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
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Lütjering, G. and Williams, J.C., 2007. Titanium. Springer Science & Business Media. ↩
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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. ↩
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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. ↩
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Hearmon, R.F.S., 1984. The elastic constants of crystals and other anisotropic materials. Landolt-Bornstein Tables, III/18, 1154. ↩
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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
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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). ↩
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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
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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. ↩
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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. ↩