Mixed peridynamic formulations for compressible and incompressible finite deformations

Bode, T.; Weißenfels, Christian; Wriggers, Peter

doi:10.1007/s00466-020-01824-2

Cited by 16 publications

(12 citation statements)

References 37 publications

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“…For a brief description of PD together with a review of its applications and related studies in different fields to date, see [59]. Very recently, Bode et al [60,61] proposed a mixed PD formulation as a generalization of PD theory that offers a stable alternative suitable for finite deformations, also referred to as Peridynamic Petrov-Galerkin method. Fundamental works on PD are growing in number but are still relatively limited, see e.g.…”

Section: Introductionmentioning

confidence: 99%

The computational framework for continuum-kinematics-inspired peridynamics

et al. 2020

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Peridynamics (PD) is a non-local continuum formulation. The original version of PD was restricted to bond-based interactions. Bond-based PD is geometrically exact and its kinematics are similar to classical continuum mechanics (CCM). However, it cannot capture the Poisson effect correctly. This shortcoming was addressed via state-based PD, but the kinematics are not accurately preserved. Continuum-kinematics-inspired peridynamics (CPD) provides a geometrically exact framework whose underlying kinematics coincide with that of CCM and captures the Poisson effect correctly. In CPD, one distinguishes between one-, two- and three-neighbour interactions. One-neighbour interactions are equivalent to the bond-based interactions of the original PD formalism. However, two- and three-neighbour interactions are fundamentally different from state-based interactions as the basic elements of continuum kinematics are preserved precisely. The objective of this contribution is to elaborate on computational aspects of CPD and present detailed derivations that are essential for its implementation. Key features of the resulting computational CPD are elucidated via a series of numerical examples. These include three-dimensional problems at large deformations. The proposed strategy is robust and the quadratic rate of convergence associated with the Newton–Raphson scheme is observed.

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Section: Introductionmentioning

confidence: 99%

The computational framework for continuum-kinematics-inspired peridynamics

et al. 2020

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“…In contrast to the classical correspondence theory, the kinematic and kinetic quantities in the neighborhood are assumed to be nonconstant and evaluated at every point in the neighborhood. An extension for weakly compressible or incompressible material behavior can be found in [7].…”

Section: Introductionmentioning

confidence: 99%

Peridynamic Galerkin method: an attractive alternative to finite elements

2022

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This work presents a meshfree particle scheme designed for arbitrary deformations that possess the accuracy and properties of the Finite-Element-Method. The accuracy is maintained even with arbitrary particle distributions. Mesh-based methods mostly fail if requirements on the location of evaluation points are not satisfied. Hence, with this new scheme not only the range of loadings can be increased but also the pre-processing step can be facilitated compared to the FEM. The key to this new meshfree method lies in the fulfillment of essential requirements for spatial discretization schemes. The new approach is based on the correspondence theory of Peridynamics. Some modifications of this framework allows for a consistent and stable formulation. By applying the peridynamic differentiation concept, it is also shown that the equations of the correspondence theory can be derived from the weak form. Likewise, it is demonstrated that special moving least square shape functions possess the Kronecker-$$\delta $$ δ property. Thus, Dirichlet boundary conditions can be directly applied. The positive performance of this new meshfree method, especially in comparison to the Finite-Element-Method, is shown in the calculation of several test cases. In order to guarantee a fair comparison enhanced finite element formulations are also used. The test cases include the patch test, an eigenmode analysis as well as the investigation of loadings in the context of large deformations.

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“…• existence of numerical instabilities of various types, such as the so-called tensile instability (when using a non-Lagrangian description of the problem) [6,11,21,22], the appearance of zero-energy modes due to the rank-deficiency introduced as a result of using particle (reduced nodal) integration [10,23], pressure spurious oscillations in the vicinity of near incompressibility [1,24,25] and the possible development of long term instabilities [12], and…”

Section: Introductionmentioning

confidence: 99%

An entropy-stable Smooth Particle Hydrodynamics algorithm for large strain thermo-elasticity

Ghavamian¹,

Lee²,

Gil³

et al. 2021

Computer Methods in Applied Mechanics and Engineering

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This paper presents a novel Smooth Particle Hydrodynamics computational framework for the simulation of large strain fast solid dynamics in thermo-elasticity. The formulation is based on the Total Lagrangian description of a system of first order conservation laws written in terms of the linear momentum, the triplet of deformation measures (also known as minors of the deformation gradient tensor) and the total energy of the system, extending thus the previous work carried out by some of the authors in the context of isothermal elasticity and elasto-plasticity [1-3]. To ensure the stability (i.e. hyperbolicity) of the formulation from the continuum point of view, the internal energy density is expressed as a polyconvex combination of the triplet of deformation measures and the entropy density. Moreover, and to guarantee stability from the spatial discretisation point of view, consistently derived Riemann-based numerical dissipation is carefully introduced where local numerical entropy production is demonstrated via a novel technique in terms of the time rate of the so-called ballistic free energy of the system. For completeness, an alternative and equally competitive formulation (in the case of smooth solutions), expressed in terms of the entropy density, is also implemented and compared. A series of numerical examples is presented in order to assess the applicability and robustness of the proposed formulations, where the Smooth Particle Hydrodynamics scheme is benchmarked against an alternative in-house Finite Volume Vertex Centred implementation.

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Mixed peridynamic formulations for compressible and incompressible finite deformations

Cited by 16 publications

References 37 publications

The computational framework for continuum-kinematics-inspired peridynamics

The computational framework for continuum-kinematics-inspired peridynamics

Peridynamic Galerkin method: an attractive alternative to finite elements

An entropy-stable Smooth Particle Hydrodynamics algorithm for large strain thermo-elasticity

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