Contents lists available atScienceDirect
Journal
of
Mathematical
Analysis
and
Applications
www.elsevier.com/locate/jmaa
Plane
stress
problems
in
nonlocal
elasticity:
finite
element
solutions
with
a
strain-difference-based
formulation
P. Fuschi, A.A. Pisano∗, D. De Domenico
Dept.PAU,UniversityMediterraneaofReggioCalabria, viaMelissari,I-89124ReggioCalabria,Italy
a r t i c l e i n f o a b s t r a c t
Article history:
Received 23 December 2014 Available online 9 June 2015 Submitted by W.L. Wendland Keywords:
Nonlocal strain-integral elasticity Eringen-type strain-difference-based model
Nonlocal finite element method
Anenhancedcomputationalversionofthefiniteelementmethodinthecontextof nonlocalstrain-integralelasticityofEringen-typeisdiscussed.Thetheoreticalbases ofthemethodareillustratedfocusingthe attentiononnumericalandcomputational aspects as well as on the construction of the nonlocal elements matrices. Two numerical examplesof planestress nonlocal elasticityare presented to show the potentialsandthelimitsofthepromotedapproach.
© 2015ElsevierInc.All rights reserved.
1. Problemposition,motivationsandgoals ofthepresentstudy
Constitutive material models established in the field of solid mechanics for classicalcontinuous media cannot describe problems where micro- and nano-effects play a crucial role in the mechanical behavior. Typical examples, among many others, are the singular stress field predicted at a sharp crack-tip in a continuum fracture mechanics problem (see e.g. [14]), or the inability of classical continuum mechanics theory indescribingdeformationphenomenaofnanotubes orother nanoscalestructures (seee.g.[21]),or, also, wave dispersion, strainsoftening, concomitant size effects (see e.g. [8]). The simplest approaches to overcome such inherent limitations of classical theories are the so-called nonlocal continuum approaches based onanenrichment ofthe classicalmodelingbykeeping thehypothesisof continuity butintroducing an internal length material scale able to take into account the phenomena imputable to the micro- or nano-structure.Thereareseveralwaystoactinthisdirectionaswitnessedbythebroad relevantliterature. The gradientapproach, promotedby Aifantis inthe1980s inasimplifiedversion (withonly three con-stants, including the Lamé ones) ofprevious gradientformulations thatcan be tracedback to the sixties (seee.g.[3,6]andthe referencestherein),aswell astheintegralapproach,proposedbyEringen[12,13],are the most widely used. Peridynamic models, first conceived by Silling [33], or continualization procedures
* Corresponding author.
E-mailaddress:aurora.pisano@unirc.it(A.A. Pisano).
http://dx.doi.org/10.1016/j.jmaa.2015.06.005
(see e.g.[5])arealsowellestablishedandverypromisingapproaches.Theremarkableworks[32,9,8,19],for conditionsensuringtheexistenceoffundamentalsolutionsandfornonlocalmodelsofintegral-typeframedin thebroadercontextofplasticityanddamage,canbealsoreferenced.Nonlocalvariationalformulationsand unifyingthermodynamicallyconsistenttreatmentscanbefinallyfoundinthelandmarkpapersofPolizzotto
[26–29]. Thelist of relevantcontributions,without any pretense ofcompleteness, is only meant to fix the backgroundofthepresentstudywhichpromotes,from acomputationalpoint ofview,anonlocalelasticity of integraltype carriedonbytheso-callednonlocal finiteelement method (NL-FEM).
The theoretical bases of what is discussed further down have been given in [26,30,31], while the first tentatives of computational nature are those given in [24,25]. In the former three theoretical papers the NL-FEMwasconceivedtogetherwithanEringen-typenonlocalintegralelasticitymodelinwhichthestress is expressed as the sum of two contributions:one is the standardlocal stress and the other, of nonlocal nature,isgivenintermsofanaveragedstraindifferencefield.InthelattercoupleofpaperstheNL-FEMwas implementedandtestedwithreferencetosimple,buteffective,examplesalsodealingwithnonhomogeneous 2Dnonlocalproblems.
On the base of such previous studies, the main goals of the present study are: i) to eliminate some computational drawbacks of the previous NL-FEM formulation due to the excessive dimensions of the nonlocaloperators relatedto theones ofthe wholeanalyzedstructure;ii)to tackle abenchmarkproblem solvedbyotherResearcherswithdifferent,oralternative,theoreticalaswellasnumerical,approaches;iii) to point outpotentialitiesand limitsofthe promotedNL-FEMhaving also alookon possible improvements andfuturefieldofapplication.
Theplanofthepaperis thefollowing.Afterthis introductorysection,inSection2thestrain-difference basednonlocalmodelandtheNL-FEM theoreticaldevelopmentsare giveninanabridgedform.Section3
goes into the details of the numerical implementation, explaining how to build the requested nonlocal operators givingalso aflow-chart of the numericalprocedure. Section4isdevoted both to the numerical results obtained for a couple of problems and to a critical investigation on the effects of some material parameters.Concludingremarksarefinally giveninSection5whichclosesthepaper.
Notation.Acompactnotationisusedthroughout,withbold-facelettersforvectorsandtensors.The“dot” and“colon” productsbetweenvectorsand tensorsdenotesimpleand doubleindex contractionoperations, respectively.Forinstance:u· v = uivi, σ:ε = σijεij, σ· n = {σijnj},D:ε ={Dijhkεhk}.Thesymbol :=
meansequalitybydefinition.Othersymbolswillbedefinedinthetextwheretheyappearforthefirsttime.
2. Eringen-typenonlocalintegral elasticity:strain-difference-basedmodelandNL-FEMformulation
2.1. The strain-difference-based nonlocalmodel
Letusconsideranonlocallinearelasticmaterialoccupying,initsundeformedstate,athree-dimensional Euclideandomain ofvolumeV referredtoorthogonalCartesiancoordinatesx = (x1,x2,x3).Thenonlocal
featureofthematerial isexpressedbythefollowingconstitutiverelation:
σ(x) = D(x) : ε(x)− α
V
J (x, x) : [ε(x)− ε(x)] d V ∀ (x, x)∈ V. (1)
Equation (1), proposed by Polizzotto et al. [31], simply states that thestress response, σ(x),to agiven strainfield,ε(x),isthesumoftwocontributions.Thefirstone,oflocalnature,isgovernedbythestandard symmetric and positivedefinite elastic moduli tensor D(x) assumed variable inspace so that, as in the quoted paper, nonhomogeneous materials can, if necessary, be considered. The second one, of nonlocal nature,dependsonthestraindifferencefield[ε(x)− ε(x)] throughthesymmetricnonlocaltensorJ (x, x)
definednext.Amaterialparameterα alsoentersEq.(1)tocontroltheproportion ofthenonlocaladdition. α has to be calibrated by means of suitable laboratory experiments or identification procedures but, as addressedinSection4,somehintstofix itsvaluecanbe givenbyasensitivityanalysis.Itisworthnoting thatforany uniformstrainfieldthenonlocalcontributionvanishesandthestress recoversthelocalvalue. Such circumstance, in agreement with some experimental findings on thin wires in tension executed by Fleck et al.[15],is obviously dueto the(averaged)strain difference appearing inEq. (1) whichmotivates themodel’snamethathoweverhastobeconsideredjustasanenhancedversionoftheEringenmodel[11]. Somenumericalinstabilitiesorincoherenciesgivenbythelattermodel,seee.g.[23],areactuallyeliminated bymodel(1),[31].
ThenonlocaltensorJ (x, x) isdefinedas:
J (x, x) := [γ(x)D(x) + γ(x)D(x)] g(x, x)− q(x, x) ∀ (x, x)∈ V, (2) with: γ(x) := V g(x, x) d V; (3) q(x, x) := V g(x, z)g(x, z)D(z) d Vz. (4)
Byinspectionofequations(2)–(4),itisevidentthatmaterialinhomogeneities,ifany,affectalsothe nonlo-cal partofthestressthroughthespatiallyvariableelasticmodulitensorD(x).Inalltheaboveoperators, g(x,x) denotes a positive, scalar attenuation function depending, by hypothesis, on an internal length material scale,say,as wellasontheEuclidean distance,|x − x|,betweenpointsx andx inV .g(x,x) is thekernel function of the Eringen model (see again [11]) and it simply assigns a“weight” to the non-local effects induced at thefield point x byaphenomenonacting at thesource point x. Theattenuation function hasapeakat|x − x|= 0 and rapidlydecreases withincreasing distance,i.e. it vanishesbeyond the so-called influence distance, say LR, the latter being a multiple of the internal length . Moreover,
g(x,x) isbi-symmetric, i.e. g(x,x) = g(x,x); it turns into aDirac deltafor → 0, i.e. inthe limit of alocal treatment;itsatisfies thenormalizationconditionV
∞g(x,x
)dV = 1 in whichV
∞ istheinfinite
three-dimensional Euclidean space inwhich V isembedded. The integral is madewith respect to x and its valueis independent of the field point x in V∞. Typical choicesfor the attenuation function g(x,x) are, forexample,theoneknownas “errorfunction”,g(x,x)= λexp−|x − x|2/2, ortheso-called “bi-exponential”, g(x,x)= λexp (−|x − x|/). Inboth expressions λ denotesaconstantto be evaluated by enforcingtheabovementionednormalization condition.
Thechoice, oridentification,ofthematerial parameter is obviouslyacrucialpoint ofallthenonlocal continuum approaches and, in the context of the integral ones, as the one here adopted, it is strictly relatedtothechoiceoftheattenuationfunction.Thismatteractuallyconcernstheconnectionbetweenthe (macroscopic) nonlocalcontinuum theories and thereal material behavior at small (atomistic)scale. The analyticalformoftheattenuationfunctiong(x,x) isusuallyderivedfromphonondispersionrelationsand itsabilityincapturingthenonlocalphenomenaisthentestedbyatomisticsimulations(seee.g.[22,34,16]). Indeed, thefunctional form ofthe kernelfunction g(x,x), foragiven material, hasto berelated both to the spatial distributionof atoms andto the interatomicinteractions i.e. tomicrostructure’s features.The internal length as well as theinfluence distance LR, also knownas cut-off radius for kernels withouta
compact support,are devoted to interpret such physical circumstances and together define the extent of nonlocality inthe continuummodel. Thenumericaldevelopments presented inSection4willaddress this point atleastforwhatconcernsomecomputationalaspects.
2.2. NL-FEM formulation
Following the rationale given in[31], a structure made of anonlocal elastic material obeying Eq. (1), occupying the volume V and subjected to given body forces and surface tractions, say b (x) and t (x)
respectively is considered.Moreover, the unknown displacement field,say u (x), satisfies given kinematic boundaryconditions,u (x)= ¯u (x) onSu= S− St,whereS denotestheboundarysurfaceofV andStthe
portion where tractions t (x) act. Thepertinent boundary-value-problemis governed, besides the stress– strainlaw (1), bythestandardequilibriumand compatibilityequations. Assuming thefurtherhypothesis ofinfinitesimaldisplacementsandloadsactinginaquasi-staticmannertherelatednonlocaltotalpotential energy functional, whose optimality conditions are the above quoted set of governing equations, can be giventheshape:
Π [u(x)] := 1 2 V ∇u(x) : D(x) : ∇u(x) d V + +α 2 V ∇u(x) : γ2(x)D(x) :∇u(x) d V + −α 2 V V ∇u(x) : J (x, x) :∇u(x) d Vd V + − V b(x)· u(x) d V − St t(x)· u(x) d S. (5)
The variational treatment of functional (5) is given in [31] being indeed a straightforward extension to thestrain-difference-basednonlocalmodelof thegeneralvariationalprincipleconceivedin[26]. Thesetwo papersarereferredforadeepercomprehension,attentionishereafterfocused onthestrain-basedNL-FEM formulationarising,obviously,fromadiscretizedform offunctional(5).
Precisely,ifthedomainV isdiscretizedinto n= 1,2,. . . ,Nefiniteelements (FEs)ofvolumeVn,within
then-th elementthedisplacementandthestrainfieldsareexpressedintermsofnodedisplacementsvector, saydn,as:
u(x) = Nn(x) dn, ε(x) = Bn(x) dn; ∀ x ∈ Vn (6)
where Nn(x) and Bn(x) denote the matrices of the element interpolation (shape) functions and their
Cartesianderivatives,respectively.BysubstitutingEqs.(6)into(5)thediscreteformofΠ[u(x)],say Π[dn],
reads: Π [dn] := 1 2 Ne n=1 dTn ⎡ ⎣ Vn BTn(x) D(x) Bn(x)d Vn ⎤ ⎦ dn + +α 2 Ne n=1 dTn ⎡ ⎣ Vn BTn(x) γ2(x) D(x) Bn(x)d Vn ⎤ ⎦ dn + −α 2 Ne n=1 Ne m=1 dTn ⎡ ⎣ Vn Vm BTn(x)J (x, x) Bm(x) d Vmd Vn ⎤ ⎦ dm+ − Ne n=1 dTn Vn NTn(x) b(x) dVn− Ne n=1 dTn St(n) NTn(x) t(x) d Sn. (7)
Expression(7)canbenotablysimplifiedas: Π [dn] = 1 2 Ne n=1 dTnklocn dn + α 2 Ne n=1 dTnknonlocn dn + −α 2 Ne n=1 Ne m=1 dTnknonlocnm dm − Ne n=1 dTnfn, (8)
where thefollowingpositionsholdtrue:
klocn := Vn BTn(x) D (x) Bn(x) d Vn, (9a) fn := Vn NTn(x) b(x) d Vn + St(n) NTn(x) t(x) d Sn, (9b)
withklocn andfn denotingthestandard(local)elementstiffnessmatrixandequivalentnodalforcesvector, respectively.Twomorepositionsdefine thematricesknonlocn andknonlocnm appearinginEq. (8),precisely:
knonlocn := Vn BTn(x) γ2(x) D (x) Bn(x) d Vn, (10a) knonlocnm := Vn Vm BTn(x)J (x, x) Bm(x) d Vmd Vn. (10b)
Theelement matrices(10a),(10b),ofnonlocal nature,deservesomecomments.
Matrixknonlocn accountsfortheinfluenceexertedonthen-thelementbythenonlocaldiffusiveprocesses over the wholedomain and this by thepresence of γ2(x) with γ(x) given by Eq. (3). Inparticular, from
Eqs. (2)to(4)itiseasyto showthat:
V
J (x, x) d V = γ2
(x) D(x) ∀ x ∈ V. (11)
Indeed, withinaFE context and aGaussian quadrature integration rule, thefield point x isthe position vector (in the Cartesian absolute coordinates system) of the current Gauss point (GP) of the current element #n, while thesource point x (x again denoting aposition vectorin thesame absolute system) willrangeoverall theGPsoftheFEmesh.Forclaritywecanstatethatx isthegenericGPofthegeneric element #m,thelatter ranging, inprinciple, overthe wholemesh (i.e. m= 1,2,. . . ,Ne). Matrixknonlocnm ,
besidesthe(nonlocal)operatorJ (x,x) definedbyEqs.(2)–(4),accountsexplicitlyforthenonlocaleffects exertedbythem-thelement onthen-thoneandthisbythepresenceofBn(x) and Bm(x) relatedtothe
element #n and#m,respectively.knonlocnm isasetofnonlocalmatrices pertainingtoelement#n,precisely: a self-stiffness matrix, obtainedfor m= n, plusallthe cross-stiffness matricesgiven bym = 1,2,. . . ,Ne
with m= n.
Followingastandardrationale,hereomittedforsakeofbrevity(seee.g.[18]),matrices/vectors(9a),(9b)
and (10a), (10b) can be rephrased with reference to the global DOFs,say U . An expression of the total potential energyfunctional(8)intermsofglobaloperatorsandDOFs,canalsobeobtained.Thelatter,by minimizingwith respectto U ,willgive asolvinglinearequationsystemresultinginallsimilar totheone of thestandard FEM,except foranonlocal global matrix,whichproves to be symmetric, positivedefinite and formallygivenby:
K = Ne n=1 CTnklocn Cn + α Ne n=1 CTnknonlocn Cn − Ne m=1 CTnknonlocnm Cm (12)
where Cn and Cm denote theconnectivitymatrices enlargingthe element matricesto global dimensions.
Equation(12),withthesumoftheenlargedelements’matrices,expressestheassemblingprocedureyielding aglobal matrix K that reflects allthe nonlocality features of the constituent material. K is bandedbut withaband-widthlargerthaninthestandardFEMdueto theelements’cross-stiffness matrices.
Remark1.Thematerialparameterα entersexpression(12),forα = 0 theglobalstandard(local)matrixis obviously recovered.Asassertedin[31],negative valuesofα shouldbe avoidedforatwofoldreason: i)to preventthelossofpositivedefinitenessofthefreeenergypotentialgenerating theconstitutivestrain-based relation (1) andthe consequentnumerical instabilities exhibited,insuch acircumstance, bythe solution; ii)to takeintoaccount thephysicalmeaningofthematerial parameterα,i.e. aparametercontrolling the proportion (whichhastobe greateror equalto zero)ofthenonlocaladditionpostulatedbyEq.(1).
Remark2. Byinspection of thenonlocal matrices (10a), (10b), taking into account the positions (2)–(4)
and thepeculiarproperty ofthe attenuationfunction g(x,x) thatvanishes for|x − x|≥ LR, itisworth
notingthatallthecrossintegrations betweenelementsare,inpractice,vanishing whentwoelements–and sothecurrentGPs(assaidx belongingtoElem.#n andx belongingtoElem.#m)–aretoofarfromeach otherwithrespecttotheinfluencedistanceLR.Thiscircumstancereducesdramaticallythecomputational
efforts.Thecrossintegrationspertainingtoanelement,say#n,willinvolveonlyacertainnumberofother elements,say#m,with#m= 1, 2,. . . ,Me,andMe Ne,neighbors of#n.Theneighborsofelement #n,
besides theadjacentelements,arealltheelementswhose GPs(oneat least)fall withintheinfluencezone of oneof theGPs belonging to #n.The influencezone ofa GPwill be a circle or asphere, in2Dor 3D respectively,ofradiusequaltotheinfluencedistanceLRandcenteredattheGP.TheexactnumberMeof
neighborselements,inadditiontothepositionofcurrentelement#n withintheFEmesh(thenumberMe
isreduced forelementsinproximity oftheborders),will thenbe relatedto thechosenLR butalsoto the
elements’size,sayh.Hereafteritismeantthat#m (= 1,2,. . . ,Me)denotesboth acounterranging over
theset of elements neighborsof acurrentelement #n and justoneof suchneighbors.Therefore #m= 1 for example is the first element of theset and not element #1 in the FE mesh.The sensitiveness of the numerical solution to the model parameters LR and h (related to the chosen kernel g(x,x) and to the
FE mesh,respectively)or to the material parameters α and (depending onthe nonlocalfeatures of the consideredmaterial)aswellas totheirratioswillbe investigatedatSection4.
Remark3.Thenonlocaloperatorq(x,x),given byEq.(4),isreferredto two currentGPs(x∈ #n with #n= 1,2,. . . ,Neandx∈ #m with#m= 1, 2,. . . ,Me);ifΓ#nandΓ#mdenotethesetsoftheneighbors
elementsof#n and#m,respectively,itisworthnotingthattheintegralonther.h.s.ofEq.(4)hastobe computed onlyfor theGPs (z points) belongingto the intersectionΓ#n∩ Γ#m. Such shrewdness,taking
intoaccount Eqs.(2)and(10b), yieldstothefollowingnotable relation:
knonlocnm ≡
knonlocmn
T
, (13)
whichfurtherreducesthecomputationalburdens.Onlytheupper(orthelower)triangle,plusthediagonal terms,ofthecross-stiffnesselement matrixknonlocnm haveto becomputed.
Remark4.Afinalremarkconcernstheevaluationofthestressfield.Oncethe(nonlocal)solutionhasbeen computedintermsofnodaldisplacements,thedisplacementsandthestrainfieldsaregivenbyequations(6). Both fields,given ∀x ∈ V , arelocal quantities but theypossessanonlocal nature, inthesense thattheir
(local) values on solution are influenced by the nonlocal constitutive nature of the material. The nodal displacementsfromwheretheyoriginateareindeedthesolutionofanonlocallinearequationsystemwhose global (stiffness)coefficientsmatrixisthematrix K.Inturn,theassociatedstresses,thatbytheprinciple of minimumtotalpotentialenergyareequilibratedwiththeloads,arenonlocal and aregivenbyEq.(1).
3. NL-FEMimplementation
3.1. Thenonlocal operators
AttentionisfocusedonthenonlocaloperatorsgivenbyEqs.(2)–(4)aswellasontheelement’snonlocal matrices (10a),(10b).Theelement’s matrices(9a),(9b)havetheformatof thestandardFEManddonot needcomments.Thecomputationaldetailsaregivenwithreferenceto8-nodes,C0-quadraticisoparametric Serendipityelementswith2DOFspernode.Tothisconcern,followingthestandardisoparametric formula-tion,anatural(intrinsic)coordinatesystem,referredtoaparentelement,isintroduced,sayξ := (ξ,η).The coordinatestransformationwillbedenotedbyx(ξ):= x(ξ,η)={x(ξ, η) y(ξ, η)}T beingJ (ξ) theJacobian matrix.Whenreferringtothegenericelement#m ortooneofitsGPsofCartesiancoordinates x,the nat-uralcoordinatesystemwillbedenoted,onlyforclarity,ξ:= (ξ,η) andconsistentlytheotherrelationswill be writtenwithprimedsymbols.Finally,allthenumericalintegrationsareperformedwithstandardGauss quadrature ruleon 3× 3 Gausspoints perelement.Inthefollowing,to gaingeneralityofthe expressions, thesymbol NG willdenotethetotalnumberofGPsperelement.
Considerfirstthenonlocaloperatorγ(x) definedbyEq.(3)andtobecomputed ateachx∈ V . Within the adoptedFE formulation γ(x) willthen be computed at each GP,of Cartesiancoordinates x,of each finiteelement,#n.Moreover,ontakingintoaccountthepropertiesoftheattenuationfunctiong(x,x),the integralonther.h.s.ofEq. (3)hasto beconfinedonlytotheGPs(ofCartesiancoordinates x)belonging to theMe elementsneighbors of#n.Equation(3)canthenbewrittenas:
γ(x) = Me m=1 Vm g(x, x) dVm ∀x ∈ #n with #n = 1, 2, . . . , Ne ∀x∈ #m with #m = 1, 2, . . . , M e (14)
Enforcing theisoparametricformulation,Eq.(14)yields:
γ [x(ξ)] = Me m=1 1 −1 1 −1 g |x(ξ)− x(ξ)|t detJ (ξ) d ξd η, (15)
wheredVm:= tdxdy= tdetJ (ξ)dξ dη,t beingtheelements’thickness,whiletheattenuationfunction
g(x,x) hasbeenassumedasdependentontheEuclidean distance betweenthefieldpointx andthesource point x.TheGaussianquadraturerulefinally yields:
γ [x(ξh, ηg)] = Me m=1 NG r=1 NG s=1 g |x(ξ r, ηs)− x(ξh, ηg)| wrwst detJ (ξr, ηs) , (16)
where (ξh,ηg) and (ξr,ηs) denote the natural coordinates of the Gauss points x ∈ #n and x ∈ #m,
respectivelyandwr,ws aretheGaussweightsat x(ξ).
Following a rationale as above and taking into account Remark 3 concerning the nonlocal operator
q(x,x),definedbyEq.(4),thelatter,withintheadoptedisoparametricFEformulation,canbegiventhe shape:
q(x, x) = Me μ=1 Vμ g(x, z)g(x, z)D(z) d Vμ ⎧ ⎪ ⎨ ⎪ ⎩ ∀x ∈ #n, with #n = 1, 2, . . . , Ne ∀x ∈ #m, with #m = 1, 2, . . . , M e ∀z ∈ #μ, with #μ = 1, 2, . . . , Me (17)
whereMe denotesthenumberofelements,sayelements#μ= 1,2,. . . ,Me,belongingtoΓ#n∩ Γ#mandz
expresses theCartesianabsolute coordinatesof ageneric GPof agenericelement #μ.Theisoparametric treatmentgives: qx(ξ), x(ξ) = Me μ=1 1 −1 1 −1 g |x(ξ) − z(ˆξ)| g |x(ξ)− z(ˆξ)| × D z(ˆξ) t detJ (ˆξ) d ˆξ d ˆη (18)
whereξ := ( ˆˆ ξ,η);ˆ z(ˆξ):= z( ˆξ,η);ˆ dVμ:= tdetJ (ˆξ)dξ dˆ η.ˆ ByGaussianquadraturerule Eq.(18)yields:
q [x(ξh, ηg), x(ξr, ηs)] = Me μ=1 N G p=1 NG q=1 g |x(ξh, ηg)− z(ˆξp, ˆηq)| × g |x(ξ r, ηs)− z(ˆξp, ˆηq)| D z( ˆξp, ˆηq) ˆ wpwˆqt detJ ( ˆξp, ˆηq) (19)
where ( ˆξp,ηˆq) denote the natural coordinates of the Gauss point z ∈ #μ and wˆp,wˆq the related Gauss
weights.
TheFE computationof the nonlocaloperatorJ (x,x),given by Eq. (2),does notneed any comment beingstraightforwardonceγ(x) and q(x,x) arebuilt.
For whatconcern thecomputation of the nonlocal element matrices (10a), (10b)the following canbe stated. Letstartwith knonlocn definedby Eq.(10a). Ifi andj denote twogeneric nodes(with2DOFsper node)ofelement #n,therelevant(2× 2) submatrix ofknonlocn ,sayknonloc(n)ij ,reads:
knonloc(n)ij =
Vn
BT(n)i(x) γ2(x) D(x) B(n)j(x) dVn ∀x ∈ #n. (20)
Byisoparametricformulationandapplyingaterminologyalreadyspecifiedthelattercanberephrasedas:
knonloc(n)ij = 1 −1 1 −1 BT(n)i[x(ξ)] γ2[x(ξ)] D [x(ξ)] B(n)j[x(ξ)] t detJ (ξ) dξ dη, (21)
whereB(n)i denotesthesubmatrixofBn,definedbyEq.(6),associatedtonode#i.TheGaussian
quadra-turerulethenyields:
knonloc(n)ij = NG h=1 NG g=1 BT(n)i[x(ξh, ηg)] γ2[x(ξh, ηg)] D [x(ξh, ηg)] × B(n)j[x(ξh, ηg)] whwgt detJ (ξh, ηg) . (22)
For whatconcern the set of nonlocalelement matrices knonlocnm definedby (10b),besides theadvantage given by Eq. (13) (i.e. it is required only the computationof diagonal terms plusthose of the upper (or lower) triangle),byRemark 2onlytheMe neighborelements ofcurrentelement#n havetobeconsidered
in the cross integrations. If i and v denote two generic nodes of element #n and #m, respectively, the relevant(2× 2) submatrixofknonlocnm ,sayknonloc(nm)iv,reads:
knonloc(nm)iv := Vn Vm BT(n)i(x)J (x, x) B(m)v(x) d Vmd Vn ∀x ∈ #n ∀x ∈ #m with #m = 1, 2, . . . , M e (23)
Thelattercanbegiventheshape:
knonloc(nm)iv = 1 −1 1 −1 1 −1 1 −1 BT(n)i[x(ξ)]Jx(ξ), x(ξ)B(m)v x(ξ) × t detJ(ξ)d ξ d η t detJ (ξ) d ξ d η (24)
which, asbefore,canbenumericallyintegratedas:
knonloc(nm)iv = NG h=1 NG g=1 NG r=1 NG s=1 BT(n)i[x(ξh, ηg)]J [x(ξh, ηg), x(ξr, ηs)] × B(m)v[x(ξr, ηs)] whwgwr wrt2detJ (ξr, ηs)detJ (ξh, ηg) . (25)
Theassemblageofknonlocn andknonlocnm ,beside klocn ,intotheglobalnonlocalmatrixK formally givenby
Eq.(12)followsastandardFEprocedurebasedontheidentificationoftheglobalDOFswiththeelements’ DOFsanddoesnotdeservespecificcomments.Fromacomputationalpointofviewitisworthnotingthat, asalreadypointedout,matrix K evenifitisdense,duetothecontributionsoftheelements’cross-stiffness terms,issymmetricandpositivedefinite.Suchpropertiesallowstandardtechniqueseitherforstoragemode and for equations solution, in particular triangular storage mode (or column-wise storage) and classical Gauss elimination or Cholesky decomposition methods (no pivoting is necessary) canbe effectively used; see e.g.[20].
Remark 5. It is worth noting that a standard displacement-based FE formulation has been expounded. Moreover,continuityoftheprimaryunknowns,namelythedisplacements,butnotoftheirspatialderivatives isrequired.C0-continuousshapefunctions andstandardGaussianquadrature rulefornumericalintegration can thenbe employed. Finally,no specific precautions are needed toenforce static or kinematic boundary conditions,which,bytheway,isoneoftheseriousobstaclestothedisseminationof(nonlocal)gradient-based numericalapproaches.Besidestheobviousadvantagesofferedbysuchcircumstancesitisalsoworthnoting thatfew appositesubroutines,computing thenonlocaloperators,sufficeto transformastandardFE code into aNL-FEone.
Remark6.Thenonlocaloperators,storedasvectorsormatrices,havebeencomputedontakingadvantage from the peculiarities of the attenuation function, so taking into account the influence distance LR and
of the whole structure.The vector γ(x) for example, collecting the γ values at each GP (i.e. at each x) within the FE mesh, is a vector with (NG× Ne) terms. This obviously implies a greater computational
burdenwithrespect tothestandardFEM.
3.2. Flow-chart andcomputationaldetails
Aflow-chartofthewholemethodisgiveninthenextpagefocusingtheattentionontheconstructionof nonlocaloperatorsandnonlocalelementmatrices.
Fromacomputationalpointofviewoneofthecrucialstepsistheselectionoftheneighborsofanelement. Thenumber (Me) ofneighborsaffects all the“nonlocaloperations” and, consequently,thecomputational
efficiencyoftheNL-FEM.TothisconcernitisworthspecifyingfirsthowtheinfluencedistanceLRisfixed.
As stated inSection1, LR establishes at whichextent the nonlocalprocesses takeplace and it is related
to the analyticalshape of the attenuation functiong(x,x). Beyond LR, g(x,x) ispractically vanishing.
Precisely,ifg(x,x) possessesacompactsupport(circularin2D),LR isgivenbytheradiusofsuchsupport
andg(x,x) isidenticallyzerooutsidethesupport,ifnot,asoftenthecase,LRhastobefixed.Remembering
thatg(x,x) hastosatisfythenormalizationconditiononV∞,i.e.V
∞g(x,x
)dV = 1,awaytofixL Rcan
be givenbytheconditionthattheintegralofg(x,x) (normalizedinV∞)evaluated overVR,namelyover
acircular influencezoneof radiusLR in2D, be closeto unity(see e.g.[16]). Thecomputationalinfluence
distance hasbeenthenevaluated byimposingthecondition:
V
g(x, x) d V= 1− Tol (26)
withTol = fixed tolerance value.
Toselect theneighboringelements,at thebeginningof theelement loop#n,theboundaryGPsof the currentelement#n (i.e. theGPsclosertotheelement’sboundary)areconsidered.AlltheGPsofthemesh whichareawaylessthanLRfromsuchboundaryGPsof#n belongtoaneighborof#n,theMeneighbors
aretheneasily identifiedforeach#n.
A final remark concerns thechoiceof the analyticalshape of g(x,x).As noted in[4],within nonlocal elasticity of integral type, many are the “legitimate choices”of the attenuation function, or kernel, each one yielding adifferent nonlocal approach. To the authors’opinion, thechoice of the analytical shape of the kernel, within anonlocalapproach of integraltype, is amodeling choice asthat of theFE mesh-size (related to the concept of h-refinement) or of the FE shape functions’ polynomial degree (related to the concept of p-refinement). From this point of viewthe choiceof the kernel cannot be unique and mustbe drivenfromexperimentalevidences ontherealnonlocalmaterialunderconsideration.Severalstudieshave been made concerning this, mainlyoriented to choose akernel able to fit experimentaldata obtained,in general,fromphonondispersionrelations.Thelatter,intheformofone-dimensionalfrequencyversuswave numbercurvesforgivenwavedirection,arethenusedtobuild2Dand/or3Dkernels,seee.g.[22,34,17,16]. Thispointisoutoftheaimsofthispaperandtheanalyticalshapesofthekernelschosenforthenumerical applications expoundedinthenextsectionhavebeenselectedto showtheeffectivenessoftheNL-FEMin catchingthenonlocalelasticbehaviorbutwithoutreferringto aspecificrealmaterial.
4. Numericalexamples
4.1. Nonhomogeneous squareplate undertension
Thefirst exampleconcernsasquare plateundertensionwhose geometry, material data, boundaryand loadingconditionsaregiveninFig. 1.Theplateisfixedalongtheedgeatx= 0,withassigneddisplacements ¯
Thiscase-study,alreadytackledin[25],isconsideredforatwofoldreason:i)toremarkthegeneralvalidity oftheadoptedstrain-difference-basedmodelwhich,intheshapegivenbyEq.(1),referstononhomogeneous nonlocal elastic material, the fourth order elasticity tensor being defined as D = D(x). The adopted constitutive model, as shown in the quoted paper, enables both to eliminate undesired boundary effects congenital to Eringen model and to address piecewise-homogeneous plane problems. The latter recall a typical problem of elasticity, namely the (elastic or rigid) inclusion within an anisotropic body, but of courseheretheproblemisjustmentioned,itismoreapotentialityoftheexpoundedNL-FEMthanareal capacityatthisstage.Thestudyofsoft/rigidinclusionsinplanestressistheobjectofanongoingresearch;
ii)withrespecttotheresultsgivenpreviouslyforthisexample,theenhancedversionoftheNL-FEMcode herepresented (see e.g.Remark 3and Eq. (13)) allowsto workwith much finerFE meshes so givingthe possibility to show themesh independency of the obtainednumerical solution, to gain somehints on the
Fig. 1. Nonhomogeneous square plate under tension with piecewise constant Young modulus – geometry, boundary and loading conditions, material data.
influence of the material parameters α and entering the constitutive relation, as well as to verify the capacity of theNL-FEM tocapture theso-calledsizeeffects.
Theanalyticalshapeoftheassumedattenuation functiong(x,x) forthis exampleisthefollowing:
g(x, x) = 1 2 π 2texp −|x − x| , (27)
thatisa2Dbi-exponentialfunction.ATol = 10−4hasbeenalsoassumedtoenforcecondition(26)obtaining acomputationalinfluencedistanceLR= 11 with aloss ontheunitvalueequalto 0.02%.
TheabilityoftheimplementedNL-FEMapproachtogivesmoothsolutionswherethevalueoftheelastic modulusabruptlychangesortheeliminationofundesiredboundaryeffectsareresultsofthepreviousstudy bytheauthorsandarenotconsideredtoavoidrepetitions.Toshowthemesh-independency ofthenumerical solutiondifferentuniformFEmesheshavebeenconsideredfortheexaminedsquareplate.Thecoarsemesh wasobtainedwith20subdivisionsbothalongx andy soutilizing400FEs;thefinermeshwasobtainedwith 60subdivisions foratotalnumberofFEsequalto 3600.Itisworthnotingthatthemeshsize isrelatedto thevalueofthematerialinternallength,smallervaluesof requirefinermeshes.Threevaluesof (in cm) havebeenconsidered:= 0.02,0.1,0.5,thethirdvaluebeingobviouslyphysicallymeaninglessbeingofthe sameorderoftheplatethickness.Figs. 2a–c showthemid-platesectionsofthestraindistributionalongthe directionoftheprescribeddisplacementu¯x,i.e. theprofilesofthestrainsεxversusx aty¯ 2.5 cm,α = 50
and= 0.02,0.1, 0.5 cm,respectively.Byinspectionoftheplottedresultsitisworthnotinghowthemesh independency ofthesolutionisalwaysaccomplishedbut“itstarts”fromafinermesh (40× 40 FEs) when thematerialinternallengthissmall, namelyfor= 0.02,appearingsincethebeginning(i.e. forthecoarse mesh of20× 20 FEs)for= 0.1and 0.5.
Theeffectsofthematerialparameters andα areillustratedinFigs. 3 and 4.Fig. 3gives,forafixedmesh of40×40 FEsandα = 50,themid-platestrainprofilesεxversusx aty¯ 2.5 cm for= 0.0,0.02,0.1,0.5 cm.
Thesolutionfor= 0 isobviously thelocalone,showingajumpattheabscissawheretheYoungmodulus abruptly changes.Theprofilefor= 0.5 is anonlocalonebutphysicallymeaninglessbecausetheinternal length is equal to the platethickness. It is worth noting thatinsuch circumstance the smoothing of the nonlocal approachisso extendedthatthe strainprofileflattens becoming inthelimitof → ∞ uniform. For theintermediate physicallysounding valuesof , thenonlocalsolution isvery differentfrom the local one at the sections where the Young moduluschanges becoming close to the local one near to the plate boundaries,i.e. noboundaryeffectsappear,asalreadyknown(seee.g.[25]).
The influence exerted by the material parameter α on the nonlocal solutionhas been investigated for = 0.1 cm andamesh of30× 30 FEs.Fig. 4showsthemid-platestrainprofilesεxversusx aty¯ 2.5 cm
for α = 0,1,10,50,100,200; α = 0 yielding again the local elastic solution. Figs. 5a–f give the same information of Fig. 4but plottingthe straindistribution εx on the wholeplate. As notedat Section 2.1,
Fig. 2. Nonhomogeneous plate under tension. Strain profiles εxversus x aty = 2.5 cm,α = 50, and for different values and FE
meshes: a) = 0.02 cm; b) = 0.1 cm; c) = 0.5 cm.
Eq. (1), the value of α controls the proportion of the nonlocal addition in the constitutive model. As expected,ifα increasesalsothe“nonlocalbehavior”increasesasexhibitedbytheplotsofFigs. 4 and 5a–f obtained for higher values of α that spread the nonlocal effects smoothing the strain distribution. Such
Fig. 3. Nonhomogeneous plate under tension. Strain profiles εx versus x aty = 2.5 cm, α = 50, mesh of 40 × 40 FEs for =
0.0, 0.02, 0.1, 0.5 cm.
Fig. 4. Nonhomogeneous plate under tension. Strain profiles εx versus x at y = 2.5 cm, = 0.1 cm, mesh of 30 × 30 FEs for α = 0, 1, 10, 50, 100, 200.
results were predictable but confirm a theoretical constitutive conjecture. α is a material parameter, as theinternallength, andithasto bedetectedexperimentally.Itsmeaning isindeeddifferentfrom ;ina nanocomposite for exampleit shouldbe related to thevolumetric percentage, or to theconcentration, of nanoparticlesinthepolymermatrix.
The capacity to capture the well known size effects, i.e. the size dependent mechanical response of the analyzed specimenbecomingdominant as thespecimen orstructural element size decreases, is finally investigated. Takinginto account thesketchand thedataof Fig. 1,theplatesize hasbeenproportionally variedassuminga= 0.5 cm anda= 2 cm;a= 1 cm isthevaluealreadyconsidered.Thethreegeometrical dimensions (a= 0.5,1,2) define three proportionalsizedplates with theboundaryconditionssketchedin
Fig. 1andsufferingthedisplacementu¯xthathasbeenaccordinglyproportionallyvariedwiththedimension
a such thatu¯x = a/1000.Moreover, thevaluesE1= 0.4E2,E2 = 210 MPa andt= 0.5 cm, alreadyused,
havebeenkeptwhile= 0.1 andameshof40× 40 FEshavebeenconsidered.Theelementsizeissovaried proportionally with the geometric dimensions of the plate. The mid-plate strain profiles εx versus x/L
obtainedwith astandardlocal FEelasticanalysis andanonlocalelasticoneonthethree plates aregiven in Fig. 6. Precisely, the three solutions obtainedwith local elasticity coincide,each FE modelis a scaled
Fig. 5. Nonhomogeneous plate under tension. Strain distribution εx= εx(x, y) for = 0.1 cm, mesh of 30 × 30 FEs: a) α = 0;
b) α = 1; c) α = 10; d) α = 50; e) α = 100; f) α = 200.
versionoftheothertwo.Thelocalsolutionsarethesameasexpected.Incontrast,thethreenonlocalelastic solutionsgivenbytheNL-FEMshowadecreasing–aflattening–fordecreasingspecimendimensions.The constitutivemodelendowedwithamaterialinternallengthcaptures thesize effects.
Fig. 6. Nonhomogeneous plate under tension. Strain profiles εxversus x/L aty = 2.5 cm, = 0.1 cm, α = 50, mesh of 40 × 40 FEs
and for three different proportional sized plates given by a = 0.5, 1 and 2 cm. Local (lines without markers) and nonlocal (lines with markers) solutions.
Fig. 7. Strip with notch (after Askes and Gutiérrez [7]) – geometry, mechanical model, boundary, loading conditions and material
data.
Itisworthnotingthatadecreaseofthespecimensizewithconstantinternallengthoranincreaseofthe internallength withconstantspecimensize,producesthesameeffect,i.e. a flatteningofthestrainprofile as deduciblebycomparison oftheplotsofFig. 3 for= 0.5 andFig. 6 fora= 0.5.Inthelimitfora→ 0 or → ∞ (i.e. g(x,x)= 1)auniform strainprofile(¯εx 2× 10−4 intheconsideredexample)isobtained
as ithastobe.
4.2. Stripwith notch
ThestripwithnotchofFigs. 7a,b showinggeometry,mechanicalmodel,boundaryandloadingconditions plus material data, analyzed by Askes and Gutiérrez [7] via a gradient approach has been addressed as benchmark problem.The symmetry of geometry and loadingallows the analysis of the top-right quarter shown inFig. 7b.The main goalis to compare thesolutionobtainable with thepresent nonlocalintegral approachwiththeonegivenbyawellestablishedgradientformulation.Tothisaimapremiseisnecessary. Itiswellknownthatnonlocalintegralelasticitymodelsandgradientelasticityonesarerelatedbymutual relationships(seee.g.[11,13];[2]).Tothisconcern,referringtotheremarkableandilluminatingcontribution of Borino and Polizzotto [10], see also Polizzotto [28,29], aclass of “associated models”canbe envisaged where “the kernel function characterizing the integral operator of the nonlocalmodel coincides with the
Green function of thedifferential operatorof the gradientmodel”. Borrowingfrom Borinoand Polizzotto
[10] the stress gradient model expressed byEq. (7) of Askes and Gutiérrez [7] is associated to the strain integral modelexpressedbyEq.(1)hereadopted.Moreover,theGreenfunctionofthedifferentialoperator L= 1− 2∇2 appearinginthequotedstressgradientmodelisgivenby(seee.g.[13]):
g(x, x) = 1 2 π 2K0 |x − x| , x, x∈ V, (28)
whereK0(z) isthemodifiedBesselfunctionofthesecondkindoforderzerowhichcanbegiventheexpression
(seee.g.[1]). K0(z) = ∞ 0 cos(z sinh(t))dt = ∞ 0 cos(zt) √ 1 + t2dt. (29)
Thelatter,incontrastwiththeoscillatingnatureofthestandardBesselfunctions,isexponentiallydecaying butexhibitsasingularityatz = 0.Suchcircumstancerendersthekernel(28),saytheBessel-likekernel,not applicableas it isintheNL-FEM context;to buildthe nonlocalelementmatrices knonlocn and knonlocnm the kernelhasindeedtopossessafinitevalueatz = 0.Ontheotherhandtheanalyticalshapeof(28)guarantees theconsistencyofthecomparisonbetweentheintegralandthegradientapproach.Toovercometheproblem a modified bi-exponential is then utilized having ageometrical shape very close to the Bessel-like kernel
(28)exceptforx→ 0 whereafinitevalueisassumed, namely:
g(x, x) = 1 2 π (β)2texp −|x − x| β . (30)
Inexpression(30)thescalarβ,with0< β ≤ 1,simplyrescalesthevalueoftheinternallength pertaining tothegivenmaterialifthebi-exponentialkernelisadoptedinsteadoftheBessel-likeone.β hastobefixed insuchaway thatfor agiven material(or agiven) thegeometricalshapesof theBessel-like kernel(28)
andthemodifiedbi-exponentialkernel(30)areascloseaspossibletoeachother.Theapproachisdefinitely heuristicbuttakes into accountthephysicalmeaning ofthekernel thatresidesintheway itcaptures the nonlocaleffectsbyweightingthem.Thelatteractionbeingstrictlyrelatedtoitsgeometricalshape.Avalue of β = 0.8 has been detected as the onegivingthe closer shapesof the two kernels. Such value hasbeen evaluatedconsideringdifferentvaluesof,inFig. 8thetwokernelsareplottedfor= 0.1 andβ = 0.8,the other valuesarenotshownforbrevity.
The results obtained for the strip with notch of Fig. 7 are given next in terms of the three strain componentsεx,εy,γxyandfordifferentFEmeshesrangingfrom20×20 to50×50 FEs.Precisely,Figs. 9a–c
showthedistributionsoftheabovestraincomponentsontheanalyzedquarterofstripobtainedforamesh of40× 40= 1600 FEs.Theobtainedresultsappearinagreementwiththeonesof AskesandGutiérrez[7]
andthere plottedfortheso-calledreferencecase.Figs. 10a–fshow againthethreestraincomponents,but aty 0,i.e. onasectioncloseto thenotch, andfor4different FEmeshes.Themesh independenceofthe nonlocal solution is confirmed. Figs. 11a–c finally show the nonlocal strain componentsagainst the local onesforafixedmeshof40× 40 FEstohighlight,atthesameplottingscale,thesmoothingachievedbythe nonlocalapproach.
5. Concludingremarks
The general validity and computational efficiency of a nonlocalelasticity version of the finite element method, knownas NL-FEM,implemented in conjunction with a strain-difference-basednonlocal integral
Fig. 8. Plots of Bessel-like kernel of Eq.(28)(solid line) against modified bi-exponential kernel of Eq.(30)(dashed line) for = 0.1 cm and β = 0.8.
modelof Eringentype hasbeenproved. Such enhanced(nonlocal)formulation oftheFEM wasconceived bythesenior Authorsin[31]andappliedforafirstvalidationin[25] toanalyzesimpleexamples.
The latterweremainlyoriented to showthe abilityof theconstitutivemodelto dealwith nonhomoge-neous nonlocalelasticproblemsortoavoidnumericalinstabilities arisingwithalocaltreatmentor,also,to eliminate undesired boundary effects properof theEringenintegralmodel.TheNL-FEM hereexpounded, while keeping theabove abilities, takes advantage from a moreaccurate construction of the nonlocal op-erators involved at element level, namely the element nonlocal matrices, so overcoming one of the main drawback evidenced in the above quoted previous studies, i.e. the excessive dimensions of the nonlocal cross-stiffness element matrices.The possibilityto manageFE mesheswithahigh numberofelements,at least as manyas thoseused inthestandardFEM,is straightforwardand haspermittedtodemonstrate a numberofpotentialitiesofthemethodthatcanbesummarized asfollows:
1) The capacityofremovalofsingularities inthoseproblemswherethestandardFEM,basedonthelocal elasticapproach, cannot givethe right solutionshowing numericalinstabilities and mesh-dependency. Smoothingofthelocalelasticsolutionwheretheelasticmodulusabruptlychanges,asintheexample #1, oratthenotchend,as intheexample#2,provestheassertionshowingalsothemeshindependency of thenonlocalsolution.
2) The capacity to model at a macro-scale level and within a standard FE scheme the nonlocal effects arising at a micro-scale level. To this concern, the influence of two material parameters, namely the internallength and theparameterα,hasbeenshownfortheexample#1.TheFEnonlocalresponse is indeedclearlyinfluenced bythe valuesof suchparametersentering theconstitutive relations.With referencetoexample#1 thecapacitytocapture thesizedependentmechanicalresponseoftheanalyzed structure,knownassize-effect, hasalsobeendemonstrated.
3) The advantageofferedbyaformulationthatrequiresatmostsecond-orderderivativesofthe fundamen-talunknowns(thedisplacementsinthepresentedNL-FEM),namelythepossibilitytouseC0-continuous
elementshape functions.AnimmediatefirstconsequenceisthepossibilitytoenrichstandardFEcodes withappositesubroutinesdevotedtotheconstructionofthenonlocalelementmatrices.Other, nontriv-ial,consequences concerntheassembling operationor the enforcingof boundary conditions remaining unaltered withrespect tothestandardFEMformulation.
Fig. 9. Strip with notch. Distribution of strain components obtained with a mesh of 40 × 40 FEs: a) εx= εx(x, y); b) εy= εy(x, y);
c) γxy= γxy(x, y).
4) Even ifthedimensionsofthenonlocalcross-stiffnesselementmatricesareconfinedtoareducednumber ofelementsinthemesh,namelytheneighboringelements,suchelementmatricesstillinvolveanumber of DOFsbigger than inthestandardFEM. This mightconstitute still adrawback whendealing with problems requiringveryfineFEmeshes.
Fig. 10. Strip with notch. Strain components profiles εx, εy, γxy, versus x aty = 0 for different FE meshes. Plots on the left side
a), c), e) refer to nonlocal solution; plots on the right side b), d), f) refer to the local solution.
5) The presenceofnonlocaloperators tobeevaluatedonthewholemeshatthebeginningoftheanalysis, evenifitcanbeperformedwithstandardtoolsastheGaussianquadratureheresuggested,itmightbe anotherlimitifmeshesofthousandsofelementsarerequired.
6) The needoftwo nonlocalmaterial parameters,namelytheinternallength plusaparameterα which controls the proportion of nonlocality within the constitutive relation, is a third weak point of the
Fig. 11. Strip with notch. Strain components profiles εx, εy, γxy, versus x aty = 0 for a mesh of 40 × 40 FEs. Local solution (solid
lines), nonlocal solution (dashed lines).
NL-FEM. This drawback is indeed not strictly related to theoretical or computational matters but rather totheexperimentalvalidationoftheadoptedconstitutivenonlocalmodel.
Overalltheobtained resultsseemquite encouraging andwitnessthe effectivenessof thewhole procedure, theyalsohighlighttheneedoffurtherstudiestobefocused,intheauthors’opinion,atleastintwodirections: i) toinvestigateonexperimentaltechniquesorientedtotheidentificationofthenonlocalparametersα and ; ii) to compare the results obtainable with the promoted NL-FEM with other well established nonlocal
approaches (as the gradient-basedones) analyzing otherbenchmark problems. Both are theobjects ofan ongoingresearch.
References
[1]M.Abramowitz,I.A. Stegun,Handbook of MathematicalFunctionswithFormulas, GraphsandMathematicalTables, Appl.Math.Ser.,vol. 55,NationalBureauofStandards,1972(tenthprinting).
[2]E.C.Aifantis,Ontheroleof gradientsinthelocalization ofdeformation andfracture, Internat.J.Engrg.Sci.30 (10) (1992)1279–1299.
[3]E.C.Aifantis,Updateonaclassofgradienttheories,Mech.Mater.35 (3–6)(2003)259–280.
[4]G.Allegri,F.L.Scarpa,Ontheasymptoticcrack-tipstressfieldinnonlocalorthotropicelasticity,Int.J.SolidsStruct.51 (2014)504–515.
[5]I.V.Andrianov,J.Awrejcewicz,D.Weichert,Improvedcontinuousmodelsfordiscretemedia,Math.Probl.Eng.(2010), ArticleID 986242,HindawiPublishingCorporation.
[6]H.Askes,E.C.Aifantis,Gradientelasticityinstaticsanddynamics:anoverviewofformulations,lengthscaleidentification procedures,finiteelementimplementationsandnewresults,Int.J.SolidsStruct.48(2011)1962–1990.
[7]H.Askes,M.A.Gutiérrez,Implicitgradientelasticity,Internat.J.Numer.MethodsEngrg.67(2006)400–416.
[8]Z.P.Bažant,L.Cedolin,StabilityofStructures:Elastic,Inelastic,FractureandDamageTheories,WorldScientific Pub-lishingCompany,2010.
[9]Z.P.Bažant,M.Jirásek, Nonlocalintegralformulations ofplasticityanddamage:surveyofprogress,J. Eng.Mech. 11 (2002)1119–1149.
[10]G.Borino, C.Polizzotto, Amethodtotransform anonlocal modelintoa gradientonewithin elasticityandplasticity, Eur.J.Mech.ASolids46(2014)30–41.
[11]A.C.Eringen,Ondifferentialequationsofnonlocalelasticityandsolutionsofscrewdislocationandsurfacewaves,J.Appl. Phys.54 (9)(1983)4703–4710.
[12]A.C.Eringen,Theoryofnonlocalelasticityandsomeapplications,Res.Mechanica21(1987)313–342. [13]A.C.Eringen,NonlocalContinuumFieldTheories,Springer-Verlag,NewYork,2002.
[14]A.C.Eringen,B.S.Kim,Stressconcentrationatthetipofacrack,Mech.Res.Commun.1(1974)233–237.
[15]N.A.Fleck,G.M.Muller,M.F.Ashby,J.W.Hutchinson,Straingradientplasticity:theoryandexperiments,ActaMetall. Mater.42(1994)475–487.
[16]S.Ghosh, A. Kumar, V. Sundararaghavan, A.M. Waas, Non-local modelingof epoxy using an atomistically-informed kernel,Int.J.SolidsStruct.50(2013)2837–2845.
[17]S.Ghosh,V. Sundararaghavan,A.M. Waas, Construction ofmulti-dimensional isotropic kernels for nonlocal elasticity basedonphonondispersiondata,Int.J.SolidsStruct.51(2014)392–401.
[18]E.Hinton,D.R.J.Owen,FiniteElementProgramming,AcademicPress,HarcourtBraceJovanovichPublishers,London, 1989.
[19]G.A.Maugin,A.V.Metrikine,MechanicsofGeneralizedContinua:OneHundredYearsaftertheCosserats,Springer-Verlag, NewYork,2010.
[20]M.Okrouhlík(Ed.),NumericalMethodsinComputationalMechanics,InstituteofThermodynamics,Prague,2012. [21]J.Peddieson, G.R. Buchanan,R.P.McNitt,Applicationof nonlocal continuummodelstonanotechnology,Internat.J.
Engrg.Sci.41(2003)305–312.
[22]R.C.Picu,Onthefunctionalformofnon-localelasticitykernels,J.Mech.Phys.Solids50 (9)(2002)1923–1939. [23]A.A.Pisano,P.Fuschi,Closedformsolutionforanonlocalelasticbarintension,Int.J.SolidsStruct.41(2003)13–23. [24]A.A.Pisano,A.Sofi,P. Fuschi,Nonlocalintegral elasticity:2Dfinite elementbasedsolutions, Int.J.SolidsStruct.46
(2009)3836–3849.
[25]A.A. Pisano, A. Sofi, P. Fuschi, Finite element solutions for nonhomogeneous nonlocal elastic problems, Mech. Res. Commun.36(2009)755–761.
[26]C.Polizzotto,Nonlocalelasticityandrelatedvariationalprinciples,Int.J.SolidsStruct.38(2001)7359–7380.
[27]C.Polizzotto, Unifiedthermodynamic frameworkfor nonlocal/gradient continuumtheories,Eur. J.Mech. ASolids22 (2003)651–668.
[28]C.Polizzotto,Stressgradientversusstraingradientconstitutivemodelswithinelasticity,Int.J.SolidsStruct.51(2014) 1809–1818.
[29]C.Polizzotto,Aunifyingvariationalframeworkforstressgradientandstraingradientelasticitytheories,Eur.J.Mech.A Solids49(2015)430–440.
[30]C.Polizzotto,P.Fuschi,A.A.Pisano,Astrain-difference-basednonlocalelasticitymodel,Int.J.SolidsStruct.41(2004) 2383–2401.
[31]C.Polizzotto, P.Fuschi,A.A. Pisano,Anonhomogeneousnonlocal elasticitymodel, Eur.J. Mech.ASolids25(2006) 308–333.
[32]D.Rogula,Introductiontononlocaltheoryofmaterialmedia,in:D.Rogula(Ed.),NonlocalTheoryofMaterialMedia, Springer-Verlag,Berlin,1982,pp. 125–222.
[33]S.A.Silling,Reformulationofelasticitytheoryfordiscontinuitiesandlong-rangeforces,J.Mech.Phys.Solids48 (1)(2000) 175–209.
[34]V.Sundararaghavan,A.M.Waas,Non-localcontinuummodelingofcarbonnanotubes:physicalinterpretationofnon-local kernelsusingatomisticsimulations,J.Mech.Phys.Solids59 (6)(2011)1191–1203.
