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

摘要

An inverse coefficient problem related to identification of the plasticity function g(η) from a given torque τ is studied for a circular section bar. Within the deformation theory of plasticity the mathematical model of torsion leads to the nonlinear Dirichlet problem -??(g(| ?u| ~2)?u)=2φ, xεΩ?R~2; u(s)=0, sε?Ω. For determination of the unknown coefficient g(η)εG, an integral of the function u(x) over the domain Ω, i.e. the measured torque τ>0, is assumed to be given as an additional data. This data τ=τ(φ), depending on the angle of twist φ, 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→T is uniquely invertible. Moreover, an explicit formula relating the plasticity function g(η) and the torque τ 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.