Indentation fracture resistance of brittle materials using irregular cracks: A review

The equation of the Vickers indentation fracture (VIF) proposed by Moradkhani et al. (J. Adv. Ceram. 2, 87–102 (2013)) has been used for the determination of indentation fracture resistance from indented area containing irregular and branch cracks formed around Vickers diamond indentation impressions. The equation is based on the concept that the surface of the micro-cracks is related to the depth of the micro-cracks. Results of the published experimental data have been compared and analyzed with similar and dissimilar methods in all different tested materials, and the accuracy, advantages and disadvantages have been presented and discussed. The results show that, in 90% of the performed tests, the accuracy of the equation is more than 90% compared to methods based on conventional fracture mechanics (CFM) and Vickers indentation fracture (VIF). The results show a high degree of accuracy to date, but the style and method of analysis need to be explained more clearly. Also, there is a requirement for a more comprehensive test range for other brittle materials.

The IF group of techniques is classified such as Cube-Corner Indentation Fracture (CCIF) [26], Vickers crack opening displacement (VCOD) [31], interface indentation fracture (IIF) and Vickers indentation fracture (VIF) [37] tests.The need for less equipment and raw materials, lower cost, simplicity and high speed of sample preparation are some of the advantages of the IF method [28,37,38].In the VIF technique, direct crack length measurements are often used around the indented areas in the sample, made by the Vickers diamond [26,[39][40][41].Usually, the K IFR of the samples is determined by the crack length parameters, quality and hardness of materials, and the amount of applied load [26,31].The relationships in this method are varied, and some studies have reported high accuracy [41,42] while others have detected many errors, even up to 48%, in them [43][44][45][46].
In the present study, an effort has been made to provide an overview of fracture resistance determination techniques such as CFM and VIF to analyze the advantages and disadvantages of equation proposed by Moradkhani et al. [47], of the VIF technique.In this equation, the surface and thickness of micro-cracks created in a sample appear to improve the accuracy of the equation according to the cost parameter.Accuracy of the results obtained by the conventional fracture resistance and VIF techniques have been compared with the published literature for a range of composites.The results often show a high-reliability index of the equation and a low cost of sample testing.Therefore, by comparing the obtained results, ambiguities and strengths of the equation are mentioned and criticized to obtain a clear picture of fracture resistance.

Theoretical background
The emergence of the idea that it is possible to obtain K IFR in brittle materials using IF was formed by Palmqvist [48] In the 1950s; however, it was ignored until the mid-1970s when Evans and Charles [49] applied the VIF technique for single crystal oxides to cemented carbides and presented functional relations and classified them into two models.Later, this technique quickly became popular due to the reduction of testing costs and was widely used for brittle materials [26,[29][30][31]50,51].
In general, there are two models based on Palmqvist (Pl) and median/half-penny (M) cracks in the VIF technique [42,52].Fig. 1  (a-b) shows differences between the two models, and Fig. 1c shows a schematic presentation of the Vickers diamond indenter in the sample.As can be seen, in the Pl model, the length of the formed cracks is considered only from the end of the indented area [53].However, in the M-crack model, the length of the cracks is considered from the center of the indented section [42].It is assumed [54][55][56][57] that by increasing the Vickers diamond loading, the approach will change from the Palmqvist model to the median crack model.The results of some studies suggest that a simple way to separate these two models is sample surface polishing.After the polishing process, in the Pl model, the crack will be separated from the rhombic part left by the Vickers indenter; but in the median model, it will be connected to the remaining portion of the indenter [58,59].Another way to distinguish between these two models is the ratio c/a of the width c of median crack to the half-width a of indenter diagonal length.If the ratio is c/a < 2.5, it is assumed as Palmqvist and if it is c/a ≥ 2.5, it will be considered an M crack.This ratio is often considered for brittle materials [60].Some workers have considered 3 and 2 to separate the relationships in some materials and alloys [42,59].
After Evans and Charles [47,61], several relations have been proposed to calculate K IFR [47,62,63], some of which are given in Table 1.As can be seen, Curve Fitting Technique (CFT) has been used in some relations.The basis of their formation is the use of trial and error and linking K IFR to crack morphology [47].Some of the relations in Table 1, relations 2, 3, 6, and 7, are simple to calculate K IFR because they only depend on the parameters in Fig. 1.But some others, relations 9 and 10, depend on determining Vickers hardness in the sample.Finally, in other existing relations, including relations 1, 4, 5, 8, 11, 12, 13, and 14, in addition to the need to calculate the Vickers hardness, Young's modulus value of the samples must be calculated.Young's modulus of specimens is often measured based on the ASTM 769 [64] standard and the variation of sound velocity in the samples according to equation 1.
where ρ is sample density and v is the sample's sound velocity variation.This slows down the calculation process and increases the required equipment.But in these cases, K IFR is more closely related to parameters dependent on the sample type and often increases accuracy [30,47,[65][66][67].
In Table 1, the dimensionless values of F and y in relations 11 and 13 in Table 1 are as follows [55]: The value of x in equations ( 2) and (3) equals equation ( 4).

Discussion
The presence of irregular and branched cracks in the samples causes errors in the K IFR results obtained from the VIF relations and researchers have always tried to solve this issue [52,74].The main reasons for these errors are high indentation loading and the presence of micro-cracks in the microstructure of the samples in brittle materials, which cause irregular cracks [72].Even three-dimensional (3D) cracks and complexes for materials with ceramic coatings, formed in some composites, have been considered, and efforts have been made to resolve this issue [75][76][77].One of these investigations was conducted by Moradkhani et al. [47] in 2013, in which they presented a semi-experimental equation (5) to minimize these errors.They believed that consideration of the mean penetration depth and effective area of formed micro-cracks and their involvement in the calculations would reduce the incidence of error.In addition, the depth and penetration of micro-cracks in the sample are directly related to the average surface and thickness of micro-cracks, and the involvement of these two parameters in the K IFR calculation reduces the number of errors.On the other hand, this method significantly reduces testing costs because it prevents a large number of tests on the sample from achieving natural cracks [47,78].Based on fitting the available data the following empirical equation was proposed [47]: K IFR = 0.00366 where t av is the average penetration depth of microcracks formed around the indented section (mm), and A is the surface area of the effect of the microcracks formed around the indented section (mm 2 ).The coefficient in the equation is also obtained by performing mathematical and experimental calculations for Al 2 O 3 -nanoSiC nanocomposites with different volume percentages of nanoSiC.To determine this coefficient, the results of Single Edge Laser Notch Beam (SELNB) and Single Edge Notch Beam (SENB) methods have been used, and to ensure its value, the results of K IFR have been compared with the Chevron Notch Beam (CNB) method.Fig. 2 (a-c) shows the schematics of three methods CNB, SELNB, and SENB to determine fracture toughness.Despite all three methods being highly accurate and having reliable results, the costs of making samples, especially for brittle materials and testing them, are always high [11,12,15,18].As can be seen, the difficulty in making accurate samples leads workers to other methods such as Vickers indentation fracture [3].Equation ( 5) is similar to relations 5, 8, and 12 of Table 1, which differ among themselves by the numerical factor equal to 0.016, 0.014, and 0.018, respectively.These relations follow from stress residual theory and the numerical factor in the case of half-penny cracks.According to this theory, the residual stress remaining around an indentation impression after removing the applied load on the indenter is the driving force for the creation of cracks [50,73].Indentation pressure is produced by a disc-shaped indenter and can be calculated as p = P/A* in the surface area underneath the indenter, where P is the applied load and A* is the surface area underneath the indenter.An increase in indentation load is expected to result in an increase in the volume of the pressurized area, as calculated by V = A*h, where h is the penetration depth and A* is dependent on A, the area deformed around the indentation.
Assuming that the average depth t of cracks formed in the vicinity of indentation is equal to h one finds that both the depth t and the area A = πC 2 /4, where C is the diameter of the deformed area containing cracks [50].This is the explanation of the inverse relation between t and A. For example, by comparing relation 8 of Table 1 with the constant 0.014 based on residual stress theory and equation ( 5), t and C can be expressed as C = 8.9t.However, it cannot be adopted for many emerging brittle materials, such as smart composites and biological hard tissues, which have complex microstructures.Moradkhani et al. [47] in their initial study, believed that the first condition for using equation ( 5) is the absence of chipping phenomenon in the samples.This phenomenon generally occurs at high loadings [72].Also, the secondary condition is the minimum brittleness of the samples, which is achieved by establishing nc/a ≥ 20, c max = 1.4n, if n ≤ 7 where c is the length of median crack (m), a is the diagonal half-length of Vickers impression (m), and n is the result of the number of cracks emitted from each side of the indented rhombus area.This claim indicates the non-necessity of having only four cracks without any branch, which distinguishes equation ( 5) from other relations of the same family and expands its efficiency in practice [47].Fig. 3 shows the Scanning Electron Microscope (SEM) images of the residual effect of the Vickers indenter in Al 2 O 3 samples.In Fig. 3a, the Vickers effect can be seen without branch cracks.In Fig. 3b, the loading value is so high that it causes chipping phenomenon in the sample, and Fig. 3c     A. Moradkhani et al. opaque area at the center in the dashed line in Fig. 4b.In Fig. 4c and d the lateral crack system is less than or equal to 29.4 N [77].
The Image Analyzer Software is commonly used to determine the parameters t av and A based on the applied load to the diamond Vickers and the hardness values of the samples.For example, Fig. 5 (a-d) illustrates the process of using software to calculate these values for the formed cracks around the Vickers indenter in the Al 2 O 3 -15%nanoSiC sample [47].In Fig. 5c and d, the software is being used to analyze and measure the images and area values of cracks, respectively.Fig. 6 (a-c) shows t av , A, and H V values for calculating K IFR in Al 2 O 3 samples containing 0, 2.5, 5, 7.5, 10, and 15 vol% nanoSiC by measuring seven times.In this experiment, a 150 N force was applied to the samples for 15 s using an indentation test, and the thickness and area of microcrack tracks were measured [47].However, the applied load values should be determined based on the hardness of the samples to create regular or irregular cracks without causing the chipping phenomenon, which allows for a wide range of applied loads to determine t av and A.
In the following years, the accuracy and efficiency of equation ( 5) for determining fracture resistance in materials such as B 4 C-nanoTiB 2 [79], B 4 C-nanoSiB 6 [80], B 4 C-nanoTiB 2 -Fe/Ni [81], Al 2 O 3 -SiC [82], Al 2 O 3 -SiC-MgO [83], B 4 C-nano/microSiC [84], 3Y-TZP [85], Al 2 O 3 /3Y-TZP [86], W-ZrC [87], and Si particle [88] were tested.The mechanical and microstructural properties of these materials are diverse and have medical and industrial applications.Hence, the range of accurate measurement of the equation   became relatively wide, but it seems that this equation is still not widely known and implemented.Fig. 7 compares the accuracy of K IFR obtained from equation ( 5) for some of the materials made at their optimal sintering temperature.The comparison includes the most practical VIF relations mentioned in Table 1 (relations 5 and 10) and some conventional fracture mechanics techniques such as CNB, SELNB, and SENB methods [79][80][81][82][83][84][85][86][87][88].As can be seen, in 90% of the samples, more than 90% accuracy was reported.However, in some cases, such as Al 2 O 3 -3-YTZP composites made by tape/slip casting, the accuracy has been reduced to 87% compared to the SELNB method [86].In some materials, the accuracy is close to 100% [80,84].This accuracy has been observed in both CFM and VIF groups.From the results of articles published since the beginning of this equation, it seems that their accuracy ensures their reliability and no studies have been presented to refute them.In addition, equation (5) has been used by researchers in a wide range of other sintering temperatures as well as the various fabrication methods of the materials mentioned in Fig. 7, and similar accuracy has been reported for them [79][80][81][82][83][84][85][86][87][88].Considering these two parameters, a wide range of sintering temperatures and various fabrication methods, equation (5) has considered up to one hundred composites with other ranges of mechanical properties, especially K IFR .Despite the microstructural defects and other mechanical properties resulting from different fabrication methods and sintering temperatures, accuracy is still reported to be high.
It should be noted that the range of brittle materials with other properties and microstructures is much broader than the scope of research accuracy and can still challenge the accuracy of the equation.There seems to be a need for further exploration of the method, a more precise and more accurate presentation of how the test was performed, the tools and software used, and a more precise description of the mathematical calculations.Except for a limited number of studies, this equation has not been used.There is also an urgent need to test on standard materials with a wide range of K IFR .Some workers, such as Anstis et al. [72], did this process to establish the accuracy of their relation.Nevertheless, so far, despite testing and using the equation on a range of brittle materials, high accuracy in the output of the equation has been reported.This could lead to applying the equation in a broader range of brittle materials to reach a greater consensus among researchers.
Another critical point claimed in equation ( 5) is that there is no need to identify the type of Palmqvist/median crack model [80].For example, in 3Y-TZP Dental Ceramics, the c/a ratio was calculated based on 29.42, 49.03, 147.09, 196.13, and 245.15 loading amounts, and this ratio ranged from 1.5 to nearly 4 for pure and silica-doped TZ, and 0.1/0.2A-TZsamples.Finally, the output of the equation for all ratios had an accuracy of over 90% compared to the relationships in Table 1 [85].Also, in other studies, the amount of loading in B 4 C-nanoTiB 2 samples was considered a variable parameter, and no effect on the accuracy of the equation was observed before chipping [89,90].However, there seems to be a need for extensive research in this regard.

Summary and outlook
The present study reviews the published research results of the equation proposed by Moradkhani et al. [47] to estimate the K IFR value using irregular and branch micro-cracks.It is based on measuring the K IFR /K IC of brittle materials, at the lowest cost and, at the same time, the simplicity of the test steps [28].In the last decade, ranges of brittle materials were tested with the aim of measuring the accuracy and benchmarking the accuracy of the equation outputs with various fabrication conditions, and satisfactory results were obtained [42].However, so far, the scope of the practical application of this equation is limited to a few investigations [80].There might be different reasons for its no widespread use; Failure to test it on known standard materials, duration and amount of loading [90], how exactly the crack effect level is determined [47], the probability of creating complex 3D cracks [77], lack of clear and accurate presentation of how the tests were performed and how the tools and software were used, and lack of accurate description of mathematical calculations and the existence of other different reliable relations [47,80].However, so far, there have been no reports of inaccuracies in the equation outputs.In several studies, the accuracy for different materials has generally been more than 90% compared to CFM and VIF [79][80][81][82][83][84][85][86][87][88][89][90].However, there is a need for more theoretical and practical use which can prove its accuracy, this will reduce the cost of testing and could be a new approach to determining the K IFR /K IC .Moreover, sufficient attention should be paid to the range of test materials and other parameters that may be problematic during the tests.
In the last two decades, the theory of peridynamic has investigated the behavior and growth of cracks using mathematical models and simulations for brittle materials [91][92][93][94][95].In this method, models and simulations explain the role Van der Waals forces play in the prediction of cracks behavior of brittle materials [96][97][98][99].The combination of the two methods, peridynamics theory and VIF [100,101], based on Griffith theory could be a new approach and a vast evolution in determining K IFR /K IC for various materials, composites, and biological hard tissues.

Fig. 1 .
Fig. 1.Schematic of a) Palmqvist crack model, b) Median/half penny crack model, and c) Vickers diamond indenter in brittle material.

A
.Moradkhani et al.

Fig. 5 .
Fig. 5. a) Formed crack in Al 2 O 3 -15%nanoSiC sample, b) preparation of the crack to be calculated by Image Analyzer Software, c) separation of cracks by software in order to measure the required values, and d) calculation the area values of cracks [47].

Fig. 7 .
Fig. 7. Comparison of the accuracy of the fracture resistance, Moradkhani et al. equation values with irregular cracks with other Vickers indentation fracture (VIF) relations and some conventional fracture mechanics (CFM) techniques in some brittle materials.

Table 1
Some relations used in the calculation of fracture resistance (K IFR )/fracture toughness (K IC ).