Evading the strength–ductility trade-off dilemma in steel through gradient hierarchical nanotwins

The strength–ductility trade-off has been a long-standing dilemma in materials science. This has limited the potential of many structural materials, steels in particular. Here we report a way of enhancing the strength of twinning-induced plasticity steel at no ductility trade-off. After applying torsion to cylindrical twinning-induced plasticity steel samples to generate a gradient nanotwinned structure along the radial direction, we find that the yielding strength of the material can be doubled at no reduction in ductility. It is shown that this evasion of strength–ductility trade-off is due to the formation of a gradient hierarchical nanotwinned structure during pre-torsion and subsequent tensile deformation. A series of finite element simulations based on crystal plasticity are performed to understand why the gradient twin structure can cause strengthening and ductility retention, and how sequential torsion and tension lead to the observed hierarchical nanotwinned structure through activation of different twinning systems.

(c) Stress-strain curves of samples subject to pre-torsion to different angles. While enhancement in their 0.2% yielding strength is clearly seen, the pre-torsion treatment also results in degradation in ductility due to the lack of a hierarchical nanotwin network. (d) The hardening modulus as a function of tensile strain on the pre-torsioned samples. The pre-torsion decreases the hardenability as strain increases, and causes reduction in tensile ductility.

(a) (b)
(c) (d) 10 Slip system Twinning system Note that dislocations may slip in both directions, as long as the absolute resolved shear stress is greater than the critical value; however, twinning can only occur in one direction due to its pole nature.

Supplementary Note 1 | Finite element modeling on gradient structure
We performed a series of finite element (FEM) simulations to address why the combination of gradients and twin structures would be beneficial for both strength and tensile ductility. In particular, the simulations were aimed to investigate (a) the strengthening mechanism by gradient structure, (b) the importance of gradient structure for ductility improvement; and (c) the activation of different twinning systems during torsion and subsequent tension that leads to the formation of twin hierarchy and improved ductility. The plastic deformation is described by von Mises flow (also referred as J2-flow) rule. For the cylindrical sample with a hard shell and soft core with distinct stress-strain behavior (Fig. 4a),material in the hard shell has high yield strength and starts to soften at hard f ε . In contrast, material in the core hardens until 40% strain, beyond which softening begins. In their respective softening regions, the same softening modulus s  1.75GPa h is used for the core and the shell. The representative two-dimensional polycrystalline microstructure used in the calculations is composed of uniform voronoi grains. 2 Each grain, depending on its radial location, is embedded with an array of parallel twin boundaries. Regions with higher twin densities also have higher yielding strength. In our 2D axi-symmetric samples, since the isotropic plasticity model is sufficient to illustrate the essential physics, grains in each strength regime are not differentiable. The commercial software Abaqus 6.11 standard 3 is used to simulate plasticity in 2D axi-symmetric samples.
The 2D sample contains 824 grains and is meshed with 332787 CAX3 and CAX4 axi-symmetrical elements.
To explore the localization condition in an axi-symmetric, gradient twin sample subjected to tension, we simulated the response of a cylindrical sample with fixed mechanical properties in the hard shell and soft core regions, while changing the volume fraction f of the soft core. We adopted the same structure shown in Fig. 4b. Material properties in the soft core is kept the same but the softening point of the hard core is now fixed at This assumption could be too conservative as deformation twinning by torsion in TWIP steel may not necessarily degrade the ductility. As seen from the TEM/SEM pictures (Fig.   2), after torsion the deformation twins in individual grains are very clean and parallel.
During subsequent tension, such preferential twinning systems may give their way to other twinning systems that can be further activated. 14

Supplementary Note 2 | Crystal plasticity modeling on active slip/twinning systems
We employ the classical framework of rate-independent single-crystal plasticity which considers both dislocation slip and deformation twinning in face-centred cubic metals. 7 For self-consistency, we include here some of the key ingredients in the model. In such framework, the deformation gradient F is decomposed into elastic ( e F ) and plastic ( p F ) parts as where the superscript ' T ' stands for the transpose of a tensor and ' C ' is the fourth order tensor of elastic moduli. The resolved shear stress τ on slip/twinning system ( 00 , mn) is calculated as The evolution of plastic deformation gradient, on multiple i-/-th slip/twin systems, is given where 0 i S is the Schmid tensor for the i-th slip system, and 0 α S is the Schmid tensor for the α -th twin system; γ is the incremental shear strain due to slip/twinning, which is determined by using the consistency condition in rate-independent plasticity scheme given the resolved shear stress τ and resistance S on the slip/twinning system. Twinning is considered as pseudo-slip, and the lattice in the region of the crystal that has gone 15 through twinning rotates with respect to a characteristic axis [8][9][10] The twelve slip systems in FCC crystals are  {111} 110 , and the twelve twin systems in FCC crystal are  {111} 112 . The initial resistance to slip is taken to be 121MPa and that to twin 70MPa. These numbers reflect the fact that the nucleation of leading partial dislocations for twins is easier than the generation of complete dislocations. 12 Initially a random texture was assigned to the grains. The reader is referred to Ref. 13 for more detailed information about the application of the crystal plasticity model to plastic deformation in TWIP steel. Two different boundary value problems, similar to the experimental set up, were considered. For the first case, we apply simple tension to the bar until about 40% strain. In another independent simulation, we twist the bar first, and then apply uniaxial tension. During each stage of deformation, we calculate the equivalent plastic strain rates contributed respectively by individual cryptographic slip systems and twinning systems in an element as follows 13 where i covers all the slip systems, and where  spans over all the twinning systems. In the above equations,