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Shear stress relaxation of dental ceramics determined from creep behavior
Dental Materials (2004) 20, 717–725
www.intl.elsevierhealth.com/journals/dema
Shear stress relaxation of dental ceramics determined from creep behavior Paul H. DeHoffa,*, Kenneth J. Anusaviceb a
Department of Mechanical Engineering and Engineering Science, University of North Carolina at Charlotte, Charlotte, NC 28223, USA b Department of Dental Biomaterials, University of Florida, Gainesville, FL, USA Received 2 August 2002; received in revised form 3 September 2003; accepted 9 October 2003
KEYWORDS Relaxation functions; Dental ceramics; Relaxation times; Viscoelasticity; ANSYS; Laplace transforms; Discrete model; Maxwell model
Summary Objective. To test the hypothesis that shear stress relaxation functions of dental ceramics can be determined from creep functions measured in a beam-bending viscometer. Methods. Stress relaxation behavior was determined from creep data for the following materials: (1) a veneering ceramic-IPS Empress2w body ceramic (E2V); (2) an experimental veneering ceramic (EXV); (3) a low expansion body porcelain-Vita VMK 68 feldspathic body porcelain (VB); (4) a high expansion body porcelain-Will Ceram feldspathic body porcelain (WCB); (5) a medium expansion opaque porcelain-Vita feldspathic opaque porcelain (VO); and (6) a high expansion opaque porcelain-Will Ceram feldspathic opaque porcelain (WCO). Laplace transform techniques were used to relate shear stress relaxation functions to creep functions for an eight-parameter, discrete viscoelastic model. Nonlinear regression analysis was performed to fit a fourterm exponential relaxation function for each material at each temperature. The relaxation functions were utilized in the ANSYS finite element program to simulate creep behavior in three-point bending for each material at each temperature. Results. Shear stress relaxation times at 575 8C ranged from 0.03 s for EXV to 195 s for WCO. Significance. Knowledge of the shear relaxation functions for dental ceramics at high temperatures is required input for the viscoelastic element in the ANSYS finite element program, which can used to determine transient and residual stresses in dental prostheses during fabrication. Q 2004 Academy of Dental Materials. Published by Elsevier Ltd. All rights reserved.
Introduction Residual tensile stresses in ceramic caused by thermal contraction mismatch between materials are an important contributing factor to failures of metal – ceramic and all-ceramic dental prostheses. It is generally understood that the residual stress state *Corresponding author. Tel.: þ1-704-687-4324; fax: þ1-704687-6069. E-mail address: [email protected]
in metal – ceramic and ceramic – ceramic dental prostheses depends on many factors, including contraction mismatch, cooling rate, firing temperature, geometry, and fabrication technique. Recently, commercial finite element programs such as ANSYS (ANSYS, Inc., Canonsburg, PA) have included a viscoelastic option that now makes it possible to include time-dependent properties of dental ceramics at high temperature for the calculation of residual stresses in dental prostheses.
0109-5641/$ - see front matter Q 2004 Academy of Dental Materials. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.dental.2003.10.005
718
P.H. DeHoff, K.J. Anusavice
However, the ANSYS program requires a shear stress relaxation function as input and we are unaware of any experimental studies that report the measurement of time-dependent shear stress properties of dental ceramics. Previous studies related to time-dependent property measurements of dental ceramics were primarily concerned with the measurement of shear viscosities1 – 3 or uniaxial stress relaxation functions.4 – 6 Recently, DeHoff and Anusavice7 used a beam-bending viscometer (BBV) to measure creep behavior from which creep functions were calculated for two glass–ceramics and four feldspathic porcelains at temperatures ranging from 450 to 650 8C. While creep functions cannot be used directly in the ANSYS program, it is possible to determine the required shear relaxation functions from the creep data through the use of Laplace transform methods and linear viscoelastic theory. Therefore, the objective of this study was to characterize the high temperature viscoelastic properties of several dental ceramics by the determination of relaxation functions based on creep functions measured in a BBV. The validity of the procedures was confirmed by using the relaxation properties in the ANSYS finite element program to simulate a BBV-type creep test for each of the dental ceramics.
Materials and methods Viscoelastic theory. The three-dimensional stress – strain relationships for a linear isotropic viscoelastic material are given by:8 ðt
›1ij ðtÞ sij ðtÞ ¼ cðt 2 tÞ dt ›t 0 Sij ðtÞ ¼ 2
QðtÞ ¼ 3
ðt
ðt
Kðt 2 tÞ
0
1ij ¼
ðt
›eij ðtÞ dt ›t
ð2Þ
›uðtÞ dt ›t
ð3Þ
Gðt 2 tÞ
0
Jðt 2 tÞ
0
ð1Þ
›sij dt ›t
ð4Þ
where sij and 1ij are the stress and strain tensors, respectively (i and j take on the values 1, 2, and 3 corresponding to the coordinate axes (x; y; and z)); t is any arbitrary time between zero and the present time, t, and is called the past time; Sij is the deviatoric stress tensor and is determined from the stress tensor by: Sij ðtÞ ¼ sij ðtÞ 2
QðtÞ d 3 ij
ð5Þ
where Q is the first stress invariant ðQ ¼ s11 þ s22 þ s33 Þ and dij is the Kronecker delta (dij ¼ 1 if i ¼ j and 0 if i – j). Likewise, eij ðtÞ ¼ 1ij ðtÞ 2
uðtÞ d 3 ij
ð6Þ
eij ðtÞ is the deviatoric strain tensor and u is the first strain invariant. cðtÞ is the uniaxial stress relaxation function, GðtÞ is the shear stress relaxation function, KðtÞ is the hydrostatic (or bulk) stress relaxation function, and JðtÞ is the creep function. Laplace transform methods are used to develop interrelationships among the material functions in transform space. The Laplace transform of a function FðtÞ is defined by the integral:9 ð1 fðpÞ ¼ e2pt FðtÞdt ð7Þ 0
The Laplace transforms of Eqs. (1) –(4), which are in convolution form, are given by:
sij ðpÞ ¼ pcðpÞ1ij ðpÞ
ð8Þ
e ij ðpÞ S ij ðpÞ ¼ 2pGðpÞ
ð9Þ
uðpÞ Q ðpÞ ¼ 3pKðpÞ sij ðpÞ 1ij ðpÞ ¼ pJðpÞ
ð10Þ ð11Þ
In a uniaxial test, s11 ¼ s11 ; and all other stress components are zero. Therefore, S 11 ¼ 2=3ðs11 Þ and e 11 ¼ 111 2 1=3ðuÞ from which it is possible to obtain the following relationships from Eqs. (8) to (11): 1 p2 JðpÞ cðpÞ 3KðpÞ GðpÞ ¼ 9KðpÞ 2 cðpÞ
cðpÞ ¼
ð12Þ ð13Þ
where p represents the transform variable and the bar above the variable indicates that the variables are in the transform space.9 Eqs. (12) and (13) are then used to determine the transformed shear stress relaxation function from the transformed uniaxial stress, creep and bulk relaxation functions, followed by Laplace transform inversion techniques to determine the shear stress relaxation function in the time domain. The process to determine the shear stress relaxation coefficients from experimental BBV creep data is described in the following section.
Shear stress relaxation functions An eight-parameter discrete spring-dashpot viscoelastic model (Fig. 1) was proposed by DeHoff and Anusavice7 to characterize the creep behavior measured in a BBV for several commercial dental ceramics. The creep function in the time domain
Shear stress relaxation of dental ceramics determined from creep behavior
b1 ¼ h0 h1 h2 h3 ðE0 2 9K0 Þ
719
ð19eÞ
b2 ¼ E0 h0 ðE1 h2 h3 þ E2 h1 h3 þ E3 h1 h2 Þ 2 9K0 E0 ðh0 h1 h2 þ h0 h1 h3 þ h0 h2 h3 þ h1 h2 h3 Þ 2 9K0 h0 ðE1 h2 h3 þ E2 h1 h3 Figure 1 Eight-parameter discrete viscoelastic model.
1 1 1 1 t þ þ þ þ E0 E1 E2 E3 h0 E1 t E2 t exp 2 exp 2 h1 h2 2 2 E1 E2 E t exp 2 3 h3 2 E3
2 9K0 E0 {h0 ½E1 ðh2 þ h3 Þ þ E2 ðh1 þ h3 Þ þ E3 ðh1 þ h2 Þ þ E1 h2 h3 þ E2 h1 h3 þ E3 h1 h2 } 2 9K0 h0 ðE1 E2 h3 þ E1 E3 h2 þ E2 E3 h1 Þ ð14Þ
Using standard tables, the Laplace transform of JðtÞ is given by:
2
1 1 1 1 1=h þ þ þ þ 20 E0 p E1 p E2 p E3 p p 1=E1 1=E2 1=E3 2 2 ðp þ E1 =h1 Þ ðp þ E2 =h2 Þ ðp þ E3 =h3 Þ
ð15Þ
Scherer10 indicates that dilatational relaxation has a minor effect on stress calculations for most stress states and therefore the dilatational behavior of the dental ceramics has been assumed to be purely elastic so that: KðtÞ ¼ K0
ð16Þ
where K0 is the bulk modulus at room temperature. The Laplace transform of the bulk modulus is then: K KðpÞ ¼ 0 p
ð17Þ
From Eqs. (12),(13),(15) and (17) it can be shown that the shear stress relaxation function in the transform space can be put in the form: GðpÞ ¼
a1 p3 þ a2 p2 þ a3 p þ a4 ðb1 p4 þ b2 p3 þ b3 p2 þ b4 p þ b5 Þ
ð19gÞ
b4 ¼ E0 E1 E2 E3 h0 2 9K0 {E0 ½E1 E3 ðh0 þ h2 Þ
9
JðpÞ
ð19fÞ
b3 ¼ E0 h0 ðE1 E2 h3 þ E1 E3 h2 þ E2 E3 h1 Þ
for this model is given by: JðtÞ ¼
þ E3 h1 h2 Þ
ð18Þ
where ai and bi are given in terms of the model coefficients and K0 by: a1 ¼ 23K0 E0 h0 h1 h2 h3
ð19aÞ
a2 ¼ 23K0 E0 h0 ðE1 h2 h3 þ E2 h1 h3 þ E3 h1 h2 Þ
ð19bÞ
a3 ¼ 23K0 E0 h0 ðE1 E2 h3 þ E1 E3 h2 þ E2 E3 h1 Þ
ð19cÞ
a4 ¼ 23K0 E0 E1 E2 E3 h0
ð19dÞ
þ E1 E2 ðh0 þ h3 Þ þ E2 E3 ðh0 þ h1 Þ þ E1 E2 E3 h0 } b5 ¼ 29K0 E0 E1 E2 E3
ð19hÞ ð19iÞ
Although Laplace inversion of Eq. (18) requires numerical values for the discrete model parameters, it is known that Laplace inversion of a function with a fourth degree polynomial in the denominator (four roots) will lead to a shear stress relaxation function in the time domain in the general form: t t GðtÞ ¼ G1 exp 2 þ G2 exp 2 t t 1 2 t t þ G3 exp 2 þ G4 exp 2 ð20Þ t3 t4 Numerical values for the coefficients ai and bi were first determined by substitution of numerical values for the discrete model parameters7 into Eqs. (19a)–(19i). Numerical values of Gi and ti were then determined by substituting the numerical values of ai and bi into Eq. (18), followed by Laplace inversion of the resulting equation. All Laplace transform manipulations were accomplished by use of the Maplew symbolic mathematics program (Waterloo Maple Inc., Waterloo, Ont., Canada). Values for the input coefficients ai and bi are given in Table 1 for each of the dental ceramics.
Finite element creep In order to test the validity of the transform procedures for calculating shear stress relaxation
720
Table 1 Polynomial coefficients for the shear stress relaxation function in the Laplace transform space, GðpÞ; (Eq. (18)). Tempa (8C)
a1 (N6 s4/mm12)
a2 (N6 s3/mm12)
a3 (N6 s2/mm12)
a4 (N6 s/mm12)
b1 (N5 s4/mm10)
b2 (N5 s3/mm10)
b3 (N5 s2/mm10)
b4 (N5 s/mm10)
b5 (N5/mm10)
E2V
525 550 575 600 475 500 525 550 575 550 575 600 625 650 550 575 600 625 650 550 575 600 625 650 575 600 625 650 675
7.208 £ 1035 1.021 £ 1034 2.381 £ 1031 3.634 £ 1029 1.445 £ 1037 5.862 £ 1033 4.776 £ 1030 2.701 £ 1028 7.116 £ 1025 7.392 £ 1037 6.275 £ 1035 2.684 £ 1033 9.226 £ 1030 2.168 £ 1029 6.590 £ 1037 3.329 £ 1034 2.451 £ 1033 7.458 £ 1030 6.780 £ 1028 1.239 £ 1038 1.051 £ 1037 2.948 £ 1034 3.257 £ 1032 2.750 £ 1030 3.745 £ 1033 3.730 £ 1035 4.450 £ 1033 3.199 £ 1031 5.999 £ 1029
1.128 £ 1035 1.260 £ 1033 3.652 £ 1030 4.756 £ 1028 5.867 £ 1035 5.250 £ 1032 7.253 £ 1029 2.723 £ 1027 3.141 £ 1025 9.464 £ 1036 8.802 £ 1034 3.774 £ 1032 2.293 £ 1030 4.923 £ 1028 1.066 £ 1037 3.399 £ 1034 1.936 £ 1032 1.072 £ 1030 1.326 £ 1028 2.002 £ 1037 1.498 £ 1036 4.865 £ 1033 1.880 £ 1031 1.057 £ 1029 9.806 £ 1036 8.005 £ 1034 8.764 £ 1032 3.388 £ 1030 6.555 £ 1028
1.796 £ 1033 2.549 £ 1031 8.487 £ 1028 1.491 £ 1027 4.185 £ 1033 1.481 £ 1031 2.264 £ 1028 6.135 £ 1025 2.578 £ 1024 1.614 £ 1035 1.463 £ 1033 1.060 £ 1031 1.158 £ 1029 3.038 £ 1027 1.935 £ 1035 5.593 £ 1032 1.811 £ 1030 2.580 £ 1028 2.180 £ 1026 9.394 £ 1035 5.459 £ 1034 1.177 £ 1032 1.972 £ 1029 3.524 £ 1026 6.079 £ 1035 2.707 £ 1033 2.192 £ 1031 5.297 £ 1028 5.830 £ 1026
7.398 £ 1030 7.757 £ 1028 1.700 £ 1026 6.993 £ 1023 8.122 £ 1030 1.338 £ 1029 1.129 £ 1026 2.038 £ 1023 8.245 £ 1021 4.468 £ 1032 6.047 £ 1030 5.393 £ 1028 5.321 £ 1026 2.731 £ 1025 7.098 £ 1032 2.254 £ 1030 3.743 £ 1027 7.780 £ 1025 5.399 £ 1023 1.010 £ 1034 3.836 £ 1032 3.358 £ 1029 3.178 £ 1026 2.433 £ 1023 8.633 £ 1033 1.710 £ 1031 1.185 £ 1029 1.157 £ 1026 1.323 £ 1024
2.970 £ 1031 4.229 £ 1029 9.910 £ 1026 1.518 £ 1025 5.571 £ 1032 2.271 £ 1029 1.856 £ 1026 1.051 £ 1024 2.783 £ 1021 2.864 £ 1033 2.442 £ 1031 1.050 £ 1029 3.620 £ 1026 8.546 £ 1024 2.405 £ 1033 1.225 £ 1031 9.013 £ 1028 2.754 £ 1026 2.516 £ 1024 5.310 £ 1033 4.529 £ 1032 1.274 £ 1030 1.416 £ 1028 1.202 £ 1026 1.614 £ 1033 1.612 £ 1031 1.934 £ 1029 1.398 £ 1027 2.631 £ 1025
8.443 £ 1030 1.891 £ 1029 1.584 £ 1027 8.550 £ 1025 5.667 £ 1031 9.699 £ 1028 4.183 £ 1026 8.518 £ 1024 1.065 £ 1023 4.756 £ 1032 6.358 £ 1030 5.949 £ 1028 7.330 £ 1026 4.025 £ 1025 5.262 £ 1032 3.219 £ 1030 5.469 £ 1028 6.119 £ 1026 1.740 £ 1025 9.979 £ 1032 8.588 £ 1031 5.149 £ 1029 9.525 £ 1027 2.405 £ 1026 4.785 £ 1032 5.381 £ 1030 9.820 £ 1028 1.876 £ 1027 9.269 £ 1025
3.235 £ 1029 1.247 £ 1028 1.701 £ 1026 9.609 £ 1024 8.780 £ 1029 5.904 £ 1027 4.234 £ 1025 7.003 £ 1023 3.571 £ 1022 1.344 £ 1031 2.750 £ 1029 4.652 £ 1027 1.122 £ 1026 7.158 £ 1024 1.496 £ 1031 1.236 £ 1029 1.982 £ 1027 4.982 £ 1025 1.995 £ 1024 5.527 £ 1031 4.319 £ 1030 3.035 £ 1028 3.385 £ 1026 5.950 £ 1024 3.582 £ 1031 3.374 £ 1029 8.459 £ 1027 9.787 £ 1025 5.126 £ 1024
2.913 £ 1027 1.740 £ 1026 3.157 £ 1024 2.615 £ 1023 3.630 £ 1027 1.136 £ 1026 8.876 £ 1023 1.205 £ 1022 2.030 £ 1021 8.102 £ 1028 2.570 £ 1027 7.996 £ 1025 3.748 £ 1024 3.339 £ 1023 1.004 £ 1029 1.102 £ 1027 1.102 £ 1025 6.470 £ 1023 1.764 £ 1022 7.987 £ 1029 5.606 £ 1028 3.101 £ 1026 2.132 £ 1024 1.308 £ 1022 7.165 £ 1029 4.890 £ 1027 1.233 £ 1026 9.203 £ 1023 2.978 £ 1022
6.433 £ 1024 4.246 £ 1023 5.200 £ 1021 1.134 £ 1020 3.837 £ 1024 6.082 £ 1023 3.079 £ 1021 3.057 £ 1019 5.584 £ 1018 7.530 £ 1025 5.416 £ 1024 2.601 £ 1023 1.149 £ 1022 2.168 £ 1021 1.092 £ 1026 2.081 £ 1024 1.021 £ 1022 8.709 £ 1020 2.281 £ 1019 1.143 £ 1027 1.046 £ 1026 4.419 £ 1023 2.142 £ 1021 5.406 £ 1018 1.378 £ 1027 9.225 £ 1024 3.064 £ 1023 8.128 £ 1020 3.931 £ 1019
EXV
VB
VO
WCB
WCO
a
Temperatures indicated are those recorded by the furnace thermocouple in the BBV.
P.H. DeHoff, K.J. Anusavice
Ceramic
Shear stress relaxation of dental ceramics determined from creep behavior
721
representing the BBV loading was applied to the model at the mid-span of the beam. Nodes at each end were constrained in the vertical direction to simulate a simply supported beam. The fourterm shear relaxation function for each ceramic at each temperature (Table 2) was used for the creep calculations.
Figure 2 Finite element model of the three-point bending specimen used in the beam-bending viscometer.
functions from creep data, the viscoelastic element in ANSYS was used to simulate a BBV-type creep test of each of the dental ceramics at each temperature. Shown in Fig. 2 is the finite element model of the BBV three-point bending specimen (5.5 mm £ 2.5 mm £ 48 mm). The model consisted of 192 20node isoparametric three-dimensional viscoelastic elements and 1285 nodes. Because of symmetry along the length, only one half of the beam was modeled with imposition of appropriate symmetry displacement boundary conditions. A line load
Uniaxial stress relaxation A second procedure for checking the validity of the inversion procedure is a comparison of calculated uniaxial stress relaxation functions for the feldspathic porcelains with measured uniaxial stress relaxation data for the same porcelains reported by DeHoff et al.4 The uniaxial stress relaxation function in the transform space is determined by substituting Eq. (15) into Eq. (12) to obtain: E E E E1 E2 E3 p þ 1 p þ 2 p þ 3 h1 h2 h3 cðpÞ ¼ D
ð21Þ
Table 2 Coefficients at the high temperature range for the four-term shear stress relaxation function, GðtÞ; (Eq. (20)). Ceramic
Tempa (8C)
G1 (MPa)
G2 (MPa)
G3 (MPa)
G4 (MPa)
t1 (s)
t2 (s)
t3 (s)
t4 (s)
ts b (s)
E2V
525 550 575 600 475 500 525 550 575 550 575 600 625 650 550 575 600 625 650 550 575 600 625 650 575 600 625 650 675
11516 20640 23355 23870 18079 24251 25158 25633 25499 5254 12789 22550 24270 25116 7100 19133 25157 26301 26634 5421 7061 15679 22244 22729 2151 8748 16605 21942 22451
10011 2684 551 33 5343 261 317 25 38 9273 10166 1746 671 2.3 8443 5400 1615 457 262 48 2703 4380 447 135 2998 7751 5141 555 312
187 720 107 40 197 27 205 27 32 6018 502 917 492 218 4414 691 137 233 36 4120 6375 2363 241 10 5110 3324 466 306 2
2555 102 11 0.2 2315 1274 52 5 0.6 6268 2234 356 55 29.1 7441 1954 286 82 20 13744 7072 708 76 12 12952 3307 796 83 40
4.16 2.70 0.67 0.18 11.90 2.80 0.47 0.13 0.03 7.56 4.79 2.10 0.54 0.22 5.36 4.58 1.76 0.47 0.15 10.68 8.55 2.98 1.57 0.51 5.16 3.89 2.48 0.78 0.29
31.5 17.1 11.0 14.4 83.0 27.0 12.8 16.6 3.8 38.5 25.2 14.9 8.5 9.2 43.7 30.8 31.5 14.6 9.4 13.7 18.5 17.7 33.9 44.2 13.8 17.5 11.5 23.8 20.1
118 67 47 22 242 44 44 51 14.6 148 112 55 23.5 14.9 126 108 185 70 126 51 61 93 154 543 35 57 69 92 225
299 323 548 2269 610 494 231 327 345 881 332 236 294 130 744 386 861 658 638 623 448 588 806 1833 466 452 319 1016 512
15.70 3.38 0.76 0.19 21.40 3.10 0.48 0.13 0.03 79.50 13.10 2.53 0.58 0.09 85.00 12.00 2.00 0.49 0.15 293.00 62.30 5.59 1.66 0.51 195.00 17.70 3.82 0.83 0.30
EXV
VB
VO
WCB
WCO
a b
Temperatures indicated are those recorded by the furnace thermocouple in the BBV. ts is the overall relaxation time defined as the time required for the shear stress relaxation function to decay to 1=e of its initial value.
722
P.H. DeHoff, K.J. Anusavice
where (
! 1 1 1 1 1 þ þ þ þ D ¼p E1 E2 E3 E0 p E1 p E2 p E3 p h0 p2 E1 E2 E3 E2 pþ pþ pþ 2 E2 E3 p þ h1 h2 h3 h2 E E E p þ 3 2 E1 E3 p þ 1 p þ 3 h3 h1 h3 E1 E2 2E1 E2 p þ pþ ð22Þ h1 h2 2
Substitution of the appropriate discrete model constants7 into Eqs. (21) and (22), followed by Laplace inversion, leads to the uniaxial relaxation function in the form:
Figure 3 Calculated shear stress relaxation function versus log10 ðtÞ for E2V at four temperatures.
cðtÞ ¼ c1 expð2t=l1 Þ þ c2 expð2t=l2 Þ þ c3 £ expð2t=l3 Þ þ c4 expð2t=l4 Þ
ð23Þ
Sr ¼
N X
ðyi 2 yðxi ÞÞ2
ð25Þ
i¼1
Numerical values for ci and li were calculated for each of the feldspathic porcelains at three temperatures.
The coefficients for the four-term shear relaxation functions for all groups at each applicable furnace temperature are presented in Table 2. Also presented is the overall relaxation time, i.e. the time at which the normalized relaxation function reaches 1=e of its initial value. In general, the glass – ceramics relax more rapidly (smaller relaxation times) than the feldspathic porcelains. The shear stress relaxation behavior of E2V as a function of log10 ðtÞ at four temperatures is illustrated in Fig. 3. The shift of the relaxation curves to the left as the temperature increases is an indication that the rate of relaxation increases with increasing temperature. Relaxation times of 15.7 and 0.19 s were calculated for E2V at 525 and 600 8C, respectively (Table 2). Shown in Fig. 4 are mid-span deflection versus time data for three E2V specimens at a temperature of 550 8C. Also shown are mid-span deflections (solid curve) calculated by means of the ANSYS finite element program. The coefficient of correlation was calculated to determine the goodness of fit of the calculated curve to the experimental data using the following equations: N X i¼1
ðy~ 2 yi Þ2
N 1 X y N i¼1 i
ð26Þ
Sr St
ð27Þ
R2 ¼ 1 2
Results
St ¼
y~ ¼
ð24Þ
where N is the number of data points, xi and yi represent the x and y values of a typical datum point, yðxi Þ is the y value calculated from the regression equation, and R2 is the correlation coefficient. We used Eqs. (24) –(27) to calculate a correlation coefficient of 0.9985 for the regression fit of the experimental data and a correlation coefficient of 0.9976 for the curve calculated with ANSYS. A comparison of the correlation coefficients
Figure 4 Mid-span deflection versus time for three specimens of E2V at 550 8C. The solid curve represents deflection calculated using ANSYS software.
Shear stress relaxation of dental ceramics determined from creep behavior
723
Table 3 Correlation coefficients for a nonlinear regression fit to the BBV mid-span deflection data compared with those calculated for the ANSYS results. Ceramic
E2V
EXV
VB
VO
WCB
WCO
a
b
c
Tempa (8C)
525 550 575 600 475 500 525 550 575 550 575 600 625 650 550 575 600 625 650 550 575 600 625 650 575 600 625 650 675
Correlation coefficient Regression fitb
ANSYSc
Difference
0.9938 0.9985 0.9976 0.9978 0.9935 0.9957 0.9943 0.9929 0.9668 0.9456 0.9895 0.9982 0.9966 0.9977 0.9227 0.9640 0.9968 0.9660 0.9995 0.9853 0.9952 0.9919 0.9742 0.9975 0.9874 0.9766 0.9927 0.9587 0.9772
0.9778 0.9976 0.9961 0.9975 0.9891 0.9884 0.9911 0.9632 0.9537 0.9396 0.9842 0.9959 0.9938 0.9939 0.9178 0.9640 0.9958 0.9658 0.9971 0.9792 0.9932 0.9863 0.9621 0.9815 0.9846 0.9742 0.9905 0.9498 0.9574
0.0160 0.0009 0.0015 0.0003 0.0044 0.0073 0.0032 0.0297 0.0131 0.0060 0.0053 0.0023 0.0028 0.0038 0.0049 0.0000 0.0010 0.0002 0.0024 0.0061 0.0020 0.0056 0.0121 0.0160 0.0028 0.0024 0.0022 0.0089 0.0198
Temperatures indicated are those recorded by the furnace thermocouple in the BBV. Correlation coefficient based on a nonlinear regression fit to the BBV mid-span deflection data of Ref. 7. Correlation coefficient based on mid-span deflections calculated by use of the ANSYS finite element program.
for the regression fit and ANSYS calculations are presented in Table 3 for all of the ceramics at each temperature. Shown in Fig. 5 is the normalized uniaxial stress relaxation function versus log10 ðtÞ for VB determined from BBV creep data at three temperatures. Also shown are uniaxial stress relaxation data for VB obtained in a previous study.4 Similar data were also available for VO, WCB, and WCO. Coefficients for a four-term uniaxial stress relaxation function based on inversion of BBV data (Eq. (23)) are presented in Table 4 for the four feldspathic porcelains at three temperatures. Also presented are comparisons of correlation coefficients calculated for nonlinear regression fits to the relaxation data4 and the inversion of BBV data.
Figure 5 Normalized uniaxial relaxation function versus log10 ðtÞ for VB at three temperatures. Curves indicated as BBV were determined by Laplace inversion of Eq. (21) after substitution of numerical values for the discrete model coefficients. Points indicated as data are average values (^SD) for three samples tested under uniaxial conditions.4
Discussion There are two indications that the procedures used to invert creep data can yield shear relaxation functions that are reasonable. The first indicator is the fact that the finite element simulation of a creep test in the BBV using the relaxation functions for all dental ceramics results in mid-span deflections that fit the experimental data quite well for most cases. The differences in the correlation coefficients between the regression fit and the ANSYS calculation varies from the best agreement (0.0) for VO at 575 8C to the worst agreement (0.0297) for EXV at 550 8C. The second procedure for validating of the inversion procedure is to compare the calculated uniaxial stress relaxation functions for the feldspathic porcelains with uniaxial stress relaxation data for the same porcelains reported by DeHoff et al.4 The best agreement between a regression fit and the inverted curve was obtained for VB at 575 8C as indicated by a difference of only 0.0010 between the calculated correlation coefficient (Eq. (27)) for the regression fit to the data (0.9988) compared with that of the inverted curve (0.9978). The worst agreement was obtained for VO at 550 8C for which a difference of 0.3498 was calculated. It is our view that the normalized shear relaxation functions based on inversion of the creep data are more reliable than those based on the uniaxial stress relaxation tests. In a relaxation test it is
724
P.H. DeHoff, K.J. Anusavice
Table 4 Coefficients at three temperatures for a four-term uniaxial stress relaxation function, cðtÞ;, for the feldspathic porcelains (Eq. (23)). Ceramic
VB
VO
WCB
WCO
a b c
Tempa (8C)
550 575 600 550 575 600 550 575 600 550 575 600
c1 (MPa)
10574 27169 53709 14716 43455 60769 11117 14152 35302 4514 14152 35302
c2 (MPa)
21108 27811 5336 19265 15518 4861 143 6503 12443 9483 6503 12443
c3 (MPa)
16292 1503 2780 11907 2070 413 9381 16140 7018 1728 16140 7018
c4 (MPa)
15525 6716 1074 21513 5858 858 36759 20305 2136 41575 20305 2136
l1 (s)
7.84 5.25 2.49 5.62 5.23 2.11 11.13 9.03 3.39 3.00 9.03 3.39
l2 (s)
41.2 27.4 15.1 46.4 32.1 31.9 13.7 18.9 18.5 43.3 18.9 18.5
l3 (s)
156 113 55 131 109 185 53 65 95 62 65 95
l4 (s)
922 339 236 789 392 863 937 478 592 937 478 592
Correlation coefficients Fit R2b
BBV R2c
Difference
0.9996 0.9988 0.9995 0.9981 0.9999 0.9994 0.9912 0.9996 0.9995 0.9994 0.9997 0.9999
0.9755 0.9978 0.9483 0.6483 0.9196 0.8468 0.9396 0.8883 0.9921 0.9675 0.9977 0.8436
0.0241 0.0010 0.0512 0.3498 0.0803 0.1526 0.0516 0.1113 0.0074 0.0319 0.0020 0.1563
Temperatures indicated are those recorded by the furnace thermocouple in the BBV. Correlation coefficient based on a nonlinear regression fit to the uniaxial stress relaxation data of Ref. 4. Correlation coefficient based on inversion of the BBV creep data of Ref. 7.
necessary to impose a step input in strain and then monitor the load as a function of time while the strain is held constant. While measurement of load through a load cell is not a problem, measuring and maintaining constant strain is difficult to achieve at elevated temperatures. In a creep test in the BBV, a constant load is suddenly applied while the midspan deflection is measured as a function of time through a linear variable differential transformer (LVDT). This process is easier to control and more reliable data should be possible. A major goal of this research was to provide reliable material properties that can be used as input for the viscoelastic element in the ANSYS finite element program. The inclusion of viscoelasticity in the ANSYS program greatly increases the power of finite element analyses to determine stress states in dental prostheses. Finite element analyses that are based strictly on elastic behavior cannot properly account for residual stresses which could play an important role in the service life of dental prostheses. Another important area that is greatly enhanced by the inclusion of viscoelasticity is the definition of thermal compatibility of metal – ceramic and ceramic – ceramic dental systems. In spite of the fact that most dental researchers agree that thermal compatibility of dental systems is a complex issue depending on many factors, it is still not unusual to find statements in the dental literature suggesting the existence of an allowable mismatch in the average coefficients of thermal expansion/contraction between materials that will result in compatible systems. Such
statements are generally based on elastic equations developed by Timoshenko11 for compound strips, cylinders, and spheres. Analyses based on elastic behavior cannot properly account for fabrication variables such as cooling rates. Thus, erroneous conclusions concerning acceptable levels of thermal expansion/contraction mismatch are likely. In addition to the expansion/contraction behavior, a proper analysis of thermal compatibility must include additional variables such as geometry, cooling rates, and number of firing cycles. We believe that viscoelastic finite element analyses with shear relaxation functions based on creep behavior measured in a BBV can help identify and quantify the factors defining compatibility of dental prostheses.
Acknowledgements This study was supported by NIH-NIDCR grant DE06672 and the Mechanical Engineering Department at UNC Charlotte.
References 1. Bertolotti RL, Shelby JE. Viscosity of dental porcelain as a function of temperature. J Dent Res 1979;58:2001—4. 2. Twiggs SW, Hashinger DT, Fairhurst CW. Viscosities of porcelains formulated from the Weinstein patent. J Am Ceram Soc 1990;73(2):446—9.
Shear stress relaxation of dental ceramics determined from creep behavior
3. Asaoka K, Kon M, Kuwayama N. Viscosity of dental porcelains in glass transition. Dent Mater J 1990;9:193—202. 4. DeHoff PH, Vontivillu SB, Wang Z, Anusavice KJ. Stress relaxation behavior of dental porcelains at high temperatures. Dent Mater 1994;10:178—84. 5. DeHoff PH, Anusavice KJ, Hojjatie B. Thermal incompatibility analysis of metal—ceramic systems based on flexural displacement data. J Biomed Mater Res 1998;41:614—23. 6. DeHoff PH, Anusavice KJ. Viscoelastic stress analysis of thermally compatible and incompatible metal—ceramic systems. Dent Mater 1998;14:237—45.
725
7. DeHoff PH, Anusavice KJ. Creep functions of dental ceramics measured in a beam-bending viscometer. Dent Mater 2004; 20:313—20. 8. Christensen RM. Theory of viscoelasticity. New York: Academic Press; 1982. pp. 3—9. 9. Churchill RV. Operational mathematics. New York: McGrawHill; 1958. p. 3, see also p. 324. 10. Scherer GW. Relaxation in glass and composites. New York: Wiley; 1986. p. 22 see also p. 84. 11. Timoshenko S. Analysis of bimetal thermostats. J Opt Soc Am 1925;11:233—55.
www.intl.elsevierhealth.com/journals/dema
Shear stress relaxation of dental ceramics determined from creep behavior Paul H. DeHoffa,*, Kenneth J. Anusaviceb a
Department of Mechanical Engineering and Engineering Science, University of North Carolina at Charlotte, Charlotte, NC 28223, USA b Department of Dental Biomaterials, University of Florida, Gainesville, FL, USA Received 2 August 2002; received in revised form 3 September 2003; accepted 9 October 2003
KEYWORDS Relaxation functions; Dental ceramics; Relaxation times; Viscoelasticity; ANSYS; Laplace transforms; Discrete model; Maxwell model
Summary Objective. To test the hypothesis that shear stress relaxation functions of dental ceramics can be determined from creep functions measured in a beam-bending viscometer. Methods. Stress relaxation behavior was determined from creep data for the following materials: (1) a veneering ceramic-IPS Empress2w body ceramic (E2V); (2) an experimental veneering ceramic (EXV); (3) a low expansion body porcelain-Vita VMK 68 feldspathic body porcelain (VB); (4) a high expansion body porcelain-Will Ceram feldspathic body porcelain (WCB); (5) a medium expansion opaque porcelain-Vita feldspathic opaque porcelain (VO); and (6) a high expansion opaque porcelain-Will Ceram feldspathic opaque porcelain (WCO). Laplace transform techniques were used to relate shear stress relaxation functions to creep functions for an eight-parameter, discrete viscoelastic model. Nonlinear regression analysis was performed to fit a fourterm exponential relaxation function for each material at each temperature. The relaxation functions were utilized in the ANSYS finite element program to simulate creep behavior in three-point bending for each material at each temperature. Results. Shear stress relaxation times at 575 8C ranged from 0.03 s for EXV to 195 s for WCO. Significance. Knowledge of the shear relaxation functions for dental ceramics at high temperatures is required input for the viscoelastic element in the ANSYS finite element program, which can used to determine transient and residual stresses in dental prostheses during fabrication. Q 2004 Academy of Dental Materials. Published by Elsevier Ltd. All rights reserved.
Introduction Residual tensile stresses in ceramic caused by thermal contraction mismatch between materials are an important contributing factor to failures of metal – ceramic and all-ceramic dental prostheses. It is generally understood that the residual stress state *Corresponding author. Tel.: þ1-704-687-4324; fax: þ1-704687-6069. E-mail address: [email protected]
in metal – ceramic and ceramic – ceramic dental prostheses depends on many factors, including contraction mismatch, cooling rate, firing temperature, geometry, and fabrication technique. Recently, commercial finite element programs such as ANSYS (ANSYS, Inc., Canonsburg, PA) have included a viscoelastic option that now makes it possible to include time-dependent properties of dental ceramics at high temperature for the calculation of residual stresses in dental prostheses.
0109-5641/$ - see front matter Q 2004 Academy of Dental Materials. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.dental.2003.10.005
718
P.H. DeHoff, K.J. Anusavice
However, the ANSYS program requires a shear stress relaxation function as input and we are unaware of any experimental studies that report the measurement of time-dependent shear stress properties of dental ceramics. Previous studies related to time-dependent property measurements of dental ceramics were primarily concerned with the measurement of shear viscosities1 – 3 or uniaxial stress relaxation functions.4 – 6 Recently, DeHoff and Anusavice7 used a beam-bending viscometer (BBV) to measure creep behavior from which creep functions were calculated for two glass–ceramics and four feldspathic porcelains at temperatures ranging from 450 to 650 8C. While creep functions cannot be used directly in the ANSYS program, it is possible to determine the required shear relaxation functions from the creep data through the use of Laplace transform methods and linear viscoelastic theory. Therefore, the objective of this study was to characterize the high temperature viscoelastic properties of several dental ceramics by the determination of relaxation functions based on creep functions measured in a BBV. The validity of the procedures was confirmed by using the relaxation properties in the ANSYS finite element program to simulate a BBV-type creep test for each of the dental ceramics.
Materials and methods Viscoelastic theory. The three-dimensional stress – strain relationships for a linear isotropic viscoelastic material are given by:8 ðt
›1ij ðtÞ sij ðtÞ ¼ cðt 2 tÞ dt ›t 0 Sij ðtÞ ¼ 2
QðtÞ ¼ 3
ðt
ðt
Kðt 2 tÞ
0
1ij ¼
ðt
›eij ðtÞ dt ›t
ð2Þ
›uðtÞ dt ›t
ð3Þ
Gðt 2 tÞ
0
Jðt 2 tÞ
0
ð1Þ
›sij dt ›t
ð4Þ
where sij and 1ij are the stress and strain tensors, respectively (i and j take on the values 1, 2, and 3 corresponding to the coordinate axes (x; y; and z)); t is any arbitrary time between zero and the present time, t, and is called the past time; Sij is the deviatoric stress tensor and is determined from the stress tensor by: Sij ðtÞ ¼ sij ðtÞ 2
QðtÞ d 3 ij
ð5Þ
where Q is the first stress invariant ðQ ¼ s11 þ s22 þ s33 Þ and dij is the Kronecker delta (dij ¼ 1 if i ¼ j and 0 if i – j). Likewise, eij ðtÞ ¼ 1ij ðtÞ 2
uðtÞ d 3 ij
ð6Þ
eij ðtÞ is the deviatoric strain tensor and u is the first strain invariant. cðtÞ is the uniaxial stress relaxation function, GðtÞ is the shear stress relaxation function, KðtÞ is the hydrostatic (or bulk) stress relaxation function, and JðtÞ is the creep function. Laplace transform methods are used to develop interrelationships among the material functions in transform space. The Laplace transform of a function FðtÞ is defined by the integral:9 ð1 fðpÞ ¼ e2pt FðtÞdt ð7Þ 0
The Laplace transforms of Eqs. (1) –(4), which are in convolution form, are given by:
sij ðpÞ ¼ pcðpÞ1ij ðpÞ
ð8Þ
e ij ðpÞ S ij ðpÞ ¼ 2pGðpÞ
ð9Þ
uðpÞ Q ðpÞ ¼ 3pKðpÞ sij ðpÞ 1ij ðpÞ ¼ pJðpÞ
ð10Þ ð11Þ
In a uniaxial test, s11 ¼ s11 ; and all other stress components are zero. Therefore, S 11 ¼ 2=3ðs11 Þ and e 11 ¼ 111 2 1=3ðuÞ from which it is possible to obtain the following relationships from Eqs. (8) to (11): 1 p2 JðpÞ cðpÞ 3KðpÞ GðpÞ ¼ 9KðpÞ 2 cðpÞ
cðpÞ ¼
ð12Þ ð13Þ
where p represents the transform variable and the bar above the variable indicates that the variables are in the transform space.9 Eqs. (12) and (13) are then used to determine the transformed shear stress relaxation function from the transformed uniaxial stress, creep and bulk relaxation functions, followed by Laplace transform inversion techniques to determine the shear stress relaxation function in the time domain. The process to determine the shear stress relaxation coefficients from experimental BBV creep data is described in the following section.
Shear stress relaxation functions An eight-parameter discrete spring-dashpot viscoelastic model (Fig. 1) was proposed by DeHoff and Anusavice7 to characterize the creep behavior measured in a BBV for several commercial dental ceramics. The creep function in the time domain
Shear stress relaxation of dental ceramics determined from creep behavior
b1 ¼ h0 h1 h2 h3 ðE0 2 9K0 Þ
719
ð19eÞ
b2 ¼ E0 h0 ðE1 h2 h3 þ E2 h1 h3 þ E3 h1 h2 Þ 2 9K0 E0 ðh0 h1 h2 þ h0 h1 h3 þ h0 h2 h3 þ h1 h2 h3 Þ 2 9K0 h0 ðE1 h2 h3 þ E2 h1 h3 Figure 1 Eight-parameter discrete viscoelastic model.
1 1 1 1 t þ þ þ þ E0 E1 E2 E3 h0 E1 t E2 t exp 2 exp 2 h1 h2 2 2 E1 E2 E t exp 2 3 h3 2 E3
2 9K0 E0 {h0 ½E1 ðh2 þ h3 Þ þ E2 ðh1 þ h3 Þ þ E3 ðh1 þ h2 Þ þ E1 h2 h3 þ E2 h1 h3 þ E3 h1 h2 } 2 9K0 h0 ðE1 E2 h3 þ E1 E3 h2 þ E2 E3 h1 Þ ð14Þ
Using standard tables, the Laplace transform of JðtÞ is given by:
2
1 1 1 1 1=h þ þ þ þ 20 E0 p E1 p E2 p E3 p p 1=E1 1=E2 1=E3 2 2 ðp þ E1 =h1 Þ ðp þ E2 =h2 Þ ðp þ E3 =h3 Þ
ð15Þ
Scherer10 indicates that dilatational relaxation has a minor effect on stress calculations for most stress states and therefore the dilatational behavior of the dental ceramics has been assumed to be purely elastic so that: KðtÞ ¼ K0
ð16Þ
where K0 is the bulk modulus at room temperature. The Laplace transform of the bulk modulus is then: K KðpÞ ¼ 0 p
ð17Þ
From Eqs. (12),(13),(15) and (17) it can be shown that the shear stress relaxation function in the transform space can be put in the form: GðpÞ ¼
a1 p3 þ a2 p2 þ a3 p þ a4 ðb1 p4 þ b2 p3 þ b3 p2 þ b4 p þ b5 Þ
ð19gÞ
b4 ¼ E0 E1 E2 E3 h0 2 9K0 {E0 ½E1 E3 ðh0 þ h2 Þ
9
JðpÞ
ð19fÞ
b3 ¼ E0 h0 ðE1 E2 h3 þ E1 E3 h2 þ E2 E3 h1 Þ
for this model is given by: JðtÞ ¼
þ E3 h1 h2 Þ
ð18Þ
where ai and bi are given in terms of the model coefficients and K0 by: a1 ¼ 23K0 E0 h0 h1 h2 h3
ð19aÞ
a2 ¼ 23K0 E0 h0 ðE1 h2 h3 þ E2 h1 h3 þ E3 h1 h2 Þ
ð19bÞ
a3 ¼ 23K0 E0 h0 ðE1 E2 h3 þ E1 E3 h2 þ E2 E3 h1 Þ
ð19cÞ
a4 ¼ 23K0 E0 E1 E2 E3 h0
ð19dÞ
þ E1 E2 ðh0 þ h3 Þ þ E2 E3 ðh0 þ h1 Þ þ E1 E2 E3 h0 } b5 ¼ 29K0 E0 E1 E2 E3
ð19hÞ ð19iÞ
Although Laplace inversion of Eq. (18) requires numerical values for the discrete model parameters, it is known that Laplace inversion of a function with a fourth degree polynomial in the denominator (four roots) will lead to a shear stress relaxation function in the time domain in the general form: t t GðtÞ ¼ G1 exp 2 þ G2 exp 2 t t 1 2 t t þ G3 exp 2 þ G4 exp 2 ð20Þ t3 t4 Numerical values for the coefficients ai and bi were first determined by substitution of numerical values for the discrete model parameters7 into Eqs. (19a)–(19i). Numerical values of Gi and ti were then determined by substituting the numerical values of ai and bi into Eq. (18), followed by Laplace inversion of the resulting equation. All Laplace transform manipulations were accomplished by use of the Maplew symbolic mathematics program (Waterloo Maple Inc., Waterloo, Ont., Canada). Values for the input coefficients ai and bi are given in Table 1 for each of the dental ceramics.
Finite element creep In order to test the validity of the transform procedures for calculating shear stress relaxation
720
Table 1 Polynomial coefficients for the shear stress relaxation function in the Laplace transform space, GðpÞ; (Eq. (18)). Tempa (8C)
a1 (N6 s4/mm12)
a2 (N6 s3/mm12)
a3 (N6 s2/mm12)
a4 (N6 s/mm12)
b1 (N5 s4/mm10)
b2 (N5 s3/mm10)
b3 (N5 s2/mm10)
b4 (N5 s/mm10)
b5 (N5/mm10)
E2V
525 550 575 600 475 500 525 550 575 550 575 600 625 650 550 575 600 625 650 550 575 600 625 650 575 600 625 650 675
7.208 £ 1035 1.021 £ 1034 2.381 £ 1031 3.634 £ 1029 1.445 £ 1037 5.862 £ 1033 4.776 £ 1030 2.701 £ 1028 7.116 £ 1025 7.392 £ 1037 6.275 £ 1035 2.684 £ 1033 9.226 £ 1030 2.168 £ 1029 6.590 £ 1037 3.329 £ 1034 2.451 £ 1033 7.458 £ 1030 6.780 £ 1028 1.239 £ 1038 1.051 £ 1037 2.948 £ 1034 3.257 £ 1032 2.750 £ 1030 3.745 £ 1033 3.730 £ 1035 4.450 £ 1033 3.199 £ 1031 5.999 £ 1029
1.128 £ 1035 1.260 £ 1033 3.652 £ 1030 4.756 £ 1028 5.867 £ 1035 5.250 £ 1032 7.253 £ 1029 2.723 £ 1027 3.141 £ 1025 9.464 £ 1036 8.802 £ 1034 3.774 £ 1032 2.293 £ 1030 4.923 £ 1028 1.066 £ 1037 3.399 £ 1034 1.936 £ 1032 1.072 £ 1030 1.326 £ 1028 2.002 £ 1037 1.498 £ 1036 4.865 £ 1033 1.880 £ 1031 1.057 £ 1029 9.806 £ 1036 8.005 £ 1034 8.764 £ 1032 3.388 £ 1030 6.555 £ 1028
1.796 £ 1033 2.549 £ 1031 8.487 £ 1028 1.491 £ 1027 4.185 £ 1033 1.481 £ 1031 2.264 £ 1028 6.135 £ 1025 2.578 £ 1024 1.614 £ 1035 1.463 £ 1033 1.060 £ 1031 1.158 £ 1029 3.038 £ 1027 1.935 £ 1035 5.593 £ 1032 1.811 £ 1030 2.580 £ 1028 2.180 £ 1026 9.394 £ 1035 5.459 £ 1034 1.177 £ 1032 1.972 £ 1029 3.524 £ 1026 6.079 £ 1035 2.707 £ 1033 2.192 £ 1031 5.297 £ 1028 5.830 £ 1026
7.398 £ 1030 7.757 £ 1028 1.700 £ 1026 6.993 £ 1023 8.122 £ 1030 1.338 £ 1029 1.129 £ 1026 2.038 £ 1023 8.245 £ 1021 4.468 £ 1032 6.047 £ 1030 5.393 £ 1028 5.321 £ 1026 2.731 £ 1025 7.098 £ 1032 2.254 £ 1030 3.743 £ 1027 7.780 £ 1025 5.399 £ 1023 1.010 £ 1034 3.836 £ 1032 3.358 £ 1029 3.178 £ 1026 2.433 £ 1023 8.633 £ 1033 1.710 £ 1031 1.185 £ 1029 1.157 £ 1026 1.323 £ 1024
2.970 £ 1031 4.229 £ 1029 9.910 £ 1026 1.518 £ 1025 5.571 £ 1032 2.271 £ 1029 1.856 £ 1026 1.051 £ 1024 2.783 £ 1021 2.864 £ 1033 2.442 £ 1031 1.050 £ 1029 3.620 £ 1026 8.546 £ 1024 2.405 £ 1033 1.225 £ 1031 9.013 £ 1028 2.754 £ 1026 2.516 £ 1024 5.310 £ 1033 4.529 £ 1032 1.274 £ 1030 1.416 £ 1028 1.202 £ 1026 1.614 £ 1033 1.612 £ 1031 1.934 £ 1029 1.398 £ 1027 2.631 £ 1025
8.443 £ 1030 1.891 £ 1029 1.584 £ 1027 8.550 £ 1025 5.667 £ 1031 9.699 £ 1028 4.183 £ 1026 8.518 £ 1024 1.065 £ 1023 4.756 £ 1032 6.358 £ 1030 5.949 £ 1028 7.330 £ 1026 4.025 £ 1025 5.262 £ 1032 3.219 £ 1030 5.469 £ 1028 6.119 £ 1026 1.740 £ 1025 9.979 £ 1032 8.588 £ 1031 5.149 £ 1029 9.525 £ 1027 2.405 £ 1026 4.785 £ 1032 5.381 £ 1030 9.820 £ 1028 1.876 £ 1027 9.269 £ 1025
3.235 £ 1029 1.247 £ 1028 1.701 £ 1026 9.609 £ 1024 8.780 £ 1029 5.904 £ 1027 4.234 £ 1025 7.003 £ 1023 3.571 £ 1022 1.344 £ 1031 2.750 £ 1029 4.652 £ 1027 1.122 £ 1026 7.158 £ 1024 1.496 £ 1031 1.236 £ 1029 1.982 £ 1027 4.982 £ 1025 1.995 £ 1024 5.527 £ 1031 4.319 £ 1030 3.035 £ 1028 3.385 £ 1026 5.950 £ 1024 3.582 £ 1031 3.374 £ 1029 8.459 £ 1027 9.787 £ 1025 5.126 £ 1024
2.913 £ 1027 1.740 £ 1026 3.157 £ 1024 2.615 £ 1023 3.630 £ 1027 1.136 £ 1026 8.876 £ 1023 1.205 £ 1022 2.030 £ 1021 8.102 £ 1028 2.570 £ 1027 7.996 £ 1025 3.748 £ 1024 3.339 £ 1023 1.004 £ 1029 1.102 £ 1027 1.102 £ 1025 6.470 £ 1023 1.764 £ 1022 7.987 £ 1029 5.606 £ 1028 3.101 £ 1026 2.132 £ 1024 1.308 £ 1022 7.165 £ 1029 4.890 £ 1027 1.233 £ 1026 9.203 £ 1023 2.978 £ 1022
6.433 £ 1024 4.246 £ 1023 5.200 £ 1021 1.134 £ 1020 3.837 £ 1024 6.082 £ 1023 3.079 £ 1021 3.057 £ 1019 5.584 £ 1018 7.530 £ 1025 5.416 £ 1024 2.601 £ 1023 1.149 £ 1022 2.168 £ 1021 1.092 £ 1026 2.081 £ 1024 1.021 £ 1022 8.709 £ 1020 2.281 £ 1019 1.143 £ 1027 1.046 £ 1026 4.419 £ 1023 2.142 £ 1021 5.406 £ 1018 1.378 £ 1027 9.225 £ 1024 3.064 £ 1023 8.128 £ 1020 3.931 £ 1019
EXV
VB
VO
WCB
WCO
a
Temperatures indicated are those recorded by the furnace thermocouple in the BBV.
P.H. DeHoff, K.J. Anusavice
Ceramic
Shear stress relaxation of dental ceramics determined from creep behavior
721
representing the BBV loading was applied to the model at the mid-span of the beam. Nodes at each end were constrained in the vertical direction to simulate a simply supported beam. The fourterm shear relaxation function for each ceramic at each temperature (Table 2) was used for the creep calculations.
Figure 2 Finite element model of the three-point bending specimen used in the beam-bending viscometer.
functions from creep data, the viscoelastic element in ANSYS was used to simulate a BBV-type creep test of each of the dental ceramics at each temperature. Shown in Fig. 2 is the finite element model of the BBV three-point bending specimen (5.5 mm £ 2.5 mm £ 48 mm). The model consisted of 192 20node isoparametric three-dimensional viscoelastic elements and 1285 nodes. Because of symmetry along the length, only one half of the beam was modeled with imposition of appropriate symmetry displacement boundary conditions. A line load
Uniaxial stress relaxation A second procedure for checking the validity of the inversion procedure is a comparison of calculated uniaxial stress relaxation functions for the feldspathic porcelains with measured uniaxial stress relaxation data for the same porcelains reported by DeHoff et al.4 The uniaxial stress relaxation function in the transform space is determined by substituting Eq. (15) into Eq. (12) to obtain: E E E E1 E2 E3 p þ 1 p þ 2 p þ 3 h1 h2 h3 cðpÞ ¼ D
ð21Þ
Table 2 Coefficients at the high temperature range for the four-term shear stress relaxation function, GðtÞ; (Eq. (20)). Ceramic
Tempa (8C)
G1 (MPa)
G2 (MPa)
G3 (MPa)
G4 (MPa)
t1 (s)
t2 (s)
t3 (s)
t4 (s)
ts b (s)
E2V
525 550 575 600 475 500 525 550 575 550 575 600 625 650 550 575 600 625 650 550 575 600 625 650 575 600 625 650 675
11516 20640 23355 23870 18079 24251 25158 25633 25499 5254 12789 22550 24270 25116 7100 19133 25157 26301 26634 5421 7061 15679 22244 22729 2151 8748 16605 21942 22451
10011 2684 551 33 5343 261 317 25 38 9273 10166 1746 671 2.3 8443 5400 1615 457 262 48 2703 4380 447 135 2998 7751 5141 555 312
187 720 107 40 197 27 205 27 32 6018 502 917 492 218 4414 691 137 233 36 4120 6375 2363 241 10 5110 3324 466 306 2
2555 102 11 0.2 2315 1274 52 5 0.6 6268 2234 356 55 29.1 7441 1954 286 82 20 13744 7072 708 76 12 12952 3307 796 83 40
4.16 2.70 0.67 0.18 11.90 2.80 0.47 0.13 0.03 7.56 4.79 2.10 0.54 0.22 5.36 4.58 1.76 0.47 0.15 10.68 8.55 2.98 1.57 0.51 5.16 3.89 2.48 0.78 0.29
31.5 17.1 11.0 14.4 83.0 27.0 12.8 16.6 3.8 38.5 25.2 14.9 8.5 9.2 43.7 30.8 31.5 14.6 9.4 13.7 18.5 17.7 33.9 44.2 13.8 17.5 11.5 23.8 20.1
118 67 47 22 242 44 44 51 14.6 148 112 55 23.5 14.9 126 108 185 70 126 51 61 93 154 543 35 57 69 92 225
299 323 548 2269 610 494 231 327 345 881 332 236 294 130 744 386 861 658 638 623 448 588 806 1833 466 452 319 1016 512
15.70 3.38 0.76 0.19 21.40 3.10 0.48 0.13 0.03 79.50 13.10 2.53 0.58 0.09 85.00 12.00 2.00 0.49 0.15 293.00 62.30 5.59 1.66 0.51 195.00 17.70 3.82 0.83 0.30
EXV
VB
VO
WCB
WCO
a b
Temperatures indicated are those recorded by the furnace thermocouple in the BBV. ts is the overall relaxation time defined as the time required for the shear stress relaxation function to decay to 1=e of its initial value.
722
P.H. DeHoff, K.J. Anusavice
where (
! 1 1 1 1 1 þ þ þ þ D ¼p E1 E2 E3 E0 p E1 p E2 p E3 p h0 p2 E1 E2 E3 E2 pþ pþ pþ 2 E2 E3 p þ h1 h2 h3 h2 E E E p þ 3 2 E1 E3 p þ 1 p þ 3 h3 h1 h3 E1 E2 2E1 E2 p þ pþ ð22Þ h1 h2 2
Substitution of the appropriate discrete model constants7 into Eqs. (21) and (22), followed by Laplace inversion, leads to the uniaxial relaxation function in the form:
Figure 3 Calculated shear stress relaxation function versus log10 ðtÞ for E2V at four temperatures.
cðtÞ ¼ c1 expð2t=l1 Þ þ c2 expð2t=l2 Þ þ c3 £ expð2t=l3 Þ þ c4 expð2t=l4 Þ
ð23Þ
Sr ¼
N X
ðyi 2 yðxi ÞÞ2
ð25Þ
i¼1
Numerical values for ci and li were calculated for each of the feldspathic porcelains at three temperatures.
The coefficients for the four-term shear relaxation functions for all groups at each applicable furnace temperature are presented in Table 2. Also presented is the overall relaxation time, i.e. the time at which the normalized relaxation function reaches 1=e of its initial value. In general, the glass – ceramics relax more rapidly (smaller relaxation times) than the feldspathic porcelains. The shear stress relaxation behavior of E2V as a function of log10 ðtÞ at four temperatures is illustrated in Fig. 3. The shift of the relaxation curves to the left as the temperature increases is an indication that the rate of relaxation increases with increasing temperature. Relaxation times of 15.7 and 0.19 s were calculated for E2V at 525 and 600 8C, respectively (Table 2). Shown in Fig. 4 are mid-span deflection versus time data for three E2V specimens at a temperature of 550 8C. Also shown are mid-span deflections (solid curve) calculated by means of the ANSYS finite element program. The coefficient of correlation was calculated to determine the goodness of fit of the calculated curve to the experimental data using the following equations: N X i¼1
ðy~ 2 yi Þ2
N 1 X y N i¼1 i
ð26Þ
Sr St
ð27Þ
R2 ¼ 1 2
Results
St ¼
y~ ¼
ð24Þ
where N is the number of data points, xi and yi represent the x and y values of a typical datum point, yðxi Þ is the y value calculated from the regression equation, and R2 is the correlation coefficient. We used Eqs. (24) –(27) to calculate a correlation coefficient of 0.9985 for the regression fit of the experimental data and a correlation coefficient of 0.9976 for the curve calculated with ANSYS. A comparison of the correlation coefficients
Figure 4 Mid-span deflection versus time for three specimens of E2V at 550 8C. The solid curve represents deflection calculated using ANSYS software.
Shear stress relaxation of dental ceramics determined from creep behavior
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Table 3 Correlation coefficients for a nonlinear regression fit to the BBV mid-span deflection data compared with those calculated for the ANSYS results. Ceramic
E2V
EXV
VB
VO
WCB
WCO
a
b
c
Tempa (8C)
525 550 575 600 475 500 525 550 575 550 575 600 625 650 550 575 600 625 650 550 575 600 625 650 575 600 625 650 675
Correlation coefficient Regression fitb
ANSYSc
Difference
0.9938 0.9985 0.9976 0.9978 0.9935 0.9957 0.9943 0.9929 0.9668 0.9456 0.9895 0.9982 0.9966 0.9977 0.9227 0.9640 0.9968 0.9660 0.9995 0.9853 0.9952 0.9919 0.9742 0.9975 0.9874 0.9766 0.9927 0.9587 0.9772
0.9778 0.9976 0.9961 0.9975 0.9891 0.9884 0.9911 0.9632 0.9537 0.9396 0.9842 0.9959 0.9938 0.9939 0.9178 0.9640 0.9958 0.9658 0.9971 0.9792 0.9932 0.9863 0.9621 0.9815 0.9846 0.9742 0.9905 0.9498 0.9574
0.0160 0.0009 0.0015 0.0003 0.0044 0.0073 0.0032 0.0297 0.0131 0.0060 0.0053 0.0023 0.0028 0.0038 0.0049 0.0000 0.0010 0.0002 0.0024 0.0061 0.0020 0.0056 0.0121 0.0160 0.0028 0.0024 0.0022 0.0089 0.0198
Temperatures indicated are those recorded by the furnace thermocouple in the BBV. Correlation coefficient based on a nonlinear regression fit to the BBV mid-span deflection data of Ref. 7. Correlation coefficient based on mid-span deflections calculated by use of the ANSYS finite element program.
for the regression fit and ANSYS calculations are presented in Table 3 for all of the ceramics at each temperature. Shown in Fig. 5 is the normalized uniaxial stress relaxation function versus log10 ðtÞ for VB determined from BBV creep data at three temperatures. Also shown are uniaxial stress relaxation data for VB obtained in a previous study.4 Similar data were also available for VO, WCB, and WCO. Coefficients for a four-term uniaxial stress relaxation function based on inversion of BBV data (Eq. (23)) are presented in Table 4 for the four feldspathic porcelains at three temperatures. Also presented are comparisons of correlation coefficients calculated for nonlinear regression fits to the relaxation data4 and the inversion of BBV data.
Figure 5 Normalized uniaxial relaxation function versus log10 ðtÞ for VB at three temperatures. Curves indicated as BBV were determined by Laplace inversion of Eq. (21) after substitution of numerical values for the discrete model coefficients. Points indicated as data are average values (^SD) for three samples tested under uniaxial conditions.4
Discussion There are two indications that the procedures used to invert creep data can yield shear relaxation functions that are reasonable. The first indicator is the fact that the finite element simulation of a creep test in the BBV using the relaxation functions for all dental ceramics results in mid-span deflections that fit the experimental data quite well for most cases. The differences in the correlation coefficients between the regression fit and the ANSYS calculation varies from the best agreement (0.0) for VO at 575 8C to the worst agreement (0.0297) for EXV at 550 8C. The second procedure for validating of the inversion procedure is to compare the calculated uniaxial stress relaxation functions for the feldspathic porcelains with uniaxial stress relaxation data for the same porcelains reported by DeHoff et al.4 The best agreement between a regression fit and the inverted curve was obtained for VB at 575 8C as indicated by a difference of only 0.0010 between the calculated correlation coefficient (Eq. (27)) for the regression fit to the data (0.9988) compared with that of the inverted curve (0.9978). The worst agreement was obtained for VO at 550 8C for which a difference of 0.3498 was calculated. It is our view that the normalized shear relaxation functions based on inversion of the creep data are more reliable than those based on the uniaxial stress relaxation tests. In a relaxation test it is
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P.H. DeHoff, K.J. Anusavice
Table 4 Coefficients at three temperatures for a four-term uniaxial stress relaxation function, cðtÞ;, for the feldspathic porcelains (Eq. (23)). Ceramic
VB
VO
WCB
WCO
a b c
Tempa (8C)
550 575 600 550 575 600 550 575 600 550 575 600
c1 (MPa)
10574 27169 53709 14716 43455 60769 11117 14152 35302 4514 14152 35302
c2 (MPa)
21108 27811 5336 19265 15518 4861 143 6503 12443 9483 6503 12443
c3 (MPa)
16292 1503 2780 11907 2070 413 9381 16140 7018 1728 16140 7018
c4 (MPa)
15525 6716 1074 21513 5858 858 36759 20305 2136 41575 20305 2136
l1 (s)
7.84 5.25 2.49 5.62 5.23 2.11 11.13 9.03 3.39 3.00 9.03 3.39
l2 (s)
41.2 27.4 15.1 46.4 32.1 31.9 13.7 18.9 18.5 43.3 18.9 18.5
l3 (s)
156 113 55 131 109 185 53 65 95 62 65 95
l4 (s)
922 339 236 789 392 863 937 478 592 937 478 592
Correlation coefficients Fit R2b
BBV R2c
Difference
0.9996 0.9988 0.9995 0.9981 0.9999 0.9994 0.9912 0.9996 0.9995 0.9994 0.9997 0.9999
0.9755 0.9978 0.9483 0.6483 0.9196 0.8468 0.9396 0.8883 0.9921 0.9675 0.9977 0.8436
0.0241 0.0010 0.0512 0.3498 0.0803 0.1526 0.0516 0.1113 0.0074 0.0319 0.0020 0.1563
Temperatures indicated are those recorded by the furnace thermocouple in the BBV. Correlation coefficient based on a nonlinear regression fit to the uniaxial stress relaxation data of Ref. 4. Correlation coefficient based on inversion of the BBV creep data of Ref. 7.
necessary to impose a step input in strain and then monitor the load as a function of time while the strain is held constant. While measurement of load through a load cell is not a problem, measuring and maintaining constant strain is difficult to achieve at elevated temperatures. In a creep test in the BBV, a constant load is suddenly applied while the midspan deflection is measured as a function of time through a linear variable differential transformer (LVDT). This process is easier to control and more reliable data should be possible. A major goal of this research was to provide reliable material properties that can be used as input for the viscoelastic element in the ANSYS finite element program. The inclusion of viscoelasticity in the ANSYS program greatly increases the power of finite element analyses to determine stress states in dental prostheses. Finite element analyses that are based strictly on elastic behavior cannot properly account for residual stresses which could play an important role in the service life of dental prostheses. Another important area that is greatly enhanced by the inclusion of viscoelasticity is the definition of thermal compatibility of metal – ceramic and ceramic – ceramic dental systems. In spite of the fact that most dental researchers agree that thermal compatibility of dental systems is a complex issue depending on many factors, it is still not unusual to find statements in the dental literature suggesting the existence of an allowable mismatch in the average coefficients of thermal expansion/contraction between materials that will result in compatible systems. Such
statements are generally based on elastic equations developed by Timoshenko11 for compound strips, cylinders, and spheres. Analyses based on elastic behavior cannot properly account for fabrication variables such as cooling rates. Thus, erroneous conclusions concerning acceptable levels of thermal expansion/contraction mismatch are likely. In addition to the expansion/contraction behavior, a proper analysis of thermal compatibility must include additional variables such as geometry, cooling rates, and number of firing cycles. We believe that viscoelastic finite element analyses with shear relaxation functions based on creep behavior measured in a BBV can help identify and quantify the factors defining compatibility of dental prostheses.
Acknowledgements This study was supported by NIH-NIDCR grant DE06672 and the Mechanical Engineering Department at UNC Charlotte.
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Shear stress relaxation of dental ceramics determined from creep behavior
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