Consistent tangent stiffness for nonlocal damage models

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A new damage model based on non‐local displacements (2005, background): “Consistent tangent matrix for integral-type models Integral-type models pose a substantial difficulty: due to non-locality, there is interaction between non-adjacent nodes, and the consistent tangent matrix exhibits a larger bandwidth (with respect to the sparsity pattern of the elastic or secant ma”
Efficient and Reliable Nonlocal Damage Models (2004, background): “For nonlocal damage models, this poses a substantial difficulty: due to nonlocality, there is interaction between nonadjacent nodes, and the consistent tangent matrix exhibits a larger bandwidth (with respect to the sparsity pattern of the elastic or secant matrices) [17,8], as discussed next.”
Modeling 3D crack propagation in unreinforced concrete using PUFEM (2005, unknown): “Numerical failure simulations of this problem, which are based on a smeared representation of the crack are given in, for example, [45,43,44,47,9,20,26,25], and simulations based on a discrete representation of the failure zone are provided in, for example, [46,43,44,47,60,62,63,1,38].”
Nonlocal anisotropic damage model and related computational aspects for quasi-brittle materials (2007, background): “One has also in mind nonlocal integral computations for which it even more difficult to define a tangent operator [56] and this key point for convergency acceleration with risk of loss of robustness is left to further developpments.”