Simultaneous pure and simple shear: the unifying deformation matrix

Abstract

Any simultaneous combination of finite simple shear and finite pure shear is a linear transformation which can be expressed as a single transformation matrix. For two dimensions, the matrix is upper triangular with an off-diagonal term, Γ, called the effective shear strain. Γ is a simple function of the pure and simple shear components. For three dimensions, a simultaneous combination of thrusting in the x direction, thrusting in the y direction, and a wrench in the x direction, in addition to 3 orthogonal components of coaxial strain, can also be represented by a 3 × 3, upper triangular matrix. Here, three off-diagonal terms ( Γ xy , Γ x, y , and Γ y, z ) occur. Γ xy is a simple function of the horizontal coaxial strain values and thrusting in the x direction, Γ yz depends on the coaxial strain components in the y and z directions and the thrusting in the y direction, while Γ xz is related to all six strain components. The matrix also allows for volume change, either homogeneously or preferentially in a single direction. A method of decomposing the deformation matrix into a series of incremental deformation matrices, where each incremental deformation records the same kinematic vorticity number as the finite deformation is shown. The orientation and magnitude of the finite-strain ellipsoid (ellipse) is easily and accurately found at any increment during the deformation.