Bifurcation and creep effects in a viscoelastic non-local damageable continuum

The conditions for localization in a material described by a non-local damage-based constitutive relation coupled with a Kelvin type creep relation are derived in a closed form. The inception of a localized mode is considered as a bifurcation into a harmonic mode. The criterion of bifurcation is reduced to the classical form of singularity of a pseudo acoustic tensor; this tensor involves the elasto-damage strain and the total one at the inception of localization and the wavelength of the bifurcation mode through the Fourier transform of the weight function used in the definition of non-local damage. A geometrical approach was adopted to analyze localization for loading paths such that the elastic strain tensor is a fraction of the total strain tensor. Such loading paths include the general triaxial ones for which changes in the loading state occur only under time-independent processes (negligible creep strain during these changes of the stress state) and the uniaxial loading. The proposed coupled model preserves the properties of localization limiters; the minimum wavelength of the localization modes cannot be zero. The critical wavelength which is related to the width of the localization zone increases when the material parameter α (0⩽α⩽1), which is the fraction of creep strain entering into the evolution of damage, is decreasing. Under a certain condition on the growth of the loading function of damage and the initial state of deformation the critical wavelength decreases as the creep effect (creep strains) increases in accordance with experimental observations—increase of brittleness due to creep. In uniaxial tension, and for a specific yield function of concrete considered in this paper, this condition is fulfilled whenever the initial damage is in a region near the first occurrence of localization.