Brian R. Lawn

Fracture of brittle solids

This is an advanced text for higher degree materials science students and researchers concerned with the strength of highly brittle covalent–ionic solids, principally ceramics. It is a reconstructed and greatly expanded edition of a book first published in 1975. The book presents a unified continuum, microstructural and atomistic treatment of modern day fracture mechanics from a materials perspective. Particular attention is directed to the basic elements of bonding and microstructure that govern the intrinsic toughness of ceramics. These elements hold the key to the future of ceramics as high-technology materials - to make brittle solids strong, we must first understand what makes them weak. The underlying theme of the book is the fundamental Griffith energy-balance concept of crack propagation. The early chapters develop fracture mechanics from the traditional continuum perspective, with attention to linear and nonlinear crack-tip fields, equilibrium and non-equilibrium crack states. It then describes the atomic structure of sharp cracks, the topical subject of crack-microstructure interactions in ceramics, with special focus on the concepts of crack-tip shielding and crack-resistance curves, and finally deals with indentation fracture, flaws, and structural reliability.

Indentation fracture: principles and applications

The basic principles and practical applications of indentation fracture are reviewed.

Equilibrium penny-like cracks in indentation fracture

A study is made of the mechanics of two basic types of indentation fracture, cone cracks (“blunt” indenters) and median cracks (“sharp” indenters). The common feature which forms the central theme in this work is that both crack types, in their well-developed stages of growth, may be regarded as essentially “penny-like”. On this basis a universal similarity relation is derived for equilibrium crack dimension as a function of indentation load. Experimental measurements confirm the general form of this relation. A more detailed fracture mechanics analysis is then given, to account for additional, contact variables evident in the data. Notwithstanding certain analytical limitations, the study serves as a useful basis for investigating a wide range of contact-related problems, both fundamental and applied, in brittle solids.

Microfracture beneath point indentations in brittle solids

The microfracture patterns observed around point indentations in brittle solids are investigated. A description is first given of the stress field in an elastic half-space loaded normally at a point in its surface. This field is then used as a basis for analysing the crack geometry. A localized zone of irreversible deformation forms about the contact point, thereby removing a singularity in the elasticity solutions and providing nucleation centres for the ensuing microcracks. Generally, two main types of ‘vent’ cracks are observed to propagate from the deformation zone: median vents, formed during indenter loading, spread downward below the point of contact on planes of symmetry, and lateral vents, formed during unloading, spread sideways toward the specimen surface. Of these, the median vent is relatively well-behaved, and is amenable to standard fracture-mechanics analysis. From such an analysis we derive the means for predetermining, in principle, the depth of fracture damage under given point loading conditions. The significance of the results in relation to important practical applications, such as glass cutting and surface fragmentation processes, is discussed.

A model for crack initiation in elastic/plastic indentation fields

A model is proposed for the initiation of microfracture beneath sharp indenters. Using a simple approximation for the tensile stress distribution in the elastic/plastic indentation field, in conjunction with the principle of geometrical similarity, fracture mechanics procedures are applied to determine critical conditions for the growth of penny-like “median cracks” from sub-surface flaws. The analysis provides a functional relationship between the size of the critical flaw and the indentation load necessary to make this flaw extend. Initiation is well defined (unstable) only if the critical flaw lies within a certain size range; outside this range, large flaws can extend stably but small flaws can not extend at all. No flaws can extend below a characteristic minimum load, values of the indentation variables at this load accordingly providing useful threshold parameters. These quantities involve the intrinsic deformation/fracture parameters, hardness and toughness, in a fundamental way, thereby establishing a basis for materials selection in fracture-sensitive applications.

On the theory of Hertzian fracture

The fracture of a brittle solid under a spherical indenter is the best studied case of fracture in a strongly inhomogeneous, well defined, stress field. Two principal topics are discussed, the path of a crack in a field of non-uniformly directed stress, and the stability of cracks of various length when the prior stress on the crack path is non-uniform. For the first, it is shown that the crack growth should, to a first approximation, be orthogonal to the most tensile principal stress, and thus correspond, in a torsion-free stress field, to a surface delineated by the trajectories of the other two principal stresses: while, to a second approximation, the crack should deviate from this path by having a larger radius of curvature at every bend, thus exhibiting a pseudonertia even in slow growth. This is in accordance with the known experimental facts about the Hertzian crack, particularly the fact that the crack at the surface forms systematically outside the edge of the circle of contact, at which the maximum tensile stress occurs. On the second question, it is found that there are four crack lengths, c0, c1, c2, c3, corresponding to stationary values of energy. c0 and c2 represent unstable equilibria, and diminish with increasing load; cx and c3 represent stable equilibria and increase with increasing load. With small indenters, c0 soon becomes less than the size of pre-present surface flaws, and an unobserved shallow ring crack of depth c1 is produced: the critical condition for observed fracture is then the merging of c1 with c2, allowing unstable growth to the cone crack of depth c3. This explains Auerbach’s law, that the critical load for production of a cone crack is proportional to the radius, r, of the indenter sphere. With larger indenters, of several centimetres radius for a typical case, c1, and c2 merge and disappear before c0 exceeds the size of pre-present flaws. The critical load for cone fracture then becomes nearly propor¬tional to r2, as observed. The previous calculations of Roesler (1956 a, b) relate to the second stable crack dimension, c3, though his energy scaling principle is also applicable to the critical condition at which c1 and c2 merge. The Hertzian fracture test, within the validity range of Auerbach’s law, affords a means of measuring surface energy at the fracture surface independent of knowledge about the pre-present flaws.

Residual stress effects in sharp contact cracking

A study is made of residual stress effects in the mechanics of median fracture in sharp indenter contact. Starting with a simplistic treatment of the elastic-plastic indentation field, the problem is conveniently resolved into two separable parts, involving reversible (elastic) and irreversible (residual) components. The assumption of geometrical similarity in the residual field about the deformation zone, later backed up by stress birefringence measurements, leads to a stress intensity factor for median crack propagation containing the elastic and residual parts as the sum of two terms. The resulting formulation for equilibrium fracture shows some differences in the crack response during the loading and unloading half-cycles. By imposing certain stress states on the specimen surface during indentation the residual component of the field may actually cause the median crack to continue in downward extension as the indenter is withdrawn, a response which is especially amenable to experimental investigation. Direct observations of median crack evolution in soda-lime glass confirm this and other essential predictions of the fracture mechanics theory. The contribution of the residual component to the crack growth is found to be by no means secondary in importance to that of the elastic component.

Indentation deformation/fracture of normal and anomalous glasses

Abstract Vickers deformation/fracture indentations have been investigated in six silicate glasses. The characteristic damage patterns fall into two distinct groups, according to whether the glass shows “normal” or “anomalous” mechanical behaviour. Observations of the damage morphology during and after contact, of the scales of the deformation and fracture zones, and of the residual stress intensity about the impressions, all point to a basic difference in the local stress/strain micromechanics. This difference is discussed in relation to the factors which control the brittleness of glass.

Role of interfacial grain-bridging sliding friction in the crack-resistance and strength properties of nontransforming ceramics

Abstract A grain-bridging model of crack-resistance or toughness (R-curve, or T-curve) properties of nontransforming ceramics is developed. A key new feature of the fracture mechanics treatment is the inclusion of internal residual (thermal expansion mismatch) stresses in the constitutive stress-separation relation for pullout of interlocking grains from an embedding matrix. These internal stresses play a controlling role in the toughness properties by determining the scale of frictional tractions at the sliding grain-matrix interface. By providing a physical account of the underlying micromechanics of the bridging process the analysis allows for predetermination of the material factors in the constitutive relation, thereby reducing parametric adjustments necessary in fitting the theoretical toughness curve to experimental data. The applicability of the model is illustrated in a case study on indentation-strength data for a “reference” polycrystalline alumina with particularly strong T-curve characteristics. From theoretical fits to these data the constitutive relation, and thence the entire T-curve, can be deconvolved. This “parametric calibration”, apart from demonstrating the plausibility of the model, allows for quantitative predictions as to how the toughness and strength characteristics of ceramics depend on such microstructural variables as grain size and shape, grain boundary energy, level of internal stress and sliding friction coefficient. An indication of this predictive capacity is provided by a preliminary calculation of the grain-size dependence of strength, using some existing data for other aluminas as a basis for comparison.

Elastic recovery at hardness indentations

The mechanics of hardness indentation are considered. On the basis of a cycle in which the loading is elastic-plastic and the unloading (and subsequent reloading) elastic, an expression is derived for the relative depth recovery of the impression as a function of hardness/modulus,H/E. Experimental observations on indented surfaces of selected materials, mostly ceramics, using a tilting procedure in the scanning electron microscope to measure the residual depths, confirm the predicted trends. The analysis offers a simple means of characterizing the deformation properties of materials and should provide a basis for evaluating a range of contact-related properties, particularly surface damage phenomena in sharp-particle impact.