Ti Modelling Parameters

The model presented uses a constitutive law based on a phenomenological crystal plasticity model described by Pierce et al. (Pierce, 1983) as part of the DAMASK framework (Roters, 2019). A phenomenological model attempts to predict the response one variable has on another, but is not derived from first principles. The DAMASK full-field crystal plasticity model considers a representative volume element as a continuous body $\mathcal{B}$ consisting of material points $\textbf{x}$ located in reference configuration $\mathcal{B}$₀ which move to the current configuration $\textbf{y}$ in $\mathcal{B}$_t with deformation. An infinitesimal line segment $d\textbf{x}$ is moved by the application of a deformation gradient tensor $F$, which maps $d\textbf{x}$ in the reference configuration to $d\textbf{y}$ in the current configuration. $d\textbf{y} = F(\textbf{x})\cdot d\textbf{x}$. Multiplicative decomposition of the deformation gradient tensor splits $F$ into the elastic deformation gradient tensor $F_{e}$ and plastic deformation gradient tensor $F_{p}$

\[F = F_{e} \cdot F_{p}\]

The elastic deformation gradient tensor $F_{e}$ is calculated from the Green-Lagrange strain $E$ which is itself determined using Hooke’s law

\[S = \mathbb{C}:E\]

Where $S$ is the Cauchy stress tensor and $\mathbb{C}$ is the elastic stiffness tensor. $E$ may then be expressed

\[E = \frac{(\boldsymbol{F_{e}}^{T}\boldsymbol{F_{e}}-\boldsymbol{I})}{2}\]

The plastic deformation gradient tensor $F_{p}$ is calculated using constitutive equations. A constitutive equation describes the response of a specific material to external stimuli. The slip rate $\dot{\gamma}^i$ of polycrystal slip plane $i$ for a given load case is determined as follows

The phenomenological power law by which the slip rate $\dot{\gamma}^i$ on some slip system $i$ is dependant upon the initial shear rate $\dot{\gamma_{0}}^i$ , ratio between resolved shear stress $\tau^i$ and critical resolved shear stress (CRSS) $\xi^i$, inverse of strain rate sensitivity $n_{sl}=\frac{1}{m}$ (also known as the stress exponent), and resolved shear stress on the slip system $\tau^{i}$, is given as

\[\dot{\gamma}^i = \dot{\gamma_{0}}^{i}\displaystyle\left\vert\frac{\tau^i}{\xi^i}\right\vert^{n_{sl}}\text{sgn}(\tau^i)\]

The CRSS $\xi^i$ in Equation is analogous to the yield of slip system $i$. When the resolved shear stress becomes greater than that of the CRSS of the slip system, $\dot{\gamma}^{i}\neq0$ and the slip system begins to slip. The resolved shear stress on the system $i$, $\tau^i$, is the second piola-kirchoff stress tensor, $\boldsymbol{S}$, projected by the corresponding schmid tensor, itself given by the dyadic product of the unit vectors along the slip direction, $\boldsymbol{b}^i$, and the slip plane normal, $\boldsymbol{n}^i$

\[\tau^i = \boldsymbol{S}\cdot\boldsymbol{b}^i\otimes\boldsymbol{n}^i\]

The following power law is used to determine the change of CRSS $\xi^i$ from its initial value $\xi_0^i$, to the defined saturated CRSS $\xi_\infty^{i^{\prime}}$ with flow hardening as shown in Equation numb

\[\dot{\xi}^i = \dot{h}_0^{s-s}\sum_{i^{\prime}=1}^{N_s} \displaystyle\left\vert\dot{\gamma}^i\right\vert \displaystyle\left\vert1-\frac{\xi^{i^{\prime}}}{\xi_\infty^{i^{\prime}}}\right\vert^{w}sgn(1-\frac{\xi^{i^{\prime}}}{\xi_\infty^{i^{\prime}}})h^{ii^{\prime}}\]

Where $\dot{h_0}^{s-s}$ is the initial hardening rate, $w$ is a fitting parameter and $h^{ii^{\prime}}$ is the components of the slip-slip interaction matrix. $h_{ij}$ is 1.0 for self-hardening and 1.4 for latent hardening.

Please find below a collection of single crystal property parameters for titanium and its alloys from a variety of literature sources. Please add to this list should your literature review include these parameters, to aid future work into modelling of titanium and its alloys.

Alpha phase (Ti-α)

Ti-$\alpha$ phase possesses a hexagonal-close packed (HCP) unit cell with $c/a$ ratio 1.587, smaller than the ideal ratio of 1.633 (Lutjering, 2007)¹.

Elastic properties

$C_{11}$	$C_{12}$	$C_{13}$	$C_{33}$	$C_{44}$	$C_{66}$	Source	Comments
162.4	92.0	69.0	180.7	46.7	-	(Fisher, 1964)2	Ultrasonic wave interference tests of CP-Ti at room temperature. Measurement of $C_{33}$ was interrupted but still included.
160.0	95.0	45.0	181.0	55.0	55.0	(Naimon, 1974)3	‘pulse superposition’ of Ti64 at room temperature.
160.0	90.0	66.0	181.0	46.5	-	(Hearmon, 1984)4	Collection of crystal parameters from legacy papers

Plastic properties - Initial and saturated Critical Resolved Shear Stresses (CRSS)

Be aware some are given as ratios.

Slip system	{ 0002 }< 11-20 >	{ 10-10 }< 11-20 >	{ 10-11 }< 11-23 >	Source	Comments
CRSS ratio	1.5	1	3	(Dunst, 1996)5	texture predictions validated eagainst experimental.
CRSS	420.0	370.0	590.0	(Bridier, 2006)6	In-situ fatigue tests of room temperature Ti64.
CRSS	349.0	150.0	1107.0	(Zambaldi, 2012)7	‘Simplex algorythym’ used to matchup results of MARC CPFE model with compression tests of room temperature CP-Ti.
CRSS	444.0	392.0	631.0	(Jones, 1981)8	‘Ball model’ - estimations of CRSS were validated against uniaxial compression and tensile tests of CP-Ti at room temperature.

Plastic properties - Hardening equation parameters

$a$	$n_{sl}=\frac{1}{m}$	$dot{\gamma}_0$	$h_0$	$h_{ij}$	Source	Comments
2.0	0.05	0.001	200.0	[1.4 (non-coplanar), 1.0 (coplanar)]	(Zambaldi, 2012)7	Set arbritrarily for use in ‘simplex algorythim to determine CRSS’

Beta phase (Ti-β)

Because single-crystal properties of the beta phase cannot be determined directly at room temperature, some assumptions may be made in order to model its deformation response. E.g. Interstitial free steel is a good approximation for the $\beta$-phase, due to possessing similar slip modes.

Elastic properties

$C_{11}$	$C_{12}$	$C_{44}$	Source	Comments
97.7	87.2	37.5	(Ledbetter, 2004)9	non-contacting electromagnetic-acoustic resonance at 1030C.

Plastic properties - Initial and saturated Critical Resolved Shear Stresses (CRSS)

Pencil glide	{ 110 }< 111 >	{ 112 }< 111 >	{ 123 }< 111 >	Source	Comments
CRSS ratio	1/3	1/3	1/3	(Dunst, 1996) 5	Hot Texture predictions compared against experimental.

Plastic properties - Hardening equation parameters

$w$	$n_{sl}$	$dot{\gamma}_0$	$h_0$	$h_{ij}$	Source	Comments

Contact

This code is maintained by the Microstructure Modelling Group at the University of Manchester. For questions, comments, bug-reports or contributions please email Dr. Adam Plowman at Adam.plowman@manchester.ac.uk or Guy Bowker at guy.bowker@postgrad.manchester.ac.uk.

References