Imbricate Continuum and its Variational Derivation

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Abstract
The one‐dimensional imbricate nonlocal continuum, developed in a previous paper in order to model strain‐softening within zones of finite size, is extended here to two or three dimensions. The continuum represents a limit of a system of imbricated (overlapping) elements that have a fixed size and a diminishing cross section as the mesh is refined. The proper variational method for the imbricate continuum is developed, and the continuum equations of motion are derived from the principle of virtual work. They are of difference‐differential type and involve not only strain averaging but also stress gradient averaging for the so‐called broad‐range stresses characterizing the forces within the representative volume of heterogeneous material. The gradient averaging may be defined by a difference operator, or an averaging integral, or by least‐square fitting of a homogeneous strain field. A differential approximation with higher order displacement derivatives is also shown. The theory implies a boundary layer wh...

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