The deformation matrix for simultaneous simple shearing, pure shearing and volume change, and its application to transpression-transtension tectonics

Abstract

Simultaneous simple shearing and pure shearing, with or without additional volume change, can be combined into a single, upper triangular deformation matrix. The off-diagonal term, Γ, is named the effective shear strain, and is a simple function of the pure shearing and simple shearing components. A three-dimensional deformation matrix for the simultaneous combination of coaxial deformation, with or without additional volume change, and up to three simple shearing systems with mutually orthogonal shear planes is also presented. By using this matrix, one can easily extract the various properties of incremental as well as finite strain, and the progressive as well as finite rotation of passive markers during deformation. The case of transpression-transtension is revised, using the unified deformation matrix. The orientation of the major axis of the strain ellipsoid ( λ 1) is always horizontal if the deformation is transtensional, switches from horizontal to vertical during transpressional wrenching (1 > W k > 0.81 for constant vorticity deformations), and is always vertical for highly transpressional deformations ( W k ⩽ 0.81). For transpression, material lines initially rotate towards the horizontal shearing direction, but generally turn to rotate towards the vertical axis after a certain strain. For transtension, all material lines rotate towards a direction in the horizontal plane which is oblique to the shearing direction.