Successive linear approximation for boundary value problems of nonlinear elasticity in relative-descriptional formulation

A method of successive linear approximation in current configuration for solving boundary value problems of large deformation in finite elasticity is proposed. Instead of using Lagrangian or Eulerian formulation, we can also formulated the problem relative to the current configuration, and linearize the constitutive function at the present state so that it leads to a linear boundary value problem for an incremental time step. Therefore, as linearization at present state proceed in time, problem for large deformation can be formulated. The idea is similar to the Euler’s method for differential equations. As an example for the proposed method, numerical simulation of bending a rectangular block into a circular section for Mooney–Rivlin material is given for comparison with the exact solution, which is one of the well-known universal solutions in finite elasticity.