Nonlocal Finite Element Analysis of Strain‐Softening Solids

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Abstract
A two‐dimensional finite element formulation for imbricate non‐local strain‐softening continuum is presented and numerically demonstrated. The only difference from the usual, local finite element codes is that certain finite elements are imbricated, i.e., they regularly overlap while skipping the intermediate mesh nodes. The element imbrication is characterized by generating proper integer matrices that give the numbers of the nodes for each finite element and the numbers of the imbricate elements overlapping each local element. The number of unknown displacements remains the same as for a local finite element code, while the number of finite elements approximately doubles. Numerical results show that stable two‐dimensional strain‐softening zones of multiple‐element width can be obtained, and that the solution exhibits proper convergence as the mesh is refined. The convergence is demonstrated for the load‐displacement diagrams, for the strain profiles across the strain‐softening band, and for the total energy dissipated by cracking. It is also shown that the local formulations exhibit incorrect convergence; they converge to solutions for which the energy dissipation due to failure is zero, which is physically unacceptable. Stability prob?ems due to strain‐softening are avoided by making the loading steps so small that no two mutually nonoverlapping elements may enter the strain‐softening regime within the same load step.

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