Coupling Peridynamic Continuum Mechanics with an Analytical Solution
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Abstract Peridynamics is a nonlocal theory of continuum mechanics expressing the dynamic equilibrium of forces by using integro‐differential equations instead of partial differential equations. Thus, the equilibrium equations are still valid in case of discontinous displacement fields. In this study, we investigate the coupling between a harmonically excited peridynamic rod with a rod based on classical continuum mechanics by using the Arlequin‐method. The peridynamic region is solved by finite elements whereas for the classical region an analytical solution can be used.Keywords:
Peridynamics
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Peridynamics
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The traditional multiscale approach couples two models operating at different scales. An alternative modelling strategy, called peridynamics originally developed by [1], is to continualize the molecular dynamic models, thus replacing inhomogeneities present on smaller length scales by an enhanced continuum description on larger length scales resulting in a nonlocal reformulation of continuum mechanics. Peridynamics is a single multiscale model valid over wide range of length scales and can be considered as an upscaling of molecular dynamics. Therefore peridynamics models should recover the same dynamics and preserve all characteristic properties of molecular dynamics, which are lost by classical continuum mechanics models. The advantage of the peridynamic models is that they can be solved more cheaply than the corresponding molecular dynamic models. Peridynamics is a generalized continuum theory employing a nonlocal model of force interaction. Each material point interacts with its neighborhood within a sphere, called the horizon that serves as an internal length scale in the model. The interaction between the material points is described by a bond force which is not an electrostatic force but can be related to the strain energy of classical continuum mechanics. The objective of this study is to investigate whether the horizon in peridynamics can model the size dependence of the Young’s modulus at the nanoscale, shown experimentally and with molecular dynamic simulation in many previous works. Peridynamic simulations of copper beams with different sizes under traction have been performed employing the molecular dynamics code LAMMPS, see [2]. The force-displacement curves have been compared directly to molecular dynamics simulation and an appropriate value for the horizon has been chosen. The results showed that the Young’s modulus changes with the size and seems to tend to a limit which is the macroscopic value of the Young’s modulus for copper. (Less)
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This paper describes an elegant statistical coarse-graining of molecular dynamics at finite temperature into peridynamics, a continuum theory. Peridynamics is an efficient alternative to molecular dynamics enabling dynamics at larger length and time scales. In direct analogy with molecular dynamics, peridynamics uses a nonlocal model of force and does not employ stress/strain relationships germane to classical continuum mechanics. In contrast with classical continuum mechanics, the peridynamic representation of a system of linear springs and masses is shown to have the same dispersion relation as the original spring-mass system.
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Statistical Mechanics
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Abstract We propose, for the first time, a thermodynamically consistent formulation for open system (continuum-kinematics-inspired) peridynamics. In contrast to closed system mechanics, in open system mechanics mass can no longer be considered a conservative property. In this contribution, we enhance the balance of mass by a (nonlocal) mass source. To elaborate a thermodynamically consistent formulation, the balances of momentum, energy and entropy need to be reconsidered as they are influenced by the additional mass source. Due to the nonlocal continuum formulation, we distinguish between local and nonlocal balance equations. We obtain the dissipation inequality via a Legendre transformation and derive the structure and constraints of the constitutive expressions based on the Coleman–Noll procedure. For the sake of demonstration, we present an example for a nonlocal mass source that can model the complex process of bone remodelling in peridynamics. In addition, we provide a numerical example to highlight the influence of nonlocality on the material density evolution.
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Peridynamics
Classification of discontinuities
Continuum hypothesis
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Displacement field
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