A new method for the modeling of geometric nonlinearities in structures

Abstract

New concepts of local displacements, local engineering stresses and strains, and virtual local rotations are used to derive geometrically exact theories of structures undergoing large rotations and displacements but small strain vibrations. The vector expressions, objectivity, and relationships of several work-conjugate pairs of stress and strain measures are discussed and limitations of their use in the formulation of geometrically nonlinear theories of structures are investigated. The Green-Lagrange strains and second Piola-Kirchhoff stresses are objective (i.e. invariant under rigid-body motions) and complete nonlinear measures, but they are not geometric measures and hence difficult to use. On the other hand, the engineering stresses and strains are easy to use because they are geometric measures and the material stiffnesses obtained from small-strain experiments can be directly used. However, the engineering stresses and strains are not objective measures when finite rotations are involved. To make the engineering measures objective, we employ a coordinate transformation to remove the rigid-body displacements from the total displacements. Then, the remaining, small local displacement field is used to derive objective local engineering strains, which are equivalent to the Jaumann-Biot-Cauchy strains. Moreover, a new interpretation and manipulation of virtual local rotations is introduced, which makes the derivation of geometrically nonlinear structural theories straightforward. A structural formulation using these concepts fully correlates the Newtonian and energy approaches. These new concepts are fully illustrated by deriving a three-dimensional nonlinear Euler-Bernoulli beam theory.