# Cold dwell behaviour of Ti6Al alloy: Understanding load shedding using
digital image correlation and crystal plasticity simulations

Yi Xiong  Department of Materials, University of Oxford, Parks Road, Oxford,
OX1 3PH, United Kingdom<EMAIL_ADDRESS>Nicolò Grilli  Department
of Engineering Science, University of Oxford, Parks Road, Oxford, OX1 3PJ,
United Kingdom  Phani S. Karamched  Bo-Shiuan Li  Edmund Tarleton  Angus J.
Wilkinson

###### Abstract

Digital image correlation (DIC) and crystal plasticity simulation were
utilised to study cold dwell behaviour in a coarse grain Ti-6Al alloy at 3
different temperatures up to 230 ∘C. Strains extracted from large volume
grains were measured during creep by DIC and were used to calibrate the
crystal plasticity model. The values of critical resolved shear stresses
(CRSS) of the two main slip systems (basal and prismatic) were determined as a
function of temperature. Stress along paths across the boundaries of two grain
pairs, (1) a ‘rogue’ grain pair and (2) a ‘non-rogue’ grain pair, were
determined at different temperatures. Load shedding was observed in the
‘rogue’ grain pair, where a stress increment during the creep period was found
in the ‘hard’ grain. At elevated temperatures, 120 ∘C was found to be the
worst case scenario as the stress difference at the grain boundaries of these
two grain pairs were found to be the largest among the three temperatures.
This can be attributed to the fact that the strain rate sensitivity of both
prismatic and basal slip systems is at its greatest in this worst case
scenario temperature.

###### keywords:

Cold dwell fatigue , Digital image correlation , Crystal plasticity , Load
shedding , Titanium alloy

##  1 Introduction

Titanium alloys with predominantly $\alpha$-phase (HCP crystalline structure)
are widely used in aero engines and gas turbines at moderate temperatures, due
to their excellent specific strength and corrosion resistance [1]. In such
applications, titanium alloy components often subjected to extremely
complicated stress state. For instance, typical flight cycles comprise of
takeoff (load up), cruise (stress hold) and landing (load off), combined with
a varying temperature from ambient conditions and engine heating. During the
cruise phase, titanium fan blades and compressor disks experience long hours
of high stress hold (dwell), which often results in a drastic lifetime
reduction for these titanium components [2, 3]. This phenomenon that seems to
happen at relatively lower temperatures (typically below 200 ∘C) has been
termed ‘cold dwell fatigue’.

Due to its significance in the aerospace industry, several efforts have been
made to study this complicated problem. Cold dwell fatigue mainly occurs in
$\alpha$-Ti alloys [4]. The deformation of HCP crystal structure is highly
anisotropic due to its low symmetric elasticity and different slip system
strengths [5, 6]. During a stress dwell, local time dependent plasticity could
occur in grains well orientated for basal and prismatic slip (‘soft’ grains).
This leads to load shedding onto grains badly orientated for slip (i.e. ‘hard’
grains with crystallographic $c$-axis of the $\alpha$-Ti grain nearly parallel
to the loading direction) [7]. The ‘hard’ oriented grain, due to higher
elastic modulus, encounters a stress increase within the grain. Dunne _et al_
[8, 9] used a crystal plasticity finite element (CPFE) model to investigate
the worst case scenario, in a combination of a ‘soft’ and ‘hard’ grain, also
known as ‘rogue’ grain pair. Upon deformation, large stresses accumulated at
the grain boundary and also into the ’hard’ grain. Sinha _et al_ [10, 11, 12]
suggested that dwell fatigue crack nucleation is usually related to {0002}
facets which lie almost perpendicular to the loading direction. This facet
formation within the ‘hard’ grain is initiated by dislocation pile-up in the
adjacent ‘soft’ grain [2, 13]. However, it has also been demonstrated that in
some samples dwell sensitivity still occurs without the presence of a ‘rogue’
grain pair [14], but the mechanism remains not well explained. The authors
believe that understanding this behaviour of easy slip systems (basal and
prism) during stress dwell will improve understanding of the complicated cold
dwell fatigue problem.

Although a considerable number of researchers have worked on this long-
standing problem, a large proportion of the work has been performed at ambient
temperatures. It has been suggested that ‘Cold dwell fatigue’ is strongly
affected by temperature and dwell fatigue is suppressed by increasing
temperature and disappears when temperature is above 200 ∘C [15]. Zhang _et
al_ [16] investigated the temperature sensitivity of load shedding and cold
dwell fatigue using the crystal plasticity model on a model Ti-6Al alloy. It
was shown that at a temperature above 230 ∘C thermally activated creep results
in a rapid stress redistribution and elimination of local load shedding
between ‘rogue’ grain pairs. An intermediate temperature of about 120 ∘C
exists at which the cold dwell debit is at its worst. However, the mechanism
has not yet been verified by experimental observations.

In this work, we combine a crystal plasticity finite element model and
experimental methods to investigate the temperature effect on cold dwell
fatigue in a coarse grain Ti-6Al alloy. An optical digital image correlation
(DIC) approach has been used to allow in-situ measurement of strain in
different grains while the samples were under load-up and load-hold (creep)
type of deformation at 3 different temperatures up to 230 ∘C. True sample
microstructures, captured by electron back scattered diffraction (EBSD), were
reconstructed for crystal plasticity simulations. Critical resolved shear
stresses (CRSS) of the two main slip systems (basal and prism) were determined
as a function of temperature by matching the experimentally measured strain to
the simulated strain in different grains. Load shedding was studied in both
‘rogue’ and ‘non-rogue’ grain pairs at different temperatures.

##  2 Material and methods

###  2.1 Material

The material used in this study is a Ti-6Al alloy bar, supplied by Timet UK
Ltd. with composition shown in Table.1. Test samples were $5\times 5\times 5$
mm3 cubes cut from the bar. In order to reduce the influence of pre-processing
(e.g. residual stress and lattice defects) and homogenise the grain size,
samples were annealed at 980 ∘C in a vacuum tube furnace for 12 h, followed by
slow cooling for a period of more than 24 h. The samples were
metallographically prepared using SiC papers (up to 4000 grit), and a final
polish with a $\approx$ 50 nm colloidal silica suspension. EBSD orientation
maps were obtained using a Zeiss Merlin scanning electron microscope (SEM)
equipped with a Bruker e-flash detector, operating at an accelerating voltage
of 20 kV and a probe current of 20 nA. Figure 1 shows the microstructure of
these samples, the mean grain size after the heat treatment was found to be
approximately 140$\pm$30 $\mu$m.

Table 1: Composition of the Ti6Al alloy. Ti | Al (wt. %) | Fe (wt. %) | O (wt. %) | N (wt. %) | Others total (wt. %)
---|---|---|---|---|---
Balance | 5.79 |  $<$0.05 | 0.07 | 0.001 | 0.2
Figure 1: EBSD maps of the Ti6Al samples (IPF colour map along the loading
direction of the samples): (a) Sample 1 tested at room temperature; (b) Sample
2 tested at 120 ∘C; (c) Sample 3 tested at 230 ∘C and (d) Sample 4 tested at
room temperature. Numbered grains 1-8 were selected to calibrate the CPFE
model.

###  2.2 Digital image correlation and creep tests

In order to measure strain on the sample surface, an optical DIC technique was
used. This requires a speckle pattern to be applied to monitor pattern motion
and subsequent strain estimation. This was achieved by spraying a polyurethane
paint onto the polished surfaces of the cubic samples. Black paint was used
first to generate a uniform background, then a white paint diluted with
isopropyl alcohol was applied to the surfaces using an airbrush kit to
generate speckle patterns. An example of the speckle pattern is shown in
Figure 2, the size of the white spots vary from 1-10 $\mu$m. The images were
recorded at 16-bit depth using a high speed camera (Hamamatsu Photonics Ltd -
ORCA-Flash 4.0) with an image resolution of 2048$\times$2048 pixels. A Questar
QM100 lens attachment was employed to enable imaging the sample through a
furnace window for elevated temperature experiments. With the working
distances of $\approx$0.5 m used in this experiment, the pixel size
corresponds to 1.5 $\mu$m.

Figure 2: Example image for DIC measurement, EBSD scanned area was located by
matching a mark pre-scratched on the sample. Grid of nodes were generated
within the EBSD scanned area. Loading along the vertical direction of this
image.

Figure 3 shows the experiment setup. Samples with thermocouples spot welded on
one face, were placed in an environmental chamber (manufactured by Severn
Thermal Solutions). DIC images were recorded by the high speed camera through
the furnace window. An LED light source was placed outside the furnace
chamber, illuminating the sample surfaces with speckle patterns, so that the
exposure time of the camera could be reduced down to 20 ms without sacrifice
the quality of the images. The acquisition rate was set to be constant at 1
frame per second for all the tests. Sample 1 to sample 3 were tested at room
temperature, 120 ∘C and 230 ∘C (ambient temperature, intermediate temperature
and temperature where cold dwell fatigue diminished). Compressive loads were
applied to the samples using a Shimadzu mechanical test frame (AGS-X series)
after the temperature of each sample was stabilised at each target temperature
for 5 minutes.

Figure 3: Sketch of the experiment setup.

Samples were loaded in compression at constant stress rate of 5 MPa/s, so that
the first gradient changes observed in the strain vs. time plots (see Figure
4(a)-(c)) represent the macroscopic yield. Sample 1-3 were designed to study
the behaviour of the two major slip systems (basal and prism slip) during
creep and thus were loaded beyond the macroscopic yield stresses to activate
both the desired slip systems. After clear gradient changes were observed in
strain vs. time data, the macroscopic stresses were held constant for a period
of 10 minutes to mimic the dwell effect. The compression strain of these
samples was found to increase during the stress dwell periods. In the first
half of this dwell period, strain appears to change faster than that of the
subsequent half. This also shows that stress redistribution is more active in
the first 5 minutes of the stress hold. Sample 4 was designed to scope into
this first 5 minutes of creep under a stress below the macroscopic yield
stress, which is a better representation of a real load condition of Ti alloys
in-service. The loading curve of sample 4 is shown in Figure 4(d) and this
sample starts to creep with strain still in elastic region, and lasts for 5
minutes.

Figure 4: Strain vs. time curves of (a) Sample 1 loaded beyond macroscopic
yield stress at room temperature; (b) Sample 2 loaded beyond macroscopic yield
stress at 120 ∘C; (c) Sample 3 loaded beyond macroscopic yield stress at 230
∘C and (d) Sample 4 loaded below macroscopic yield stress at room temperature.

As shown in Figure 2, identification marks were scratched on the painted
surfaces so that the area of the EBSD scan can be correlated to the DIC strain
measurements. A grid of nodes were generated within the EBSD scanned area,
with intervals of 25 pixels in the optical images ($\approx$37.5 $\mu$m). DIC
strain measurements were performed on the images captured by the camera via a
modified version of the Python based DIC code [17] so that the in-plane strain
components at the nodes can be calculated. The $\varepsilon_{yy}$ components
of the nodes in the central regions of the selected grains (dismissing the
nodes near the grain boundaries) were averaged to represent the averaged
strain along the load direction within particular grains. The selected grains
have a relatively larger area and contain more nodes and thus the average
values have a higher confidence when correlating areas between EBSD and DIC.
The strains within these selected grains were then used to calibrate the CPFE
model.

###  2.3 Crystal plasticity model and simulations

A crystal plasticity hypoelastic formulation was used as a constitutive model
for the finite element simulations [8]. It is implemented as a UMAT for Abaqus
based on the user-defined elements (UEL) subroutine by Dunne et al [8] and it
is based on the finite strain theory. The temperature dependent elastic
constants of the Ti-6Al alloy are shown in Table 2. The constitutive model
calculates the Cauchy stress $\bm{\sigma}$ based on the total strain increment
$\Delta\bm{\varepsilon}$ at each time step, which is calculated from the
deformation gradient $\mathbf{F}$ provided by the finite element solver. The
plastic strain $\bm{\varepsilon}_{p}$ is the internal variable that quantifies
the irreversible plastic deformation induced by the slip movement of
dislocations. The time evolution of the plastic strain takes into account
$N_{\textrm{slip}}=$30 slip systems in the HCP crystal [18, 19]:

$\dot{\bm{\varepsilon}}_{\textrm{p}}\left(\bm{\sigma}\right)=\frac{1}{2}\sum_{\alpha=1}^{N_{\textrm{slip}}}\dot{\gamma}_{\alpha}\left(\bm{\sigma}\right)\left(\bm{s}_{\alpha}\otimes\bm{n}_{\alpha}+\bm{n}_{\alpha}\otimes\bm{s}_{\alpha}\right)\
,$

where $\dot{\gamma}_{\alpha}$ is the plastic strain rate, $\bm{s}_{\alpha}$
the slip direction and $\bm{n}_{\alpha}$ the slip plane normal of the slip
system $\alpha$. $\bm{s}_{\alpha}$ and $\bm{n}_{\alpha}$ are expressed in the
sample reference frame.

Table 2: Elastic constants (GPa) of Ti6Al in Voigt notation [8, 16]. Temperature |  $\mathbb{C}_{11}$,$\mathbb{C}_{22}$ | $\mathbb{C}_{33}$ | $\mathbb{C}_{12}$ |  $\mathbb{C}_{13}$,$\mathbb{C}_{23}$ |  $\mathbb{C}_{44}$,$\mathbb{C}_{55}$ | $\mathbb{C}_{66}$
---|---|---|---|---|---|---
20 ∘C | 139.21 | 162.57 | 81.95 | 68.78 | 42.50 | 30.02
120 ∘C | 126.48 | 147.63 | 74.45 | 62.46 | 36.10 | 26.20
230 ∘C | 112.52 | 131.35 | 66.23 | 55.57 | 31.90 | 23.20

The slip systems used are reported in Table 3. $\bm{s}_{\alpha}^{0}$ and
$\bm{n}_{\alpha}^{0}$ are the slip directions and normals in the crystal
reference frame.

Table 3: Slip systems used in the model. Slip directions and normals are expressed in the crystal reference frame. The vectors are normalised. Slip system | $\bm{s}_{\alpha}^{0}$ | $\bm{n}_{\alpha}^{0}$
---|---|---
$<$a$>$ basal slip | $\left[-0.5,\pm 0.866,0\right]$ | $\left[0,0,1\right]$
| $\left[1,0,0\right]$ | $\left[0,0,1\right]$
$<$a$>$ prismatic slip | $\left[-0.5,\pm 0.866,0\right]$ | $\left[\mp 0.866,-0.5,0\right]$
| $\left[1,0,0\right]$ | $\left[0,-1,0\right]$
$<$a$>$ 1st order pyramidal slip | $\left[-1,0,0\right]$ | $\left[0,\mp 0.7559,0.6547\right]$
| $\left[-0.5,-0.866,0\right]$ | $\left[\pm 0.6547,\mp 0.3780,0.6547\right]$
| $\left[-0.5,0.866,0\right]$ | $\left[\mp 0.6547,\mp 0.3780,0.6547\right]$
$<$c+a$>$ 1st order pyramidal slip | $\left[-0.3536,0.6124,\pm 0.7071\right]$ | $\left[\pm 0.6547,\mp 0.3780,0.6547\right]$
| $\left[-0.3536,-0.6124,\mp 0.7071\right]$ | $\left[-0.6547,-0.3780,\pm 0.6547\right]$
| $\left[0.7071,0,\pm 0.7071\right]$ | $\left[\mp 0.6547,\pm 0.3780,0.6547\right]$
| $\left[\mp 0.7071,0,0.7071\right]$ | $\left[\pm 0.6547,\pm 0.3780,0.6547\right]$
| $\left[-0.3536,-0.6124,\pm 0.7071\right]$ | $\left[0,\pm 0.7559,0.6547\right]$
| $\left[\pm 0.3536,\mp 0.6124,0.7071\right]$ | $\left[0,\pm 0.7559,0.6547\right]$
$<$c+a$>$ 2nd order pyramidal slip | $\left[-0.3536,0.6124,\pm 0.7071\right]$ | $\left[\pm 0.3536,\mp 0.6124,0.7071\right]$
| $\left[-0.3536,-0.6124,\pm 0.7071\right]$ | $\left[\pm 0.3536,\pm 0.6124,0.7071\right]$
| $\left[0.7071,0,\pm 0.7071\right]$ | $\left[\mp 0.7071,0.0000,0.7071\right]$

The plastic strain rate on each slip system is calculated using a physics-
based law (Orowan’s equation) that is a function of the resolved shear stress
(RSS) $\tau_{\alpha}=\bm{s}_{\alpha}\cdot\bm{\sigma}\cdot\bm{n}_{\alpha}$ on
the respective slip system [8, 20, 21, 22]:

$\dot{\gamma}_{\alpha}\left(\bm{\sigma}\right)=\rho_{m}b_{\alpha}^{2}\nu\exp\left(-\frac{\Delta
F_{\alpha}}{kT}\right)\sinh\left(\frac{\left(\left|\tau_{\alpha}\right|-\tau_{\alpha}^{c}\right)\Delta{V}_{\alpha}}{kT}\right)\mathrm{sign}(\tau_{\alpha})H\left(\left|\tau_{\alpha}\right|-\tau_{\alpha}^{c}\right)\
.$ (1)

$\rho_{m}$ is a constant, representing the average value of the mobile
dislocation density. $b_{\alpha}$ is the Burgers vector magnitude and $\nu$ is
the characteristic frequency at which a dislocation attempts to overcome the
Helmoltz free energy barrier $\Delta F_{\alpha}$. Once this barrier is
overcome, the dislocation movement can induce plastic slip.
$\Delta{V}_{\alpha}$ is the activation volume of the process. $k$ is the
Boltzmann constant and $T$ is the temperature. $\tau_{\alpha}^{c}$ is the
critical resolved shear stress (CRSS) of the slip system $\alpha$ and $H$ is
the Heaviside function. Therefore, plastic deformation on one slip system is
induced only when the magnitude of RSS exceeds the CRSS.

At each time increment $\Delta t$, the increment of the Cauchy stress in the
sample reference frame is calculated as [23, 24]:

$\displaystyle\Delta\bm{\sigma}$
$\displaystyle=\mathbb{C}\Delta\bm{\varepsilon}_{e}+\left(\bm{W}_{e}\bm{\sigma}_{0}-\bm{\sigma}_{0}\bm{W}_{e}\right)\Delta
t$ (2)
$\displaystyle=\mathbb{C}\left(\Delta\bm{\varepsilon}-\dot{\bm{\varepsilon}}_{\textrm{p}}\left(\bm{\sigma}\right)\Delta
t\right)+\left(\bm{W}_{e}\bm{\sigma}_{0}-\bm{\sigma}_{0}\bm{W}_{e}\right)\Delta
t\ ,$

where $\bm{\sigma}_{0}$ is the Cauchy stress at the previous increment and
$\bm{W}_{e}$ is the elastic continuum spin [25]. Reorientation of the grains
due to deformation is also included in the model. The constant parameters
input into the model are summarised in Table. 4.

Table 4: Constant parameters in the model [8, 26, 27]. Mobile dislocation density | $\rho_{m}$ | 5 $\mu$m-2
---|---|---
Burgers’ vector (basal and prismatic slip) | $b_{\alpha}$ | 0.295 nm
Burgers’ vector (pyramidal slip) | $b_{\alpha}$ | 0.5533 nm
Characteristic attempt frequency | $\nu$ |  $1\times 10^{11}$ s-1
Boltzmann constant | $k$ | 1.38$\times 10^{-23}$ JK-1

The other three terms, $\Delta F_{\alpha}$, $\Delta{V}_{\alpha}$ and
$\tau_{\alpha}^{c}$ are temperature dependent. Values of $\Delta F_{\alpha}$
and $\Delta{V}_{\alpha}$ input into the model can be found in Figure 6,
following the previous work [28]. $\tau_{\alpha}^{c}$ for prismatic and basal
slip were the target values we seek to obtain in the optimisation procedure
carried out in this work. $<c+a>$ Pyramidal slip is much more difficult to
activate due to its higher $\tau_{\alpha}^{c}$, which is about three times the
value of basal slip at temperature between 20 ∘C and 300 ∘C [5, 16, 29]. The
values of $\tau_{\alpha}^{c}$ for pyramidal slip was set to be three time the
value of basal slip in this work.

The geometrically necessary dislocation (GND) density is calculated using a
least square minimisation [30]. The sum of the squares of the GND densities on
the different slip systems:
$\sum_{\alpha=1}^{N_{\textrm{slip}}}\rho_{\alpha}^{2}$ is minimised. A
constrained minimisation algorithm is applied. Given the simulated plastic
deformation gradient $\mathbf{F}_{p}$, the GND densities $\rho_{\alpha}$ must
satisfy the condition:

$\left(\nabla\times\mathbf{F}_{p}\right)^{T}=\sum_{\alpha=1}^{N_{\textrm{slip}}}\rho_{\alpha}\bm{b}_{\alpha}\otimes\bm{l}_{\alpha}\
,$ (3)

where $\bm{b}_{\alpha}$ is the Burgers vector and $\bm{l}_{\alpha}$ is the
dislocation line unit vector for the $\alpha^{th}$ slip system. The right-hand
side of equation (3) is known as Kröner-Nye tensor [31]. The dislocation
energy is minimised while respecting the condition in equation (3), therefore
the different $\rho_{\alpha}$ can be calculated at each time step of the
simulation. More details of the algorithm are given in [21]. Analysing the GND
density is useful to understand the relative activation of the slip systems
and the incompatibility of the plastic deformation between neighbouring
grains, as described in section 3.2. The crystal plasticity code is available
in the following repository [32].

Figure 5: Example of the meshed model of sample 4, the depth of the model was
2 elements. Arrows show the boundary condition of the model. Grains were
rendered with different colours so that the microstructure was found to
consistent with the EBSD map. Figure 6: Input parameters of (a) activation
volume ($\Delta{V}$) and (b) activation energy ($\Delta{F}$) as a function of
temperature [28].

Figure 5 shows the representative volume of sample 4, which is generated using
the EBSD map and a MATLAB function called ebsd2abaqus [33]. Grain morphology
and orientations were the same as the real samples. The element type was set
to be quadratic with reduced integration (C3D20R) and meshed as a regular,
equisized grid of size matched to 8 $\mu$m. The model was a 3D model with a
depth of 2 elements, the 2 layers of elements were identical (validation can
be found in the Appendix). The movement of the left face, bottom face and back
face were constrained along the X, Y and Z direction respectively.
Displacement was applied to the top face so that the overall strain of the
model matches with the averaged strain of the sample (as shown in Figure 4).
Within each grain, nodes on the front surface in the central region of the
grain (again the nodes near the grain boundaries were dismissed, to be a
closer estimate to DIC measurements) were grouped to represent the grain.
Strain component $\varepsilon_{yy}$ of these nodes was averaged and matched
with the experimentally measured DIC strain, as described in the next section.

##  3 Results

###  3.1 Critical resolved shear stress

The CRSS for the two main slip systems (prism and basal) were obtained by
calibrating the CPFE model. The optimisation was achieved using a Scipy
package [34], by use of the Nelder-Mead method [35, 36]. In a typical loop, a
set of initial guesses of the CRSS values were input into the model and the
simulation job was submitted. Once the simulation was complete, strains from
the 8 selected grains in each sample were compared with the experimental
strain results of these grains. The differences between simulated strains and
experimental strains were calculated and was aimed to be minimised. The
Nelder-Mead algorithm then generated a new set of CRSS to re-run the
simulation. This iterative process was carried out until the difference
between simulated and experimental strain in the grains was minimal.

Figure 7 shows the comparison of the strains within the 8 chosen grains
between experiments and simulations. The optimisation method was applied to
samples 1-3, after which the difference between simulated and experimental
strain in the grains becomes lower than 10%. All 3 samples show good agreement
in experimental and simulated strains. The CRSS values of prismatic and basal
slip systems were extracted at these three temperatures (as shown in Figure
8(a)). CRSS values at room temperature were then used for sample 4. As shown
in Figure 7(g) and Figure 7(h), the simulated strains still fit well with the
experimental strains.

The relationship between $\tau_{\alpha}^{c}$ and temperature can be fitted
using an empirical power law function [37, 38]. The resulting temperature
dependence of the CRSS of the prismatic and basal slip systems are given by:

$\tau_{basal}^{c}=1054.6{T}^{-0.472}\ \textrm{(MPa)}$ (4)
$\tau_{prism}^{c}=733.8{T}^{-0.4622}\ \textrm{(MPa)}$ (5)

where $T$ is the temperature in ∘C.

There are also other workers fitted the CRSS vs. temperature relationship with
quadratic functions [16, 39]. However, for our data-set, false minima were
found at relatively low temperatures (150-200 ∘C as shown in Figure 8(b)).
Although quadratic function can provide a better fitting, the trend of CRSS
vs. temperature predicted by quadratic functions cannot reflect the true
material behaviour.

###  3.2 Slip activity and trace analysis

Figure 9 shows the geometrically necessary dislocations (GND) density of the
three prism and the three basal slip systems in sample 4 at the end of creep.
These maps were meant to identify the active slip systems within different
grains. At the stress level of sample 4, the basal slip was barely activated.
By contrast, prism slip was activated in some specific grains, as shown by the
Prism #1 and Prism #3 GND density in Figure 9(d) and (f). There are two
regions of interests: pair 1 (grain 5 & 9) represents the ‘rogue grain pair’
where only one grain (grain 9) activates plastic slip; pair 2 (grain 6 & 7)
represents the ‘non-rogue grain pair’ where both grains activate plastic slip.

The simulated slip systems activities matches well with the experimental
observations. Figure 10 shows the comparison of grain morphology of the two
grain pairs before and after the experiment. Slip traces were observed in
grain 6, grain 7 and grain 9 but not in grain 5. All the newly generated slip
lines were found to be parallel within each grain, indicating only one slip
system was activated for each grain. This agreed well with the simulated slip
activity, where Prism 1 was activated in grain 7, Prism 3 was activated in
grain 6 and grain 9. Slip lines simulated by the CPFE model were plotted for
better comparison with the experimental observations. To do this, we used the
method introduced by Guery et al [40], where the slip plane normal,
$\textbf{{n}}_{\alpha}$ of the active slip system $\alpha$ was rotated into
the sample coordinate system by the grain rotation matrix, R. The slip line
vector, $\textbf{{t}}_{\alpha}$ was obtained by the cross product of the
rotated slip plane normal $\textbf{{n}}_{\alpha}^{s}$ and the sample surface
normal $\textit{{e}}_{z}$ as:

$\textbf{{t}}_{\alpha}=\textbf{{n}}_{\alpha}^{s}\times\textit{{e}}_{z}$ (6)

These line segments from every element within grain 6, grain 7 and grain 9
were then plotted (as shown in Figure 10). There is good agreement between
alignment of slip traces calculation in the model and experimental
observation. The length of the segment was proportional to the magnitude of
the averaged GND density of each element. The ratio of the averaged GND
density in grain 6, grain 7 and grain 9 was approximately 8:10:5 (i.e. the
slip activity was the highest in grain 7 and the lowest in grain 9). This is
also consistent with the slip line density observed in these grains.

###  3.3 Load shedding

As shown in Figure 11(a) and 11(b), along path A and path B in grain pair 1
and 2 respectively, the load shedding effect in ‘rogue’ and ‘non-rogue’ grain
pairs has been studied. Figure 11(c)-(h) show the stress evolution along the
paths at 3 different temperatures. At room temperature, load shedding was
obvious in the ‘rogue’ grain pair (pair 1). Compared to the start of the creep
loading, the stress at the end of creep within the ‘hard’ grain was found to
increase while the stress within the ‘soft’ grain remained approximately at
the same level. As a result, the stress difference near the grain boundary
increased to a higher level of approximately 100 MPa. In comparison, load
shedding was not as significant in the ‘non-rogue’ grain pair, where stress
changes in both grains before and after the creep are smaller and the stress
difference at the grain boundary after creep was only approximately 50 MPa.

Higher temperature simulations have been performed on the model of sample 4
for better comparison, the parameters (CRSS, $\Delta{F}$, $\Delta{V}$ and
elastic constants) were changed to parameters at 120 ∘C and 230 ∘C but the
boundary condition remained the same. At higher temperatures, the stresses in
all these four grains were found to relax after creep, indicating plasticity.
This is also confirmed by the increase of GND density in these grains as
temperature rises and shown in Figure 11(i) and 11(j). GND densities at the
grain boundaries were significantly higher than that within the grains,
showing a dislocation pile-up [7, 41] at the grain boundaries. For the rogue
grain pair, the initial stress difference across the grain boundary increases
slightly for $T=120^{\circ}$C compared to room temperature, then drops at
higher temperature. But during the load hold, the stress difference does not
increase as some plasticity is now also possible in the hard grain (non-zero
GND density in the hard grain in Figure 11(i)).

Figure 7: Strain vs. time plots of experiments and simulations of (a) sample 1
grain 1-4; (b) sample 1 grain 5-8; (c) sample 2 grain 1-4; (d) sample 2 grain
5-8; (e) sample 3 grain 1-4; (f) sample 3 grain 5-8; (g) sample 4 grain 1-4;
(h) sample 4 grain 5-8 (Note that the scales of the vertical axes are
different the plots). Figure 8: Calibrated CRSS values of prismatic and basal
slip systems as a function of temperature: (a) fitted with power law
functions; (b) fitted with quadratic functions.

Figure 9: Slip activity of prism and basal slip in sample 4 at room
temperature, clear contrast were observed in (d) and (e), where Prism #1 was
activated in grain 7 and Prism #3 slip was activated in grain 6 and grain 9.

Figure 10: Images of the two grain pairs before and after the creep test and
simulated slip traces from the CPFE model.

Figure 11: Path within the grain pairs of (a) pair 1 (path A); (b) pair 2
(path B); S22 component (along the loading direction) of (c) path A at room
temperature; (d) path B at room temperature; (e) path A at 120 ∘C; (f) path B
at 120 ∘C; (g) path A at 230 ∘C; (h) path B at 230 ∘C; GND density at the end
of creep of (i) path A and (j) path B. (Note that the vertical axes are not
the same in each plot).

##  4 Discussion

The CRSS values obtained in this work are in sound agreement with the values
reported in literature [5, 16, 29, 42, 43, 44], but are perhaps slightly lower
particularly at elevated temperatures. Two reasons for these difference could
be: (1) the Ti6Al alloy tested in this work had lower Al content ($\approx$
5.8wt%) compared to that in the literature, (2) strain rates in our work
extend to lower values. Aluminium, as a common substitutional alloying
addition to Ti, has proved to have strengthening effect on both basal and
prism slip [29, 43, 44, 45]. The strengthening is more marked for prism slip
however so that the strengths of prism and basal slip systems are more similar
in Ti6Al than in pure Ti.

The grain orientations of the two grain pairs are shown in Figure 12. The
angles between the $c$-axis and the loading direction were 56∘ for grain 5,
83∘ for grain 9, 74∘ for grain 6 and 73∘ for grain 7. Although the grain pair
1 behaved as a ‘rogue’ grain pair in this experiment, it is not a typical
hard-soft grain combination. In grain pair 1, grain 9 was in ‘soft’
orientation but grain 5 was not a typical ‘hard’ grain and in fact it yields
in the simulations for 120∘C and 230∘C. This is consistent with the
observation by Brandes et al [14], where dwell fatigue facet was observed in a
material lacking ‘hard’ orientated grains (with their $c$-axis nearly parallel
to the loading direction). These non-typical ‘rogue’ grain pair could also
lead to dwell fatigue failure. Due to the stress level being quite low to
activate basal slip in this experiment, grains oriented favourably for prism
slip systems were classified as ‘soft’ grains. Xu et al [7] suggested that the
cold dwell fatigue and the dwell debit is sensitive to the dwell stress, which
agrees well with the observations in this work. The origin of the dwell debit
is the activation of slip systems, resulting in an accumulation of plastic
strain during the stress dwell, which leads to load shedding from ‘soft’
grains to neighbouring ‘hard’ grains. High stress difference at the soft-hard
grain boundary and stress increase in the ‘hard’ grain were observed and
believed to lead to a nucleation of facets within the ‘hard’ grain [5, 11,
46]. As external stress changes, the activation of slip systems changes, thus
the definition of ‘soft’ and ‘hard’ grains will vary dynamically. In reality
the binary hard-soft nomenclature is in fact a continuum of varied relative
grain strengths which are affected by external conditions such as temperature
and macroscopic stress level.

Schmid factor analysis of these grains can be found in Table. 5. The Schmid
factor of the active slip systems are marked as red in the table. It is found
that the actual slip systems activated were not always the ones with the
highest Schmid factor. This implies that the local stress states is different
from the macroscopic stress state arising from the loading direction on the
sample itself. The activation of slip systems is thus not always dependent on
only that particular grain orientation with respect to the macroscopic load
direction, but the local stress states plays a more crucial role. As a result,
predicting the ‘hard’ and ‘soft’ grains based on macroscopic load direction
and grain orientations in polycrystal samples is not always viable.

In Figure 10, grain 6 and the grain at its bottom right have very similar
crystallographic orientations (refer to Figure 1 along the loading direction).
Slip transfers easily across the boundaries between these similar orientated
grains, which behave as if the boundary between them was not present.
Partially correlated group of grains often arrive in Ti alloys and are known
as macrozones and increase the effective grain or structural unit size [47,
48]. Creep can happen in a ‘soft’ macrozone that is orientated favourably for
slip, which leads to load shedding to a neighbouring ‘hard’ macrozone
orientated poorly for slip and crack can initiate within the ‘hard’ macrozone
[49, 50, 51].

Figure 12: Grain orientations of the two grain pairs. Table 5: Schmid factor of the three prism slip systems in the two grain pairs, the active slip systems were marked as red. | Grain 5 | Grain 9 | Grain 6 | Grain 7
---|---|---|---|---
Prism 1 | 0.026 | 0.356 | 0.374 | 0.299
Prism 2 | 0.095 | 0.170 | 0.110 | 0.024
Prism 3 | 0.121 | 0.185 | 0.264 | 0.061

Load shedding was found to be sensitive to temperature. As the energy required
to activate basal and prismatic slip decreases when temperature increases
[28], more plasticity was activated at higher temperatures. Stresses were
found to relax in both ‘rogue’ and ‘non-rogue’ grain pairs. Although the
stresses were relaxed, the differences in stress at the grain boundaries still
exist. Since the stress level in each grain pair was different, direct
comparison of the stress difference was not possible. Instead, the percentage
of stress difference was calculated by dividing the averaged stress along the
paths. Figure 13(a) shows the percentage of stress difference at the grain
boundaries at the end of the creep against temperature. At room temperature
and 120 ∘C, stress differences in grain pair 1 were significantly higher than
that in grain pair 2. At 230 ∘C, the stress differences in the two pairs
altered but both of them reduce to relatively low levels. It was found that in
both grain pairs, the percentage stress difference increased as temperature
increased to 120 ∘C and decreased as the temperature further increased to 230
∘C. Among the three temperature assessed, 120 ∘C was found to be the worst
case scenario. To reveal the mechanism leading to this observation, the
activity of slip systems need to be quantified.

Figure 13: (a) Percentage of stress difference at the grain boundaries of the
two grain pairs at the end of the creep vs. temperature; (b) Strain rate
sensitivity of basal and prism slip vs. temperature.

Strain rate sensitivity (SRS) are often used to express the slip activity in
literature and it is considered to be a significant factor that controls load
shedding [52]. The strain rate sensitivity exponent, $m$, is calculated by:

$m=d(log(\sigma))/d(log(\dot{\varepsilon}))$ (7)

which corresponds to the gradient of a $log(\sigma)$ vs.
$log(\dot{\varepsilon})$ plot. The $m$ value is crystallographic orientation
dependent [42], where $m$ values of ‘soft’ grains are typically higher than
that of ‘hard’ grains at room temperature [52, 53]. This phenomenon is deemed
to be related to the different SRS in different slip systems, which is
confirmed by Jun et al [54], who reported that in $\alpha$-Ti prism slip had
higher SRS ($m=0.07$) over basal slip ($m=0.03$) at room temperature. A more
recent study [28] shows that the SRS is also temperature dependent and higher
SRS of a slip system leads to more creep strain accumulation during a stress
dwell (i.e. higher activity of the slip system). We attempted to evaluate the
SRS for the two major slip systems (prism and basal) at different temperatures
in order to interpret the worst case scenario temperature.

The three temperature dependent key parameters $\Delta{V}$, $\Delta{F}$ and
$\tau_{c}$ were fitted with quadratic, linear and power law functions [16, 28,
39, 55, 37, 38] so that the values can be predicted at different temperatures
other than experimentally tested (recall Figure 6 and Figure 8). The three
parameters were calculated for every 10 ∘C between the range of 20 ∘C to 300
∘C and were inserted into the slip law (recall equation. 1). This allowed
applied shear stress, $\tau_{\alpha}$ against shear strain rate,
$\dot{\gamma}_{\alpha}$ to be plotted at log scale over a strain rate range of
10-7 to 10-5 s-1 (the creep strain rate range observed in this work). Linear
fitting was performed on the $\dot{\gamma}_{\alpha}$ vs. $\tau_{\alpha}$
curves and the gradients were obtained to represent the $m$ values. The linear
fitting errors are shown by the error bars. The strain rate sensitivity
variation against temperature is shown in Figure 13(b). More details can be
found in the previous work [28]. Prism slip has higher SRS over basal slip
between room temperature and 300 ∘C. The SRS as a function of temperature for
both prism and basal slip follow the same trend as the stress difference at
grain boundaries (see Figure 13(a)). The peak values of $m$ for both prism and
basal slip can be expected at approximately 80 ∘C. The peak temperature is in
remarkable agreement with the worst case scenario temperature for dwell debit
from the experience of aero engine industry (between 90 ∘C and 120 ∘C) [16,
56]. The worst dwell debit was caused by the highest SRS of the two major slip
systems (i.e. the highest slip activities). During stress dwell at the worst
case scenario temperature, high slip activity results in a greater creep
strain accumulation in the ‘soft’ grains which in turn cause dislocation pile-
up at the soft-hard grain boundaries and load shedding which generates a high
stress difference at the grain boundaries. At higher temperatures (e.g. above
200 ∘C), the SRS of both prism and basal slip systems reduced to lower levels
compared to room temperature. The slip activity of the two major slip systems
were low at higher temperatures, which explained the phenomenon that cold
dwell fatigue diminished at temperatures above 230 ∘C [16].

##  5 Summary

This work utilised DIC technique and CPFE modelling to study the temperature
effect on cold dwell fatigue in Ti6Al alloy. CRSS of the two major slip
systems (prism and basal) of $\alpha$-Ti were quantified as a function of
temperature by calibrating the CPFE model from experimentally measured strain
evolution in individual grains during creep. Load shedding was clearly
observed in the ‘rogue’ grain pair, resulting in an increase in stress within
the ‘hard’ grain and high stress difference at the grain boundary. As
temperature increased, the stress difference became larger at 120 ∘C. At
higher temperatures (above 230 ∘C), the stress difference dropped to a
reasonably low level.

Grain orientation analyses showed that the ‘soft’ and the ‘hard’ grain would
dynamically vary depending on the applied stress and operating temperature,
which results from the activation of different slip systems. Local stress
state plays a more important role in the activation of slip systems compared
to grain orientations with respect to macroscopic load direction in large
polycrystals.

The strain rate sensitivity for prism slip is higher than that of basal at
temperatures between room temperature and 300 ∘C. The SRS of both prism and
basal slip show an increase-and-decrease trend as temperature increases. The
peak SRS is expected to be at approximately 80 ∘C for both prism slip and
basal slip. The worst case scenario temperature for dwell debit (between 90 ∘C
and 120 ∘C) can be explained by the highest SRS and slip activity of the two
major slip systems at these temperatures.

## Data Availability

The EBSD maps and DIC images recorded during this experiment will be made
openly available on the website https://zenodo.org/. Scripts for post-
processing are openly available on Github: https://github.com/sann6001

## Author Contributions

Yi Xiong: Data Curation, Formal Analysis, Methodology, Investigation,
Software, Validation, Visualisation, Writing – Original Draft, Writing –
Review & Editing

Nicolò Grilli: Data Curation, Formal Analysis, Methodology, Software, Writing
– Original Draft, Writing – Review & Editing

Phani S Karamched: Data Curation, Investigation, Methodology, Software,
Validation, Writing – Review & Editing

Bo-Shiuan Li: Investigation, Methodology, Writing – Review & Editing

Edmund Tarleton: Data Curation, Formal Analysis, Software, Supervision,
Writing – Review & Editing

Angus J Wilkinson: Conceptualization, Funding Acquisition, Methodology,
Investigation, Project Administration, Supervision, Visualisation, Writing –
Review & Editing

## Acknowledgements

We are grateful for use of characterisation facilities within the David
Cockayne Centre for Electron Microscopy, Department of Materials, University
of Oxford, which has benefitted from financial support provided by the Henry
Royce Institute (Grant ref EP/R010145/1). YX expresses gratitude to the
financial support of China Scholarship Council (CSC) and ET acknowledges EPSRC
for support through Fellowship grant (EP/N007239/1).

## Appendix A: Justification of the depth of the model

Three models with depth of 1 layer of elements, 2 layers of elements and 4
layers of elements were created for sample 1. Simulations were performed using
same set of parameters (not the final optimised parameters). Figure A1 shows
the comparison of the strains of the grain 5 and the grain 7 from the three
models and experimental results. It is found that as the depth of the models
increased from 1 element to 4 element, the simulated strain decreased.
However, the difference in strains of the three models is relatively small,
which did not affect their fittings with the experimental strains. To ensure
enough degree of freedom of the elements on the top layer (where simulation
results output from) and efficiency of the simulations, the model depth was
set to be 2 elements for simulations presented in this paper.

Figure A1: Comparison of the strain in the models with 1 element, 2 elements
and 4 elements in depth and the experimental strain of (a) grain 5 and (b)
grain 7 in sample 1.

## Appendix B: Macroscopic stress

Figure B1 shows the comparison of macroscopic stresses between experiments and
simulations for the four samples. The experimental stress was estimated by
dividing the initial sample cross-section area from the force recorded by the
mechanical test frame. In simulations, forces were applied by displacement and
we simplified the curved strain vs. time curves with several straight line
segments (recall Figure 4(a)-(c)). This results in the observation of stress
relaxation periods in the simulation stress, particularly for sample 1 and
sample 2. Generally speaking, the macroscopic stress for experiments and
simulations are in good agreement.

Figure B1: Comparison of the macroscopic stresses of experiments and
simulations for (a) sample 1 at room temperature; (b) sample 2 at 120∘C; (c)
sample 3 at 230∘C and (d) sample 4 at room temperature.

## Appendix C: Fitting errors and optimisation efficiency

The fitting errors and number of iterations are summarised in Table C1. The
optimisation procedures were performed on a high function PC with an Intel
Xeon(R) CPU E5-2687W v2 @ 3.40GHz with 8 cores. It took approximately 80
minutes for one iteration of the Nelder-Mead method to complete. Therefore,
the completion of the optimisation procedure for one sample could take 2-4
weeks. Using the automatic optimisation tool could save time for people but
could also occupy the computer resources. The choice of the initial parameters
that input into the Nelder-Mead method is extremely important. Sample 4, for
example, using the calibrated CRSS values as input has the lowest averaged
error among these four samples. The efficiency of optimisation could be
improved by a more accurate estimation of the materials properties in our
case. More advanced machine learning method [57] could also be applied to
improve the efficiency.

Table C1: Averaged difference in strain between the simulations and experiments of the 8 selected grains in four samples and approximate number of iterations of the Nelder-Mead method. | Averaged errors | No. of iterations
---|---|---
Sample 1 | 4.3% |  $\approx$300
Sample 2 | 7.6% |  $\approx$500
Sample 3 | 6.8% |  $\approx$500
Sample 4 | 3.1% | NA

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