Elastic strain energy of deformation twinning in tetragonal crystals

Abstract

The aspect of elastic strain for a deformation twin with a pure shear strain is studied through Eshelby’s inclusion theory. Beta-Sn, TiO₂, and TiAl of tetragonal structures are considered. As the aspect ratio of a twin approaches zero, its elastic strain energy vanishes since the stress components coupled with the twin shear strain vanish, suggesting that the twin habit plane cannot be determined solely from the shear strain energy viewpoint, for any twin mode would provide a vanishingly small strain energy for a thin twin. The application of Johnson and Cahn’s shape bifurcation theory predicts that the transition from a circular to an elliptic shape would occur when the linear dimension of a lenticular twin is only in the order of 10 nm, indicating that most twins with a substantial aspect ratio should be influenced by growth kinetics. Under an applied stress, the free energy change is found to depend strongly on the orientation and the sense of the applied stress. The extreme condition of the free energy change usually occurs when the resolved shear stress becomes extreme in the direction of the twin shear strain, thus following the relationship of Schmid’s law. The analysis of the matrix stress field immediately outside a twin plate shows a bimodal stress distribution around the lateral tip of the lenticular plate. The locations of stress concentrations depend on both the twin aspect ratio and the elastic anisotropy. As the twin aspect ratio approaches zero, however, the two exterior stress concentrations merge together at the lateral tip of the lenticular plate, yielding a maximum stress value in the order of μg, where μ andg are shear modulus and twin shear strain, respectively.