  Nonlinear bending phenomena of thin elastic structures arise in various
modern and classical applications. Characterizing low energy states of elastic
rods has been investigated by Bernoulli in 1738 and related models are used to
determine configurations of DNA strands. The bending of a piece of paper has
been described mathematically by Kirchhoff in 1850 and extensions of his model
arise in nanotechnological applications such as the development of externally
operated microtools. A rigorous mathematical framework that identifies these
models as dimensionally reduced limits from three-dimensional hyperelasticity
has only recently been established. It provides a solid basis for developing
and analyzing numerical approximation schemes. The fourth order character of
bending problems and a pointwise isometry constraint for large deformations
require appropriate discretization techniques which are discussed in this
article. Methods developed for the approximation of harmonic maps are adapted
to discretize the isometry constraint and gradient flows are used to decrease
the bending energy. For the case of elastic rods, torsion effects and a
self-avoidance potential that guarantees injectivity of deformations are
incorporated. The devised and rigorously analyzed numerical methods are
illustrated by means of experiments related to the relaxation of elastic knots,
the formation of singularities in a M\"obius strip, and the simulation of
actuated bilayer plates.
