Closed form solutions for the strain localization problem in a softening circular bar in pure torsion with the continuum damage and the embedded discontinuity models

The boundary value problems and closed form solutions for the strain localization in a softening bar under pure torsion with the continuum damage and the embedded discontinuity formulations are developed. In this formulation, quasi-brittle materials such as concrete or mortar are considered. The boundary value problem is developed from a variational formulation and the corresponding Euler-Lagrange differential equation. These formulations include not only the continuum damage model, in which the twist angle jump and the angular strain concentration are smeared into the volume of the bar, but also the embedded discontinuity model, in which the twist angle jump and the angular strain localization are lumped into a zero thickness localization zone. The non-linear behaviour of the materials is described by a continuous constitutive model for the continuum damage model and by a discrete constitutive model for the embedded discontinuity model; both constitutive relationships include linear or exponential softening. Solutions to overcome the problem of crack bandwidth dependency when modelling strain localization with continuous models are developed, these solutions are based on a rational analysis of the kinematics of the strain model and on the fracture energy per unit area. To validate the developed closed form solutions for the strain localization problem in a softening bar under torsion, three examples of concrete cylindrical specimens under torsion are presented. The relationship between the total area under the twisting moment-twist angle curves and the fracture energy of the material guarantees the correctness of the developed closed form solutions.