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Peridynamic Petrov-Galerkin method: a generalization of the peridynamic theory of correspondence materials. (English) Zbl 1441.74016
Summary: The Peridynamic Petrov-Galerkin (PPG) method is a meshfree approach based on the peridynamic integro-differential form of the momentum equation. The spurious oscillations in the common peridynamic correspondence formulation are investigated. They occur due to an inadmissible linearized mapping of the family deformation field. This leads to a generalized correspondence formulation, which contains the common formulation as a special case. It is based on the weak form of the peridynamic momentum equation. Test and trial function requirements are examined which ensure an exact imposition of Dirichlet and Neumann boundary conditions and Weighted Least Square (WLS) shape functions as well as Local Maximum Entropy (LME) approximants are utilized to examine the PPG Method. A consistent linearization is provided, which can also be used to speed up common implicit peridynamic correspondence codes. It is used in an implicit quasistatic framework to investigate the impact of different shape function combinations. Test cases show that low-energy modes can be prevented by the PPG Method and highlight the fast convergence and stability.

MSC:
74A45 Theories of fracture and damage
74S05 Finite element methods applied to problems in solid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
Software:
AceFEM
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