Homogenization of the 1D Peri-static/dynamic Bar with Triangular Micromodulus
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Homogenization of the 1D Peri-static/dynamic Bar with Triangular Micromodulus ¨ 1,2 Kjell Eriksson1 · Christer Stenstrom Received: 28 December 2019 / Accepted: 16 September 2020 / © The Author(s) 2020
Abstract In peridynamics, boundary effects generally appear due to nonlocality of interparticle forces; in particular, end effects are found in 1D bars. In a previous work by Eriksson and Stenstr¨om (J Peridyn Nonlocal Model 2(2):205–228, 2020), a simple method to remove end effects in certain types of 1D bars, or to homogenize such bars, was presented for bars with constant micromodulus. In this work, which is a continuation of Eriksson and Stenstr¨om (J Peridyn Nonlocal Model 2(2):205–228, 2020), the homogenizing procedure is applied to bars with a linear, or “triangular,” micromodulus. For the examples studied, common in practice, the linear elastic behavior of a homogenized bar, is identical to that of a corresponding classical continuum mechanics bar, independently of the interparticle force range and total number of material points of the bar. Keywords Peridynamics · Peristatics · Homogenization · Nonlocal methods
1 Introduction Peridynamic theory, introduced by Silling [2], is a nonlocal extension of solid mechanics, in which each material point is connected to its neighboring material points through pairwise forces acting inside a closed horizon. The formulation allows handling of long-range forces in general and unguided modeling of fractures in particular [2, 3]. Peridynamics is based on integral equations that only involve the displacement field, thereby avoiding spatial derivatives that do not exist across discontinuities in classical continuum mechanics. The pairwise forces of two material points are related to displacement through the stiffness of the bond between the points. The bond stiffness is computed under the assumption that a material point has a complete neighborhood. However, the neighborhood of a point Kjell Eriksson
[email protected] Christer Stenstr¨om [email protected] 1
Mechanics of Solid Materials, Lule˚a University of Technology, Lule˚a, Sweden
2
Luossavaara-Kiirunavaara Aktiebolag (LKAB), Lule˚a, Sweden
Journal of Peridynamics and Nonlocal Modeling
near a boundary is incomplete, and using the complete neighborhood bond stiffness in a boundary region, results in a softer material and larger strains in such regions. For a 1D peri-static/dynamic bar, this effect appears near the ends of the bar, but affects the entire bar. In general, a very large number of material points, in the order of 100–1000, is required to simulate a homogeneous bar, i.e., to achieve a displacement field that does not deviate notably from the analytical solution of a classical continuum mechanics bar. In a previous work [1], we derived the parameters necessary to remove the end effects in 1D bars with a constant micromodulus c, or to homogenize such bars. The micromodulus is the elastic stiffness of the bond between two material points. In this work, the homogenizing procedure is applied to b
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