Crack propagation and toughness

Size reduction or comminution is carried out by a process of crack propagation, whereby localized stresses produce strains in the particles that are large enough to cause bond rupture and thus propagate the crack. In general, cracks are propagated through regions of a material that possess the most flaws or discontinuities. Crack propagation is related to the strain energy in specific regions according to Griffith’s theory. The stress in a material is concentrated at the tip of a crack and the stress multiplier can be calculated from an equation developed by Inglis:

(10.1)

where σ_K is the multiplier of the mean stress in a material around a crack, L is the length of the crack and r is the radius of curvature of the tip of the crack. For a simple geometric structure such as a circular discontinuity L = 2r and the stress multiplier σ_K will have a value of 3.

In the case of a thin disc-shaped crack, shown in cross-section in Figure 10.1, the crack is considered to have occurred at molecular level between atomic surfaces separated by a distance of 2 × 10⁻¹⁰ m for a crack 3 µm long, which gives a stress multiplier of approximately 245. The stress concentration diminishes towards the mean stress according to the distance from the crack tip (Fig. 10.1). Once a crack is initiated, the crack tip propagates at a velocity approaching 40% of the speed of sound in the solid. This crack propagation is so rapid that excess energy from strain relaxation is dissipated through the material and concentrates at other discontinuities, where new cracks are propagated. Thus a cascade effect occurs and almost instantaneous brittle fracture occurs.

Not all materials exhibit this type of brittle behaviour and some can resist fracture at much larger stresses. This occurs because these tougher materials can undergo plastic flow, which allows strain energy relaxation without crack propagation. When plastic flow occurs, atoms or molecules slip over one another and this process of deformation requires energy. Brittle materials can also exhibit plastic flow and Irwin and Orowan suggested a modification of Griffiths’ crack theory to take this into account. This relationship has a fracture stress, σ, which varies inversely with the square root of crack length, L:

(10.2)

where E_p is the energy required to form unit area of double surface.

It can therefore be seen that the ease of comminution depends on the brittleness or plasticity of the material because of their relationship with crack initiation and crack propagation.

Surface hardness

In addition to the toughness of the material described above, size reduction may also be influenced by the hardness of the material. Hardness can be described empirically by its position in a scale devised by a German mineralogist called Mohs. Mohs’ scale is a table of minerals; at the top of the table is diamond, with Mohs hardness >7, and this has a surface that is so hard that it can scratch anything below it. At the bottom of the table is talc, with Mohs hardness <3, and this is soft enough to be scratched by anything above it.

A quantitative measurement of surface hardness was devised by Brinell. This involves placing a hard spherical indenter (e.g. hardened steel or sapphire) in contact with the test surface and applying a known constant load to the sphere. The indenter will penetrate into the surface until and when the sphere is removed, the permanent deformation of the sample is measured. From this, the hardness of the material can be calculated. Hardness has the dimensions of stress (force applied to the indenter divided by the area of test material that will support the load, example units are MPa). A similar Vickers hardness test employs a square-pyramidal diamond as the indenter tip.

Such determinations of hardness are useful as a guide to the ease with which size reduction can be carried out because, while it appears to be a surface assessment, the test actually quantifies the deformation characteristics of the bulk solid. In general, harder materials are more difficult to comminute and can lead to abrasive wear of metal mill parts, which can then result in product contamination. Conversely, materials with a large elastic component, such as rubber, are extremely soft yet difficult to size reduce.

Materials such as rubber that are soft under ambient conditions, waxy substances such as stearic acid that soften when heated, and ‘sticky’ materials such as gums are capable of absorbing large amounts of energy through elastic and plastic deformation without crack initiation and propagation. This type of material, which resists comminution at ambient or elevated temperatures, can be more easily size reduced when temperatures are lowered below the glass transition point of the material. At these lower temperatures the material undergoes a transition from plastic to brittle behaviour and crack propagation is facilitated.

Other factors that influence the process of size reduction include the moisture content of the material. In general, a material with a moisture content below 5% is suitable for dry grinding and one with greater than 50% will generally require wet grinding to be carried out.

Energy requirements of size reduction process

Only a very small amount of the energy put into a comminution operation actually effects size reduction. This has been estimated to be as little as 2% of the total energy consumption, the remainder being lost in many ways, including:

A number of hypotheses and theories have been proposed in an attempt to relate energy input to the degree of size reduction produced.

Rittinger’s hypothesis relates the energy, E, used in a size reduction process to the new surface area produced, S_n, or:

(10.3)

where S_i is the initial surface area and κ_R is Rittinger’s constant, expressing energy per unit area.

Kick’s theory states that the energy used in deforming or fracturing a set of particles of equivalent shape is proportional to the ratio of the change in size, or:

(10.4)

where κ_K is Kick’s constant of energy per unit mass, d_i is the initial particle diameter and d_n the new particle diameter.

Bond’s theory states that the energy used in crack propagation is proportional to the new crack length produced, which is often related to the change in particle dimensions according to the following equation:

(10.5)

Here κ_B is known as Bond’s work index and represents the variation in material properties and size reduction methods, with dimensions of energy per unit mass.

Walker proposed a generalized differential form of the energy–size relationship that can be shown to link the theories of Rittinger and Kick, and in some cases that of Bond:

(10.6)

where κ_W is Walker’s constant and d is a size function that can be characterized by an integrated mean size or by a weight function, n is an exponent. When n = 1 for particles defined by a weight function, integration of Walker’s equation corresponds to a Kick-type theory, when n = 2 a Rittinger-type solution results and when n = 1.5 Bond’s theory is given.

When designing a milling process for a given particle, the most appropriate energy relationship will be required in order to calculate energy consumptions. It has been considered that the most appropriate values for n are 1 for particles larger than 1 µm where Kick-type behaviour occurs, and 2 for Rittinger-type milling of smaller particles of less than 1 µm. The third value of n = 1.5 is the average of these two extremes and indicates a possible solution where neither Kick’s nor Rittinger’s theory is appropriate. Other workers have found that n cannot be assumed to be constant, but varies with particle size.