Analytical stiffness matrices with Green-Lagrange strain measure

Separating the dependence on material and stress/strain state from the dependence on initial geometry, we obtain analytical secant and tangent stiffness matrices. For the case of a linear displacement triangle with uniform thickness and uniform constitutive behaviour closed-form results are listed, directly suited for coding in a finite element program. The nodal positions of an element and the displacement assumption give three basic matrices of order three. These matrices do not depend on material and stress/strain state, and thus are unchanged during the necessary iterations for obtaining a solution based on Green–Lagrange strain measure. The approach is especially useful in design optimization, because analytical sensitivity analysis then can be performed. The case of a three node triangular ring element for axisymmetric analysis involves small modifications and extension to four node tetrahedron elements should be straight forward