The incremental Cauchy Problem in elastoplasticity: General solution method and semi-analytic formulae for the pressurised hollow sphere

A general solution method to the Cauchy Problem (CP) formulated for incremental elastoplasticity is designed. The method extends previous works of the authors on the solution to Cauchy Problems for linear operators and convex nonlinear elasticity in small strain to the case of generalised standard materials defined by two convex potentials. The CP is transformed into the minimisation of an error between the solutions to two well-posed elastoplastic evolution problems. A one-parameter family of errors in the constitutive equation is derived based on Legendre–Fenchel residuals. The method is illustrated by the simple example of a pressurised thick-spherical reservoir made of elastic, linear strain-hardening plastic material. The identification of inner pressure and plasticity evolution has been carried-out using semi-analytical solutions to the elastoplastic behaviours to build the error functional.