# Modelling the High Temperature Fatigue Behaviour of Titanium Alloys

Sat 11th January

**Modelling the High Temperature Fatigue Behaviour of Titanium Alloys**

**Hector Basoalto and Jeffery Brooks**

**School of Metallurgy and Materials, University of Birmingham,Edgbaston B15 2TT, UK**

The industrial application of titanium alloys requires a critical understanding of the relationship between the underlying microstructure and the mechanical properties. The aim of this work is to define a physics based microstructural model which can account for the changes in structure developed during manufacturing and service and so provide a predictive capability for the resulting mechanical properties. This paper focuses specifically on the development of a microstructure-based damage mechanics approach to model the high temperature fatigue behaviour. Within the proposed framework damage is defined in terms of the evolution of relevant microstructure parameters that are known to influence the accumulation of plastic strains and eventually limit life. The constitutive formulation is developed from a continuum description of dislocation flow that accounts for structural features such as alpha laths, grains and the dislocation network itself. The approach is general enough that it can be extended to include grain boundary damage associated with void growth. The proposed modelling framework extends the original definition of "damage" of Kachanov/Robatnov to include any rearrangement of the microstructure that affects the accumulation of plastic strain.

If plastic deformation is dominated by slip, the macroscopic flow stress behaviour can be determined by modelling the evolution of the dislocation density at a material point. The plastic strain rate is then determined by the dislocation flux through a representative volume element, which is equal to the product of the mobile dislocation density and velocity. The mobile dislocation density, *ρ _{m}* , can be decomposed into gliding,

*ρg*, and pinning segments,

*ρ*, with the gliding segments determining the shear strain rate on each active slip system. Such an evolution can be described by a continuity equation that is extended to include a dislocation source and trapping terms for the gliding segments. This framework results in the following continuity equation:

_{w}...............................(1)

where *ν _{g}* is the glide velocity and

*λ*the mean distance travelled by the gliding dislocations. The G(τ,T;μ) function represents the rate of generation of gliding dislocations and is a function of the applied shear stress, τ , the temperature, T, and details of the microstructure μ (a list of all microstructural features considered by the model). The present paper derives an expression for this function in terms of the material microstructure with the following starting assumptions: release of pinned dislocations occurs through thermally activation; and plastically hard regions provide a kinematic back stress. These assumptions provide the basic equation set for the flow stress behaviour, which together with evolution equations of the microstructure can be used to predict fatigue response of engineering alloys.

Possible length scale effects on the fatigue behaviour of titanium alloys have also been considered. These include the effect of grain size and lamellar spacing on the evolution of stress/strain curves during fatigue. The central concept in this part of the work is that of geometrically-necessary dislocations (GND), which arise from the non-homogenous plastic deformation in structural engineering alloys. By dealing with the average of the deformation gradients arising from the heterogeneous nature of an alloys microstructure, it is possible to estimate the dislocation density content to main such gradients, which in turn can be used to estimate their contribution to resistance to plastic flow. The starting point is the theory proposed by Ashby [1] on the deformation of non-homogeneous two phase materials, in which one phase undergoes plastic deformation whilst only elastic distortions occur in the other. Deformation in such materials gives rise to gradients of plastic deformation which are imposed on the microstructure and produce enhanced strain hardening behaviour compared to a homogenous material system. Kroner [2] and Bilby [3] showed the connection between the density of dislocations and the gradients of plastic deformation with distance. For the case of single slip, a gradient of slip, *∂γ ^{(a)}/∂x_{1} *, on the active slip system along the global

*1- direction requires a certain density of geometrically necessary dislocations (GNDs)*

*x**ρ*

*given by , where*

_{g}*is the magnitude of the Burger’s vector. From this estimates of the GND density a mean spacing can be calculated which can then be used to estimate their influence on deformation through a dislocation forest interaction approach.*

*b*Numerical solutions of the constitutive equations are presented. For the class of alloys under consideration the effective spacing of pining obstacles will initially be taken to be the spacing between pinning jogs along the dislocation line with the GND network limiting the flow behaviour at later stages of deformation. The effect of jog spacing on the stress-strain loop of the model material is investigated, undercyclic loading, and it can be seen that as the jog spacing decreases from 40nm to 30nm the material response shows greater resistance to deformation. The corresponding evolution of the half peak stress as a function of loading cycles are also determined. In the current calculations the jog spacing has been assumed to be constant during the loading cycle, however, this is not realistic and experiments have shown that the jog spacing decreases with increasing applied stress. However, the simulation does illustrate the influence of the effective pinning distance on the fatigue behaviour. Stress/strain curves at 500°C have been calculated for a ‘generic’ near alpha-titanium alloy and predict yield stresses on loading of the order of ~ 600-700MPa for a strain rate of 10^{-3} s^{-1}. This approach has been used to show how a finer lamellar structure can induce an increase in fatigue strength.

**References**

1. M.F. Ashby, Phil. Mag., 21, 399-424, 1970.

2. E. Kroner, Phys. Stat. Sol., 1, 3; 1961.

3. B.A. Bilby, Prog. Solid. Mech., 1, 331, 1960.

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