Forming Aligned Fiber Composites into Complex Shapes

Differential geometry theory is used to show how an ideal aligned fiber composite (incompressible, inextensible fibers, constant uniform spacing between fibers) must be deformed in order to make a part of complex curvature. Three fundamental results are obtained: the first two relate the required shears to the fiber curvatures (normal and geodesic), and the third, called the Gauss-Bonnet theorem, relates the shears to the Gaussian curvature of the part and the fiber orientation. Finally, it is shown how these results can be used to simplify experimental data.