Many modelling approaches have been postulated to account for the role of dislocations in plastic flow for different modes of continuous deformation. At the mesoscale (typically be- tween 10−8−10−5m), mathematical descriptions are formulated by considering the average values of the variables with little regard to individual dislocation behaviours or arrangements. The Kocks-Mecking model [81, 176, 177] is based on the assumption that a single internal variable, i.e. the average dislocation density ρ, can be used to determine the kinetics of plastic deformation. The dislocation density is also linked to the macroscopic stress response σ via the Taylor equation
σ = Mα µ b√ρ , (2.5)
where M is the Taylor factor, µ the shear modulus, b the Burgers vector and α a constant of order unity accounting for the strength of the dislocation/dislocation interactions [177].
The evolution of the dislocation density as a function of the plastic strain εpis
dρ dεp = dρ + dεp −dρ − dεp , (2.6)
where dρ+/dεpand dρ−/dεpare the dislocation storage and annihilation rates, respectively.
The storage of dislocations is an athermal contribution and it depends on the mean free path of the dislocations Λ in the form
dρ+ dεp
= M
bΛ. (2.7)
This accounts for the spacing of different microstructural parameters that act as obstacles to dislocation glide Λi, and for dislocation/dislocation interactions, separated by a distance Λd,
as Λ−1= k1Λ−1d +
∑
i Λ−1i , (2.8) where Λd = 1/ √ρ and k1 is a material dependent parameter called dislocation storage
coefficient [176]; empirical approximations for this term have been given via experimental datasets in pure materials [176–179].
Alternatively, dynamic recovery arises from the collapse of dislocations with opposite signs, both edge dipoles by climb and screw dislocations by cross-slip at an average separation lower than a critical distance that depends on the composition and glide behaviour of the dislocations [170]. Thus, the annihilation term in equation (2.6) is highly dependent on
temperature and strain rate, and it increases proportionally to the dislocation density, i.e.
dρ− dεp
= Mk2ρ , (2.9)
where k2is the dynamic recovery coefficient. Additional modelling on this parameter has
also been performed [169, 180], although, similarly to the case of k1, most of the literature
focuses on pure materials, which deform in a more homogeneous way than nickel-based superalloys.
One of the advantages of this model is that it can be used for tests with non-constant temperatures and strain rates, as the response of the material is a function of the immediate dislocation density. However, fatigue modelling at the mesoscopic scale requires additional considerations as dislocations travel in opposite directions during the tensile and compressive loading stages, which results in phenomena such as the Bauschinger effect. This has been addressed by considering a supplementary internal variable, the density of dislocations par- tially recoverable upon reversal loading ρr, with an evolution that depends on microstructure
and operation conditions [181].
Estrin [182], among other authors, has further extended the model to consider additional microstructures, including alloys with shearable and non-shearable precipitates, solute atoms and grain size effects. These and other modifications are essentially done by considering ad- ditional stress contribution terms in equation (2.5) or supplementary obstacles to dislocations in equation (2.8), although obvious variations in the values of k1and k2come associated with
these changes.
Precipitates in particular have a big influence on the stress response of the material and they are relevant to the study of nickel-based superalloys. According to Ashby’s theory [183], non-shearable particles are a source of geometrically necessary dislocations (GNDs), required for compatibility at the matrix/precipitate interface, which give rise to internal stresses due to the misfit and difference in shear moduli [172, 184]. The number of dislocations per particle, their influence to the flow stress and their contribution to the Bauschinger effect have been modelled within Kocks-Mecking frameworks for different alloying systems [185–187]. However, shearable precipitates have been addressed to a lesser extent, and the evolution of the particle morphology has not been incorporated into these models.
The formation of dislocation structures has been considered in the case of cell formation by acknowledging that there is a heterogeneous distribution of dislocations consisting of walls (high ρ) and cell interiors (low ρ) [188]. This is similar to treating the material as a composite with two different phases, with the difference that their volume fractions may vary with straining and they do not have well defined interfaces, so that dislocations constantly
move from one region to the other [81]. Within the composite model, equation (2.5) can be rewritten as σ = Mα1µ b √ ρ1+ Mα2µ b √ ρ2, (2.10)
where the subindices 1 and 2 correspond to the different dislocation regions, and the α coefficients are not the same due to their different dislocation interactions and effects on the stress response.
Morphology and evolution of slip bands
3.1
Introduction
Fatigue in metals has been studied for over 180 years [189]; the term "fatigue" itself dates back to 1854 [190]. For the majority of this time an empirical approach was used to identify trends in the behaviour of materials upon different loading conditions. The well known Coffin-Manson power law for the prediction of fatigue life as a function of plastic strain during low cycle fatigue is an example of this. Whilst these are useful to describe the behaviour of tested materials, research on the physics behind deformation and damage phenomena provide a clearer picture of how and why these occur.
The morphology of the slip bands that develop during low cycle fatigue has been charac- terised to some extent in a number of nickel-based superalloys. However, a lot less is known regarding how these evolve. Due to the destructive nature of the techniques, microscopy studies reveal the deformation structure only after the test has stopped. These inconveniences can be partially circumvented by characterising samples from interrupted tests, so that direct comparisons can be made between behaviours after a different number of cycles.
A combination of electron microscopy techniques is used here to investigate multiple aspects of the slip bands, including their formation, propagation, dislocation arrangements, interaction with γ′precipitates and, most importantly, the evolution of all these. Two imaging modes are used within the SEM to analyse the heterogeneity of slip across grains and the morphology of sheared precipitates. TEM is employed to observe the dislocations structures in detail. A new methodology is also introduced to measure slip band parameters such as slip line spacing and shear step length through an orientation analysis. Statistics of these parameters are obtained and compared for different conditions. The combination of these techniques provides a unique mechanistic and quantitative insight into the slip band and precipitate morphology evolution.