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Mechanics of Advanced Materials and Structures

Formerly Mechanics of Composite Materials and StructuresPublished By: Taylor & FrancisVolume Number: 16Frequency: 8 issues per yearPrint ISSN: 1537-6494Online ISSN: 1537-6532Subscribe Online |Free Sample Copy | Table of Contents Alerting |View Full PricingDetails

Aims & Scope

2008 Impact Factor: 0.857Ranking: 73/112 in Mechanics122/191 in Materials Science, Multidisciplinary; 8/21 in Materials Science,Composites; 9/28 in Materials Science, Characterization & Testing2008 5-Year Impact Factor: 1.167 2009 Thomson Reuters, 2008Journal Citation Reports

The central aim ofMechanics of Advanced Materials and Structures is to promotethe dissemination of significant developments and publish state-of-the-art reviews andtechnical discussions of previously published papers dealing with mechanics aspectsof advanced materials and structures. Refereed contributions describing analytical,numerical and experimental methods and hybrid approaches that combine theoreticaland experimental techniques in the study of advanced materials and structures will be

published along with critical surveys of the literature and discussions of papers in thefield. Contributions will range from new theories and formulations to analyses andnovel applications. Emphasis will be placed on mechanics aspects and aspects at theinterface of materials and mechanics issues. The journal will publish manuscripts

dealing with the mechanics aspects (for example, the mechanical characterization,mathematical modeling, novel applications, and numerical simulation) of advanced
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materials and structures. Contributions may range from new methods to novelapplications of existing methods to gain understanding of the material and/orstructural behavior of new and advanced. Typical topic areas are:

Materials: Adhesives, ceramics, metal-matrix composites, and polymer-basedcomposites; processing and manufacturing of composite; actuator/sensor (smart)

materials and electromagnetic materials; and damage and failure mechanisms inmaterial.

Structures: Basic structural elements such as beams, plates, and shells; structureswith actuators/sensors (smart structures); active and passive control of structures;aerospace, automotive, and underwater structures; and adhesively bondedstructures.

Methodologies: Mathematical formulation of the kinematic, constitutive, andstructural behavior of materials and structures; experimental methods directed towardmechanical characterization, damage evolution, and failures in materials andstructures; computational methods for the solution of micro-, meso-, macro-mechanics mathematical models; methods dealing with the determination of localeffects; and novel computational approaches for material and structural modeling ofnew and advanced materials.

Publication office: Taylor & Francis, Inc., 325 Chestnut Street, Suite 800,Philadelphia, PA 19106

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Mecanica materialelor avansate si Structuri

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Factor de impact 2008: 0.857Ranking: 73/112, n Mecanica122/191, n tiina Materialelor, multidisciplinare; 8 / 21 n tiina Materialelor,

compozite; 9 / 28, n tiina Materialelor, de caracterizare i TestareAnul 2008 5-Impact Factor: 1.167

2009 Thomson Reuters, 2008 Oficial Referirea Reports Obiectivul central alMecanica materialelor avansate si Structurieste de a promovadifuzarea de evoluii semnificative i public state-of-the-art comentarii i discuiitehnice ale documentelor publicate anterior n care se ocup de aspectele legate demecanic a materialelor avansate i structuri. Contribuii arbitrat descrie metodeanalitice, numerice i experimentale i abordri hibrid care combina tehnici deteoretice si experimentale in studiul materialelor avansate i structuri vor fi publicate,mpreun cu anchetele critic a literaturii i discuii de lucrri n domeniu.Contribuiile vor varia de la noi teorii i formulri la analize i aplicaii noi. Accentulva fi pus pe aspectele mecanica si aspecte la interfaa de materiale i probleme demecanica. Revista va publica manuscrise care se ocup de aspectele mecanica (de

exemplu, caracterizarea mecanic, modelare matematic, aplicaii noi, i simularenumeric) din materiale avansate i structuri. Contribuii poate varia de la noi metode

pentru a noile aplicaii ale metodelor existente pentru a obine nelegerea materialei / sau de comportament structural de noi i avansate. Zonele de subiect tipice sunt:

Materiale: Adezivi, ceramica, metal-compozite cu matrice, i-compozite pe baz depolimeri; prelucrarea i fabricarea de compozite; acionare / senzor (Smart) materialei materiale electromagnetice; i prejudiciu i a mecanismelor de eec n materialul.

Structuri: elementele structurale de baz, cum ar fi grinzi, plci, i coji; structuri cuacionare / senzori (structuri inteligente); de control active i pasive de structuri;aerospatiale, automobile, precum i a structurilor subacvatice; i a structurilor deadhesively aglomerat.

Metodologii: formularea matematic a comportamentului cinematic, constitutive, i

structural a materialelor i structurilor; metodelor experimentale ndreptat sprecaracterizarea mecanice, daune evoluie, i eecurile n materialelor i structurilor;metodelor computaionale pentru soluia de micro-, mezo-, mecanica macro - modelematematice; metode de care se ocup de determinare a efectelor locale, precum inoi abordri computaionale pentru materiale i modelarea structurii de noi materialei avansate.


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Traducere COSSRV

Metodelor experimentale deSTUDY OF COSSERAT ELASTIC SOLIDS AND OTHER STUDIU DE ELASTICSOLIDE COSSERAT I ALTEGENERALIZED ELASTIC CONTINUA Generalizate ELASTIC CONTINUAin nContinuum models for materials with micro-structure ,Modelele Continuum pentrumateriale cu structura de micro-,ed. ed. H. Mhlhaus, J. Wiley, NY Ch. H. Mhlhaus, J. Wiley, NY Cap. 1, p. 1, p. 1-

22, (1995). 1-22, (1995).Roderic LakesLacuri RodericDepartment of Engineering Physics Departamentul de Inginerie FizicEngineering Mechanics Program Programul de Inginerie MecanicaUniversity of Wisconsin-Madison Universitatea din Wisconsin-Madison147 Engineering Research Building 147 Inginerie de cercetare de constructii1500 Engineering Drive, Madison, WI 53706-1687 1500 Inginerie Drive, Madison,WI 53706-1687AbstractAbstractThe behavior of solids can be represented by a variety of continuum theories.Comportamentul de solide pot fi reprezentate de o varietate de teorii continuu. Forexample, De exemplu,Cosserat elasticity allows the points in the continuum to rotate as well as translate,and the Elasticitate Cosserat permite puncte n continuumul s se roteasc, precum itraducerea, precum icontinuum supports couple per unit area as well as force per unit area. continuumsprijin tnr pe unitatea de suprafa, precum i fora pe unitatea de suprafa.We examine experimental methods for determining the six Cosserat elastic constantsof an Noi examina metodelor experimentale de determinare a ase constanteleCosserat elastice a unuiisotropic elastic solid, or the six Cosserat relaxation functions of a Cosserat

viscoelastic solid. izotropa elastica solide, sau cele ase funcii de relaxare Cosserat aunui Cosserat solide viscoelastic. We Noialso consider other generalized continuum theories (including micromorphicelasticity, Cowin's , de asemenea, n considerare alte teorii generalizate continuum(inclusiv elasticitate micromorphic, Cowin'svoid theory, and nonlocal elasticity). Teoria nula, si elasticitatea nonlocal). Ways ofexperimentally discriminating among various Modalitati de a experimentaldiscriminatorii ntre diferitelegeneralized continuum representations are presented. reprezentari continuumgeneralizat sunt prezentate. The applicability of Cosserat elasticity to AplicabilitateaCosserat elasticitate la


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cellular solids and fibrous composite materials is considered as is the application ofrelated solide celulare i fibroase materiale compozite este considerat ca este aplicarealegate degeneralized continuum theories. teorii continuum generalizat.IEu

IntroductionPrezentareThe classical theory of elasticity is presently used in engineering analyses ofdeformable Teoria clasic de elasticitate este n prezent utilizat n analizele deinginerie a deformabilobjects at small strain. obiecte la tulpina de mici dimensiuni. However there are othercontinuum theories for linear isotropic materials. Cu toate acestea, exist teoriicontinuum alte materiale de linie izotrop.Some have more freedom, and some have less freedom than classical elasticity. Uniis-au mai mult libertate, iar unele au libertatea de mai puin de elasticitate clasice. Thevarious Diferitecontinuum theories are all mathematically self consistent. teoriile continuum sunt

toate matematic de sine coerente. Therefore a discrimination among them is Prinurmare, o discriminare ntre ele esteto be made by experiment. care urmeaz s fie fcute de experiment.It is the purpose of this article to explore the physical consequences of variouscontinuum Acesta este scopul acestui articol pentru a explora consecinele fizice alecontinuum-diversetheories, and how these consequences may be used in the design of experiments todiscriminate teorii, i modul n care aceste consecine pot fi folosite n proiectarea deexperimente de a discriminaamong the theories. printre teorii. The constitutive equations for several theories are

presented, and some of the Ecuaiile constitutiv pentru teorii sunt prezentate maimulte, iar unele dintresalient consequences of each theory are stated and discussed. consecine importantedin fiecare teorie sunt indicate i discutate. Some of the causal physical Unele dintrefizice de cauzalitatemechanisms associated with each theory are briefly discussed. mecanisme asociate cufiecare teorie sunt discutate pe scurt. Experimental methods for Metodelorexperimentale deevaluating materials as generalized continua are presented, with emphasis on Cosseratelasticity. materii ca evaluarea continua generalizate sunt prezentate, cu accent peelasticitate Cosserat.

The treatment is restricted to linearly elastic behavior; study of Cosserat plasticity andrelated issues Tratamentul este limitat la un comportament liniar elastice; studiu alCosserat plasticitate i chestiuni legate deis presented elsewhere in this volume. este prezentat n alt parte n acest volum. Adiscussion of experimental aspects of generalized O discuie aspecte experimentale degeneralizatcontinua is considered particularly appropriate in view of the fact that most of thework done thus continua este considerat adecvat, mai ales avnd n vedere faptul cmajoritatea lucrrilor realizate astfelfar in generalized continuum mechanics has been theoretical. pn n prezent nmecanica continuumului generalizate a fost teoretice.

IIIIConstitutive EquationsEcuaiile constitutiv


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Uniconstant ElasticityElasticitatea UniconstantThe early uniconstant elasticity theory of Navier is based upon the assumption thatforces Teorie timpurie uniconstant elasticitatea Navier se bazeaz pe presupunerea cforele deact along the lines joining pairs of atoms and are proportional to changes in distance

between them actul de-a lungul liniilor care unete de perechi de atomi i suntproporionale cu schimbri n distanta dintre ele(Timoshenko, 1983). (Timoenko, 1983). The constitutive equation is as follows.Ecuaia constitutiv este dup cum urmeaz.

Page 2Page 2 is stress, is strain, and G is an elastic constant, the shear modulus. este de stres, este tulpina, i G este o constant elastic , modulul de forfecare. This uniconstanttheory wasAceast teorie a fost uniconstant used by Navier, Cauchy, Poisson, and Lam during the early days of the theory ofelasticity. utilizate de ctre Navier, Cauchy, Poisson, i Lame n primele zile ale

teoriei de elasticitate. Thetheory contains less freedom than the classical theory of elasticity now in commonuse. teorie a libertii conine mai puin de teoria clasic de elasticitate n prezent n uzcomun. There is Existno length scale in uniconstant elasticity. nici o scar, n lungime de elasticitateuniconstant.Classical ElasticityClassical ElasticitateaThe constitutive equation forclassicalisotropic elasticity (Sokolnikoff, 1983; Fung,Ecuaia constitutiv pentru elasticitatea clasice izotrop (Sokolnikoff, 1983; Fung,1968), is as follows, in which there are the two independent elastic constants and G,the Lam 1968), este dup cum urmeaz, n care exist dou constante independente

elastica i G, Lameconstants. constante. kl KL= = rr rr kl KL+ 2G + 2G kl KL

(2) (2)The Poisson's ratio = /2( + G) is restricted by energy considerations to havevalues in the The Poisson's ratio = / 2 ( + G) este limitat de considerente deenergie pentru a avea valori nrange from -1 to 1/2. intervalul -1 - 1 / 2. There is no length scale in classicalelasticity. Nu exist nici o scar, n lungime de elasticitate clasice.Cosserat (micropolar) ElasticityCosserat (micropolar) ElasticitateaThe Cosserat theory of elasticity (Cosserat, 1909) incorporates a local rotation of

points as Teoria Cosserat de elasticitate (Cosserat, 1909) include o rotaie locale depuncte de cawell as the translation assumed in classical elasticity; and a couple stress (a torque per

unit area) as precum i traducerea asumate n elasticitatea clasice, precum i un cuplude stres (un cuplu pe unitate de suprafa), astfel cum


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well as the force stress (force per unit area). precum i fora de stres (fora pe unitateade suprafa). The force stress is referred to simply as 'stress' in Stres vigoare estemenionat la stres, pur i simplu ca "" nclassical elasticity in which there is no other kind of stress. elasticitate clasice, n carenu exist nici un alt tip de stres. The idea of a couple stress can be Ideea unui stres

cuplu poate fitraced to Voigt (1887,1894) during the formative period of the theory of elasticity.urmrite pn la Voigt (1887,1894) n timpul perioadei de formare a teoriei deelasticitate. In more recent n mai recenteyears, theories incorporating couple stresses were developed using the full capabilitiesof modern de ani, care ncorporeaz teorii subliniaz tnr s-au dezvoltat utilizndcapacitile complet a modernecontinuum mechanics (Ericksen and Truesdell, 1958; Grioli, 1960; Aero andKuvshinskii, 1960; Mecanica continuum (Ericksen i Truesdell, 1958; Grioli, 1960;Aero i Kuvshinskii, 1960;Toupin, 1962; Mindlin and Tiersten, 1962; Mindlin, 1965; Eringen, 1868; Nowacki,

1970). Toupin, 1962; Mindlin i Tiersten, 1962; Mindlin, 1965; Eringen, 1868;Nowacki, 1970). A Unsurvey of the interrelation between generalized continuum analysis and materialdefects, trecere n revist a relaiei dintre analiza continuum generalizate i defecte demateriale,dislocations and other inhomogeneities was presented by Kunin (1982, 1983).dislocri i a altor inhomogeneities a fost prezentat de ctre Kunin (1982, 1983).Eringen (1968) Eringen (1968)incorporated micro-inertia and renamed Cosserat elasticity micropolarelasticity.ncorporat de micro-ineria i redenumite Cosserat elasticitate elasticitatemicropolar. Here we use the Aici vom folositerms Cosserat and micropolar interchangeably. Termeni Cosserat i micropolaralternativ. In the isotropic Cosserat solid there are six elastic n izotrop Cosseratsolide exist ase elasticconstants, in contrast to the classical elastic solid in which there are two, and theuniconstant constante, n contrast cu clasice elastice solid, n care exist dou, iuniconstantmaterial in which there is one. material n care exist unul. The constitutive equationsfor a linear isotropic Cosserat elastic solid Ecuaiile constitutiv pentru o Cosseratliniar elastice izotrope solidare, in the symbols of Eringen, (1968): sunt, n simbolurile de Eringen, (1968):

The usual summation convention for repeated indices is used throughout, as is thecomma Convenia de obicei de sumare pentru indicii repetate este utilizat tot, aa cumeste virgulaconvention representing differentiation with respect to the coordinates. Conveniade la reprezentnd o difereniere n ceea ce privete coordonatele. kl KLis the force stress, este stresul vigoare,which is a symmetric tensor in equations 1 and 2 but it is asymmetric in Eq. care esteun tensor simetric n ecuaii 1 i 2, dar este asimetric n ecuaia. 3. 3. m Suntkl KLis the couple Este tnr

stress, stres, kl KL


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= (u = (Uk,l K, L+ u + Ul,k l, k)/2 is the small strain, u ) / 2 este tulpina mic, u

k Kis the displacement, and e este de deplasare, i eklm klmis the permutation este permutareasymbol. simbol. The microrotation microrotationk Kin Cosserat elasticity is kinematically distinct from the macrorotation n elasticitateaCosserat este kinematically distinct de macrorotationr rk K= (e = (Eklm klmu um,l M, L)/2 obtained from the displacement gradient. ) / 2 obinute de la gradientul dedeplasare. Components of stress and couple stress Componente de stres i de cuplu destreson a differential element of a Cosserat solid, and the corresponding increments offorce and pe un element diferenial al unui Cosserat solide, precum i creterilecorespunztoare de for imoment on the structural elements of a real material are shown in Fig. moment cu

privire la elementele structurale ale unui material real, sunt prezentate n Fig. 1. 1.Page 3Page 3In three dimensions, the isotropic Cosserat elastic solid requires six elastic constants, , , , , n trei dimensiuni, a izotrop Cosserat elastica solide necesit aseconstante elastice , , , , ,and for its description. i pentru descrierea acestuia. A comparison of symbolsused by various authors was presented by O comparaie a simbolurilor folosite dectre autori a fost prezentat de ctre diverseCowin (1970a). Cowin (1970a). The following technical constants derived from thetensorial constants are more Urmtoarele constante tehnice derivate din constanteletensorial sunt mai

beneficial in terms of physical insight. benefice n termeni de cunoatere fizice. Theseare (Eringen, 1968; Gauthier and Jahsman, 1975): Acestea sunt (Eringen, 1968;Gauthier i Jahsman, 1975):Young's modulus E = (2+)(3+2+)/(2+2+) , E modulul lui Young =(2 + ) (3 2 + ) / (2 2 + ), shear modulus G = (2+) /2, modul de forfecare G = (2 + ) / 2,Poisson's ratio = /(2+2+) , Poisson's ratio = / (2 2 + ),characteristic length for torsion l Lungimea caracteristic pentru L torsiunet T= [ (+)/(2+) ] = [( + ) / (2 + )]1/2 1 / 2, ,


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characteristic length for bending l Lungimea caracteristic pentru ndoire Lb b= [ /2(2+) ] = [ / 2 (2 + )]1/2 1 / 2, ,

coupling number N = [ /2(+) ] de cuplare numrul N = [ / 2 ( + )]1/2 1 / 2, and , i de

polar ratio = (+)/(++) . Polar raportul = ( + ) / ( + + ). When , , , vanish, the solid becomes classically elastic. Cnd , , ,disprea, solide devine clasic elastica. The case N = 1 (its upper bound) is Caz, n = 1(ITS limita de sus) este deknown as 'couple stress theory' (Mindlin and Tiersten, 1962; Cowin, 1970b).cunoscut sub numele de "teoria cuplu de stres" (Mindlin i Tiersten, 1962; Cowin,1970 b). This corresponds to Acest lucru corespunde cu

, a situation which is permitted by energetic considerations, as isincompressibility in , o situa ie care este permis de considerente energetice, a a cum este n incompressibility classical elasticity. elasticitate clasice. The case = 0 corresponds to a decoupling ofthe rotational and translational Cazul = 0 corespunde la o decuplare a de rotaie i atranslaionaldegrees of freedom. grade de libertate. Although some theoretical writers choose tosolve problems for this case since Dei unii scriitori teoretice alege pentru a rezolva

problemele de acest caz, deoarecethe analysis is simpler, the limit 0 presents physical difficulties (Lakes, 1985a).

analiza este mai simpl, limita de 0 prezint dificulti fizice (Lacuri, 1985a).Void ElasticityVoid ElasticitateThe theory of elastic materials with voids (Cowin, 1983) incorporates a change ofvolume Teoria de materiale elastice cu goluri (Cowin, 1983) include o schimbare devolumfraction, rather than rotation, as an additional kinematical variable. fraciune, maidegrab dect de rotaie, ca o variabil suplimentar cinematic. The constitutiveequations for Ecuatiile constitutiv pentruthe elastic case (no rate dependence) are as follows. n cazul elastica (nu rata dedependen) sunt dup cum urmeaz.with as the stress, h as the equilibrated stress vector, and as the classical Lam

elastic cu ca stresul, H ca vector de stres echilibrat, i ca elastica clasice Lameconstants, g as the intrinsic equilibrated body force, as the change of volumefraction, and constante, g ca for intrinsec organism echilibrat, ca schimbareafracie de volum, i ,k , kas la fel dethe gradient of the change of volume fraction. gradientul de schimbare a fracie devolum. The change of volume fraction can be interpreted as Schimbare de fracie devolum poate fi interpretat caa dilatation ofpoints in the continuum. o dilatare depuncte n continuu.

Nonlocal ElasticityNonlocal Elasticitate


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In an isotropic nonlocal solid, the points can only undergo translational motion as inthe ntr-un izotrop nonlocal solide, punctele poate s fac obiectul numai micarea detranslaie la fel ca nclassical case, but the stress at a point depends on the strain in a region near that point(Krner, caz clasic, dar stresul ntr-un punct depinde de tulpina ntr-o regiune aproape

c punctul (coroane,1967, Eringen, 1972). 1967, Eringen, 1972). The constitutive equation for stress Ecuaia constitutiv de stresij ijis, in terms of the position vectorx este, n termeni de vectorul de poziie xof points in the solid, de puncte in solide, (| x |) subject to cu kernel-ului nonlocal (| x |), sub rezerva

Page 4Page 4 V V

(| x |)dV = 1 (| x |) dv = 1(10) (10)requiring the kernel to be a member of a Dirac delta sequence. care necesit kernel-uls fie un membru dintr-o secven delta Dirac. So, in the limit of the nonlocal Deci, nlimita a nonlocaldistance of influence or characteristic length 'a' becoming vanishingly small, Hooke'slaw (Eq. 2 la distan de influen sau de lungime caracteristic "a" a devenivanishingly mici, legea lui Hooke (Eq. 2for classical elasticity) is recovered. pentru elasticitatea clasic) este recuperat.An example of a finite range kernel is (|x|) = Un exemplu de un nucleu gam finiteste (| x |) =Characteristic lengths can be defined in nonlocal elasticity, in terms of the effectiverange Lungimi de caracteristica poate fi definit n elasticitatea nonlocal, n ceea ce

privete gama eficienteassociated with a kernel. asociat cu un nucleu.Microstructure (micromorphic) ElasticityMicrostructura (micromorphic)ElasticitateaIn microstructure (Mindlin, 1965) or micromorphic (Eringen, 1968) elasticity the

points in n microstructura (Mindlin, 1965) sau micromorphic (Eringen, 1968)elasticitateapuncte nthe continuum representation of the solid can deform microscopically as well as

translate and reprezentarea continuum de solide pot deforma la microscop, precum ide a traducerotate. roti. There are 18 elastic constants in the isotropic case. Sunt 18 constanteelastice, n cazul izotrop. The constitutive equations for isotropic Ecuaiileconstitutiv pentru izotropmicrostructure elasticity are: elasticitate microstructur sunt:

pq pqis the symmetric Cauchy stress, este de stres simetric Cauchy,

pq pqis the asymmetric relative stress, and este de stres asimetrice relativ, i

pqr pqris the se


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double stress. stres dublu. The g's, b's, and a's as well as and are elastic constants.G's, b's, precum i o, precum i i constantelor sunt elastic. The antisymmetric Theantisymmetric

part (with respect to the last two indices) of the double stress represents the couplestress of parte (n ceea ce privete ultimele dou indici) din stres dublu reprezint

stresul pereche deCosserat elasticity. is strain, is the macro-deformation minus the micro-deformation, and is Elasticitate Cosserat. este tulpina, este macro-deformareminus de micro-deformare, i estethe micro-gradient of micro-deformation, the antisymmetric part of whichcorresponds to the de micro-gradientul de micro-deformare, o parte din antisymmetriccare corespunde curotation gradient in Cosserat elasticity. gradientul de rotaie n elasticitate Cosserat.Microstructure elasticity includes Cosserat elasticity and the theory of voids as specialElasticitate Microstructura include Cosserat elasticitate i teoria golurilor ca speciale

cases. de cazuri. Classical elasticity is a special case of Cosserat elasticity and of voidtheory. Elasticitatea clasic este un caz special de Cosserat elasticitate i a teorieineavenit. Uniconstant Uniconstantelasticity is a special case of classical elasticity. elasticitatea este un caz special deelasticitate clasice.Viscoelastic materialsMateriale de viscoelasticTime dependence or frequency dependence can be incorporated in any of the aboveOra dependen sau dependena de frecven pot fi ncorporate n oricare din cele demai susconstitutive equations by use of the correspondence principle of linear viscoelasticity.ecuaii constitutiv prin folosirea principiului corespondenei de viscoelasticity liniare.

For Pentruclassically viscoelastic materials, the transition is well known. clasic materialevascoelastice, tranziia este bine cunoscut. Each elastic constant becomes a Fiecareconstanta elastica devine ocomplex number in the viscoelastic case in the frequency domain. numr complex, ncazul viscoelastic n domeniul de frecven. In the time domain, the n domeniul timp,constitutive equations assume a convolution form. ecuatii constitutiv i asume o formcircumvoluie. Eringen (1967) has developed a viscoelastic Eringen (1967) adezvoltat o viscoelasticversion of the micropolar theory. versiune a teoriei micropolar. The theory of voids asoriginally presented has a simple time Teoria golurilor ca iniial a prezentat are untimp de simpludependence built in. dependena de construit inIIIIIIConsequences of constitutive equationsConsecine de ecuatii constitutivThe uniconstanttheory predicts a Poisson's ratio of 1/4 forallmaterials. Teoriauniconstantprezice un raport de 1 Poisson / 4 pentru toate materialele. Since mostDeoarece cele mai multecommon isotropic materials exhibit a Poisson's ratio close to 1/3, the uniconstanttheory was comun materiale izotrope manifesta un Poisson aproape la valoarea de 1 /3, teoria uniconstant a fost


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rejected based on experimental measurements of Poisson's ratio. a respins bazate pemsurtori experimentale ale raportului lui Poisson. Decisive experiments wereExperimentele au fost decisive

Page 5Page 5difficult to perform in the late 1800's, so the issue was not decided until well after the

introduction dificil de efectuat la sfarsitul anilor 1800, astfel nct problema nu a fostdecis pn la mult timp dup introducereaof the theories. de teorii.In several segments below we refer to size effects. n mai multe segmente de mai josne vom referi la efectele dimensiune. These size effects, and all other Aceste efectedimensiuni, precum i toate celelalte

phenomena presented in this article, are within the framework of linear (but possiblynonclassical) fenomene prezentate n acest articol, sunt n cadrul liniare (dar posibilnonclassical)elastic or viscoelastic behavior. comportament elastic sau viscoelastic. Size effects inthe current context refer to a non-classical Dimensiune efecte, n contextul actual serefer la un non-clasicdependence of the rigidity of an object upon one or more of its dimensions.dependen de rigiditate a unui obiect, la unul sau mai multe dintre dimensiunile sale.This type of size Acest tip de mrimeeffects are to be distinguished from size effects in the fracture behavior; fracture is anonlinear Efectele sunt s se fac distincie ntre dimensiunea efecte ncomportamentul rupere; fractur este o neliniare

process not considered here. proces nu luate n considerare aici.Classicalelasticity is, according to its name, the currently accepted theory ofelasticity. Elasticitatea clasic este, n funcie de numele acestuia, n prezent, teoria

acceptat de elasticitate.Several salient predictions are as follows. Mai multe predicii importante sunt dupcum urmeaz.(i) (i)The rigidity of circular cylindrical bars of diameter d in tension goes as d Rigiditateacircular bare cilindrice cu diametrul d tensiune n merge ca d2 2; in bending and ; n ndoire itorsion, the rigidity goes as d torsiune, rigiditatea merge ca d4 4. .

(ii) (ii)Plane waves in an unbounded medium propagate without dispersion (the wave speedis Valurile de avion ntr-un mediu fr limite, fr a se propaga dispersie (valul devitez esteindependent of frequency) for shear waves and dilatational waves. independente defrecven) pentru undele de forfecare i de valuri dilatational.(iii) (iii)There is no length scale in classical elasticity, hence stress concentration factors forholes or Nu exist nici o scar, n lungime de elasticitate clasice, prin urmare, factoriide stres de guri sau de concentrare pentruinclusions in an infinite domain under a uniform stress field depend only on the shape

of the incluziuni ntr-un domeniu infinit n cadrul unui cmp uniform de stres depindenumai de forma de


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inhomogeneity, not on its size. inhomogeneity, nu pe dimensiunea acesteia.(iv) (iv)Poisson's ratio can have values in the range -1 1/2. Poisson raportul poateavea valori n intervalul -1 1 / 2.Ordinary materials have a positive Poisson's ratio, so that the range from zero to 1/2

was Materiale de ordinar au un raport Poisson pozitiv, astfel ca intervalul de la zero la1 / 2 a fostconsidered for many years to be the physically acceptable range (Fung, 1968).considerat de muli ani s fie n intervalul punct de vedere fizic acceptabil (Fung,1968). Recently, a new Recent, un nouclass of cellular solids with a negative Poisson's ratio has been developed (Lakes,1987), clas de solide celulare, cu un raport de Poisson negativ a fost dezvoltat(Lacuri, 1987),extending the range for to -0.7 and below. extinderea gamei de la -0.7 i de mai

jos. Further developments in negative Poisson's ratio Evoluia viitoare a raportului

Poisson negativ'smaterials are reviewed by Lakes (1993a). materiale sunt revizuite de ctre Lacuri(1993a).Cosseratormicropolarelasticity has the following consequences. Cosseratsau deelasticitate micropolarare urmtoarele consecine.(i) (i)A size-effect is predicted in the torsion of circular cylinders of Cosserat elasticmaterials. O mrime-efect este anunat n torsiune de butelii circulare de materialeCosserat elastica.Slender cylinders appear more stiff then expected classically (Gauthier and Jahsman,1975); Fig. Cilindri Slender aprea mai rigid apoi de ateptat clasic (Gauthier i

Jahsman, 1975); Fig.2. 2. A similar size effect is also predicted in the bending of plates (Gauthier andJahsman, 1975) and Un efect similar este, de asemenea dimensiune a prezis n ndoirede plci (Gauthier i Jahsman, 1975) iof beams (Krishna Reddy and Venkatasubramanian, 1978); Fig. de grinzi (KrishnaReddy i Venkatasubramanian, 1978); Fig. 2. 2. No size effect is predicted in Nici unefect dimensiune este prevzut ntr -tension. tensiune.(ii) (ii)The stress concentration factor for a circular hole, is smaller than the classical value,and Factor de stres de concentrare pentru un orificiu circular, este mai mic dect

valoarea clasice, ismall holes exhibit less stress concentration than larger ones (Mindlin, 1963). guriexponat de concentrare stres mai mici dect cele mai mari (Mindlin, 1963). Stressconcentration Concentraia Stressaround a rigid inclusion in an flexible medium is greater in a Cosserat solid than in aclassical solid. n jurul unui includerea rigid, ntr-un mediu flexibil este mai mare ntr-un Cosserat solid dect ntr-un solid clasic.Stress concentration near cracks and elliptic holes is reduced in comparison toclassical predictions Stresul de concentrare lng fisuri si gauri eliptice este redus ncomparaie cu predicii clasice(Kim and Eringen, 1973; Itou, 1973; Sternberg and Muki, 1967; Ejike, 1969;

Nakamura et al, (Kim i Eringen, 1973; Itou, 1973; Sternberg i Muki, 1967; Ejike,1969; Nakamura et al,


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1984). 1984).(iii) (iii)Dilatational waves propagate non-dispersively, ie with velocity independent offrequency, Propagarea undelor Dilatational non-dispersively, adic cu vitezaindependent de frecven,

in an unbounded isotropic Cosserat elastic medium as in the classical case. ntr-unmediu fr limite izotrop Cosserat elastice ca i n cazul clasic. Shear waves propagateValuri Shear propagadispersively in a Cosserat solid (Eringen, 1968). dispersively ntr-o Cosserat solide(Eringen, 1968). A new kind of wave associated with the micro- Un nou tip de valasociate cu micro -rotation is predicted to occur in Cosserat solids. rotaie se anticipeaz s apar lasolide Cosserat.(iv) The mode structure of vibrating Cosserat bodies is modified from that of classicalelastic (iv) modul de structura vibrator organismelor Cosserat este modificat de cea aelastice clasice

bodies (Mindlin and Tiersten, 1975). (organisme de Mindlin i Tiersten, 1975).(v) (v)The range in Poisson's ratio is from -1 to +0.5, the same as in the classical case(Gauthier Gama n proporie Poisson este de -1 - 0.5, la fel ca i n cazul clasice(Gauthierand Jahsman, 1975). i Jahsman, 1975).(vi) (vi)In Cosserat solids which lack a center of symmetry, called noncentrosymmetric orchiral n solide Cosserat care nu sunt deloc un centru de simetrie, numitnoncentrosymmetric sau chiralmaterials, qualitatively new phenomena are predicted. materiale, calitativ fenomenenoi sunt prezis. A rod under tensile load deforms in torsion O bar n sarcin detraciune se deformeaz n torsiune(Lakes and Benedict, 1982). (Lacuri i Benedict, 1982). Wave speed for transversecircularly polarized waves depends on the Wave de vitez pentru transversal circular

polarizat valuri depinde desense of polarization. sentiment de polarizare. This leads to rotation of the principal

plane of elliptically polarized transverse Acest lucru duce la rotaie a planuluiprincipal al elliptically polarizat transversalwaves (Lakhtakia, Varadan, and Varadan, 1988; 1990). valuri (Lakhtakia, Varadan, iVaradan, 1988; 1990). Examples of chiral materials include Exemple de astfel de

materiale chiral includcrystalline materials such as sugar which are chiral on an atomic scale, as well ascomposites with Materiale de cristalin, cum ar fi de zahr, care sunt chiral la o scaratomic, precum i a materialelor compozite cuhelical inclusions or spiraling fibers. incluziuni de elicoidale sau fibre de spirala.Chirality has no mechanical effect in classical elasticity. Chiralitate nu are nici unefect mecanice n elasticitate clasice.

Page 6Page 6Voidtheory gives rise to the following. Teoria Voidd natere la urmtoarele.(i) (i)Size effects are predicted by Cowin and Nunziato (1983) in the bending of rods but

not in Dimensiune efecte sunt prezise de Cowin i Nunziato (1983), n indoire de tije,dar nu n


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torsion or tension. torsiune sau de tensiune.(ii) (ii)The stress concentration factor for a hole in a planar region under tension is greaterthan the Factor de stres de concentrare pentru o gaur ntr-o regiune planar subtensiune este mai mare dect

classical value (Cowin, 1984a). Valoarea clasice (Cowin, 1984a).(iii) (iii)Dilatational waves in an unbounded medium propagate dispersively (Puri and Cowin,Valurile Dilatational propaga ntr-un mediu fr limite dispersively (Puri i Cowin,1985). 1985). There are two kinds of such waves. Exist dou tipuri de astfel devaluri. Shear waves exhibit no dispersion. Valurile de forfecare prezint nici odispersie.

Microstructure (micromorphic) elasticity gives rise to the following.Microstructura(micromorphic) Elasticitatea d natere la urmtoarele.(i) (i)Dispersion of both dilatational waves and shear waves occurs in solids obeying Att

de dispersie a undelor dilatational i valuri de forfecare apare n ascultarea de solidemicrostructure elasticity (Mindlin, 1965). elasticitate microstructur (Mindlin, 1965).Cut-off frequencies for acoustic waves are predicted. Cut-off spectrului de frecvene

pentru undele acustice sunt prezis.(ii) (ii)The stress concentration factor for a spherical cavity can be greater than the classicalvalue Factorul de stres de concentrare pentru o cavitate sferic poate fi mai mare dectvaloarea clasice(Bleustein, 1966) (Bleustein, 1966)

Nonlocalelasticity gives rise to the following. ElasticitateNonlocald natere laurmtoarele.(i) (i)The stress concentration near a crack is alleviated (Eringen, et al, 1977). Concentraiade stres lng un crack este mbuntit (Eringen, et al, 1977).(ii) (ii)Dispersion of elastic waves is predicted (Eringen, 1972). Dispersia de unde elastice seanticipeaz (Eringen, 1972).(iii) (iii)Size effects are predicted in tension and bending, as described below. EfecteleDimensiune se prognozeaz n tensiune i de ndoire, aa cum este descris mai jos.For certain short Pentru anumite scurt

range nonlocal behaviors, these size effects can be thought of as surface or 'skin'effects. comportamente gama nonlocal, aceste efecte dimensiuni pot fi gandit casuprafa sau "efecte de piele".IVIvFurther consequences of constitutive equationsConsecine n continuare aecuatiilor constitutivIn this section we present several new results which are relevant to experimentalmethods. n aceast seciune vom prezenta mai multe rezultate noi care sunt relevante

pentru a metodelor experimentale.A slab is considered in tension and in bending for a Cosserat solid and for a nonlocalsolid. O lespede este considerat n tensiune i n indoire pentru o Cosserat solide i

pentru un solid nonlocal.Cosserat solidCosserat solide


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Consider bending of an isotropic Cosserat elastic beam of rectangular section andwidth B Luai n considerare de ndoire a unui fascicul de izotrope Cosserat elasticede seciune dreptunghiular i limea Band depth A, with the following displacement field u and microrotation field . i oadncime, cu u urmtoarele domeniul de deplasare i de microrotation teren. R is theradius of R este raza decurvature of the bent beam. curbur a fasciculului aplecat.{ u (Ux x= -(1/2R)[z = - (1/2R) [z2 2+ (x + (x2 2-y -y2 2

)], u )], Uy y= - xy/R, u = - XY / R, uz z= xz/R} = Xz / R)(17) (17){( x x= 0 = 0 y y= - z/R= - Z / Rz z= - y/R} = - Y / R)We use the semi-inverse method. Noi folosim metoda de semi-invers. The procedureis the same for Cosserat solids as it is for classical Procedura este aceeai pentrusolide Cosserat cum este pentru clasicsolids. solide. The displacement field is assumed to be the same as for the classicalelastic case, and the Cmpul de deplasare se presupune a fi aceleai ca i pentru cazulclasic elastice, precum imicrorotation field is assumed to be the same as the classical macrorotation. cmpmicrorotation se presupune a fi la fel ca macrorotation clasice. Again, size effects in

Din nou, efectele mrime nthe rigidity occur. rigiditate apar. The bending rigidity ratio (ratio of rigidity of aCosserat beam to that of its Raportul de ndoire rigiditate (raportul de rigiditate a unuifascicul de Cosserat cu cel al suclassical counterpart) for a particular rectangular cross section bar of sides A and B isfound to be: n contrapartid clasice), pentru un bar special dreptunghiular seciuneatransversal a pri A i B se dovedete a fi: = RM/E(BA = RM / E (BA3 3/12) = [1+24(l / 12) = [1 24 (L

b b

/A) / A)2 2


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(1- )]. (1 - )].(18) (18)Size effects in the rigidity are predicted, in which slender bars (with A lDimensiune efecte n rigiditatea se prognozeaz, n care baruri zvelte (cu o L

b b) are stiffer than ) Sunt mai acerb dectexpected classically. de ateptat clasic. The displacement and rotation fields give riseto an exact solution only if/ = De deplasare i cmpurile de rotaie da natere la osoluie exact numai n cazul n / = .- . If/ , then they are valid if a system of couple stresses were applied tothe lateraln cazul n care / - , atunci acestea sunt valabile n cazul n care un sistem de subliniaz cuplu au fost aplicate pentru a laterale surface. de suprafa. The exact solution for the general case is not known. Soluiaexact pentru cazul general, nu este cunoscut. If these stresses are not applied, theDac aceste subliniaz nu sunt aplicate,

displacement field will be different: the inclined lateral surfaces will exhibit somebulge rather than cmp de deplasare va fi diferit: nclinat laterale suprafeele vaexpune unele umfla, mai degrab dect

being straight as they are in the classical case, or in the Cosserat case with / = .a fi dreapt n care acestea sunt, n cazul clasic, sau, n cazul Cosserat cu / =.Hence De aici nainteCosserat elasticity predicts a change ofshape of the cross section of the bent beam,for/ . Elasticitate Cosserat prezice o schimbare aformei de seciuneatransversal a fasciculului aplecat, pentru / . The bending rigidity will also be different from the above. Rigiditatea ndoire va fi, de

asemenea, diferite de cele de mai sus.The rigidity size effects in other situations such as bending of a plate or circular rod,or Efectele rigiditate mrime n alte situaii, cum ar fi ndoirea o plac de tij sau decirculare, sau detorsion of a circular rod, have a similar form. torsiune de o bar circular, au o formsimilar. For example, Gauthier and Jahsman (1975) give, for De exemplu, Gauthieri Jahsman (1975) da, pentru acylindrical bending of a plate, cilindrice ndoirea o farfurie, = 1 + 24 = 1 + 24l L

b b

2 2(1- ) (1 - )h ore2 2(19) (19)

Page 7Page 7with h as the plate thickness. , cu h ca grosimea plcii. The length parameter ofGauthier and Jahsman was converted into the Parametru lungime de Gauthier iJahsman a fost transformat ncharacteristic length for bending defined above. lungime caracteristic pentru ndoire

definite mai sus. In plate bending, the anticlastic curvature due to n placa de ndoire,curbura anticlastic din cauza


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the Poisson effect is constrained, in contrast to beam bending. efectul Poisson esteconstrns, n contrast cu fascicul de ndoire. A similar kind of solution for Un tipsimilar de soluie pentru

bending of a void solid was constructed by Cowin, 1984b. Prelucrare de un gol solida fost construit de Cowin, 1984b.

Nonlocal solidNonlocal solideChirita (1978) has examined Saint Venant's problem in nonlocal elastic solids; thisincludes Chirita (1978) a examinat problema Saint Venant n nonlocal solide elastice;aceasta includesimple tension, bending, and torsion of a slender bar. Tensiunea simplu, ndoire,

precum i de o bar de torsiune de subire. Chirita writes the constitutive equationmuch Chirita scrie ecuaia constitutiv de multas in Eqs. la fel ca n SCM. (8,9) but with x' called z', called K, and extra termsoutside the integral, (8,9), dar cu x "numit z", numit K, iar termenii suplimentare nafara integrant,corresponding to classical elasticity. corespunztoare elasticitate clasice. For bothtension and bending, the displacement field is Pentru ambele tensiune i de ndoire,domeniul de deplasare este

predicted to be identicalto the classical displacement field. anticipat a fi identic ndomeniul clasice de deplasare. So, nonlocal elasticity predicts no Deci, elasticitateanonlocal prezice nici o

shape changes in comparison with classical elasticity for bending. modificrileforman comparaie cu elasticitate clasice de indoire. Compare the bending of a Comparaiindoire a uneiCosserat elastic bar, considered above. Cosserat bar elastica, luate n considerare demai sus. Chirita gives integral forms for the rigidities but not explicit Chirita d forme

integrant pentru rigiditile, dar nu n mod explicitforms or interpretation. forme sau interpretare.In the following, consider the origin of coordinates to be the center line of the slab,which n cele ce urmeaz, ia n considerare originea de coordonate pentru a fi linia decentru al lespede, carehas width W and breadth V >> W. are limea W i limea V>> W.In simple tension , we have uniform strain so (away from a boundary), n tensiunesimplu, ne-am att de tulpina uniform (departe de o grani), (x) (X)= E = E

x-x'=-a x-x "=- una un (x - x') dx' (x - x ') DX "(20) (20) (x) = 1 (x) = 1E E , from Eq.10. , de la Eq.10.(21) (21)The nonlocal region of influence has dimensions a and is shown in Fig. Regiuneanonlocal de influen are dimensiuni a i este prezentat n fig. 4. 4. To first orderthere is Pentru a comanda primul nu este


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no size effect in tension, however to second order there will be a size effect due tointegration over nici un efect dimensiune n tensiune, cu toate acestea la comandasecunde, va exista un efect de mrime datorit integrrii pe

part of the domain as a result of interception of a portion of the nonlocal influence bythe o parte din domeniu, ca urmare a interceptrii o parte din influena nonlocal de

ctreboundaries. limite. We remark that this effect has been neglected in prior treatmentsof crack problems in Noi remarc faptul c acest efect a fost neglijat n tratamenteanterioare de probleme crack nnonlocal elasticity (see Eringen, 1983). elasticitate nonlocal (a se vedea Eringen,1983).Let us now consider the surface effect of interception of a portion of the kernel'sregion of Haidei, acum, ia n considerare cu efect de suprafa de interceptare a unei

pri a regiunii kernel-ului deinfluence. influen. In the one dimensional slab geometry, the problem is tractable. ngeometria o lespede dimensionale, problema este uor de mnuit.

(x) (X)= E = E x-x'=-a x-x "=- una un (x - x') dx', (x - x ') DX ",for pentru- W/2 + a < x < W/2 - a - W / 2 + a


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contributes to the stress. contribuie la stres. Suppose that the kernel is positive definitethroughout its range, as has S presupunem c kernel-ul este pozitiv definit n ntreagagama sa, aa cum a

been done in several analyses of stress around cracks. a fost fcut n mai multe analizede stres n jurul crpturi. Then there is a surface layer of depth a in Apoi, exist un

strat de suprafa de la o adncimewhich the stress is less than pe care stres este mai mic dectE E . . Such a surface effect has a negligible effect on the rigidity if a


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may be circular and in three dimensions it may be spherical. poate fi circular i ntrei dimensiuni, poate fi sferice. Computation of segments intercepted Calculul desegmente interceptate

by such boundaries would be more complex, however we expect that, as in Eq. culimite ar fi mult mai complex, cu toate acestea ne ateptm ca, la fel ca n ecuaia. 26,

the rigidity 26, rigiditateawould depend on some fraction of a/W. s-ar depinde de o fractiune de / W.In cylindrical bending of a plate, = gx and we consider one dimension only. ncilindrice ndoirea o farfurie, = GX i lum n considerare doar o singurdimensiune.We consider first x = 0. Noi considerm primul x = 0.Then, (0) Apoi, (0)= E = E x'=-a x "=- un

a un (x - x')gx' dx'. (x - x ') GX "DX".(28) (28) (0) (0)= E = E x'=-a x "=- una un ( - x')gx' dx'. (- x ') GX "DX".If is an even function, then the integral is zero.ncazul n care este o func ie, chiar, atunci integrala este zero. For example,Deexemplu,

(0) (0)= E = E x'=-a x "=- una un(x' (x "2 2)gx' dx' = [(x' ) DX GX '= [(x "4 4)/4] ) / 4]

-a -oa un= 0. = 0.(29) (29)So in cylindrical bending of a plate, in which the classical displacement field givesrise to Deci, n cilindrice ndoirea o farfurie, n care cmpul clasice de deplasare dnatere lastrains varying in one direction, the one dimensional nonlocal theory predicts stressesat the origin tulpini de variabile ntr-o singur direcie, teoria un dimensional nonlocal

prezice subliniaz, la origineaidentical to the classical values. identice cu valorile clasice.

Consider now the stress distribution. Luai n considerare acum de distribuie de stres. (x) (X)


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= E = E x'=- x =- (x - x') gx' dx' (x - x ') GX "DX"

(30) (30)Suppose that the kernel is constant over a range as in Eq. S presupunem c kernel-uleste constant ntr-o gam la fel ca n ecuaia. 13. 13. (x) (X)= E = E x-x'=-a x-x "=- una ungx'/a dx' = GX '/ a DX' =E E

x'=x+a x '= x + axa xagx'/a dx' GX '/ a DX "(31) (31) (x) = (x) =gEx' GEX "2 22a 2a[ [x+a x + a

xa xa= =gE gE2a 2a((xa) ((Xa)2 2- (x+a) - (X + a)2 2) = ) =gE gE2a 2a

(x (x2 2- 2ax +a - 2AX + a2 2- (x - (X2 2+ 2ax +a 2AX + + a2 2)) (32) )) (32) (x) = (x) =gE gE2a 2a(-2ax) = -gEx (-2AX) =-GEX


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(33) (33)There is no influence of nonlocality on the stress field (away from the free surfaces);the stress Nu exist nici o influen a nonlocality pe cmpul de stres (departe de drumliber suprafee); de stresfield is purely classical for this kernel. cmp este pur clasice pentru acest kernel-ului.

Consequently there are no size effects in rigidity except for n consecin, nu existefecte mrime n rigiditate, cu excepiathose associated with surface phenomena, as was the case in tension. cele asociate cufenomene de suprafa, cum a fost cazul n tensiune. The surface effects will beEfectele de suprafa va fisimilar to those for tension, provided that a


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j J= 1. = 1.(34) (34)A wider variety of functional forms of the kernel could be accommodated by passingto the limit as O mai mare varietate de forme funcionale a kernel-ului ar putea fi

cazai prin trecerea la limit, n msuraan integral. parte integrant. Since each component of such a superposition gives riseto a classical stress field away Din moment ce fiecare component a unei astfel desuprapunere d natere la un domeniu clasic de stres departefrom the edges, then the superposed kernel as well gives rise to a classical stress field.de la marginile, apoi kernel-ului suprapuse, precum i d natere la un cmp clasice destres.

Page 9Page 9We remark that for a Poisson's ratio of zero, the displacement field for bending of a

bar is Ne remarc faptul c, pentru un raport de zero a lui Poisson, domeniuldeplasarea de indoire de un bar esteidentical to that for cylindrical bending of a plate (which we have considered), but fornonzero identic cu cea pentru cilindrice ndoirea o plac (pe care le-am considerat),dar pentru nenulPoisson's ratio, strains in a bent bar are nonzero in all three coordinate directions.Raportul lui Poisson, tulpini ntr-un bar aplecat sunt nenul n toate cele trei direcii decoordonate.Observe that the Cosserat bending equation differs from Eq. Se observ faptul cCosserat de ndoire ecuaia difer de ecuaia. 26 for nonlocal size effects in 26 pentruefecte dimensiunea nonlocal nthat the latter has a linear term in the length scale ratio, for simple kernels. c aceasta

din urm are un termen liniar n raport lungime de scar, pentru nucleele de simplu.Consequently nonlocal n consecin nonlocaland Cosserat solids can be distinguished by the functional form of the size effects. isolide Cosserat se distinge prin forma funcional a efectelor dimensiune. Acomparison O comparaie

between predicted size effects is shown in Fig. ntre prezis efectele mrime esteprezentat n fig. 6. 6. Observe that the Cosserat and nonlocal curves Se observfaptul c curbele Cosserat i nonlocalcross each other and have different shape. ncruciate ntre ele i au diferite forme.VVPhysical causes of mechanical behaviorCauzele fizice de comportament

mecaniceContinuum theories make no reference to structural features, however they areintended to Teorii Continuum nu fac nicio referire la caracteristicile structurale, cutoate acestea ele sunt destinate srepresent physical solids which always have some form of structure. reprezint solidefizice care au ntotdeauna o form de structur.The ultimate origin of elastic behavior is the electromagnetic force between atoms ina solid. Originea final de comportament elastic este fora electromagnetic dintreatomi ntr-un solid.The uniconstant theory was derived assuming that such interatomic forces werecentral forces Teoria uniconstant a fost derivat presupunnd c o astfel de fore au fost

interatomic centrale forele


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along lines connecting pairs of atoms, and that the movement of atoms was affine. de-a lungul liniilor care conecteaz de perechi de atomi, i c micarea a fost de atomi deafine. Since the Avnd n vedere cPoisson's ratio for most materials differs experimentally from 1/4, it can be concludedthat in most Raportul lui Poisson pentru cele mai multe materiale difer experimental

de la 1 / 4, se poate concluziona c, n majoritateamaterials either the interatomic forces are non-central, or non-affine deformationoccurs, or both. materiale, fie forele interatomic sunt non-centrale, sau non-deformareafine are loc, sau ambele.The couple stresses in Cosserat and microstructure elasticity represent spatialaverages of Cuplul subliniaz, n Cosserat i elasticitatea microstructur reprezintmediile spaial adistributed moments per unit area, just as the ordinary (force) stress represents aspatial average of momente distribuite pe unitate de suprafa, la fel cum ordinare(vigoare), stresul reprezinta o medie spaial aforce per unit area. fora pe unitatea de suprafa. Such moments can occur as a result

of the fact that the interatomic forces Astfel de momente poate aprea ca urmare afaptului c forele interatomic

propagate further than one atomic spacing (Krner, 1963). propaga mai departe decto spaiere atomice (coroane, 1963). Such effects will occur in allsolids, Aceste efectese vor produce n toate solide,

but the corresponding characteristic lengths would be of atomic scale and notamenable to dar lungimi corespunztoare caracteristic ar fi de scar atomic, i nu ifactorii de succesmacroscopic mechanical experiment. Experimentul macroscopice mecanice. Momentsmay be also transmitted on a much larger scale Momente pot fi transmise, deasemenea, pe o scar mult mai marethrough fibers in fiber-reinforced materials, or in the cell ribs or walls in cellularsolids. prin fibre in fibre-materiale ranforsate, sau n coaste de celule sau perei nsolide celulare. TheCosserat characteristic lengths would then be associated with the physical size scalesin the Lungimi Cosserat caracteristic ar fi apoi asociate cu baremele dimensiuneafizic nmicrostructure, and be sufficiently large to observe experimentally. microstructura, is fie suficient de mare pentru a observa experimental.The nonlocal theory incorporates long range interactions between particles in acontinuum Teoria nonlocal ncorporeaz interaciunile dintre particule pe distane

lungi ntr-un continuummodel. model. Such long range interactions occur between charged atoms ormolecules in a solid. Astfel de interaciuni Long Range s apar ntre atomi perceputesau molecule ntr-un solid. Long Lungrange forces may also be considered to propagate along fibers or laminae in acomposite material Forele gam pot fi, de asemenea, considerate a se propaga de-alungul fibrelor sau lamel ntr-un material compozit(Ilcewicz, et. al, 1981, 1981). (Ilcewicz, et. Al, 1981, 1981).Analytical predictions of Cosserat characteristic lengths have been developed for avariety Predicii analitic de lungimi Cosserat caracteristici au fost dezvoltate pentru ovarietate


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of structures. de structuri. In fibrous composites, the characteristic length l may be theon the order of the n compozite fibroase, l caracteristic lungime poate fi pe ordineadespacing between fibers (Hlavacek, 1975); in cellular solids it may be comparable tothe average cell Spaierea ntre fibre (Hlavacek, 1975); n solide celulare, poate fi

comparabil cu celula mediesize (Adomeit, 1967); in laminates it may be on the order of the lamination thickness(Herrmann mrime (Adomeit, 1967); n laminatelor acesta poate fi pe ordinea degrosime laminare (Herrmannand Achenbach, 1967). i Achenbach, 1967). Structure, however, does not necessarilylead to Cosserat elastic effects. Structura, cu toate acestea, nu conduce neaprat laCosserat efecte elastica.Composite materials containing elliptic or spherical inclusions are predicted to have acharacteristic Materiale compozite care conin eliptice sferice sau incluziuni se

prognozeaz de a avea o caracteristiclength ofzero (Hlavacek, 1976; Berglund, 1982). lungime dezero (Hlavacek, 1976;

Berglund, 1982).A schematic diagram of force increments upon ribs (in the structural view)corresponding Un Diagrama schematic a incremente vigoare la coaste (din punctulde vedere structurale), care corespundto stress (in the continuum view) and moment increments corresponding to couplestress is shown la stres (n ecranul continuu), precum i creteri momentcorespunztoare cuplu de stres este afiatin Fig. n fig. 1. 1.One may distinguish the continuum view from the structural view. Se poate facedistincia de vedere continuum de la ecranul structurale. The continuum view is Viewcontinuum este deuseful for making engineering predictions and for visualizing global response ofmaterials. utile pentru a face previziuni de inginerie i pentru vizualizarea de rspunsla nivel mondial de materiale. Thestructural view is relevant to the underlying causes of the behavior. vedere structuraleste relevant pentru cauzele profunde ale comportamentului. One may link theseviews by Se poate link-ul aceste puncte de vedere de ctredeveloping an analytical model of the material microstructure, and obtainingapproximations dezvoltarea unui model de analiz a microstructurii materialului,

precum i obinerea de aproximri(possibly by series expansions for local deformation fields) in order to obtain average

values. (eventual prin dezvoltrile n serie pentru domenii de deformare locale), nscopul de a obine valori medii.Retention of only the lowest order terms in such analysis gives classical elasticity as acontinuum Pstrarea doar termenii cel mai sczut ordinea n astfel de analiz oferelasticitate clasice ca un continuumrepresentation. de reprezentare. When higher order terms are retained, a generalizedcontinuum representation (such n cazul n termeni de ordin superior sunt pstrate, oreprezentare generalizat continuum (cum aras Cosserat elasticity) is obtained. ca elasticitatea Cosserat) se obine. In either casethe predicted elastic constants are functions of the n orice caz, a prezis constanteleelastice sunt funcii ale


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structure and properties of the constituents. structur i proprieti ale componentelor.This is how the microphysics is introduced. Acesta este modul n care microphysicseste introdus.V Experimental Procedures for Cosserat elastic solidsV proceduriexperimentale pentru solide Cosserat elastice

Methods based on size effectsMetode bazate pe efecte dimensiuneaThe bending and torsion rigidities of a classically elastic rod are proportional to thefourth De ndoire i a rigiditii torsiune de o bar clasic elastice sunt proporionalecu al patrulea

power of the diameter. de putere al diametru. In a thin classical elastic plate thebending rigidity is proportional to the ntr-o plac subire clasic elastice rigiditatea dendoire este proporional cu

Page 10Page 10third power of the thickness. puterea a treia grosime. The rigidity depends on size in adifferent way in Cosserat elastic Rigiditatea depinde de dimensiunea ntr-un moddiferit n Cosserat elasticematerials as discussed above. Materiale de cum sa discutat mai sus. Thin specimensare more rigid than would be expected classically. Exemplarele Thin sunt mai rigidedect s-ar fi de ateptat clasic. It El / eais possible to determine one or more of the Cosserat elasticity constants bymeasurements of este posibil s se stabileasc una sau mai multe dintre constantelorde elasticitate Cosserat prin msurtori despecimen rigidity vs size. fa de mrimea specimen rigiditate. This approach, whichwe call the method of size effects, has been used Aceast abordare, pe care o numimmetoda de efectele dimensiuni, a fost utilizatto experimentally determine Cosserat elastic constants. pentru a determina

experimental constante Cosserat elastica. The method of size effects makes use ofMetoda de efecte mrimea face uz deanalytical solutions for the dependence of rigidity upon size. soluii analitice pentrudependena de rigiditate la dimensiune. Most of these solutions have dealt Cele maimulte dintre aceste soluii s-au ocupatwith isotropic materials. cu materiale izotrop. Specifically, Gauthier and Jahsman(1975) demonstrated that no size n mod specific, Gauthier i Jahsman (1975) ademonstrat c nu dimensiuneaeffects occur in tension, so that E and are determined from a tension test as in theclassical case. efectele s apar n tensiune, astfel nct E i sunt determinate de laun test de tensiune ca i n cazul clasic.

Size effects occur in torsion, and the isotropic Cosserat constants G, l Dimensiuneefecte apar la torsiune, precum i constantele izotrop Cosserat G, Lt T, N, can be obtained from , N, pot fi obinute de lasize effect data for torsion of rods of circular section. Datele dimensiunea efectului detorsiune de tije de sectiune circulara. Bending of circular section rods of differentPrelucrare de tije sectiune circulara din diferitesize (Krishna Reddy and Venkatasubramanian, 1978) gives E, l mrime (KrishnaReddy i Venkatasubramanian, 1978) d E, L

b b

, N. Cylindrical bending of a , N. cilindrice indoire a uneiplate gives E and l plac d E i L


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b b, according to Gauthier and Jahsman (1975). , n conformitate cu Gauthier i Jahsman(1975). Bending of a circular plate with a ndoirea o plac circular, cu oclamped edge depends on E, l marginea fixata depinde de E, L

b b

, N, according to Ariman (1968). , N, n conformitate cu Ariman (1968). Gauthier andJahsman (1976) Gauthier i Jahsman (1976)show that bending of a curved bar depends on E and l arat c indoire de o bar decurbe depinde de E i L

b b. . Park and Lakes (1987) show that size Park i a lacurilor (1987) arat c mrimeaeffects in torsion of a square cross section bar depend on G, l efectele n torsiune de-un bar ptrat seciunii transversale depind G, Lt T,N. , N.Instrumentation capable of determining bending and/or torsion rigidity may be used in

the Instrumentation capabile de determinare a ndoire i / sau rigiditate la torsiune potfi utilizate nmethod of size effects. Metoda de efecte dimensiune. Rigidity is to be determinedover a considerable range, so it is necessary to Rigiditate este de a fi stabilit ntr-ogam considerabil, astfel nct este necesar s semake sure that parasitic errors such as those due to instrument friction are minimizedor eliminated. asigurai-v c erorile de parazitare, cum ar fi cele din cauza frecriiinstrument sunt minimizate sau eliminate.Load may be applied electromagnetically or by dead weights. De ncrcare poate fiaplicat electromagnetic sau prin greuti mort. Cantilever bending is appropriateConsol de ndoire este adecvatsince there is no friction associated with dead weight loading. deoarece nu exist nicio frecare asociat cu ncrcare greutate mort. If dead weights are used in torsion, ncazul n care ponderile mori sunt utilizate n torsiune,the pulleys used to redirect the load could introduce errors due to friction. Roata detransmisie utilizate pentru a redireciona sarcinii ar putea introduce erori din cauzafrecrii. Such errors would be Astfel de erori ar fimore important for thin specimens and would obscure the size effects. mult maiimportant pentru specimenele subiri i ar obscur efectele dimensiune. Frictionalerrors could be Erori antiusura ar putea fieliminated by the use of air bearings. eliminate prin utilizarea de rulmenti de aer.

Deformation can be measured by holographic interferometry, Deformare poate fimsurat prin interferometrie holografic,other optical methods, strain gages, and free core LVDT's without friction error. altemetode optice, GAGES tulpina, i gratuit de baz LVDT fr erori de frecare. Springloaded De primvar ncrcateLVDT's with sleeve bearings would by contrast introduce friction errors which aremore LVDT cu rulmeni maneca ar fi, prin contrast introduc erorile de frecare, caresunt mai

problematical in thin specimens. problematice cu exemplare subire.Size effect methods have been used by several authors to obtain only one Cosseratelastic Metodele Dimensiune scop au fost folosite de mai muli autori pentru a obine

un singur Cosserat elastice


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constant. constant. A particular form of the method of size effects was found to beuseful by the present O form specific de metoda de efectele dimensiunea sa dovedita fi util prin prezentaauthor. autor. The rigidity of the same circular rod specimen was tested in both torsionand pure bending Rigiditatea modelului aceeai tija circular a fost testat n ambele

torsiune pur i de ndoireusing the same apparatus, which makes use of electromagnetic torque generation andutiliznd aceeai aparatur, care face uz de generare de cuplu electromagnetice iinterferometric detection of angular displacement. detectarea interferometric dedeplasarea unghiular. Each specimen was then cut to a smaller size andthe rigidities again determined. All six of the Cosserat elastic constants can bedetermined this way.Moreover, cross verification of the results is possible, in a manner similar to themeasurement ofE, G, and in classical elasticity, with verification of their interrelation.Specimen preparation for size effect methodIn the method of size effects a set of specimens of different diameter or thickness isused. If Dacthe characteristic lengths are small, thin specimens must be studied. Stiff materials,such as boneand the stiffer polymer foams, may be successfully cut on a lathe with conventionalcutting tools.Circularly cylindrical specimens thinner than about 3 mm in diameter down to 0.2 -0.5 mm are

prepared on a lathe by an abrasive machining technique. The lathe is operated at highspeed and a

strip of abrasive cloth is applied to the surface using a small force. Rectangularsection specimensmay be cut with a low speed saw, and the surfaces polished with graded abrasives.Flexible Flexibil

polymer foams can be cut into circular cylinders by use of a coring tool driven by apower drill.The coring tool consists of a metal tube with a sharpened end and thin walls.Rectangular sectionspecimens of foam can be cut from these by compressing the foam between platensand cuttingwith a scalpel.

Since surface damage to the specimen would cause a softening size effect, theopposite ofthat expected in Cosserat solids, considerable care should be taken to avoid surfacedamage. In the Incase of cellular solids, a layer of damaged or incomplete cells has been shown tocause such a

Page 11Page 11softening size effect by Brezny and Green (1990). To minimize the effect of surfacedamage,removal of the damaged layer by polishing is recommended if it is possible.Data reduction for size effect methodData analysis in the method of size effects makes use of the exact analytical solutionsfor


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the geometry used in the experiments. For torsion of a circular cylinder (Gauthier andJahsman,1975), the ratio of rigidity to its classical value is = 1 + 6(lt T

/r)2 2[1 - 4 /3)/(1 - )],(35) (35)with r as the rod radius, = ( + )/( + + ) and = I1 1(pr)/pr I0 0(pr), and p2 2

= 2 /( + + ).IEu1 1and I si eu0 0are the modified Bessel functions of the first kind. A special case of interest, referredto as 'couple stress elasticity', is for N = 1 ( ) in which the [] bracket in equation4 becomesunity. unitate. If the rod diameter is large, the corresponding rigidity ratio is 1 +6(lt T

/r)2 2. .For bending (Krishna Reddy and Venkatasubramanian, 1978) of a circular section rodofradius r, the rigidity ratio is = 1 + 8(l

b b/r)2 2

(1 - ( / )2 2) + [ (8N2 2( / + )2 2/ ( ( a) +N2 2(1 - ))(1 + ) ](36) (36)with

(

r) = (

r)

2 2


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[( r I0 0( r ) - I1 1( r)) / ( r I0 0( r) - 2I1 1( r))], and = N/l

b b. .Data may be plotted as vs radius; however one may also plot rigidity divided by thesquare of the diameter vs. the square of the diameter for torsion (Fig. 2) and bending(Fig. 3).Such plots are useful since the characteristic lengths can be extracted from the

intercept of theextrapolated straight portion of the curve upon the ordinate. They are not stress straincurves.In classical elasticity, the torsion size effect plot is a straight line through the originwithslope proportional to the shear modulus G. The Young's modulus E is obtained fromthe slope inthe bending case. Comparison of experimental plots and theoretical curves permit thedeterminationof the Cosserat elastic constants. The characteristic lengths are obtained by interceptsas described

above, and as indicated in the figures, either graphically or by numerical procedures.The shape ofthe torsion plot is then used to extract the coupling number N. A large value of N (theupper boundis 1) leads to a large apparent stiffening for slender specimens. The structure of thetorsion plot inthe vicinity of the origin is used to determine ; this is difficult since it requires verythinspecimens. specimene. In our laboratory, a numerical algorithm has been used tominimize the mean-square

deviation between the experimental data and the theoretical graphs, to extract theelastic constants.For some combinations of elastic constants, the apparent modulus tends to infinity asthe

bar or plate size goes to zero. Large stiffening effects might be seen in compositematerialsconsisting of very stiff fibers or laminae in a compliant matrix. However, infinitestiffening effectsare unphysical. For very slender specimens, it is likely that a continuum theory moregeneral thanCosserat elasticity; or use of a discrete structural model, would be required to deal

with theobserved phenomena.


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In the bending of long rods, one can invoke Saint Venant's principle in order toeliminatethe need to consider end effects. For that reason, we consider rod bending to be amore attractiveexperimental modality than plate bending. Moreover, the same rod can be used for

bending,torsion, and tension experiments.As for the cross sectional shape of the rod, circular sections currently have theadvantagethat the available analytical solutions are exact and not excessively complex. Howeversquare crosssection rods can also be studied and the results interpreted using the analysis

presented by Park andLakes (1987).

Page 12Page 12Field methodsDiscrimination among generalized continuum theories can also be accomplished byexamining the distribution of strain in deformed objects. This is in contrast to theabove method ofsize effects in which the rigidity, a global quantity, is measured. For example, Parkand Lakes(1987) presented analysis of the distribution of strain in a bar of rectangular crosssection undertorsion. The surface strain does not vanish at the corner of the cross section, incontrast to the caseof classical elasticity. A screening method based on this prediction was developed by

Lakes, et al(1975): a holographic image of a small notch in the corner discloses any motion of thenotch undertorsion. Such motion would occur in a Cosserat solid but not in a classical one. Alecturedemonstration based on displacement of a corner notch was presented by Lakes(1985b). Another Altfield method involves measuring the distribution of strain around a stress concentratorsuch as acircular hole, for which analytical solutions are available in classical elasticity(Sokolnikoff, 1983;

Fung, 1968) and for Cosserat elasticity (Eringen, 1968, Cowin, 1970; Mindlin, 1963).Wave methodsThe propagation of stress waves can be used as a probe into the constitutive equationgoverning materials with structure. Plane waves in an unbounded classically elasticmaterial

propagate without dispersion: their speed is independent of frequency. The elasticmoduli can beextracted from the speeds of transverse and longitudinal waves and from the materialdensity.Dispersion of waves is predicted to occur in Cosserat solids (Eringen, 1968), inmicromorphic/

microstructure solids (Mindlin, 1964), in nonlocal solids (Eringen, 1972), and in voidsolids (Puri


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and Cowin, 1985). Measurement of wave dispersion could be used to determinegeneralizedcontinuum characteristics of a material. A wave method was used by Gauthier (1982)to examinethe particulate composite which had appeared classical in the size effect studies of

Gauthier andJahsman (1975). The micro-inertial characteristics were determined, and it was foundthat N2 2= =0.0039, so small that the static behavior would indeed appear classical.A drawback of wave methods is that wave dispersion also arises from viscoelasticityofmaterials. materiale. Therefore wave methods are most suitable if the waveattenuation due to viscoelasticityis small enough to be neglected. By contrast, the method of size effects can be made

independentof viscoelasticity by conducting all measurements at the same time following loading,or at thesame frequency. If a material exhibits Cosserat viscoelasticity, the time or frequencydependence ofthe six Cosserat coefficients can be extracted from size effect plots generated atdifferent times orfrequencies. spectrului de frecvene. This cannot be done in wave methods, since thewavelength which governs the straingradient cannot be decoupled from the frequency. However wave methods can beused for largescale structures such as layered rock formations, which are too large to study in thelaboratory.Resolution of the characteristic length lIn all methods there is a limit to the smallest value of characteristic length l which can

beresolved. In the method of size effects, preparation of very thin specimens can presentdifficulties.In field methods the specimen can be large but the effects of generalized continuummechanicsdepends on strain gradients, and measurement of strain in the presence of the large

strain gradientsrequired to reveal a small l is difficult. One can examine the strain at the corner of asquare section

bar in torsion, as a null experiment, but again the resolution is limited by the fact thatstrain can bemeasured only over a nonzero length. In wave methods, the resolution of small lrequires waves ofhigh frequency, but in many materials the attenuation of stress waves becomes largeat sufficientlyhigh frequency.VIVI

Experimental Results: a reviewCosserat elastic constants


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The results of some published experimental studies of materials as Cosserat solids arepresented in Table 1. The relationship between Cosserat characteristic lengths and thestructure sizeis evident. Characteristic lengths on the millimeter scale are not observed unless thereare

corresponding structural features of a similar size scale. For example polymethylmethacrylate(PMMA) is an amorphous polymer for which the relevant structure scale is atomicand molecular.

Page 13Page 13It was used as a control in experiments upon bone (Yang and Lakes, 1981), which is anaturalcomposite in which the largest structural elements are large fibers up to 250 m indiameter. No Numacroscopic evidence of Cosserat elasticity was expected or found in PMMA.Although structureappears to be necessary to produce Cosserat elastic effects, it is not sufficient. Particlereinforcedcomposites exhibit Cosserat characteristic lengths of zero (Gauthier and Jahsman,1975). That Acelaobservation is in harmony with micromechanical analysis (Hlavacek, 1976; Berglund,1982)which predicts characteristic lengths of zero. We remark that the syntactic foamwhich was foundto be nearly classical, is composed of glass micro-balloons in an epoxy matrix, astructure which is

particulate in nature.Several authors have used 'couple stress theory' for interpretation. This corresponds toaCosserat solid for which N = 1. Since the characteristic lengths are defineddifferently, thecharacteristic length in Cosserat elasticity is 3 times the length in couple stresstheory. Results Rezultatelehave been converted to the Cosserat form in Table 1. In the study of the PVC foam, aresonantthickness shear size effect approach was used. Viscoelasticity could be a confoundingvariable here

since as the layer thickness was reduced, the resonant frequency increased. Inviscoelasticmaterials, stiffness increases with frequency even if they are classical. The graphitewas nonlinearin its stress-strain relation, but most of the others were studied in the linear domain.The studies of

bone and dense polyurethane and syntactic foams incorporated error analysis fromwhich ameaningful discrimination between Cosserat and classical behavior was achieved.Cosserat viscoelasticity has been studied in human bone. In a Cosserat viscoelasticsolid,the characteristic lengths as well as the stiffnesses can depend on time or frequency.The torsional


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characteristic length in bone was a factor of 1.6 larger under quasistatic conditionsequivalent to afrequency of about 0.1 Hz (Yang and Lakes (1981) than at 32 kHz (Lakes, 1982).This wasattributed to the viscoelastic attributes of the cement substance between the large

fibers (osteons) inbone. osoase.Study of predictive powerThe experimental determination of Cosserat elastic constants is useful if it permits oneto

predict stresses and strains under conditions which differ from those in theexperiments used tofind the elastic constants. Several experimental tests of predictive power have beenconducted.Cosserat elastic constants (based on isotropic theory) for bone, which is actuallyanisotropic, were

used to predict the strain distribution around a hole in a strip under tension, and theresults werecompared with experiment. Reasonable agreement was found by Lakes and Yang,(1983) eventhough the anisotropic solution was not available. The same Cosserat elasticconstants, derivedfrom size effect studies, were used to predict strain distributions in a square bar undertorsion.Comparison with experimental results by Park and Lakes (1986) was favorable. Inthis case, n acest caz,anisotropy is less of a problem, since the same elastic constantsl Lt Tand N are relevant in thisgeometry as in the torsion size effect study. Consequently, with the specimen alignedthe sameway in both cases, the same elastic constants appear in both cases.In torsion of a Cosserat square cross section bars a small notch in the corner of thecrosssection is predicted to displace as the bar is twisted. The displacement should be zeroin a classical

solid, since the stress is zero at the corner (Park and Lakes, 1985). Holographicstudies wereconducted, and corner notch displacement was zero in solid polymethyl methacrylate(PMMA), butwas nonzero in dense polyurethane foam (Lakes, et al, 1985) which was shown bysize effectstudies to be Cosserat elastic (Lakes, 1986). Similar displacements were easilyobserved visuallyin large cell foams (Lakes, 1986) which had been identified as Cosserat elastic(Lakes, 1983).Interpretation via void, nonlocal, and microstructure theories

If we attempt to interpret the size effect results in Table 1 with the void theory ofCowin


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and Nunziato (1983), there arises the difficulty that the theory predicts size effects inbending butnot in torsion. Among the materials studied, none exhibits a size effect only in

bending. We Noiconclude that a material describable by the void theory has not yet been found.

Materials with asmall volume fraction of voids have not, however been studied in this context.Page 14Page 14

As for nonlocal elasticity, some cases of wave dispersion have been interpreted viathattheory by Ilcewicz et al (1981, 1985). In a particle board composite, the characteristiclength wasfound to be about 0.3 mm, and that value was linked to the fracture toughness.As for microstructure elasticity, little comparison has been made with experimentsince fewanalytical solutions are available for this theory. However, wave dispersion and cut-off frequencieswere observed in dynamic studies of foams, including foams with negative Poisson'sratios (Chenand Lakes, 1989), and the results interpreted in view of microstructure elasticity.Wave dispersionhas been observed in several other structured materials (Sutherland and Lingle, 1972;Kinra, et. al,1980), without interpretation via generalized continuum mechanics.Studies of fibrous compositesThe fracture strength of graphite epoxy plates with holes depends on the size of the

hole(Karlak, 1977). Moreover the strain around small holes and notches in fibrouscomposites well

below the yield point is smaller than expected classically (Whitney and Nuismer,1974; Daniel,1978), while for large holes, the strain field follows classical predictions (Rowlands etal, 1973).Further results are given in a review by Awerbuch and Madhukar (1985). Such resultsare inharmony with the predictions of generalized continuum mechanics. However, th

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