Closed form solutions for the strain localization problem in a softening bar under tension with the continuum damage and the embedded discontinuity kinematics

The boundary value problems and closed form solutions for the strain localization in a softening bar with the continuum damage and the embedded discontinuity formulations are developed. The boundary value problem is developed from a variational formulation and the corresponding Euler-Lagrange differential equation. These formulations include not only the classical continuum damage model, in which the displacement jump and the strain concentration are smeared into the bar length, but also the embedded discontinuity model, in which the displacement jump and the strain concentration are lumped into a zero thickness localization zone. For the classical continuum damage model, the non-linear behaviour of the materials is described by a continuum constitutive model and for the embedded discontinuity model by a discrete constitutive model; both constitutive relationships may consider linear or exponential softening. Solutions to overcome the problem of crack bandwidth dependency when modelling strain localization with continuum models are developed. These solutions are based on a rational analysis of the kinematics of the continuum damage 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, two examples of cylindrical specimens of concrete under tension are presented. The relationship between the total area under the load-displacement curves and the fracture energy of the material guarantees the correctness of the developed closed form solutions.