Exotic buckling patterns in fiber-reinforced materials: Numerical simulations of Cosserat elasticity

This paper discusses the three-dimensional finite element implementation of a nonlinear constrained Cosserat model for fibrous materials that accounts for extensional, flexural, and torsional fiber stiffness. Fiber bending and twisting effects are recorded by a rotation field introduced as an additional kinematic variable, the gradient of which is included in the pointwise constitutive response. A fiber-materiality constraint, enforced implicitly through Lagrange multipliers, convects the fibers as material curves. Two independent length scales corresponding to the fiber embedding, in the present case fiber cross sectional radius and fiber spacing, are explicitly included in the model. The open-source finite element code FEniCS is utilized for the numerical implementation, with which we study compression-induced flexural buckling of an elastic rectangular cantilever. Simulations reveal that fiber flexural stiffness affects out-of-plane buckling modes, and that by modifying the fiber size, modulus, and embedding pattern, unusual deformation patterns and global force-displacement responses can be achieved. These results are expected to facilitate more effective analysis of these materials and ultimately guide their design in the spirit of the metamaterial paradigm.