This paper describes the effect of geometric non-linearity on the deformation analysis for linear elastic grounds using the FE analysis. This FE analysis has been formulated by the finite deformation theory based on the up-dated Lagrangian scheme. A deformation characteristic is discussed in comparison the finite deformation theory and the infinitesimal deformation theory to study the effect of geometric nonlinearity. As a result, three main conclusions have been obtained. 1) At the time of small deformation, the both analyses lead the same results, 2) At the time of large deformation, because of the effect of surface expansions, the infinitesimal deformation analysis overestimates lateral displacements when compared with the finite deformation one, and 3) At the time of large deformation, because of the rotation effect, the finite deformation analysis overestimates vertical displacements on a place where the load concentrates when compared with the infinitesimal deformation one. To this end, careful choices are required to use deformation theory in treating a large deformation problem.
In continuum mechanics, a "non-linearity" is divided into two phenomena. One is "material non-linearity", the other is "geometric non-linearity". The former is popular and its characteristic has been expressed by using constitutive model like an elasto-plastic or elastvisco plastic model since the old days. On the other hand, the latter is often neglected because of its complexity. Instead of considering geometric non-linearity, the infinitesimal deformation theory is often used. This theory supposes that deformation during loading is very small, as if the body doesn't deform before and after loading. So, even if an elasto-plastic model is used, deformation obtained from analyses using this theory is geometrically linear (deformation gradient is linear). However, when the actual body deformed during loading, the geometric nonlinearity appeared. This theory is not suitable to investigate the deformation strictly in the deformation analysis, though the governing equations are simple. Under this situation, "Finite deformation theory (Finite strain theory)" is necessary.
