Extensional Elastica in large deformation as Gamma-limit of a discrete 1D mechanical system

The present paper deals with the rigorous homogenization of a discrete system consisting of extensible rods linked by rotational springs. Specifically, a Γ-convergence result is proven for a sequence of discrete measure functionals En, describing the energy of the discrete system, toward the continuous energy functional for the extensible Euler beam model (Elastica) in large deformation regime. A relative compactness result for the sequence En is also proven. Moreover, numerical results are shown on the deformed shape and on the total energy of the system when the number of elements of the discrete system increases. The numerical convergence of the energy to a definite value is shown in two cases. The results provide rigorous justification of a very commonly used algorithm for the discretization of the extensible Euler beam, namely Hencky-type beam model.