使用了广义的Maxwell模型描述的应力松弛行为的固体状的食物.docx
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使用了广义的Maxwell模型描述的应力松弛行为的固体状的食物.docx
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使用了广义的Maxwell模型描述的应力松弛行为的固体状的食物
UseofthegeneralizedMaxwellmodelfordescribingthestressrelaxationbehaviorofsolid-likefoods
∙M.A.DelNobilea,b,
∙S.Chillob,
∙A.Mentanab,
∙A.Baianob
∙aDepartmentofFoodScience,UniversityofFoggia,ViaNapoli,25-71100Foggia,Italy
∙bIstitutoperlaRicercaeleApplicazioniBiotecnologicheperlaSicurezzaelaValorizzazionedeiProdottiTipiciediQualità,UniversitàdegliStudidiFoggia,ViaNapoli,25-71100Foggia,Italy
Received5March2005
Accepted2December2005
Availableonline2February2006
∙http:
//dx.doi.org/10.1016/j.jfoodeng.2005.12.011,HowtoCiteorLinkUsingDOI
∙Permissions&Reprints
Abstract
AcomprehensivestudyontheabilityofthegeneralizedMaxwellmodeltodescribethestressrelaxationbehavioroffoodmatricesispresentedinthiswork.Fivedifferentfoodmatriceswerechosenasrepresentativeofawiderangeoffoods:
agargel,meat,ripenedcheese,“mozzarella”cheeseandwhitepanbread.Theabovefoodsmatriceswereselectedonthebaseoftheirmacro-structureandcompressivestressstrainbehavior.Stressrelaxationtestswererunontheabovefoodmatricesatroomtemperature.Therelaxationtimesdistributionfunctionusedinthisinvestigationtodescribetheviscoelasticbehavioroftheinvestigatedfoodswasobtainedbysummingtwonormaldistributionfunctionswiththemeansequaltozero.Resultsshowthattheproposedmodelsatisfactorilyfitstheexperimentaldata.Moreover,itwasalsofoundasubstantialdifferencebetweentherelaxationtimesdistributioncurvesoftheinvestigatedbulkyandspongyfoods.
Keywords
∙Modeling;
∙Generalizedmaxwellmodel;
∙Stressrelaxation;
∙Mechanicalproperties
1.Introduction
Foodproductsarecomplexmatricesshowingawiderangeofdifferentmechanicalbehaviors(Gunasekaran&Mehmet,2000)evenifsomeofthemcouldbedescribedwithacertainapproximationonthebaseofsimplifiedmodelssuchastheidealsolid(elastic),theidealliquid(viscous),theidealplastic(elasticpropertiesbelowaspecificappliedstressandviscousbehaviorabovethisvalue)andcombinationoftwoormoreofthem.Mostfoodproductsareneitherpureliquidsnorpuresolids,simultaneouslyshowingviscousandelasticbehaviors.Forinstance,whenaforceisappliedtoaviscoelasticfoodmatrixitneedsacertaintimetoacquireitsnewdimensionsand,whentheforceisremoved,thematerialcouldremainpermanentlydeformed.
Toquantitativelydeterminetheviscoelasticbehaviorofamatrix,transient(quasi-static)anddynamictestscanbeperformed.Themoretypicaltransienttestsarerepresentedbycreepandstressrelaxationexperiments.Increeptests,aconstantstressisappliedtoaspecimenandthecorrespondingstrainismeasuredasafunctionoftime.Theconsideredparametersarethe“compliance”(thatisthestrain–stressratio)andthe“relaxationtime”.Instressrelaxationtests,aconstantstrainisappliedandthestressrequiredtomaintainthedeformationismeasuredasafunctionoftime.Whenastressrelaxationtestisperformed,differentbehaviorscanbeobserved:
idealelasticmaterialsdonotrelaxwhereasidealviscousmaterialsinstantaneouslyshowarelaxation.Viscoelasticsolidsgraduallyrelaxandreachanequilibriumstressgreaterthan0,whereasforviscoelasticfluids,instead,theresidualstressvanishestozero(Steffe,1992).Thedynamicexperimentsdifferfromtransienttestsbecauseasinusoidalstressorstrainisappliedinsteadofaconstantstressorstrain.
Inordertodescribetheviscoelasticbehaviorofamaterial,masslessmechanicalmodelsareused.Thesemodelsarecomposedofsprings(consideredidealsolids,theyaccountfortheelasticbehaviorofviscoelasticmaterials)anddashpots(representingidealfluids,theyaccountfortheviscousbehavior)combinedinmanydifferentways.Threearethemostcommonlymechanicalanalogsused:
theKelvin–Voigtmodel,theMaxwellmodel([BrunoandMoresi,2004] and [CorreiaandMittal,2002]),andthestandardlinearsolidone.
TheKelvin–Voigtmodel,composedofaspringandadashpotinparallel,representsthestartpointforthedevelopmentofmechanicalanalogsdescribingthecreepbehavior.Infact,theKelvinmodelisnotsufficienttodescribecreepsinbiologicalmaterialsthatarebettermodeledbytheBurgersmodel(aKelvinandaMaxwellelementinseries)(Steffe,1992).AmodelconsistingofoneMaxwellelementinserieswithtwoKelvin–Voightelementsisabletodescribealiquid-likeviscoelasticbehavior(Mitchell&Blanshard,1976).
Stressrelaxationdataareveryimportancesincetheysupplyinformationaboutphenomenainvolvingfoodproductssuchasfruitripening(Hassa,Alhamdan,&Elansari,2005),fruitfirmness(Blahovec,1996),stalingofcerealproducts(Limanond,Castell-Perez,&Moreira,2002),checkingphenomenon(Kim&Okos,1999).
TheMaxwellmodel,consistingofaHookeanspringandaNewtoniandashpotinseries(Mohsenin&Mittal,1977),issuitableforunderstandingstressrelaxationdata,butdoesnotconsidertheequilibriumstress.Forthisreason,theviscoelasticbehavioroffoodcanbebetterdescribedbyusingageneralizedMaxwellmodelconsistingofseveralelementsinparallelwithaspring(Steffe,1992).Inasimilarmodel,ifthesystemissubjectedtoaconstantstrain,thetotalstressisthesumofthestressofeachelement.Sinceeachelementmayhaveadifferentrelaxationtime,arelaxationspectracanbeobtainedforaviscoelasticmaterial.ModelscontainingmoreexponentialcomponentsandaresidualtermshavebeenusedtodescribethestressrelaxationbehaviorofCheddarcheese(Hort,1997)and“pastafilata”cheese(Masi,1989).Mancini,Moresi,andRancini(1999)wereabletodescribetheviscoelasticbehaviorofseveralalginategelsdifferingfortheeffectivealginateconcentrationbymeansofageneralizedMaxwellmodelconsistingoffiveelements.AmodelincludingonespringandtwoMaxwellelementswassuccessfullyusetodescribethestressrelaxationoflipidssuchasbeeswax,candelillawax,carnaubawaxandahigh-meltingmilkfatfraction(Shellhammer,Rumsey,&Krochta,1997).Limanondetal.(2002)modeledthekineticsofcorntortillastalingthroughstressrelaxationdataandfoundthataseven-elementgeneralizedMaxwellmodelfitsthedatabetterthanthethreeandfiveones.TheviscoelasticpropertiesofsomeporkhammuscleswerebetterfittedtoageneralizedMaxwellmodelconsistingofaparallelcouplingofaHookesbodyandtwoMaxwell’sbodies(Lachowicz,Sobczak,Gajowiecki,&Zych,2003).ThecompressionstressrelaxationdataobtainedforpotatoesexposedtopulsifiedelectricfieldweremodeledwithafiveparametergeneralizedMaxwellmodel(Finca&Dejmek,2003).Hassaetal.(2005)comparedthreepopularstressrelaxationmodels,namelythegeneralizedMaxwell,Nussinovitch,andPelegmodel,inordertodescribetheviscoelasticpropertiesofeightdatecultivarsattheirkhalal(balah)andrutabstagesofmaturity.TheyfoundthatallconsideredmodelswerevalidforquantifyingtherelaxationbehavioroftheproductsbutthegeneralizedMaxwellmodelwasthebestinpredictingexperimentaldata.
Thestandardlinearsolidmodel,alsocalledZenermodel,canconsistintwodifferentmechanicalanalogs:
aspringinserieswithaKelvinmodeloraspringinparallelwithaMaxwellmodel.
Inadditiontotheusedmodel,theequilibriummodulus,thedecaymodulus,timeofrelaxation,andspecificviscosityintherelaxationmodelareaffectedbythespecimenorientationanditslocationwiththeproduct(Wang,2003).
InthisstudythepossibilitytousethegeneralizedMaxwellmodelfordescribingthestressrelaxationbehavioroffoodmatricesisaddressed.Fivedifferentfoodmatriceswereselectedwiththecriteriatoberepresentativeofawideclassoffoods.StressrelaxationtestswererunatroomtemperatureandtheposedmodelwasfittotheexperimentaldatatoestablishifthegeneralizedMaxwellmodelcanbeusedtodescribetheviscoelasticbehavioroffoods.
2.Materialsandmethods
Fivedifferentmatriceswereselectedasrepresentativeofawideclassoffoodproducts:
agargelandfourcommerciallyavailablefoodssuchas“mozzarella”cheese,meat,ripenedcheeseandwhitepanbread.Themeancompositionforeachoftheconsideredproducts(Carnovale&Marletta,1997)isreportedbelow:
“mozzarellacheese”—water58.8%,proteins18.7%,lipids19.5%,sugars0.7%,mineralsaltsandvitamins;
meat(beef)—76.9%water,proteins20.7%,lipids1.0%,mineralsalts;
ripenedcheese—water31.5%,proteins33.9%,lipids28.5%,sugars3.7%,mineralsaltsandvitamins;
whitepanbread—water33.5%,proteins9%,lipids8.7%,sugars48.2%,mineralsaltsandvitamins.
Thepreparationofthefoodmatrices,performedimmediatelybeforedynamic-mechanicalanalyses,consistedincutting13 mm-diameterspecimensbymeansofacylindricalmould.Theexternalportionofeachcylindricalsamplewascoveredwithsomegreasetoavoiddehydrationduringthedynamic-mechanicalanalysis.
Agargelwaspreparedbydissolvingagar(Oxoid,Milan,Italy)indistilledwater(finalconcentration1.0%w/v).Theobtainedsolutionwasthenheattreatedat121 °Cfor15 min.Cylindricalspecimens(11 mm-diameter)wereobtainedbyinjectioninamouldandcoolingatroomtemperature.Beforeanalyses,thesespecimenwererefrigeratedat4 °Cfor2 h.Gelsrepresentgoodsystemsforbasicrheologicalstudiessincetheyareusuallyisotropicandhomogeneous.
2.1.Elasticmodulusdetermination
Theinvestigatedfoodmatricesweresubmittedtocompressiontestsbymeansofadynamicmechanicalanalyser(DMA-Q800,TAInstruments,NewCastle,DE,USA)equippedwithp
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