Deformation of a Peridynamic Bar

Silling, Stewart A.; Zimmermann, Markus; Abeyaratne, Rohan

doi:10.2172/809105

Cited by 76 publications

(141 citation statements)

References 7 publications

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“…(10) and (11) are integro-differential equations and, due to the differential term, they yield continuous displacement fields under concentrated forces. This is not the case in the peridynamic solution (Silling et al, 2003).…”

Section: A Model Of 1d Elastic Continuum With Long-range Forcesmentioning

confidence: 87%

“…(10) and (11) involve an integral term analogous to the integral term accounting for longrange forces in a peridynamic bar (Silling et al, 2003). However Eqs.…”

Section: A Model Of 1d Elastic Continuum With Long-range Forcesmentioning

confidence: 99%

“…On the other hand, approaches involving non-local weighted integrals lead to non-convex potential energy functions, while approaches involving non-local gradients may experience inherent difficulties in deriving the boundary conditions. As an alternative to modified constitutive relations, microstructural effects have been accounted for in the equilibrium equation (i) by including non-local forces in an integral form (Silling, 2000;Silling et al, 2003); (ii) by the socalled continualization, where a continuum model is built based on higher-order differential operators ensuring that the continuum behaviour approximates the behaviour of the discrete lattice (Metrikine and Askes, 2002a,b;Andrianov and Awrejcewicz, 2005).…”

Section: Introductionmentioning

confidence: 99%

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Solution strategies for 1D elastic continuum with long-range interactions: Smooth and fractional decay

Failla

Santini

Zingales

2010

Mechanics Research Communications

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Keywords:Non-local elasticity Long-range interactions Weak formulation of elastic problems Fractional calculus Fractional finite differences a b s t r a c t An elastic continuum model with long-range forces is addressed in this study within the context of approximate analytical methods. Such a model stems from a mechanically-based approach to non-local theory where long-range central forces are introduced between non-adjacent volume elements. Specifically, long-range forces depend on the relative displacement, on the volume product between interacting elements and they are proportional to a proper, material-dependent, distance-decaying function. Smooth-decay functions lead to integro-differential governing equations whereas hypersingular, fractional-decay functions lead to a fractional differential governing equation of Marchaud type. In this paper the Galerkin and the Rayleigh-Ritz method are used to build approximate solutions to the integro-differential and the fractional differential governing equations. Numerical applications show the accuracy of the proposed approximate solutions as compared to the finite difference approximation and to the fractional finite difference approximation.

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Section: A Model Of 1d Elastic Continuum With Long-range Forcesmentioning

confidence: 87%

“…(10) and (11) involve an integral term analogous to the integral term accounting for longrange forces in a peridynamic bar (Silling et al, 2003). However Eqs.…”

Section: A Model Of 1d Elastic Continuum With Long-range Forcesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Solution strategies for 1D elastic continuum with long-range interactions: Smooth and fractional decay

Failla

Santini

Zingales

2010

Mechanics Research Communications

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“…Eringen and Edelen [22], Eringen [23], Kunin [24]), as well as important differences. In particular, the peridynamic method does not require averaging of some measure of infinitesimal strain over a neighbourhood of the material particle (see [21,[25][26][27]). …”

mentioning

confidence: 99%

“…The peridynamic method offers new promise in connecting existing material models developed in the classical continuum mechanics literature with nonlocal continuum models that can approximate the dispersion of short elastic waves, for example. The direct connection between peridynamics and classical elasticity is explained in [21,25,33] and nonlinear elasticity in [26]. Work is currently under way to allow peridynamic formulations (PFs) to use existing constitutive models in plasticity, thermoelasticity, etc.…”

mentioning

confidence: 99%

Influence of van der Waals forces on increasing the strength and toughness in dynamic fracture of nanofibre networks: a peridynamic approach

Bobaru

2007

Modelling Simul. Mater. Sci. Eng.

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The peridynamic method is used here to analyse the effect of van der Waals forces on the mechanical behaviour and strength and toughness properties of three-dimensional nanofibre networks under imposed stretch deformation. The peridynamic formulation allows for a natural inclusion of long-range forces (such as van der Waals forces) by considering all interactions as 'long-range'. We use van der Waals interactions only between different fibres and do not need to model individual atoms. Fracture is introduced at the microstructural (peridynamic bond) level for the microelastic type bonds, while van der Waals bonds can reform at any time. We conduct statistical studies to determine a certain volume element for which the network of randomly oriented fibres becomes quasi-isotropic and insensitive to statistical variations. This qualitative study shows that the presence of van der Waals interactions and of heterogeneities (sacrificial bonds) in the strength of the bonds at the crosslinks between fibres can help in increasing the strength and toughness of the nanofibre network. Two main mechanisms appear to control the deformation of nanofibre networks: fibre reorientation (caused by deformation and breakage) and fibre accretion (due to van der Waals interaction). Similarities to the observed toughness of polymer adhesive in the abalone shell composition are explained.

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Wavelet‐based numerical techniques for 1D peristatic problems in infinite domain

Singh

Panja

2020

Math Meth Appl Sci

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Peristatics is an important branch of continuum mechanics dealing with nonlocal effects in solid structures. Some 1D peristatic problems with four different micromodulus functions have been studied here. A wavelet‐based collocation method has been developed to find multiscale approximate solutions of the peristatic problems that avoid cumbersome evaluation of singular type integrals present in analytical solution available in literature. The obtained results offer important insights into applications and simulations of peristatic models and seem to be useful for investigation in the field of peridynamics in one or higher dimensions.

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Deformation of a Peridynamic Bar

Cited by 76 publications

References 7 publications

Solution strategies for 1D elastic continuum with long-range interactions: Smooth and fractional decay

Solution strategies for 1D elastic continuum with long-range interactions: Smooth and fractional decay

Influence of van der Waals forces on increasing the strength and toughness in dynamic fracture of nanofibre networks: a peridynamic approach

Wavelet‐based numerical techniques for 1D peristatic problems in infinite domain

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