012 * T) x 10-6/OC Alloy
equal to the sum of the strains due to thermal contraction, normal internal stress, bending, and viscous relaxation. Transient stresses were calculated by an incremental time-step method, as shown in Fig. 1, and with the temperature from Ts (heat-soak temperature) to To (30'C) divided into stages, 10C apart. Stresses were computed for every stage. The average coefficients of thermal expansion for the alloy and the porcelain are shown in Fig. 2. Here, the temperature is represented on the horizontal axis, and the reference temperature is 40'C. The coefficients for both of the materials are clearly dependent on the temperature. In general, the coefficient of thermal expansion for a material in a temperature range which has no phase transformation has been found to fit the following Eq. to a high degree:
C1 + C2T (2) where (x is the thermal coefficient of expansion; C1 and C2 are a =
-10
-20
-30
0 100 300 500 700 900 Temperature, T. 'C
0100 300 500 700 900 Temperature, T 'C Fig. 6-Transicnt stress developed in the opaque porcelain during cooling. Cooling rates are 1PC/s (left) and 50'C/s (right). constants, characteristic of each material; and T is the absolute temperature. The true coefficients of thermal contraction for
the porcelain below Tg and for the alloys could be represented by Eq. (2). The constants Cl and C2 for the porcelains were computed from the data on heating curves from 40 to 400'C, for porcelains that had been fired three times and cooled slowly (0.017'C/s). Fig. 3 shows the true coefficients for the same porcelain and the alloy as in Fig. 2. Measurement of the thermal expansion of porcelain at high temperatures is difficult because of deformation of specimens near the softening temperature (-500'C). A reasonable assumption is then needed for the porcelain in this temperature range. In this case, a constant expansion was assumed, as shown by the solid line for the porcelain. Unique values for Tg do not exist. Tg is defined as a single temperature; however, as seen from thermal expansion data (Figs. 2 and 3), it is a fictitious property. It is well-known that there is not a unique temperature at which the thermal expansion changes abruptly during the transition from the glass to the liquid state, or vice versa. This change occurs
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466
ASAOKA & TESK
Dent Res Februan, 1990
CO,
0t-CL -E50
Interface
5 X |XInterface
I0
I0
x
x a
CU:
'.) CIO
0
0 L)
/
-
0
0
-150 -
-150
0 100 300 500 700 900o -C Temperature, T
-1 6.0 ' 0100 300 500 700 900 0100 300 500 700 900 Temperature, T 'C Temperature, T 'C Fig. 9-Strain contraction of the body porcelain during cooling. Solid lines are thermal strain contraction only, depending on the thermal-expansion coefficients. Dashed lines are the net strains with strain relaxations for the body porcelain at the surface and at the opaque interface. Cooling rates are 1 C/s (left) and 50'C/s (right).
-200
0100 300 500 700 900
eC Temperature, T Fig. 7-Transient stress developed in the alloy during cooling. Cooling rates are 1'C/s (left) and 50'C/s (right).
TV
_ E E
40
0.0
E
0.0-
-
-2.0-
-2.0
x
-4.0 x
C,,
10 x
-6.0
-6.0
a
CU
Alloy
-4.0
Alloy
V)
?5 6I
0
1=
-8.0
.2
-8.0-
C.)
CU
0 100 300 500 700 900 Temperature, T 'C
Temperature, T 'C Fig. 8-Curvature of the composite beam during cooling. Cooling rates are 1'C/s (left) and 50'C/s (right). over a temperature range with a continually changing value off the coefficient of thermal expansion. As will be shown, the consideration of this behavior has pronounced effects on the calculated values of transient and residual stress. The moduli of elasticity for the porcelains and alloys were taken from the data by Kise et al. (1985) and Kase and Tesk (1984). The following viscosities of body and opaque porcelain, as found by Tesk et al. (1981), were used: viz., -q 5.3 x 10-9 exp (38,200/T) Pa s, and by Bertolotti and Shelby (1979), viz., m = 3.6 x 10-9 exp (40,400/T) Pa s, respec=
tively.
Creep relaxation was assumed to follow a Voigt model. The incremental time-step, At, was determined from the cooling rate at the surface for 10C temperature increments. Fig. 4 shows creep relaxation for two cooling rates calculated according to the following Eq. E/E0 = 1 - exp (-G A t/lr ) (3)
where EO is the normal strain due to internal stress from the previous time-step; G and A, shear modulus and shear viscosity, respectively; At, time-step.
Results. Transient stresses developed in a three-layered composite beam during cooling from a heat-soak temperature to room
-10.0
52-10.0 -12.0-
0o -12.0
/::/
Opaque Porcelain
Opaque -14.0
/
Porcelain
-16.0
-14.0/
-16.0
0 100 300 500 700 900
0 100 300 500 700 900
Temperature, T 'C
Temperature, T 'C Fig. 10-Strain contraction of the opaque porcelain during cooling. Solid lines are thermal strain contraction only. Dashed lines are the net strains with strain relaxations for the opaque porcelain at the body-porcelain interface and at the alloy interface. Cooling rates are 10C/s (left) and 50TC/s (right).
temperature, and the residual stresses were calculated. In these calculations, the cooling rate was taken as constant. Under this condition, the temperature distribution was not changed during cooling to room temperature. The properties of specific body and opaque porcelains and the nonprecious alloy (for porcelain veneers) that were used in the model are shown in the Table. Transient stresses in the body, opaque, and alloy are shown in Figs. 5 to 7, respectively. Here, the cooling rate was chosen for two cases, that is, 10C/s for slow cooling and 50'C/s for rapid cooling. Above the T the coefficient of thermal expansion of the porcelain was higher than that of the alloy, and below the Tg it was less. When the porcelain had low viscosity, strain relaxation of each layer compensated for the thermal contraction mismatch between the porcelain and the alloy. In this temperature range, the porcelains and alloy were almost stress-free. The temperature range of stress build-up (or creep relaxation) is shown in Fig. 4. The transient stress that developed depended highly on the thermal contraction mismatch.
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40-
30
30-
20
20 -I
1c
4)
4
40
I/ y
nn
a6,
-lC
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0 I-
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0. en
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-20 -I
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-30 -I
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-40 -I
-sc
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467
TRANSIENT AND RESIDUAL STRESS IN PFM STRIP
Vol. 69 No. 2
0.5 1.0 1.5 Body Opaque Alloy Thickness, mm
-50 -I 0.0 0.5 1.0
3 2
-
0
1 1.5
Body Opaque Alloy Thickness, mm
Fig. 11-Residual stress distribution in composite beam after cooling rates of 10C/s (left) and 50'C/s (right).
100
0
0
I
1
I
I
.- l
5 10
Cooling Rate, q
50
-
Interface
(Tension)
f-I
50 100 Cs-1
Fig. 13-Curvature of the composite beam at the room temperature related to the cooling rate. tures within the porcelain when flaws are present, for example, due to incomplete fusing of the frit during firing. For the low temperature range, strain relaxation was negligible. Transient stress was then due to the thermal contraction
in
10
0-
X)5
Surface
- At
(Compression)
1,. .. 1 5 10 50100
Cooling Rate, q 'Cs-1 Fig. 12-Maximum residual stress in the body porcelain related to cooling rate.
The stress in the alloy was due mostly to the bending of the composite beam. During the cooling process, the progressively increasing stresses reached a peak at the temperature at which the coefficient of thermal expansion of the alloy met that of the porcelain, as shown in Fig. 3. A high tensile stress developed in the porcelain at the porcelain-alloy interface. It can be speculated that this transient tensile stress may result in frac-
mismatch of the materials and their elastic moduli. In this region, the coefficient of thermal expansion of the alloy was greater than that of the porcelain, and the transient stress in the composite beam decreased during cooling at temperatures below the glass-transition range. When an alloy with a low coefficient of thermal expansion was used, the surface of the porcelain was in tension at room temperature. The stress in the alloy was almost entirely caused by the bending of the composite beam. These stresses were well below the yield stresses of dental porcelain alloys by a factor of almost four in most instances. With rapid cooling, Tg shifted to a higher temperature, as did the temperature of interest for stress buildup, as shown in Fig. 4. The temperature gradient in the porcelain was also higher. Rapid cooling generated higher transient stresses, which led to higher residual stresses in the composite beam. Fig. 8 shows the thermal deflection behavior of the porcelainized strip during cooling. Insufficient stress relaxation of the porcelain produced initial bending of the composite beam; this was caused by the temperature gradient. After that, the thermal contraction mismatch of the materials also became an important factor in determining the curvature. The general behavior, as shown in Fig. 8, is in good agreement with the experimental results of Tuccillo and Nielsen (1972). Here, rapid cooling produced a large curvature, which resulted in surface compressive stress. Dimensional changes of the body and the opaque porcelain during cooling are shown in Figs. 9 and 10. Solid lines represent thermal contraction related to the coefficient of the thermal expansion. Dashed lines are the total contraction (thermal
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J Dent Res
ASAOKA & TESK
468
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February 1990
a
E 3.0
E
60
un
40
0-
20
0
q=500C/s
a -,
le
0 T" x
2.0
3.,
q=1C/s * 1.0o
0
-20
7
8
9
10
C1 X1O-6oC-1
-40
13
-60
I
13
9
8 Ci
7
X10-60C-l
I
14
10
I
15
a
16
10-60C-I
Fig. 14-Maximum transient tensile stress at body-opaque interface (BO) and opaque-alloy interface (OA), and residual stress at the surface of the porcelain (S) related to the coefficient of thermal expansion of the alloy. C1 is the constant, as shown in Eq. (2), and (x is the average thermalexpansion coefficient, ranging from 40 to 500'C. Cooling rates are 1 C/s (solid lines) and 50'C/s (dashed lines).
contraction and creep relaxation) at the porcelain-surface and the opaque-porcelain interface. For rapid cooling from the heatsoak temperature to room temperature, there was a smaller thermal contraction than for slow cooling. This was because the Tg shifted to a higher temperature, and glassy porcelain (below Tg) had a lower coefficient of thermal expansion than that of liquid porcelain (above Tg). However, rapid cooling produced a larger net contraction value because of reduced creep relaxation in porcelain during cooling. Fig. 11 shows the residual stress at room temperature. The body porcelain contained stress over a wide range of 7 MPa (compression) to 24 MPa (tension) for a cooling rate of 10C/ s, and from -35 MPa to 42 MPa for cooling at 50'C/s. Tensile strength of the body porcelain was previously estimated by Asaoka (1986) at about 70 MPa. On this basis, the surface compressive residual stress produced a strengthening of about -
14
15 16 a x10 6oC-1
Fig. 15-Curvature of the composite beam at room temperature related to the coefficient of thermal expansion of the alloy. Cooling rates are 10C/ s (solid line) and 50TC/s (dashed line).
50% by rapid cooling. In the alloy interface region, the opaque porcelain had a lower residual tensile stress than the stress in the body. The body porcelain may be more prone than the opaque to cracking under the application of an external load. The factors considered in determining the transient and residual stress in the composite beam are the coefficients of thermal expansion for alloys and porcelains, elastic moduli, cooling rate, thermal diffusivities, viscosities of porcelains, and dimensions of the composite. From a practical sense, isolation of the effects of all of these factors is difficult. For example, the thermal expansion of porcelain, which is important in development of the internal stress, is itself a phenomenon related to many factors, i.e., thermal history, phase transition, and viscosity as affected by temperature and the internal-phase structures. It is a complex topic in itself and is beyond the scope of this paper.
Discussion. Precise effects of cooling rate on the residual stress in PFM restorations are an interesting problem for dental laboratories. Fig. 12 shows the relation between the residual stress in the body porcelain and cooling rates. The compressive residual stress at the surface of the porcelain and tensile residual stress at the opaque interface were raised with an increasing cooling rate, according to the following empirical Eq. (4):
oa
-
k(q/qo)n
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(4)
TRANSIENT AND RESIDUAL STRESS IN PFM STRIP
Vol. 69 No. 2
Here, Co is the residual stress; q, cooling rate in 'C/s; qO = l0C/s; and k and n are constants. For the input data used, the constants for the surface of the body porcelain and opaque interface were k = - 4.2 MPa, n = 0.56, and k = 21 MPa, n = 0.16, respectively. Fig. 13 shows the relation between the curvature of the composite beam and the cooling rate. The following empirical Eq. (5) was found to apply: i/p = D(q/qo)M (5) Here, 1/p is curvature; (q/qo) is as previously defined; and D and m are constants. The constants D and m were calculated as 1.9 x 10-3 /mm and 0.08, respectively. Combining Eqs. (4) and (5) shows: a= k(pD)-(n/m) (6) As can be seen from either Eq. (6) or Fig. 13, the residual stress is highly dependent on curvature; here, curvature is not a sensitive indication of differences in residual stress developed from different cooling rates (n/m 2 7). Too rapid cooling and high net contraction differences (thermal contraction plus strain relaxation) can set up tensile stresses high enough to cause cracking at the alloy interface. Clinically, occlusal forces may add stresses sufficient to cause cracking at the surface of the porcelain, if the residual compressive surface stress is not high enough. The two factors that have the greatest effect on the incompatibility of PFM restorations are: (1) the difference of the coefficients of thermal expansion for the alloy and porcelains, and (2) the cooling rate. The relationship of these factors was calculated; that is, the transient and residual stresses were simulated when alloys with different coefficients of thermal expansion were used. Fig.14 shows the maximum tensile transient stresses in the body and the opaque porcelains, and the residual compressive stress at the surface of the body porcelain. Here, the abscissa represents the average thermal expansion coefficient of the alloy, c, from 40 to 500°C or the intercept C1, in Eq. (2). The solid lines and the dashed lines are for cooling at 1°C/s and 50°C/s, respectively. The designations S, BO, and OA are for the surface of the porcelain (S), body-opaque interface (BO), and opaque-alloy interface (OA). A high transient tensile stress develops at the opaque-alloy interface with rapid cooling, or when an alloy with a low thermal expansion coefficient is used. In these cases, processing cracks may occur easily. The surface of the porcelain has a tensile stress when slow cooling and an alloy with a high thermal-expansion coefficient are used. It is important to note that a PFM restoration with an alloy of high thermal-expansion coefficient is highly desirable for use with rapid cooling. On the other hand, a PFM restoration with an alloy of low thermalexpansion coefficient is desirable for use with slow cooling. The cooling rate can make up for some for the thermal-expansion mismatch between the alloy and the porcelain, up to -2 x 10-6/OC. Whitlock et al. (1980) reported average thermal expansion coefficients between 14.1 and 15.7 x 10-6/0C over a range of 40 to 5000C during cooling. -
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Fig. 15 shows the relation between the curvature of the composite beam and the coefficient of the thermal-expansion coefficient for the alloy. Results indicate less distortion of a PFM restoration after firing when an alloy with high thermal expansion or slow cooling is used. These simulation results suggest that cracks in PFM restorations can be prevented when the proper cooling rate is chosen for a given porcelain-alloy system. REFERENCES ASAOKA, K. (1986): Estimation on Residual Stress in Porcelain/ Alloy System by Thermal Shock Test, Dent Mater J 5:145-157. ASAOKA, K. and TESK, J.A. (1987): Residual Stress in Porcelain Related to Cooling Rate, J Dent Res 66:270, Abst. No. 1306. ASAOKA, K. and TESK, J.A. (1989): Transient and Residual Stresses in Dental Porcelains as Affected by Cooling Rates, Dent Mater J 8:9-25. BERTOLOTTI, R.L. and FUKUI, H. (1982): Measurement of Softening Temperatures in Dental Bake-on Porcelains, J Dent Res 61:1066-1069. BERTOLOTTI, R.L. and SHELBY, J.E (1979): Viscosity of Dental Porcelain as a Function of Temperature, J Dent Res 58:20012004. DEHOFF, P.H. and ANUSAVICE, K.J. (1986): An Analytical Model to Predict the Effects of Heating Rate and Applied Load on Glass Transition Temperatures of Dental Porcelain, J Dent Res 65:643647. FAIRHURST, C.W.; ANUSAVICE, K.J.; HASHINGER, D.T.; RINGLE, R.D.; and TWIGGS, S.W. (1980): Thermal Expansion of Dental Alloys and Porcelains, J Biomed Mater Res 14:435446. FAIRHURST, C.W.; HASHINGER, D.T.; and TWIGGS, S.W. (1981): Glass Transition Temperature of Dental Porcelain, J Dent Res 60:995-998. KASE, H. and TESK, J.A. (1984): Elastic Constants of Three Representative Nonprecious Dental Alloys, J Dent Res 63:258, Abst. No. 791. KASE, H.R.; TESK, J.A.; and CASE, E.D. (1985): Elastic Constants of Two Dental Porcelains, J Mater Sci 20:524-531. KINGERY, W.D.; BOWEN, H.K.; and UHLMANN, D.R. (1975): Introduction to Ceramics, New York: John Wiley and Sons, p. 759. MOYNIHAN, C.T.; EASTEAL, A.J.; WILDER, J.; and TUCKER, J. (1974): Dependence of the Glass Transition Temperature on Heating and Cooling Rate, J Phys Chem 78:2673-2677. TIMOSHENKO, S. (1925): Analysis of Bimetal Thermostats, J Opt Soc Am 11:233-255. TESK, J.A.; HINMAN, R.W.; WHITLOCK, R.P.; HOLMES, A.; and PARRY, E.E. (1981): Temperature Dependence of Shear Viscosity for Several Dental Porcelains, IADR Prog & Abst 59: No. 839. TUCCILLO, J.J. and NIELSEN, J.P. (1972): Shear Stress Measurements at a Dental Porcelain Gold Bond Interface, J Dent Res 51: 626-633. WHITLOCK, R.P.; TESK, J.A.; WIDERA, G.E.O.; HOLMES, A.; and PARRY, E.E. (1980): Consideration of Some Factors Influencing Compatibility of Dental Porcelains and Alloys, Proc Int Precious Metals Conf 4:273-282. WILLIAMSON, E.D. and ADAMS, L.H. (1919): Temperature Distribution in Solids During Heating or Cooling, Phys Rev 14:99114.
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