A Compact Formula for the Derivative of a 3-D Rotation in Exponential Coordinates

We present a compact formula for the derivative of a 3-D rotation matrix with respect to its exponential coordinates. A geometric interpretation of the resulting expression is provided, as well as its agreement with other less-compact but better-known formulas. To the best of our knowledge, this simpler formula does not appear anywhere in the literature. We hope by providing this more compact expression to alleviate the common pressure to reluctantly resort to alternative representations in various computational applications simply as a means to avoid the complexity of differential analysis in exponential coordinates.

Automatic quantification of matrix cracking and fiber rotation by X-ray computed tomography in shear-deformed carbon fiber-reinforced laminates

The deformation and damage mechanisms of carbon fiber-reinforced epoxy laminates deformed in shear were studied by means of X-ray computed tomography. In particular, the evolution of matrix cracking, interply delamination and fiber rotation was ascertained as a function of the applied strain. In order to provide quantitative information, an algorithm was developed to automatically determine the crack density and the fiber orientation from the tomograms. The investigation provided new insights about the complex interaction between the different damage mechanisms (i.e. matrix cracking and interply delamination) as a function of the applied strain, ply thickness and ply location within the laminate as well as quantitative data about the evolution of matrix cracking and fiber rotation during deformation