An inversion method for identification of elastoplastic properties of a beam from torsional experiment

A new method of determining elastoplastic properties of a beam from an experimentally given value T≔T(φ)T≔T(φ) of torque (or torsional rigidity), during the quasistatic process of torsion, given by the angle of twist φ∈[φ*,φ*]φ∈[φ*,φ*], is proposed. The mathematical model leads to the inverse problem of determining the unknown coefficient g=g(ξ2)g=g(ξ2), ξ≔|∇u|ξ≔|∇u|, of the non-linear differential equation −∇(g(|∇u|2)∇u)=2φ−∇(g(|∇u|2)∇u)=2φ, x∈Ω⊂R2x∈Ω⊂R2. The inversion method is based on the parametrization of the unknown coefficient, according to the discrete values of the gradient ξ≔|∇u|ξ≔|∇u|. Within the range of J2-deformation theory, it is shown that the considered inverse coefficient problem is an ill-conditioned one. A numerical reconstruction algorithm based on parametrization of the unknown coefficient g=g(ξ2)g=g(ξ2), with optimal selection of the experimentally given data Tm≔T(φm)Tm≔T(φm), is proposed as a new regularization scheme for the considered inverse problem. Numerical results with noise free and noisy data illustrate applicability and high accuracy of the proposed method.