Characteristics of the Double-torsion test to Determine the R-curve of Ceramic Materials

Double-torsion tests were carried out on a commercial ceramic floor tile to verify whether this test is suitable for determining the R-curve of ceramics. The instantaneous crack length was obtained by means of compliance calibration, and it was found that the experimental compliance underestimates the real crack length. The load vs. displacement curves were also found to drop after maximum loading, causing the stress intensity factor to decline. The R-curves were calculated by two methods: linear elastic fracture mechanics and the energetic method. It was obtained that the average values of crack resistance, R, and the double of the work of fracture, 2 ⋅ γ wof , did not depend on notch length, a 0 , which is a highly relevant finding, indicating that these parameters were less dependent on the test specimen’s geometry. The proposal was to use small notches, which produce long stable crack propagation paths that in turn are particularly important in the case of coarse microstructures.


Introduction
The correct selection of a material for mechanical application requires that its intrinsic characteristics should be taken into account, as well as the theoretical conception that introduces the parameters describing its behavior and the relation between those parameters and the material's microstructural characteristics.
When dealing with ceramics, one must keep in mind that they are brittle, fail catastrophically and possess low toughness.The brittle nature of ceramics usually derives from the types of chemical bonds these materials present (an ionic-covalent combination), which also gives the material high tensile strength (because of the high bond strength) and low plastic strain (due to the lack of slip resulting from the high shear modulus and Burgers vector values) 1 .
The mechanical behavior of these materials can be described by the theory of linear elastic fracture mechanics (LEFM), which quantitatively describes the transformation of an intact structural component into a fractured one in response to crack growth.Fracture mechanics refers mainly to the beginning and propagation of one or several cracks subjected to a particular stress field.Crack propagation may be rapid and incontrollable, this means unstable propagation, or slow and stable one.
One way to make a detailed characterization of the crack propagation behavior of a given material is by determining its R-curve.An increasing crack resistance or toughness during crack extension, namely R-curve or K R -curve behavior, is a direct consequence of energy-dissipating toughening mechanisms which reduce the crack driving force at the crack tip 2 .To obtain this curve experimentally, the stable crack propagation condition is required.
Several geometries can be used to obtain the R-curve 3 , but no standard has so far been defined for this test applied to ceramic materials.and for the determination of fracture toughness, K IC .Some of the researchers who have employed the double-torsion test to characterize the R-curve are Vekinis et al. 20 and Ebrahimi et al. 22,26 .
The purpose of the present study was to apply the double-torsion test to obtain the R-curve through linear elastic fracture mechanics (LEFM) and the energetic method (EM), comparing them to each other.For the determination of R-curve it is necessary to obtain the instantaneous crack length, which in this work has been obtained by experimental compliance calibration of the test specimen.The double-torsion geometry offers the advantages of propagating the crack along the test specimen's longest length.The literature makes no mention of this technique applied to determine the R-curve by the energetic method.

Material
The material used in tests was a commercial ceramic floor tile whose chemical composition is given in Table 1.
The ceramic floor tiles (hereinafter referred to as "tiles") were ground to even their surfaces.Longitudinal notches with nominal length of 20, 70, 95, 120 and 180 mm and 5 mm width were made at the tiles' mid-span, using a diamond disc.The notch tip had a slop of 45°.This was made to impose a notch similar to the crack profile, which is not straight through the thickness, but extend further along the tensile side of the plate to form a curved crack front.In addition to the notch, the specimens had a 1 mm deep, 5 mm wide longitudinal groove on their lower face (surface under tensile stress), starting from the tip of the notch.The purpose of this groove was to keep the crack on an approximately straight propagation path from the notch tip.The nominal dimensions of the test specimens were: thickness -8 mm, t, width -107 mm, W, and length -240 mm, L, with the proportion t:W:L equal to 1:13:30.These proportions are in agreement with the work of Pletka et al. 19 and Evans et al. 11 .

Experimental Procedure
All the tests were conducted in an MTS, series 810/458, servohydraulic mechanical testing machine.For double-torsion tests it was used constant displacement rate condition with 1 µm/min.
Tests were also conducted to determine Young's modulus, E, using the three-point bending configuration.The calculations were made following the procedure proposed by Hübner and Schuhbauer 36 .
The physical properties of apparent porosity, apparent specific mass of the solid portion and apparent specific mass of the material were obtained by the Archimedes method 37 .
Steel devices were fabricated to fix test specimens for the doubletorsion test.Figure 1 illustrates the basic geometry of the experimental setup, showing the two points where lower loads are applied and the two points of upper loads.These last points are formed by the contact of the upper sphere with the two edges of the notch, thus providing four-point loading.
Figure 2 shows the propagation front of a crack, which propagated up to a certain point under double-torsion.This front, which displays a curvature, is not parallel to the applied load, but extends further along the lower face of the plate (tensile stress) than along the upper face (compressive stress).In Figure 2, the crack is represented by the dark-gray area, which is revealed with red dye.The dark strip along the entire lower portion of the specimen is the groove.The lighter region on the right is the part where the dye did not penetrate, indicating the intact portion of the sample that was fractured manually.The notch is shown at the extreme left of the figure.
The stress intensity factor for the double-torsion test is given by the following Equation 12: where P is the applied load, W m is the lever arm of the torsion moment, W is the width of the specimen, t is the thickness of the specimen, t n is the thickness minus the groove length, ν is the Poisson's coefficient and ε is a correction factor for the plate thickness, which is given by 16 :

Determination of the instantaneous crack length
To find the instantaneous crack length, a i , along its propagation path, the specimen's compliance was calibrated as a function of the initial crack length, a 0 (notch length).
In the double-torsion test, the compliance is linearly related with the crack length, according to the following Equation 19 : where C i is the compliance for the crack length a i , B is a constant that can be found by compliance calibration 18 or analytically, C 0 is the initial compliance for a sample with notch of the length a 0 and Da i = a i -a 0 is the increasing in the crack length.
In this work, B-value was determined experimentally by compliance calibration.So, load-displacement curves (P-d curve) were obtained for specimens with different notch length values as shown in Figure 3a.The slope of the curves in their elastic portion that is given by tg(β) (that is equal P/d) defines the specimen's stiffness.The compliance, C = 1/tg(β), is the reciprocal of stiffness.From line like that one shown in Figure 3b is obtained the B-value, which is the slope of the C-a curve, that means B = tg (φ).
The instantaneous crack length can be obtained then by the following equation: where C 0 was calculated for each test from the initial slope of P-d curve in its elastic portion, C i was obtained from lines radiating from the origin (Figure 4), where C i is the reciprocal of tg(ξ i ) = d i /P i .The analytical value for B is given by the following Equation 18: where G is the shear modulus.

Relation between the average crack propagation velocity, V, and K I
Using a simple experimental procedure also enabled us to evaluate the relation between the average crack propagation velocity, V, and the average stress intensity factor, K I , acting at the tip of the crack during its propagation.The experimental propagation velocity was calculated considering the time it took for the stable crack to propagate.After the natural failure of the specimen, which occurred catastrophically at a critical crack length, a c , the total stable propagation path, Da c , (it does not include the notch length, a o ) was divided by the time interval recorded by the test machine.The value of a c was estimated based on the compliance calibration curve.The theoretical velocity was calculated based on the following equation 38 : (6)   where P is the average load throughout the stable propagation, d .
is the displacement velocity of the actuator, and B was calculated based on Equation 5.

Linear Elastic Fracture Mechanics Method (LEFM)
The resistance to quasi-static crack propagation, R, was calculated by two methods: LEFM and EM.The LEFM method gives the value of R by the following equation 39 : where the following expression is given by introducing Equation 1: where P i is the instantaneous load obtained of P-d curve, under constant displacement rate and R i is the crack propagation resistance at point (P i ,d i ).

Energetic Method (EM)
The energetic method, EM 40 , determines the consumed energy for small increments in the length of the crack propagation based on the concept that the work made by the test machine to cause the fracture is the area under P-d curve.It is also supposed that all this work is converted in fracture energy under the condition of quasestatic propagation.
For a crack extension, Da i , from a i-1 to a i , the energy necessary for the formation of crack surface, U(Da i ), is given by the area between the lines drawn from the origin to the points (P i-1 ,a i-1 ) and (P i ,a i ).In Figure 5 a i represents a crack length corresponding to position d i of the actuator.In the same way, Da i corresponds to Dd i .
Using the EM method, R was calculated based on the following equation 40 : The denominator t n ⋅ Da i represents the corresponding increment in the area of propagation.

The criterion to check the R-values
The criterion to check if the R-values are reliable is to compare the average R value, R, with the double of the work of fracture, 2 ⋅ γ wof , according to the following equation [40][41][42] : R = 2 ⋅ γ wof (10)   To apply this criterion, the value of R was calculated for the energetic method, R(EM), and the LEFM method, R(LEFM), using the following relation: (11)   where a c is the last crack length still under stable propagation.The integral that appear in Equation 11was calculated numerically by the trapezoid method.To determine the work of fracture γ wof , the work performed by the test machine on the sample had to be calculated, according to Nakayama 43 , in relation to the entire stable crack propagation path.The area under the P-d curve up to the catastrophic propagation point was divided by the corresponding projected fracture surface area.The elastic energy stored in the system at the catastrophic propagation point was also subtracted from the work performed by the test machine.Figure 6 illustrates the integrated area of the P-d curve.
Based on Figure 6, the value of γ wof is given by the following equation: where P c and d c are, respectively, the actuator's force and displacement at the point where catastrophic propagation began, and Da c = a c -a 0 , with a c obtained from the compliance calibration curve.

Results and Discussion
The average value of Young's modulus of the material used in this work was 58 GPa ± 4 GPa.The values of apparent porosity, ap-parent specific mass, and apparent specific mass of the solid portion, were 12.1 ± 0.5% -vol, 3.15 ± 0.05 g/cm 3 and 3.56 ± 0.06 g/cm 3 , respectively.
Figure 7 depicts the compliance calibration curve.Its equation, obtained from the linear regression of the points in Figure 7, with units of "m/N" for C and "m" for a, are shown below: C = 4.16 x 10 -6 ⋅ a + 3.05 x 10 -7 (13)   compliance calibration curve found through the experimental method (Figure 7) provided a slope (B-value) of 4.16 x 10 -6 N -1 .The theoretical average value of B (Equation 5) calculated for all samples was 5.5 x 10 -6 N -1 .
The literature normally shows the P-d graph for the double-torsion test with the elastic region (load increase), the crack propagation portion (approximately constant load) and the rupture (sudden load drop) under constant displacement rate condition.In the double-torsion test, the crack propagates catastrophically starting from a given length, defined here as the critical crack length, a c .
Figure 8 shows the critical crack length, ac, vs. notch length, clearly showing that the notch length did not influence the critical crack length.The straight horizontal line in this figure indicates an average value of 17.7 cm for a c , which represents 71% of the total length of the specimen.Thus, the smaller the notch length, the longer the stable propagation of the crack, Da c .This is a favorable aspect of the double-torsion test in terms of the interest in determining the R-curve, since it requires long stable crack propagation lengths, which can be achieved with small notches.In this context, we highlight another interesting aspect of the double-torsion test in comparison with three-or four-point bending tests: for the latter tests, the shorter the notch, the higher the tendency for unstable crack propagation, thus hindering control of the propagation.The opposite holds true in double-torsion: long stable crack propagations are obtained even with small notches (Figure 8).Although the double-torsion test is, in theory, a test with a constant K I value, two different types of P-d curve were observed in this work, as indicated in Figure 9.In type 1, there was almost no stable crack propagation, i.e., the specimen failed shortly after maximum load point.This occurred with all the specimens whose notch length exceeded a c .
The type 2 curve showed a progressively decreasing load as the crack length increased along its stable propagation.Pletka et al. 19 also observed a decrease in P, indicating that K I is not effectively constant in the double-torsion test.Based on the V x K I graph, those authors confirmed that K I decreased in subsequent relaxation tests after sections of crack propagation in the same specimen using the double-torsion geometry.This behavior, however, was not observed with glass.
Figure 10 shows the average crack propagation velocity, V, as a function of notch length.The K I value used here was the arithmetic average of all the instantaneous K I -values after initiation of the crack propagation in each test.This figure indicates that the experimental crack propagation velocity showed a significant dispersion when compared with the theoretical value (Equation 6). Figure 10 clearly shows that, for small notch sizes, the congruence between experimen-tal and theoretical propagation velocities was quite good due to the more precise determination of the length of stable crack propagation (Da C = a C -a 0 ) in this condition.
Based on the greater length of crack propagation with shorter notch lengths, the lesser scattering in crack propagation velocities with shorter notch lengths, and the greater similarity between experimental and theoretical propagation velocities using shorter notch lengths which were found in this study, we recommend the use of smaller notch lengths in the double-torsion test.This represents a significant advantage for interest in R-curve studies, for one has a sample with a longer stable propagation path than that achieved with other, more commonly used geometries.This point is particularly important when ceramics with coarse microstructures are involved, as in the case of refractories.
Figure 11 gives three examples of R-curves for notch lengths of 0.02 m, 0.07 m and 0.095 m obtained by the LEFM and EM methods.The a-values of the R-curves were calculated according to Equation 4and the description in section 3.1.For the longest notch lengths, the number of points of the P-d curve after the maximum load was very few due to the short extension of Da c , so the R-curves for these samples were not considered representative and are not shown here.
The first point to note about the R-curves in Figure 11 is that they show no variation in shape when calculated by the LEFM and EM methods.The maximum R-value is given for the same crack length by both methods, but the R-values obtained with EM are smaller.
The R-curves found by the two methods showed declining values after reaching the highest peak (Figure 11).This decrease was associated with the shape of the P-d curves, which displayed the same characteristics.As mentioned earlier, when they applied the doubletorsion geometry to other materials, Pletka et al. 19 also observed a decrease in load and, hence, a drop in K I as the crack propagated.Using a bending test, Saadaoui et al. 44 also observed a drop in the R-curve for partially stabilized zirconia.The authors showed that the drop in that material occurred due to the low velocity at which the load was applied, which caused the crack propagation to occur predominantly by corrosion.This was confirmed when the velocity at which the load was applied was increased and the drop no longer occurred.In the present work, tests were conducted to check if this phenomenon occurred with the material under study here.Our results indicated that this was apparently not the case, since our material continued to show a declining load up to 20-fold higher at the actuator's displacement velocity, as depicted in Figure 12.
According to Sakai 45 , a material composed of large-size grains shows an increasing rather than a decreasing R-curve.Moreover, various ceramics with coarse microstructures show that the behavior of the R-curve is related to the bridging mechanism in the crack wake, involving interlocking and friction between grains 26 .Therefore, this material should show an upward R-curve.
Chevalier et al. 21propose an empirical correction factor as a function of the crack's relative length, a/a 0 , elevated to an exponent, x, which depends on the specimen's material and dimensions.Thus, K I is now dependent on the crack length.According to Chevalier et al. 21, double-torsion tests with constant loading and relaxation tests would be required to determine the value of x.However, these tests were not our objective.The aim was to study the double-torsion test to obtain the R-curve by the two methods EM and LEFM and make comparison one each another.Beside this, it was not possible to do the correction for the energetic method because the error has its origin in the experimental P-d curve and the correction in LEFM method introduced by Chevalier et al. 21, has been made on the stress intensity factor equation and not on the P-d curve.In this way would be impossible compare the value of R(EM) and R(LEFM).
Ciccotti 24 , has also proposed corrections for K I , however it was not possible to use them in this work because they are corrections for K I and not for P-d curve.Beside this, the corrections proposed by Ciccotti are valid only for specimens with length of 17 cm (6 and 10 cm of width) and of 25 cm (6 and 10 cm of width).These were not the case of the specimens of the present work.
It was found that, after reaching its peak, the P-d curve showed an increasing downward tendency as a 0 increased (Figure 11).This was probably because the length of stable propagation, Da C , decreased and the catastrophic fracture effect became more evident in the P-d curve, which would force a more intense correction factor, according to procedure of Chevalier et al. 21.Therefore, the most reliable R-values should be the ones calculated with low a 0 -values and high Da C -values.
Table 2 displays the values of R(EM), R(LEFM) and 2 ⋅ γ wof .From this table, it can be seen that, generally speaking, R(EM) presents a value more congruent with 2 ⋅ γ wof than R(LEFM).The value of R(LEFM) was 19% higher than 2 ⋅ γ wof , while R(EM) was 9% lower.It can also be observed that R(EM), R(LEFM) and 2 ⋅ γ wof did not change with a 0 .
The average R-value was also calculated considering only increasing values of R up to the highest point, R HP .The value of R HP (LEFM) was 57 ± 7 J/m 2 and R HP (EM) was 44 ± 5 J/m 2 .Therefore, the values of R HP were 7% higher than R 0 for LEFM and 9% higher for EM, revealing the occurrence of wake buildup mechanisms, i.e., an increasing R-curve.
The values of initial crack resistance, R 0 , were also obtained and were 53 ± 8 J/m 2 and 40 ± 6 J/m 2 , respectively, by the LEFM and EM methods.The R 0 value corresponds to the first point where the crack begins to propagate.
Figure 13 indicates that R 0 also did not change with a 0 , except with a notch length of 0.18 m, when the R 0 value was lower.However, no stable propagation occurred with this notch length, which was already greater than the critical length, a c , and should therefore not be considered.

Conclusions
• Double-torsion is a suitable method to determine the R-curve, for the crack propagates along the longest dimension of the sample; • The double-torsion method presents a crack length where catastrophic propagation occurs, herein called critical crack length, a c , under constant displacement rate condition.This length did not vary with notch length and occurred, on average, over 71% of the specimen's length.Because the critical crack length did not vary with notch length, the samples with smaller notches showed longer stable crack propagation paths, Da c ; • The analytical and experimental values of B were in agreement;   • The longer length of stable propagation with shorter notch lengths, the lesser scattering in crack propagation velocities with shorter notch lengths, and the greater similarity between experimental and theoretical propagation velocities using shorter notch lengths suggest the validity of using shallower notches in the double-torsion test.This is a significant advantage in R-curve studies, for one has a specimen with a longer stable propagation path than that achieved with other, more commonly used geometries.This point is particularly important in the case of ceramics with coarse microstructures; • P vs. d curves showed a decline, which was reflected in the R-curve.For the material used here, the low testing velocity did not cause this drop, which, albeit increased 20-fold, was unable to prevent this decline; • The R-curves showed the same shape when obtained through the LEFM and EM methods, however, although the LEFM method led to higher values; • The criterion adopted to evaluate the reliability of the R-values was the comparison of the average values of R, R, and 2 ⋅ γ wof .
The result indicated the relation R(EM) < 2 ⋅ γ wof < R(LEFM).The ceramic material used here presented an R(EM) more similar to 2 ⋅ γ wof ; and • The double-torsion method proved suitable for determining the R-curve of ceramic materials, easily reaching stable crack propagation.Therefore, this method deserves further studies aimed at reaching the R-curves of materials with coarse microstructures, as is the case of refractory castables, particularly in order to understand the drop in the P vs. d curve and, hence, the R-curve after it reaches its highest point.

Figure 2 .Figure 3 .Figure 4 .
Figure 2. Crack profile of a specimen tested under double-torsion.A black line highlights the crack border.

Table 1 .
Chemical composition of the ceramic floor tile.
W t Figure 1.Double-torsion test arrangement showing devices fabricated to carry out the test and some important dimensions.