Locking mitigation in geometrically nonlinear solid beams through isogeometric analysis and a generalized constitutive approach

This work proposes a novel geometrically nonlinear isogeometric solid-beam element, based on a one-dimensional interpolation of the displacement field. More specifically, the latter is approximated by a 12-parameter polynomial expansion in terms of displacement control variables and the cross-section parametric directions up to a bilinear term. To keep a beam-like modeling approach, the displacements are function of the beam's axis coordinate only through the control variables. This way, the model allows to flexibly capture the structure behavior with respect to e.g., bending and torsion, and a straightforward application of boundary conditions. The nature of the expansion is retained in the covariant axial and shear strains. Conversely, the transverse normal strains are kept constant and Poisson-thickness locking is treated by modifying the constitutive material matrix with exact integration within the beam's cross-section. This operation requires a proper representation of the strains that is carried out through the appropriate analytical expansion of the Jacobian without spurious terms. Additionally, by exploiting the concept of patch-wise reduced integration, namely employing cubic (C2) isogeometric shape functions in an S14 approximation space, shear and membrane locking are resolved with high efficiency, whereas the use of an isogeometric analysis framework naturally resolves curvature-thickness locking. The chosen expansion allows for a straightforward and efficient implementation given the possibility to construct the approximate strains via quadratic matrices and the performance of the proposed formulation is assessed by extensive numerical testing.