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A computational approach to obtain nonlinearly elastic constitutive relations of special Cosserat rods. (English) Zbl 1441.74034
Summary: We present a computational framework to obtain nonlinearly elastic constitutive relations of one-dimensional continua modeled as special Cosserat rods. The kinematics of the recently proposed Helical Cauchy-Born rule is used to construct a family of six-parameter (corresponding to the six strain measures of rod theory) helical rod configurations which are subjected to uniform strain field along their arc-length. This uniformity along the rod’s arc-length results in the reduction of three-dimensional equations of elasticity to just the rod’s cross-section which further allows us to obtain the induced force, moment and stiffnesses of the rod by solving a nonlinear cross-sectional warping problem for every state of strain. The formulation is general in that the rod’s material could obey arbitrary three-dimensional hyperelastic constitutive relations. A nonlinear finite element formulation is presented to solve the cross-sectional warping problem and further obtain the induced force, moment and stiffnesses numerically. Several numerical examples are presented illustrating warping due to bending, shearing and torsion in rectangular as well as circular rods and how the warping affects stiffnesses. We also obtain all the stiffnesses of helically reinforced tubes and show the variation in their stiffnesses with the tube’s fiber angle.
##### MSC:
 74B20 Nonlinear elasticity 74K10 Rods (beams, columns, shafts, arches, rings, etc.) 74S99 Numerical and other methods in solid mechanics
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