## Abstract

Based on the stress fields formed upon Vickers indentation, the radial crack and half-penny crack systems are analysed separately using finite-element analyses. The crack length is correlated with the stress intensity factor (SIF) at the crack front and material elastoplastic properties via explicit relationships, from which a reverse analysis can be carried out such that the critical SIF (fracture toughness) can be readily derived once the crack length is measured. The proposed technique is validated by comparing with experimental data in the literature as well as parallel experiments of Vickers indentation-induced cracking.

## 1. Introduction

Comparing with traditional testing techniques (e.g. tension, bending, etc.), instrumented indentation is relatively easy to perform since it requires minimum sample preparation and thus it is widely used to characterize the mechanical properties of materials (Chen *et al*. 2007), including advanced small material structures (Cao & Chen 2006; Xiang *et al*. 2006). Sharp indentation can be employed to reveal the material elastoplastic properties (Ogasawara *et al*. 2006*a*,*b*) and residual stresses (Chen *et al*. 2006*b*; Yan *et al*. 2007*a*,*c*), as well as the fracture characteristics of brittle materials. The analysis of the results obtained from an indentation test, however, is often more complicated than the conventional tests owing to the large local deformation and three-dimensional stress fields involved during the process.

Several types of cracks may appear upon indentation, depending on the indenter shape (conical, spherical, flat punch, Berkovich, Vickers, etc.) and on the properties of the material. Among them, the lateral and cone cracks have been studied using both experimental and theoretical approaches (Lawn & Evans 1977; Chen *et al*. 2005; Yan *et al*. 2007*b*). Various forms of cone cracks were observed in brittle materials when an axisymmetric indenter was used. Lawn and co-workers have investigated cone cracks developed from the free surface in spherical indentation (Lawn & Fuller 1975; Lawn 1993, 1998), and the inner cone cracks developed in brittle solids submerged in water (Chai & Lawn 2005). Cone cracks induced by conical indentation were studied in Yan *et al*. (2007*b*). The elastic recovery during unloading leads to stress redistribution, which is responsible for lateral cracking (Marshall *et al*. 1982; Chen *et al*. 2005). Lateral cracks at the interface between thin films and substrate can be used to estimate the interface toughness (Drory & Hutchinson 1996; Vasinonta & Beuth 2001). Such cracks are also responsible for the abrasive wear and erosion of ceramics and ceramic coatings (Chen *et al*. 2004*a*,*b*).

When sharp indenters (Vickers, Knoop and Berkovich) penetrate into brittle materials, surface cracks often nucleate due to the remnant ‘plastic’ deformation and propagate along the radial direction (figure 1*a*; Cook & Pharr 1990; Boyce *et al*. 2001; Peters *et al*. 2002). Two types of such surface cracks were widely observed in experiments (Palmqvist 1957; Hagan & Swain 1978; Cook & Pharr 1990): the half-penny crack (HPC) system and radial crack (RC) distinguished by their cross-section morphology (figure 1*b*). The RC, whose shape can be approximated as a semicircle, is a shallow surface crack that emanates from the corner of the wedge and expands radially. The half-penny crack is generated beneath the plastic deformation zone and also extends in the radial direction upon unloading. Since it is easier to measure the geometry of the surface crack (especially for non-transparent materials), both HPCs and RCs become the basis for estimating fracture toughness (Evans & Charles 1976; Lawn & Evans 1977; Lawn *et al*. 1980; Anstis *et al*. 1981; Lankford 1982; Marshall 1983; Niihara 1983).

One of the most comprehensive set of experiments was carried out by Cook & Pharr (1990) who investigated cracks in brittle materials induced by Vickers indentation. They argued that for a given material RCs form at lower loads whereas HPCs form at higher loads; both RCs and HPCs grow to full length after complete unloading (also at the time when their lengths can be measured), and they are independent crack systems (i.e. the RC is not the precursor of the HPC above a critical load). It is therefore useful to investigate the two crack systems separately. The crack systems were found to be material dependent, in particular the ratio between elastic modulus and hardness, which brings the importance of varying material elastoplastic properties during the indentation cracking analysis. Although the indentation force–displacement curve was widely used in extracting elastoplastic properties (Ogasawara *et al*. 2005, 2007, 2008; Chen *et al*. 2006*a*, 2007), they were found to be continuous during the event of cracking and they were the same even when different crack systems were produced (Cook & Pharr 1990). Thus, it was suggested that the use of the indentation force–displacement curve is not a reliable method. The understanding of the fracture mechanism should be based on the stress analysis.

Since the indentation behaviour is coupled with the complex stress field arisen from finite deformation, it is implicitly related with the material elastoplastic and fracture behaviour, and thus an effective theoretical framework must be established to correlate the fracture toughness with the indentation response, crack morphology and other material properties. In the literature, simple analytical stress solutions (e.g. Boussinesq solution for elastic solid or expanding cavity for elastoplastic solid) were used to establish a dimensionless or empirical relationship between the relevant variables, and key parameters were fitted from extensive experiment. Evans & Charles (1976) proposed a simple relationship between fracture toughness and crack length for HPCs. By fitting experimental results, Niihara (Niihara *et al*. 1982) argued that the Evans–Charles equation was not very accurate at lower loads where RCs seem to take over; Niihara then proposed a revised form of the Evans–Charles equation to distinguish RCs from HPCs. In a follow-up study, Niihara (1983) established a simple model for RCs and developed a new equation for RCs. Meanwhile, dimensionless empirical equations were also fitted by Lankford according to different material properties and different crack lengths (without distinguishing HPCs and RCs; Lankford 1982).

Although these methods are quite simple for probing the material fracture toughness, the analytical stress field may be overly simplified; due to the non-proportional loading condition occurring beneath the indenter, numerical simulations based on the flow theory of plasticity could lead to a more reliable indentation stress field (Begley *et al*. 1999). Many of these previous numerical studies (Zeng *et al*. 1995; Begley *et al*. 1999; Zhang & Subhash 2001; Muchtar *et al*. 2003; Pachler *et al*. 2007), however, focused primarily on the stress field obtained from finite-element analyses and they did not couple the fracture analysis with indentation stress field. Analysis of the crack stress intensity factor (SIF) upon indentation is scarce, in particular, because three-dimensional analyses are required for Vickers indentation-induced RCs and HPCs.

The present paper aims to fill in such gap by presenting detailed stress intensity analyses of both RCs and HPCs upon Vickers indentation experiment. The crack length is related to the indent size and material properties through the fracture toughness of the film. The results are also compared with the empirical equations (e.g. Niihara *et al*. 1982). A reverse analysis can be carried out such that for a given set of material elastoplastic properties, once the crack length of the RC or HPC is measured, the fracture toughness can be derived. The results are compared with experiments in the literature (Evans & Charles 1976; Lankford 1982; Niihara *et al*. 1982) as well as our own experiments on a few brittle material specimens. In addition to extracting fracture toughness from Vickers indentation experiment, a better understanding of the surface cracking mechanism (such as HPCs and RCs) is also useful for controlling the crack patterns or crack lengths, where the fractured specimen may be used for microfluidic channels or as a template for micro-fabrication.

## 2. Model and computation method

As shown in figure 1, a Vickers indenter is applied whose apex angle is 136°. Upon maximum penetration (the maximum indentation load is *P*), the total diagonal length of the impression is 2*a*. Both the RC and HPC are taken as ideal, semicircular geometry to simplify the analysis; the radius of the HPC is *c*, and the radius of the RC is *l*/2. One end of the RC is pinned at the corner of impression determined at the maximum load, and it is assumed that with the presence of the indenter, this end of the RC cannot propagate inward, and only the far end of the RC may grow outward radially. The expansion of the HPC is assumed uniform in all directions. For RCs and HPCs, the polar coordinates are *θ* and *ψ*, respectively—it is expected that along the circular crack front, the SIF is a function of *θ* or *ψ*. For brittle materials upon indentation, elastic-perfectly plastic model is often used to describe its constitutive property, which could effective describe the permanent penetration (Sun *et al*. 1995; Care & Fischer-Cripps 1997; Chen *et al*. 2004*b*, 2005; Yan *et al*. 2007*b*). Indeed, compression tests on many brittle ceramic-like materials show that these materials can be approximated by the ideal elastic perfectly-plastic behaviour (Francois *et al*. 1988; Rabier & Demenet 2000; Gei *et al*. 2004; Jiang *et al*. 2006; Yao *et al*. 2006). Most brittle ceramics and glasses can be assumed isotropic and homogeneous and, in this paper, the material's Young's modulus is *E* and the yield stress is *σ*_{y}; the plastic behaviour is characterized by the von Mises yield criterion and with the elastic-perfectly plastic behaviour. The Poisson ratio *ν*, which is a minor factor during indentation cracking (Chen *et al*. 2005; Yan *et al*. 2007*b*), is set to be 0.2, a typical value for brittle materials. We do not consider residual stress in the specimen prior to indentation although that could affect the indentation behaviour (Chen *et al*. 2006*b*; Zhao *et al*. 2006*b*).

The commonly used superposition technique (Anderson 1995; Chen & Hutchinson 2001, 2002; Chen *et al*. 2005; Yan *et al*. 2007*b*) is employed to determine the stress intensity of the system, which is suitable when the crack-tip plasticity can be ignored (e.g. for brittle material). The principle is illustrated in figure 2. The stress field of case (1) is the summation of those of cases (2) and (3). In cases (1) and (2), the elastic body with and without crack is subjected to the prescribed displacement and traction boundaries (e.g. the indentation load). The crack surface in case (1) is traction-free, whose SIF is of our interest. A residual stress field is generated in case (2) but in order to keep the crack face closed, a negative stress needs to be applied on the crack position. From superposition, a distributed residual stress *σ*(** x**) is applied over the crack surface in case (3), whose SIF equals that of case (1). Therefore, the indentation cracking problem can be divided into two steps; in the first step, the residual stress field of elastoplastic indentation is derived in absence of a crack (case (2); see §3

*a*). As introduced earlier, both RCs and HPCs grow to full length after unloading according to previous studies, thus only the indentation residual stress field after unloading is considered. In the second step, a crack (either a RC or HPC) is embedded in the deformed specimen, and the distributed residual stress

*σ*(

**) obtained from the first step is acting on the surface of the crack, and the SIF is computed for a given crack geometry (which is case (3); see §3**

*x**b*,

*c*). Apparently, the SIF value depends on the location at the crack front, material property and indentation load (or impression size). The critical condition is reached when the maximum SIF equals the fracture toughness of the material, and for the crack to propagate on the surface (and thus become measurable) the SIF at the surface needs to exceed the material fracture toughness. Owing to the symmetry, both RCs and HPCs are of mode I fracture.

Motivated by the work of Evans and Niihara (Evans & Charles 1976; Niihara *et al*. 1982), the SIF *K* is normalized by the yield stress *σ*_{y} and the square root of the half diagonal length of the impression, *a*. The normalized *K* depends on the non-dimensional crack length (*c*/*a* for HPCs or *l*/*a* for RCs), the position on the crack front (*ψ* for HPCs and *θ* for RCs) and the material property *E*/*σ*_{y},(2.1)(2.2)

The dimensionless functional forms *f* and *g* can be established during the forward analysis using numerical simulations based on the finite-element method (FEM), as the crack length and the material property are varied over a large range. Numerical simulations are carried out using ABAQUS (2006) with finite deformation. The rigid analytical surface option is used to represent the Vickers indenter and the Coulomb friction coefficient is taken to be 0.1. Eight-node brick elements are used and the mesh is more refined near the indenter and around the crack. The indentation simulation and crack analysis are carried out separately, both in three dimensions. Followed by a mesh convergence study, a typical mesh comprises more than 30 000 elements. The maximum indentation depth is controlled through the desired value of *a*.

## 3. Numerical results and discussion

### (a) Stress field upon Vickers indentation

Upon indentation, the radius of the plastic zone is about twice the average radius of impression, and it increases slightly when *E*/*σ*_{y} is higher (i.e. for more ductile materials). The normal stress field (*σ*_{22}) is responsible for both RC and HPC growth; here, *σ*_{22} is the stress component perpendicular to the crack plane, see the axis definition in figure 1. The contour plots of representative residual *σ*_{22} (after unloading) are given in figure 3*a* for *E*/*σ*_{y}=60 and in figure 3*b* for *E*/*σ*_{y}=200. The maximum residual tensile stress occurs just around the corner of impression; however, there is a subtle difference where in the more ductile material, the exact location of the maximum residual *σ*_{22} is below the surface (figure 3*b*), whereas it is on the surface for material with smaller *E*/*σ*_{y} (figure 3*a*).

At point D, which is just outside the impression zone on the specimen surface (figure 3*a*), the *σ*_{22} history when *E*/*σ*_{y}=60 is given in figure 3*c*. It can be seen that the stress grows to its maximum value after unloading, which justifies that the RC and HPC reach maximum length after the indenter is withdrawn, consistent with previous experiments (Cook & Pharr 1990).

According to figure 2, after the indentation residual stress field is obtained, it is applied to a given crack geometry to compute the SIF. The exact indentation geometry (residual profile of the impression zone) is taken into account in the three-dimensional fracture analysis using the FEM; the results are reported in §3*b*,*c* for HPCs and RCs, respectively.

### (b) Analysis of SIFs of half-penny cracks

For a given material (*E*/*σ*_{y}) and crack length of a HPC (*c*/*a*), the normalized SIF varies along the circular crack front (when 0°≤*ψ*≤90°). Figure 4 shows such variations for several values of *c*/*a* (from 2 to 6) and for three different ratios of *E*/*σ*_{y} (from 20 to 200). It can be readily seen that for all cases, the normalized SIF is monotonically increased. In other words, the SIF is at its maximum at the specimen surface (*ψ*=90°), which justifies that the HPC propagates along the surface. At the final surface crack length (*c*) for HPC, the maximum SIF corresponds to the (mode I) critical SIF (or fracture toughness) of the material, *K*_{C}. The normalized SIF is higher when the crack length *c*/*a* is smaller; that is, for a given material, once nucleated the small HPC would grow but its stress intensity would keep decreasing, until it is arrested when its SIF equals *K*_{C}. In addition, the normalized SIF is higher in more ductile material (in part due to the smaller yield stress).

In figure 5, the critical SIF is plotted as a function of *c*/*a* and for selected *E*/*σ*_{y} values; the results are shown in square symbols. The normalized *K*_{C} decreases with the increase in crack length and/or decrease in *E*/*σ*_{y}, which means that if the arrested (measured) crack length is longer the material is more brittle, and such effect is more prominent in harder materials (with smaller *E*/*σ*_{y}). We carry out extensive numerical analyses when *c*/*a* is varied between 2 and 6 and when *E*/*σ*_{y} is varied between 20 and 200, from which *K*_{C} of HPCs can be fitted as a dimensionless function of *c*/*a* and *E*/*σ*_{y}:(3.1)The coefficients are given in table 1.

According to the experiments in Niihara *et al*. (1982), when the apparent surface crack length *c*/*a* is larger than approximately 2.5, HPCs are expected whose behaviours can be fitted into the empirical Evans–Charles equation(3.2)where *ϕ*≡*H*/*σ*_{y} is a constraint factor (Evans & Wilshaw 1976) and *H* is the hardness; for elastic perfectly-plastic materials *H*/*σ*_{y} is a function of *E*/*σ*_{y} (Johnson 1985). The relationship above can be rewritten as(3.3)

The Evans–Charles equation is represented by lines in figure 5 for 2<*c*/*a*<6 and for selected materials. There is an excellent agreement between the numerical results presented in this paper and the Evans–Charles equation summarized from previous experiments (Niihara *et al*. 1982). The difference between the present study (equation (3.1)) and Evans–Charles equation (3.3) is approximately 5% when *E*/*σ*_{y}=100, and the discrepancy of the normalized fracture toughness is approximately 20% when *E*/*σ*_{y}=40. In other words, equations (3.1) and (3.3) give close results when the material is not too hard. Although in figure 5 there is a relatively large discrepancy between the current approach and the Evans–Charles equation when *E*/*σ*_{y}=20, we note that in Niihara *et al*. (1982), all materials employed to derive equation (3.2) have 40<*E*/*σ*_{y}<200. In other words, it remains arguable whether the empirical fitting functions, equations (3.2) or (3.3), could be applied to the harder materials, such as those with *E*/*σ*_{y}=20. On the other hand, note that some brittle materials (e.g. soda-lime glass) have *E*/*σ*_{y}<40, thus, the present analysis may be applicable to a wider range of brittle materials compared with empirical formulae in the literature (Niihara *et al*. 1982).

### (c) Analysis of SIFs of radial cracks

The above analyses are extended to the RC and the trends are somewhat different. For selected *l*/*a* values between 0.4 and 2.5, and *E*/*σ*_{y}=20, 60, 120 and 200, the normalized SIF along the crack front is given as a function of −90°≤*θ*≤90° shown in figure 6. Since only the crack length at the surface (*θ*=90°) can be measured in an experiment, at the final crack length (*l*) of the RC system, the SIF at *θ*=90° equals to the material fracture toughness, *K*_{C}.

When *E*/*σ*_{y}=20, the normalized SIF is highest at the crack initiation site (*θ*=−90°, which is pinned at the impression corner at the maximum load), and it gradually decreases with *θ* until it reaches the other crack tip on the surface (*θ*=90°). In addition, the SIF becomes smaller as the crack length is increased.

As the material gets more plastic, where *E*/*σ*_{y}=60, the normalized SIF oscillates a little with respect to *θ*, especially when the crack is small. This is due to the different indentation stress fields formed in the different materials. At *θ*=90°, the SIF still decreases monotonically with crack growth.

For more ductile materials, where *E*/*σ*_{y}=120 and 200, the oscillation of the SIF along the crack is more obvious, due to the complicated stress field (e.g. figure 3*b*, where for more plastic materials the maximum tensile stress appears below the surface, causing a peak in the SIF in that region). In figure 6*b*, at the surface, the SIF is largest at about *l*/*a*=0.9 and then it decreases with further growth. Such variation is also due to the prominent residual *σ*_{22} formed beneath the surface, which has also affected the SIF at the surface. Note that in figure 3*b*, the maximum normal stress *σ*_{22} also peaks at about *l*/*a*=0.9 on the surface, which is consistent with the trend of the SIF.

In figure 7, the computed critical SIF, *K*_{C} (square symbol), is shown as a function of *l*/*a* and for selected *E*/*σ*_{y} values. The general trend is that the fracture toughness decreases with increasing crack length, except for the more plastic materials with larger *E*/*σ*_{y} and when the crack is too short (which may be strongly influenced by the highly inhomogeneous stress field near the impression corner, see figure 3*b*). In addition, the normalized *K*_{C} is smaller when the material gets harder. From extensive numerical analyses where *l*/*a* is varied between 0.4 and 2.5, and *E*/*σ*_{y} is changed between 20 and 200, the calculated *K*_{C} of the RC can be fitted as a dimensionless function of *l*/*a* and *E*/*σ*_{y}(3.4)The coefficients are given in table 1.

Niihara proposed that according to experimental data (Niihara *et al*. 1982), RCs are preferred when *l*/*a* is smaller than approximately 2.5. The following empirical equation was proposed, which is also represented by lines in figure 7 (when 0.3<*l*/*a*<2.5) for selected values of *E*/*σ*_{y}(3.5)which can be rewritten as(3.6)The Niihara equation (3.6) is monotonic with respect to *l*/*a* and *E*/*σ*_{y}. When compared with the present numerical results in figure 7 (or equation (3.4)), a good match can be found for the more plastic materials, when *l*/*a* is larger than approximately 0.8 (see the next paragraph) and when 60<*E*/*σ*_{y}<200, the discrepancy is smaller than approximately 25%. Once again, we note that the Niihara equation may not be applicable to the harder (more elastic) materials since only the relatively more ductile materials were used to derive equation (3.5) (eqn (7) in Niihara *et al*. (1982)).

The trends of variation of the normalized critical SIF in figure 7 also suggest that there is a minimum crack length for RCs formed in more plastic materials. For example, when *E*/*σ*_{y}=120 the minimum crack length is 0.8*a*, and when *E*/*σ*_{y}=200, the RC has a minimum length of approximately 0.9*a*. Any crack length shorter than the minimum threshold is unstable, which would be perturbed and advanced by the peak stress located at about *l*/*a*=0.9 at the surface.

### (d) Comparison with experiments in literature

Equations (3.4) and (3.1) represent a theoretical framework (technique) for determining the fracture toughness via Vickers indentation for RC and HPC, respectively. A variety of experimental data are available in literature (Evans & Charles 1976; Lankford 1982; Niihara *et al*. 1982). In those experiments, the material's Young's modulus and hardness were reported, from which the yield stress can be derived using the relationship between *H*/*σ*_{y} and *E*/*σ*_{y} (Johnson 1985). For soda-lime glass, ZnS and a WC/Co alloy, their material properties are tabulated in table 2. In those experiments, the fracture toughness (critical SIF) was measured by the double torsion technique. Indentations with variable impression sizes (*a*) were applied and the surface crack lengths were measured as a function of *a*. Since during the indentation experiments it may be difficult to distinguish HPCs and RCs (especially for non-transparent materials), we use the *apparent* surface crack radius (*c*) as a uniform measure in this subsection; for HPCs, the definition of *c* is unchanged (figure 1), and the length of the RC becomes *l*=*c*−*a* (figure 1). Niihara (Niihara *et al*. 1982) proposed that RCs are likely to form when *l*/*a*<2.5 and HPCs are preferred when *c*/*a*>2.5. Thus, there is an overlapping region of the apparent surface crack length (2.5<*c*/*a*<3.5) where both cracks are possible.

For the three different materials, the experimental relationships between the normalized critical SIF and crack length are given as symbols in figure 8. By contrast, for a given material with properties specified in table 2, such relationship can be obtained from (3.4) or (3.1) for RCs or HPCs, respectively; the results are represented by lines in figure 8. The good agreement between them has validated the current approach. The only notable difference is that equation (3.4) seems to have underestimated the fracture toughness of soda-lime glass. Next, we examine the proposed technique via parallel experiments on several brittle materials.

## 4. Experimental verification and discussion

Three materials (table 3) are chosen for the experiment: soda-lime glass; Si_{3}N_{4}; and TiB_{2}. Their elastic modulus and yield strength are measured from established indentation techniques (Chen *et al*. 2007), which are also close to the values reported in literature. In addition, the fracture toughness *K*_{C} of these materials is measured from the average of standard Chevron notch tests (Yonezu *et al*. 2002), denoted as the reference fracture toughness.

For each material, different indentation loads are used to indent the material with the Vickers pyramidal indenter (the loading rate is 0.02 N s^{−1}), resulting in different sized indents and lengths of surface cracks. For each load value, multiple tests (approx. 5–10) are carried out, leading to a scatter of the measured impression size (*a*) and apparent surface crack length. The crack lengths are measured by laser microscopy; after unloading in each indentation experiment, the crack lengths in all four directions are averaged and reported. An example of the indentation-induced surface crack is shown in the insert of figure 9, for Si_{3}N_{4} (with a maximum load of 490 N). Similar to §3*d*, we do not distinguish the RC from the HPC in the measurement of the apparent surface crack length. In figure 9, the normalized apparent surface crack length *c*/*a* is plotted as a function of *P*/*c* (*P* is the maximum indentation load); both *c* and *a* are averaged values measured from a given indent load and the error bar denotes the data scatter. For HPCs, the crack length is *c*, and for RCs, the crack length should be *l*=*c*−*a* in the current framework.

For each indentation test, assuming the RC is formed, the measured crack length and impression size are substituted in equation (3.4) along with relevant material properties (table 3), from which *K*_{C} is derived. The results are given in figure 10 as solid symbols, where the error bar corresponds to the multiple tests carried out at each given peak indentation load. The process is repeated assuming an HPC is formed (where equation (3.1) is used), and the identified critical SIF is given as open symbols in figure 10.

The comparison with the reference values of *K*_{C} obtained from notch tests (table 3, given as lines in figure 10) shows that, for soda-lime glass, RCs are probably formed at small loads (where *c*/*a*=1.9); at higher loads, HPCs are formed (where *c*/*a*=4.3). For TiB_{2}, at small loads, the results of RCs are closer to the reference *K*_{C} (where *c*/*a*=1.6), and at higher loads, the results of HPCs are closer to the reference fracture toughness (where *c*/*a*=3.9). For Si_{3}N_{4}, it seems HPCs are dominant, and the corresponding crack length range, *c*/*a*=2.1–2.6, is close to the empirical RC–HPC transition range (when *c*/*a* is approx. 2.5) found by Niihara *et al*. (1982).

In all cases, the identified fracture toughness from Vickers indentation is reasonably close to the reference value, which validates the proposed theoretical framework. During practical application, we still recommend the empirical criterion proposed by Niihara (Niihara *et al*. 1982) that if the crack is relatively long, or when *c*/*a* is larger than approximately 2–2.5, the HPC formulation should be favoured (equation (3.1)); otherwise the crack is probably radial, and the RC equation (3.4) can be employed.

We caution that the present analysis assumes no delamination (or lateral cracking). In practice, sometimes longer cracks are desired for easier observation and measurement, therefore, larger indentation load may be applied and an HPC is likely to occur. However, that may also induce lateral cracking, which will cause delamination of surface material. Such an event would not only make it difficult to observe HPCs, but also the proposed theory cannot be directly applied. In addition, when the current method is applied to brittle coatings, usually a shallow indent is desired so as to avoid the substrate effect (which is also not accounted for in the present paper), where RCs may dominate. If shallow indentation makes it difficult to observe cracks accurately and deeper penetrations are needed, the present framework also needs to be modified to incorporate the substrate effect, by following the framework we proposed elsewhere (Zhao *et al*. 2006*a*, 2007).

## 5. Conclusion

Indentation cracking is an effective way of measuring the material fracture toughness, and the RC and HPC are the most common yet independent crack systems. The analysis of these crack mechanics needs to be coupled with indentation stress field, which strongly depends on material elastoplastic properties.

In this paper, by using three-dimensional finite-element analysis, the indentation stress field is obtained and used to compute the SIF of RCs and HPCs. Explicit relationships (equations (3.4) and (3.1) for RCs and HPCs, respectively) are established between the critical SIF (fracture toughness), crack geometry, indentation size and material properties. The formulations are validated via experimental data in literature, and it can be applied to a wide range of materials. Furthermore, we carry out Vickers indentation experiments on three different materials, and by measuring the surface crack length the material fracture toughness is derived, which agrees well with that obtained from the standard notch test. It is found that for a given material, RCs are preferred at lower loads, and HPCs are more likely at higher indentation depths.

Unlike the previous studies of the evaluation fracture toughness from indentation cracking experiment, where empirical fitting formulae were established based on approximate analytical solutions, the studies in this paper are based on extensive three-dimensional finite-element analyses of the indentation and fracture problems, which may improve the accuracy. The proposed is expected to be applicable to a wide range of brittle materials, through which the fracture toughness can be easily extracted upon Vickers indentation experiment.

## Acknowledgments

The work of A.Y. was supported in part by Grant-in-Aid for Young Scientist of (B) (no. 19760075) of the Ministry of Education, Culture, Sports, Science and Technology, Japan, and Research Grant (General research for Electricity & Energy) of TEPCO research foundation. The work of X.C. was supported in part by the National Science Foundation CMMI-0407743 and CMMI-CAREER-0643726.

## Footnotes

- Received February 25, 2008.
- Accepted May 20, 2008.

- © 2008 The Royal Society