Two-dimensional finite deformations evaluated from pre- and post-deformation markers: Application to balanced cross sections

A method is proposed for determining the 2D deformation gradient tensor that represents the deformations of pre- and post-deformation markers with arbitrary shapes. The deformation is not necessarily coaxial. The tensor is evaluated in a least-square sense. Therefore, the method can deal with heterogeneous deformations, and calculate their average tensor. The inverse method has the residuals that can be directly converted to the logarithmic strain needed to transform the calculated post-deformation shapes to observed ones. In addition, we propose the measure of heterogeneity for finite deformations. The method was applied to artificial and natural data from balanced cross sections.