The effect of groove and notch tip angles on testing fracture toughness by SEVNB method: models and experimental validation

ABSTRACT

The single-edged V-notch beam (SEVNB) method is considered as an effective method for evaluating the fracture toughness values of brittle materials. In this method, it is assumed that the V-notch is a natural crack. However, this assumption may cause an overestimation of the fracture toughness due to the “notch passivation effect”. To investigate the effects of the V-notch and groove tip angles on the fracture toughness testing of ceramic materials, three typical models were established in this work. The stress intensity factors of these models were calculated using a J-integral based on the linear finite element method (LFEM). The results indicated that the measured fracture toughness values could be overestimated by 0.5%- 13.7% when the angle of the V-notch tip increased from 10° to 60°. Increasing the angle formed by the V-notch and groove from 10° to 60°, fracture toughness was overevaluated by about 0% – 2.0%. When the angle formed by the V-notch and groove increased to 120°, the fracture toughness was overevaluated by about 31%. Finally, two equations were fitted to assess the angles effects on fracture toughness, and the results have been validated by experiments. An important reference for the SEVNB method can be found in this work.

KEYWORDS:

• Fracture toughness
• ceramic materials
• J-integral
• LFEM

1. Introduction

As an index of the mechanical properties of brittle ceramic materials, fracture toughness reflects the maximum resistance during crack growth, and it plays a vital role in the damage and destruction of ceramics [1–3]. Accurate testing of the fracture toughness has been continuously studied by many researchers in the past few decades. Single-edged V-notch beam (SEVNB) method shows great application potential in this field [4–6]. The SEVNB method was developed from the single-edged notch beam (SENB) method to improve the overestimation of the SENB method due to the “notch passivation effect” [5]. In this method, a sharp V-notch at the root of the groove is prefabricated to eliminate the effect. Nishida et al [7] studied the effect of notch root radius on the fracture toughness of Al₂O₃ ceramic with fine grain size. It is concluded that fracture toughness can be measured when the radius of the notch tip is less than 10 μm. Unfortunately, the traditional SEVNB method still overestimates the fracture toughness values of some fine-grained ceramic materials due to notch passivation effect [6–11]. One modified method is to use an ultra-short pulsed laser to ablate a sharper V-notch at the groove root. According to the previous reports [12–14], the tip radius of the V-notch ablated by a laser is less than 0.5 μm, which is less than size of the microstructure feature (average grain size) of most ceramics. However, the V-notch produced by a laser is still not a natural crack, the notch tip angle may have a significant effect on the fracture toughness test, for larger angle will shield the induction and propagation of the crack. In addition, the formula used in the SEVNB method by assuming the notch is a crack may be inappropriate. Thus, the applicability of the SEVNB method should be determined.

Fett [15] first studied the SEVNB method with a model of a groove whose root was inserted in a real crack, and he recommended the following function:

(1) $\frac{K_{exp}}{K_{th}}=tanh\left(2Y\sqrt{\frac{l}{\rho }}\right)$(1)

where K_exp is the experimental stress intensity factor of the limited crack length (l) compared to the root radius of the groove root (ρ), K_th is the true stress intensity factor with a crack that is sufficiently long compared to the root radius of the groove root, and Y is the appropriate geometric correction factor. Y = 1.12 for a through-thickness edge crack. Fett [10] further studied the relationship of real crack length and the groove root radius. It showed that if the groove root radius ρ > ρ_c (groove critical radius), spontaneous crack extension is possible. However, the ideal crack inserted at the root of groove may be not the same as a V-notch. Recently, Wang et al. [16] studied the effect of the angle of the V-notch on fracture toughness testing with the linear finite element method (LFEM). They concluded that the fracture toughness would be overestimated when the V-notch tip angle was higher than 60°. However, they did not provide the accurate relationship between the angle and the stress intensity factor. Wang et al. [17] also studied the angle of the V-notch during fracture toughness testing by assuming the V-notch was a natural crack. In addition, the angle of the notch tip is difficult to be processed and measured accurately in three-dimensional samples. Thus, the results may be not accurate enough for fracture toughness testing. In order to analyze the impact of the angle on the fracture toughness test in the SEVNB method in more detail, the width of the groove, the tip angle and length of the V-notch should be analyzed separately to better guide the fracture toughness test.

In this work, we aimed to study the relationship between the angle of the groove tip with a V-notch and the fracture toughness by the J-integral based on the LFEM. Based on this relationship, it is anticipated that a more accurate fracture toughness can be obtained when using the SEVNB method.

2. Experimental procedure

Zirconia powder with 2.3 mol% Y₂O₃ (2.3Y-TZP, Fan Meiya commercial materials Co. Ltd, Jiangxi, China), zirconia powder with 3 mol% Y₂O₃ (3Y-TZP Fan Meiya commercial materials Co. Ltd, Jiangxi, China), and Al₂O₃ (99.5% pure, Showa Denko, Japan) powder were used as the starting materials. Granulated raw materials were pressed into disks at 20 MPa in a steel die, and cold-pressed isostatically at 250 MPa for 3 min to obtain the green bars. The green bars of the 2.3Y-TZP and 3Y-TZP were sintered in air at 1450°C for 2 h in a MoSi₂ resistance furnace (LHT 08/17, Nabertherm, Germany) at a heating rate of 5°C /min. The green bars of the Al₂O₃ were sintered in air at 1300°C for 2 h with a heating rate of 5°C /min. The densities of the sintered disks were measured in distilled water according to Archimedes’ principle, and the theoretical densities were assumed to be 6.08 g/cm³ for the TZP ceramic and 3.98 g/cm³ for the Al₂O₃ ceramic. All the measured relative densities of the investigated samples were larger than 98%. The sintered samples were ground into test bars with dimensions of 3 × 4 × 35 mm. A groove was machined by a diamond wheel with thickness of 200 μm at the center of 3 × 35 mm surfaces, and sharp V-notches with different lengths were fabricated at the bottom of the groove by a femtosecond laser (Chameleon ultra; Coherent, USA) with a power and scanning speed of 70 mW and 100 μm/s, respectively. Scanning electron microscopy (SEM; ZEISSEVO 18, Oberkochen, Germany) was used to measure the V-notch length (l) on the fractured surface, while optical microscopy (SZX 10, Olympus, Japan) was used to measure the depth of the groove (u) and the groove shape. All the test bars were tested in a mechanical test machine (Model 5567, Instron, USA) using a three-point flexure test configuration with a span of 30 mm and a constant loading rate of 0.05 mm/min. The fracture toughness values were calculated according to the ASTM C-1421 standard [18], which assume the notch as a natural crack:

(2) $K_{IC}=Y\left[\frac{F\times s}{B\times h^{\frac{3}{2}}}\right]\left[\frac{3\times {\left(\frac{a}{h}\right)}^{\frac{1}{2}}}{2\times {\left(1-\frac{a}{h}\right)}^{\frac{3}{2}}}\right]$(2)

where a is the notch length which is equal to the V-notch length (v_n) plus the depth of the groove (u); F is the fracture load; S is the span width; B is the sample width (3 mm); h is the sample height (4 mm); and Y is the dimensionless correction factor which can be calculated by the following formula:

$Y=\frac{1.99-\left(\frac{a}{h}\right)\times \left(1-\frac{a}{h}\right)\times \left[2.15-3.93\times \left(\frac{a}{h}\right)+2.7\times {\left(\frac{a}{h}\right)}^2\right]}{\left[1+2\times \left(\frac{a}{h}\right)\right]\times {\left(1-\frac{a}{h}\right)}^{\frac{3}{2}}}$

The microstructures of the sintered samples were observed by scanning electron microscopy (SEM) on the polished surfaces. The phase compositions of the samples were analyzed by X-ray diffraction (XRD, X’pert PRO, Panalytical, Netherlands). The monoclinic phase volume fractions (V_m) of the 2.3Y-TZP and 3Y-TZP were calculated using the formula proposed by Toraya et al. [19].

3. Linear finite element modeling (LFEM)

3.1. Geometric models

Figure 1 shows the SEM image of test bar with a groove and a V-notch for the SEVNB method test [12]. The tip angle of the V-notch and the width of the groove have significant effects on the stress concentration at the tip of the V-notch. When the tip angle of the V-notch is sharper and the groove is narrower, the stress at the V-notch tip becomes more concentrated. Therefore, the tip angle of the V-notch, θ₁, was used to study the effect of the V-notch morphology on the stress intensity factor testing, and θ₂ was used to study the effect of the groove morphology and V-notch length on the stress intensity factor testing. The interaction of the laser and the material around the V-notch tip was studied by previous researchers [20–22]. The results showed that although a thin molten region with a width of about 2 μm was produced on the V-notch surface by a laser, the effect of the molten region on the fracture toughness testing could be ignored.

Figure 1. SEM image of the groove and V-notch tilted by 30° [12]. Two angles, θ₁, and θ₂, were used to control the morphology of the groove and V-notch, respectively.

In order to analyze the influences of θ₁ and θ₂ on the accuracy of the fracture toughness testing by the SEVNB method in the three-point flexure test, three two-dimensional (2D) models were constructed. Figure 2a shows a schematic of the model with an ideal crack, that was used to calculate the theoretical stress intensity factor (K _I/th) of an ideal crack tip. This model is referred to as the ideal crack model. Figure 2b shows the schematic diagram of the model of a groove with a V-notch root, which was marked as a non-crack model. The groove length and V-notch length are denoted by u and v_n, respectively. This model was used to calculate the stress intensity factors (K_I/θ1) corresponding to different θ₁ values. The stress intensity factor of this model is only related to θ₁ and does not depend on θ₂, and it can be used to guide the processing of the V-notch. Figure 2c shows the schematic of the model of a groove with a semi-circle root, and a crack was implanted in the middle of the semi-circle root. This model is marked as groove model. The groove and crack lengths are denoted as u and l. This model is used to calculate the stress intensity factors (K_I/θ2) corresponding to different θ₂ values. The stress intensity factor of this model is only related to θ₂ and does not depend on θ₁, and it can be used to guide the processing of the groove and V-notch length.

Figure 2. Schematic diagrams of the three models; (a) ideal crack model, (b) non-crack model, (c) groove model. u is the groove length, l is the crack length, V_n is the V-notch length, a is the overall length of the flaw, defined as a = u + l + V_n. In the ideal crack model (a), u = 0, V_n = 0, and a = l; In the non-crack model (b), l = 0, a = u + V_n; In the groove model (c), V_n = 0, a = u + l.

3.2. Stress intensity factor

In this work, the stress intensity factor was calculated using the J-integral. The J-integral is an independent path line integral along a freely selected counterclockwise contour, Г, surrounding the crack tip. The J-integral is defined as follows:

(3) $J={\oint }_{\mathrm{\Gamma }}{\hspace{0.17em}}^{\hspace{0.17em}}Wdy-T_i\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }x}ds={\int }_{\mathrm{\Gamma }}{\hspace{0.17em}}^{\hspace{0.17em}}\left(W{n}_x-T_i\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }x}\right)ds$(3)

where W is the strain energy density, which can be evaluated as:

$W=\frac{1}{2}\left({\sigma }_x\cdot {\epsilon }_x+{\sigma }_y\cdot {\epsilon }_y+{\sigma }_{xy}\cdot 2\cdot {\epsilon }_{xy}\right)$

T is the traction vector, defined as

$T=\left[\begin{array}{c}{\sigma }_x\cdot n_x+{\sigma }_{xy}\cdot n_y\\ {\sigma }_{xy}\cdot n_x+{\sigma }_y\cdot n_y\end{array}\right]$

where σ_ij represents the stress components, ε_ij represents the strain components, and n_i represents the normal vector components. For a plane strain case and linear elastic materials, the stress intensity factor can be calculated based on the J-integral, as follows:

(4) $K_I=\sqrt{\frac{J\cdot E}{1-{\nu }^2}}$(4)

where E is Young’s modulus, and ν is the Poisson’s ratio.

3.3. Convergence study

The J-integral values of the models were determined by 2D finite element (FE) analysis which was performed by using ABAQUS (Dassault Systemes, USA). The ideal crack model is shown in Figure 3a and its crack tip is enlarged in Figure 3b. A circumference C with the crack tip as the center and radius R is proposed to facilitate the subsequent refinement of the grid near the crack tip. The typical meshed ideal crack model is shown in Figure 3c. The refined mesh within the circumference C is shown in Figure 3d. The type of the mesh within the circumference C was a four-node bilinear plane-strain quadrilateral with reduced integration, while that outside the circumference C was four-node bilinear plane strain quadrilateral without reduced integration. As Figure 3d shows, the number of nodes (red point) on the radius R was controlled to adjust the length L between the crack tip and the first point while the number of nodes (cyan point) on the circumference C was controlled to determine the average degree (η) of the circumference C. Global elements were used to limit the element size of the entire model. The convergence study results for the mesh refinement are is shown in Table 1. As shown by item B to H, the J-integral values decreased with the mesh refinement within the circumference C. The details are shown in Figure 4. The J-integral decreased with the decrease in the angle η, and the J-integral did significantly decrease until item E. Although the angle of item E was reduced by half, the J integral was only reduced by 0.05%. Further reducing the angle by half, the J integral only increased by 0.002% (item H). Thus, the error of the J-integral calculated by the ideal crack model with the element parameters of item F was less than 0.1%. In order to study the effect of the global element size on the J-integral calculation, the smaller global element sizes (items I and J) were used to determine the J-integral values, which were less than that of item F by 0.014% and 0.026%, respectively. Therefore, in this work, the number of nodes on the circle C, the number of nodes on the radius R, and the global element size were 100, 50, and 0.05 mm, respectively. The corresponding length L and angle η were 1.603 μm and 3.616°, respectively. The uncertainties in the calculated J-integral values were assumed to be less than 0.1%. Convergence studies of the non-crack model and the groove model were also performed, but the details are not shown here. Results show that the uncertainties in the calculated J-integral value are less than 0.1%. The number of nodes on the circumference C, the number of nodes on the radius R, and global size were also 100, 50, and 0.05 mm, respectively, for the non-crack model and the groove model. The corresponding length L an angle η were 1.603 μm and 3.616°, respectively. The model material property parameters were those reported by Quinn et al. [23], and they are listed in Table 2. Figure 5 shows the typical stress cloud (S11, along the length of the beam) images of the ideal crack model.

Figure 3. Grid-independent test of ideal crack model, (a) model diagram and node setting method and (b) global mesh generation map and enlarged mesh distribution map of the crack tip.

Figure 4. Variation of J-integral for an ideal crack model with varying angle (η).

Figure 5. Global stress cloud image of ideal crack model and its stress cloud image enlarged at the crack tip.

Table 1. Grid-independent test of ideal crack model.

Item	Number of nodes on circumference C	Number of nodes on radius R	Length of L (μm)	Degree η (°)	Global element size (mm)	Number of global elements	Value of J-integral ×10^−3
A	10	5	15.296	36	0.10	19,906	0.4671870
B	10	5	15.296	36	0.05	78,173	0.4672230
C	20	10	4.894	18	0.05	78,707	0.4562888
D	30	20	2.504	12	0.05	80,313	0.4521192
E	50	30	1.628	7.231	0.05	83,747	0.4500180
F	100	50	1.603	3.616	0.05	92,223	0.4489621
G	200	80	1.003	1.852	0.05	112,833	0.4487524
H	400	200	0.402	0.816–1.088	0.05	198,819	0.4487597
I	100	50	1.603	3.616	0.04	136,051	0.4488960
J	100	50	1.603	3.616	0.03	242,055	0.4488455

Table 2. Reference values of model material parameters [23].

Mechanical property	parameters
Young’s modulus, E (GPa)	210
Poisson’s ratio, ν	0.31
Density (g/cm^3)	6.05

As shown in Figure 6, the fitted LFEM results of the ideal crack model with different crack lengths were almost consistent with the ASTM C-1421 standard based on a natural crack (Equation (2)(2) $K_{IC}=Y\left[\frac{F\times s}{B\times h^{\frac{3}{2}}}\right]\left[\frac{3\times {\left(\frac{a}{h}\right)}^{\frac{1}{2}}}{2\times {\left(1-\frac{a}{h}\right)}^{\frac{3}{2}}}\right]$(2) ).

Figure 6. Comparison of LFEM results with Equation (2)(2) $K_{IC}=Y\left[\frac{F\times s}{B\times h^{\frac{3}{2}}}\right]\left[\frac{3\times {\left(\frac{a}{h}\right)}^{\frac{1}{2}}}{2\times {\left(1-\frac{a}{h}\right)}^{\frac{3}{2}}}\right]$(2) based on the ideal crack model for three-point bending configuration.

4. Results and discussion

4.1. Effect of V-notch tip angle (θ₁) on fracture toughness measurement

Figure 7 shows the stress distribution along the length of the beam (S11) in the crack tip of the ideal crack model and the V-notch tip of the non-crack model. As expected, the tensile stress distribution at the crack tip was highly concentrated in the ideal crack model, and its stress contour resembled a dumbbell. Near the crack surface, an equilateral triangle area with a stress close to zero was found, which is consistent with the result reported by Wang et al. [16]. However, when the crack tip was enlarged, the two sides of the triangle near the crack tip became cambered, and closer to the crack tip, the angle (θ₁) was smaller. This observation was not found in the report of Wang et al. [16]. Figure 7b shows the non-crack model with a V-notch tip angle θ₁ of 90°.The stress contour still resembled a dumbbell, although a part of the volume was removed compared to the ideal crack model. In addition, at the V-notch tip, no compressive stress area was found. Compared with the fracture load of sample with real crack, the large-angle V-notch will lead to the loss of part of the region with high elastic strain energy, so it will be unfavorable for crack induction and early propagation, and the fracture toughness of the material will be overestimated.

Figure 7. Typical tensile stress distribution of (a) ideal crack model and (b) non-crack model with a V-notch tip angle θ₁ of 90°in a two-dimensional three -point bending beam by LFEM.

The non-crack model can be used to assess the effect of the angle of the V-notch tip θ₁ on the fracture toughness testing by the SEVNB method. Before the assessment, the effect of the u/a value on the J-integral of the V-notch tip must be evaluated. Figure 8 shows the J-integral of the non-crack model with different u/a values and a constant V-notch tip angle of 60°. The stress cloud images (S11, along the beam length) of the non-crack model with u/a = 0, 0.5, 0.95 are shown in Figure 8a–c, respectively. The stress distributions at the V-notch tip were similar. Moreover, the J-integral value almost remained constant as the u/a value increased. Angles of the V-notch tip θ₁ of 30°, 90°, and 120° were also studied, and the conclusions were the same as those with 60°. Thus, they are not shown in Figure 8. Therefore, the effect of the u/a value on the J-integral of the V-notch tip in the non-crack model can be ignored. When u was almost equal to a, the width of the groove became very thin similar to the results shown in Figure 8b. Thus, the J-integral of the non-crack model only depended on the angle of the V-notch tip, and it is feasible to study the effect of the θ₁ on the fracture toughness testing using the SEVNB method.

Figure 8. J-integral values of non-crack model with different u/a values and the stress cloud near the V-notch tip with (a) u/a = 0, (b) 0.5 and (c) 0.95, where the angle of the V-notch tip was maintained at 60°. The results when the angle of the V-notch tip θ₁ was 30°, 90°, and 120° were the same as those at 60°, and they are not shown here.

The influence of θ₁ on the fracture toughness testing based on the non-crack model with u/a = 0 is shown in Figure 9. K_Ic/th is the fracture toughness of ideal crack model. K_Ic/θ1 is the fracture toughness of the non-crack model with u/a = 0. K_Ic/θ1 also kept increasing as θ₁ increased. To study the effect of the notch length (a) on the fracture toughness testing, three non-crack models with a/h values of 0.25, 0.5, and 0.75 were constructed. The results are shown in Figure 9a. The three non-crack models followed the same trend with the increase in θ₁. This implied that θ₁ had the same effect on the fracture toughness testing when a/h was in the range of 0.25–0.75. To further analyze the impact of θ₁ on the fracture toughness testing using the SEVNB method, the results obtained by the non-crack model were fitted, as shown in Figure 9b. The relationship between θ₁ and the relative fracture toughness, K_Ic/θ1/K_Ic/th can be described by the following equation:

Figure 9. K_Ic/θ1/K_Ic/th for various angles θ₁ (a) and (b) fitted curve. Very good agreement was achieved with the data obtained from the non-crack model.

(5) $\frac{K_{Ic/\theta 1}}{K_{Ic/th}}=0.01493\times EXP\left(\frac{{\theta }_1}{25.70964}\right)+0.98309$(5)

K_Ic/th can be calculated using Equation (5)(5) $\frac{K_{Ic/\theta 1}}{K_{Ic/th}}=0.01493\times EXP\left(\frac{{\theta }_1}{25.70964}\right)+0.98309$(5) , and the test error of the fracture toughness corresponding to different θ₁ values can also be evaluated. For example, K_Ic/θ1 was higher than the K_Ic/th by 13.71% when θ₁ was 60°. K_Ic/θ1 was higher than the K_Ic/th by 3.10% when θ₁ was 30°. K_Ic/θ1 was just higher than K_Ic/th by 0.51% whenθ₁ was 10°. This result was consistent with that of Palacios et al. [24]. In their work, the fracture toughness values of tungsten alloy samples with notches produced by a laser were studied. The results showed that the actual fracture toughness values of the samples with θ₁ values of about 5° could be obtained. The fracture toughness would be overevaluated by 21% when θ₁ was 70°, which is consistent with the value calculate by Equation 5(5) $\frac{K_{Ic/\theta 1}}{K_{Ic/th}}=0.01493\times EXP\left(\frac{{\theta }_1}{25.70964}\right)+0.98309$(5) . Wang et al. [20] also studied the fracture toughness values of ZrB₂ and ZrB₂-SiC ceramic samples with θ₁ value of 20° and 3°, respectively. They found that the fracture toughness values were overevaluated by 0.38% and 0.01% for ZrB₂ and ZrB₂-SiC ceramic samples, respectively, which was consistent with the result calculated using the non-crack model (1.56% and 0%, respectively). However, the results of this work are different from those reported by Wang et al. [16]. In their work, K_Ic/θ1 was equal to K_Ic/th when the θ₁ was less than 60°. The difference may have resulted from the assumption of perfect dumbbell-shaped stress contours. However, this assumption was not precise. As shown by the stress cloud of an ideal crack tip in Figure 10a, when θ₁ was about 30°, the stress included in the V-notch region was close to zero. However, when θ₁ reached 60°, although the stress was still close to zero in the region away from the V-notch tip, the stress near the V-notch tip reached higher levels. The stress cloud images of the non-crack model with θ₁ value of 10°, 30°, 60°, 90°, and 120° are shown in Figures 10b-f, respectively. For θ₁ value of 10° and 30°, an area with high stress concentration were reserved, and the stress in the removed area was close to zero when comparing with the stress cloud of natural crack (Figure 10a). However, the area with high stress concentration near the crack tip was eliminated when θ₁ was greater than 60°. The fracture toughness of the V-notch with the tip at 60° may be overestimated due to the removal of the high stress area. This observation was basically consistent with the results of the simulation (Equation (5)(5) $\frac{K_{Ic/\theta 1}}{K_{Ic/th}}=0.01493\times EXP\left(\frac{{\theta }_1}{25.70964}\right)+0.98309$(5) ).

Figure 10. (a) Stress cloud image of a typical ideal crack tip including two marked angles. The stress in the area for a 30° angle was close to zero, while the 60° angle included a large local stress area. Stress cloud image of non-crack models with θ₁ value of (b) 10°, (c) 30°, (d) 60°, (e) 90°, and (f) 120°.

4.2. Effect of groove tip angle (θ₂) on fracture toughness measurement

To study the influence of θ₂ on the fracture toughness test, a groove model that was not affected by θ₁ was established. In most cases, the V-notch is assumed to be a crack when testing the fracture toughness by the SEVNB method. Fett, et al [15]. determined that the actual fracture toughness can be obtained when the length of the V-notch is longer than the sum of three-grain size scales of the sample materials. However, the fracture load increases due to the geometric shielding effect at the V-notch tip, leading to an overestimation of fracture toughness. Thus, different widths of the groove may need different lengths of the V-notch [5,15]. In this analysis, grooves with different groove widths were built, as shown in Figure 2c. As shown in Figure 11, although the groove widths of 100 μm, 200 μm, and 500 μm were different, θ₂ had the same effect on the fracture toughness testing results. In addition, different groove lengths were also studied. As shown in Figure 11a, the relative length a/h (0.2–0.6) had the same effect on the fracture toughness testing. To study the effect of θ₂ on the fracture toughness testing, the data in Figure 11a were fitted, and the result is shown in Figure 9b and can be expressed as follows:

Figure 11. K_Ic/θ2/K_Ic/th for various values of angle θ₂ (a) and the (d) fitted curve. Excellent agreement with the data was obtained by the groove model.

(6) $\frac{K_{Ic/\theta 2}}{K_{Ic/th}}=0.00223\times EXP\left(\frac{{\theta }_2}{24.19072}\right)+0.99337$(6)

Equationn (6)(6) $\frac{K_{Ic/\theta 2}}{K_{Ic/th}}=0.00223\times EXP\left(\frac{{\theta }_2}{24.19072}\right)+0.99337$(6) shows that K_{Ic /θ2} is higher than K_Ic/th by 0.5% when θ₂ is 40°, and K_Ic/θ2 is higher than K_Ic/th by 2.0% when θ₂ is 60°, while K_Ic/θ2 is higher than K_Ic/th by 31.1% when θ₂ is 120°. Equation (1)(1) $\frac{K_{exp}}{K_{th}}=tanh\left(2Y\sqrt{\frac{l}{\rho }}\right)$(1) predicts that K_Ic/θ2 is higher than the K_Ic/th by 2.20% when θ₂ is 60° (l = ρ), while K_Ic/θ2 is higher than K_Ic/th by 0.40% whenθ₂ is 40° [15]. These results were almost consistent with the results obtained by LFEM in this work. In addition, these results indicated that θ₁ had more influence on the fracture toughness testing than θ₂.

The stress cloud images of the groove with θ₂ values of 0, 19°, 60°, 91°, 113°, and 131° are shown in Figure 12. Regardless of the value of θ₂ of the crack tip, the contour of the stress cloud images at the crack tip remained almost the same. Meanwhile, the stress distribution almost remained constant due to the large distance between the crack tip and the groove tip. This is the reason that θ₁ had a greater impact on the fracture toughness test than θ₂. The stress did not accumulate significantly for the groove tip far from the crack tip. Thus, θ₂ has a weaker effect on the fracture toughness testing when using the SEVNB method. It seems that θ₂ has a weaker effect than θ₁ on the fracture toughness test. Although the width of the groove is usually less than 1 mm, the length of V-notch is generally 100 μm (the process of longer V-notch will lead to a decrease in the sharpness of the V-notch). Thus, a relatively narrow groove may lead to a larger θ₂ due to the too small length of the V-notch. Even the V-notch is sharp enough, the large θ₂ may shield the induction and propagation of the crack at the tip of the notch. Therefore, Both the length of V-notch and the width of the groove affect the stress concentration at the tip.

Figure 12. (a) Stress cloud images of a typical ideal crack tip. Stress cloud image of groove models with θ₂ of (b) 19°, (c) 60°, (d) 91°, (e) 113°, and (f) 131°.

4.3. Experimental results and Discussion

Figure 13 shows the typical thermally etched microstructures of the investigated samples. The grains in all the samples had fairly equiaxed shapes and the average grain sizes were found to be about 500 nm, 550 nm and 1200 nm for 2.3Y-TZP, 3Y-TZP, and Al₂O₃, respectively. No residual pores were detected. These three ceramic materials have a dense and uniform microstructure, which is suitable for the research on linear elastic fracture of materials.

Figure 13. SEM images of the samples of (a) 2.3Y-TZP, (b) 3Y-TZP and (c) Al₂O_3.

The relationship between the fracture toughness test values of the three ceramic materials and the angle is shown in Figure 14. This indicates that the fracture toughness tested by the SEVNB method increases as θ₂ increases. In addition, the results predicted using Equation (6)(6) $\frac{K_{Ic/\theta 2}}{K_{Ic/th}}=0.00223\times EXP\left(\frac{{\theta }_2}{24.19072}\right)+0.99337$(6) are plotted in Figure 14 to compare with the experimental results. It is clear that the predicted curve fit well with the experimental values for the Al₂O₃ sample, while the fracture toughness values were slightly different for 2.3Y-TZP and 3Y-TZP in the high-angle area. The difference can be explained by the different R-curves of the three samples. As shown in Figure 15, the volume fractions of the m-ZrO₂ on the polished surface were zero, and the phase transformations on the fractured surface were 10 vol%, and 30 vol% for 2.3Y-TZP, and 3Y-TZP, respectively. Therefore, R-curve was steep to prevent the crack growth in the TZP samples, in contrast to the R-curves of Al₂O₃ samples reported in the literature [25–27]. The cracks in the 2.3Y-TZP, and 3Y-TZP samples grew stably before fracture. Thus, the actual θ₂ before fracture of the samples became larger. As shown in Figure 14 the predicted curves of 2.3Y-TZP and 3Y-TZP had light differences from the experimental data, but the predictions were the same as the experimental data in the Al₂O₃ samples for larger θ₂ value. In addition, the fracture toughness values of the 3Y-TZP and Al₂O₃ samples were measured by the SEVNB method in our previous work [28–30]. The morphologies of the V-notch and groove for 3Y-TZP and Al₂O₃ samples were also studied, θ₁ was in the range of 20°- 30°, while θ₂ was about 105° (Figure 16). The fracture toughness was compared with the predicted result of Equation (6)(6) $\frac{K_{Ic/\theta 2}}{K_{Ic/th}}=0.00223\times EXP\left(\frac{{\theta }_2}{24.19072}\right)+0.99337$(6) in Figure 14. Although the fracture toughness reported in our previous work was overestimated because of the larger θ₂, the relationship between the fracture toughness and θ₂ was consistent with the results of this work, as shown in Figures 14b and 14c.

Figure 14. Experimental curve and curve predicted by Equationn (6)(6) $\frac{K_{Ic/\theta 2}}{K_{Ic/th}}=0.00223\times EXP\left(\frac{{\theta }_2}{24.19072}\right)+0.99337$(6) of the fracture toughness: (a) 2.3Y-TZP, (b) 3Y-TZP, and (c) Al₂O_3.

Figure 15. The XRD patterns of the (a) 2.3Y-TZP and(b) 3Y-TZP for the polished surface and fractured surface after fracture toughness testing. V_{m, polished surface} denotes the volume fraction of the phase transformation on the polished surface of the sample. V_{m, fractured surface 1} is the volume fraction of the phase transformation on the fractured surface of the samples with small θ₂, while V_{m, fractured surface 2} is the volume fraction of the phase transformation on the fractured surface of the sample with a large θ₂. m: monoclinic phase of zirconia, t: tetragonal phase of zirconia.

Figure 16. SEM images of (a) 3Y-TZP [27], (b) Al₂O₃ [29], and (c) 3Y-TZP [28] samples with grooves and V-notches.

As listed in Table 3, the fracture toughness values of the 2.3Y-TZP, 3Y-TZP, and Al₂O₃ predicted by Equation (6)(6) $\frac{K_{Ic/\theta 2}}{K_{Ic/th}}=0.00223\times EXP\left(\frac{{\theta }_2}{24.19072}\right)+0.99337$(6) were 4.68 MPa•m^1/2, 3.71 MPa•m^1/2, and 3.50 MPa•m^1/2, respectively. In this work, θ₁ was in the range of 20°- 30°, and thus, the fracture toughness would be over-evaluated by about 1.56% – 3.10% according to the Equation (5)(5) $\frac{K_{Ic/\theta 1}}{K_{Ic/th}}=0.01493\times EXP\left(\frac{{\theta }_1}{25.70964}\right)+0.98309$(5) . Therefore, the theoretical fracture toughness values of the 2.3Y-TZP, 3Y-TZP, and Al₂O₃ were 4.54–4.61 MPa•m^1/2, 3.59–3.65 MPa•m^1/2, and 3.39–3.45 MPa•m^1/2, respectively. The average experimental fracture toughness values of the samples with θ₁ less than 30° and θ₂ less than 50° were 4.74 ± 0.06 MPa•m^1/2, 3.76 ± 0.07 MPa•m^1/2, and 3.54 ± 0.09 MPa•m^1/2 for the 2.3Y-TZP, 3Y-TZP, and Al₂O₃ samples, respectively. The experimental fracture toughness values were very close to the predicted values. Thus, the fracture toughness values can be considered to be 4.6 MPa•m^1/2, 3.6 MPa•m^1/2, and 3.4 MPa•m^1/2 for 2.3Y-TZP, 3Y-TZP, and Al₂O₃, respectively.

Table 3. Fracture toughness of 2.3Y-TZP, 3Y-TZP and Al₂O₃ samples.

Sample	Experimental fracture toughness values with θ2 less than 50°MPa•m^1/2		Fracture toughness predicted by Equation (6)(6) $\frac{K_{Ic/\theta 2}}{K_{Ic/th}}=0.00223\times EXP\left(\frac{{\theta }_2}{24.19072}\right)+0.99337$(6) MPa•m^1/2	Fracture toughness predicted by Equation (5)(5) $\frac{K_{Ic/\theta 1}}{K_{Ic/th}}=0.01493\times EXP\left(\frac{{\theta }_1}{25.70964}\right)+0.98309$(5) and Equation (6) /MPa•m^1/2
	Average	Range
2.3Y-TZP	4.74 ± 0.06	4.66–4.82	4.68	4.54–4.61
3Y-TZP	3.76 ± 0.07	3.66–3.88	3.71	3.60–3.65
Al2O3	3.54 ± 0.09	3.41–3.70	3.50	3.40–3.45

Wang et al. [17] studied the effect of the notch tip angle on the fracture toughness testing through experiments with typical brittle ZrB₂ and ZrB₂-SiC ceramics. When the notch tip angle was small, they used samples with only θ₁ (typical samples are shown in Figure 17b,c). When the notch tip angle was large, they used samples with θ₁ and θ₂ (typical samples are shown in Figure 17d,e), and ignored the effect of θ₁ on the fracture toughness testing. In addition, they used a non-crack model to compare with the experimental data. The results are shown in Figure 16a. The match between the experimental data and the prediction curve was poor when the notch tip angle was large. This deviation mainly arose from ignoring the influence of θ₁ when the angle was large. As shown in Figure 17d, when θ₂ was 50°, θ₁ was only 9° which leads a limited effect for the fracture toughness values. However, as shown in Figure 16h, when θ₂ was 140°, θ₁ reached 21° which should not be ignored. The fracture toughness of the ZrB₂ sample with θ₁ and θ₂ values of 21° and 140° predicted by its non-crack model was 4.5 MPa•m^1/2, but those predicted by Equation 5(5) $\frac{K_{Ic/\theta 1}}{K_{Ic/th}}=0.01493\times EXP\left(\frac{{\theta }_1}{25.70964}\right)+0.98309$(5) (non-crack model) and Equation 6(6) $\frac{K_{Ic/\theta 2}}{K_{Ic/th}}=0.00223\times EXP\left(\frac{{\theta }_2}{24.19072}\right)+0.99337$(6) (groove model) were both 4.7 MPa•m^1/2. The result is marked in Figure 17a. The fracture toughness values predicted by Equations 5(5) $\frac{K_{Ic/\theta 1}}{K_{Ic/th}}=0.01493\times EXP\left(\frac{{\theta }_1}{25.70964}\right)+0.98309$(5) and 6(6) $\frac{K_{Ic/\theta 2}}{K_{Ic/th}}=0.00223\times EXP\left(\frac{{\theta }_2}{24.19072}\right)+0.99337$(6) were closer to the average of the experiment values. The experiment data and the predictions by Equations 5(5) $\frac{K_{Ic/\theta 1}}{K_{Ic/th}}=0.01493\times EXP\left(\frac{{\theta }_1}{25.70964}\right)+0.98309$(5) and 6(6) $\frac{K_{Ic/\theta 2}}{K_{Ic/th}}=0.00223\times EXP\left(\frac{{\theta }_2}{24.19072}\right)+0.99337$(6) are listed in Table 4. The fracture toughness (K_IC/ref) can be obtained from the SEPB method, which can be considered to be the ideal crack model. For the non-crack model in the ZrB₂ and ZrB₂-SiC ceramics, the rate of the fracture toughness (K_IC/SEVNB/K_IC/Ref) obtained from the experiment were basically consistent with the prediction by Equationn 5(5) $\frac{K_{Ic/\theta 1}}{K_{Ic/th}}=0.01493\times EXP\left(\frac{{\theta }_1}{25.70964}\right)+0.98309$(5) (K_IC/SEVNB/K_IC/th). In addition, in tungsten alloys, Equation 5(5) $\frac{K_{Ic/\theta 1}}{K_{Ic/th}}=0.01493\times EXP\left(\frac{{\theta }_1}{25.70964}\right)+0.98309$(5) was also verified to yield good predictions [24]. When θ₁ was 50°, the fracture toughness of the test was overevaluated by 20%, and that predicted by Equation 5(5) $\frac{K_{Ic/\theta 1}}{K_{Ic/th}}=0.01493\times EXP\left(\frac{{\theta }_1}{25.70964}\right)+0.98309$(5) was overevaluated by 21%. When θ₁ was 3°, the experimental test could obtain the real fracture toughness, which was also consistent with that predicted by Equation 5(5) $\frac{K_{Ic/\theta 1}}{K_{Ic/th}}=0.01493\times EXP\left(\frac{{\theta }_1}{25.70964}\right)+0.98309$(5) . For the ZrB₂ material with the mixed model notch (non-crack model and groove model), when θ₁ and θ₂ were 9° and 50°, respectively, the test predicted that the fracture toughness would be overestimated by about 2%, and the fracture toughness values predicted by Equations 5(5) $\frac{K_{Ic/\theta 1}}{K_{Ic/th}}=0.01493\times EXP\left(\frac{{\theta }_1}{25.70964}\right)+0.98309$(5) and 6(6) $\frac{K_{Ic/\theta 2}}{K_{Ic/th}}=0.00223\times EXP\left(\frac{{\theta }_2}{24.19072}\right)+0.99337$(6) would be overestimated by 1.5%. When θ₁ and θ₂ were 21° and 140°, the overestimation was 77.2%, as predicted by Equations 5(5) $\frac{K_{Ic/\theta 1}}{K_{Ic/th}}=0.01493\times EXP\left(\frac{{\theta }_1}{25.70964}\right)+0.98309$(5) and 6(6) $\frac{K_{Ic/\theta 2}}{K_{Ic/th}}=0.00223\times EXP\left(\frac{{\theta }_2}{24.19072}\right)+0.99337$(6) , respectively, which was almost consistent with the average value of the test data as shown in Figure 17a. In general, different notch shapes may be used to assess the fracture toughness by the SEVNB method. Both Equations 5(5) $\frac{K_{Ic/\theta 1}}{K_{Ic/th}}=0.01493\times EXP\left(\frac{{\theta }_1}{25.70964}\right)+0.98309$(5) and 6(6) $\frac{K_{Ic/\theta 2}}{K_{Ic/th}}=0.00223\times EXP\left(\frac{{\theta }_2}{24.19072}\right)+0.99337$(6) can be used to analyze the excessive fracture toughness values of samples with only V-notches and samples with mixed notches (V-notch + groove). This can be served as a reference for testing the real fracture toughness values of brittle materials.

Figure 17. (a) Experimental and predicted curves of K_Ic in ZrB₂ and ZrB₂-SiC specimens with various θ₁ and θ₂ values [17]. Images of the notched test bars with θ₁ of (b) 3°, (c) θ₁ of 20°, (d) θ₁ and θ₂ of 9° and 50°, respectively, and (e) θ₁ and θ₂ of 21° and 140° [17], respectively.

Table 4. Fracture toughness values of materials with different θ₁ and θ₂ values.

Sample	Model type for SEVNB Method	θ1	θ2	Test method for$K_{IC/SEVNB}$	Test method for ${\mathrm{K}}_{IC/Ref}$	Fracture toughness rate		Ref.
						$\frac{K_{IC/SEVNB}}{{\mathrm{K}}_{IC/Ref}}$	$\frac{K_{IC/SEVNB}}{{\mathrm{K}}_{IC/th}}$
Tungsten alloy	1	3		SEVNB	SENB-S	1	1	[24]
	1	70		SEVNB	SENB-S	1.2	1.21
ZrB2	1	3		SEVNB	SEPB	1	1	[17,20]
	1	20		SEVNB	SEPB	~1	1.016
	1 + 2	9	50	SEVNB	SEPB	~1.02	1.015
	1 + 2	21	140	SEVNB	SEPB	1.679	1.772
ZrB2/SiC	1	3		SEVNB	SEPB	~1	1	[17,20]

1* non-crack model; 2* groove model; the SEPB method can be considered as the ideal crack model.

5. Conclusion

In this work, two important angles in the fracture toughness testing of ceramic materials by the SEVNB method were proposed to determine the shape of the prefabricated notch and groove. The effects of the two angles on the measured fracture toughness were analyzed, and quantitative relationships (Equation (5)(5) $\frac{K_{Ic/\theta 1}}{K_{Ic/th}}=0.01493\times EXP\left(\frac{{\theta }_1}{25.70964}\right)+0.98309$(5) and (6(6) $\frac{K_{Ic/\theta 2}}{K_{Ic/th}}=0.00223\times EXP\left(\frac{{\theta }_2}{24.19072}\right)+0.99337$(6) )) were obtained by LFEM and validated by experiments. The V-notch tip angle (θ₁) had a significant influence on the tested fracture toughness, which was overevaluated by 13.7%, when the angle of the V-notch was 60°. When the angle formed by the V-notch and groove (θ₂) was 60°, the fracture toughness was only overevaluated by 2.0%. The fracture toughness was overevaluated by about 31% when θ₂ increased to 120°. As a result of this research, we can more accurately and reliably test the fracture toughness values of ceramic materials using the SEVNB method.

Acknowledgments

The authors also acknowledge the Northwestern Polytechnical University High-Performance Computing Center for the allocation of computing time on their machines. We thank LetPub (www.letpub.com) for its linguistic assistance during the preparation of this manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

We thank the Fundamental Research Funds for the Central Universities (Grant No. 2022ZYGXZR026), the National Natural Science Foundation of China (Grants No. 51972268, 51702100), Science and Technology Planning Project of Guangdong Province (Grant No. 2021B1212050004), China Postdoctoral Science Foundation (Grant No. 2018M643075) and 2016 Year YangFan Innovative & Entrepreneurial Research Team Project of Guangdong Province of China(Grant No. 2016YT03C025) for financial support.