Derivation of a linear prestressed elastic rod model from three-dimensional elasticity

Abstract

We derive a one-dimensional model for the displacement of a longitudinally stressed elastic rod starting from a cylindrical three-dimensional linearized prestressed elastic body with a small diameter. The prestress is due to the prior elastic deformation of an isotropic, homogeneous, elastic body. We deduce the scaling of forces by a formal asymptotic expansion. Then we prove that the family of solutions of three-dimensional problems converges to the solution of the prestressed rod model. The energy of the limit model is the sum of the classical bending energy of the elastic rod and the energy of the prestressed string.