Incompressible and locking-free finite elements from Rayleigh mode vectors: quadratic polynomial displacement fields

Under pure bending, with an arbitrary patch of plane four-node finite elements, the exact analytical algebraic expressions of deformation, strain and stress fields are numerically captured by a computer algebra program for both compressible and incompressible continua. Linear combinations of Rayleigh displacement vectors yield the Ritz test functions. These coupled fields model pure bending of an Euler-Bernoulli beam with appropriate linearly varying axial strains devoid of shear. Such Courant admissible functions allow an undeformed straight side to curve in flexure. Since these displacement vectors satisfy equilibrium conditions, they are necessarily functions of the Poisson’s ratio. Applications in bio-, micro- and nano-mechanics motivated this formulation that blurs the frontier between the finite and the boundary element methods. Exact integration yields the element stiffness matrix of a compressible convex or concave quadrilateral, or a triangular element with a side node. For the generic energy density integral, the paper furnishes an analytical expression that can be incorporated in Fortran or C ⁺⁺. In isochoric plane strain problems, the Rayleigh kinematic mode of dilatation is replaced by a constant element pressure. The equivalent nodal loadings are calculated according to the Ritz variational statement. Subsequently, without assembling the global stiffness matrix, nodal compatibility and equilibrium equations are solved in terms of Rayleigh modal participation factors.