SIMULTANEOUS PURE AND SIMPLE SHEAR - THE UNIFYING DEFORMATION MATRIX

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, GAMMA, called the effective shear strain. GAMMA 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 X 3, upper triangular matrix. Here, three off-diagonal terms (GAMMA(xy), GAMMA(xy), and GAMMA(yz)) occur. GAMMA(xy) is a simple function of the horizontal coaxial strain values and thrusting in the x direction, GAMMA(yz) depends on the coaxial strain components in the y and z directions and the thrusting in the y direction, while GAMMA(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.