STRAIN LOCALIZATION AND BIFURCATION IN A NONLOCAL CONTINUUM

The conditions for localization and wave propagation in a strain softe ning material described by a nonlocal damage-based constitutive relati on are derived in closed form. Localization is understood as a bifurca tion into a harmonic mode. The criterion for bifurcation is reduced to the classical form of singularity of a pseudo ''acoustic tensor''; th is tensor is not a material property as it involves the wavelength of the bifurcation mode through the Fourier transform of the weight funct ion used in the definition of the nonlocal damage. A geometrical solut ion is provided to analyse localization. The conditions for the onset of bifurcation are found to coincide in the nonlocal and in the corres ponding 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 analys is in dynamics exhibits the well-known property of wave dispersion. In some instances, i.e. for large wavelength modes, wave celerities beco me imaginary, but waves with a sufficiently short wavelength are found to propagate during softening in all the situations.