Eight-node brick, PN340, represents constant stress fields exactly

We develop a new eight-node brick element, PN340, whose interpolation functions exactly satisfy the Navier equation. We begin with the Papcovitch–Neuber solutions in polynomial form, to the Navier equation and consider all the terms necessary to represent cubic displacement fields. We derive constraints on the unknown polynomial coefficients to make the eight-node brick element represent exactly every constant stress field. We provide explanations for the occurrence of kinematic modes. Based on this understanding, we develop a systematic procedure to identify the maximum independent degrees of freedom which the cubic displacement field will have while satisfying the Navier equation. Kinematic modes will never occur if the newly identified dof are used. The newly developed element PN340, based on our present procedure, predicts both stresses and displacements accurately at every point in the element in all the constant stress fields. In tests involving higher order stress fields the element is assured to converge in the limit of discretisation.