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R-Curve Behavior of Dental Ceramic Materials

Department of Dental Prosthetics, Section of Dental Materials, University of Technology Aachen, Pauwelsstrasse 30, D-52074 Aachen, Germany;

Correspondence: ^* corresponding author, h.fischer{at}rwth-aachen.de

	ABSTRACT

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Some technical ceramics exhibit the R-curve effect, i.e., an increasing fracture resistance with crack extension which is a desirable material property because more energy is necessary to propagate a microscopic crack. The objective of this study was to prove whether dental ceramic materials exhibit R-curve behavior. Nine dental ceramics were examined by the indentation-strength method. It was found that all of the tested ceramic materials exhibt a rising R-curve with crack extension. The R-curve behavior was more pronounced for the high-strength materials In-Ceram Alumina, monolithic alumina, and especially Empress 2. We conclude from our results that the mechanical behavior of a dental ceramic material can be judged more comprehensively, if the R-curve of the respective material is known.

Key Words: ceramics • R-curves • indentation-strength method • fracture resistance • crack extension

	INTRODUCTION

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Ceramic materials are suitable for the dental field because of their outstanding esthetics and their excellent biocompatibility. Problematic aspects are their brittleness and thereby their sensitivity to tensile stresses. Tensile stresses can force crack extension. Three types of crack extension may occur in ceramic materials: subcritical crack extension, stable crack extension, and unstable crack extension (Munz and Fett, 1999). The subcritical crack extension can occur at stress intensities below a critical value referred to as ‘crack-tip toughness’, K_I0 (Rödel et al., 1990; Seidel and Rödel, 1997; Fett et al., 2000). The subcritical crack extension is the reason for the well-known time-dependent strength decrease of ceramic materials (Ritter, 1995). Reaching K_I0 by load increase, the crack propagates stably and finally unstably until the component fails. The point of failure is characterized by the critical stress intensity factor, better known as ‘fracture toughness’ value K_Ic.

Some technical ceramic materials show an increase of fracture resistance with crack extension under stable crack growth (Buresch and Papst, 1973; Steinbrech and Schmenkel, 1988; Fett and Munz, 1993). This phenomenon is called R-curve behavior. This increase of fracture resistance with crack extension may occur due to several effects. The most serious influence is friction at the border of the crack tip, which can cause so-called bridging effects. Further possible reasons for the increase of fracture resistance with crack extension are all energy-consuming effects, for example, crack branching. Another reason for R-curve behavior is phase transformation effects, which are characteristic of zirconia ceramics.

R-curve behavior of ceramic materials is a desirable mechanical effect. Due to the increase of fracture resistance with crack extension, successively increasing additional energy is necessary for a crack to propagate until failure of a ceramic component occurs (Green, 1998). The R-curve behavior is more dominant for larger than for smaller cracks, because the friction at the border of the crack tip increases with the increase in crack size. Besides the positive strength effect, the scatter-in-strength will be reduced by the R-curve behavior as well (Shetty and Wang, 1989). Finally, the increase in fracture resistance with crack extension can also positively influence the subcritical crack growth, i.e., the long-term strength behavior (Steinbrech et al., 1983; Okada and Hirosaki, 1990; Fett and Munz, 1992).

Different techniques have been developed to evaluate the R-curve characteristics of ceramic materials (Marshall and Lawn, 1979; Chantikul et al., 1981). A widely used technique, which was chosen for this study, is the indentation flexural strength (IF) method (Krause, 1988). Bar specimens are indented by the Vickers indentation method. The indented specimens are then flexural-strength-tested. While the bending strength is a function of the indentation load, the R-curve can be examined from the bending strength results of specimens which were indented with different loads.

The objective of this study was to prove whether dental ceramic materials exhibit R-curve behavior. The answer to this question can help us better understand the fracture mechanics of dental ceramic materials. The consequences of the results of this study for the dental field will be discussed.

	MATERIALS & METHODS

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Experimental Procedure
The investigations were performed on 9 ceramic materials: dense alumina (Al₂O₃, Frialit F99.7, Friatec, Mannheim, Germany); Cerec Mark II (Vita, Bad Säckingen, Germany); Duceram Opaker (Degussa, Rosbach, Germany); Empress 1 and Empress 2 (both Ivoclar, Schaan, Liechtenstein); and In-Ceram Alumina Celay, Vitadur Alpha Opaker, Vita Omega Opaker, and VMK 98 (all from Vita, Bad Säckingen, Germany).

Fifteen bar specimens of each material of the dimensions 1.5 mm x 3.0 mm x 30.0 mm were made. The Empress 1 and 2 specimens were made in investment molds formed by PMMA bars (lost forms). The specimens of Duceram Opaker, Vitadur Alpha Opaker, Vita Omega Opaker, and VMK 98 were made in slip casting as recommended by the manufacturers. The finished specimens of Al₂O₃ and Cerec Mark II and the green-body specimens of In-Ceram Alumina Celay were cut from ceramic monoblocks by a diamond-charged cutting-off machine (Isomet, Buehler, Lake Bluff, IL, USA). The cut green-body specimens of In-Ceram Alumina Celay were glass-infiltrated as recommended by the manufacturer.

The surfaces of the specimens were ground and polished (R_max < 2 µm). All specimens were annealed to minimize residual stresses which were possibly induced into the surfaces of the specimens by the manufacturing, grinding, and polishing processes. The individual annealing temperatures are listed in the second column of the Table. The annealed specimens were indented by the Vickers indentation method. The indenter was centered on the prospective tensile surface of the flexural strength specimens. The ranges of the indentation loads which were used for the respective ceramic materials are given in the third column of the Table. The indented specimens were strength-tested according to DIN EN 843-1 (1995) in a four-point bending test arrangement (roller spans, 12 mm and 24 mm; roller diameters, 4 mm) with the use of a universal testing machine (Z030, Zwick, Ulm, Germany).

(1)

which defines the critical stress intensity factor, i.e., the fracture toughness K_c (Fig. 3A). To describe the fracture resistance as a function of crack extension, one can use a fractional power law in the form (Krause, 1988)

View larger version (18K): [in this window] [in a new window]   Figure 3. Stress intensity factor K (MPam0.5) as a function of crack extension a (mm) for Vita Omega Opaker. The R-curve (bold line, KR) is calculated according to Eq. 2. The stress intensity curve due to the bending stress Kb, the stress intensity curve due to the indentation stress Kr, and the superposed [Kb+Kr] curve in the left diagram (A) are plotted for an indentation load of P = 70.6 N. The point of tangency (Kc) denotes the onset of crack-extension instability, which defines the fracture toughness value. The 6 [Kb+Kr] curves in the right diagram (B) result from the observed data at different indentation loads (6.9-70.6 N). The dashed segments of the curves denote extrapolations beyond the upper limit of Eq. 2.

(2)

where K_R is the fracture resistance, a is the crack extension normal to the surface, and k and q are constants. Note that K_R will be invariant with crack extension, i.e., the material will exhibit no R-curve behavior, if q is zero. After indentation, semicircular radial cracks will develop under the surface of the material. The resulting residual stress intensity factor can be written as

(3)

where P is the indenter load, a_I is the half-length of surface crack, and is a constant. When an indented test material is subjected to a bending stress, an additional ‘bending stress intensity factor’ will be superposed upon K_r. This bending stress intensity factor K_b can be written as

(4)

where is the bending stress, a is the crack length, and Y is a configuration coefficient (Y = 1.174; Krause, 1988). This leads to the equation

(5)

Failure occurs when the criterion of Eq. 1 is satisfied. Therefore, Eq. 5 can be differentiated with respect to crack length. Additional mathematical substitutions (Krause, 1988) finally give us the formula

(6)

where _F is the bending strength of indented specimens, P is the indenter load, and and β are material-specific parameters. The mathematical correlation between the parameter β and the exponent q (Eq. 2) is given by

(7)

Note that q is zero (no R-curve effect) for β equal to 1/3. The constant k (Eq. 2) and the parameter can be correlated as

(8)

with

(9)

It can be shown that log() is invariant over the range of indenter load P. Therefore, a mean value of calculated from measurements of indentation crack length according to Eq. 9 can be used to evaluate the constant k.

Plotting the bending stress over the indentation load in a double-logarithmic diagram and creating a regression curve through the data points result in a least-squares fit of the logarithm of Eq. 6. The slope of this fit line is –β, and the intercept with the abscissa is . The parameters q and k can be calculated from Eqs. 7 and 8.

We analyzed the random errors in the strength measurements to judge the statistical variability of the observed data. The uncertainty for each indentation strength parameter value is given by its standard deviation. The standard deviations of the parameters β and log() were calculated following the statistical analysis of a straight line (Eq. 6). The standard deviations of the parameters k and q were calculated following the law of propagation of errors (Mandel, 1984; Krause, 1988).

	RESULTS

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The evaluated indentation strength parameters log(), β, q, and k of the observed ceramic materials and its standard deviations are listed in the Table. All tested materials exhibit β-values smaller than 1/3. The values for the exponent q used for the R-curve formula (Eq. 2) cover a range between 0.1308 and 0.2592; the values for the parameter k are in the range between 3.58 and 21.05. Fig. 1 shows the observed bending strength as a function of indentation load. The solid lines are least-squares fits of the logarithm of Eq. 6. The dashed line in the Fig. has a slope of -1/3 (β = 1/3). This slope represents a material without R-curve behavior. The fracture resistance K_R as a function of crack extension a, i.e., the R-curves, is plotted in Fig. 2. The R-curves of the observed ceramic materials appear as straight lines in the double-logarithmic presentation. The R-curve steepness is displayed as the straight-line slope, and the R-curve level can be judged by its value of fracture resistance, K_R, at a specific value of crack extension, a, respectively.

View larger version (17K): [in this window] [in a new window]   Figure 1. Observed bending strength (MPa) as a function of indentation load P (N). The solid lines are least-squares fits of the logarithm of Eq. 6. The dashed line has a slope of -1/3 (β = 1/3). This slope represents a material without R-curve behavior.

View larger version (20K): [in this window] [in a new window]   Figure 2. Fracture resistance KR (MPam0.5) as a function of crack extension a (mm). The R-curves of the observed ceramic materials appear as straight lines in the double-logarithmic presentation. The R-curve steepness is displayed as the straight-line slope, and the R-curve level can be judged by its value of fracture resistance KR at a specific value of crack extension a, respectively. The dashed segments of the R-curves denote extrapolations beyond the upper limit of Eq. 2.

	DISCUSSION

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The present study revealed that the well-established dental ceramic materials which were examined exhibit R-curve characteristics without exception. This conclusion can be made based on the interpretation of the indentation strength parameters (Table) and from the graphs in Figs. 1 and 2. All ceramic materials have values for the indentation strength parameter β smaller than 1/3. For β = 1/3, the parameter q is zero (Eq. 7). For q = 0, the fractional power is a constant, i.e., the fracture resistance K_R is invariant with crack extension (Eq. 2). This means that no R-curve effect occurs for β = 1/3 and for q = 0.

The analysis of the propagation of errors due to the random errors of the strength measurements indicates that the R-curve behavior is significantly different for the investigated dental ceramic materials. The high strength-level (core) ceramics Al₂O₃, Empress 2, and In-Ceram Alumina Celay exhibt a very high R-curve level compared with the other tested low-strength dental ceramics. This is visualized by the respective value of fracture resistance K_R at a specific value of crack extension a. Moreover, the steepness of the R-curve is very high for the material Empress 2. This can be taken from the slope of the straight-line of this material and from the indentation strength parameter value q (Table, q = 0.2483). The pronounced R-curve effect of Empress 2 may be explained by the microstructure of this material. Empress 2 consists of a glassy matrix in which about 70% of lithium-disilica crystals are embedded. The crystalline phase forms along this structures (length approx. 10 µm, diameter approx. 1 µm ) (Fischer et al., 2001). The interlocked crystals will probably cause the pronounced R-curve effect.

A very interesting finding is that the low-strength ceramics Cerec Mark II, Duceram Opaker, and VMK 98 exhibit very steep R-curves (q = 0.2498 for Cerec Mark II, q = 0.2286 for Duceram Opaker, and q = 0.2592 for VMK 98), although its respective R-curve level is low (k < 10) (Fig. 2, Table). The significance of the greater steepness of the R-curves of these 3 dental ceramic materials compared with the other 3 low-strength ceramics (Empress 1, Vitadur Alpha Opaker, and Vita Omega Opaker) was confirmed by Student’s t tests, which were performed at a confidence level of 95% based on the respective standard deviations of the parameter q (Table).

To confirm the course of the R-curve function given by the fractional power law (Eq. 2), we calculated the [K_r+K_b] functions at different indentation loads (Eqs. 3, 4, 5). The [K_r+K_b] function (Fig. 3A) must be tangential with the R-curve at the respective point of fracture toughness, K_Ic (see Eq. 1). The envelope of the respective [K_r+K_b] functions describes the course of the R-curve as well. The goodness of the fit of this envelope function to the R-curve calculated by the power law (Eq. 2; Fig. 2) is a measure of the reliability of the method which was used to evaluate the R-curves of the dental ceramic materials in this study. The plot in Fig. 3B shows the R-curve for Vita Omega Opaker calculated according to Eq. 2 (bold line). Furthermore, the [K_r+K_b] functions at different indentation loads (6.9-70.6 N) are plotted. The example confirms that the envelope of the [K_r+K_b] functions fits well to the R-curve.

The R-curves (Figs. 2, 3) calculated based on the empirical expression of Eq. 2 are plotted as dashed lines for greater values of crack extension (a > 500 µm). This reflects that Eq. 2 is limited. The lower limit is constrained by the inert strength of a material, namely, when a specimen fails from the most severe natural flaws instead of from an indentation crack. The upper limit is constrained by the specimen boundary. When a bending moment is superposed upon an indentation crack considered to be like a half-penny, it can be considered to become semi-elliptical (Krause, 1994). Considering that the indented surface crack must be shorter than the width of the specimen, the crack depth normal to the surface must be less than 500 µm for the specimens used.

It should be noted that the real R-curves may be slightly flatter than the calculated ones evaluated in this study (Fig. 2). Two principal effects may influence the results, which are not considered in Eqs. 2 and 5. These are possible environmentally assisted crack growth during bend loading and deviations from the ideal half-penny crack shape (Krause, 1994). It was shown on technical glasses that these influences can lead to an overestimation of the R-curve effect (Smith and Scattergood, 1992). Environmentally assisted crack growth may of course also occur on the materials tested in this study, but the effects will be less for glass ceramics and especially for monolithical ceramics like dense alumina compared with glasses, i.e., materials without crystalline phase.

It is an important finding of this study that the intensity of the R-curve behavior correlates to the strength level of the respective dental ceramic material. This is helpful for a better understanding of the mechanical behavior of this class of materials. It is known that the high-performance ceramic material Al₂O₃ not only exhibits a better short-term strength behavior but also has a better long-term strength prognosis as well (Munz and Fett, 1999). Furthermore, a rising fracture resistance with crack extension occurs not only during stable crack growth but also during subcritical crack growth (Steinbrech et al., 1983). This means that a pronounced R-curve effect may help to improve the short- as well as the long-term strength behavior of a dental ceramic material. This is important because most of the well-established dental ceramic materials exhibit a strong tendency to subcritical crack growth (Marx et al., 2000,2001). The R-curve behavior must therefore be understood as a basic material property and should be evaluated as a standard for every new dental ceramic material.

	ACKNOWLEDGMENTS

 
The authors are grateful to Ralph F. Krause Jr., National Institute of Standards and Technology, Gaithersburg, MD, and to Theo Fett, Forschungszentrum Karlsruhe, Germany, for helpful discussions on the R-curve topic. We thank the dental companies for providing the test materials. This work was funded by the LuFG Zahnärztliche Werkstoffkunde/RWTH Aachen, Germany (internal funds).

Received for publication November 28, 2001. Revision received May 9, 2002. Accepted for publication May 23, 2002.

	REFERENCES

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Journal of Dental Research, Vol. 81, No. 8, 547-551 (2002)
DOI: 10.1177/154405910208100809


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This Article Abstract Figures Only Full Text (PDF) Alert me when this article is cited Alert me if a correction is posted Services Email this article to a friend Similar articles in this journal Similar articles in PubMed Alert me to new issues of the journal Add to Saved Citations Download to citation manager Request Permissions Request Reprints Add to My Marked Citations Citing Articles Citing Articles via Google Scholar Citing Articles via Scopus Google Scholar Articles by Fischer, H. Articles by Marx, R. Search for Related Content PubMed PubMed Citation Articles by Fischer, H. Articles by Marx, R. Social Bookmarking                         What's this?