Finite Gradient Models with Enriched RBF-Based Interpolation

Pedro Areias (), Rui Melicio, Fernando Carapau and José Carrilho Lopes
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Pedro Areias: IDMEC, Instituto de Engenharia Mecânica, Universidade de Lisboa, 1649-004 Lisboa, Portugal
Rui Melicio: IDMEC, Instituto de Engenharia Mecânica, Universidade de Lisboa, 1649-004 Lisboa, Portugal
Fernando Carapau: CIMA, Centro de Investigação em Matemática e Aplicações, Universidade de Évora, 7000-667 Évora, Portugal
José Carrilho Lopes: Departamento de Geociências, Universidade de Évora, 7000-671 Évora, Portugal

Mathematics, 2022, vol. 10, issue 16, 1-19

Abstract: A finite strain gradient model for the 3D analysis of materials containing spherical voids is presented. A two-scale approach is proposed: a least-squares methodology for RVE analysis with quadratic displacements and a full high-order continuum with both fourth-order and sixth-order elasticity tensors. A meshless method is adopted using radial basis function interpolation with polynomial enrichment. Both the first and second derivatives of the resulting shape functions are described in detail. Complete expressions for the deformation gradient F and its gradient ∇ F are derived and a consistent linearization is performed to ensure the Newton solution. A total of seven constitutive properties is required. The classical Lamé parameters corresponding to the pristine material are considered constant. From RVE homogenization, seven properties are obtained, two homogenized Lamé parameters plus five gradient-related properties. Two validation 3D numerical examples are presented. The first example exhibits the size effect (i.e., the stiffening of smaller specimens) and the second example shows the absence of stress singularity and hence the convergence of the discretization method.

Keywords: gradient elasticity; radial basis functions; size effect; substitution models (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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