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(η)g(η) from a given torque ττ 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−∇⋅(g(|∇u|2)∇u)=2φ, x∈Ω⊂R2x∈Ω⊂R2; u(s)=0u(s)=0, s∈∂Ωs∈∂Ω. For determination of the unknown coefficient g(η)∈Gg(η)∈G, an integral of the function u(x)u(x) over the domain ΩΩ, i.e. the measured torque τ>0τ>0, is assumed to be given as an additional data. This data τ=τ(φ)τ=τ(φ), depending on the angle of twist φφ, 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↦TT:G↦T is uniquely invertible. Moreover, an explicit formula relating the plasticity function g(η)g(η) and the torque ττ is derived. The well-known formula between the elastic shear modulus G>0G>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.