Cleavage and the ductile–brittle transition

7.4.3 Cleavage and the ductile–brittle transition

The fracture toughness versus strength chart, shown in Figure 7.32, indicates the wide spread of values for the different classes of material. Metals dissipate much energy in the plastic zone, which accounts

for the difference between the fracture energy G 1c and the true surface energy γ. The larger the zone, the more energy is absorbed; G 1c is high and so is K 1c ( ≈100 MN m −3/2 ). Ceramics and glasses fracture without much plastic flow to blunt the cracks by simply breaking atomic bonds, leading to

cleavage; for these materials K 1c is less than 10 MN m −3/2 .

At low temperatures some metals, notably steels, also become brittle and fracture by cleavage. Since they are ductile at room temperature this transition to brittle cleavage behavior is quite spectacular and has led to several engineering catastrophes. In general, brittle cleavage can occur in metals with bcc and cph under the appropriate conditions, while in fcc materials it does not. The most important factor link- ing these three different structures is the Peierls stress and the way the yield stress varies with tempera- ture. In steel, for example, the yield stress increases rapidly with lowering of temperature below room temperature such that plastic deformation at the crack tip is minimized and the fracture mechanism changes from plastic tearing to cleavage. Even in these materials some plastic deformation does occur.

Several models have been suggested for the process whereby glide dislocations are converted into microcracks. The simplest mechanism, postulated by Stroh, is that involving a pile-up of dislocations against a barrier, such as a grain boundary. The applied stress pushes the dislocations together and

a crack forms beneath their coalesced half-planes, as shown in Figure 7.33a. A second mechanism of crack formation, suggested by Cottrell, is that arising from the junction of two intersecting slip planes. A dislocation gliding in the (1 0 1) plane coalesces with one gliding in the (1 0 ¯1) plane to form a new dislocation which lies in the (0 0 1) plane according to the reaction:

a/2[¯1 ¯1 1] + a/2[1 1 1] → a[0 0 1]. The new dislocation, which has a Burgers vector a[0 0 1], is a pure edge dislocation and, as shown in

Figure 7.33b, may be considered as a wedge, one lattice constant thick, inserted between the faces of the (0 0 1) planes. It is considered that the crack can then grow by means of other dislocations in the

430 Physical Metallurgy and Advanced Materials (1 0 1) and (1 0 ¯1) planes running into it. Although the mechanism readily accounts for the observed

(1 0 0) cleavage plane of bcc metals, examples have not been directly observed. While these dislocation coalescence mechanisms may operate in single-phase materials, in two- phase alloys it is usually easier to nucleate cracks by piling up dislocations at particles (e.g. grain boundary cementite or cementite lamellae in pearlite). The pile-up stress then leads to cracking of either the particle or the particle/matrix interface. A brittle–ductile transition can then be explained on the basis of the criterion that the material is ductile at any temperature, if the yield stress at that temperature is smaller than the stress necessary for the growth of these microcracks, but if it is larger the material is brittle. If cleavage cracks are formed by such a dislocation mechanism, the Griffith formula may be rewritten to take account of the number of dislocations, n, forming the crack. Thus, rearranging Griffith’s formula we have

σ (2cσ/E)

= γ = 2cσ 2 / E,

where the product of length 2c and strain σ/E is the maximum displacement between the faces of the crack. This displacement will depend on the number of dislocations forming the cleavage wedge and may be interpreted as a pile-up of n edge dislocations, each of Burgers vector a, so that equation (7.23) becomes

σ (na) =γ (7.25) and gives a general criterion for fracture. Physically, this means that a number of glide dislocations, n,

run together and in doing so cause the applied stress acting on them to do some work, which for fracture to occur must be at least sufficient to supply the energy to create the new cracked faces, i.e. (γ +γ p ).

Qualitatively, we would expect those factors which influence the yield stress also to have an effect on the ductile–brittle fracture transition. The lattice ‘friction’ term σ i , dislocation locking term k y , and grain size 2d should also all be important because any increase in σ i and k y , or decrease in the grain size, will raise the yield stress with a corresponding tendency to promote brittle failure.

These conclusions have been put on a quantitative basis by Cottrell, who considered the stress needed to grow a crack at or near the tensile yield stress, σ y , in specimens of grain diameter 2d. Let us consider first the formation of a microcrack. If τ y is the actual shear stress operating, the effective shear stress acting on a glide band is only (τ y –τ i ), where, it will be remembered, τ i is the ‘friction’ stress resisting the motion of unlocked dislocations arising from the Peierls–Nabarro lattice stress, intersecting dislocations or groups of impurities. The displacement na is then given by

na = [(τ y −τ i )/μ]d, (7.26) where μ is the shear modulus and d is the length of the slip band containing the dislocations (assumed

here to be half the grain diameter). Once a microcrack is formed, however, the whole applied tensile stress normal to the crack acts on it, so that σ can be written as (τ y × constant), where the constant is included to account for the ratio of normal stress to shear stress. Substituting for na and σ in the Griffith formula (equation (7.25)), then fracture should be able to occur at the yield point when σ =σ y and

τ y (τ y −τ i )d = Cμγ, (7.27) where C is a constant. The importance of the avalanche of dislocations produced at the yield drop

can be seen if we replace τ y by (constant ×σ y ), τ i by (constant ×σ i ) and use the Petch relationship σ y =σ i +k y d −1/2 , when equation (7.27) becomes

(σ i d 1/2 +k y )k y = βμγ, (7.28)

Mechanical properties II – Strengthening and toughening 431 where β is a constant which depends on the stress system; β

≈ 1 for tensile deformation and β ≈ 1

for a notched test. This is a general equation for the propagation of a crack at the lower yield point and shows what factors are likely to influence the fracture process. Alternative models for growth-controlled cleavage fracture have been developed to incorporate the possibility of carbide particles nucleating cracks. Such models emphasize the importance of yield parameters and grain size as well as carbide thickness. Coarse carbides give rise to low fracture stresses, thin carbides to high fracture stresses and ductile behavior.