Fracture Behavior of Twin Boundaries in Pure Titanium Under Biaxial Loading

Abstract

Six different twin boundary interface models were constructed by molecular dynamics simulations to investigate the effect of biaxial load ratio on the fracture behavior of titanium twin boundaries. Analysis of microstructural evolution indicates that twin boundaries exhibit a dual role during crack propagation. On one hand, they serve as preferential sites for void nucleation, promoting crack propagation along the twin boundary; on the other hand, they provide favorable sites for dislocation nucleation, inducing local plastic deformation at the crack tip, altering the crack path, and thereby hindering crack propagation. The crack propagation behavior in the ($\overline{1}0$11) and ($\overline{1}0$13) twin boundary models shows evident asymmetry: the crack on the left side mainly propagates through the void nucleation mechanism and exhibits a faster growth rate, while the right-side twin boundary inhibits crack propagation by favoring dislocation nucleation. In contrast, the crack propagation behavior in the ($\overline{1}0$12), ($\overline{2}1$11), ($\overline{2}1$12) and ($\overline{2}1$14) twin boundary models is largely symmetric on both sides, showing no significant difference in propagation rate. Stress field analysis further reveals that the differences in crack propagation behavior among the various twin boundary models mainly originate from the disparity in dislocation activity on both sides of the crack, resulting in different levels of stress concentration at the crack tip. When void nucleation occurs at the twin boundary interface, the stress concentration between the main crack and the void intensifies, promoting their coalescence and further propagation. Meanwhile, with an increase in biaxial load ratio, the stress concentration at the crack tip becomes more pronounced, further accelerating crack propagation.

1. Introduction

In HCP metals and their alloys, twinning is one of the dominant deformation modes during plastic deformation, which has a significant influence on the mechanical properties and fracture behavior of materials. The formation of deformation twins introduces twin boundaries into the matrix, and these special interfaces exert a strong hindering effect on dislocation motion. However, twin boundaries can also act as potential sites for void nucleation and crack initiation. Consequently, the controversial issue remains regarding whether twin boundaries in HCP metals and their alloys are beneficial or detrimental to crack propagation. Taking the typical {$\overline{1}$012} twin as an example: Zhang [1] found through fatigue experiments on bicrystalline copper that the tendency of cracks to propagate along twin boundaries is governed by the loading direction. At specific angles, cracks tend to propagate along twin boundaries, thereby significantly reducing the fracture toughness of the material. Jang [2] confirmed that twin boundaries can serve as cleavage planes in brittle fracture. Thompson [3] observed that fatigue cracks preferentially nucleate at twin boundaries. In thermomechanical fatigue tests of nickel-based alloys, Sun [4] found that twins accelerate failure by promoting the propagation of main cracks along twin boundaries. Nevertheless, other studies have suggested that {$\overline{1}$012} twin boundaries can suppress crack propagation [5].

Atomic-scale simulations and in situ characterization techniques have been widely applied to investigations on the ductile-to-brittle mechanism of grain boundaries in FCC materials [6,7]. For instance, Kou [5] combined in situ tensile tests in transmission electron microscopy and molecular dynamics simulations, revealing the dual role of twin boundaries in the dynamic propagation of cracks: dislocation slip near twin boundaries accelerates the fracture process, whereas twins in Al can induce crack blunting by alleviating stress concentration. In mode-I loaded cracks in magnesium, Wu [8] found that crack propagation along {$\overline{1}$012} twin boundaries and basal-prismatic interfaces is dominated by brittle fracture. Zu [9] further verified via atomic simulations that the plastic mechanism at crack tips in {$\overline{1}$011} twin boundaries and basal-prismatic interfaces is synergistically controlled by pyramidal-I dislocations and basal-prismatic phase transformations, and {$\overline{1}$012} twin boundaries exhibit brittle cracking characteristics at the initial stage of propagation. For the α-titanium system, MD studies by Wang et al. [10] demonstrated that twin boundaries provide preferential propagation paths for cracks, and microvoid nucleation promotes the propagation process. Meanwhile, the ductile-to-brittle transition induced by temperature elevation shifts the plasticity-dominated mechanism from grain boundary migration to dislocation slip.

According to the above literature, most existing studies have focused on the fracture behavior of twin boundary crack models under uniaxial loading. In hexagonal close-packed (HCP) metals, {$\overline{1}$011}, {$\overline{1}$012}, {$\overline{1}$013}, {$\overline{2}$111}, {$\overline{2}$112} and {$\overline{2}$114} twin boundaries are typical symmetric tilt grain boundaries (symmetric tilt grain boundaries, STGBs). However, there is still a lack of systematic research on the anisotropic fracture mechanism of twin boundaries under biaxial loading. Therefore, in this paper, six types of twin boundary models are established based on atomic-scale simulation methods to systematically investigate the evolution of the fracture behavior of titanium twin boundaries.

2. Models and Methods

2.1. Crack Models of Different Twin Boundaries

According to the crystallographic orientation difference between adjacent grains, grain boundaries can be classified into low-angle grain boundaries (misorientation < 10°) and high-angle grain boundaries (misorientation > 15°). High-angle grain boundaries are composed of dislocation arrays and are divided into two categories based on the relative rotation mode of grains: tilt grain boundaries (formed by tilting two grains around a parallel axis) and twist grain boundaries (generated by rotating around a perpendicular axis). As the simplest type of symmetric grain boundaries, STGBs can be fully characterized by only the tilt axis and tilt angle. Twin boundaries belong to a type of STGBs with strict mirror symmetry, serving as the interface between the matrix and twins during twinning deformation. Due to their low grain boundary energy and excellent atomic coherency, twin boundaries exhibit significantly higher structural stability than conventional grain boundaries, making them a widespread interface type in metallic materials.

Specific construction method of STGBs: a specific crystal orientation is selected as the tilt axis. By constructing an HCP titanium single-crystal model with periodic boundary conditions, a tilting operation (e.g., 5°, 10°, etc.) is performed along the selected orientation to generate symmetric crystal regions. The grain boundary position is determined and atomic positions nearby are optimized. The model is relaxed using an energy minimization algorithm (e.g., conjugate gradient method) combined with a potential function suitable for titanium. Finally, visualization tools are used to observe the grain boundary structure. Figure 1 shows a schematic diagram of a symmetric tilt grain boundary, where z is the tilt axis and the y-axis is the normal direction of the STGB plane. The top HCP crystal A and the bottom HCP crystal B are separated by the x-z plane. The two crystals are rotated around the z-axis by a certain angle: crystal A is rotated clockwise by θ_U, and crystal B is rotated counterclockwise by θ_L, thus constructing the STGB.

Table 1. Tilt axis, tilt angle and crystal orientation for different TB models.

TB Models	Crack Plane	Tilt Axis z	Tilt Angles θ	x′	y′	z′ = z
$(\overline{1}011)$TB	$(\overline{1}011)$	$[1\overline{2}10]$	62°	$[10\overline{1}0]$	$\left[0001\right]$	$[1\overline{2}10]$
$(\overline{1}012)$TB	$(\overline{1}012)$		43.41°	$[10\overline{1}0]$	$\left[0001\right]$	$[1\overline{2}10]$
$(\overline{1}013)$TB	$(\overline{1}013)$		32.15°	$[10\overline{1}0]$	$\left[0001\right]$	$[1\overline{2}10]$
$(\overline{2}111)$TB	$(\overline{2}111)$	$[10\overline{1}0]$	72.8°	$[11\overline{2}0]$	$\left[0001\right]$	$\left[10\overline{1}0\right]$
$(\overline{2}112)$TB	$(\overline{2}112)$		58.51°	$[11\overline{2}0]$	$\left[0001\right]$	$[10\overline{1}0]$
$(\overline{2}114)$TB	$(\overline{2}114)$		39.3°	$[11\overline{2}0]$	$\left[0001\right]$	$[10\overline{1}0]$

lists the tilt axes, tilt angles, and crystal orientations of six different twin boundary models. For the ($\overline{1}$011), ($\overline{1}$012), and ($\overline{1}$013) twin models with [1$\overline{2}$10] as the tilt axis, the coordinate systems are x′-[10$\overline{1}$0], y′-[0001], z′-[1$\overline{2}$10], with tilt angles θ of 62°, 43.41°, and 32.15° respectively, and the crack planes are ($\overline{1}$011), ($\overline{1}$012) and ($\overline{1}$013) planes. For the ($\overline{2}$111), ($\overline{2}$112) and ($\overline{2}$114) twin models with [10$\overline{1}$0] as the tilt axis, the coordinate systems are x′-[11$\overline{2}$0], y′-[0001], z′-[10$\overline{1}$0], with tilt angles θ of 72.8°, 58.51°, and 39.3° respectively, and the crack planes are ($\overline{2}$111), ($\overline{2}$112) and ($\overline{2}$114) planes. The microstructures of the ($\overline{1}$01K) twin crack models (K = 1, 2, 3) and ($\overline{2}$11K) twin crack models (K = 1, 2, 4) are shown in Figure 2 and Figure 3, respectively.

2.2. Simulation Methods and Parameter Settings

Based on the ATOMSK Beta 0.13.1 [11], the procedure for constructing twin boundary models with a central crack is as follows: (1) Initial grain boundary modeling: first, crystallographic orientation parameters are defined and a unit cell matrix is generated. Then, the model dimensions satisfying periodic conditions and the tilt angle are set. Taking the ($\overline{1}$011) twin boundary as an example, a unit cell with initial orientation x′-[10$\overline{1}$0], y′-[0001], z′-[1$\overline{2}$10] is established, rotated 62° around the z′-axis to form a symmetric grain boundary, with the X-axis along [$\overline{1}$011] and the Y-axis along [$\overline{1}$012]. (2) Energy minimization: combined with atomic position gradient optimization and selective atom removal, the conjugate gradient algorithm is adopted to minimize the interfacial energy, and the relaxed configuration with optimal energy is selected. (3) Model expansion: periodic supercell expansion is performed on the lowest-energy configuration to obtain the grain boundary model with target dimensions (80 × 80 × 2 nm³). (4) Potential function selection: based on the discussion of potential functions in the previous paper, the Mendelev-II (Ti2) potential is employed. The Mendelev-II (Ti2) EAM potential, optimized for α-titanium, has been validated against first-principles calculations for stacking fault energies and twin boundary energies. It has been successfully applied in previous MD studies of titanium fracture [10,12], confirming its reliability for modeling crack–twin boundary interactions. (5) Predefined central crack: the grain boundary model is imported into LAMMPS 2025, with free boundary conditions (s) in the X/Y directions and periodic boundary conditions (p) in the Z direction. A 24 nm central crack is created by removing atoms in the twin boundary region, and interatomic interactions across the crack are disabled. (6) Relaxation and tensile loading: after sufficient relaxation under the NPT ensemble to reach equilibrium, tension is applied using the fix addforce command. By setting different loading rates along the X and Y axes, the tensile fracture behavior of twin boundaries under uniaxial and equi-biaxial loading is simulated respectively. (6) Biaxial force-controlled loading is realized by ‘fix addforce’. The Y-direction force loading rate is dFy/dt = h/2,000,000, and the X-direction rate equals α times the Y-direction rate, α = 0 corresponds to uniaxial loading and α = 1 corresponds to equal biaxial force loading.

The atomic stress is calculated using the virial stress formulation:

$${\sigma }_{\alpha \beta }^{\left(i\right)}={\displaystyle \frac{1}{V_i}}\left(-m_i{v}_i^{\alpha }{v}_i^{\beta }+{\displaystyle \frac{1}{2}}{\sum }_{j\ne i}{r}_{ij}^{\left(\alpha \right)}{f}_{ij}^{\left(\beta \right)}\right)$$

(1)

where $V_i$ is the local atomic volume of atom $i$, $m_i$ is the atomic mass, $v_i^{\alpha }$ is the $\alpha$-component of atomic velocity, $r_{ij}^{\left(\alpha \right)}$ is the $\alpha$-component of the distance vector between atoms $i$ and $j$, and $f_{ij}^{\left(\beta \right)}$ is the $\beta$-component of the interatomic force between atoms $i$ and $j$.

The atomic strain is computed via the local deformation gradient method:

$$\mathbf{F}=\left({\sum }_j{\mathbf{r}}_{ij}{\mathbf{r}}_{ij}^{0T}\right){\left({\sum }_j{\mathbf{r}}_{ij}^0{\mathbf{r}}_{ij}^{0T}\right)}^{-1}$$

(2)

$${\mathsf{\epsilon }}_{\alpha \beta }={\displaystyle \frac{1}{2}}\left[{\left({\mathbf{F}}^{\mathbf{T}}\mathbf{F}\right)}_{\alpha \beta }-{\delta }_{\alpha \beta }\right]$$

(3)

where ${\mathbf{r}}_{ij}^0$ and ${\mathbf{r}}_{ij}$ are the position vectors of neighbor atom $j$ relative to atom $i$ in the reference and current configurations, respectively, and ${\delta }_{\alpha \beta }$ is the Kronecker delta.

Dislocation analysis is performed using the Dislocation Extraction Algorithm (DXA) in OVITO. Basal ⟨a⟩-type dislocations are identified by Burgers vectors b = 1/3⟨$11\overline{2}0$⟩ on (0001) planes. Pyramidal dislocations are distinguished by their ⟨c + a⟩ Burgers vectors (1/3⟨$11\overline{2}3$⟩) and their operation on {$10\overline{1}1$} or {$11\overline{2}2$} planes, verified through the angle between the dislocation line and the [0001] direction. The identification procedure involves: (i) DXA extraction, (ii) Burgers vector classification, (iii) slip plane determination via crystal analysis, (iv) cross-referencing with HCP slip systems, and (v) geometric validation against the twin boundary orientation.

The crack tip position is tracked using a coordination-number-based criterion. Atoms with coordination number < 8 (deviating >30% from the perfect HCP value of 12) are identified as surface atoms. The crack tip is defined as the foremost such atom along the crack propagation direction within ±2 nm of the original twin boundary plane. Crack length is calculated as the distance between the left and right crack tips. During extensive dislocation emission, this plane-restricted search excludes emitted dislocations that have glided away from the crack front. The results are cross-validated against a local stress threshold method (σyy < 0.5 GPa) and visual inspection, ensuring reliability even under complex dislocation activity.

3. Results

3.1. Stress–Strain Curves of Crack Models with Different Twin Boundaries

Figure 4 shows the stress–strain curves of six different twin boundary crack models under various loading ratios. The stress increases steadily during the elastic stage of the material. After reaching the peak stress, the curve begins to decline and enters the plastic deformation stage. For different models, the peak stress under biaxial loading is higher than that under uniaxial loading. In the later stage of plastic deformation, the stress–strain curve decreases significantly faster under biaxial loading than under uniaxial loading.

It should be noted that the gradual strain softening observed in Figure 4 under force-controlled loading is fundamentally distinct from the catastrophic failure typically expected in macroscopic samples. In our atomic-scale twin boundary models, the presence of twin boundaries provides multiple energy dissipation channels: (i) dislocation emission from crack tips blunts the crack and reduces stress concentration; (ii) void nucleation at twin boundaries accommodates plastic deformation without immediate catastrophic crack extension; and (iii) twin boundary migration and dislocation-twin boundary interactions induce local plasticity and crack path deflection. These sequential deformation mechanisms allow the applied force to be progressively redistributed, resulting in the gradual stress decrease from 1/2 to 1/10 of the peak value, rather than an abrupt stress drop.

3.2. Microstructural Evolution During Twin Boundary Crack Propagation Under Biaxial Loading

3.2.1. In the ($\overline{1}$011) Twin Boundary Model

Figure 5 shows the microstructural evolution of the ($\overline{1}$011) twin boundary model at different times under a loading ratio of 0. Distinct differences in crack propagation modes between the left and right sides can be clearly observed. As shown in Figure 5a, at 20,000 loading steps with a strain of 0.018, cracks on both sides propagate along the twin boundary. With continued loading, at 40,000 steps and a strain of 0.036, vacancy nucleation occurs near the left crack and gradually expands into voids, leading to rapid crack growth, while basal partial dislocations begin to emerge at the right crack tip, as illustrated in Figure 5b. As loading proceeds, continuous dislocation emission generates multiple dislocation slip bands near the right crack tip, accompanied by a significant increase in dislocation density and intensified dislocation tangling, resulting in crack-tip blunting, as shown in Figure 5c. The interaction between dislocations and the twin boundary further promotes microstructural evolution, producing substantial plastic deformation in local regions and impeding crack propagation, as presented in Figure 5d.

Significant differences also exist in the crack propagation modes on the left and right sides of the ($\overline{1}$011) twin boundary model at a loading ratio of 1. Figure 6 shows the microstructural evolution during crack propagation on both sides of the ($\overline{1}$011) twin boundary model at a loading ratio of 1. Microstructural evolution of the left crack: as shown in Figure 6a, no dislocation activity is observed at the initial stage of crack propagation. At 50,000 loading steps with a strain of 0.044, pronounced stress concentration appears at the crack tip, accompanied by vacancy nucleation around the tip. These vacancies gradually develop into voids, accelerating crack propagation along the -X direction, as illustrated in Figure 6b. Subsequently, at 70,000 loading steps and a strain of 0.062, basal dislocations begin to nucleate and emit at the twin boundary, and the glide of these basal <a>-type dislocations further promotes the formation of stacking faults, as shown in Figure 6c,d. Overall, the left crack propagates very rapidly, with a significantly higher growth rate compared with that at a loading ratio of 0.

Microstructural evolution of the right crack: at 50,000 loading steps with a strain of 0.044, basal <a>-type dislocations are activated on the right side of the crack tip, as shown in Figure 6b. Subsequently, continuous emission of basal dislocations leads to gradual blunting of the crack tip. Meanwhile, multiple dislocation slip bands formed near the crack increase the density of dislocation tangles, significantly enhancing the degree of crack blunting. The interaction between dislocations and the twin boundary induces grain boundary migration and considerable plastic deformation, forcing the crack propagation path to change, as illustrated in Figure 6c,d. Under this loading mode, the existence of twin boundaries increases the resistance to crack propagation. Extensive plastic deformation occurs near twin boundaries, and the crack path deflects, which hinders crack growth.

3.2.2. In the ($\overline{1}$012) Twin Boundary Model

Figure 7 shows the microstructural evolution during crack propagation in the $(\overline{1}012)$ twin boundary model at a loading ratio of 0. The crack propagation modes and mechanisms on the left and right sides are basically identical. At 20,000 loading steps with a strain of 0.020, the crack initiates propagation, and no dislocation activity is detected by DXA analysis, with the corresponding atomic microstructure shown in Figure 7a. At 40,000 loading steps and a strain of 0.039, white disordered atoms appear at the crack tip, and vacancies begin to form on both sides of the crack tip, as shown in Figure 7b. When the strain rises to 0.049, a void forms ahead of the crack tip, as can be seen in Figure 7c. The void is mainly formed by the fracture of atomic bonds and the gradual increase in interatomic spacing in local regions [12]. The formation of microvoids accelerates crack growth and enlarges the crack opening displacement. At 60,000 loading steps with a strain of 0.064, a small number of basal dislocations emit from the crack tip, as shown in Figure 7d. In addition, basal stacking fault structures left by basal dislocation slip are also observed near the voids, a phenomenon also reported in single-crystal nickel [13] and BCC-structured iron [14].

Figure 8 shows the microstructural evolution during crack propagation in the ($\overline{1}$012) twin boundary model at a loading ratio of 1. The crack propagation processes on the left and right sides are basically identical. Vacancy nucleation is first observed at the crack tips on both sides without any dislocation activity, and a large number of disordered atoms appear at the crack tips, as shown in Figure 8a. Subsequently, with continuous loading, vacancies expand and form voids, and the crack begins to propagate rapidly along the twin boundary, as shown in Figure 8b. Dislocation activity near the crack is only observed in the late stage of crack propagation, as illustrated in Figure 8c,d.

In addition, the coalescence of the main crack with voids ahead can be clearly seen, accelerating crack growth. Compared with the case at a loading ratio of 0, dislocation activity at the crack tip is less and crack propagation is faster at a loading ratio of 1, and the ($\overline{1}$012) twin boundary exhibits more brittle characteristics.

3.2.3. In the ($\overline{1}$013) Twin Boundary Model

Figure 9 shows the microstructural evolution of the ($\overline{1}$013) twin boundary model at different loading steps under a loading ratio of 0. The crack propagation behavior on the left and right sides is slightly different. As shown in Figure 9b, at 40,000 loading steps with a strain of 0.025, pyramidal dislocations are activated on the right side of the crack tip, while the left crack propagates along the twin boundary. With further loading, at 60,000 steps and a strain of 0.048, many basal partial dislocations are observed near the right crack tip. The interaction between dislocations and the twin boundary causes an obvious deflection of the right crack plane, thus hindering crack propagation, as shown in Figure 9c,d.

Figure 10 shows the microstructural evolution during crack propagation on both sides of the ($\overline{1}$013) twin boundary model at a loading ratio of 1. Microstructural evolution of the left crack: at 20,000 loading steps with a strain of 0.021, the crack is in the linear elastic stage, as shown in Figure 10a. At 60,000 loading steps and a strain of 0.066, a considerable number of white disordered atoms appear near the left crack tip, and the crack begins to propagate along the twin boundary, as illustrated in Figure 10b. With further loading, a small number of basal partial dislocations are activated on the left side of the crack tip. Overall, the left crack propagates relatively rapidly along the twin boundary, and few dislocations are activated at the crack tip. The twin boundary exhibits brittle characteristics under this mode, with the corresponding microstructures shown in Figure 10c,d.

Microstructural evolution of the right crack: pyramidal <a>-type dislocations are first activated on the right side of the crack tip and emit continuously as loading proceeds, as shown in Figure 10a,b. Subsequently, at 70,000 loading steps with a strain of 0.073, basal dislocations begin to nucleate and emit at the crack tip. With increasing applied strain, new dislocations nucleate at the phase transformation region, as illustrated in Figure 10c. Extensive dislocation activity enhances the interaction between dislocations, resulting in large plastic deformation. Atoms near the crack tip become more disordered and rearrange, deviating from the ideal lattice structure. The crack tip deflects, which lengthens the propagation path, consumes more energy, and thus hinders crack growth. Figure 10d clearly shows that crack propagation is suppressed, with the crack extending only a short distance along the twin boundary.

3.2.4. In the ($\overline{2}$111) Twin Boundary Model

Figure 11 shows the microstructural evolution during crack propagation in the ($\overline{2}$111) twin boundary model at a loading ratio of 0. As shown in Figure 11a, no dislocation activity is observed at the initial stage of crack propagation. At 50,000 loading steps with a strain of 0.038, significant stress concentration appears in the crack-tip region, and the crack then begins to propagate along the twin boundary, as illustrated in Figure 11b. When the loading steps increase to 70,000 and the strain further rises to 0.074, basal dislocations emit from the crack tip, as shown in Figure 11c. In this twin boundary model, the density of emitted dislocations remains below 0.05 nm⁻² throughout the loading process, and no continuous dislocation emission events are observed until the late stage of crack propagation. The limited dislocation activity allows the crack to propagate smoothly along the twin boundary. Numerous unidentifiable defect types by DXA analysis only appear on both sides of the model in the late stage of crack propagation, as presented in Figure 11d.

Figure 12 shows the microstructural evolution at a loading ratio of 1. The propagation modes on the left and right sides of the crack are basically identical. As shown in Figure 12a, no dislocation activity is detected at the onset of crack propagation. At 20,000 loading steps with a strain of 0.011, significant stress concentration occurs at the crack tip, accompanied by more disordered atomic arrangement around the tip, and the crack propagates rapidly along the twin boundary, as illustrated in Figure 12b. The twin boundary has low interfacial energy and weak bonding strength, leading to stress concentration near the boundary. Crack propagation tends to follow the path with the least energy consumption, which is exactly provided by the twin boundary. With increasing strain, a small amount of basal dislocation activity is observed. Compared with the case at a loading ratio of 0, the crack propagates faster with less plastic deformation at a loading ratio of 1 under the same loading steps, exhibiting typical brittle fracture characteristics, as shown in Figure 12c,d.

3.2.5. In the ($\overline{2}$112) Twin Boundary Model

Figure 13 shows the microstructural evolution during crack propagation in the ($\overline{2}$112) twin boundary model at a loading ratio of 0. As shown in Figure 13b, at 20,000 loading steps with a strain of 0.023, stress concentration occurs at the crack tip, and the crack begins to propagate along the twin boundary. At 45,000 loading steps and a strain of 0.049, a small number of basal dislocations form and emit at the right crack tip, while many white disordered atoms appear around the left crack tip, leading to rapid crack propagation along the twin boundary, as illustrated in Figure 13c,d.

Figure 14 shows the microstructural evolution during crack propagation in the ($\overline{2}$112) twin boundary model at a loading ratio of 1. Compared with the case at a loading ratio of 0, the crack propagates faster in the in-plane direction at a loading ratio of 1, and the twin boundary fractures in a brittle manner. In contrast to Figure 13, the number of disordered atoms at the crack tip during propagation in the ($\overline{2}$112) twin boundary model is significantly increased in Figure 14, while the number of dislocations is considerably reduced.

3.2.6. In the ($\overline{2}$114) Twin Boundary Model

Figure 15 shows the microstructural evolution during crack propagation in the ($\overline{2}$114) twin boundary model at a biaxial loading ratio of 0. At 20,000 loading steps with a strain of 0.020, the crack initiates propagation, and no dislocation activity is observed at the initial stage, as shown in Figure 15a. A considerable number of defective atoms exist on both sides of the crack tip, accompanied by local vacancy nucleation. When the interatomic bonding force is lost under external loading, bond fracture occurs in local regions accompanied by increased lattice spacing. These lattice defects gradually aggregate under stress drive and eventually evolve into macroscopic void structures. In addition, stacking fault structures left by basal dislocation slip are also observed near the twin boundary, as illustrated in Figure 15b,c. With continuous loading and increasing strain, the crack propagates rapidly, as shown in Figure 15d.

Figure 16 shows the microstructural evolution during crack propagation in the ($\overline{2}$114) twin boundary model at a biaxial loading ratio of 1. Vacancies form at the twin boundaries on both sides of the crack, which expand continuously with loading and eventually develop into voids. New vacancies keep nucleating at the twin boundaries, propagate, and finally coalesce with the main crack, accelerating crack growth and leading to final fracture, as shown in Figure 16a–d. Obviously, the crack propagates faster in Figure 16 than in Figure 15 at the same loading steps.

Through the above microanalysis of different twin models, it can be found that the crack propagation mechanisms of different twin boundary models under biaxial loading are different. Twin boundaries exhibit a dual effect on crack propagation, and even distinct crack propagation behaviors may occur on the left and right sides of the same twin boundary model. For the ($\overline{1}$011) and ($\overline{1}$013) twin boundary models, dislocation nucleation and emission near the right crack give rise to dislocation tangles and strong interactions between dislocation slip bands and the twin boundary. This causes twin boundary migration and cracking, crack blunting and deflection, so that the existence of twin boundaries effectively hinders crack propagation. For the left crack, however, the twin boundary provides favorable sites for void nucleation, and the growth of voids guides crack propagation, ultimately promoting crack growth. For the ($\overline{1}$012), ($\overline{2}$111), ($\overline{2}$112) and ($\overline{2}$114) twin boundary models, cracks propagate rapidly along the twin boundaries, and only a small number of dislocations are observed in the late propagation stage in some models. The twin boundaries act as preferential sites for void nucleation, which in turn induces rapid crack propagation. Therefore, twin boundaries show bidirectional regulatory characteristics during crack propagation: they can either act as preferential propagation paths to accelerate crack growth, or induce crack path deflection through dislocation nucleation, resulting in a retarding effect on crack propagation.

3.2.7. Atomic-Scale Mechanisms of Gradual Strain Softening Under Force-Controlled Loading

Under force-controlled conditions, the competition between crack driving forces and energy dissipation mechanisms at the atomic scale determines the macroscopic mechanical response. When the applied stress reaches the peak value, crack initiation triggers intensive dislocation emission from the crack tip. The emitted dislocations glide along slip planes and interact with twin boundaries, forming dislocation pile-ups and tangled networks that consume elastic strain energy. Simultaneously, void nucleation at twin boundaries creates additional free surfaces that accommodate deformation. The coalescence of voids with the main crack further delays rapid failure by distributing the crack tip stress field over a broader region. This sequential activation of multiple deformation mechanisms ensures that the stress decreases gradually rather than abruptly, consistent with the progressive strain softening observed in Figure 4.

3.3. Quantitative Statistics of Deformation Defects in Different Twin Boundary Models

During the microstructural evolution, deformation defects exist throughout the entire process. Figure 17 shows the variation curves of dislocation density for different twin boundary crack models under various loading ratios. It can be seen from the figure that the dislocation density of all crack models increases with the increase in loading steps under different loading ratios. In addition, the dislocation density of each model decreases significantly with the increase in loading ratio, indicating that the plastic deformation at the crack tip also weakens as the loading ratio rises.

3.4. Crack Length Curves of Different Twin Boundary Models

The crack propagation length is calculated by recording the position variation in the crack tip during loading, enabling a quantitative comparison of crack propagation rates among different twin boundary models. The relationship between crack length and loading steps for various twin boundary models is presented in Figure 18. Investigations on the ($\overline{1}$011) and ($\overline{1}$013) twin boundary models reveal asymmetric crack propagation behavior on the two sides of the material, as shown in Figure 18a,c. The left crack is dominated by the void nucleation mechanism and exhibits a relatively high propagation rate. In contrast, many dislocations form in the twin boundary region near the right crack and interact strongly with the twin boundary. The presence of the twin boundary significantly suppresses crack propagation, causing crack blunting and path deflection, thus remarkably reducing its propagation velocity. For the ($\overline{1}$012), ($\overline{2}$111), ($\overline{2}$112) and ($\overline{2}$114) twin boundary models, the crack propagation rates on the left and right sides are almost identical, as shown in Figure 18b,d–f. For these models, twin boundaries provide favorable sites for void nucleation, and void coalescence accelerates crack propagation. It can therefore be concluded that twin boundaries exert a dual effect on crack propagation. By comparing the crack length curves of different models under uniaxial and equal biaxial loading, it is found that for all models, the crack length increases more rapidly with loading steps at a loading ratio of 1. This indicates that an increase in the loading ratio significantly promotes crack propagation.

3.5. Stress Field Analysis at Crack Tips for Different Twin Boundary Models

From the above analysis of the micro-mechanisms of crack propagation in different twin boundary models, different crack models exhibit distinct propagation mechanisms. Dislocation nucleation and emission at the crack tip suppress crack growth, whereas the coalescence of voids formed by vacancy nucleation and propagation with the main crack accelerates crack growth. These deformation behaviors at the crack tip are related to the evolution of the crack-tip stress field [15,16,17], which in turn influences the crack propagation rate. Therefore, this section focuses on the stress field distribution at the crack tip in different twin boundary crack models.

Figure 19 shows the atomic stress σ_yy contour maps at the left and right crack tips of the ($\overline{1}$011) twin boundary model under different loading ratios. At a loading ratio of 0, as shown in Figure 19a,c, stress concentration appears at the crack tips, initiating crack propagation. With ongoing loading, dislocations emit continuously from the right crack tip, weakening the stress concentration and causing obvious crack blunting. As a result, the propagation length of the right crack is smaller than that of the left crack. At a loading ratio of 1, significant stress concentration can also be observed on both sides of the initial crack, as shown in Figure 19b. As the loading steps increase, dislocation emission occurs at the right crack tip, relieving the stress concentration and leading to distinct blunting of the crack tip, as illustrated in Figure 19d. In contrast, the left crack tip shows weak dislocation activity, limited blunting, and more severe stress concentration, resulting in faster crack propagation on the left side.

In summary, within the same model, distinct dislocation activities at the left and right crack tips lead to different degrees of crack blunting, which further cause asymmetric stress concentration and thus different crack propagation rates. In addition, by comparing the stress field distributions at loading ratios of 0 and 1, it is found that the stress concentration at the crack tip is more pronounced at a loading ratio of 1, leading to faster crack propagation under this condition.

Figure 20 shows the atomic stress σ_yy contour maps at the crack tip of the ($\overline{2}$114) twin boundary model at a loading ratio of 1. As shown in Figure 20a, stress concentration occurs at the crack tip, initiating crack propagation. With increasing loading, when the number of loading steps reaches 60,000, vacancies nucleate near the crack tip. The accumulation of vacancy atoms forms obvious voids, accompanied by significant stress concentration around the voids, as illustrated in Figure 20b.

When the loading steps increase to 80,000, continuous nucleation and growth of vacancies near the crack tip result in the formation of voids, and stress concentration persists around both the voids and the crack, as shown in Figure 20c. With further loading, the coalescence of the main crack and the voids drive rapid crack propagation, as presented in Figure 20d.

Figure 21 shows the atomic stress σ_yy distribution at the crack tip of the ($\overline{2}$112) twin boundary model. As shown in Figure 21a, at 50,000 steps and a loading ratio of 0, significant stress concentration appears at the crack tip, leading to rapid crack propagation. When the loading steps increase to 60,000, the crack length extends to 290.6 Å, as presented in Figure 21c. It can be seen from Figure 21b that at a loading ratio of 1, pronounced stress concentration also occurs at the crack tip, resulting in fast crack growth. When the loading steps reach 60,000, the crack length increases to 330.4 Å, as illustrated in Figure 21d. The stress concentration at the crack tip at a loading ratio of 1 is consistently stronger than that at 0. This indicates that an increasing loading ratio elevates the crack-tip stress level, enhances the driving force for crack propagation, and thus accelerates crack growth, making the material exhibit more obvious brittleness.

4. Conclusions

In this paper, molecular dynamics simulations were performed to systematically investigate the fracture behavior of six twin boundary crack configurations under biaxial loading. The atomic arrangements and defect evolution mechanisms near the crack tip were analyzed in detail. Based on quantitative evaluation of crack length variation and dislocation density evolution, the fracture modes of different twin boundaries were clarified. The main conclusions are summarized as follows:

(1) For the ($\overline{1}$011) and ($\overline{1}$013) twin boundaries, crack propagation on the left side is mainly promoted by void nucleation, while the right twin boundary provides nucleation sites for dislocations and thus hinders crack growth. For the ($\overline{1}$012), ($\overline{2}$111), ($\overline{2}$112) and ($\overline{2}$114) twin boundary models, crack propagation is dominated by void nucleation at the twin boundaries, which accelerates crack growth.

(2) Twin boundaries exhibit a dual effect during crack propagation: on the one hand, they act as preferential sites for void nucleation and tend to promote crack propagation along the twin boundary; on the other hand, twin boundaries can serve as dislocation nucleation sources, inducing local plastic deformation in the crack-tip region, thereby changing the crack propagation path and impeding crack growth.

(3) For the ($\overline{1}$011) and ($\overline{1}$013) twin boundary models, the left crack propagates faster than the right one. For the ($\overline{1}$012), ($\overline{2}$111), ($\overline{2}$112) and ($\overline{2}$114) twin boundary models, the crack propagation rates on the left and right sides are nearly identical.

(4) Stress field analysis reveals that the asymmetric dislocation activity on the two sides of the twin boundary leads to different levels of stress concentration at the crack tips, thereby affecting the respective crack propagation rates. When void nucleation occurs at the twin boundary, the stress concentration near the main crack and voids becomes more significant, accelerating their coalescence and promoting crack growth. As the biaxial loading ratio increases, stress concentration at the crack tip becomes more pronounced, further accelerating crack propagation.

(5) Under force-controlled loading, the atomic-scale deformation mechanisms (dislocation emission, void nucleation, and twin boundary migration) in our twin boundary models act as energy dissipation channels that prevent catastrophic failure and result in gradual strain softening. This behavior is fundamentally distinct from conventional macroscopic samples where constrained plasticity leads to abrupt failure under force control.

However, the Z-direction thickness of the simulation model was set to 2 nm to reduce computational cost, and this thin size with periodic boundaries may constrain out-of-plane deformation. To eliminate size effects, future work will adopt thicker models with Z-direction thicknesses of 6 nm and above, while maintaining computational feasibility.