Determination of the R-Curve from the Strength—Indentation Load Relation

Strength as a function of indentation load has been measured to evaluate toughness versus crack size curves (i.e., R -curves) of ceramics. In the past, this evaluation is achieved by generating a family of global stress intensity versus crack size curves (i.e., K -curves) at each indentation load and its corresponding strength, and then fitting an envelope of tangency points to these curves. This envelope yields the R -curve. A methodology is proposed in the present study to define the point in the K -curve at which the R -curve intersects. This methodology complements the conventional "tangency condition" approach in determining the R -curve from the experimental strength-indentation load relation.     

R-curves of lead zirconate titanate (PZT)

R-curves were measured for ferroelectric ceramic lead zirconate titanate (PZT) using surface cracks in flexure (SCF) in a single composition of unpoled, ferroelastic PZT. The effects of several parameters on the R-curves were experimentally determined. These included grain size, indentation load, polishing away the residual stress zone associated with the Knoop indentation, and thermal depolarization after indentation. The larger grain size resulted in a higher plateau value of the R-curve, a result consistent with the larger amount of ferroelastic switching observed in the stress/strain curve. Increasing the indentation load from 10 to 50 N resulted in larger initial crack sizes. This had some effect on the early part of the R-curve, but did not much affect the plateau value. Polishing away the residual stress zone eliminated the residual stress contribution of the Knoop indentation to the stress intensity factor. This resulted in the most accurate measurement of the intrinsic toughness (0.4 MPa m^1/2). Thermal depolarization to remove any potential ferroelastic crack tip switching zone associated with the indentation had little or no effect on the measured R-curves.     

R-Curve Behavior and Strength for In-Situ Reinforced Silicon Nitrides with Different Microstructures

R -curves for two in-situ reinforced silicon nitrides A and B of nominally the same composition are characterized using the Griffith equation and indentation fracture mechanics. These R -curves are calibrated against fine-grained silicon nitrides which have a known chevron-notch (long-crack) toughness and with a nearly flat R -curve behavior. Silicon nitride A, with its coarser microstructure and higher chevron-notch toughness, shows lower resitance to crack growth than silicon nitride B if the crack size is less than ∼200 μm. These results are consistent with the indentation–Strength measurements which show a crossover of strength between the two materials at an indentation load between 49 and 98 N, and below the crossover A has a lower strength. The toughening behavior is explained using an elastic-bridging model for the short crack, and a pullout model for the long crack. The effects of R -curve properties on design are discussed.     

A Critical Evaluation of Indentation Techniques for Measuring Fracture Toughness: I, Direct Crack Measurements

The application of indentation techniques to the evaluation of fracture toughness is examined critically, in two parts. In this first part, attention is focused on an approach which involves direct measurement of Vickers-produced radial cracks as a function of indentation load. A theoretical basis for the method is first established, in terms of elastic/plastic indentation fracture mechanics. It is thereby asserted that the key to the radial crack response lies in the residual component of the contact field. This residual term has important implications concerning the crack evolution, including the possibility of post indentation slow growth under environment-sensitive conditions. Fractographic observations of cracks in selected "reference" materials are used to determine the magnitude of this effect and to investigate other potential complications associated with departures from ideal indentation fracture behavior. The data from these observations provide a convenient calibration of the Indentation toughness equations for general application to other well-behaved ceramics. The technique is uniquely simple in procedure and economic in its use of material.     

Indentation Studies on Y2O3-Stabilized ZrO2: II, Toughness Determination from Stable Growth of Indentation-Induced Cracks

Stable indentation cracks were grown in four-point bend tests to study the fracture toughness of two Y₂O₃-stabilized ZrO₂ ceramics containing 3 and 4 mol% Y₂O₃. By combining microscopic in situ stable crack growth observations at discrete stresses with crack profile measurements, the dependence of toughness on crack extension was determined from crack extension plots, which graphically separate the crack driving residual stress intensity and applied stress intensity factors. Both materials exhibit steeply rising R -curves, with a plateau toughness of 4.5 and 3.1 Mpa·m^1/2 for the 3- and 4-mol% materials, respectively. The magnitude of the plateau toughness reflects the fraction of tetragonal grains contributing to transformation toughening.     

Rising Crack-Growth-Resistance (R-Curve) Behavior of Toughened Alumina and Silicon Nitride

R -curves for a sinter/HIPed SiC(whisker)-reinforced alumina and a sintered silicon nitride were assessed by direct measurements of lengths of cracks associated with Vickers indentation flaws. The fracture toughness measurements based on (a) initial (as-indented) crack lengths, (b) equilibrium growth of cracks during increasing far-field loading, and (c) crack lengths corresponding to unstable fracture showed definitive trends of R -curves for both materials. The fracture mechanics analyses employed an indenter-material constant that was independently estimated using a physical model for the residual driving force and a free surface correction factor that accounted for the effects of size and shape of the cracks on stress intensity. It is shown that R -curve estimations based on crack length measurements have the intrinsic advantage that crack length dependence of fracture toughness is not assumed a priori as is done in conventional analysis based on strength. The measured fracture toughness of SiC(whisker)-reinforced alumina was in agreement with the prediction of a toughening model based on crack bridging by partially debonded whiskers.     

Flat R-Curve from Stable Propagation of Indentation Cracks in Coarse-Grained Alumina

The fracture toughness of coarse-grained A1₂O₃, known for pronounced "Iong"-crack R-curve behavior, was studied in the "short"-crack regime utilizing the stable propagation of indentation cracks in bending. A combination of in situ microscopic crack growth observations and mechanical testing enabled measurement of crack extension curves. They reflect the contributions of residual indentation stress intensity and applied bending stress intensity on the total crack driving stress intensity and allow determination of the residual stress factor χ and the toughness K_R. The results indicate that χ depends on indentation load and A_R is surprisingly constant rather than increasing. To resolve the latter contradiction with long-crack R-curve behavior, combined short/long-crack fracture tests were performed with the same specimens. Starting with stable indentation crack growth and continuing with stable long-crack extension, the previous toughness results were confirmed, i.e., constant toughness from indentation cracks and increasing toughness from long cracks. The influence of crack-opening behavior on bridging-controlled R-curve toughening can qualitatively explain the observed discrepancies.     

Ferroelastic domain switching and R-curve behavior in lead zirconate titanate (Zr/Ti = 52/48)-based ferroelectric ceramics

In this work, ferroelastic domain switching and R-curve behavior in lead zirconate titanate (Nb/Ce co-doped Pb(Zr_0.52Ti_0.48)O₃, ab. PZT-NC)-based ferroelectric ceramics were investigated, using the indentation-strength-in-bending (ISB) method. Firstly, Vickers indentation test examined the notable fracture anisotropy of PZT-NC ceramics between the poling direction and its perpendicular direction, and the crack open displacement (COD) profiles in the two directions were also theoretically calculated from the indentation fracture mechanics. And then two kinds of ferroelastic domain switching modes (in-plane and out-of-plane) were used for explaining such anisotropic propagation behavior of indentation cracks. The subsequent three-point bending test illustrated the dependence of fracture strength on indentation load and the rising crack growth resistance curves (R-curves) in two directions. The resulted R-curves were fitted by the Hill's type Growth Function successfully, giving the reasonable values of crack extension exponential (n), plateau fracture toughness (K_max), and initial fracture toughness (K_ini). The in-plane ferroelastic domain switching was identified as a more significant toughening mechanism for PZT-NC ceramics than the out-of-plane switching due to more switchable domains.     

Measurement of Crack Tip Toughness in Alumina as a Function of Grain Size

Crack profile measurements near the crack tip in the SEM were used to measure crack tip toughness of alumina as a function of grain size (average grain size 0.9–16 μm). For comparative tests, two crack configurations were included in the present study: straight cracks (CT specimen) loaded with an in situ device; and radial indentation cracks. The measured crack tip toughness values were independent of crack geometry, and no grain size dependence could be discerned. A mean crack tip toughness of 2.3 MPam^1/2 was evaluated. The crack tip toughness determined from crack profile measurements is significantly lower than the toughness evaluated with conventional indentation techniques (e.g., indentation strength bending).     

Toughness-Curve Behavior of an Alumina-Mullite Composite

Toughness-curve ( T - or R -curve) behavior of a composite of 30 vol%, polycrystalline, coarse-grained, spherical alumina agglomerates dispersed throughout a fine-grained, 50/50 vol% alumina-mullite matrix, and that of its microstructural end-members (100% matrix and 100% alumina), were studied using the indentation-strength-in-bending technique. T -curves were deconvoluted from indentation-strength data using an indentation fracture mechanics model. The monolithic matrix and alumina exhibited an invariant toughness and a moderate T -curve, respectively. In comparison, the composite exhibited a pronounced T -curve. The T -curve of the composite is best explained as deriving from the interaction of a propagating crack with the alumina agglomerates: crack propagation experiments revealing two possible toughening mechanisms-intra-agglomerate frictional grain bridging and elastic bridging ligaments in the matrix that appeared to be associated with alumina agglomerates. Rule-of-mixtures toughness calculations indicated that intra-agglomerate bridges could account for only a fraction of the toughening exhibited by the composite. It is suggested that the extra toughening arises from the elastic bridging ligaments.     

Flat and Rising R-Curves for Elliptical Surface Cracks from Indentation and Superposed Flexure

Flat and rising R -curves, fracture resistance versus crack extension, were determined for a sintered 99%α-silicon carbide and for a hot-pressed composite of 25 wt% silicon carbide whiskers and alumina, respectively. The R -curves were evaluated from a combination of measured crack lengths, which were produced over a range of Vickers indentation loading, and of measured strengths, which were correlated either with the indentation flaws or with the most severe natural flaws on flexure specimens. A published analysis of the stress-intensity factor for a surface crack in flexure was interpreted to show that the crack front takes the form of a semiellipse where both the ratio of the minor to major radii and the configuration coefficient itself decrease with increasing crack extension. A power-law function of the indentation load was fitted to the product of an effective configuration coefficient and the flexural strength to evaluate the R -curves. When the configuration coefficient is assumed constant, a customary practice, the R-curves appear to have steeper rises. The assumed constancy of the coefficient of the indentation driving force may also have an effect on R-curves, but the effect would be much less.     

Methodology for measuring fracture toughness of ceramic materials by instrumented indentation test with vickers indenter

Virtual crack closure technique and elastoplastic finite element method were employed to calculate the stress intensity factors (SIF) of ceramic materials on the tip of both half‐penny crack (HPC) and radial crack (RC) induced by Vickers indenter and the value of fracture toughness (K_IC) was extracted by the design of equi‐SIF contour of HPC and RC crack front. Through dimensional theorem and regressive analysis, a functional relationship between instrumented indentation parameters, crack length of Vickers impression and fracture toughness of ceramic materials was established, thus a novel methodology has been presented for measuring fracture toughness of ceramic materials by instrumented Vickers indentation. Both numerical analysis and experiments have indicated that this methodology enjoys higher measurement precision compared with other available indentation methods. The methodology is universally suitable for HPC, RC as well as transition cracks and capable of determining fracture toughness and elastic modulus in a single indentation test. In addition, it saves the effort of measuring the diagonal length of Vickers impression in case that the impression remains unclear.     

Effect of Temperature on Toughness Curves in Alumina

The effect of temperature on fracture strength and toughness curves ( T -curves) in the short-crack region of a polycrystalline alumina was studied. The indentation-strength technique was used to measure strength and T -curve behavior in the temperature range of 25° to 1300°C. Grain-localized crack bridging resulted in improved flaw tolerance and rising T -curves in this alumina. Both strength and toughness were observed to decrease with increasing temperature. A theoretical grain-bridging model was used to calculate the T -curves and fit to the experimental data. This allowed the evaluation of the temperature dependence of important bridging parameters.     

Crack-Growth Resistance of in Situ-Toughened Silicon Nitride

The fracture toughness of a commercial, hot-pressed, in situ -toughened silicon nitride with an elongated grain structure is determined by four different testing methods. The fracture toughness is found to be 5.76 ± 0.27, 8.48 ± 0.50, 10.16 ± 0.66, and 10.68 ± 0.39 Mpa.m^1/2, respectively, by indentation crack size measurement, indentation strength, single-edge-precracked-beam, and chevron-notched-beam methods. The discrepancy in fracture toughness between the testing methods is related to R -curve behavior, as measured using the indentation strength technique. These results indicate that there is no unique fracture toughness value and that a fracture toughness testing method with appropriate qualifiers is needed for rising R -curve materials. Therefore, care should be taken in interpreting and utilizing fracture toughness values evaluated from different testing methods if a material exhibits a rising R -curve. Complete characterization of the R -curve may be a prerequisite.     

Rising Fracture Toughness from the Bending Strength of Indented Alumina Beams

The analytical function of crack extension to a fractional power is used to represent the fracture resistance of a vitreous-bonded 96% alumina ceramic. A varying flaw size, controlled by Vickers indentation loading between 3 and 300 N, was placed on the prospective tensile surfaces of four-point bend specimens, previously polished and annealed. The lengths of surface cracks were measured by optical microscopy. Straight lines were fitted to the logarithmic functions of observed bending strength versus indentation load in two series of experiments: (I) including the residual stress due to indentation and (II) having the residual stress annealed out at an elevated temperature. Within the precision of measurement these lines have the same slope, being about 32% less than the -1/3 slope which a fracture toughness independent of crack extension would indicate. Considering the criteria for crack extension and specimen failure, the fracture mechanics equations were solved for the conditions of the two series of experiments. Approximately the same values of fracture toughness, rising as a function of indentation flaw size, were obtained from both series of experiments.     

Effect of Indenter Geometry on Controlled-Surface-Flaw Fracture Toughness

Fracture toughness values obtained using both Knoop and Vickers-indentation-produced controlled surface flaws were compared as a function of indentation load for a well-characterized glass-ceramic material. At the same indentation load, Knoop cracks were larger than Vickers. As-indented K_c values calculated from fracture mechanics expressions for surface flaws were higher for Knoop flaws than Vickers, but both types gave low K_c values due to indentation residual stress effects. Analysis suggested that theoretical formalisms for indentation residual stress effects based on fracture mechanics solutions for a center-loaded penny crack in an infinite medium should apply to both indentation types. K_c values calculated using the residual stress approach were identical for Knoop and Vickers controlled surface flaws when a "calibration" value for a constant term in the expression for K_c was used for both indentation types.     

Influence of Surface Stress on indentation Cracking

A simple, two-dimensional fracture mechanics analysis was used to determine the influence of nonuniform residual surface stresses on the formation of radial indentation cracks. The indentation behavior depends on the depth of the compressive stresses, such that the apparent fracture toughness passes through a maximum with increasing indentation load. The analysis was used to estimate the surface stress from indentation data for a zirconia-toughened ceramic and was compared to previous X-ray diffraction measurements of this stress. The comparison gives only fair agreement; the sources of possible error are discussed. Such surface stresses also influence the accuracy of K _{I C} measurements when an indentation crack length technique is used; surface preparation is a critical factor in the measurement. Finally, the K _{I C} values obtained from indentation crack sizes were compared with those obtained by the double-cantilever-beam technique.     

Measurement of Indentation Residual Stresses in Materials Susceptible to Slow Crack Growth

Knowledge of the size and distribution of the indentation residual stress field is important when interpreting slow crack growth data for indented ceramic materials. A technique based on compressively loading indentation cracks has been used to measure the wedging residual stresses at radial indentation cracks. The method also gives information on the fatigue limit and can be applied on any ceramic material susceptible to slow crack growth. Soda–lime glass specimens were indented and the resulting residual stresses, wedging the radial cracks, were measured as a function of indentation load. Calculations of K ₀, the fatigue limit, were made for both virgin indentation cracks and cracks aged until saturation. The magnitude of closing stress needed to prevent slow crack growth was found to depend linearly on the indentation load. For example, for indentation loads of 20 and 60 N, the corresponding closing stresses were 14 and 26 MPa, respectively.     

Alumina Agglomerate Effects on Toughness-Curve Behavior of Alumina–Mullite Composites

Toughness-curve ( T -curve) behavior of composites of spherical, polycrystalline, coarse-grained, alumina agglomerates dispersed throughout a constant-toughness, fine-grained, 50–50 vol% alumina–mullite matrix has been evaluated as a function of agglomerate content for the range 15 to 45 vol%. T -curve behavior was evaluated using the indentation-strength method. Increasing alumina agglomerate content resulted in a progressive increase of large indentation load strengths with negligible change of plateau strength levels at small indentation loads. This behavior is consistent with underlying T -curves that rise to greater values and are shifted toward longer crack lengths with increasing agglomerate content, suggesting that both bridge spacing and bridge potency increase with increasing agglomerate content over the range tested. The proposed relationships between bridge spacing and agglomerate content, and bridge potency and agglomerate content, are rationalized in terms of residual stress considerations. The indentation-strength data also demonstrated that the composite containing the greatest alumina agglomerate content, 45 vol%, exhibited the greatest flaw tolerance.     

Statistical investigation of the influence of maximum shear stresses induced by thermal treatments on fracture toughness of soda-lime silica glasses: Comparison between Hertzian and Vickers indentations

Hertzian and Vickers indentation tests have been performed to estimate the hardness and the fracture toughness of a soda-lime silica glass fabricated by the float process. A comparison between as-prepared glass, annealed glass (90 min at 680°C), and tempered glass (quenched from 660°C to 25°C) has been conducted to investigate the influence of thermal treatments on fracture toughness. In this study, a new method based on acoustic emission, recorded during Hertzian indentation tests, has been used in order to determine precisely the minimum load for fracture of these glasses having various thermal histories. Experimental results have shown the existence of a threshold load below which no crack can be propagated in glass. These critical loads have been used to determine Weibull’s fracture laws as a function of surface quality and maximum shear stresses. It has been also shown that the presence of residual stresses induced by quenching leads to a shift of this threshold load and modifies Weibull’s laws. Therefore, this method, which requires no measurement of any crack length, can be used to accurately estimate residual stresses induced by quenching in soda-lime silicate glasses.