text: A deformation twin embryo forms in BCC metal by accumulating stacking faults, with a variant selection governed by the local stress state. Variation of the stress field close to twins inferred from HR-EBSD experimental and crystal plasticity finite element ([https://damask.mpie.de/index.html CPFE]) simulation data indicated that twins nucleate on sites with maximum strain energy density and twin resolved shear stress; thus, reducing the total elastic energy after formation. This relaxation depends on the twin thickness and is a deciding factor in the spacing between twins. Experimental and three-dimensional analysis has focussed on the (stored) strain energy density measured along a path. This highly localised stress field can provide a sufficient driving force for concurrent twin nucleation and inter/intra-granular crack nucleation.
  Deformation twin growth can be perceived as a two-step process of i) thickening that is mediated by the interaction between the residual and mobile twin partials at the coherent twin-parent interface, and ii) dislocation mobility along the twin shear direction. The twin propagates when the homogeneous shear stress reaches a critical value, and a twin-parent interface advances inside the parent grain [240]. The propagating deformation twin generates a stress field due to its confinement by the surrounding parent crystal, and deformation twins develop a 3D oblate spheroid shape (which appears in 2D sections as a bi-convex lens) with a mixed coherent and non-coherent interface (Figure b).
  Kannan et al. found, using in-situ ultra-high-speed optical imaging, that twin nucleation in single-crystal magnesium is stress-driven accompanied by instantaneous propagation at a speed of 1 km/s (initially) that prioritises volume lateral thickening over forward propagation, past a critical width where growth is then become faster along the shear direction. Barnett also indicated that growth is due to twin tip extension. Furthermore, elastic simulations of the local stress field surrounding the ellipsoidal twin tip find that the field can be described using its lens angle () and that the stress field magnitude increases with twin thickness.
  In practice, plastic accommodation occurs in the parent crystal; thus, it also depends on the material’s yield stress, the anisotropic elastic stiffness of the parent crystal lattice, and the deformation twinning shear magnitude. This can also be accompanied by long-range diffusion of elements and elemental segregation (e.g., Cr and Co in single crystal Ni-based superalloy MD2), which occurs at the twin boundary to facilitate twin growth by lowering the critical stacking fault energy. A linear variation has been observed between twin thickness, stacking fault energy and grain size, and to a lesser degree, the stress state of the twinning grain (Schmid Factor). The twin thickness saturated once a critical residual dislocations’ density reached the coherent twin-parent crystal boundary.
  Significant attention has been paid to the crystallography, morphology and macro mechanical effects of deformation twinning. Although the criterion for deformation twin growth is not entirely understood, it is a tip-controlled phenomenon linked to the interaction between the residual and mobile twin partials at the twin interface; thermodynamically, this involves the elastic energy of the strained lattice, the interface and volume free-energy of the twin, and the dissipated energy of the growth  mechanism. To fully understand the interactions between microstructure (i.e., grain size, texture), temperature and strain rate on deformation twinning, it is crucial to characterise the (high) local stress and strain field associated with twin thickening and propagation. This is especially important for materials where cleavage fracture can be initiated by twinning (e.g., iron-silicon, the ferrite phase of age-hardened duplex stainless-steel, and single-crystal magnesium) as a stress-relieving mechanism.
  Early studies of deformation twins arrested within grains of niobium and iron visualised the highly local strain concentration at the twin tip using an etch-pit procedure. More recently, high-resolution electron backscatter diffraction (HR-EBSD) has been used to investigate the strain singularity ahead of a twin tip in hexagonal close-packed (HCP) zirconium alloy. A deformation twin in commercial purity titanium was characterised similarly and then quantified using a local Schmid factor (LSF) at the twin tip, as described in equation below.
  where σ is the stress tensor, S is the Schmid tensor, P is its symmetric part, d is the shear direction and n is the shear plane normal for ith slip system. The authors concluded that conditions at the twin tip control thickening and propagation in a manner analogous to the operation of dislocation sources ahead of a crack-tip. In the analysis, a broad region of high LSF ahead of the twin tip favoured propagation, whereas a narrow region of high LSF promoted thickening. Since then, it has been argued that the LSF firmly controls the twin variant selection, as twinning has strong polarity.
  The LSF novelty – compared to other criteria to describe conditions at the twin – lies in combining a geometrical criterion with the deformation field in the parent grain to provide an approximate indication of the local twin mode (i.e., thickening or propagation). However, the LSF analysis does not take advantage of the available full-field data, relies on global information on the applied stress, and does not consider the energy balance that drives twin growth. There have been few in-situ experiments to quantify the strain field ahead of a propagating deformation twin. Such observations might validate geometrical or hybrid geometrical-energy-based criteria for growth. Nanoscale testing (i.e., transmission electron microscopy) may not represent the behaviour in bulk samples due to plasticity starvation, i.e., large surface area to volume ratio, so a suitable analysis method is needed.
  Lloyd described the stress concentration field ahead of the twin tip using a two-dimensional dislocation-based model within a single magnesium grain. Wang and Li, who considered microscopic phase-field (MPF) models of cracks, noted that the stress fields were similar for dislocations, deformation twinning and martensitic transformations, with differences only in the traction of the created surface, i.e., there is 100% traction recovery for dislocations and a traction-free surface for a crack. They highlighted that the stress field singularity regulates the advancement of the crack-tip and dislocations. This stress concentration can be characterised using a path-independent line integral, as shown by Eshelby for dislocations considering the contribution from the surface traction and ellipsoidal inclusions, and Rice for cracks and stress concentrations with traction-free surfaces. Furthermore, Venables noted that the oblate spheroid shape of the twin tip is the ideal example of an ellipsoid inclusion or a notch.