Imbricate Continuum and its Variational Derivation

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 which requires special treatment. The blunt crack band model, previously used in finite element analysis of progressive fracturing, is extended by the present theory into the range of mesh sizes much smaller than the characteristic width of the crack band front. Thus, the crack band model is made part of a convergent discretization scheme. The nonlocal continuum aspects are captured by an imbricated arrangement of finite elements of the usual type.