A.G. Atkins

Modelling metal cutting using modern ductile fracture mechanics: quantitative explanations for some longstanding problems

The assumption that negligible work is involved in the formation of new surfaces in the machining of ductile metals, is re-examined in the light of both current Finite Element Method (FEM) simulations of cutting and modern ductile fracture mechanics. The work associated with separation criteria in FEM models is shown to be in the kJ/m2 range rather than the few J/m2 of the surface energy (surface tension) employed by Shaw in his pioneering study of 1954 following which consideration of surface work has been omitted from analyses of metal cutting. The much greater values of surface specific work are not surprising in terms of ductile fracture mechanics where kJ/m2 values of fracture toughness are typical of the ductile metals involved in machining studies. This paper shows that when even the simple Ernst–Merchant analysis is generalised to include significant surface work, many of the experimental observations for which traditional ‘plasticity and friction only’ analyses seem to have no quantitative explanation, are now given meaning. In particular, the primary shear plane angle φ becomes material-dependent. The experimental increase of φ up to a saturated level, as the uncut chip thickness is increased, is predicted. The positive intercepts found in plots of cutting force vs. depth of cut, and in plots of force resolved along the primary shear plane vs. area of shear plane, are shown to be measures of the specific surface work. It is demonstrated that neglect of these intercepts in cutting analyses is the reason why anomalously high values of shear yield stress are derived at those very small uncut chip thicknesses at which the so-called size effect becomes evident. The material toughness/strength ratio, combined with the depth of cut to form a non-dimensional parameter, is shown to control ductile cutting mechanics. The toughness/strength ratio of a given material will change with rate, temperature, and thermomechanical treatment and the influence of such changes, together with changes in depth of cut, on the character of machining is discussed. Strength or hardness alone is insufficient to describe machining. The failure of the Ernst–Merchant theory seems less to do with problems of uniqueness and the validity of minimum work, and more to do with the problem not being properly posed. The new analysis compares favourably and consistently with the wide body of experimental results available in the literature. Why considerable progress in the understanding of metal cutting has been achieved without reference to significant surface work is also discussed.

Mechanisms and causes of wear in tooth enamel: implications for hominin diets

The wear of teeth is a major factor limiting mammalian lifespans in the wild. One method of describing worn surfaces, dental microwear texture analysis, has proved powerful for reconstructing the diets of extinct vertebrates, but has yielded unexpected results in early hominins. In particular, although australopiths exhibit derived craniodental features interpreted as adaptations for eating hard foods, most do not exhibit microwear signals indicative of this diet. However, no experiments have yet demonstrated the fundamental mechanisms and causes of this wear. Here, we report nanowear experiments where individual dust particles, phytoliths and enamel chips were slid across a flat enamel surface. Microwear features produced were influenced strongly by interacting mechanical properties and particle geometry. Quartz dust was a rigid abrasive, capable of fracturing and removing enamel pieces. By contrast, phytoliths and enamel chips deformed during sliding, forming U-shaped grooves or flat troughs in enamel, without tissue loss. Other plant tissues seem too soft to mark enamel, acting as particle transporters. We conclude that dust has overwhelming importance as a wear agent and that dietary signals preserved in dental microwear are indirect. Nanowear studies should resolve controversies over adaptive trends in mammals like enamel thickening or hypsodonty that delay functional dental loss.

Fracture in forming

Abstract The formation of cracks in metalworking processes is considered from both the metallurgical and mechanics points of view. Stress and strain states are shown to be important, and hydrostatic stress figures prominently in the coalescence of voids initiated at inclusions or hard second phases. Changes of stress or strain ratio during an operation can have a profound effect on cracking, as illustrated by fracture following neck formation in sheet materials. Conditions for cracking in triaxial stress states will be of interest to those requiring fracture criteria for large-deformation finite-element calculations.

Single‐point incremental forming and formability—failure diagrams

In recent work, the present authors constructed a closed‐form analytical model that is capable of dealing with the fundamentals of single‐point incremental forming (SPIF) and explaining the experimental and numerical results published in the literature over the past couple of years. The model is based on membrane analysis with in‐plane contact frictional forces but is limited to plane strain, rotationally symmetric conditions. The aim of the present paper is twofold: first, to extend the previous closed‐form analytical model into a theoretical framework that can easily be applied to the different modes of deformation that are commonly found in general single‐point incremental forming processes and, second, to investigate the formability limits of SPIF in terms of ductile damage mechanics and the question of whether necking does, or does not, precede fracture. Experimentation by the present authors, together with data retrieved from the literature, confirms that the proposed theoretical framework is capable of successfully addressing the influence of the major parameters of the SPIF process. It is demonstrated that neck formation is suppressed in SPIF, so that traditional forming limit diagrams are inapplicable to describe failure. Instead fracture forming limit diagrams should be employed.

The Role of Dust, Grit and Phytoliths in Tooth Wear

The threat of wear to dental enamel from hard particles of silica or silicates may have exerted great selective pressure on mammals. Increasing the hardness of enamel helps to forestall this, but capacity for variation is small because the tissue is almost entirely composed of hydroxyapatite. Hard though it is, enamel also displays considerable toughness, which is important in setting the sharpness of particles (defined as an attack angle) necessary to wear it. Added to the threat from environmental silica(tes) are phytoliths, particles of opaline silica embedded in plant tissues. We show here that phytoliths have very different properties to grit and dust and are unlikely to wear enamel. However, phytoliths would tend to fracture between teeth under similar conditions, so resembling natural agents of wear. In this context, we suggest that phytoliths could represent an example of mimicry, forming an example of a feeding deterrent operating by deceit.

Dynamic effects in progressive failure of structures

Abstract This paper considers the behaviour of loaded structures when a member (or members) breaks prematurely. This behaviour consists of a transient vibration of the remaining damaged structure as it moves towards a new state of equilibrium. Static calculations may predict that the structure, with the failed members omitted, is capable of safely bearing the external loading. This static approach to the load-bearing capacity of the damaged structure is implicit in many design codes for redundant, damage-tolerant, structures. But it is shown here that transient overloads induced by the sudden fracture of a member may cause progressive fracture of other elements before a new equilibrium state is reached. Sometimes the cascade of fractures arrests; sometimes catastrophic failure of the whole structure occurs. Experiments with a redundant radially-tied (spoked-wheel) structure confirm the prediction that there are structures which are statically safe, but which are dynamically unsafe. In spite of the simplistic nature of these experiments they shed light on several aspects of dynamic failure which will be important for real engineering structures.

Necking and radial cracking around perforations in thin sheets at normal incidence

Abstract The formation of multiple necks and cracks around perforations in ductile materials is investigated. Expressions are obtained for the number of plane-strain radial necks formed by conical penetrators (with or without starter holes in the flat target), and also by round-ended projectiles into flat targets having starter holes or, in plain targets with no starter holes, after fracture of the initial circumferential neck has detached a circular cap of material (“discing”). Likewise, expressions are derived for the number of radial cracks which form subsequently in both cases and lead to “petalling”. The number of cracks is smaller than the number of necks. Experiments seem to agree with the analyses, although accurate counting of necks in thin sheets is illusory.

Scaling in combined plastic flow and fracture

Abstract The scaling laws are given for bodies undergoing simultaneous plastic flow and crack propagation, deformations which can be adequately described by rigid-plastic fracture mechanics. The laws depend on (i) a material-dependent term given by the ratio of plastic work done/volume ∫σ de to the material fracture toughness R for the given pattern of deformation, as well as on (ii) a geometrical term given by the ratio, in the reference model structure, of the volume of material plastically deformed, V, to the crack area, A. The two contributing factors are combined in a single non-dimensional parameter ξ = (∫σ d e R )( V A ) . Energy scaling in prototype (p) and model (m) follows W p W m = λ 2 (λξ + 1) (ξ + 1) , which is the true form of λx-type empirical relations W A ) vs ( V A ) which is central to rigid-plastic fracture mechanics. The new scaling laws agree well with a wide range of quasi-static and dynamic experimental data on scaled bodies. They help to explain hitherto anomalous behaviour in the impact of scaled structures.

On the number of cracks in the axial splitting of ductile metal tubes

Abstract An analysis is given for the number of cracks which propagate in experiments on the splitting of ductile metal tubes when the ends are flared out. The number of cracks depends on the diameter of the tube (2 r 0 ), the strength-to-toughness ratio of the metal ( \ gs R ) and its hoop fracture strain e θ ∫ (which may be related to microstructure via models of microvoid growth and coalescence). Since the expression for the number of cracks contains the product of ( \ gs R ) and e θ ∫ , approximately the same number of cracks is predicted whether the tubes are annealed or work-hardened. Experiments by Reddy and Reid [ Int. J. Mech. Sci. 28, 111 (1986)] give support to the predictions.

A review of the J and I integrals and their implications for crack growth resistance and toughness in ductile fracture

The application of the J and the I-integrals to ductile fracture are discussed. It is shown that, because of the finite size of the fracture process zone (FPZ), the initiation value of the J-integral is specimen dependent even if the plastic constraint conditions are constant. The paradox that the I-integral for steady state elasto-plastic crack growth is apparently zero is examined. It is shown that, if the FPZ at the crack tip is modelled, the I-integral is equal to the work performed in its fracture. Thus it is essential to model the fracture process zone in ductile fracture. The I-integral is then used to demonstrate that the breakdown in applicability of the J-integral to crack growth in ductile fracture is as much due to the inclusion in the J-integral of progressively more work performed in the plastic zone as it is to non-proportional deformation during unloading behind the crack tip. Thus JR-curves combine the essential work of fracture performed in the FPZ with the plastic work performed outside of the FPZ. These two work terms scale differently and produce size and geometry dependence. It is suggested that the future direction of modelling in ductile fracture should be to include the FPZ. Strides have already been made in this direction.