Peridynamics Presentation

TheoryNumerical method

ImplementationSimulations

Future developments

Politecnico diTorino

Prof. Marco Di SciuvaProf. Paolo Maggiore

University ofCalifornia,Berkeley

Prof. David Steigmann

Stability and applicationsof the peridynamic method

Candidate Matteo Polleschi

Date July 21, 2010

Matteo Polleschi Peridynamics: stability and applications



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Aim of the thesis

Peridynamic method overview

Numerical method stabilization

Qualitative verification




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IntroductionEquation of motionHorizonPPF

Theory (1)

What is peridynamics?

New formulation of continuum mechanics by StewartSilling (Sandia Labs), first published in 2000

Nonlocal, as particles interact at a finite distance

Based upon integral equations, avoiding spatialderivatives = able to deal with discontinuities(especially fractures)




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Theory (2)

Physical approach:close to molecular dynamics




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Theory (3)

Equation ofmotion

Generic form

(x)u(x, t) =

Rf(u u, x x)dVx + b(x, t)




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Theory (3)

Equation ofmotion

Generic form

(x)u(x, t) =


density




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Theory (3)

Equation ofmotion

Generic form

(x)u(x, t) =


acceleration




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Theory (3)

Equation ofmotion

Generic form

(x)u(x, t) =


pairwise force function




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Theory (3)

Equation ofmotion

Generic form

(x)u(x, t) =



u u relative displacement




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Theory (3)

Equation ofmotion

Generic form

(x)u(x, t) =



u u relative displacementx x relative initial position




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Theory (3)

Equation ofmotion

Generic form

(x)u(x, t) =


body force density field




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Theory (4)

Horizon Integral is not taken over the entire body.We define a quantity ,called horizon, such that

if x x f = 0

usually assumed = 3if < 3 unnatural crackpathsif > 3 wave dispersion,fluid-like behaviour

R

f

x

x'




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Theory (5)

Pairwise forcefunction

force/volume2 on a particle at x due to a particle at x.Completely defines the properties of a material(elasticity, plasticity, yield loads...)

stretch

force

rupture

rupture

brittle failureMatteo Polleschi Peridynamics: stability and applications



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Previous approachExplicit stabilityMixed methodAlgorithm

Numerical method (1)

Previousapproach

Dominium discretization grid of nodesNo elements required method is meshlessEq. of motion discretization

uni =p

f(unp uni , xp xi)Vp + bni




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Previousapproach

Dominium discretization grid of nodesNo elements required method is meshlessEq. of motion discretizationand linearization

uni =p

C(unp uni )(xp xi)Vp + bni




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Previousapproach

Dominium discretization grid of nodesNo elements required method is meshlessEq. of motion discretizationand linearization

uni =p

C(unp uni )(xp xi)Vp + bni

subscript i - nodesuperscript n - time step




Future developments



Stability Linearized equation von Neumann stability analysis leadsto

t

Peridynamics Presentation

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