STRAIN LOCALIZATION AND BIFURCATION IN A NONLOCAL CONTINUUM

The conditions for localization and wave propagation in a strain softening material described by a nonlocal damage-based constitutive relation are derived in closed form. Localization is understood as a bifurcation into a harmonic mode. The criterion for bifurcation is reduced to the classical form of singularity of a pseudo ''acoustic tensor''; this tensor is not a material property as it involves the wavelength of the bifurcation mode through the Fourier transform of the weight function used in the definition of the nonlocal damage. A geometrical solution is provided to analyse localization. The conditions for the onset of bifurcation are found to coincide in the nonlocal and in the corresponding local cases. In the nonlocal continuum, the wavelength of the localization mode is constrained to remain below a threshold which is proportional to the characteristic length of the continuum. The analysis in dynamics exhibits the well-known property of wave dispersion. In some instances, i.e. for large wavelength modes, wave celerities become imaginary, but waves with a sufficiently short wavelength are found to propagate during softening in all the situations.