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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

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

ImplementationSimulations

Future developments

Aim of the thesis

Peridynamic method overview

Numerical method stabilization

Qualitative verification

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

ImplementationSimulations

Future developments

Aim of the thesis

Peridynamic method overview

Numerical method stabilization

Qualitative verification

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

ImplementationSimulations

Future developments

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)

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

ImplementationSimulations

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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)

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

ImplementationSimulations

Future developments

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)

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

ImplementationSimulations

Future developments

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)

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

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IntroductionEquation of motionHorizonPPF

Theory (2)

Physical approach:close to molecular dynamics

Matteo Polleschi Peridynamics: stability and applications

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IntroductionEquation of motionHorizonPPF

Theory (3)

Equation ofmotion

Generic form

ρ(x)u(x, t) =

∫Rf(u′ − u, x′ − x)dVx′ + b(x, t)

Matteo Polleschi Peridynamics: stability and applications

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IntroductionEquation of motionHorizonPPF

Theory (3)

Equation ofmotion

Generic form

ρ(x)u(x, t) =

∫Rf(u′ − u, x′ − x)dVx′ + b(x, t)

density

Matteo Polleschi Peridynamics: stability and applications

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IntroductionEquation of motionHorizonPPF

Theory (3)

Equation ofmotion

Generic form

ρ(x)u(x, t) =

∫Rf(u′ − u, x′ − x)dVx′ + b(x, t)

acceleration

Matteo Polleschi Peridynamics: stability and applications

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IntroductionEquation of motionHorizonPPF

Theory (3)

Equation ofmotion

Generic form

ρ(x)u(x, t) =

∫Rf(u′ − u, x′ − x)dVx′ + b(x, t)

pairwise force function

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

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IntroductionEquation of motionHorizonPPF

Theory (3)

Equation ofmotion

Generic form

ρ(x)u(x, t) =

∫Rf(u′ − u, x′ − x)dVx′ + b(x, t)

pairwise force function

u′ − u relative displacement

Matteo Polleschi Peridynamics: stability and applications

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IntroductionEquation of motionHorizonPPF

Theory (3)

Equation ofmotion

Generic form

ρ(x)u(x, t) =

∫Rf(u′ − u, x′ − x)dVx′ + b(x, t)

pairwise force function

u′ − u relative displacement

x′ − x relative initial position

Matteo Polleschi Peridynamics: stability and applications

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IntroductionEquation of motionHorizonPPF

Theory (3)

Equation ofmotion

Generic form

ρ(x)u(x, t) =

∫Rf(u′ − u, x′ − x)dVx′ + b(x, t)

body force density field

Matteo Polleschi Peridynamics: stability and applications

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IntroductionEquation of motionHorizonPPF

Theory (4)

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

if ‖x− x′‖ ≥ δ ⇒ f = 0

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

R

δ

f

x

x'

Matteo Polleschi Peridynamics: stability and applications

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IntroductionEquation of motionHorizonPPF

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 failure

Matteo 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

ρuni =∑p

f(unp − uni , xp − xi)Vp + bni

Matteo 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 discretizationand linearization

ρuni =∑p

C(unp − uni )(xp − xi)Vp + bni

Matteo 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 discretizationand linearization

ρuni =∑p

C(unp − uni )(xp − xi)Vp + bni

subscript i - nodesuperscript n - time step

Matteo Polleschi Peridynamics: stability and applications

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Previous approachExplicit stabilityMixed methodAlgorithm

Numerical method (2)

Stability Linearized equation von Neumann stability analysis leadsto

∆t <

√2ρ∑

p Vp|C(xp − xi)|

Drawbacks:

linearization is not always acceptable

subject to data entry mistakes

not optimal solution

Matteo Polleschi Peridynamics: stability and applications

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Previous approachExplicit stabilityMixed methodAlgorithm

Numerical method (2)

Stability Linearized equation von Neumann stability analysis leadsto

∆t <

√2ρ∑

p Vp|C(xp − xi)|

Drawbacks:

linearization is not always acceptable

subject to data entry mistakes

not optimal solution

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

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Previous approachExplicit stabilityMixed methodAlgorithm

Numerical method (3)

Mixed method Developed by Professor Zohdi of the University ofCalifornia, Berkeley, for thermo-chemical multifieldproblems

Explicit “external ”time step

At each step, implicit ∆t evaluation

Error based upon limit on particle movement

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

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Previous approachExplicit stabilityMixed methodAlgorithm

Numerical method (3)

Mixed method Developed by Professor Zohdi of the University ofCalifornia, Berkeley, for thermo-chemical multifieldproblems

Explicit “external ”time step

At each step, implicit ∆t evaluation

Error based upon limit on particle movement

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

ImplementationSimulations

Future developments

Previous approachExplicit stabilityMixed methodAlgorithm

Numerical method (3)

Mixed method Developed by Professor Zohdi of the University ofCalifornia, Berkeley, for thermo-chemical multifieldproblems

Explicit “external ”time step

At each step, implicit ∆t evaluation

Error based upon limit on particle movement

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

ImplementationSimulations

Future developments

Previous approachExplicit stabilityMixed methodAlgorithm

Numerical method (3)

Mixed method Developed by Professor Zohdi of the University ofCalifornia, Berkeley, for thermo-chemical multifieldproblems

Explicit “external ”time step

At each step, implicit ∆t evaluation

Error based upon limit on particle movement

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

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Previous approachExplicit stabilityMixed methodAlgorithm

Numerical method (4)

Algorithm Global fixed-point iteration:

for all the N nodes,

compute the new position as

uin+1,K ≈ ∆t2

mif(uin+1,K−1) + ∆tuni + uni

compute the new (internal cycle) interaction forces (storingthem in temporary variables)

compute the error measures

if tolerance met

increment time t = t + ∆t and start from the beginningconstruct new time step ∆t = ΦK∆t

if tolerance not met

construct new time step ∆t = ΦK∆trestart from time t

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

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Previous approachExplicit stabilityMixed methodAlgorithm

Numerical method (4)

Algorithm Global fixed-point iteration:

for all the N nodes,

compute the new position as

uin+1,K ≈ ∆t2

mif(uin+1,K−1) + ∆tuni + uni

compute the new (internal cycle) interaction forces (storingthem in temporary variables)

compute the error measures

if tolerance met

increment time t = t + ∆t and start from the beginningconstruct new time step ∆t = ΦK∆t

if tolerance not met

construct new time step ∆t = ΦK∆trestart from time t

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

ImplementationSimulations

Future developments

Previous approachExplicit stabilityMixed methodAlgorithm

Numerical method (4)

Algorithm Global fixed-point iteration:

for all the N nodes,

compute the new position as

uin+1,K ≈ ∆t2

mif(uin+1,K−1) + ∆tuni + uni

compute the new (internal cycle) interaction forces (storingthem in temporary variables)

compute the error measures

if tolerance met

increment time t = t + ∆t and start from the beginningconstruct new time step ∆t = ΦK∆t

if tolerance not met

construct new time step ∆t = ΦK∆trestart from time t

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

ImplementationSimulations

Future developments

Previous approachExplicit stabilityMixed methodAlgorithm

Numerical method (4)

Algorithm Global fixed-point iteration:

for all the N nodes,

compute the new position as

uin+1,K ≈ ∆t2

mif(uin+1,K−1) + ∆tuni + uni

compute the new (internal cycle) interaction forces (storingthem in temporary variables)

compute the error measures

if tolerance met

increment time t = t + ∆t and start from the beginningconstruct new time step ∆t = ΦK∆t

if tolerance not met

construct new time step ∆t = ΦK∆trestart from time t

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

ImplementationSimulations

Future developments

Previous approachExplicit stabilityMixed methodAlgorithm

Numerical method (4)

Algorithm Global fixed-point iteration:

for all the N nodes,

compute the new position as

uin+1,K ≈ ∆t2

mif(uin+1,K−1) + ∆tuni + uni

compute the new (internal cycle) interaction forces (storingthem in temporary variables)

compute the error measures

if tolerance met

increment time t = t + ∆t and start from the beginningconstruct new time step ∆t = ΦK∆t

if tolerance not met

construct new time step ∆t = ΦK∆trestart from time t

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

ImplementationSimulations

Future developments

Previous approachExplicit stabilityMixed methodAlgorithm

Numerical method (4)

Algorithm Global fixed-point iteration:

for all the N nodes,

compute the new position as

uin+1,K ≈ ∆t2

mif(uin+1,K−1) + ∆tuni + uni

compute the new (internal cycle) interaction forces (storingthem in temporary variables)

compute the error measures

if tolerance met

increment time t = t + ∆t and start from the beginning

construct new time step ∆t = ΦK∆t

if tolerance not met

construct new time step ∆t = ΦK∆trestart from time t

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

ImplementationSimulations

Future developments

Previous approachExplicit stabilityMixed methodAlgorithm

Numerical method (4)

Algorithm Global fixed-point iteration:

for all the N nodes,

compute the new position as

uin+1,K ≈ ∆t2

mif(uin+1,K−1) + ∆tuni + uni

compute the new (internal cycle) interaction forces (storingthem in temporary variables)

compute the error measures

if tolerance met

increment time t = t + ∆t and start from the beginningconstruct new time step ∆t = ΦK∆t

if tolerance not met

construct new time step ∆t = ΦK∆trestart from time t

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

ImplementationSimulations

Future developments

Previous approachExplicit stabilityMixed methodAlgorithm

Numerical method (4)

Algorithm Global fixed-point iteration:

for all the N nodes,

compute the new position as

uin+1,K ≈ ∆t2

mif(uin+1,K−1) + ∆tuni + uni

compute the new (internal cycle) interaction forces (storingthem in temporary variables)

compute the error measures

if tolerance met

increment time t = t + ∆t and start from the beginningconstruct new time step ∆t = ΦK∆t

if tolerance not met

construct new time step ∆t = ΦK∆trestart from time t

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

ImplementationSimulations

Future developments

Previous approachExplicit stabilityMixed methodAlgorithm

Numerical method (4)

Algorithm Global fixed-point iteration:

for all the N nodes,

compute the new position as

uin+1,K ≈ ∆t2

mif(uin+1,K−1) + ∆tuni + uni

compute the new (internal cycle) interaction forces (storingthem in temporary variables)

compute the error measures

if tolerance met

increment time t = t + ∆t and start from the beginningconstruct new time step ∆t = ΦK∆t

if tolerance not met

construct new time step ∆t = ΦK∆t

restart from time t

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

ImplementationSimulations

Future developments

Previous approachExplicit stabilityMixed methodAlgorithm

Numerical method (4)

Algorithm Global fixed-point iteration:

for all the N nodes,

compute the new position as

uin+1,K ≈ ∆t2

mif(uin+1,K−1) + ∆tuni + uni

compute the new (internal cycle) interaction forces (storingthem in temporary variables)

compute the error measures

if tolerance met

increment time t = t + ∆t and start from the beginningconstruct new time step ∆t = ΦK∆t

if tolerance not met

construct new time step ∆t = ΦK∆trestart from time t

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

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Implementation

Pre-processor Geometry and Mesh: SalomeConstraints, loads and initial velocities: Impact

Solver C++ solver built from scratchParallelization by use of OpenMP (shared memory)External libraries: Armadillo (linear algebra), VTK(visualization)

Post-processor Real-time visualization: VisItPicture production: Gmsh

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

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MembranePlate with holeImpactSpecimen traction

Simulations (1)

Membranedamped

obscillations

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

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MembranePlate with holeImpactSpecimen traction

Simulations (1)

Membranedamped

obscillations

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

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MembranePlate with holeImpactSpecimen traction

Simulations (1)

Membranedamped

obscillations

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

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MembranePlate with holeImpactSpecimen traction

Simulations (1)

Membranedamped

obscillations

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

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MembranePlate with holeImpactSpecimen traction

Simulations (1)

Membranedamped

obscillations

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

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MembranePlate with holeImpactSpecimen traction

Simulations (1)

Membranedamped

obscillations

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

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MembranePlate with holeImpactSpecimen traction

Simulations (1)

Membranedamped

obscillations

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

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MembranePlate with holeImpactSpecimen traction

Simulations (1)

Membranedamped

obscillations

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

ImplementationSimulations

Future developments

MembranePlate with holeImpactSpecimen traction

Simulations (1)

Membranedamped

obscillations

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

ImplementationSimulations

Future developments

MembranePlate with holeImpactSpecimen traction

Simulations (1)

Time stepsover execution

time

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

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MembranePlate with holeImpactSpecimen traction

Simulations (2)

Plate with holebrittle fracture

Matteo Polleschi Peridynamics: stability and applications

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MembranePlate with holeImpactSpecimen traction

Simulations (2)

Plate with holebrittle fracture

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

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MembranePlate with holeImpactSpecimen traction

Simulations (2)

Plate with holebrittle fracture

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

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MembranePlate with holeImpactSpecimen traction

Simulations (2)

Plate with holebrittle fracture

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

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MembranePlate with holeImpactSpecimen traction

Simulations (2)

Plate with holebrittle fracture

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

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MembranePlate with holeImpactSpecimen traction

Simulations (2)

Plate with holebrittle fracture

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

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MembranePlate with holeImpactSpecimen traction

Simulations (2)

Plate with holebrittle fracture

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

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MembranePlate with holeImpactSpecimen traction

Simulations (2)

Plate with holebrittle fracture

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

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MembranePlate with holeImpactSpecimen traction

Simulations (2)

Plate with holebrittle fracture

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

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MembranePlate with holeImpactSpecimen traction

Simulations (3)

Impact

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

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MembranePlate with holeImpactSpecimen traction

Simulations (3)

Impact

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

ImplementationSimulations

Future developments

MembranePlate with holeImpactSpecimen traction

Simulations (3)

Impact

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

ImplementationSimulations

Future developments

MembranePlate with holeImpactSpecimen traction

Simulations (3)

Impact

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

ImplementationSimulations

Future developments

MembranePlate with holeImpactSpecimen traction

Simulations (3)

Impact

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

ImplementationSimulations

Future developments

MembranePlate with holeImpactSpecimen traction

Simulations (3)

Impact

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

ImplementationSimulations

Future developments

MembranePlate with holeImpactSpecimen traction

Simulations (3)

Impact

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

ImplementationSimulations

Future developments

MembranePlate with holeImpactSpecimen traction

Simulations (3)

Impact

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

ImplementationSimulations

Future developments

MembranePlate with holeImpactSpecimen traction

Simulations (3)

Impact

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

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Future developments

MembranePlate with holeImpactSpecimen traction

Simulations (4)

Specimentraction

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

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Future developments

MembranePlate with holeImpactSpecimen traction

Simulations (4)

Specimentraction

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

ImplementationSimulations

Future developments

MembranePlate with holeImpactSpecimen traction

Simulations (4)

Specimentraction

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

ImplementationSimulations

Future developments

MembranePlate with holeImpactSpecimen traction

Simulations (4)

Specimentraction

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

ImplementationSimulations

Future developments

MembranePlate with holeImpactSpecimen traction

Simulations (4)

Specimentraction

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

ImplementationSimulations

Future developments

MembranePlate with holeImpactSpecimen traction

Simulations (4)

Specimentraction

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

ImplementationSimulations

Future developments

MembranePlate with holeImpactSpecimen traction

Simulations (4)

Specimentraction

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

ImplementationSimulations

Future developments

MembranePlate with holeImpactSpecimen traction

Simulations (4)

Specimentraction

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

ImplementationSimulations

Future developments

MembranePlate with holeImpactSpecimen traction

Simulations (4)

Specimentraction

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

ImplementationSimulations

Future developments

MembranePlate with holeImpactSpecimen traction

Simulations (4)

Specimentraction

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

ImplementationSimulations

Future developments

MembranePlate with holeImpactSpecimen traction

Simulations (4)

Specimentraction

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

ImplementationSimulations

Future developments

MembranePlate with holeImpactSpecimen traction

Simulations (4)

Specimentraction

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

ImplementationSimulations

Future developments

MembranePlate with holeImpactSpecimen traction

Simulations (4)

Specimentraction

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

ImplementationSimulations

Future developments

Future developments

Spatial discretization

Complete range of material behaviour

Fatigue (variable loads)

Maintenance support by simulations

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

ImplementationSimulations

Future developments

Future developments

Spatial discretization

Complete range of material behaviour

Fatigue (variable loads)

Maintenance support by simulations

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

ImplementationSimulations

Future developments

Future developments

Spatial discretization

Complete range of material behaviour

Fatigue (variable loads)

Maintenance support by simulations

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

ImplementationSimulations

Future developments

Future developments

Spatial discretization

Complete range of material behaviour

Fatigue (variable loads)

Maintenance support by simulations

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

ImplementationSimulations

Future developments

Conclusions

Peridynamic code from scratch

Stability

Results coherent with brittle fracture

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

ImplementationSimulations

Future developments

Conclusions

Peridynamic code from scratch

Stability

Results coherent with brittle fracture

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

ImplementationSimulations

Future developments

Conclusions

Peridynamic code from scratch

Stability

Results coherent with brittle fracture

Matteo Polleschi Peridynamics: stability and applications

TheoryNumerical method

ImplementationSimulations

Future developments

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Matteo Polleschi Peridynamics: stability and applications