where s isthestresstensorand f(q,x,t)isaforcedensity(perunitvolume squared)thataneighboringpoint q exertson x.Thisforcedensityis determinedbythedeformationaccordingtothematerialmodel.Itis alwaysrequiredasaconsequenceofNewton’slawsthat
forall x, q, t.Theregionofintegrationin Eq.(1.1) isthe family of x,which isaneighborhoodwhoseradius d > 0iscalledthe horizon (Fig.1.1).
Thehorizonisacutoffdistanceforforceinteractions.
Using Eq.(1.1),theperidynamicequationofmotionisasfollows:
where r isthemassdensity, y isthedeformationmap,and b isthe externalbodyforcedensityfield.Theattributeofthetheorythatallows interactionsbetweenpointssuchas x and q directlyacrossafinitedistanceissometimescalled strongnonlocality.
Inintroducingperidynamics,thebasic Eq.(1.3) isusuallypresentedas anassumptionthatiselaboratedupontoshowthatithasdesirable properties,suchasbeingabletosustaingrowingcracks,andnotviolating anylawsofphysics(Silling,2000).Thisleavessomepeoplewondering, “Wheredoesthiscomefrom?”Inparticular,thestronglynonlocalnature of Eq.(1.3) isperceivedaslackingmotivation.Withthisinmind,Ihope
forall k, [,and t.Itisalsoassumedthatthereisacutoffdistance d forthe atomicinteractionssuchthat Fk[ ¼ 0if jx[ xk j > d.
Additionally,particle k issubjecttoaprescribedexternalforce Bk(t).For any x˛R3,defineasmoothingfunction U(x, $)suchthatthefollowing normalizationholds:
foranypoint p.Itisconvenienttoassumethatatany x, U(x, $)vanishes outsideaneighborhoodofradius R,where R isapositivenumber.Define thesmoothedmassdensityandbodyforcedensityfieldsby
Definethesmootheddisplacementfieldby
Nowweinvestigatetheevolutionequationfor u.Theparticlesobey Newton’ssecondlaw, Mk € uk ðtÞ¼ X [ Fk[ ðtÞþ Bt ðtÞ.(1.8)
Differentiating Eq.(1.7) twicewithrespecttotimeyields rðxÞuðx; tÞ¼ X k Uðx; xk ÞMk uk ðtÞ (1.9)
From Eqs.(1.6),(1.8),and(1.9),
rðxÞ € uðx; tÞ¼ X k Uðx; xk Þ"X [ Fk[ ðtÞþ Bk ðtÞ# ¼ X k X [ Uðx; xk ÞFk[ ðtÞþ bðx; tÞ (1.10)
From Eqs.(1.5)and(1.10),
rðxÞuðx; tÞ¼ X k X [ Uðx; xk ÞFk[ ðtÞ Z Uðq; x[ Þdq þ bðx; tÞ (1.11)
Eq.(1.11) canberewrittenasfollows:
rðxÞuðx; tÞ¼ Z fðq; x; tÞdq þ bðx; tÞ (1.12)
where
and b isgivenby Eq.(1.6).Itiseasilyshownfrom Eqs.(1.4)and(1.13) that f hastheantisymmetry Eq.(1.2).Thevectordefinedby x ¼ q x (1.14) iscalleda bond.(Whentalkingaboutbonds,itisalwaysassumedthat x s 0,withoutexplicitlystatingthis.)Thefunction f iscalledthe pairwise bondforcedensity andhasdimensionsofforce/volume2.From Eq.(1.13), thepoints x and q interactonlyif
Thelength d isthehorizonforthecontinuummodel(Fig.1.1).
Insummary,wedefinedaweightingfunction U andusedittodefine smoothedfields r, b,and u.Withthesedefinitions,andNewton’ssecond lawappliedtotheparticles,thesmootheddisplacementfieldwasfound toobeytheperidynamicequationofmotion Eq.(1.12).Theperidynamic bondforcesthatappearinthisequationofmotionaredefinedby Eq. (1.13).Conceptually,theequationofmotionis,andshouldbe,nonlocal becausechangingthedisplacementofasingleparticle k directlyaffects thesmootheddisplacementsatallthepoints x whosesmoothingfunction U(x, $)havenonzerovaluesat xk.Amorecompletederivationofthe peridynamicequationswasobtainedfromstatisticalmechanicsby LehoucqandSears(2011).
Thedefinitionofthepairwisebondforcedensity Eq.(1.13) isnotvery practicalasamaterialmodelbecauseitdoesnotdirectlyrelatethevalues of u near x tothebondforces f.Morepracticalmethodsofdetermining f arediscussedin Section3.
3.Materialmodels
Thepurposeofamaterialmodelinperidynamicsistodeterminethe valuesofthepairwisebondforcedensity f(q,x,t)intermsofthesmoothed displacementsinthevicinityof x and q andanyotherphysicallyrelevant fieldssuchastemperature.(Theword“smoothed”willbeomittedfrom nowon,sincewearenolongerconcernedwiththeunderlyingatomic system.)Itisassumedthatthereisahorizon d,suchthat
Thestrainenergydensitygivenby (1.22) hasthesamemeaningasin thelocaltheory,sinceitrepresentsthepotentialenergyperunitvolume thatisstoredat x duetodeformationofthenearbymaterial.Ifthematerialisisotropicaswellaslinearmicroelastic,thisinterpretationprovidesaneasywaytocalibrate c(x)ifthegeneralformofthedependenceof c onbondlengthisgiven.Foranisotropicmaterial, c(x)dependsonlyon thebondlength, x ¼ |x|.Byrequiringtheperidynamicstrainenergy densitygivenby (1.22) toequalitsvalueinthelocaltheoryforanisotropic expansion,itiseasilyshownthat c isrelatedtothebulkmodulus k bythe followingexpression(SillingandAskari,2005):
Thus,withthesimplifyingassumption Eq.(1.24),ifthehorizon d is given,theonlyparameterinthelinearmicroelasticmaterialmodelis easilycomputedfromthebulkmodulus.
Thelinearmicroelasticisanexampleof bond-based peridynamicmaterialmodels.Thisclassofmaterialmodelshasthepropertythatthe pairwiseforcedensityinthebond x ¼ q x isfullydeterminedbythe deformationofthatparticularbond,anddoesnotdependonwhathappensinotherbonds.
Bond-basedmaterialshavetheadvantageofbeingsimpletounderstandandcalibrate.Theyallowforgeometricnonlinearity,becausethe bondsrotatealongwiththebondsastheydeform.Theyconserveangular momentum,becauseregardlessofthedeformation,eachpairwisebond forceexertszeromomentontheendpointsofthebond.Bond-based materialmodelscanincludematerialnonlinearity,aswellasgeometric nonlinearity,byusingafunction b f in Eq.(1.19) thatincludesnonlinear dependenceonthebondstrain.
where m isabinary-valuedfunctionthatstartsat1anddropsto0accordingtosomeprescribeddamagecriterion.Thesimplestsuchcriterion breaksabondwhenitsstrainexceedssomepredefinedvalue s0:
where H istheHeavisidestepfunctionand tbreak istheearliesttimeat which s(x,t) s0.Thelinearmicroelasticmaterialusing Eq.(1.24) and equippedwiththebondbreakagecriteriongivenby Eqs.(1.26)and(1.27) iscalledthe prototypemicroelasticbrittle (PMB)materialmodel.
Another random document with no related content on Scribd:
Project Gutenberg™ is synonymous with the free distribution of electronic works in formats readable by the widest variety of computers including obsolete, old, middle-aged and new computers. It exists because of the efforts of hundreds of volunteers and donations from people in all walks of life.
Volunteers and financial support to provide volunteers with the assistance they need are critical to reaching Project Gutenberg™’s goals and ensuring that the Project Gutenberg™ collection will remain freely available for generations to come. In 2001, the Project Gutenberg Literary Archive Foundation was created to provide a secure and permanent future for Project Gutenberg™ and future generations. To learn more about the Project Gutenberg Literary Archive Foundation and how your efforts and donations can help, see Sections 3 and 4 and the Foundation information page at www.gutenberg.org.
Section 3. Information about the Project
Gutenberg Literary Archive
Foundation
The Project Gutenberg Literary Archive Foundation is a nonprofit 501(c)(3) educational corporation organized under the laws of the state of Mississippi and granted tax exempt status by the Internal Revenue Service. The Foundation’s EIN or federal tax identification number is 64-6221541. Contributions to the Project Gutenberg Literary Archive Foundation are tax deductible to the full extent permitted by U.S. federal laws and your state’s laws.
The Foundation’s business office is located at 809 North 1500 West, Salt Lake City, UT 84116, (801) 596-1887. Email contact links and up to date contact information can be found at the Foundation’s website and official page at www.gutenberg.org/contact
Section 4. Information about Donations to the Project Gutenberg Literary Archive Foundation
Project Gutenberg™ depends upon and cannot survive without widespread public support and donations to carry out its mission of increasing the number of public domain and licensed works that can be freely distributed in machine-readable form accessible by the widest array of equipment including outdated equipment. Many small donations ($1 to $5,000) are particularly important to maintaining tax exempt status with the IRS.
The Foundation is committed to complying with the laws regulating charities and charitable donations in all 50 states of the United States. Compliance requirements are not uniform and it takes a considerable effort, much paperwork and many fees to meet and keep up with these requirements. We do not solicit donations in locations where we have not received written confirmation of compliance. To SEND DONATIONS or determine the status of compliance for any particular state visit www.gutenberg.org/donate.
While we cannot and do not solicit contributions from states where we have not met the solicitation requirements, we know of no prohibition against accepting unsolicited donations from donors in such states who approach us with offers to donate.
International donations are gratefully accepted, but we cannot make any statements concerning tax treatment of donations received from outside the United States. U.S. laws alone swamp our small staff.
Please check the Project Gutenberg web pages for current donation methods and addresses. Donations are accepted in a number of other ways including checks, online payments and
credit card donations. To donate, please visit: www.gutenberg.org/donate.
Section 5. General Information About Project Gutenberg™ electronic works
Professor Michael S. Hart was the originator of the Project Gutenberg™ concept of a library of electronic works that could be freely shared with anyone. For forty years, he produced and distributed Project Gutenberg™ eBooks with only a loose network of volunteer support.
Project Gutenberg™ eBooks are often created from several printed editions, all of which are confirmed as not protected by copyright in the U.S. unless a copyright notice is included. Thus, we do not necessarily keep eBooks in compliance with any particular paper edition.
Most people start at our website which has the main PG search facility: www.gutenberg.org.
This website includes information about Project Gutenberg™, including how to make donations to the Project Gutenberg Literary Archive Foundation, how to help produce our new eBooks, and how to subscribe to our email newsletter to hear about new eBooks.
