Crystal Plasticity Simulations of Dislocation Slip and Twinning in α-Ti Single and Polycrystals

Abstract

A crystal plasticity finite element model is developed and implemented to numerically study the deformation behavior of hexagonal close-packed metals using α-titanium as an example. The model takes into account micromechanical deformation mechanisms through dislocation slip along prismatic, basal, and first-order <c+a> pyramidal systems, as well as tensile twinning. Twin initiation follows a two-conditional criterion requiring that both the resolved shear stress in a twin system and the accumulated pyramidal slip simultaneously reach their critical values. Three-dimensional polycrystalline models are generated using the step-by-step packing method. The crystal plasticity constitutive model describing the deformation behavior of grains is integrated into the boundary-value problem of continuum mechanics, including dynamic governing equations. The three-dimensional problem is solved numerically using the finite element method. The micromechanical model is tested for an α-titanium single crystal along the [0001] direction and a polycrystal consisting of 50 grains. The numerical results reveal that twin propagation is controlled by the critical value of accumulated pyramidal slip, emphasizing the need for experimental calibration. The agreement between numerical and experimental results provides the model validation at the meso- and macroscales.

1. Introduction

Metals with a hexagonal close-packed (hcp) crystal lattice—such as titanium, magnesium, zirconium, and their alloys—exhibit pronounced plastic anisotropy at the grain scale [1]. This behavior stems from the inherently low symmetry of the hcp lattice, which limits the number of crystallographic slip systems available for plastic deformation. In general, five distinct slip modes can be activated in hcp metals: prismatic (three slip systems), basal (three slip systems), <a>-pyramidal (six slip systems), first-order <c+a>-pyramidal (twelve slip systems), and second-order <c+a>-pyramidal (six slip systems) (Figure 1).

A distinct characteristic of hcp metals is the significant variation in the critical resolved shear stress (CRSS) τ_CRSS required to activate different slip modes. This variation is governed by the crystal lattice c/a ratio. In α-titanium, where c/a = 1.587, prismatic slip is initiated at substantially lower CRSS values than other slip modes. Although reported τ_CRSS values for different slip systems in α-titanium vary widely across the literature (see, e.g., [2,3,4,5]), their interrelation is generally consistent: ${\mathrm{\tau }}_{CRSS}^{prismatic}<{\mathrm{\tau }}_{CRSS}^{basal}<{\mathrm{\tau }}_{CRSS}^{pyramidal}$ (e.g., [6,7,8]). Consequently, prismatic slip systems in α-titanium are considered as the primary deformation mode, while basal slip is typically a secondary one. A notable exception occurs under loading along the c-axis, corresponding to the [0001] direction (Figure 1), where activation of prismatic and basal slip systems is impossible according to Schmid’s law. Under these conditions, plastic strain can only proceed through either <c+a>-pyramidal slip or deformation twinning. The latter is particularly important in hcp metals, as it not only accommodates plastic strain but also induces localized lattice reorientation within narrow bands (Figure 2c).

Numerically, the deformation slip and twinning can be described through models based on crystal plasticity (CP) theory [9,10,11,12,13,14], which explicitly capture the elastoplastic anisotropy imposed by the crystal structure. CP-based models incorporating deformation twinning are commonly classified into three groups: (1) elastic–viscoplastic self-consistent models [7,15,16], (2) crystal plasticity finite element (CPFE) models that treat twinning as pseudo-slip [17], and (3) CPFE models based on discrete twinning representations [18,19]. A detailed comparison of these modeling strategies is presented in [20] using magnesium alloys as an example.

Many existing CP-based formulations for hcp metals introduce a large number of hardening and interaction parameters to describe the combined effect of slip and twinning [20,21,22]. The calibration of these parameters is often challenging, especially when experimental data on twin evolution and local stress–strain states are limited. In parallel, several recent studies employ coupled CP–phase-field or other hybrid approaches to explicitly represent twin morphology, though these formulations typically require additional length-scale or gradient terms or predefined twin geometries to reproduce experimental features [23,24].

Our earlier work [25] introduced the CPFE model, taking into account prismatic and basal slip modes, which was subsequently refined to include the effect of the first-order <c+a>-pyramidal slip systems on the plastic response of α-titanium [26]. These studies demonstrated that although the pyramidal slip was activated only in a few grains and contributed less to the deformation response than other slip modes, its overlooking led to the overestimation of both the macroscopic (homogenized) stress and the local stresses in unfavorably oriented grains.

The present work further extends this modeling framework by explicitly incorporating dislocation glide along prismatic, basal, and first-order <c+a>-pyramidal slip systems together with deformation twinning using a new two-conditional criterion. This criterion links twinning activation to both the resolved shear stress and the local stress–strain heterogeneity, enabling the model to reproduce the twin morphology and propagation without introducing explicit size-dependent parameters. The model is qualitatively validated at multiple scales by comparing simulated single-crystal and polycrystal deformation patterns with available analytical and experimental data.

This paper is organized as follows. Experimental background is described in Section 2. Section 3 introduces the mathematical formulation of the CP-based model description, with particular emphasis on the treatment of deformation twinning in the model and its numerical implementation. Section 4 presents the verification and validation of the model at the micro-, meso-, and macroscales. The main findings are summarized in the Conclusion.

2. Experimental Background

The experimental data necessary for constructing the microstructure-based model were obtained for α-titanium. The dog-bone-shaped specimens with 36 × 12 × 3 mm³ gauge parts (Figure 2a) were cut from a titanium rod along the rod axis. The grain shape, crystallographic orientations, and twin morphology were examined by electron backscatter diffraction (EBSD) before (Figure 2b,d,e) and after uniaxial tension (Figure 2c,f,g), using a field emission scanning electron microscope Apreo 2 S (Thermo Fisher Scientific Inc., Waltham, MA, USA), equipped with an energy-dispersive X-ray spectrometer Octane Elect Super (EDAX, Mahwah, NJ, USA) and electron backscatter diffraction detector Velocity Super (EDAX, Mahwah, NJ, USA). The specimens were subjected to quasistatic uniaxial tension using a BISS Nano 15 kN universal testing machine (BiSS-ITW India, Bengaluru, India) at 1 mm/min loading velocity (corresponding to a strain rate of 3.3 × 10⁻³ s⁻¹) to obtain the stress–strain curve used for model calibration.

The {0001} and $\left\{10\overline{1}0\right\}$ pole figures revealed that the specimen initially possessed an axial texture <0001> ∥ MM′ with a ±20° scatter angle, where MM′ is the texture axis lying in the ZY plane at an angle of 45° to the Z-axis before tension (Figure 2d). After uniaxial tension, the texture axis for the {0001} planes rotated by approximately 5° toward the Z direction (Figure 2f), while the texture became sharper (±10° scatter angle), and the <10-10> directions aligned parallel to X (Figure 2g).

Deformation twinning in α-titanium is usually observed under specific loading conditions, such as severe plastic deformation [27], cyclic loading [28], or high strain rates [15]. In [29], twinning was also reported at room temperature for strains above ~8%. Likewise, in the present study, twins were experimentally identified in the neck region of an α-titanium specimen subjected to 30% quasistatic tensile strain (Figure 2c, marked by white arrows). The literature data (e.g., [3,6,8]) suggest that the τ_CRSS required for twin initiation is generally lower than that for <c+a>-pyramidal slip, making twinning the dominant deformation mode for grains loaded along the [0001] direction.

3. Crystal Plasticity Finite-Element Model

3.1. Governing Equations

The system of governing equations is formulated in a global coordinate system (GCS) associated with the model geometry. To reduce computational cost and storage requirements [30], the three-dimensional boundary-value problem is solved in a dynamic formulation. In this setting, the numerical procedure involves solving the equations of motion instead of the equilibrium equations in the static case. This approach enables the use of an explicit time integration scheme, which offers significant computational advantages since it does not require either iterations or the inversion of the global stiffness matrix.

Accordingly, the system of governing equations comprises the equation of motion,

$$\mathrm{\rho }{\dot{u}}_i={\mathrm{\sigma }}_{ij,j},$$

(1)

the continuity equation,

$${\displaystyle \frac{\dot{V}}{V}}-u_{i,i}=0,$$

(2)

and the kinematic relations for the strain rate tensor,

$${\dot{\mathrm{\epsilon }}}_{ij}={\displaystyle \frac{1}{2}}\left(u_{i,j}+u_{j,i}\right),$$

(3)

where $\mathrm{\rho }$ is the density, u_i are the velocity vector components, V is the relative volume, σ_ij and ε_ij are the stress and total strain tensors, respectively. A dot above the symbol denotes a time derivative, while a comma between subscripts indicates the partial derivative with respect to the corresponding spatial coordinate.

Stress correction under rigid-body rotation is introduced through the Jaumann derivative:

$${\dot{S}}_{ij}^{*}={\dot{S}}_{ij}-S_{ik}{\omega }_{jk}-S_{jk}{\omega }_{ik},$$

(4)

where ${\omega }_{ij}={\displaystyle \frac{1}{2}}\left(u_{i,j}-u_{j,i}\right)$ are the spin components, and S_ij are the components of the stress deviator.

The system of Equations (1)–(4) is completed by the generalized Hooke’s law linking the stress and elastic strain rates as

$${\dot{\mathrm{\sigma }}}_{ij}=C_{ijkl}{\dot{\mathrm{\epsilon }}}_{kl}^e,$$

(5)

where C_ijkl are the components of the tensor of elastic moduli and ${\mathrm{\epsilon }}_{kl}^e$ are the components of the elastic strain tensor.

The full system (1)–(5) is complemented by the initial and boundary conditions specifying the applied load. Ref. [30] demonstrated that explicit dynamic solution of a quasistatic problem can provide highly accurate results when the loading is applied smoothly. Accordingly, the loading velocity was gradually increased to its amplitude value and then kept constant in all simulations. The ramp-up time, selected to minimize wave effects unavoidable in dynamic calculations, was adjusted according to the geometry of each model.

3.2. Constitutive Description of Hexagonal Close-Packed Grains

The constitutive equations describing the deformation behavior of hcp grains are formulated within the CPFE framework, capturing elastoplastic anisotropy dictated by crystal lattice symmetry. This naturally brings the micromechanical deformation mechanisms—dislocation slip and twinning—into the model. Notice that the motion of individual dislocations is not considered in this model; instead, their averaged contribution to the plastic deformation is incorporated for each finite element.

In terms of small strains, the total strain rate tensor ${\dot{\mathrm{\epsilon }}}_{ij}$ can be decomposed into the elastic and plastic parts:

$${\dot{\mathrm{\epsilon }}}_{ij}={\dot{\mathrm{\epsilon }}}_{ij}^e+{\dot{\mathrm{\epsilon }}}_{ij}^p,$$

(6)

where ${\dot{\mathrm{\epsilon }}}_{ij}^p$ are the components of the plastic strain rate tensor. With this decomposition, Hooke’s law (5) takes the form

$${\dot{\mathrm{\sigma }}}_{ij}=C_{ijkl}\left({\dot{\mathrm{\epsilon }}}_{kl}-{\dot{\mathrm{\epsilon }}}_{kl}^p\right).$$

(7)

For hcp metals, the tensor of elastic moduli reduces to five independent components: C₁₁₁₁, C₁₁₂₂, C₁₁₃₃, C₃₃₃₃, and C₂₃₂₃. Accordingly, Equation (7), written in a local coordinate system (LCS) with its axes aligning with the crystal directions $[10\overline{1}0]$, $[\overline{1}2\overline{1}0]$, and [0001], takes the following form:

$$\begin{array}{l}{\dot{\mathrm{\sigma }}}_{11}=C_{1111}\left({\dot{\mathrm{\epsilon }}}_{11}-{\dot{\mathrm{\epsilon }}}_{11}^p\right)+C_{1122}\left({\dot{\mathrm{\epsilon }}}_{22}-{\dot{\mathrm{\epsilon }}}_{22}^p\right)+C_{1133}\left({\dot{\mathrm{\epsilon }}}_{33}-{\dot{\mathrm{\epsilon }}}_{33}^p\right)\\ {\dot{\mathrm{\sigma }}}_{22}=C_{1122}\left({\dot{\mathrm{\epsilon }}}_{11}-{\dot{\mathrm{\epsilon }}}_{11}^p\right)+C_{1111}\left({\dot{\mathrm{\epsilon }}}_{22}-{\dot{\mathrm{\epsilon }}}_{22}^p\right)+C_{1133}\left({\dot{\mathrm{\epsilon }}}_{33}-{\dot{\mathrm{\epsilon }}}_{33}^p\right)\\ {\dot{\mathrm{\sigma }}}_{33}=C_{1133}\left({\dot{\mathrm{\epsilon }}}_{11}-{\dot{\mathrm{\epsilon }}}_{11}^p\right)+C_{1133}\left({\dot{\mathrm{\epsilon }}}_{22}-{\dot{\mathrm{\epsilon }}}_{22}^p\right)+C_{3333}\left({\dot{\mathrm{\epsilon }}}_{33}-{\dot{\mathrm{\epsilon }}}_{33}^p\right)\\ {\dot{\mathrm{\sigma }}}_{23}=2C_{2323}\left({\dot{\mathrm{\epsilon }}}_{23}-{\dot{\mathrm{\epsilon }}}_{23}^p\right)\\ {\dot{\mathrm{\sigma }}}_{13}=2C_{2323}\left({\dot{\mathrm{\epsilon }}}_{13}-{\dot{\mathrm{\epsilon }}}_{13}^p\right)\\ {\dot{\mathrm{\sigma }}}_{12}=\left(C_{1111}-C_{1122}\right)\left({\dot{\mathrm{\epsilon }}}_{12}-{\dot{\mathrm{\epsilon }}}_{12}^p\right)\end{array}.$$

(8)

Equation (3) kinematically relates the components of the total strain rate tensor ${\dot{\mathrm{\epsilon }}}_{ij}$ and velocity vector u_i, disregarding any deformation mechanisms. Contrastingly, the plastic strain rate ${\dot{\mathrm{\epsilon }}}_{ij}^p$ is attributed to the plastic deformation mechanisms specific to a particular metal or alloy. To capture the slip and twinning mechanisms typical for hcp metals, we employ the CPFE model prescribing these processes along specified lattice planes and directions.

The plastic strain rate tensor is geometrically related to the slip rate on active slip systems by

$${\dot{\mathrm{\epsilon }}}_{ij}^p={\displaystyle \sum _{\mathrm{\alpha }}{\dot{\mathrm{\gamma }}}^{(\mathrm{\alpha })}}{\mathrm{\theta }}_{ij}^{(\mathrm{\alpha })},$$

(9)

where ${\dot{\mathrm{\gamma }}}^{(\mathrm{\alpha })}$ is the slip rate in the α-th slip system, and ${\mathrm{\theta }}_{ij}^{(\mathrm{\alpha })}$ is the Schmid tensor defined through the vectors of the slip direction $s_i$ and the normal $n_i$ to the slip plane:

$${\mathrm{\theta }}_{ij}^{(\mathrm{\alpha })}={\displaystyle \frac{1}{2}}{\left(s_i{n}_j+s_j{n}_i\right)}^{(\mathrm{\alpha })}.$$

(10)

The slip activation is governed by Schmid’s law [10]:

$${\dot{\mathrm{\gamma }}}^{(\mathrm{\alpha })}=\left\{\begin{array}{l}0, \mathrm{for} {\mathrm{\tau }}^{(\mathrm{\alpha })}<{\mathrm{\tau }}_{CRSS}^{(\mathrm{\alpha })}\\ {\dot{\mathrm{\gamma }}}_0{\left|{\displaystyle \frac{{\mathrm{\tau }}^{(\mathrm{\alpha })}}{{\mathrm{\tau }}_{CRSS}^{(\mathrm{\alpha })}}}\right|}^{\mathrm{\nu }}sgn({\mathrm{\tau }}^{(\mathrm{\alpha })}), \mathrm{for} {\mathrm{\tau }}^{(\mathrm{\alpha })}={\mathrm{\tau }}_{CRSS}^{(\mathrm{\alpha })}\end{array}\right.,$$

(11)

where ${\dot{\mathrm{\gamma }}}_0$ is the reference slip rate, ${\mathrm{\tau }}^{(\mathrm{\alpha })}=s_i{\mathrm{\sigma }}_{ij}{n}_j$ is the resolved shear stress acting in the α-th slip system, ${\mathrm{\tau }}_{CRSS}^{(\mathrm{\alpha })}$ is the CRSS necessary to initiate dislocation glide in the α-th slip system, and $\mathrm{\nu }$ is the strain rate sensitivity coefficient. The parameters ${\dot{\mathrm{\gamma }}}_0$ and $\mathrm{\nu }$ were chosen to minimize the strain rate sensitivity. As shown in [30], this condition is satisfied when ${\dot{\mathrm{\gamma }}}_0$ exceeds the equivalent plastic strain rate ${\dot{\mathrm{\epsilon }}}_{eq}$ by at least one order, i.e., $0.1{\dot{\mathrm{\gamma }}}_0\approx {\dot{\mathrm{\epsilon }}}_{eq}$.

In this study, dislocation glide is assumed to occur along the prismatic, basal, and first-order <c+a>-pyramidal slip systems, with their CRSSs related by the ratio ${\mathrm{\tau }}_0^{prism}:{\mathrm{\tau }}_0^{basal}:{\mathrm{\tau }}_0^{pyr}\approx 1:2:3$. At room temperature, the two most common twinning modes in α-titanium are the $\left\{10\overline{1}2\right\}〈\overline{1}101〉$ tensile twinning and the $\left\{11\overline{2}2\right\}〈11\overline{2}\overline{3}〉$ compressive twinning (Figure 1), which accommodate plastic strain along the [0001] direction under tension and compression, respectively [6,22]. Since compression is not considered in this study, only tensile twinning is incorporated into the model.

In the calculations carried out for polycrystalline structures, the CRSS for an α-th slip system is defined as a sum of additive contributions of grain-boundary strengthening and strain hardening:

$${\mathrm{\tau }}_{CRSS}^{(\mathrm{\alpha })}={\mathrm{\tau }}_0^{(\mathrm{\alpha })}+k{D}^{-1/2}+f({\mathrm{\epsilon }}_{eq}^p).$$

(12)

Here, the first term ${\mathrm{\tau }}_0^{(\mathrm{\alpha })}$ denotes the initial CRSS activating an α-th slip system in a single crystal. The second term takes into account the grain-boundary strengthening in polycrystals according to the Hall–Petch relationship. In our implementation, the grain structure is explicitly defined, so each finite element is assigned to a specific grain with a known volume. Consequently, in Equation (12), for elements belonging to the same grain, the parameter D is introduced as the diameter of a sphere having an equivalent volume. The final term represents the strain-hardening contribution, expressed as a function of the accumulated plastic strain ${\mathrm{\epsilon }}_{eq}^p$ and calculated as

$$f({\mathrm{\epsilon }}_{eq}^p)=a\left(1-\exp \left(-{\mathrm{\epsilon }}_{eq}^p/b\right)\right),$$

(13)

where a and b are the fitting parameters calibrated according to the experimental stress–strain curve. The slip and twin systems together with the corresponding initial CRSSs are summarized in

Table 1. Slip and twin systems are considered in the model developed.

Slip/Twin Mode	Slip/Twin Plane	Slip/Twin Direction	Number of Systems	${\mathbf{\tau }}_0$, MPa
Prismatic	$\left\{10\overline{1}0\right\}$	$〈11\overline{2}0〉$	3	70
Basal	$\left\{0001\right\}$	$〈11\overline{2}0〉$	3	140
First-order <c+a>-pyramidal	$\left\{10\overline{1}1\right\}$	$〈11\overline{2}3〉$	12	210
Tensile twinning	$\left\{10\overline{1}2\right\}$	$〈\overline{1}011〉$	6	145

.

Twinning is accompanied by a 180° rotation about the twin plane normal vector $n_i^{(\mathrm{\beta })}$. Numerically, this is implemented by a mirror reorientation of the LCS associated with a finite element once a twin is nucleated. Experimental observations [31] show that twinning preferentially occurs in the regions with high dislocation density accumulated through <c+a>-pyramidal slip. Therefore, a finite element is assumed to undergo twinning only when two conditions are met simultaneously:

$${\mathrm{\tau }}^{\mathrm{\beta }}\ge {\mathrm{\tau }}_0^{twin}\mathrm{and} {\mathrm{\gamma }}^{pyr}={\displaystyle \underset{t}{\int }{\dot{\mathrm{\gamma }}}^{pyr}dt}={\mathrm{\gamma }}_{cr}^{pyr}.$$

(14)

Here, β denotes the twin system with the largest resolved shear stress, and ${\mathrm{\gamma }}_{cr}^{pyr}$ is the critical amount of accumulated pyramidal slip. Upon twin nucleation, the crystallographic vectors $n_i$ and $s_i$, defining slip systems within the finite element, are transformed according to

$$R^{\mathrm{\beta }}=I-2n^{\mathrm{\beta }}\otimes n^{\mathrm{\beta }},$$

(15)

with I being the identity matrix. Note that only a single twin system is permitted within each finite element; activation of multiple twin systems is prohibited. This simplification is commonly employed in crystal plasticity models addressing twinning, e.g., [31].

The model parameters used in all calculations were as follows: C₁₁₁₁ = 162 GPa, C₁₁₂₂ = 92 GPa, C₁₁₃₃ = 69 GPa, C₃₃₃₃ = 181 GPa, C₂₃₂₃ = 47 GPa, a = 50 MPa, b = 0.5, and ${\mathrm{\gamma }}_{cr}^{pyr}$ = 0.01.

3.3. Numerical Implementation for Single and Polycrystals

In the numerical implementation, two coordinate systems, GCS and LCS, are used for each finite element. The governing Equations (1)–(4) and boundary conditions are formulated in GCS associated with the model geometry (X_i in the index notation). The CP-based constitutive Equations (8)–(14) are calculated in LCS with its axes aligning with the crystal directions $[10\overline{1}0]$, $[\overline{1}2\overline{1}0]$, and [0001] (denoted by x_i). The transformation from the crystallographic coordinate system of the hcp lattice to the orthonormal LCS is performed according to the relations provided in Ref. [32]. Transition between the GCS and LCS is determined by the rotation matrix R specific for each grain.

The constitutive equations are incorporated into the finite element package ABAQUS/Explicit (2019, Dassault Systèmes, Vélizy-Villacoublay, France) through a user subroutine VUMAT. Generally, the computation follows the scheme shown in Figure 3. This procedure is performed for all finite elements at each time increment. Note that in every finite element, the LCS undergoes a rigid-body rotation together with the material, while remaining orthonormal.

Within the CP framework, a polycrystal is represented as a conglomerate of single crystals with different crystallographic orientations relative to the GCS. Each single crystal is represented by a set of finite elements with the same initial crystallographic orientations. The initial conditions imply that all grains possess identical material properties (density, elastic moduli, CRSSs, etc.). The only distinction between grains lies in their unique LCS orientations relative to the GCS.

Numerical stress–strain curves are obtained by averaging the stress fields over the simulated volume:

$$〈{\mathrm{\sigma }}_{eq}〉={\displaystyle \frac{{\displaystyle \sum _N{\left({\mathrm{\sigma }}_{eq}V\right)}_i}}{{\displaystyle \sum _N{V}_i}}},$$

(16)

where σ_eq is the Mises stress, $V_i$ is the volume of the i-th finite element, and N is the total number of elements in the computational domain.

The strain is calculated from the nodal displacements of loaded surfaces as

$$\mathrm{\epsilon }={\displaystyle \frac{l-l_0}{l_0}},$$

(17)

where $l_0$ and l are the initial and current length of the model along the loading axis, respectively.

4. Results

4.1. Slip and Twinning in [0001] Single Crystals

At the microscale, the model is verified for an α-titanium single crystal subjected to uniaxial tension along the [0001] direction. The FE model with the dimensions of 6 × 3 × 3 mm³ is meshed by 2 × 10⁶ hexahedral FEs (C3D8R FE type in ABAQUS [33]). The initial crystal orientation is given by [0001] || X, $[\overline{1}2\overline{1}0]$ || Y, and $[10\overline{1}0]$ || Z (see Figure 4). Kinematic boundary conditions are applied to the top (S₁) and bottom (S₂) surfaces to simulate tension along the X-axis:

$$u_i{|}_{S_1}=v_i^0,\hspace{1em}{u}_i{|}_{S_2}=-v_i^0,$$

(18)

where $v_i^0$ are the prescribed components of the velocity vector on the S₁ and S₂ surfaces. The lateral surfaces are free from external forces. In the test simulations for single crystals, the grain boundary strengthening and strain hardening terms in Equation (12) are intentionally disregarded to artificially intensify plastic strain localization.

The distributions of accumulated pyramidal and prismatic slip, as well as the twin pattern, are shown in Figure 4 for two deformation stages, and corresponding equivalent stress fields are given in Figure 5b,c. Figure 4a–c depicts the deformation stage immediately following twin initiation. According to the proposed criterion (14), this stage is preceded by the accumulation of the <c+a>-pyramidal slip. Analytically, all twelve <c+a>-pyramidal slip systems under uniaxial tension along the [0001] direction are equally loaded and, thus, should preserve a symmetry of crystal deformation. In practice, however, wave effects intrinsic to dynamic simulations generate small perturbations in the local stress, strain, and kinematic fields, disrupting the symmetry of the material response. Consequently, although all twelve pyramidal systems are activated simultaneously, some of them become dominant shortly after activation. Subsequent plastic strain localizes within narrow through-thickness bands coinciding with the slip traces of the dominant systems on the crystal surface (cf. Figure 4a,d). This deformation scenario is qualitatively consistent with the experimental observations for other single crystals (see, e.g., [34]).

According to criterion (14), twinning is initiated in a region exhibiting the largest pyramidal slip and, on further loading, propagates through the specimen as a narrow band, growing in width and length (Figure 4c,g). Ultimately, one dominating twin region occupies the entire specimen cross-section, while smaller twins cease to grow. Although the twin nucleation criterion is local for each element and does not depend on the state of any neighboring elements (unlike that in [19]), the orientation of the twin band aligns with the intersection of the twin plane and the specimen surface (cf. Figure 4g,h). The lattice reorientation within the twin band enables activating the prismatic slip unrealizable in the reference configuration. On further loading, prismatic slip accumulates rapidly, surpassing the contribution of pyramidal slip to the overall deformation response. Subsequent plastic deformation develops predominantly within the twin region, resulting in macroscopic changes in the specimen shape.

Let us examine the stress evolution in the initial stage of twinning in terms of the equivalent stress distributions and stress–strain curves (Figure 5). Before twin nucleation, stress heterogeneity is attributed to a non-uniform accumulation of pyramidal slip (Figure 5a). As twin nucleates in some regions of more intensive pyramidal slip (Figure 4c), the lattice reorientation therein is accompanied by decreasing and redistributing stresses in adjacent areas (Figure 5b), while stresses in the rest of the crystal still remain high, and so do the overall stresses in the stress–strain curve (Figure 5d). On further loading, the twin propagates across the specimen, and prismatic slip is accumulated in the reoriented lattice behind the twin front (Figure 4e,f). This process is accompanied by stress relaxation in the rest of the specimen (Figure 5c). Correspondingly, the stress–strain curve demonstrates a drop-down portion (Figure 5d).

The model is additionally validated by comparing the calculated yield stress, obtained from stress–strain curves, with the analytical values derived from Schmid’s law [35]:

$${\mathrm{\sigma }}_y={\displaystyle \frac{{\mathrm{\tau }}_0}{F}},$$

(19)

where ${\mathrm{\sigma }}_y$ is the macroscopic yield stress, F is the Schmid factor. The discrepancy between the numerical and analytical results did not exceed 1%.

To evaluate the sensitivity of the numerical solution to the critical value of accumulated pyramidal slip, additional calculations were carried out for ${\mathrm{\gamma }}_{cr}^{pyr}$ = 0.005 and ${\mathrm{\gamma }}_{cr}^{pyr}$ = 0.02. The stress–strain curves in Figure 5d indicate that the smaller value of ${\mathrm{\gamma }}_{cr}^{pyr}$ leads to a faster decrease in the overall stress. The lower the ${\mathrm{\gamma }}_{cr}^{pyr}$ values, the faster the twin propagation and the larger the twin thickness. However, the general twinning patterns in all calculations are qualitatively similar.

Therefore, these findings emphasize the need for careful calibration of the ${\mathrm{\gamma }}_{cr}^{pyr}$ parameter along with the strain hardening coefficients, using experimental data of mechanical tests and twin morphology analysis.

4.2. Deformation Pattern in Polycrystalline Grains

A polycrystalline model measuring 100 × 50 × 100 μm³ was generated using the step-by-step packing method [36] on a 300 × 100 × 300 voxelated mesh. In the generation procedure, the grain seeds were randomly distributed throughout the volume and incrementally grown according to a fourth-order ellipsoidal equation [36], providing irregular-shaped grains (Figure 6b). The resulting model comprising 50 quasi-equiaxed grains is shown in Figure 6a in IPF colors. Correspondingly, the grain crystallographic orientations are described by the {0001} pole figure in Figure 6d,e. Note that the MM’ texture axis was intentionally aligned parallel to the tensile axis X, so that a larger number of grains were favorably oriented for twinning.

The boundary conditions in terms of prescribed velocities were imposed on two YZ lateral surfaces (S₁ and S₂ in Figure 6a) to simulate uniaxial tension along the X-axis, similarly to Equation (18). The lateral XY and top XZ surfaces were free from external forces, and the symmetry conditions were applied on the bottom XZ surface (S₃ in Figure 6a) with respect to its plane:

$$Y\hspace{0.33em}symmetry:\hspace{0.33em}{U}_2=0,\hspace{0.33em}U{R}_1=0,\hspace{0.33em}U{R}_3=0,$$

(20)

where $U_i$ is the displacement along the i-th direction and $U{R}_i$ is the rotation around the i-th direction.

The explicit introduction of the grain structure enables capturing the mesoscopic processes arising from the collective contribution and interaction of grain groups. The calculated plastic strain pattern is compared in Figure 7 with the experimental results (see Figure 8b in [37]) obtained by high-resolution digital image correlation. In both cases, plastic strain localizes in the mesoscale shear bands going through the grains, irrespective of their orientations and lying at an angle of ~45 degrees to the loading direction. The highest plastic strain within the localization bands accumulates near the grain boundaries. The qualitative similarity between the experimental and numerical grain-scale strain fields (Figure 7a and Figure 8b in [37]) validates the model at the mesoscale.

The experimental and numerical stress–strain curves compared in Figure 7b closely coincide in the elastic loading and after 4% tensile strain, but demonstrate a discrepancy in the stage of elastic–plastic transition. The experimental curve has a yield drop portion attributed to twinning in multiple grains. Contrastingly, the numerical stress–strain curve has a smooth portion following the yield point, where differently oriented grains are progressively involved in plastic deformation. Of the whole set of grains, only 16 are favorably oriented for twinning with their [0001] directions being close to the tensile axis X (red-colored orientations in Figure 6d). Twin nucleation and propagation lead to stress relaxation and redistribution in these grains, like in single crystals (see Figure 5). However, their contribution to the overall deformation response is low in comparison to that from prismatic and basal slip in most grains.

The calculated twin pattern for 9% tensile strain is shown in Figure 8a. The numerical results are qualitatively similar to the experimental EBSD data obtained for the deformed α-titanium specimen (see Figure 2b and Figure 8b). Both experimentally and numerically, there are grains with a single twin, with multiple twins parallel to each other, and with a set of intersecting twin bands formed in conjugate twin systems.

Let us analyze twin-induced stress–strain fields in a selected grain. Figure 9a shows the twin pattern in IPF colors relative to the loading direction, where the regions with the reference and reoriented crystal lattice are red and blue, respectively. Twinning along the primary and conjugate systems forms a set of intersecting through-thickness lamellae. Their platy-like shape is well seen in Figure 9b, where the elements with the reference lattice are hidden.

Corresponding equivalent stress and plastic strain fields are shown in Figure 9c,d. According to criterion (14), twin activation is preceded by plastic strain accumulation via pyramidal slip. Lattice reorientation within the twin regions changes the crystallographic direction initially aligned with the tensile axis from ~[0001] to ~$[01\overline{1}0]$ (blue elements in Figure 9a). In the regions with the reoriented lattice, the prismatic slip is activated and becomes a dominating mode of plastic deformation on further loading (Figure 9a,d), similar to the observations for single crystals. This leads to redistributing local stresses in the grain, forming a sandwich-like pattern of lower stresses in the twin lamellae and higher stresses in between.

5. Conclusions

In this study, the CPFE model, describing the deformation behavior of hcp metals, was developed and numerically implemented using α-titanium as a case study. The two-conditional criterion for twin activation was proposed, requiring that both the resolved shear stress in a twin system and the accumulated pyramidal slip simultaneously reach their respective critical values.

The model was implemented for a single crystal loaded along the [0001] direction, and for a polycrystal containing 50 grains and qualitatively validated at different scales. At the microscale, the simulated plastic strain localization and twin distribution were consistent with analytically expected intersections of slip and twin systems with specimen surfaces. At the mesoscale, the calculated patterns of plastic strain localization and twin distribution reproduced the key features observed experimentally in α-titanium. At the macroscale, the experimental and numerical stress–strain curves demonstrated close agreement in the elastic stage and after 4% tensile strain. The only noticeable discrepancy appeared during elastic–plastic transition, where the experimental curve exhibited a yield drop associated with twinning in multiple grains, while the numerical curve showed a smooth transition due to progressive involvement of differently oriented grains in plastic deformation. This difference arose because only a few grains in the model were favorably oriented for twinning, whereas most grains deformed by prismatic and basal slip.

Thus, the model demonstrated its ability to reproduce the key physical features of α-titanium deformation and twin evolution and can be further extended to investigate the local stress–strain state in hcp metals under various loading conditions, provided that the model parameters are appropriately calibrated.