A peridynamic formulation for nonlocal bone remodelling

Schaller, E.; Javili, Ali; Schmidt, Ingo; Papastavrou, Areti; Steinmann, Paul

doi:10.1080/10255842.2022.2039641

Cited by 6 publications

(4 citation statements)

References 27 publications

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“…The relative density evolution for different horizon sizes δ that results from a stepwise applied deformation function highlights the influence of nonlocality on the temporal change in relative density. Further numerical examples and information on the computational implementation can be found in [12].…”

Section: Discussionmentioning

confidence: 99%

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Open system peridynamics

Schaller

Javili

Steinmann

2022

Continuum Mech. Thermodyn.

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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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Section: Discussionmentioning

confidence: 99%

“…The peridynamics review of Javili et al [9] shows that the application of PD has extended significantly in recent years, even in fields unrelated to fracture mechanics, e.g. [10][11][12][13].…”

Section: Introductionmentioning

confidence: 99%

Open system peridynamics

Schaller

Javili

Steinmann

2022

Continuum Mech. Thermodyn.

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“…It is therefore widely used in fracture mechanics, 2 but has expanded to a multitude of application fields, such as multiscale modeling, [3][4][5][6][7] multiphysics, [8][9][10] and biological systems. [11][12][13][14] In the basic version of PD, that is, bond-based PD, the Poisson ratio is restricted to 𝜈 = 1∕4 in three-dimensional problems and 𝜈 = 1∕3 in two-dimensional problems. Consequently, Silling et al 15 later extended the theory by introducing the so-called "state" that allows for a more sophisticated description of the interactions.…”

Section: Introductionmentioning

confidence: 99%

“…Due to the replacement of spatial derivatives with integral operators, PD is inherently capable of modeling discontinuities, such as cracks, within a continuum framework. It is therefore widely used in fracture mechanics, 2 but has expanded to a multitude of application fields, such as multiscale modeling, 3‐7 multiphysics, 8‐10 and biological systems 11‐14 . In the basic version of PD, that is, bond‐based PD, the Poisson ratio is restricted to

\nu =1/4

in three‐dimensional problems and

\nu =1/3

in two‐dimensional problems.…”

Section: Introductionmentioning

confidence: 99%

A nonlocal interface approach to peridynamics exemplified by continuum‐kinematics‐inspired peridynamics

Laurien

Javili

Steinmann

2022

Numerical Meth Engineering

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In this contribution, we present a novel approach on how to treat material interfaces in nonlocal models based on peridynamics (PD) and in particular continuum-kinematics-inspired peridynamics (CPD), a novel variationally consistent peridynamic formulation. Our method relies on a nonlocal interface where the material subdomains overlap. Within this region, a kinematic coupling of the two constituents is enforced. The contact is purely geometrical as interaction forces act only between points of the same material. We provide a detailed description of the computational implementation within the framework of CPD, that is in principle applicable to all formulations of PD. A variety of numerical examples for modeling bimaterial interfaces illustrate the utility of the technique for both two-dimensional and three-dimensional problems, including examples at large deformations. Our model approaches a local model when the nonlocality parameter, the horizon size, is decreased. The proposed methodology offers a viable alternative to previous approaches in PD, which are essentially imposing mixture rules for the interfacial material parameters.

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A versatile implicit computational framework for continuum-kinematics-inspired peridynamics

Firooz,

Javili,

Steinmann

2023

Comput Mech

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Continuum-kinematics-inspired peridynamics (CPD) has been recently proposed as a novel reformulation of peridynamics that is characterized by one-, two- and three-neighbor interactions. CPD is geometrically exact and thermodynamically consistent and does not suffer from zero-energy modes, displacement oscillations or material interpenetration. In this manuscript, for the first time, we develop a computational framework furnished with automatic differentiation for the implementation of CPD. Thereby, otherwise tedious analytical differentiation is automatized by employing hyper-dual numbers (HDN). This differentiation method does not suffer from round-off errors, subtractive cancellation errors or truncation errors and is thereby highly stable with superb accuracy being insensitive to perturbation values. The computational framework provided here is compact and model-independent, thus once the framework is implemented, any other material model can be incorporated via modifying the potential energy solely. Finally, to illustrate the versatility of our proposed framework, various potential energies are considered and the corresponding material response is examined for different scenarios.

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A peridynamic formulation for nonlocal bone remodelling

Cited by 6 publications

References 27 publications

Open system peridynamics

Open system peridynamics

A nonlocal interface approach to peridynamics exemplified by continuum‐kinematics‐inspired peridynamics

A versatile implicit computational framework for continuum-kinematics-inspired peridynamics

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