A GENERAL DEFORMATION MATRIX FOR 3-DIMENSIONS

A deformation that is obtained by any simultaneous combination of two steady-stare progressive deformations: simple shearing and a coaxial p rogressive deformation, involving or not a volume change, can be expre ssed by a single transformation, or deformation matrix. In the general situation of simple shearing in a direction non-orthogonal with the p rincipal strains of the coaxial progressive deformation, this deformat ion matrix is a function of the strain components and the orientation of shearing. In this example, two coordinate systems are defined: one for the coaxial progressive deformation (x(i) system), where the princ ipal and intermediate strains are two horizontal coordinate axes, and another for the simple shear (x(i)' system), with any orientation in s pace. For steady-state progressive deformations, from the direction co sines matrix that defines the orientation of shear strains in the x(i) coordinate system, an asymmetric finite-deformation matrix is derived . From this deformation matrix, the orientation and ellipticity of the strain ellipse, or the strain ellipsoid for three-dimensional deforma tions, carl be determined. This deformation matrix also can be describ ed as a combination of a rigid-body rotation and a stretching represen ted by a general coaxial progressive deformation. The kinematic vortic ity number (W-k) is derived for the general deformation matrix to char acterize the non-coaxiality of the three-dimensional deformation. An a pplication of the deformation matrix concept is given as an example, a nalyzing the changes in orientation and stretching that variously-orie nted passive linear markers undergo after a general two-dimensional de formation. The influence of the kinematic vorticity number, the simple and pure shear strains, and the obliquity between the two deformation components, on the linear marker distribution after deformation is di scussed.