[Software & Simulation]

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

  1. Lütjering, G. and Williams, J.C., 2007. Titanium. Springer Science & Business Media. 

  2. Fisher, E.S. and Renken, C.J., 1964. Single-crystal elastic moduli and the hcp→ bcc transformation in Ti, Zr, and Hf. Physical review, 135(2A), p.A482. 

  3. Naimon, E.R., Weston, W.F. and Ledbetter, H.M., 1974. Elastic properties of two titanium alloys at low temperatures. Cryogenics, 14(5), pp.246-249. 

  4. Hearmon, R.F.S., 1984. The elastic constants of crystals and other anisotropic materials. Landolt-Bornstein Tables, III/18, 1154. 

  5. Dunst, D. and Mecking, H., 1996. Analysis of Experimental and Theoretical Rolling Textures of Two-phase Titanium Alloys/Analyse von gemessenen und berechneten Walztexturen bei zweiphasigen Titanbasislegierungen. International Journal of Materials Research, 87(6), pp.498-507.  2

  6. Bridier, F., 2006. Analyse expérimentale des modes de déformation et d’endommagement par fatigue à 20° C d’alliage de titane: aspects cristallographiques à différentes échelles (Doctoral dissertation, Poitiers). 

  7. Zambaldi, C., Yang, Y., Bieler, T.R. and Raabe, D., 2012. Orientation informed nanoindentation of α-titanium: Indentation pileup in hexagonal metals deforming by prismatic slip. Journal of Materials Research, 27(1), pp.356-367.  2

  8. Jones, I.P. and Hutchinson, W.B., 1981. Stress-state dependence of slip in Titanium-6Al-4V and other HCP metals. Acta Metallurgica, 29(6), pp.951-968. 

  9. Ledbetter, H., Ogi, H., Kai, S., Kim, S. and Hirao, M., 2004. Elastic constants of body-centered-cubic titanium monocrystals. Journal of applied physics, 95(9), pp.4642-4644.