An inverse coefficient problem related to elastic-plastic torsion of a circular cross-section bar

An inverse coefficient problem related to identification of the plasticity function g(eta) from a given torque tau is studied for a circular section bar. Within the deformation theory of plasticity the mathematical model of torsion leads to the nonlinear Dirichlet problem -del . (g(vertical bar del u vertical bar(2))del u) = 2 phi, x is an element of Omega subset of R-2; u(s) = 0, s is an element of partial derivative Omega. For determination of the unknown coefficient g(eta) is an element of g, an integral of the function u(x) over the domain Omega, i.e. the measured torque tau > 0, is assumed to be given as an additional data. This data tau = tau (phi), depending on the angle of twist phi, is obtained during the quasi-static elastic-plastic torsional deformation. It is proved that for a circular section bar, the coefficient-to-torque (i.e. input-output) map T : g bar right arrow T is uniquely invertible. Moreover, an explicit formula relating the plasticity function g(eta) and the torque tau is derived. The well-known formula between the elastic shear modulus G > 0 and the torque is obtained from this explicit formula, for pure elastic torsion. The proposed approach permits one to predict some elastic-plastic torsional effects arising in the hardening bar, depending on the angle of twist. (C) 2013 Elsevier Ltd. All rights reserved.