A strain-difference-based nonlocal elasticity model

A two-component local/nonlocal constitutive model for (macroscopically) inhomogeneous linear elastic materials (but constant internal length) is proposed, in which the stress is the sum of the local stress and a nonlocal-type stress expressed in terms of the strain difference field, hence identically vanishing in the case of uniform strain. Attention is focused upon the particular case of piecewise homogeneous material. The proposed model is thermodynamically consistent with a suitable free energy potential. It constitutes an improved form of the Vermeer and Brinkgreve [A new effective nonlocal strain measure for softening plasticity. In: Chambon, R., Desrues, J., Vardulakis, I. (Eds.), Localization and Bifurcation theory for Soils and Rocks. Balkema, Rotterdam, 1994, pp. 89–100] model, and can also be considered derivable from the Eringen nonlocal elasticity model through a suitable enhancement technique based on the concept of redistribution of the local stress. The concept of equivalent distance is introduced to macroscopically account for the further attenuation effects produced by the inhomogeneity upon the long distance interaction forces. With the aid of a piecewise homogeneous bar in tension, a portion of which degrades progressively till failure, it is shown that–– under a suitable choice of a material constant––the solution procedure exhibits no pathological features (numerical instability, mesh sensitivity) in every degraded bar condition, including the limit idealized stress-free condition of the failed bar.