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We present, to the best of our knowledge, the first completethree-dimensional numerical solutions to a broad range of boundary valueproblems for a general theory of finite strain gradient elasticity. We havechosen 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. Ourframework 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 ofC1-continuity. The second is a weak treatment of the higher-order Dirichletboundary conditions in the formulation, which control the development of straingradients in the solution. The third ingredient is algorithmic (automatic)differentiation, which eliminates the need for linearization "by hand" of therather complicated geometric and material nonlinearities in gradient elasticityat finite strains. We present a number of numerical solutions to demonstratethat the framework is applicable to arbitrary boundary value problems in threedimensions. We discuss size effects, the role of higher-order boundaryconditions, and perhaps most importantly, the relevance of the framework toproblems with elastic free energy density functions that are non-convex instrain space.

Three-dimensional iso-geometric solutions to general boundary value problems of Toupin's gradient elasticity theory at finite strains