Analytical stiffness matrices for tetrahedral elements

Stiffness matrices based on the non-linear Green–Lagrange strain definition may seem too complicated for analytical treatment. However, for the case of a linear displacement tetrahedron element no numerical integrations are needed. Closed form explicit analytical results are presented, making it possible to see the influence of each individual parameter and the results are directly suited for coding in a finite element program.The analytical secant and tangent element stiffness matrices are obtained by separating the dependence of the material constitutive parameters and of the stress/strain state from the dependence of the initial geometry and of the displacement assumption. The nodal positions of an element and the displacement assumption give six basic matrices of fourth order. These matrices do not depend on the material and the stress/strain state, and are thus unchanged during the necessary iterations for obtaining a solution based on the Green–Lagrange strain measure. This basic matrix approach on the directional level diminish the order of the involved matrices from 12 to 4.The presented resulting stiffness matrices are especially useful for design optimization, because analytical sensitivity analysis can then be performed. Another aspect of the paper is that linear strains imply erroneous displacement fields when rotations are involved, and we quantify these errors in relation to results based on Green–Lagrange strains.