# Characterisation of slip and twin activity using digital image correlation
and crystal plasticity finite element simulation: Application to orthorhombic
$\alpha$-uranium

Nicolò Grilli  Philip Earp  Alan C.F. Cocks  James Marrow  Edmund Tarleton
<EMAIL_ADDRESS>Department of Engineering Science,
University of Oxford, Parks Road, OX1 3PJ, UK  Department of Materials,
University of Oxford, Parks Road, OX1 3PH, UK

###### Abstract

Calibrating and verifying crystal plasticity material models is a significant
challenge, particularly for materials with a number of potential slip and twin
systems. Here we use digital image correlation on coarse-grained
$\alpha$-uranium during tensile testing in conjunction with crystal plasticity
finite element simulations. This approach allows us to determine the critical
resolved shear stress, and hardening rate of the different slip and twin
systems. The constitutive model is based on dislocation densities as state
variables and the simulated geometry is constructed from electron backscatter
diffraction images that provide shape, size and orientation of the grains,
allowing a direct comparison between virtual and real experiments. An
optimisation algorithm is used to find the model parameters that reproduce the
evolution of the average strain in each grain as the load is increased. A
tensile bar, containing four grains aligned with the load direction, is used
to calibrate the model with eight unknown parameters. The approach is then
independently validated by simulating the strain distribution in a second
tensile bar. Different mechanisms for the hardening of the twin systems are
evaluated. The latent hardening of the most active twin system turns out to be
determined by coplanar twins and slip. The hardening rate of the most active
slip system is lower than in fine-grained $\alpha$-uranium. The method
developed in the present research can be applied to identify the critical
resolved shear stress and hardening parameters of other coarse-grained
materials.

###### keywords:

Crystal Plasticity , Dislocations , Twinning , Tensile test , EBSD , Digital
Image Correlation , Uranium

††journal: Journal of the Mechanics and Physics of Solids

##  1 Introduction

Hardening in metals is due to interactions within, and between, the slip and
twin systems (Kalidindi, 1998). The nucleation and motion of dislocations and
twin nucleation and migration occur when the critical resolved shear stress
(CRSS) for these deformation mechanisms is reached (Devincre et al., 2008;
Ojha et al., 2014). Since the seminal research of (Asaro, 1983), there has
been significant effort dedicated towards the development of crystal based
models for the inelastic deformation of engineering materials. Models of this
type can contain a large number of material parameters, particularly if there
are a number of different slip and twin systems that can be activated. A
challenge then is to develop robust experimental procedures for the
determination of these parameters. This either requires conducting a series of
different experiments on polycrystalline materials covering a range of loading
conditions and or stress states, or testing single crystals with different
crystallographic configurations with respect to the loading direction
(Franciosi et al., 1979). For a wide range of materials, particularly if they
have limited ductility, each of these approaches might be impractical or even
impossible to undertake. This is the case for the primary material of interest
here - $\alpha$-uranium (Inouye and Schaffhauser, 1969; Lander et al., 1994).

In this paper we demonstrate how the material parameters can be determined
from a single uniaxial test on a coarse grained material by comparing the
detailed strain and displacement fields within the individual grains
determined using digital image correlation (DIC) with crystal plasticity
finite element (CPFE) simulations. For $\alpha$-uranium, this allows us to
determine the 8 material parameters that describe the response of the
different slip and twin systems that are active at room temperature. The
approach can be readily generalised to other material systems.

Polycrystalline samples with columnar grains and arbitrary boundary conditions
can be modelled using the crystal plasticity finite element (CPFE) method
(Roters et al., 2018), which considers the anisotropic elastic behaviour,
plastic deformation due to slip and twinning. Given the grain orientation,
CPFE simulations provide the strain distribution in each grain (Dunne et al.,
2007) and can be compared to digital image correlation (DIC) measurements (Lim
et al., 2014).

The CPFE method has been widely used in conjunction with DIC measurements. For
instance, DIC measurements can determine the local strain in polycrystalline
samples, which is used for small scale CPFE simulations (Nellessen et al.,
2015; Grilli et al., 2018). CPFE simulations are able to predict the strain
localization (Irastorza-Landa et al., 2016) and lattice rotation (Irastorza-
Landa et al., 2017b) in single crystals and oligocrystals, as detected by high
resolution DIC (Guan et al., 2017). The DIC results can be correlated with the
presence of geometrically necessary dislocations near grain boundaries (Roters
et al., 2004). DIC has also been used to correlate strain concentrations with
fracture paths (Khan and Marrow, 2009). As shown in the present study, the
comparison between CPFE simulations and DIC measurements allows us to find the
CRSS and hardening coefficients for the plastic deformation mechanisms in
coarse-grained $\alpha$-uranium.

In the CPFE method, the CRSS and hardening of the slip systems can be
described by dislocation-based models, where the dislocation densities on the
slip systems are the state variables (Grilli et al., 2015). An increase in
dislocation density leads to hardening, usually described using a matrix
detailing the strength of interactions between slip systems (Arsenlis and
Parks, 2002). The dislocation density increase in $\alpha$-uranium during
deformation has not been systematically measured and few discrete dislocation
dynamics studies appear in the literature (Behmer, 2018), which focus on the
effect of twin thickness on dislocation pile-up and yield stress. The CRSS for
slip and twinning have been identified for single crystal (Daniel et al.,
1971) and fine-grained $\alpha$-uranium (McCabe et al., 2010), obtained by hot
or cold rolling. The interaction matrix between slip systems has been
calibrated for fine-grained $\alpha$-uranium (McCabe et al., 2010) using
polycrystal simulations and the visco-plastic self-consistent (VPSC)
homogenization scheme (Hutchinson and Hill, 1970; Lebensohn and Tomé, 1993).
However, the CRSS and hardening have not been widely studied for coarse-
grained $\alpha$-uranium.

In the CPFE method, the interaction between twins and dislocations can be
described using a CRSS for twin systems that depends on the accumulated shear
on the slip systems (Abdolvand and Daymond, 2013a). The hardening rate of
existing twins is also governed by the activity of coplanar and non-coplanar
twins (Roters et al., 2010). This is based on the observation that, in FCC
metals, deformation twinning initially produces only coplanar twins, while
non-coplanar twin variants appear at larger strains (Kalidindi, 2001). This
has led to the introduction of self hardening and latent hardening
coefficients that couple the CRSS for twinning with the twin volume fraction
(Abdolvand and Daymond, 2013a). These coefficients are set to zero in
constitutive models describing fine-grained $\alpha$-uranium, calibrated using
stress-strain curves (McCabe et al., 2010) and twin volume fraction
measurements (Knezevic et al., 2012). However, polycrystal simulations and
fine-grained samples, in which thousands of grains are considered and the twin
volume fraction is low (Inouye and Schaffhauser, 1969), are not ideal to study
twin-slip and twin-twin interactions. The same holds true for neutron
diffraction experiments (Earp et al., 2018; Grilli et al., 2019a, b), in which
the measured lattice strain is averaged over a millimetre-sized gauge volume
that contains many grains.

Coarse-grained $\alpha$-uranium, obtained by casting, has a base centred
orthorhombic crystal structure (Jacob and Warren, 1937; Lukesh, 1949; Kapoor
et al., 2015) and grains reaching a size of several millimetres (Garlea et
al., 2013). It is ideal to study slip-twin interactions because of the
propensity to form several twins after quenching or deformation (Cahn, 1951,
1953). This is because the CRSS for twinning is comparable to the CRSS for
slip. Plastic deformation mechanisms in $\alpha$-uranium are strongly
influenced by temperature (Daniel et al., 1971; Lloyd and Barrett, 1966),
strain rate (Huddart et al., 1980), grain size (Taplin, 1967; Inouye and
Schaffhauser, 1969), grain shape (Taplin and Martin, 1963), loading conditions
(Zecevic et al., 2016a), hydrogen content (Calhoun et al., 2015) and alloying
elements (Yanzhi et al., 2015; Chen et al., 2017). At room temperature, the
most active slip system is the so-called “wall” slip system, which has the
lowest CRSS up to about 400$\degree$C, followed by the “floor” slip system
(Zecevic et al., 2016b). Other slip systems exist, such as the “chimney” slip
system, with two variants, and the “roof” slip system, with four variants
(Calhoun et al., 2018; Calhoun, 2016; Calhoun et al., 2013), but are observed
only at high temperatures ($T>873$ K) when the temperature is close to the
$\alpha\rightarrow\beta$ phase transition (Daniel et al., 1971; Kapoor et al.,
2015). The most commonly observed twin system is $\\{130\\}\langle
3\bar{1}0\rangle$ (Brown et al., 2009), but a small fraction of
'$\\{172\\}$'$\langle 3\bar{1}2\rangle$ twins are also observed (Zhou et al.,
2016). The '$\\{176\\}$'$\langle 512\rangle$ twin system is observed only at
high strain rate (Ho, 2012; Rollett et al., 1991).

As shown in the present study, tensile bars with a characteristic width of a
few millimetres can be cut out from a larger polycrystalline plate made of
coarse-grained $\alpha$-uranium. The extruded area can be chosen such that
only a small number of grains are present in the tensile bar aligned along the
loading direction. High quality metallic surfaces, suitable for electron
backscatter diffraction (EBSD) measurements, can be produced (Sutcliffe et
al., 2018) and the grain orientation can be found. In-situ digital image
correlation (DIC) measurements are carried out to measure the axial strain in
each grain and the lateral displacement of the tensile bar. The stress-strain
curve of each grain is subsequently obtained and compared with the simulated
curves obtained with the CPFE method. A nonlinear optimisation procedure,
based on the Nelder-Mead algorithm (Nelder and Mead, 1965), is used to find
the CRSS and hardening rates of different slip systems. The optimisation
procedure can validate the values of 4 parameters against data from a tensile
bar containing 4 grains. Two tensile bars are used to validate the model by
comparing the measured and simulated strain field. Different models for the
hardening of the twin systems are compared (Kalidindi, 2001).

The hardening of the most active slip system (“wall” slip) in the coarse-
grained material turns out to be lower than for fine-grained $\alpha$-uranium
(McCabe et al., 2010). It will be shown that the hardening of the most active
twin system $\\{130\\}\langle 3\bar{1}0\rangle$ due to coplanar twins and
“floor” slip is able to explain the experimental results, and are therefore
important mechanisms. This is possible by choosing a grain in which one
variant of this twin system has a Schmid factor greater than for the slip
systems. The experimental technique and computational method developed in the
present research can be applied to identify the CRSS and hardening parameters
of other coarse grained materials.

In Section 2 the crystal plasticity model for slip and twinning is described.
Section 2.2 contains the three different models for hardening of the twin
systems which were tested. Experimental details are reported in section 3.
Section 4 contains the simulation results and the optimisation procedure to
find the CRSS and hardening rates. The discussion and conclusions then follow
in sections 5 and 6 respectively.

##  2 Crystal plasticity finite element modelling

The crystal plasticity framework is based on the decomposition of the
deformation gradient $\bm{F}$ into an elastic part $\bm{F}_{e}$, describing
the stretch of the crystal lattice, and a plastic part $\bm{F}_{p}$,
describing slip and twinning (Roters et al., 2012):

$\bm{F}=\bm{F}_{e}\bm{F}_{p}\ .$ (1)

The plastic deformation gradient $\bm{F}_{p}$ is an eigenstrain that does not
contribute directly to the stress tensor. It has the following time evolution
law (Kalidindi, 1998):

$\displaystyle\bm{L}_{p}=\dot{\bm{F}}_{p}\bm{F}_{p}^{-1}=$
$\displaystyle\left(1-\sum_{\beta=1}^{N_{\textrm{twin}}}f_{\beta}\right)\sum_{\alpha=1}^{N_{\textrm{slip}}}\dot{\gamma}_{\alpha}\left(\bm{\sigma}\right)\bm{s}_{\alpha}\otimes\bm{n}_{\alpha}$
$\displaystyle+\sum_{\beta=1}^{N_{\textrm{twin}}}\dot{f}_{\beta}\left(\bm{\sigma}\right)\gamma^{\textrm{twin}}_{\beta}\bm{s}_{\beta}\otimes\bm{n}_{\beta}.$
(2)

The first term on the right hand side accounts for dislocation slip in the
untwinned lattice, while the second term accounts for plastic deformation due
to the creation of new twins. $\bm{L}_{p}$ is the plastic velocity gradient
which relates $\bm{F}_{p}$ to its rate of change, $\dot{\bm{F}}_{p}$.
$\dot{\gamma}_{\alpha}\left(\bm{\sigma}\right)$ is the dislocation slip rate
on slip system $\alpha$; defined by the slip direction, $\bm{s}_{\alpha}$, and
slip plane normal $\bm{n}_{\alpha}$. $f_{\beta}$ is the twin volume fraction
of twin system $\beta$ (defined by the twin direction, $\bm{s}_{\beta}$, and
twin plane normal $\bm{n}_{\beta}$) which increases at a rate of
$\dot{f}_{\beta}\left(\bm{\sigma}\right)$ whereas the shear produced by the
twin system, $\gamma^{\textrm{twin}}_{\beta}$, is a constant. Note, that both
$\dot{\gamma}_{\alpha}\left(\bm{\sigma}\right)$ and
$\dot{f}_{\beta}\left(\bm{\sigma}\right)$ are stress dependent, as discussed
in section 2.1. The slip and twin systems used in the simulations are reported
in table 1.

The slip directions and slip plane normals are rotated from the lattice
reference frame to the sample reference frame by a rotation matrix $\bm{R}$;
which represents the grain orientation. $\bm{R}$ is updated at each time
increment using the continuum elastic spin matrix, as explained in (Clausmeyer
et al., 2011).

To model the nonlinear mechanical behaviour, at every time increment $\Delta
t$ of the simulation, the Cauchy stress increment
$\Delta\bm{\sigma}=\bm{\sigma}-\bm{\sigma}_{0}$ is calculated as a function of
the deformation gradient, where $\bm{\sigma}_{0}$ and $\bm{\sigma}$ are the
Cauchy stress at the beginning and end of the time increment respectively.
This is given by the anisotropic Hooke’s law:

$\Delta\bm{\sigma}=\mathbb{C}\Delta\bm{\varepsilon}_{\textrm{e}}+\left(\bm{W}_{e}\bm{\sigma}_{0}-\bm{\sigma}_{0}\bm{W}_{e}\right)\Delta
t\ ,$ (3)

where $\mathbb{C}$ is the fourth order elasticity tensor (Fisher and McSkimin,
1958) and $\bm{W}_{e}$ is the continuum elastic spin (Belytschko et al.,
2014). The elastic constants for $\alpha$-uranium are reported in table 2.
Details of the calculation of the elastic part of the small strain increment
$\Delta\bm{\varepsilon}_{\textrm{e}}$ are reported in (Grilli et al., 2019a).
To find the stress increment that satisfies equations (1), (2) and (3), an
implicit equation for $\Delta\bm{\sigma}$ is solved using a Newton-Raphson
algorithm, as detailed in (Dunne et al., 2007; Dunne and Petrinic, 2006). The
crystal plasticity framework is implemented in a user material subroutine
(UMAT) for the finite element software Abaqus, which solves the equilibrium
equation (Erinosho et al., 2013): $\nabla\cdot\bm{\sigma}=0$.

Slip system | $\bm{s}_{\alpha}^{0}$ | $\bm{n}_{\alpha}^{0}$
---|---|---
$\alpha=1$ (wall) | $\left[1,0,0\right]$ | $\left[0,1,0\right]$
$\alpha=2$ (floor) | $\left[1,0,0\right]$ | $\left[0,0,1\right]$
$\alpha=3$ (chimney) | $\left[0.437,-0.899,0\right]$ | $\left[0.899,0.437,0\right]$
$\alpha=4$ (chimney) | $\left[0.437,0.899,0\right]$ | $\left[0.899,-0.437,0\right]$
$\alpha=5$ (roof) | $\left[0.241,-0.495,0.835\right]$ | $\left[0.0,0.860,0.510\right]$
$\alpha=6$ (roof) | $\left[-0.241,-0.495,0.835\right]$ | $\left[0.0,0.860,0.510\right]$
$\alpha=7$ (roof) | $\left[0.241,0.495,0.835\right]$ | $\left[0.0,0.860,-0.510\right]$
$\alpha=8$ (roof) | $\left[0.241,-0.495,-0.835\right]$ | $\left[0.0,0.860,-0.510\right]$
Twin system | $\bm{s}_{\beta}^{0}$ | $\bm{n}_{\beta}^{0}$
$\beta=1$ | $\left[0.825,-0.565,0\right]$ | $\left[0.565,0.825,0\right]$
$\beta=2$ | $\left[-0.825,-0.565,0\right]$ | $\left[-0.565,0.825,0\right]$
Table 1: Slip and twin systems used in the model (McCabe et al., 2010). The directions and plane normals are expressed in Cartesian coordinates and in the lattice coordinate system. $\mathbb{C}_{11}$ | $\mathbb{C}_{12}$ | $\mathbb{C}_{13}$ | $\mathbb{C}_{22}$ | $\mathbb{C}_{23}$ | $\mathbb{C}_{33}$ | $\mathbb{C}_{44}$ | $\mathbb{C}_{55}$ | $\mathbb{C}_{66}$
---|---|---|---|---|---|---|---|---
214.74 | 46.49 | 21.77 | 198.57 | 107.91 | 267.11 | 124.44 | 73.42 | 74.33
Table 2: Elastic constants (GPa) at room temperature for the orthorhombic
structure of $\alpha$-uranium (Fisher and McSkimin, 1958; Beeler et al., 2013)
in Voigt notation.

###  2.1 Constitutive model for slip and twinning

In this section the constitutive model to calculate the shear strain rate
$\dot{\gamma}_{\alpha}\left(\bm{\sigma}\right)$ for each slip system and the
rate of increase of the twin volume fraction for each twin system is reported.
A power law relationship is used for
$\dot{\gamma}_{\alpha}\left(\bm{\sigma}\right)$ (Asaro and Needleman, 1985):

$\dot{\gamma}_{\alpha}(\bm{\sigma})=\dot{\gamma}_{0}\left|\frac{\tau_{\alpha}}{\tau_{\alpha}^{c}}\right|^{n}\textrm{sign}(\tau_{\alpha})\
,$ (4)

where $\dot{\gamma}_{0}$ and $n$ are constants that determine the strain rate
dependence and rate sensitivity of the slip activity. $\tau_{\alpha}$ is the
resolved shear stress and $\tau_{\alpha}^{c}$ is the CRSS of slip system
$\alpha$, which depends on the dislocation densities. Similarly, the rate of
increase of the twin volume fraction depends on the resolved shear stress
$\tau_{\beta}$ on the $\beta$ twin system and is also assumed to follow a
power law (Abdolvand and Daymond, 2013a):

$\gamma^{\textrm{twin}}_{\beta}\dot{f}_{\beta}\left(\bm{\sigma}\right)=\begin{cases}\dot{\gamma}_{0}\left|\frac{\tau_{\beta}}{\tau_{\beta}^{c}}\right|^{n}\
,\ \ \mbox{if}\ \tau_{\beta}>0\ ,\\\ 0\ ,\ \ \ \ \ \ \ \ \ \ \ \mbox{if}\
\tau_{\beta}<0\ .\end{cases}$ (5)

The CRSS for twinning, $\tau_{\beta}^{c}$, is weakly temperature dependent
(Christian and Mahajan, 1995) and it can depend on dislocation density
(Abdolvand and Daymond, 2013a). The present model is not able to resolve
discrete twins, but only the average volume fraction can be studied.
$\tau_{\beta}^{c}$ has to be interpreted as the stress necessary to nucleate
individual twins (Remy, 1978). The stress necessary for twin propagation (Qiao
et al., 2016) and migration (Ojha et al., 2014) is typically lower than this
value. These softening mechanisms, leading to stress drops (Patriarca et al.,
2013), are not included in the present model.

A twin system can be activated only by a positive resolved shear stress
(Kalidindi, 1998). The CRSS for slip is linked to the evolution of dislocation
densities (Beyerlein and Tomé, 2008; Mecking and Kocks, 1981; Madec et al.,
2002):

$\tau_{\alpha}^{c}=\tau_{\alpha}^{0}+0.9\
b_{\alpha}\mu_{\alpha}\sqrt{\rho_{\alpha}^{\textrm{for}}}-0.086\
b_{\alpha}\mu_{\alpha}\sqrt{\rho^{\textrm{sub}}}\log\left({b_{\alpha}\sqrt{\rho^{\textrm{sub}}}}\right)\
,$ (6)

where $\tau_{\alpha}^{0}$ is the initial slip resistance, $b_{\alpha}$ is the
Burgers vector and $\mu_{\alpha}$ is the projected shear modulus (Grilli et
al., 2019a) of the $\alpha$ slip system. $\rho_{\alpha}^{\textrm{for}}$ is the
forest dislocation density, while $\rho^{\textrm{sub}}$ represents dislocation
debris that forms stable dislocation substructures at the onset of stage IV
hardening (Beyerlein and Tomé, 2008; Madec et al., 2002). The logarithmic term
in (6) represents the stress for gliding dislocations to bow out through
arrays of locked dislocations with spacing $1/\sqrt{\rho^{\textrm{sub}}}$.
This model was originally developed by Beyerlein and Tomé (2008) for pure Zr,
who provide further details.

The hardening rate is determined by the time evolution of the dislocation
densities (McCabe et al., 2010):

$\displaystyle\dot{\rho}_{\alpha}^{\textrm{for}}$
$\displaystyle=k_{\alpha}^{1}\frac{\left|\dot{\gamma}_{\alpha}(\bm{\sigma})\right|}{b_{\alpha}}\Bigg{[}\sqrt{\rho_{\alpha}^{\textrm{for}}}-\hat{d}_{\alpha}\rho_{\alpha}^{\textrm{for}}\Bigg{]}\
,$ (7) $\displaystyle\dot{\rho}^{\textrm{sub}}$
$\displaystyle=1800k_{1}^{1}\hat{d}_{1}\rho_{1}^{\textrm{for}}\sqrt{\rho^{\textrm{sub}}}\left|\dot{\gamma}_{1}(\bm{\sigma})\right|\
,$ (8)

where $k_{\alpha}^{1}$ are dimensionless parameters and $\hat{d}_{\alpha}$ is
the annihilation distance for slip system $\alpha$.

The initial slip resistances $\tau_{\alpha}^{0}$ for $\alpha=1,2,3$ (“wall”,
“floor”, “chimney” slip systems) are three of the four parameters used in the
optimisation procedure for the CRSS, described in section 4.1, and the
dimensionless parameters $k_{\alpha}^{1}$ are three of the four parameters
used in the optimisation procedure for the hardening.

###  2.2 Constitutive model for twin systems hardening

In the present study three hardening models for the twin systems are compared.
The CRSS of the twin systems $(130)[3\bar{1}0]$ and
$(\bar{1}30)[\bar{3}\bar{1}0]$ will be indicated by $\tau_{(130)}^{c}$ and
$\tau_{(\bar{1}30)}^{c}$ in the following. The corresponding twin volume
fractions will be indicated by $f_{(130)}$ and $f_{(\bar{1}30)}$. The first
hardening model takes into account the interaction between twin and slip
systems (Abdolvand and Daymond, 2013a). The CRSS for twinning is approximated
with a linear function of the forest dislocation densities:

Twin-slip: $\displaystyle\tau_{\beta}^{c}$
$\displaystyle=\tau_{\beta}^{0}+k_{\beta}^{1}\mu_{\beta}b_{\beta}\sum_{\alpha=1}^{N_{\textrm{slip}}}b_{\alpha}\rho_{\alpha}^{\textrm{for}}\
.$ (9)

The second hardening model (Roters et al., 2010) takes into account the
interaction between the noncoplanar twin systems $(130)[3\bar{1}0]$ and
$(\bar{1}30)[\bar{3}\bar{1}0]$:

Twin noncoplanar: $\displaystyle\tau_{(130)}^{c}$
$\displaystyle=\tau_{\beta}^{0}+k_{\beta}^{1}\mu_{\beta}f_{(\bar{1}30)}\ ,$
(10) $\displaystyle\tau_{(\bar{1}30)}^{c}$
$\displaystyle=\tau_{\beta}^{0}+k_{\beta}^{1}\mu_{\beta}f_{(130)}\ .$ (11)

The third hardening model (Roters et al., 2010) also includes the interaction
between coplanar twin systems:

Twin coplanar: $\displaystyle\tau_{\beta}^{c}$
$\displaystyle=\tau_{\beta}^{0}+k_{\beta}^{1}\mu_{\beta}\left(f_{(130)}+f_{(\bar{1}30)}\right)\
.$ (12)

$\tau_{\beta}^{0}$ is the initial twin resistance, $\mu_{\beta}$ is the
projected shear modulus (Grilli et al., 2019a) on the twin system and
$k_{\beta}^{1}$ is a dimensionless interaction parameter. $\tau_{\beta}^{0}$
is the fourth parameter used in the optimisation procedure for the CRSS,
described in section 4.1. $k_{\beta}^{1}$ is the fourth parameter used in the
optimisation procedure for the hardening. The three hardening models are
referred to as the twin-slip, twin-noncoplanar and twin-coplanar model from
here on. The complete set of model parameters are reported in Table 3.

Slip law parameters | Eqn. (4)
---|---
Plastic strain rate coefficient ($\dot{\gamma}_{0}$) | 0.001 s-1
Plastic strain rate exponent ($n$) (Irastorza-Landa et al., 2017a) | 20
Twin law parameters | Eqn. (5)
Magnitude of shear due to twinning ($\gamma^{\textrm{twin}}_{\beta}$) (Cahn, 1953) | 0.299
Initial twin volume fraction $f_{\beta}\left(t=0\right)$ | 0
Slip hardening law parameters | Eqn. (6)
Constant friction stress (roof slip) ($\tau_{5}^{0}$;$\tau_{6}^{0}$;$\tau_{7}^{0}$;$\tau_{8}^{0}$) (Calhoun et al., 2013) | 235 MPa
Burgers vector (wall slip) ($b_{1}$) (Lukesh, 1949) | 0.285 nm
Burgers vector (floor slip) ($b_{2}$) (Lukesh, 1949) | 0.285 nm
Burgers vector (chimney slip) ($b_{3}$;$b_{4}$) (Lukesh, 1949) | 0.651 nm
Burgers vector (roof slip) ($b_{5}$;$b_{6}$;$b_{7}$;$b_{8}$) (Lukesh, 1949) | 1.185 nm
Projected shear modulus (wall slip) ($\mu_{1}$) (Fisher and McSkimin, 1958) | 74.330 GPa
Projected shear modulus (floor slip) ($\mu_{2}$) (Fisher and McSkimin, 1958) | 73.420 GPa
Projected shear modulus (chimney slip) ($\mu_{3},\mu_{4}$) (Fisher and McSkimin, 1958) | 92.255 GPa
Projected shear modulus (roof slip) ($\mu_{5},\mu_{6}$,$\mu_{7},\mu_{8}$) (Fisher and McSkimin, 1958) | 115.67 GPa
Dislocation density evolution law parameters | Eqns. (7),(8)
Dislocation multiplication prefactor (roof slip) ($k_{5}^{1}$;$k_{6}^{1}$;$k_{7}^{1}$;$k_{8}^{1}$) (McCabe et al., 2010) | 0.948
Dislocation annihilation length (wall slip) ($\hat{d}_{1}$) (McCabe et al., 2010) | 0.936 $\mu$m
Dislocation annihilation length (floor slip) ($\hat{d}_{2}$) (McCabe et al., 2010) | 0.429 $\mu$m
Dislocation annihilation length (chimney slip) ($\hat{d}_{3}$,$\hat{d}_{4}$) (McCabe et al., 2010) | 0.174 $\mu$m
Dislocation annihilation length (roof slip) ($\hat{d}_{5}$,$\hat{d}_{6}$,$\hat{d}_{7}$,$\hat{d}_{8}$) (McCabe et al., 2010) | 0.124 $\mu$m
Initial dislocation density $\rho_{\alpha}^{\textrm{for}}\left(t=0\right)=\rho^{\textrm{sub}}\left(t=0\right)$ |  $10^{10}$ m-2
Twin hardening law parameters | Eqns. (9)-(12)
Twin Burgers vector ($b_{\beta}$) | 0.1036 nm
Twin system projected shear modulus ($\mu_{\beta}$) | 99.537 GPa
Initial twin resistance (fitting parameter) $\tau_{\beta}^{0}$ | Table 4
Hardening coefficients (fitting parameters) $k_{\beta}^{1}$ | Table 4
Table 3: Model parameters.

##  3 Experimental details

Experiments were performed to measure the full-field surface strains during
tensile deformation of cast $\alpha$-uranium specimens with a known
microstructure, using digital image correlation (DIC). Two dog-bone specimens
were manufactured by AWE plc via wire electro-discharge machining (EDM) from a
cast plate. The gauge dimensions were 32.00 mm $\times$ 6.00 mm, with a
specimen thickness of 1.50 mm.

###  3.1 Sample preparation and EBSD

Sample preparation and EBSD microstructural characterisation were performed at
the Interface Analysis Centre (IAC), School of Physics, University of Bristol,
UK. Each specimen was mounted in Struers ClaroCit resin, and mechanically
ground with 120 grit SiC paper to remove the oxide layer. Grinding with
progressively finer papers, from 320 to 2500 grit, was followed by polishing
with 6 $\mu$m and 3 $\mu$m diamond suspension. The specimens were removed from
the resin with acetone, and electropolished at a voltage of 12 V in an
electrolyte comprising a 10:6:6 volumetric ratio of ethanol, ethylene glycol,
and phosphoric acid. The electrolyte was continuously stirred and polishing
proceeded for between 10 and 20 minutes, until a smooth, reflective surface
was observed.

EBSD analysis was performed on a Zeiss Sigma HD VP Field Emission SEM, with
EDAX EBSD camera and associated OIM software. A series of large-area maps
covering the whole gauge area were recorded with a slight overlap between each
scan region. The EBSD datasets were post-processed using the MTEX toolbox
(Bachmann et al., 2011) in MATLAB. The individual orientation maps were
stitched together to produce a single map for the gauge region of each
specimen.

Figures 1 and 2 show orientation maps from the two tensile bars, with colours
representing the crystal direction parallel to the z direction (into-page).
The microstructure is coarse, with sub-grains in excess of 200 $\mu$m in
diameter clustered together into regions of similar orientations. Grain
clusters are labelled according to their orientation; with 4 grains in the
first tensile bar and 6 in the second.

Figure 1:  EBSD orientation map of the first tensile bar. The black dots
indicate the crystal orientations chosen for the four grains. Figure 2:  EBSD
orientation map of the second tensile bar. The black dots indicate the crystal
orientations chosen for the six grains.

###  3.2 In-situ mechanical testing

A speckle pattern of black and white acrylic paint was applied to the surface
of each specimen with an airbrush. Tensile testing was performed on a Shimadzu
Autograph AGS-X universal test machine with a 5 kN load cell, connected to a
data acquisition PC running Shimadzu Trapezium-X software.

The specimen was enclosed in a bespoke sample holder to prevent contamination
of the tensile grips. A window in the front face of the holder allows viewing
of the specimen surface with the DIC cameras. The specimen was loaded in
parallel with a thin ductile copper bar which would absorb energy in the event
of specimen failure to prevent loss of radioactive material. The mechanical
properties of the copper were characterised before the test; its flow curve
was used to calculate the stress in the $\alpha$-uranium specimen from the
total load measured by the load cell. Each specimen was loaded in displacement
control at a rate of either 0.05 mm/min or 0.025 mm/min.

DIC was performed with a GOM ARAMIS 5M stereo DIC system. The camera mounting
frame was rotated by 90∘ from its usual horizontal orientation such that the
affixed cameras were positioned above one another and angled inwards towards
the specimen at an angle of 12.5∘. This allowed the gauge section of the
specimen to be viewed in a portrait orientation. A calibration process allowed
the co-ordinates in each image to be related to 3D co-ordinates in a small
calibrated volume centred on the specimen surface. Two camera DIC systems as
used here allow the separation of in-plane strains from out-of-plane
displacements, which cause strain artifacts in single-camera DIC. Illumination
was provided by polarised light from high-intensity LEDs. A pair of DIC images
was automatically acquired in the ARAMIS software every 2.5 seconds during the
test.

The DIC analysis was performed with a facet size of 20$\times$20 pixels and a
50% overlap between each facet. A rigid-body movement of the specimen before
testing was used to calculate the in-plane strain error as $\pm$0.1%, and the
error in out-of-plane displacement as $\pm$30 $\mu$m. The boundaries between
the grain clusters in figure 1 were used to segment the DIC data such that the
mean strain within each grain cluster can be calculated. This enables the
plotting of stress-strain curves for the entire gauge volume (figure 4) and
for the individual grain clusters (figure 5), for comparison with the
simulation results.

##  4 Simulation details and results

The representative volume used for the first tensile bar is shown in figure 3
(a). The bar was 32 mm long and 6 mm wide. Displacement along the x axis was
applied on the surface x = 32 mm. Zero displacement along the x direction was
imposed on the surface x = 0. The point (x,y,z) = (0, 3, 0.75) mm was fixed to
prevent translation of the surface x = 0. The points (x,y,z) = (0, 3, 1.5) mm
and (x,y,z) = (0, 6, 0.75) mm can translate only along the z and y directions.
This prevents rotation around the x axis of the surface x = 0. The texture of
the representative volume consisted of four grains with different orientations
(figure 3). This is a simplification compared to the experimental EBSD map
(figure 1) but is useful to maintain constant Schmid factors in each grain
and, therefore, to better understand the slip-twin activity in each grain.
This approximation also avoids the introduction of spurious data at points
where the experimental EBSD has larger uncertainty. An uncertainty
quantification on the grain orientations has been carried out and its effect
on the computational results is discussed in section 5.

The coarser mesh used for the optimisation procedure is shown in figure 3 (b).
It consisted of 434 elements with an average size of around 1 mm. The coarser
mesh is necessary to reduce the computational time during the optimisation
procedure, which requires hundreds of simulations. A finer mesh, of 19500
elements, shown in figure 3 (c), was used to repeat the simulation with the
optimised parameters and compare the simulated and experimental strain fields.

The maximum time increment used in the simulations was 0.04 s and the total
time was 20 s, after which a total strain of 2% was reached. Therefore, the
simulated strain rate was $10^{-3}$ s-1: this value corresponds to the
constant $\dot{\gamma}_{0}$ in equation 4 and was chosen to simulate the
regime in which the model is strain rate independent (Capolungo et al., 2009).
In this regime, the resolved shear stress on a slip or twin system remains
close to the CRSS $\tau_{\alpha}^{c}$ or $\tau_{\beta}^{c}$ respectively. At
the beginning of the simulations, the twin volume fractions were set to zero
and the dislocation densities were set to $10^{10}$ m-2.

(a) (a)

(b) (b)

(c) (c)

Figure 3: (a) Representative volume of the first tensile bar. (b) Coarse mesh
used during the optimisation procedure; different colours correspond to the
Four different grains. (c) Fine mesh used for the validation procedure.

###  4.1 Optimisation procedure

The first optimisation procedure was carried out to find the initial CRSS
parameters $\tau_{\alpha}^{0}$ and $\tau_{\beta}^{0}$ for “wall” slip
($\alpha=1$), “floor” slip ($\alpha=2$), “chimney” slip ($\alpha=3$) and the
$\\{130\\}\langle 3\bar{1}0\rangle$ twin ($\beta=1$). A nonlinear optimisation
procedure, based on the Nelder-Mead algorithm (Nelder and Mead, 1965), was
used, which is implemented in the scipy.optimize package (Jones et al., 2001).
This algorithm was used because it does not require an analytical expression
for the Jacobian. At every iteration of the algorithm, the simulation with the
coarse mesh in figure 3 (b) was run with a set of parameters
$\tau_{\alpha}^{0}$, $\tau_{\beta}^{0}$. For each grain in figure 1 the axial
strain was averaged; denoted $\bar{\varepsilon}^{\textrm{G1}}$,
$\bar{\varepsilon}^{\textrm{G2\&3}}$ and $\bar{\varepsilon}^{\textrm{G4}}$,
for grains 1, 2 and 3, 4 respectively. The strain was averaged together in
grains 2 and 3 because their mechanical behaviour was similar. The total
strain in the tensile bar is referred to as
$\bar{\varepsilon}^{\textrm{gauge}}$ and the stress was averaged over the load
surface in figure 3 (a). This post-processing allows stress-strain curves to
be extracted from the tensile bar and from the individual grains. The yield
stresses of the tensile bar, $\sigma_{y}^{\textrm{gauge}}$, and of the
individual grains, $\sigma_{y}^{\textrm{G1}}$, $\sigma_{y}^{\textrm{G2\&3}}$,
$\sigma_{y}^{\textrm{G4}}$, were defined using the 0.1% offset criterion (Pham
et al., 2013) for both the experiment and the simulation. The residual
minimised by the Nelder-Mead algorithm was:

$R_{\textrm{CRSS}}=\left(\sigma_{y,\textrm{exp}}^{\textrm{gauge}}-\sigma_{y,\textrm{sim}}^{\textrm{gauge}}\right)^{2}+\left(\sigma_{y,\textrm{exp}}^{\textrm{G1}}-\sigma_{y,\textrm{sim}}^{\textrm{G1}}\right)^{2}+\left(\sigma_{y,\textrm{exp}}^{\textrm{G2\&3}}-\sigma_{y,\textrm{sim}}^{\textrm{G2\&3}}\right)^{2}+\left(\sigma_{y,\textrm{exp}}^{\textrm{G4}}-\sigma_{y,\textrm{sim}}^{\textrm{G4}}\right)^{2}\
,$ (13)

where the subscripts exp and sim indicate experimental and simulated
quantities. After every iteration of the Nelder-Mead algorithm, the parameters
$\tau_{\alpha}^{0}$ and $\tau_{\beta}^{0}$ are updated. This first
optimisation procedure is carried out without considering the hardening of the
twin systems because this does not significantly affect the yield stress. The
CRSS parameters $\tau_{\alpha}^{0}$ and $\tau_{\beta}^{0}$ that minimise the
residual $R_{\textrm{CRSS}}$ in equation (13) are given in table 4.

CRSS parameters optimisation | Eqn. (6) and (9)-(12)
---|---
Constant friction stress (wall slip) ($\tau_{1}^{0}$) | 24.5 MPa
Constant friction stress (floor slip) ($\tau_{2}^{0}$) | 85.5 MPa
Constant friction stress (chimney slip) ($\tau_{3}^{0}$;$\tau_{4}^{0}$) | 166.5 MPa
Twin resistance ({130} twin system) ($\tau_{\beta}^{0}$) | 55.5 MPa

Hardening parameters optimisation | Eqn. (9) | (10)-(11) | (12)
---|---|---|---
Dislocation multiplication prefactor (wall slip) ($k_{1}^{1}$) | 0.0121 | 0.0088 | 0.0123
Dislocation multiplication prefactor (floor slip) ($k_{2}^{1}$) | 0.36 | 1.69 | 1.80
Dislocation multiplication prefactor (chimney slip) ($k_{3}^{1}$;$k_{4}^{1}$) | 0.136 | 0.176 | 0.0
Twin system hardening parameter ($k_{\beta}^{1}$) | 4706 | 0.0570 | 0.00097

Table 4: Parameters obtained from the optimisation procedures for the CRSS and
for the hardening rate. The three values of the parameters $k_{1}^{1}$,
$k_{2}^{1}$, $k_{3}^{1}$, $k_{4}^{1}$ and $k_{\beta}^{1}$ are obtained using
the twin-slip, twin-noncoplanar and twin-coplanar model respectively.

A second optimisation procedure was then carried out to find the hardening
parameters $k_{\alpha}^{1}$ and $k_{\beta}^{1}$ for “wall” slip ($\alpha=1$),
“floor” slip ($\alpha=2$), “chimney” slip ($\alpha=3$) and the
$\\{130\\}\langle 3\bar{1}0\rangle$ twin ($\beta=1$) using the CRSS parameters
$\tau_{\alpha}^{0}$ and $\tau_{\beta}^{0}$ obtained from the first
optimisation procedure. The hardening parameters $k_{\alpha}^{1}$ and
$k_{\beta}^{1}$ were updated after every iteration of the Nelder-Mead
optimisation algorithm, in which a simulation with the coarse mesh in figure 3
(b) up to 2% total strain was run. The maximum stress of the tensile bar
reached during the deformation is denoted $\sigma^{\textrm{gauge}}$. The
maximum total axial strain in the tensile bar is denoted
$\bar{\varepsilon}^{\textrm{gauge}}$, and the maximum average axial strains
reached during the deformation in the individual grains,
$\bar{\varepsilon}^{\textrm{G1}}$, $\bar{\varepsilon}^{\textrm{G2\&3}}$,
$\bar{\varepsilon}^{\textrm{G4}}$, were calculated after each simulation. The
residual minimised during the second optimisation procedure:

$R_{\textrm{hard}}=R_{\textrm{hard}}^{\sigma}+\left(10^{4}\
\textrm{MPa}\right)^{2}R_{\textrm{hard}}^{\bar{\varepsilon}}\ ,$ (14)

where

$R_{\textrm{hard}}^{\sigma}=4\left(\sigma_{\textrm{exp}}^{\textrm{gauge}}-\sigma_{\textrm{sim}}^{\textrm{gauge}}\right)^{2}\
,$ (15)

and

$R_{\textrm{hard}}^{\bar{\varepsilon}}=\left(\bar{\varepsilon}_{\textrm{exp}}^{\textrm{gauge}}-\bar{\varepsilon}_{\textrm{sim}}^{\textrm{gauge}}\right)^{2}+\left(\bar{\varepsilon}_{\textrm{exp}}^{\textrm{G1}}-\bar{\varepsilon}_{\textrm{sim}}^{\textrm{G1}}\right)^{2}+\left(\bar{\varepsilon}_{\textrm{exp}}^{\textrm{G2\&3}}-\bar{\varepsilon}_{\textrm{sim}}^{\textrm{G2\&3}}\right)^{2}+\left(\bar{\varepsilon}_{\textrm{exp}}^{\textrm{G4}}-\bar{\varepsilon}_{\textrm{sim}}^{\textrm{G4}}\right)^{2}\
,$ (16)

was determined for the three different hardening models for the twin systems
reported in section 2.2. The hardening parameters $k_{\alpha}^{1}$ and
$k_{\beta}^{1}$ that minimise the residual $R_{\textrm{hard}}$ in (14) are
reported in table 4.

The simulated stress-strain curves obtained using the optimised parameters and
the three different hardening models for the twin systems are shown in figure
4 alongside the experimental data.

(a) (a)

(b) (b)

(c) (c)

Figure 4:  Simulated and measured gauge stress-strain curves after the
optimisation procedure using (a) the twin-slip model; equation (9), (b) the
twin-noncoplanar model; equations (10)-(11) and (c) twin-coplanar model;
equation (12).

Figure 5 shows stress as a function of the average axial strains
$\bar{\varepsilon}^{\textrm{G1}}$, $\bar{\varepsilon}^{\textrm{G2\&3}}$,
$\bar{\varepsilon}^{\textrm{G4}}$ in the individual grains for the three
different hardening models for the twin systems. The stress-strain curves
agree with the experiment for the twin-slip (figure 5 (a)) and for the twin-
coplanar (figure 5 (c)) interaction models. In grains 2 and 3, the maximum
stress agrees with the experiment, as shown in figures 5 (a) and (c), while
the maximum strain is underestimated by the models.

(a) (a)

(b) (b)

(c) (c)

Figure 5:  Simulated and measured stress-strain curves in grain 1, grain 2&3
and grain 4 after the optimisation procedure using (a) the twin-slip model;
equation (9), (b) the twin-noncoplanar model; equations (10)-(11) and (c)
twin-coplanar model; equation (12).

The twin-noncoplanar interaction model is not able to match the stress value
reached in the experiment, as shown in figures 4 (b) and 5 (b). It
overestimates the strain accommodated in grain 4, as shown in figure 5 (b).
This will be interpreted in the following section according to the activity of
the slip and twin systems.

###  4.2 Fine mesh simulation results and model validation

Once the optimal parameters $\tau_{\alpha}^{0}$, $\tau_{\beta}^{0}$,
$k_{\alpha}^{1}$, $k_{\beta}^{1}$ were found, the simulation of the first
tensile bar was repeated using the finer mesh in figure 3 (c). The model
chosen for the hardening of the twin systems was the twin-coplanar model
described by equation (12) as this provides the best optimised fit to the
experimental data. The “wall” slip system was the most active because of the
lower CRSS, as reported in table 4. Figure 6 shows the dislocation density
$\rho_{1}^{\textrm{for}}$ in the “wall” slip system at 1% total strain. “Wall”
slip is mostly active in grain 1 and the dislocation density grows by a factor
of approximately 100 compared with the initial value.

Figure 6: Dislocation density $\rho_{1}^{\textrm{for}}$ in the wall slip
system in the first tensile bar at 1% total strain using the twin-coplanar
model of equation (12).

Figure 7 shows the twin volume fraction $f_{\beta}$ at 2% total strain. It
grows in the upper part of grain 4 and at the interface between grain 1 and
grain 4, reaching a value of about 12%. We have found that the contribution to
the twin volume fraction of the system
$\left(130\right)\left[3\bar{1}0\right]$ is much larger than the contribution
of $\left(\bar{1}30\right)\left[\bar{3}\bar{1}0\right]$. The twin-slip model
and the twin-coplanar model lead to the correct prediction of the stress-
strain curve in grain 4, as shown in figure 5 (a) and (c). The optimisation
procedure allows us to find the interaction coefficients between coplanar
twins and between slip and twin systems.

Figure 7: {130} Twin volume fraction in the first tensile bar at 2% total
strain for the twin-coplanar model of equation (12).

The detailed strain and displacement fields within the grains provide
additional information that can be used to validate the model resulting from
the optimisation procedure. The axial strain concentrates in the upper part of
grain 1 due to the activity of the “wall” slip system, as shown in figure 8.
This agrees with the DIC measurement.

(a) (a)

(b) (b)

Figure 8: Axial strain $\varepsilon_{xx}$ in the first tensile bar at 1% total
strain: (a) DIC measurement, (b) simulation.

The same slip activity leads to the concentration of the shear strain
component $\varepsilon_{xy}$ in the upper part of grain 1, as shown in figure
9.

(a) (a)

(b) (b)

Figure 9: Shear strain $\varepsilon_{xy}$ in the first tensile bar at 1% total
strain: (a) DIC measurement, (b) simulation.

The measured and predicted out-of-plane displacement $u_{z}$ is shown as a
surface plot in figure 10. The simulation results can be interpreted by
studying the direction of the Burgers vectors of the “wall” slip system and of
the $\\{130\\}$ twin system. The largest component of these vectors is the
$u_{z}$ component, which is positive. This is found by multiplying the
rotation matrices $\bm{R}$ of grains 1 and 4 and the slip and twin directions
$\bm{s}_{\alpha=1}^{0}$ and $\bm{s}_{\beta=1}^{0}$ in the lattice reference
system. Therefore, the large activity of the “wall” slip system in the upper
part of grain 1 leads to positive $u_{z}$ in that region. The same can be
stated for the large twin activity on the bottom part of the grain boundary
between grain 1 and 4, shown in figure 7. This agrees with the DIC measurement
in figure 10 (a).

The twin-noncoplanar model described by equations (10)-(11) does not predict
strong hardening of the most active twin system
$\left(130\right)\left[3\bar{1}0\right]$ in grain 4. This explains the
underestimation of the maximum stress reached in grain 4 using the twin-
noncoplanar model, as shown in figure 5 (b).

(a) (a)

(b) (b)

(c)

Figure 10: Out-of-plane displacement $u_{z}$ in the first tensile bar at 1%
total strain: (a) DIC measurement, (b) simulation.

An independent model validation was obtained by simulating the second tensile
bar with 6 grains shown in figure 2. A fine mesh with 15984 elements was used.
The boundary conditions were the same as for the first tensile bar, shown in
figure 3 (a). The same model parameters $\tau_{\alpha}^{0}$,
$\tau_{\beta}^{0}$, $k_{\alpha}^{1}$, $k_{\beta}^{1}$, optimised for the first
tensile bar, were used. The model chosen for the hardening of the twin systems
was again the twin-coplanar model described by equation (12). The measured and
predicted axial strains are shown in figure 11. The “wall” slip system is
active in grains 1 and 5, leading to strain concentration in those grains.

(a) (a)

(b) (b)

Figure 11: Axial strain $\varepsilon_{xx}$ in the second tensile bar at 1%
total strain: (a) DIC measurement, (b) simulation.

The same slip activity leads to the concentration of the shear strain
component $\varepsilon_{xy}$ in grain 1 and 5, as shown in figure 12.

(a) (a)

(b) (b)

Figure 12: Shear strain $\varepsilon_{xy}$ in the second tensile bar at 1%
total strain: (a) DIC measurement, (b) simulation.

The measured and predicted out-of-plane displacement $u_{z}$ is shown in
figure 13. A rotation of about $1\degree$ around the x axis is added to the
simulation results to match the experimental inclination on the surface $x=0$,
which was not present in the first tensile bar. Both the measurement and the
simulation show a negative region of $u_{z}$ corresponding to grain 1.

(a) (a)

(b) (b)

(c)

Figure 13: Out-of-plane displacement $u_{z}$ in the second tensile bar at 1%
total strain: (a) DIC measurement, (b) simulation.

##  5 Discussion

The optimisation method presented in section 4.1 shows that it is possible to
use a single tensile bar experiment and DIC measurements to calibrate the
yield strength and hardening rate of a crystal plasticity constitutive model
for $\alpha$-uranium. This material has slip and twin systems with very
different properties. The reliability of the method is confirmed by the
correct prediction of the strain and displacement fields, as shown in figures
8-13. This optimisation technique can be readily adapted to other materials.

The stress-strain curves in figure 5 (a)-(c) show that the {130} twin system
has a significant strain hardening response. For instance, the yield strength
of grain 4 in the first tensile bar increases as the bar is loaded even though
most plastic deformation in that grain is provided by a single twin variant
$\left(130\right)\left[3\bar{1}0\right]$, as shown in figures 6 and 7. Early
studies on twin initiation and growth assumed a constant CRSS for twinning
(Christian and Mahajan, 1995). Recent experimental studies (Patriarca et al.,
2013) and molecular dynamics simulations (Ojha et al., 2014), however, show
that both slip activity and twins affect the CRSS for twinning. This is
consistent with the present work, in which the optimisation algorithm
automatically identifies the extent of the twin-slip interaction and the
interaction between coplanar twins. The comparison between the twin-slip model
and twin-coplanar hardening models, reported in section 2.2, shows that both
models can describe the mechanical behaviour of the first tensile bar. It is
likely that the twin variant $\left(130\right)\left[3\bar{1}0\right]$
interacts with both slip systems and coplanar twins. The present approach is
not able to distinguish between these two interactions, but it can provide
values for the interaction coefficients $k_{\beta}^{1}$.

The determination of the coefficients $k_{\beta}^{1}$ was not possible in
previous works on fine-grained $\alpha$-uranium, based on tensile and
compression tests of textured samples (McCabe et al., 2010), and on neutron
diffraction (Brown et al., 2009). This is because these measurements include
information from thousands of grains and are not able to discriminate between
the slip-slip and slip-twin interactions. Therefore, the interaction
coefficients between the slip systems and the $\\{130\\}$ twin system were set
to zero (see table 2 in (McCabe et al., 2010)).

Values for the twin hardening coefficients $k_{\beta}^{1}$ of $\alpha$-uranium
are not available in the literature, but a quantitative comparison with other
materials can be made. (Abdolvand and Daymond, 2013b) used a Voce hardening
law to describe Zircaloy, in which the CRSS for tensile twins was given in the
small strain approximation:

$\tau_{\beta}^{c}\approx\tau_{\beta}^{0}+\theta_{\beta}^{0}\Gamma\ ,$ (17)

where $\Gamma$ is the accumulated shear on all slip/twin systems and
$\theta_{\beta}^{0}=50$ MPa. Assuming that the $\\{130\\}$ twin system is the
only plastic deformation mechanism, equation (12) for the twin-coplanar
hardening model becomes:

$\tau_{\beta}^{c}=\tau_{\beta}^{0}+\left(\frac{k_{\beta}^{1}\mu_{\beta}}{\gamma_{\beta}^{\textrm{twin}}}\right)\Gamma\approx\tau_{\beta}^{0}+323\Gamma\
,$ (18)

where the stress is expressed in MPa. This shows that the values obtained for
$k_{\beta}^{1}$ have the same order of magnitude as in other materials.
Generally, the hardening rate found in the present experiments is lower than
that found for fine-grained $\alpha$-uranium. For instance, the dislocation
multiplication prefactor of the “wall” slip system reported by McCabe et al.
(McCabe et al., 2010) is $k_{1}^{1}/b_{1}=200$ $\mu$m-1 while the present
optimisation algorithm finds a maximum value of $k_{1}^{1}/b_{1}=43$ $\mu$m-1,
as reported in tables 3 and 4. This is consistent with the finding that the
average dislocation density grows faster in fine-grained materials (Haouala et
al., 2018; De Sansal et al., 2010).

Mesoscale models for FeCr single crystals, based on molecular dynamics
simulations (Ojha et al., 2014), show that the twin migration stress grows
linearly with the residual Burgers vector left at the twin interface during
twin-dislocation interaction (see figure 14 in (Ojha et al., 2014)). Those
simulations are carried out in a single crystal with a characteristic
dimension of $500w$, where $w$ is the dislocation core width. A migration
stress of 167 MPa is found if the residual Burgers vector is
$\left|b_{r}\right|=a$, where $a=0.286$ nm is the lattice constant (Ojha,
2013). Assuming $w\approx a$, the residual Burgers vector in those simulations
corresponds to a dislocation density $\rho\approx
1/\left(500a\right)^{2}\approx 49$ $\mu$m-2. The linear relationship between
the dislocation density and the twin migration stress gives an increase of
$167$ MPa $/$ $49$ $\mu$m-2 $\approx$ $3.4$ MPa $/$ $\mu$m-2. Also equation
(9) predicts an increase of the CRSS $\tau_{\beta}^{c}$, above which twin
volume fraction grows, that depends linearly on the dislocation density; the
factor of proportionality is
$k_{\beta}^{1}\mu_{\beta}b_{\beta}b_{\alpha}\approx 14$ MPa $/$ $\mu$m-2. This
comparison shows the consistency between the present model, calibrated without
imaging individual twins, and atomistic models, which determine the
interaction between dislocations and twins at the sub-micron length scale.

The uncertainty in the present optimisation procedure arises from the
approximation that the four grains of the first tensile bar in figure 1 each
have uniform crystal orientations. An uncertainty quantification procedure was
carried out: the Euler angles representing the crystal orientation were
sampled at different points in the 4 grains; for instance, this leads to an
uncertainty of about 20% on the Schmid factor of the “wall” slip system. The
optimisation procedure has been repeated with different Euler angles. The
uncertainty is small, about 2%, on the constant friction stress $\tau_{1}^{0}$
and on the dislocation multiplication prefactor $k_{1}^{1}$ of the “wall” slip
system, and it is about 30% for $\tau_{\beta}^{0}$ and $k_{\beta}^{1}$ of the
twin system.

The constant friction stress of the “chimney” slip system has a large
uncertainty: if the value is changed, the residual $R_{\textrm{CRSS}}$ of the
optimisation procedure does not change significantly. This is because of the
low Schmid factor of the “chimney” slip system in the 4 grains and the higher
CRSS.

##  6 Conclusions

An optimisation procedure was developed to find parameters of a dislocation-
based constitutive model, implemented in a crystal plasticity finite element
framework. DIC measurements were made during tensile tests on coarse-grained
$\alpha$-uranium bars containing a small number of grains. This allows us to
find the average strain in a grain and to plot the stress-strain curves of
individual grains, which can be directly compared to the simulations.

The constitutive model includes the activity of 8 slip systems and 2 twin
variants, as previously identified in $\alpha$-uranium (Zhou et al., 2016).
Three different models for the hardening of the twin systems were compared.
Models which account for the interaction between twin and slip systems and the
interaction between coplanar twins can explain the experimental data. The
optimisation procedure is able to find the values of the interaction
coefficients, which have not been identified in previous studies on
$\alpha$-uranium.

The optimisation procedure is based on consideration of the average response
of the grains within the coarse-grained specimen. The resulting model has been
validated by comparing the detailed experimental and simulated strain and
displacement fields within the same specimen and for a second tensile specimen
with a different grain structure. These are interpreted by analysing the
activity of twin/slip systems and the direction of their Burgers vectors.

The method presented can be applied to find the CRSS and hardening properties
of plastic deformation mechanisms of other coarse-grained materials. The
method has the advantage that the preparation of single crystal samples is not
required and a large number of parameters can be determined from a single
test.

## Acknowledgements

The authors acknowledge financial support from AWE plc for this research,
program manager: Dr John Askew. ET acknowledges support from the Engineering
and Physical Sciences Research Council under Fellowship grant EP/N007239/1.
The authors kindly acknowledge Dr Keith Hallam, Christopher Jones and Joseph
Sutcliffe for their assistance with the experimental work.

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