1. Background

Near-α titanium alloys such as TA31 are widely used in aerospace structures and deep-sea equipment because of their high specific strength, excellent fracture toughness, good corrosion resistance, and favorable weldability. In service, these components are frequently subjected to complex multiaxial stress states. For example, the pressure hulls of deep-sea submersibles experience typical biaxial loading rather than simple uniaxial deformation. Nevertheless, structural design and performance evaluation still rely predominantly on uniaxial tensile data, and the deformation behavior and underlying microstructural mechanisms of titanium alloys under biaxial loading remain insufficiently understood.

In hexagonal close-packed (HCP) metals, the low crystallographic symmetry and the limited number of independent slip systems make deformation twinning crucial for accommodating strain along the c-axis. Previous studies have shown that biaxial loading may alter the competition between slip and twinning and may even activate unconventional twin modes. However, reports on twinning behavior in titanium alloys under biaxial loading are contradictory, and the formation of high-index, high-energy twin boundaries and their role in strain accommodation remain poorly understood.

This work systematically investigates the mechanical response and microscopic deformation mechanisms of an industrial-grade TA31 near-α titanium alloy under biaxial tension, with particular focus on the high-energy {13‾41} twin boundary identified for the first time and the hierarchical twinning behavior it induces. The aim is to reveal a new mechanism by which biaxial stress states regulate twin activation and strain partitioning.

2. Materials and Experiments

A hot-rolled and subsequently annealed TA31 titanium alloy plate with a thickness of approximately 42 mm was selected. The material exhibits a pronounced duplex microstructure, in which the α phase accounts for ~96% of the volume fraction and the average grain size is approximately 20 μm, together with a strong basal texture. Cruciform specimens were designed and optimized for biaxial tensile testing. Stress-relief holes were introduced to reduce edge stress concentration, and local thinning and a shallow central recess were implemented to promote fracture initiation from the specimen center.

Biaxial tensile tests were conducted on a vertical biaxial testing machine. Loading was applied along the rolling direction (RD) and transverse direction (TD) at a force ratio of 1:1 until final fracture. Full-field strain evolution in the central region was continuously acquired using digital image correlation (DIC).

Because the cruciform specimen does not have a uniform gauge section, the true stress–strain response under biaxial loading cannot be measured directly. An inverse analysis method was therefore employed. The experimentally measured load–time histories were applied as boundary conditions in a finite element (FE) model, and the strain evolution in the specimen center obtained by DIC was compared with the simulation. The material constitutive parameters were iteratively updated until the simulated and experimental strain–load curves converged. This approach overcomes the difficulty of directly measuring stress and strain in biaxial tests and is of considerable practical value.

Microstructural characterization was performed using electron backscatter diffraction (EBSD), transmission Kikuchi diffraction (TKD), and transmission electron microscopy (TEM). First-principles density functional theory (DFT) calculations were used to quantify the twin boundary energies. Interfacial disconnections and step structures were analyzed by high-resolution TEM (HRTEM) combined with Burgers circuit mapping.

3. Results and Discussion

3.1 Mechanical Response: Anisotropy Markedly Amplified under Biaxial Loading

Fig. 2(a) Comparison of true stress–strain curves (uniaxial vs. biaxial).

The true stress–strain curves obtained by inverse analysis (Fig. 2a) show that under biaxial loading the yield strengths along RD and TD increase to approximately 950 MPa and 1070 MPa, respectively, substantially higher than those under uniaxial tension. Meanwhile, the plastic strain along TD is only about 2.1%, much lower than the 11.7% along RD, indicating that biaxial loading markedly amplifies the intrinsic anisotropy of the material. FE simulations and DIC results consistently show that the high-stress region in the specimen center exhibits a band-like distribution along TD, accompanied by stress redistribution, which ultimately triggers early fracture along TD. This amplification of anisotropy originates from the suppression of transverse plastic relaxation by biaxial tension and from the higher hydrostatic stress component (stress triaxiality of about 0.66) that constrains plastic flow.

3.2 Evolution of Twin Types: From Uniaxial to Biaxial Loading

Fig. 3 EBSD twin distributions and statistics after uniaxial and biaxial deformation ({10‾12} in green, {11‾22} in red).

Under uniaxial tension, {10‾12} tensile twins and {11‾22} compression twins coexist, with an interfacial density ratio of approximately 3:1. Under biaxial loading, the density of {10‾12} twins increases markedly, whereas {11‾22} compression twins are almost absent. This difference is not primarily caused by differences in stress path reported in the literature, but by the change in Schmid factor determined by the initial texture. In the initial basal texture, the c-axes of most grains deviate from the ND. Under biaxial loading, the stress in the ND direction in the central region is predominantly compressive, which is unfavorable for the activation of compression twinning.

3.3 First Identification of High-Energy {13‾41} Twins and Hierarchical Twinning Structures

Fig. 4 TKD maps and pole figures (primary {13‾41} twin and secondary/tertiary {10‾12} twins).

In the high-strain region near the fracture of the biaxially loaded specimen, TKD characterization identified for the first time a {13‾41} twin (denoted T_H). This twin exhibits a 63° [55‾103] orientation relationship with the matrix, and the boundary is straight with a serrated morphology. More importantly, a clear multilevel twinning structure was observed within this high-energy twin: secondary {10‾12} twins (T′) nucleate preferentially along the primary twin boundary, and tertiary {10‾12} twins (T″) subsequently form within the secondary twins. Pole figure and orientation analyses confirm that the interface between the secondary twin and the matrix is also close to a {10‾11} twin relationship.

Selected-area electron diffraction (SAED) further verified the mirror symmetry of the {13‾41} interface and the 85° [11‾20] orientation relationship between the secondary and tertiary {10‾12} twins. HRTEM shows that the angle between the (0001) planes on both sides of the secondary twin/matrix interface is approximately 59°, very close to the theoretical value of 57.2° for a {10‾11} twin. In addition, a high density of dislocations and step structures is present near the interface, indicating that the interface is undergoing stress-driven crystallographic relaxation.

3.4 DFT and Interfacial Dislocation Analysis: Nature of the High-Energy Interface and Its Evolution Driving Force

Table 1 Comparison of twin boundary energies from DFT calculations.

DFT calculations show that the {13‾41} twin boundary energy is 0.52 J/m², significantly higher than those of common twins such as {10‾12} (0.29 J/m²) and {11‾22} (0.37 J/m²), confirming its high-energy nature. However, under biaxial loading, the hydrostatic tensile stress, approximately twice that under uniaxial loading, significantly lowers the nucleation barrier, thereby allowing this high-energy twin to be activated.

Furthermore, HRTEM combined with Pond circuit mapping was used to resolve the interfacial dislocations on the {13‾41} boundary. The spatial direction of the Burgers vector was determined to be<56‾12>, with an in-plane component of<18‾9‾11>. The generalized Schmid factor (GSF) was used to evaluate the driving effect of different stress states on interface migration. It was found that biaxial loading not only increases the upper limit of the GSF for the most favorable variant (from 0.43 to 0.49), but also provides actual stress levels much higher than those under uniaxial loading, thus supplying a stronger mechanical driving force for the nucleation and propagation of the {13‾41} twin.

3.5 Interface Evolution Mechanism: From Nucleation to a Near-{10‾11} Stable Interface

Figs. 8–9 TEM comparison and dislocation analysis of secondary twin nucleation and growth interfaces.

Statistical analysis shows that the interfacial angle between the secondary twin and the matrix is generally larger than 60° at the nucleation stage, and gradually decreases below about 60° during growth, approaching the symmetric geometry of a {10‾11} twin. HRTEM further shows that the grown interface already exhibits an atomic arrangement close to that of a {10‾11} twin, and contains b₃/₂-type interfacial dislocations that can decompose into mobile b₂/₂ twinning dislocations and

Fig. 10 Schematic of crystallographic relationships and the formation process of hierarchical twinning.

Based on the crystallographic analysis, the authors propose the following evolution path: a primary {13‾41} twin nucleates under high stress → its high-energy boundary becomes a preferential nucleation site for secondary {10‾12} twins → among the twelve {13‾41} variants and six {10‾12} variants, only a specific variant combination satisfies the geometric matching and energy-minimization requirements, so that the interface between the secondary twin and the matrix gradually approaches the stable {10‾11} twin relationship. This path of “high-energy interface → selective nucleation → stress-driven relaxation → near-stable interface” represents a previously unrecognized strain accommodation mechanism that has not been observed during room-temperature deformation.

4. Conclusions

This study systematically reveals the mechanical response and microscopic twinning behavior of a near-α titanium alloy under biaxial tension. The main conclusions are as follows:

1. Mechanical behavior: True stress–strain curves under biaxial loading were accurately obtained by inverse analysis. Biaxial loading significantly amplifies anisotropy, with yield strengths along RD and TD reaching 950 MPa and 1070 MPa, respectively, and very low ductility along TD (ε_f ≈ 2.1%).

2. Transition in twinning modes: Under uniaxial tension, {10‾12} and {11‾22} twins coexist. Under biaxial loading, {11‾22} compression twins are suppressed, while the density of {10‾12} tensile twins increases markedly, reflecting the strong coupling between texture, stress state, and twin selection.

3. First identification of high-energy {13‾41} twins: Under biaxial loading, the high hydrostatic stress lowers the nucleation barrier, enabling the otherwise unfavorable {13‾41} twin (γ = 0.52 J/m²) to be activated and to participate in local strain accommodation.

4. New mechanism of hierarchical twinning and interfacial relaxation: The {13‾41} twin boundary preferentially induces the nucleation of secondary {10‾12} twins and, through stress-driven interface evolution, causes the secondary twin/matrix interface to gradually approach the stable {10‾11} twin boundary. This multilevel twinning path provides a new understanding of strain accommodation in HCP metals under complex stress states.

This work not only provides high-quality experimental data and a physical mechanism for the mechanical behavior of titanium alloys under biaxial loading, but also demonstrates that under complex stress states materials may activate deformation modes traditionally regarded as “unlikely.” These findings have important theoretical significance and engineering value for constitutive modeling, life assessment, and service reliability design of advanced structural materials.