Compressive Surface Strengthening of Brittle Materials by a Residual Stress Distribution

A theoretical approach is presented for predicting the strengthening of brittle materials subjected to a residual stress distribution represented by a polynomial series. In the approach, the stress intensity factor for a surface crack is derived incorporating the effect of crack closure. The crack-closure distance is then calculated using an approximate approach which allows the strengthening due to the residual stresses to be estimated. Illustrating the approach using residual stresses typical of tempering, it was found the approach agreed well with previous work. The influence of partial crack closure was found to give higher values of the stress intensity factor than would be calculated if the crack were assumed to be open. This effect decreases the amount of strengthening predicted and gives a wide range of conditions for which subcritical crack-growth processes can occur. For the example of tempering it was also found that these are conditions when weakening or spontaneous failure of the body can occur.     

Determination of Surface Residual Stresses in Brittle Materials by Hertzian Indentation: Theory and Experiment

Hertzian indentation has been used to determine the surface residual stress levels in brittle materials. In this method, a hard sphere is pressed into the surface of the material: at a critical load a preexisting surface-breaking crack in the neighborhood of the contact will propagate. There is a threshold load below which no such crack, of whatever size, can be propagated. The presence of a residual stress in the surface will lead to a shift in this threshold load. The effects of residual stresses on the minimum load to produce Hertzian fracture are predicted for alumina and glass, assuming that the variation of the residual stress over the length of the crack is small. Two methods of analysis (one approximate, one more general) are presented that enable the residual stress to be calculated from the shift in threshold load; the only further information required is a knowledge of the radius of the sphere, the elastic constants of the sphere and substrate, and also the fracture toughness of the substrate (or use of a stress-free specimen as a reference). No measurement of any crack length is necessary. Experimental results are presented for the residual stress levels determined in glass strengthened by ion exchange. Indenting balls of a variety of materials with a range of elastic mismatch to the glass substrate were used, so as to evaluate the effects of elastic mismatch and interfacial frictional tractions on the results obtained. The results obtained by Hertzian indentation are consistent with residual stress levels determined by differential surface refractometry. We also present results on alumina specimens with induced surface stresses.     

On Compressive Brittle Fragmentation

Dynamic brittle fragmentation is typically described using analytical and computational approaches for tensile stress‐states. However, most fragmentation applications (e.g., impact, blast) involve very large initial compressive stresses and deformations. In this study, the compressive fragmentation of brittle materials is investigated experimentally across a range of materials: silicon carbide, boron carbide, spinel, basalt and a stony meteorite. Analysis of our experimental results suggests that there exists two different regimes in the fragment size distributions, based on two brittle fragmentation mechanisms. The first is a mechanism that produces larger fragments and is associated with the structural failure of the sample being tested. This mechanism is influenced by the loading conditions (rate, stress state) and sample geometry. The second fragmentation mechanism produces comparatively smaller fragments and arises from the coalescence of fractures initiating and coalescence between defects in regions of large stresses and contact forces (e.g., between two fractured surfaces from the larger fragments). A framework is developed for comparing experimental compressive fragmentation results with tensile fragmentation theories. The compressive experimental results are shown to be adequately described by the theories using the new framework.     

Damage-Resistant SiC–AlN Layered Composites with Surface Compressive Stresses

Three-layer samples in the SiC-AlN system with outer layers richer in SiC, as well as monolithic samples of uniform composition, were fabricated by hot-pressing. The strain gage technique, previously described, was used to estimate residual compressive stress in the outer layers developed because of the difference in coefficients of thermal expansion between inner and outer layers. Bar-shaped samples were indented using a Vickers indenter under loads as high as 20 kg (196 N) on outer layers and were fractured in three-point bending. Three-layer samples with outer layers under compression exhibited increased resistance to the contact-induced damage in comparison to the monoclinic samples.     

A Statistical, Micromechanical Theory of the Compressive Strength of Brittle Materials

A general theory of the compressive strength of brittle materials is presented. This theory proposes that failure is brought about by structural weakening from accumulated crack damage which increases with the stress level. The statistics of the flaw distribution and the mechanism of crack initiation and extension are important. A sample calculation using the theory is given to demonstrate its application.     

Statistical Determination of Surface Flaw Density in Brittle Materials

The density of surface flaws in brittle materials can be determined from the loads in many different kinds of tests. A general method is derived and applied to indentation experiments on crown, plate, window, lead, and commercial heat-resistant and heat-absorbing glasses. Knowledge of flaw density is important for estimating size effect, and differences in flaw density indicate the possibility of significant improvements in strength by eliminating a few major sources of flaws.     

Compressive Surface Stresses Developed in Ceramics by an Oxidation-Induced Phase Change

Silicon nitride/zirconium oxide hot-pressed materials contain ZrO₂-_{2 x} N_{4 x /3} (0.25≤ x ≤0.43), which readily oxidizes at ≥ 500°C to monoclinic ZrO₂. The molar volume increase (∼4 to 5%) of this reaction was used to develop compressive surface stresses. The development of these useful surface stresses was demonstrated with an indentation technique used to measure the apparent critical stress intensity factor ( K_a ) on the surfaces of materials fabricated with 5 to 30 vol% ZrO₂ and subsequently oxidized at 600° to 800°C. The increase in K_a was directly related to the oxidation kinetics and the initial volume content of the unstable phase.     

Development of Surface Compressive Stresses in Zirconia–Alumina Composites by an Ion-Exchange Process

A two-step ion-exchange technique was developed for introducing compressive stresses on the surface of ZrO₂–Al₂O₃ composites. In the first step, a thin layer (∼250 μm) of Na-β"-Al₂O₃ was formed on the surface of the composite by a vapor-phase process at ∼1400°C. In the second step, Na⁺ ions were replaced by K⁺ ions by a heat treatment at ∼385°C for 2 h in a molten KNO₃ bath. Replacement of sodium by potassium led to the creation of surface compressive stresses. The flexural strength and Weibull modulus of ZrO₂–Al₂O₃ composite were ∼915 MPa and 10, respectively, for the as-sintered samples. By contrast, the flexural strength and Weibull modulus were ∼1140 MPa and 26, respectively, for the ion-exchanged samples. A residual surface compressive stress of ∼480 MPa was measured by a strain-gauge technique in K⁺-ion-exchanged samples. The presence of surface compressive stresses also was confirmed using an indentation technique. The technique developed here can be used to introduce compressive stresses on components of virtually any shape.     

Effect of Residual Surface Stress on the Strenght Distribution of Brittle Materials

The introduction of residual surface stresses into a material can change the strength distribution simply due to variability in the stress intensity factor of surface cracks. This variability arises from the distribution in surface (critical) crack lengths. A fracture mechanics approach was used to determine the influence of surface compression on the average strength and the standard deviation of the strength distribution. The strengths of the stress-free samples were assumed to fit a twoparameter Weibull distribution, and the increases in strength resulting from the surface compression were determined using fracture mechanics. It was determined from the analysis that the standard deviation/average strength ratio (coefficient of variation) initially increases, passes through a maximum, and ultimately reaches a plateau value below the initial value, as one increases the depth of surface compression. The maximum increase in the strength scatter occurs when the depth of the surface compression is approximately equal to the characteristic crack size. These changes were found to be dependent on the magnitude of the surface stress, as well as the characteristic strength, Weibull modulus, and fracture toughness of the stress-free material.     

A Technique for Introducing Surface Compression into Zirconia Ceramics

A technique is presented whereby compressive surface stresses are introduced into transformation-toughened ZrO₂ ceramics by removing the stabilizing oxides, such as Y₂O₃, from the surface. The advantages of this approach over other techniques are discussed, and experimental data confirming the presence of compressive surface stresses and subsequent improved resistance to indentation cracking are described .     

Crack-Growth Resistance of Microcracking Brittle Materials

A mechanics model of microcrack toughening is presented. The model predicts the magnitude of microcrack toughening as well as the existence of R -curve effects. The toughening is predicated on both the elastic modulus diminution in the microcrack process zone and the dilatation induced by microcracking. The modulus effect is relatively small and process-zone-size-independent. The dilatational effect is potentially more substantial, as well as being the primary source of the R curve. The dilatational contribution is also zone-size-dependent. The analysis demonstrates that microcrack toughening is less potent than transformation toughening.     

Mechanics of Transformation-Toughening in Brittle Materials

Particles which undergo a stress-induced martensitic transformation are known to toughen certain brittle materials. The enhanced toughness can be considered to originate from the residual strain fields which develop following transformation and tend to limit the crack opening. The increased toughness can estimated from the crack-tip stress-intensity change induced by the transformation of a volume of material near the crack tip. It is found that the initial zone, prior to crackgrowth, provides no change in stress intensity. As the crack grows, the zone (associated with a positive transformation strain) induces a stress-intensity reduction that rises to a maximum level after some crack propagation. The influence of particle-size distribution on the stress-intensity reduction is also discussed.     

Sandwiched-Beam Procedure for Precracking Brittle Materials

A technique for precracking brittle materials is presented. This procedure, which is called the sandwiched-beam (SB) technique, allows the production of sharp through-thickness cracks with predetermined length in specimens with a rectangular section. A bar, in which an initial notch is produced by using a conventional saw, is inserted between two supporting beams and the sandwich assembly is loaded in three-point bending. Conditions can be defined that allow the stable propagation of a sharp flaw from the notch as the applied load is increased. Then, the cracked bar can be used to determine the fracture toughness. The SB technique is applied to different brittle materials, including soda-lime-silica glass, alumina, Si₃N₄, a SiC _w -Si₃N₄ composite, graphite, a Ti-Al intermetallic, and Carrara marble.     

Fracture of Brittle Solids in the Presence of Thermoelastic Stresses

A micromechanical model for strength behavior as a function of grain size in two-phase materials with thermal expansion mismatch and in single-phase materials with thermal expansion anisotropy is presented. The strength vs grain-size plot is interpreted in terms of internal stresses and the ratio of grain to flaw size. The strength of thermally isotropic material is predicted to exhibit a weak grain-size dependence with a negative grain-size exponent normally different from 0.5. In thermally anisotropic poly crystalline solids, before the critical grain size for spontaneous cracking is reached, there will be a region of decreasing strength with increasing grain size due to an increase in the grain-to-flaw size ratio. When a critical grain size is reached, the ratio of grain to flaw size will decrease instantaneously and the strength will decrease in the same fashion. Very good agreement was obtained between predicted and experimentally observed strength behavior for TiO₂ and MgO ceramics.     

Crack-Tip-Bridging Stresses in Ceramic Materials

A simple approach is presented for the determination of crack-tip-bridging stresses in ceramic materials. The technique utilizes the difference in compliance predicted for a crack of known size and that measured. The bridging stresses for an alumina material that displays considerable R -curve behavior are calculated from measured compliance and crack-length observations. The results are almost identical with those recently reported in the literature. Additional measurements on a more complex "duplex" ceramic that exhibits a very substantial R -curve due to a crack-branching chain reaction are also presented.     

Uniaxial Strength Behavior of Brittle Cellular Materials

Two types of brittle reticulated materials were evaluated under uniaxial tensile and compressive loading and analyzed in terms of the Gibson and Ashby model for brittle open-cell solids. The samples consisted of an open-cell alumina–mullite material which was tested as a function of density at a constant cell size and a reticulated vitreous carbon tested at one density and two cell sizes. The samples were mounted such that only the loading direction was varied in the tests. A combination of video photography and acoustic emission was critical to interpreting the results. The model assumes that identical deformation modes, bending failure of the struts, are responsible for failure of the bulk foam in tension and compression. The results of this work indicate a significant difference between the density dependence in tension and compression. Tensile failure in both materials appeared to be characterized by the catastrophic propagation of a single crack. Compressive failure was significantly different between the alumina and glassy carbon foams. The alumina foam failed by a damage accumulation process, whereas the carbon foam failed by the catastrophic collapse of a band of cells perpendicular to the loading direction.     

Statistics of Flaw Interaction in Brittle Materials

The problem of elastic interaction of an array of microcracks is considered. The solution is based on an efficient surface integral method using distribution of edge dislocations to represent the microcracks in the mathematical model. From the analysis of the interaction between two identical cracks, it is seen that interaction is most likely to produce a slight enhancement in the stress intensity factor. By performing computer experiments on a random array of microcracks, the effect of crack interaction is studied statistically as a function of the density of the microcracks. The significance of the interaction effects for fracture in ceramics is discussed.