problems Continuum-kinematics-inspired peridynamics. Mechanical Journal of the Mechanics and Physics of Solids

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ContentslistsavailableatScienceDirect

Journal of the Mechanics and Physics of Solids

journalhomepage:www.elsevier.com/locate/jmps

Continuum-kinematics-inspired peridynamics. Mechanical problems

A. Javili

^a^,^∗

, A.T. McBride

^c

, P. Steinmann

^b^,^c

a Department of Mechanical Engineering, Bilkent University, Ankara, 06800, Turkey

b Chair of Applied Mechanics, University of Erlangen-Nuremberg, Egerland Str. 5, Erlangen,91058, Germany

c Glasgow Computational Engineering Centre, School of Engineering, University of Glasgow, Glasgow, G12 8QQ, United Kingdom

a r t i c l e i n f o

Article history:

Received 15 January 2019 Revised 12 May 2019 Accepted 26 June 2019 Available online 2 July 2019 Keywords:

Peridynamics Continuum kinematics Thermodynamic consistency

a b s t r a c t

The main objective of this contribution is to develop a novel continuum-kinematics- inspiredapproachforperidynamics(PD),andtorevisitPD’sthermodynamicfoundations.

Wedistinguish betweenthreetypesofinteractions, namely,one-neighbourinteractions, two-neighbourinteractionsandthree-neighbourinteractions. Whileone-neighbourinter- actionsareequivalenttothebond-basedinteractionsoftheoriginalPDformalism,two- andthree-neighbourinteractionsarefundamentallydifferenttostate-basedinteractionsin thatthe basicelements ofcontinuum kinematicsarepreserved exactly.Inaddition, we proposethatan externallyprescribed tractiononthe boundaryofthe continuum body emergesnaturallyand need not vanish.Thisis incontrast to, butdoes not necessarily violate, standard PD. We investigate the consequencesof the angular momentum bal- anceandprovideasetofappropriateargumentsfortheinteractionsaccordingly.Further- more,weelaborateonthermodynamicrestrictionsontheinteractionenergiesandderive thermodynamically-consistentconstitutivelawsthroughaColeman–Noll-likeprocedure.

ThisisanopenaccessarticleundertheCCBYlicense.

(http://creativecommons.org/licenses/by/4.0/)

1. Introduction

Peridynamics(PD)isan alternativeapproach toformulatecontinuum mechanics(Silling,2000)the rootsofwhichcan be tracedback to the pioneeringworks ofPiola (dell’Isola etal., 2015; 2016; 2017) which prepared the foundationsfor nonlocalcontinuum mechanicsand peridynamics.PD hasexperienced proliﬁc growthasan area of research,with asig- niﬁcantnumberofcontributions inmultipledisciplines. PD isa non-localcontinuum mechanics formulation.However, it isfundamentally differentfromcommonnon-localelasticity (e.g.Eringen,2002) inthat the conceptsof stressandstrain arenotpresent.Asanon-localtheory,thebehaviourofeachmaterialpointinPDisdictatedbyitsinteractionswithother materialpointsinitsvicinity.Furthermore,incontrasttoclassicalcontinuummechanics,thegoverningequationsofPDare integro-differentialequationsappropriateforproblemsinvolvingdiscontinuitiessuchascracksandinterfaces.

Whilethe discretized format ofPD bearsa similarity todiscrete mechanics formulations such asmolecular dynamics (MD), it is still a continuum formulation and only takes advantage of basic MDconcepts such as the cutoff radius and

∗ Corresponding author.

E-mail address: ajavili@bilkent.edu.tr (A. Javili).

https://doi.org/10.1016/j.jmps.2019.06.016

( http://creativecommons.org/licenses/by/4.0/ )

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Table 1

Major applications and selected key contributions of PD.

PD application Important contributions

Quasi-static problems Dayal and Bhattacharya (2006) , Mikata (2012) , Breitenfeld et al. (2014) , Huang et al. (2015) , and Madenci and Oterkus (2016)

Coupled problems Gerstle et al. (2008) , Bobaru and Duangpanya (2010) , Oterkus et al. (2014a, 2014b, 2017)

Multiscale modeling Bobaru et al. (2009) , Shelke et al. (2011) , Rahman and Foster (2014) , Talebi et al. (2014) , Ebrahimi et al. (2015) , Tong and Li (2016) , and Xu et al. (2016)

Structural mechanics Silling and Bobaru (2005) , Diyaroglu et al. (2016) , O’Grady and Foster (2014) , Taylor and Steigmann (2015) , Chowdhury et al. (2016) , and Li et al. (2016)

Constitutive models Aguiar and Fosdick (2014) , Sun and Sundararaghavan (2014) , Tupek and Radovitzky (2014) , Silhavý (2017) , and Madenci and Oterkus (2017)

Material failure Kilic and Madenci (2009) , Foster et al. (2011) , Silling et al. (2010) , Agwai et al. (2011) , Dipasquale et al. (2014) , Chen and Bobaru (2015) , Han et al. (2016) , Emmrich and Puhst (2016) , De Meo et al. (2016) , Sun and Huang (2016) , and Diyaroglu et al. (2016)

Biomechanics Taylor et al. (2016) , Lejeune and Linder (2017a, 2017b, 2018a, 2018b)

Wave dispersion Zingales (2011) , Vogler et al. (2012) , Wildman and Gazonas (2014) , Bazant et al. (2016) , Nishawala et al. (2016) , Silling (2016) , and Butt et al. (2017)

Fig. 1. Schematic illustration and comparison between the standard PD formulation (left) and the proposed continuum-kinematics-inspired alternative (right). One-neighbour interactions in our framework are identical to bond-based interactions in the PD formulation of Silling (20 0 0) . Two and three- neighbour interactions corresponding to Eq. (4) and Eq. (5) , respectively, are alternatives to state-based interactions. The difference between the bond- based, ordinary state-based, and non-ordinary state-based PD formulations lies in the magnitude and direction of the interaction forces (green arrows) between the materials points. In our approach, the difference between the one-, two- and three-neighbour interactions lies in their kinematic descriptions.

(For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

point-wiseinteractions.ForfurtherconnectionsanddifferencesbetweenPDtheory,continuummechanicsandparticlesys- temsseethefundamentalcontributionsbyFried(2010),Murdoch(2012),Fosdick(2013),andPodio-Guidugli(2017),among others.PDinherentlyaccountsforgeometricaldiscontinuities,henceitisreadilyemployedinfracturemechanicsandrelated problems.However, theapplicationsofPD extendfarbeyondfracture anddamage.Foran extensivestudyofthebalance laws,applications,andimplementations,seeMadenciandOterkus(2014),andforabriefdescriptionofPDtogetherwitha review ofitsapplicationsandrelatedstudiesindifferentﬁeldsto date,seeJavilietal.(2018).Table1categorisesvarious PDapplicationsandtheassociatedkeycontributionsintheliterature.Itisclearthat therangeofPDapplicationsisbroad andnotlimitedtofracturemechanics.

Theoriginal PDtheoryofSilling(2000) wasrestrictedtobond-basedinteractions.Thislimiteditsapplicabilityforma- terialmodelling,includingtheinabilitytoaccountforPoisson’sratiootherthan1/4forisotropicmaterials.Thisshortcom- ing wasaddressedinvarious contributions andfinally rectifiedby Silling etal.(2007)via theintroduction ofthenotion of state andcategorisingthe interactions asbond-based, ordinary state-based andnon-ordinary state-based asschemati- cally illustratedinFig.1(left).DespitethelargeamountofresearchonPD,itsthermodynamic foundationshavenotbeen fullyinvestigated.FundamentalworksonPDarelimitedinnumberbutincludethoseofSillingandLehoucq(2010),Ostoja- Starzewskietal.(2013),andOterkusetal.(2014a).ThestartingpointofthesecontributionsisthePDtheoryandconstitutive formulationofSillingetal.(2007).Thegoalhereistoadoptacontinuum-kinematics-inspiredapproachandtherebybridge the gapbetweenclassical continuum thermodynamicsandPD.More precisely, wepropose an alternative PDformulation whose underlyingconcepts are reminiscentofclassical continuum mechanics.In particular,we firstly propose todecom- posetheinteraction potentialsintothreepartscorresponding toone-neighbour interactions,two-neighbour interactionsand

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three-neighbourinteractionswithinthehorizon,asillustratedinFig.1(right).Note,one-neighbourinteractionsareidentical to bond-basedinteractions in the PD formulationof Piola(dell’Isola etal., 2015) andSilling (2000).Secondly, we derive the balance of linear and angular momentum corresponding to our interaction potentials and identify the fundamental propertiesofthesepotentialssuchthatangularmomentumbalanceisapriorifulﬁlled.Finally,wederivethedissipationin- equalityandproposethermodynamically-consistentconstitutivelaws.Crucially,wepostulatethevirtualpowerequivalence asthekeyrequirementofourapproachandbuildourentireframeworksolelyonthisvariationalassumption.

Remark. Before proceeding,we revisit thenotions ofa “localizationprocedure” and a “point-wise equation” since inthe currentcontext they serve a broader purposethan they usually do inclassical continuum mechanics. Localization refers to the process of deriving a point-wiserelation from an integral form over a domain. The resulting point-wise relation itselfmayormaynotbe anintegralform.Applying thelocalizationprocedureonglobalformsinCCMrenderspoint-wise relationsateachXthatarenotintegralsandthusarelocal.Onthecontrary,point-wiseequationsateachXinCPDinclude integralsoverthehorizonandarehencenon-local.Itispossibletoapplyalocalizationprocedureonthesenon-localforms toderive neighbour-wise equationsthat are point-wiseforms ateach neighbouring particle’slocation X^|.Henceforth,we usetheterm“localform” exclusivelytoindicatethepoint-wisequantitiesandequationsofCCM.Theterm“non-localform”

ontheotherhandreferstopoint-wiseintegralformsassociatedwithCPD.Finally,theterm“neighbour-wiseform” refersto non-integralquantitiesandrelationsinCPDobtainedvialocalizationoftheirnon-localforms.

Themanuscriptisorganizedasfollows.Section2introduces thenotation,elaboratesonthekinematicsoftheproblem andpresentsthegeometricalaspectsoftheproposedframework.Herethenoveltyistointroducetwo-andthree-neighbour interactionsinspiredbybasicelementsofclassicalcontinuumkinematics.Firstly,asamotivation,wederivethegoverning equationsusingtheDirichletprincipleinSection3viaminimizingthetotalenergyfunctional,forthespecialcaseofaquasi- static,conservativeproblem.Next,forthegeneralcase,thermodynamicbalancelawsarediscussedinSection4.Inparticular, we detail the kinetic energy,energy and entropy balance equations. Afterwards,through a Coleman–Noll-likeprocedure basedon the dissipation inequality, we providethermodynamically-consistent constitutive laws.Section 5 concludes this work.

2. Kinematics

Consider acontinuum body that occupiesthe materialconﬁguration B0∈R³ attime t=0andthat is mappedto the spatialconﬁgurationBt∈R³ viathenonlineardeformationmapyas

x=y

(

^X,t

)

^: B0× R+→Bt ⇒ Bt=y

(

B0

)

inwhichXandxidentifythepointsinthematerialandspatialconfigurations,respectivelyillustratedinFig.2.Centralto thePDtheory,andincontrasttostandardlocalcontinuum mechanics,isthenon-localityassumptionthat anypoint Xin thematerialconfigurationcaninteractwithpointswithinitsfiniteneighbourhoodH0(^X)^.^TheneighbourhoodH0isreferred toasthehorizoninthematerialconfiguration.Themeasureofthehorizoninthematerialconfigurationisdenoted

δ

0and isgenerallytheradiusofasphericalneighbourhood atX.ThespatialhorizonHt istheimage ofthematerialhorizonH0

underthedeformationmapyanditsmeasureisdenoted

δ

^t^,^that^is

Ht=y

(

H0,t

)

^with

δ

⁰^:=^meas

(

H0

)

^and

δ

^t^:=^meas

(

Ht

)

=y

( δ

⁰

)

.

NotethatthehorizonH0coincideswiththepointXinthelimitofaninﬁnitesimalneighbourhoodandtherefore δlim0→0H0→X and lim

δ0→0Ht→x

recoveringthekinematicsofthelocalcontinuummechanicsformalism.

TobemorepreciseandtobetterdistinguishthePDformalismfromconventionalcontinuummechanics,weidentifythe points(neighbours)within thehorizonbyasuperscript.ForinstancethepointX^|∈H0(^X)^denotes^a^neighbour^of^point^X

Fig. 2. Motion of a continuum body. Illustration of classical continuum mechanics formalism (left) and the peridynamics formulation (right). The continuum body that occupies the material conﬁguration B 0 ∈ R ³ at time t = 0 is mapped to the spatial conﬁguration B t ∈ R ³ via the nonlinear deformation map y .

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inthematerialconﬁguration.Thepointx^|withinthehorizonofxisthespatialcounterpartofthepointX^|deﬁnedthrough thenonlineardeformationmapyas

x^|:=y

(

^X^|,t

)

. (1)

Forourproposedframework,weidentifytheneighboursetofpointXas

X^|,X^||,X^|||

∀

^X^|^∈^H⁰

(

^X

)

, X^||∈H0

(

^X

)

, X^|||∈H0

(

^X

)

.

These neighbours ofX denoted X^|, X^||, X^||| are mapped onto x^|, x^||, x^|||,respectively. The relative positions, i.e. the finite lineelements,inthematerialandspatialconfigurationsaredenotedas^{^•^}ând

ξ

^{^•^}^,respectively,wherethesuperscript{•} identiﬁestheneighbour,thatis

^|^:=X^|− X and

ξ

^|^:=x^|− x where

ξ

^|=

ξ (

^X^|; X

)

=y

(

^X^|

)

− y

(

^X

)

,

^||^:=X^||− X and

ξ

^||^:=x^||− x where

ξ

^||=

ξ (

^X^||; X

)

=y

(

^X^||

)

− y

(

^X

)

,

^|||^:=X^|||− X and

ξ

^|||^:=x^|||− x where

ξ

^|||=

ξ (

^X^|||; X

)

=y

(

^X^|||

)

− y

(

^X

)

.

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Inaddition,wedeﬁnetheconventionalinﬁnitesimallineelements,byalimitoperation,as dX^|:=lim

δ^o→0

^|^, ^d^x^|^:=^lim

δ^o→0

ξ

^|^, ^d^X^||^:=^lim

δ^o→0

^||^, ^d^x^||^:= ^lim

δ^o→0

ξ

^||^, ^d^X^|||^:=^lim

δ^o→0

^|||^, ^d^x^|||^:= ^lim

δ^o→0

ξ

^|||^.

Inordertoovercomethe bond-basedrestrictionsofearlyPD formulations,andinthe spiritofclassicalconstitutivemod- elling, we ﬁrst recallthe three local kinematicmeasures ofrelative deformation, namely the deformation gradient F, its cofactorKanditsdeterminantJ,where

F:=Grady and K:=CofF and J:=DetF. (3)

We nowintroducethree non-localPDkinematic measuresofrelative deformationchosen to resemblethe localmeasures (3).

(i) The ﬁrst relative deformation measure

ξ

dx^| = lim δ0→0[x^|− x]

= lim δ⁰→0

ξ

^|

= lim δ0→0

F

X·

^|+¹₂G

X:

^|

+...

≈ F· dX^|,

whereGisthesecondgradientofthedeformationmapy.InviewofourproposedPDformalism,therelativedeformation measurex^|− xisthemainingredienttodescribeone-neighbourinteractions,seeFig.3.

(ii) Similar to finite line elements, we introduce finite area elements constructed from two finite line elements. For instance,thevectorialareaelementA^|^/^||inthematerialconfigurationcorrespondstothevectorproductofthelineelements

^|ând^||âsÂ^|^/^||^:=^|×^|| ^withîtscounterpartinthespatialconfigurationdenotedasa^|^/^||:=

ξ

^|×

ξ

^||,i.e.

A^|^/^||:=

^|×

^|| ^and ^a^|^/^||:=

ξ

^|×

ξ

^|| ^where ^a^|^/^||=a

(

^X^|,X^||; X

)

. (4)

Fig. 3. Illustration of finite line elements within the horizon in the material and spatial configurations corresponding to one-neighbour interactions. The finite line elements are the relative positions between points.

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Fig. 4. Illustration of ﬁnite area elements within the horizon in the material and spatial conﬁgurations corresponding to two-neighbour interactions.

Thesecondrelative deformationmeasurea^|^/^||mimics thelinearmapfromtheinﬁnitesimal(vectorial)areaelementdA^|^/^||

inthematerialconfigurationtoitsspatialcounterpartda^|^/^||.Aninfinitesimalareaelementisconstructedfromthreepoints withinthehorizoninthelimitofinfinitesimalhorizonmeasureas

da^|^/^|| = lim

δ⁰→0a^|^/^||=lim δ⁰→0

[x^|− x]× [x^||− x]

= lim δ0→0

ξ

^|×

ξ

^||

=

F· dX^|

×

F· dX^||

=K· dA^|^/^||.

ThisisessentiallytheNanson’sformulafrequentlyusedinconventionalcontinuumkinematics.Inourproposedframework, the relative area measure [x^|− x]× [x^||− x] is the main ingredient to describe two-neighbour interactions,see Fig. 4.(iii) Inasimilarfashiontofinitelineelementsandareaelements,wedefinefinitevolumeelementsformedbythreefiniteline elements.LetV^|^/^||^/^|||denotethefinitevolumeelementinthematerialconfigurationwithitsspatialcounterpartbeing

v

^|^/^||^/^|||. ThevolumeelementsV^|^/^||^/^|||and

v

^|^/^||^/^|||areobtainedbyascalartripleproduct,alsoreferredtoasamixedproduct,oftheir edgesas

V^|^/^||^/^|||:=

^|×

^||

·

^||| ^and

v

^|^/^||^/^||:=

ξ

^|^×

ξ

^||

·

ξ

^||| ^where

v

^|^/^||^/^||=

v (

^X^|,X^||,X^|||; X

)

. (5)

Thethirdandlastdeformation measure

v

^|^/^||^/^||| mimicsthelinearmap Jfromtheinﬁnitesimalvolume elementdV^|^/^||^/^||| in thematerial conﬁguration to its spatialcounterpart d

v

^|^/^||^/^|||. However unlike Jthat must be strictly positive, the volume elements

v

^|^/^||^/^||| andV^|^/^||^/^||| canbe positiveornegativeaslongasthey areconsistent inthesense that

v

^|^/^||^/^|||/V^|^/^||^/^|||>0 musthold.Theinﬁnitesimalvolumeelementsare formedfromfourpointswithinthehorizoninthelimit ofinﬁnitesimal horizonmeasureas

d

v

^|^/^||^/^||| = lim

δ⁰→0

v

^|^/^||^/^|||=lim δ⁰→0

[x^|− x]× [x^||− x]

· [x^|||− x]

= lim δ0→0

ξ

^|^×

ξ

^||

·

ξ

^|||

=

F· dX^|

×

F· dX^||

·

F· dX^|||

=JdV^|^/^||^/^|||.

The relative volume measure [[x^|− x]× [x^||− x]· [x^|||− x]]is the main ingredient todescribe three-neighbour interactions, seeFig.5.

3. Dirichletprinciplesetting

Togain insightintothethermodynamicbalance lawsbefore investigatingthe generalcaseinSection4,we beginwith the specialcase of a quasi-staticconservative problem. Thus, in order to set the stage andto motivate the structure of thegoverningequationsfortheimportantproblemofaconservativesystemthatisequippedwithatotalpotentialenergy functional,weconsider theDirichletprinciple. Moreprecisely,we obtain thegoverningequationsby minimizing thecor- respondingtotal potential energyfunctionalvia settingits ﬁrst variation tozero.The total potential energyfunctional

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Fig. 5. Illustration of ﬁnite volume elements within the horizon in the material and spatial conﬁgurations corresponding to three-neighbour interactions.

consistsofinternalandexternalcontributions,denotedasînt ândêxt^,respectively,andisgivenby

=

^int+

^ext. (6)

The internalandexternal contributionsare detailedin Sections3.1and3.2,respectively. InSections3.3thegoverning equationsare derivedandtheir connectiontoclassical(local)Cauchy continuum mechanicsishighlighted.The discussion on the variational setting in thissection is entirelyrestricted to non-dissipative processes. As outlined by dell’Isola and Placidi (2011), however, this variational setting can be extended to more generic dissipative cases using the Hamilton–

Rayleighvariationalprinciple,aswillbeexploredinaseparatecontribution.

3.1. Internalpotentialenergy

Theinternal potential energyofthe systemînt îsâssumed^without^loss ôf^generality ^to^be ^separable,î.e.^to^be ^com- posedoftheinternalpotentialenergyduetoone-neighbourinteractionsînt₁ ,two-neighbourinteractions2int andthree- neighbourinteractions3int,thatis

^int⁼

1

int+

int2 +

3int,

wherethenumberinthesubscriptindicatesthetypeofinteraction.Thesecontributionstotheinternalpotentialenergyare nowexplored.

3.1.1. One-neighbourinteractions

Toproceed, wedeﬁne the one-neighbourinteractionenergy densityper volume squaredinthe materialconﬁguration w₁^|asafunctionoftherelativeposition

ξ

^|^between^two^points,^that^is

w1|:=w1

( ξ

^|

)

=w1

( ξ (

^X^|; X

))

≡ w1

( ξ

^|;

^|,X

)

^with ^[^w1]=N.m/m⁶

wherethesemi-colondelineatesargumentsofafunctionfromitsparametrisation.Furthermore,wedeﬁnethemorefamiliar energydensitypervolumeashalfoftheintegralofw₁ overthehorizonH0,thatis

W1:=1 2

H0

w1dV^| with [W1]=N.m/m³

whereinthefactorone-halfisintroduced toprevent doublecountingsincewevisit eachpoint twiceduetotheresulting double-integrationinthe nextstep. Consequently,theinternal potential energyduetoone-neighbourinteractions ₁înt îs definedby

1 int:=

B0

W1dV = 1 2

B0

H0

w1

( ξ

^|

)

^d^V^|^d^V ^with

1 int

=N.m

≡ 1 2

B0

w₁

( ξ

^|

)

^d^V^|^dV.

ThelaststepholdssinceatanypointXone-neighbourinteractionswithpointsoutsidethehorizonvanish.Next,thevaria- tionof1

int canbeexpressedas

δ

1 int =

B0

∂

^w1

∂ ξ

^| ^·

δξ

^|^d^V^|^d^V

=

B0

H0

∂

^w1

∂ ξ

^| ^·

δξ

^|^d^V^|^d^V ⁽⁷⁾

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inwhichthepreviouslyintroducedfactorone-halfdisappearsduetothevariationrulesonmultipleintegrals.Motivatedby thestructureofEq.(7),wedeﬁnetheforcedensitypervolumesquaredduetoone-neighbourinteractionsby

p₁^|:=

∂

^w1

∂ ξ

^| ^with

p₁^|

=N/m⁶ (8)

andthereforethevariationof1

int,using

δξ

^|=

δ

^y^|−

δ

^y^from^Eqs.⁽¹⁾^and⁽²⁾^,^reads

δ

1

int=

B0

H0

p₁^|·

δξ

^|^d^V^|^d^V

=

B0

H0

p₁^|·

δ

^y^|^d^V^|−

H0

p₁^|dV^|·

δ

^y

dV

=

B0

p₁^|·

δ

^y^|^d^V^|^d^V−

B0

H0

p₁^|dV^|·

δ

^y^d^V. (9)

Weidentifytheinternalforcedensitypervolumeinthematerialconfigurationduetoone-neighbourinteractions bînt₀₁as bînt₀1:=

H0

p1|dV^| with

b^int₀1

=N/m³. (10)

Note,we recognize the right-hand side of Eq.(10) as an internal force density since it is the virtual power conjugated quantityto

δ

^y âccording^toÊq.⁽⁹⁾^.^Finally,^the^variationôf^theînternal^potentialênergy^due^toone-neighbourinteractions

₁^int ^reads

δ

1

int=

B0

p₁^|·

δ

^y^|^d^V^|^d^V⁻

B0

b^int₀₁·

δ

^y^d^V^.

3.1.2. Two-neighbourinteractions

Next,wedeﬁnethetwo-neighbourinteractionenergydensitypervolumecubedinthematerialconﬁgurationw₂^|^/^||asa functionoftheareaelementa^|^/^|| betweenthreepoints,thatis

w2|/||=w2

(

^a^|^/^||

)

=w2

ξ (

^Xı; X

)

×

ξ (

^X^||; X

)

≡ w2

(

^a^|^/^||; A^|^/^||,X

)

^with ^[^w2]=N.m/m⁹.

Furthermore,we deﬁne the morefamiliar energydensityper volume asone third ofthe doubleintegral ofw₂ over the horizonH0,thatis

W2:=1 3

H0

w2dV^||dV^| with [W2]=N.m/m³.

Thefactorone-thirdisintroducedtopreventtriplecountingduetotheresultingtriple-integralsthatcomenext.Notethat thesequence of integrationmaybe exchanged.The internal potential energy dueto two-neighbourinteractions denoted

2int isdeﬁnedby

int2 :=

B0

W2dV = 1 3

B0

H0

w2

(

^a^|^/^||

)

^d^V^||^d^V^|^d^V ^with ^[

2int]=N.m

≡ 1 3

B0

w₂

(

^a^|^/^||

)

^d^V^||^d^V^|^d^V.

Again,thelaststepholdssinceatanypointXtwo-neighbourinteractionswithpointsoutsidethehorizonvanish.Next,the variationof2int canbewrittenas

δ

int2 =

B0

∂

^w2

∂

^a^|^/^||^·

δ

^a^|^/^||^d^V^||^d^V^|^d^V=: 1 2

B0

m^|^/^||·

δ

^a^|^/^||^d^V^||^d^V^|^dV,

inwhichthepreviously introducedfactorone-third disappearsandthefactorone-halfis introducedforconvenience.The doubleforcedensitypervolumecubedisdeﬁnedbym^|^/^|| where

m^|^/^||≡ m

ξ

^|×

ξ

^||

:=2

∂

^w2

∂

^a^|^/^|| ^with

m^|^/^||

=N/m¹⁰.

Importantly,¹misassumedtobehomogeneousofdegreeoneina^|^/^|| sothat m^||^/^|=m

ξ

^||×

ξ

^|

=m

−

ξ

^|×

ξ

^||

=−m

ξ

^|×

ξ

^||

=−m^|^/^||. (11)

1 This is not only a model assumption but also requirement to satisfy suﬃciently the balance of angular momentum, as will be shown in the discussion after Eq. (35) .

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Usingtherelation

δ

^a^|^/^||=

δξ

^|×

ξ

^||+

ξ

^|×

δξ

^||^from^Eq.⁽⁴⁾^,^the^variation^of2int reads

δ

2int= 1 2

B0

ξ

^||× m^|^/^||

·

δξ

^|+

m^|^/^||×

ξ

^|

·

δξ

^||

dV^||dV^|dV

= 1 2

B0

ξ

^||^{× m}^|^/^||

·

δξ

^|⁻

ξ

^|^{× m}^|^/^||

·

δξ

^||

dV^||dV^|dV.

Toproceed,wechangetheorderofintegrationforthesecondtermandrelabelthequantities,whichyields

δ

2int= 1 2

B0

ξ

^||× m^|^/^||

·

δξ

^|−

ξ

^||× m^||^/^|

·

δξ

^|

dV^||dV^|dV

= 1 2

B0

ξ

^||× m^|^/^||

·

δξ

^|+

ξ

^||× m^|^/^||

·

δξ

^|

dV^||dV^|dV

=

B0

ξ

^||^{× m}^|^/^||

·

δξ

^|^d^V^||^d^V^|^d^V

=

B0

H0

ξ

^||× m^|^/^||

·

δξ

^|^d^V^||^d^V^|^dV. (12)

Motivatedbythestructure ofEq.(12),wedeﬁne theforcedensitypervolumesquaredduetotwo-neighbourinteractions by

p₂^|:=

H0

ξ

^||^{× m}^|^/^||^d^V^|| ^with

p₂^|

=N/m⁶. (13)

This result should be compared with the force density per volume squared due to one-neighbour interactions (8). The variationof2int with

δξ

^ı=

δ

^y^ı−

δ

^y^reads

δ

2int=

B0

H0

p2|·

δξ

^|^d^V^|^d^V

=

B0

H0

p2|·

δ

^y^|^d^V^|−

H0

p2|dV^|·

δ

^y

dV

=

B0

p2|·

δ

^y^|^d^V^|^d^V−

B0

H0

p2|dV^|·

δ

^y^dV, (14)

wherewe identify theinternal force densityper volume inthe materialconﬁguration duetotwo-neighbour interactions b^int₀₂ as

b^int₀2:=

H0

p2|dV^| with

b^int₀2

=N/m³. (15)

Again, we recognize the right-hand side of Eq.(15) asan internal force densitysince it is thevirtual powerconjugated quantityto

δ

^yâccording^toÊq.⁽¹⁴⁾^.^Finally,^the^variationôf^theînternal^potentialênergy^due^totwo-neighbourinteractions

2intreads

δ

2int=

B0

p2|·

δ

^y^|^d^V^|^d^V⁻

B0

b^int₀2·

δ

^y^d^V.

3.1.3. Three-neighbourinteractions

Thethree-neighbourinteractionenergydensitypervolumetothefourthpowerinthematerialconﬁgurationw₃^|^/^||^/^|||is afunctionofthevolumeelement

v

^|^/^||^/^|||betweenfourpointsandreads

w3|/||/|||=w3

( v

^|^/^||^/^|||

)

=w3

ξ (

^Xı; X

)

×

ξ (

^X^||; X

)

·

ξ (

^X^|||; X

)

≡ w3

( v

^|^/^||^/^|||;V^|^/^||^/^|||,X

)

with

w3|/||/|||

=N.m/m¹².

Wedeﬁnethemorefamiliarenergydensitypervolumeasonequarterofthetripleintegralofw3overthehorizonH0by W3:=1

4

H0

w3dV^|||dV^||dV^| with [W3]=N.m/m³

(9)

withthefactorone-fourthpreventingquadruplecountingduetothefollowingquadrupleinterchangeableintegrals.Conse- quentlytheinternalpotentialenergyduetothree-neighbourinteractionsdenoted3int reads

int3 :=

B0

W3dV = 1 4

B0

H0

w3

( v

^|^/^||^/^|||

)

^d^V^|||^d^V^||^d^V^|^d^V ^with

3int

=N.m

≡ 1 4

B0

w3

( v

^|^/^||^/^|||

)

^d^V^|||^d^V^||^d^V^|^dV.

Next,thevariationof3intcanbewrittenas

δ

int3 =

B0

∂

^w3

∂v

^|^/^||^/^|||

δv

^|^/^||^/^|||^d^V^|||^d^V^||^d^V^|^d^V^=: ¹₃

B0

p^|^/^||^/^|||

δv

^|^/^||^/^|||^d^V^|||^d^V^||^d^V^|^d^V

whereinthe previously introduced factorone-fourth disappears due to the variation rules on multiple integralsand the factorone-third on thelast termis introduced forconvenience. Thetriple force densityper volume tothe fourthpower p^|^/^||^/^|||isdeﬁnedby

p^|^/^||^/^|||≡ p

ξ

^|×

ξ

^||

·

ξ

^|||

:=3

∂

^w3

∂v

^|^/^||^/^||| ^with

p^|^/^||^/^|||

=N/m¹⁴.

Wenotethatpisinvariantwithrespecttoevenpermutationsin

ξ

^|^,

ξ

^||^and

ξ

^|||^since

ξ

^|^×

ξ

^||

·

ξ

^|||⁼

ξ

^||^×

ξ

^|||

·

ξ

^|⁼

ξ

^|||^×

ξ

^|

·

ξ

^|| ^⇒

v

^|^/^||^/^|||=

v

^||^/^|||^/^|=

v

^|||^/^|^/^|| ⇒ p^|^/^||^/^|||=p^||^/^|||^/^|=p^|||^/^|^/^||. (16)

Weemphasizethatm wasassumedto behomogeneous of degreeonesuch thatthe propertym^|^/^||=−m^||^/^| holds.However, pisinvariant withrespect toeven permutationsbydeﬁnition.Using therelation

δv

^|^/^||^/^|||=[

ξ

^||×

ξ

^|||^]·

δξ

^|+[

ξ

^|||×

ξ

^|^]·

δξ

^||+ [

ξ

^|×

ξ

^||^]·

δξ

^|||^from^Eq.⁽⁵⁾^,^the^variation^of3int reads

δ

int3 = 1 3

B0

p^|^/^||^/^|||

[

ξ

^||×

ξ

^|||^]·

δξ

^|+[

ξ

^|||×

ξ

^ı]·

δξ

^||+[

ξ

^|×

ξ

^||^]·

δξ

^|||

dV^|||dV^||dV^|dV

=

B0

p^|^/^||^/^|||

[

ξ

^||×

ξ

^|||^]·

δξ

^|

dV^|||dV^||dV^|dV

=

B0

H0

p^|^/^||^/^|||

[

ξ

^||^×

ξ

^|||^]^·

δξ

^|

dV^|||dV^||dV^|dV, (17)

inwhichinthesecond stepwechangedtheorderofintegrationandrelabelledthequantities.Motivatedby thestructure ofEq.(17),wedeﬁnetheforcedensitypervolumesquaredduetothree-neighbourinteractionsas

p3|:=

H0

p^|^/^||^/^|||[

ξ

^||^×

ξ

^|||^]^d^V^|||^d^V^|| ^with

p3|

=N/m⁶.

Thisshouldbe compared withthe force densityper volume squaredduetoone-neighbourinteractions (8)andtheforce densitypervolumesquaredduetotwo-neighbourinteractions(13).Thevariationof3int with

δξ

^|=

δ

^y^|−

δ

^y^reads

δ

^int³ ⁼

B0

H0

p3|·

δξ

^|^d^V^|^d^V

=

B0

H0

p3|·

δ

^y^|^d^V^|⁻

H0

p3|dV^|·

δ

^y

dV

=

B0

p3|·

δ

^y^|^d^V^|^d^V⁻

B0

H0

p3|dV^|·

δ

^y^d^V ⁽¹⁸⁾

inwhichweidentifytheinternalforcedensitypervolumeinthematerialconﬁgurationduetothree-neighbourinteractions b₀₃^int as

b^int₀₃ :=

H0

p3|dV^| with

b^int₀₃

=N/m³. (19)

The right-hand side ofEq.(19) is againan internal force densitysince it is thevirtual power conjugated quantity to

δ

^y

accordingtoEq.(18).Finally,thevariationoftheinternalpotentialenergyduetothree-neighbourinteractions3intreads

δ

int3 =

B0

p₃^|·

δ

^y^|^d^V^|^d^V−

B0

b^int₀

3·

δ

^y^d^V.

problems Continuum-kinematics-inspired peridynamics. Mechanical Journal of the Mechanics and Physics of Solids

Journal of the Mechanics and Physics of Solids

Continuum-kinematics-inspired peridynamics. Mechanical problems

A. Javili

, A.T. McBride

, P. Steinmann

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