  We present, to the best of our knowledge, the first complete
three-dimensional numerical solutions to a broad range of boundary value
problems for a general theory of finite strain gradient elasticity. We have
chosen for our work, Toupin's theory [Arch. Rat. Mech. Anal., 11(1), 385-414,
1962]-one of the more general formulations of strain gradient elasticity. Our
framework has three crucial ingredients: The first is iso-geometric analysis
[Hughes et al., Comp. Meth. App. Mech. Engrg., 194(39-41), 4135-4195, 2005],
which we have adopted for its straightforward and robust representation of
C1-continuity. The second is a weak treatment of the higher-order Dirichlet
boundary conditions in the formulation, which control the development of strain
gradients in the solution. The third ingredient is algorithmic (automatic)
differentiation, which eliminates the need for linearization "by hand" of the
rather complicated geometric and material nonlinearities in gradient elasticity
at finite strains. We present a number of numerical solutions to demonstrate
that the framework is applicable to arbitrary boundary value problems in three
dimensions. We discuss size effects, the role of higher-order boundary
conditions, and perhaps most importantly, the relevance of the framework to
problems with elastic free energy density functions that are non-convex in
strain space.
