Continuous-assumed-strain (CAS) elements have been recently developed to overcome locking in \(C^1\) -continuous quadratic NURBS-based discretizations. In this work, we generalize CAS elements to overcome locking in \(C^1\) -continuous cubic NURBS-based discretizations. Both symmetric and non-symmetric versions of CAS elements are developed. Linear plane Kirchhoff rods, linear plane Timoshenko rods, and nearly-incompressible plane-strain linear elasticity are used as model problems to study how to overcome membrane locking in fourth-order structural theories, membrane and shear locking in second-order structural theories, and volumetric locking, respectively. We solve benchmark problems with known exact solutions so that we can compute the relative errors in \(L^2\) norm of all the quantities of interest. Our results show that CAS elements effectively overcome locking, namely, the numerical approximations of the unknowns become more accurate for coarse meshes and the numerical approximations of the quantities of interest that depend on the derivatives of the unknowns are free from spurious oscillations. The symmetric and non-symmetric versions of CAS elements result in essentially the same accuracy when solving fourth-order theories while the non-symmetric version of CAS elements is more accurate than its symmetric counterpart when solving second-order theories. Both the symmetric and non-symmetric versions of CAS elements are computationally efficient since these locking treatments can be applied by only modifying how the element stiffness matrices are computed. 3 Gauss-Legendre quadrature points per direction can be used to speed up the simulations without sacrificing accuracy.
