Ductile failure

7.4.7 Ductile failure

Ductile failure was introduced in Chapter 3 because of the role played by voids in the failure processes, which occurs by void nucleation, growth and coalescence. The nucleation of voids often takes place at

Mechanical properties II – Strengthening and toughening 435

local necking

Figure 7.35 Schematic representation of ductile fracture. (a) Voids nucleate at inclusions. (b) Voids elongate as the specimen extends. (c) Voids coalesce to cause fracture when their length 2h is about equal to their separation (after Ashby et al., 1979).

inclusions. The dislocation structure around particle inclusions leads to a local rate of work hardening higher than the average, and the local stress on reaching some critical value σ c will cause fracture of the inclusion or decohesion of the particle/matrix interface, thereby nucleating a void. The critical nucleation strain ε n can be estimated and lies between 0.1 and 1.0 depending on the model. For dispersion-hardening materials where dislocation loops are generated the stress on the interface due to the nearest prismatic loop, at distance r, is μb/r, and this will cause separation of the interface when it reaches the theoretical strength of the interface, of order γ w /

b. The parameter r is given in terms of the applied shear strain ε, the particle diameter d and the length k equal to half the mean particle spacing as r = 4kb/εd. Hence, void nucleation occurs on a particle of diameter d after a strain ε, given by ε = 4kγ w /μ

db. Any stress concentration effect from other loops will increase with particle size, thus enhancing the particle size dependence of strain to voiding. Once nucleated, the voids grow until they coalesce to provide an easy fracture path. A spherical- shaped void concentrates stress under tensile conditions and, as a result, elongates initially at about C( ≈2) times the rate of the specimen, but as it becomes ellipsoidal the growth rate slows until finally the elongated void grows at about the same rate as the specimen. At some critical strain, the plasticity becomes localized, the voids rapidly coalesce and fracture occurs. The localization of the plasticity is thought to take place when the voids reach a critical distance of approach, given when the void length 2h is approximately equal to the separation, as shown in Figure 7.35. The true strain for coalescence is then

ε = (1/C) ln[α(2l − 2r v )/2r v ]

(7.30) ≈ (1/C) ln[α(1/f v − 1)],

where α ≈ 1 and f v is the volume fraction of inclusions. Void growth leading to failure will be much more rapid in the necked portion of a tensile sample following instability than during stable deformation, since the stress system changes in the neck from uniaxial tension to approximately plane strain tension. Thus, the overall ductility of a specimen will

436 Physical Metallurgy and Advanced Materials depend strongly on the macroscopic features of the stress–strain curves which (from Considère’s

criterion) determine the extent of stable deformation, as well as on the ductile rupture process of void nucleation and growth. Nevertheless, an equation of the form of (7.30) reasonably describes the fracture strain for cup and cone failures.

The work of decohesion influences the progress of voiding and is effective in determining the overall ductility in a simple tension test in two ways. The onset of voiding during uniform deformation depresses the rate of work hardening, which leads to a reduction in the uniform strain, and the void density and size at the onset of necking determine the amount of void growth required to cause ductile rupture. Thus, for matrices having similar work-hardening properties, the one with the least tendency to ‘wet’ the second phase will show both lower uniform strain and lower necking strain. For matrices with different work-hardening potential but similar work of decohesion, the matrix having the lower work-hardening rate will show the lower reduction prior to necking but the greater reduction during necking, although two materials will show similar total reductions to failure.

The degree of bonding between particle and matrix may be determined from voids on particles annealed to produce an equilibrium configuration by measuring the contact angle θ of the matrix surface to the particle surface. Resolving surface forces tangential to the particle, then the specific

interface energy γ I is given approximately in terms of the matrix surface energy γ m and the particle surface energy γ P as γ I =γ P −γ m cos θ. The work of separation of the interface γ w is then given by

(7.31) Measurements show that the interfacial energy of TD nickel is low and hence exhibits excellent

γ w =γ P +γ m −γ I =γ m (1 + cos θ).

ductility at room temperature. Specific additions (e.g. Zr to TD nickel, and Co to Ni–Al 2 O 3 alloys) are also effective in lowering the interfacial energy, thereby causing the matrix to ‘wet’ the particle and increase the ductility. Because of their low γ I , dispersion-hardened materials have superior mechanical properties at high temperatures compared with conventional hardened alloys.