Derivation of the Linear Elastic String Model from Three-Dimensional Elasticity

We derive a one-dimensional model for the displacement and torsion of an elastic string starting from a cylindrical three-dimensional linearized prestressed elastic body with small diameter. The prestress is due to the prior elastic deformation of an isotropic, homogenous, 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 a limit that is the unique solution of the string model. Coefficients of the string model depend on the three-dimensional elasticity coefficients and the tension due to the predeformation.