Molecular Approaches for MultifieldContinua: origins and current developments

Patrizia Trovalusci*

*Department of Structural and Geotechnical Engineering,Sapienza – University of Rome

Article published in T. Sadowski, P. Trovalusci (Eds.), Multiscale Modelingof Complex Materials: Phenomenological, Theoretical and ComputationalAspects, CISM (International Centre for Mechanical Sciences) series, No.556, Springer-Verlag, Berlin, 2014. (DOI:10.1007/978−3−7091−1812−23).PREPRINT.

Abstract The mechanical behaviour of complex materials, charac-terised at finer scales by the presence of heterogeneities of signif-icant size and texture, strongly depends on their microstructuralfeatures. Attention is centred on multiscale approaches which aimto deduce properties and relations at a given macroscale by bridg-ing information at proper underlying microlevel via energy equiva-lence criteria. Focus is on physically–based corpuscular–continuousmodels originated by the molecular models developed in the 19th

century to give an explanation per causas of elasticity. In particu-lar, the ‘mechanistic–energetistic’ approach by Voigt and Poincarewho, when dealing with the paradoxes deriving from the search ofthe exact number of elastic constants in linear elasticity, respec-tively introduced molecular models with moment and multi–bodyinteractions is examined. Thus overcoming the experimental dis-crepancies related to the so–called central–force scheme, originallyadopted by Navier, Cauchy and Poisson.

Current research in solid state physics as well as in mechanics ofmaterials shows that energy equivalent continua obtained by defin-ing direct links with lattice systems are still among the most promis-ing approaches in material science. This study aims at emphasizingthe suitability of adopting discrete–continuous approaches, basedon a generalization of the so–called Cauchy–Born rule used in crys-tal elasticity and in the classical molecular theory of elasticity, toidentify continua with additional degrees of freedom (micromorphic,

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multifield, etc.), which are essentially ‘non–local’ models with inter-nal length and dispersive properties. By lacking in internal lengthparameters, the classical continuum does not always seem appropri-ate to describe the macroscopic behaviour of such materials, takinginto account the size, orientation and disposition of the microhetero-geneities. Within the general framework of the principle of virtualpower, it is described as the selection of a correspondence map, re-lating the finite number of degrees of freedom of discrete models tothe kinematical fields of equivalent continua, provides a guidancefor non-standard continuous approximations of heterogeneous me-dia by–passing the intrinsic limits of scale separation of classicalcontinua formulations. The circumstances in which, not very dif-ferently than in the past, empirical inadequacies still call for theneed of removal of the local character of the classical hypothesis oflattice mechanics (central-forces or homogeneous deformations) arealso discussed. A sample application of discrete–continuum homog-enization approach leading to multifield description is finally shownwith reference to microcracked composite materials, which can berepresentative of fiber–reinforced composites, ceramic matrix com-posites or porous metal–ceramic composites, as well as concrete andmasonry–like materials.

1 Introduction. Multiscale approaches: a short review

A material can be defined as complex because of the presence of heteroge-neous and discontinuous internal structure, which can be detected at dif-ferent (meso, micro/nano, atomic, electronic) length scales, and becauseof non–linear constitutive behaviours, such as plasticity, damage, fracture,growth, etc. Based on their internal structure these materials can be classi-fied as fiber–reinforced composite, materials with voids or defects, granular,rock, masonry, etc. A basic issue of mechanics of complex materials, frommodern nanoscience to structural engineering, is the definition of constitu-tive models suitable to account for the presence of the internal structurebalancing accuracy of the description with computational burden. Thepossibility of designing and/or testing materials with internal structure,addressing the wider technological applications in engineering, is closelyrelated to the ability to derive their constitutive relationships taking intoaccount the internal structure: shape, spatial distribution, orientation ofthe constituents and size, which may presents several orders of magnitude,starting from the submicron scale up to larger meso and macro scales.

A common feature of these materials is their intrinsic discrete nature,not only at the smallest scales but also at mesoscales, because interfaces(grain boundaries, dislocations, disclinations, joints, etc.) frequently domi-

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nate the gross behaviour of the whole. For many years direct modelling atsmaller scales than the macroscopic, such as methods of molecular or dislo-cation dynamics simulations have been widely used for composite materials(Amodeo and Ghoniem, 1990; Rapaport, 1995; Devincre and Roberts, 1996;Rhee et al., 1998; Ariza and Ortiz, 2006; Yao et al., 2007). Also for otherkinds of discontinuous and heterogeneous materials, such as jointed rocksystems or block masonry, distinct elements methods or limit analysis ofrigid blocks interacting through no–tension and frictional interfaces havebeen used (Cundall and Strack, 1979; Baggio and Trovalusci, 1998, 2000;Camborde et al., 2000; Ferris and Tin-Loy, 2001).

Such approaches, although accurate and representative of the actual dis-crete nature of the materials, have often been revealed as computationallycumbersome for systems with many degrees of freedom so that, in mostcases, continuum approximation is to be preferred. In addition, when ma-terials are made of particles of significant size and disposition (distribution,orientation), it is necessary to adopt non–classical (enhanced) continuousmodels which exploit the advantages of the field description at the coarselevel without forsaking the memory of the fine organization of the mate-rial (Eringen, 1964a, 1999; Mindlin, 1964, 1965; Kunin, 1982, 1983; Capriz,1989; Gurtin, 2000; Maugin, 1993, 2011). All these models are essentiallynon–local because of the presence of internal length scales in the field equa-tions and because they show dispersion properties, i.e. dependency of thephase velocity of travelling waves on the wave–length or frequency. Thesefeatures reveal the presence of the internal structure providing an effectivemacroscopic description of the microstructure, necessary when the fabricdimension is not small compared to the macroscopic dimensions. Althoughthe nomenclature in literature is varied, these models here are called non–classical or, in accordance with the definition of complex continua by Caprizand Podio-Guidugli (2004), non–simple continua.1 In particular, when thesemodels are characterized by additional deformation fields with respect tothe standard fields, we also use the term multifield continua. Within a vir-tual power description these fields can be seen as primal variables, powerconjugated to dual variables, accounting for the presence of some types ofinternal structure, in the defined non–local meaning.

For many years theoretical and computational tools for the coarse scale

1 Generally, continua with additional degrees of freedom (generalized continua, micro-

continua, continua with configurational forces, etc.) are distinguished from continua

called ‘explicitly’ or ‘strongly’ non–local (Eringen, 1999, 2002; Maugin, 1979, 2010).

The use of this term here intends to point out the common feature of non–locality

of all these size–dependent continua formulations which will be treated below in this

chapter.

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modelling of complex materials have been developed, within a truly mul-tidisciplinary framework, by resorting to multiscale modelling (Ortiz andPhillips, 1999; Phillips, 2001; Finel et al., 2003; Guo, 2007; Liu et al., 2006).This approach aims at deducing properties and relations at a given scaleby bridging information at proper underlying levels and offers useful newinsights into complex materials modelling, opening the way to various chal-lenging applications in material science and engineering.2

While condensed matter physicists consider multiscale modelling as thedialogue among several different material scales, in structural mechanics andmaterial engineering, it generally suffices to link only two scales: a fine scale,conventionally defined microscopic, and a gross continuum scale referred asmacroscopic. Over the last decades the possibility of providing a macro-scopic description for heterogeneous media, accounting for the geometry ofthe microstructure, has been largely investigated within the framework ofthe homogenization or coarse–graining theories. Among various approaches,we can grossly distinguish the two categories mentioned below.

Homogenization: from heterogeneous continua to homogenenized

continua One category concerns the homogenization procedures (includ-ing asymptotic, variational, averaging methods) which are based on the so-lutions at a conventional micro–level of hierarchies of boundary value prob-lems, or of one single boundary value problem on a Representative VolumeElement (RVE). The procedures most frequently used to obtain numeri-cal solutions are consistent with macrohomogenity conditions of the Hill–Mandel type (Hill, 1963). These procedures allow the replacement of a fine–grained continuous anisotropic description with a lower resolution homoge-neous continuous coarse–grained model (Benoussan et al., 1978; Sanchez-Palencia, 1980; Hashin, 1983; Bakhvalov and Panasenko, 1989; Nemat-Nasserand Hori, 1993) and have often been applied to describe the macroscopicbehaviour of heterogeneous materials of various kinds. To account for mi-crostructure evolution they have also been extended to the constitutive non–linear case and framed within the ambit of finite inelasticity (Suquet, 1985;Geymonat et al., 1993; Ponte Castaneda and Suquet, 1998; Feyel, 2001;Miehe et al., 2002). The literature related to complex constitutive modelsfor heterogeneous materials produced up to the present day is extremelywide. We here defer to the discussion contained in Chapter 3 of this vol-ume by Ghosh, who proposed and developed a very efficient computationalmethod for multiscale analyses based on Voronoi tessellation (Ghosh et al.,

2See, e.g., Trovalusci (2007); Trovalusci and Ostoja-Starzewski (2011); Trovalusci and

Schrefler (2012).

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1995; Ghosh, 2011).In order to model complex spatial interaction effects or describe ma-

terials in which internal length scales are not negligible when compared tostructural length scales, homogenization techniques have also been extendedto non–simple continua. The latter circumstance becomes significant whendealing with complex constitutive behaviours dependent on the microstru-ture size, such as damage concentration, and the field equations of the simple(Grade 1) classical continuum become ill–posed. To this regard, non–localor higher order deformation gradient descriptions, specifically addressed tomultiscale computational homogenization, have been proposed (Sluys et al.,1993; Kouznetsova et al., 2002, 2004; Peerlings and Fleck, 2004; Massartet al., 2007; Bacigalupo and Gambarotta, 2010, 2011).3 Moreover, homoge-nization procedures developed for micromorphic continua, in particular withrigid local structure, have been satisfactorily applied to various composites(Forest and Sab, 1998; Forest et al., 1999; Ostoja-Starzewski et al., 1999;Bouyge et al., 2001; Forest et al., 2001; Onck, 2002; Tekoglu and Onck,2008; Addessi and Sacco, 2012). These procedures are based on macroho-mogenity conditions of Hill–Mandels type and consider both classical andnon–classical continua at the micro–level. It is worth noting that in theformer case, when periodic boundary conditions are assigned to a represen-tative volume element some drawbacks arise (Forest and Trinh, 2011; Trinhet al., 2012); while in the latter case, a non–trivial definition of boundaryconditions consistent with a generalized Hill–Mandel’s condition is required(Li and Liu, 2009; Liu, 2013; Ostoja-Starzewski, 2011).

Finally, a specific mention should be made of the statistical homoge-nization of materials with random microstructure based on size–dependentprocedures. The problem of the determination of the RVE size, well estab-lished in periodicity–based homogenization techniques resorting to singlecell concept, as in most works mentioned below, for random media is stillopened (Ostoja-Starzewski, 2008).4 These procedures are employed withinlimit processes involving several finite–scale continuous descriptions (Sta-tistical Volume Elements), relative to the microstructural length scale. Thesolution of series of Dirichlet and Neumann boundary value problems at sev-eral mesoscales deduced from Hill–Mandel type macrohomogeneity condi-tion, also valid for non–periodic and non classical media (Ostoja-Starzewski,2011), provides two hierarchies of bounds for the material properties as well

3When dealing with non–simple formulations, a mention should also be made of non–

local explicit solutions obtained for specific cases of elastic composites (Drugan and

Willis, 1996; Luciano and Willis, 2000; Smyshlyaev and Cherednichenko, 2000; Bacca

et al., 2013).4See also Chapter 4 of this volume.

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as the microstructural minimal size of the RVE for performing homoge-nization (Ostoja-Starzewski, 2006; Khisaeva and Ostoja-Starzewski, 2006;Ostoja-Starzewski et al., 2007). In order to deal with materials with in-ternal lengths, these procedures have been recently extended to micropolarmedia by exploiting generalized macro–homogeneity conditions (Trovalusciet al., Submitted).

Coarse–graining: from lattice systems to homogenized continua

The other homogenization category concerns approaches based on latticedescriptions at the micro–level and integral equivalences with classical ornon–classical continuous models. These corpuscular–continuous approachesare based on the assumption that at a selected microscopic level the struc-ture of the matter can be described as discontinuous, and that the transitionfrom the fine to the coarse scale is governed by an a priori map betweenthe large set of degrees of freedom of discrete systems to the deformationfields of the continuum. The homogenization is guaranteed by the validityof the localization theorem.

Formerly, averaging processes aimed at deriving macroscopic quantitiesfrom Lagrangian systems have been used to provide coarse scale descriptions(Muncaster, 1983; Murdoch, 1985). Many of these discrete–continuum ap-proaches have been specifically addressed to derive non–local or generalizedcontinua (Askar, 1985(1943); Aero and Kuvshinskii, 1961; Kroner, 1963;Kunin, 1968; Kroner, 1968; Wozniak, 1969; Capriz and Podio-Guidugli,1983; Kunin, 1982, 1983; Pitteri, 1990). Furthermore, corpuscular (atom-istic/molecular) information have been widely employed to develop materialbehaviour descriptions and to formulate tractable boundary–value prob-lems. The classical theory of crystal lattices and molecular theory of elas-ticity are basic examples of such approaches, based on the hypothesis thatthe lattice points undergo the same deformation of the macroscopic model.(Stakgold, 1949; Born and Huang, 1954; Maradudin et al., 1971; Ericksen,1977, 1984).

Discrete to continuum modelling is also currently addressed to derivesize–dependent non–simple, physically–based, constitutive models, mainlyapplying to materials with microstructure detectable at length scales of sev-eral orders of magnitude smaller than the macroscopic scale; such as nano–composites, atomic–scale defected materials, dislocated bodies (Sunyk andSteinmann, 2003; Pyrz and Bochenek, 2007; Lee et al., 2010; Steinmannet al., 2011). However, refined discrete–continuum correspondence mapshave been also used for deriving non–classical continuous models of dis-crete systems of various kind, such as composite and masonry–like materials(granular, porous or jointed rocks, concrete, block masonry, etc.), always ac-

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counting for phenomena dependent on the microstructure size (Chang andLiao, 1990; Di Carlo et al., 1990; de Borst and Muhlhaus, 1992; Changand Ma, 1992; Vardoulakis and Frantziskonis, 1992; Bardenhagen and Tri-antafyllidis, 1994; Masiani et al., 1995; Masiani and Trovalusci, 1996; Sulemand Muhlhaus, 1997; Trovalusci and Masiani, 1999; Suiker et al., 2001;Trovalusci and Masiani, 2003; Ehlers et al., 2003; Goddard, 2005, 2007;Trovalusci and Masiani, 2005; Sansalone et al., 2006; Trovalusci et al., 2010;Stefanou et al., 2010; Dos Reis and Ganghoffer, 2011; Pau and Trovalusci,2012; Trovalusci and Pau, 2014).

Various other approaches searching for direct links between discrete andcontinuum solid mechanics have been proposed for several years and arestill a topic of current research in material science, often adopted to de-scribe non–linear constitutive behaviours dominated by deformation pro-cesses at the micro–level (Curtin and Miller, 2003; Devincre et al., 2003;Mesarovic, 2005; Di Paola and Zingales, 2008; Mesarovic et al., 2010, 2011;Coenen et al., 2011; Nguyen et al., 2012; Evers et al., 2002). Other examplesare mathematical coarse–graining processes via variational, asymptotic andgamma–convergence methods (Braides et al., 1999; Braides and Gelli, 2000;Paroni, 2000; Blanc et al., 2002; Cacace and Garroni, 2009). Finally, othercontinuum theories resorting to atomistic are the so–called quasi–continuumtheories, which essentially rely on adopting interpolating functions of dis-crete arguments by special classes of analytical functions in order to guaran-tee a one–to–one correspondence between continuum and discrete variables(Tadmor et al., 1996; Miller and Tadmor, 2002; Fish and Schwob, 2003; Fishand Chen, 2004; Fago et al., 2005). Moreover, the so–called cohesive–zonetheories, originally proposed to study plasticity (Peierls–Nabarro models)and then extended to describe fracture (Barenblatt–Dugdale models), alsoexploit a mixed continuous–discontinuous description where the constitutivenon–linearity is confined to interfacial planes with cohesive potential, whichcan be built–up by resorting to atomistics. The computational aspects ofcohesive zone model, particularly addressed to strain localization and frac-ture phenomena, have been widely studied by de Borst (2003); de Borstet al. (2006). In this volume de Borst (Chapter 1) also presents an efficientand challenging computational approach to fracture behaviour by linkingcontinuum to atomistic within a proper intermediate domain, in which thediscrete and continuous kinematical fields are coupled, providing that theloss of energy due to the coupling scheme is minimized.

The discrete to continuum approaches follows a tradition which datesback to the first molecular models in elasticity of the 19th century (Navier,1827; Cauchy, 1828; Poisson, 1829; Voigt, 1887, 1900, 1910; Poincare, 1892;

7

Marcolongo, 1904) adopted to derive the constitutive equations of classi-cal continua starting from the description of lattice systems. The origi-nal molecular models by Navier and Cauchy, were models made of parti-cles interacting in pairs depending on their mutual distance through forcesdirected along this distance. This so–called central–force scheme led toexperimental discrepancies concerning the number of elastic constants de-rived (15 coefficients in the general anisotropic case), that are well knownto matter physicists who study triangular lattices with pair–potentials only(central–force interactions) (Palla et al., 2007). In this framework, Voigtfirst, starting from a refined lattice model made of rigid particles interact-ing through forces and moment of forces depending on their distance as wellas on their orientation, derived a classical model with the correct number ofelasticities (21 coefficients for anisotropic materials). Successively, Poincareproposed a different refinement of the ‘central–force’ molecular model byintroducing potentials not only of pairs but also of triplets of point–likemolecules leading to a description which in current terms would be call‘multibody’ potential description (Liu et al., 2006; Palla et al., 2007; Blancet al., 2002). This model, corresponding to an angular potential model,also enabled the derivation of the exact number of elasticities. Althoughthe significant work of Voigt and Poincare led to results in agreement withexperimental data, the mechanistic-molecular approach was abandoned infavour of the energetistic–continuum approach based on Green’s macro-scopic criterium of the deformation work as an exact differential (Green,1839, 1842; Love, 1906).

Today these ideas enjoy a renewed interest, not only in condensed matterphysics where such approaches are standard (i.e. in crystal elasticity), butalso in mechanics of complex materials. To develop methods and concep-tual guidelines for continuous field descriptions, by linking continuum anddiscrete solid mechanics, as widely investigated in the molecular theories of19th century, is still a topic of crucial importance in material science andengineering. This is due to the fact that the gross behaviour of the mate-rial is greatly influenced by the fine scale structure, which is too complexto be captured through a coarse–scale direct modelling or a weak principlegoverning constitutive variables and relations.

In this framework, the key issue becomes: what physically–based latticesystem has to be defined to derive proper macroscopic models with non–local properties, which intrinsically retain ‘memory’ of the material finescale organization? In other words: what is the most appropriate contin-uum approximation for a material with a given microstructure? A possibleanswer, described in this chapter, is inspired by the original idea of Voigt,who first proposed an enhanced mechanistic–energetistic homogenization

8

approach work based on the enrichment of the kinematics of the discretesystem. Generalization of this approach to various (complex) lattice systemsby endowing the particles with ‘structure’, such as extension or propertiesrepresenting any kind of material microstructure (defects, voids, rigid in-clusions, etc.), naturally leads, within the general framework of the virtualpower equivalence, to the identification of effective size–dependent multi-field continua.

This chapter is organized as follows. In Section 2, we briefly recallthe mechanistic molecular model by Cauchy and the refined mechanistic–energetistic molecular models by Voigt and Poincare, whereby the experi-mental discrepancies, related to the so–called central force scheme adoptedby Navier, Cauchy and Poisson, can be by-passed. The actual validity ofsuch an approach, at least from the epistemological point of view, is then re-marked with a mention of modern discrete–continuum theories. In Section 3,some basic features of continua with additional field descriptors (multifieldcontinua) are recalled, with a view to the development of the concepts re-lated to the description of the material microstructure. In Section 4, withina generalized virtual power framework, a molecular–multifield approach isadopted to derive a constitutive model for a fibre reinforced composite ma-terial made of short, stiff and strong fibres embedded in a more deformablematrix with distribution of microcracks (fibre reinforced polymer/ceramicmatrix composite, cellular material, masonry, jointed rock, etc.). Finally, inSection 6 some short remarks point out the advantages of coarse–grainingapproaches combined with non–simple continuum modelling.

2 The Nineteenth century molecular models with a

glance at modern discrete–continuum theories

The genesis of multiscale approaches, which aim at deducing properties andrelations at a given macroscale by bridging information at proper underlyingmicrolevel via equivalence criteria, historically coincides with the genesis ofcontinuum mechanics. The molecular theory of elasticity, as developed byNavier (1827); Cauchy (1828) and Poisson (1829), represents one of the firststeps in this direction.This theory was introduced to justify the ‘causes’ ofelasticity, which were presumed to stem from the natural attractive or repul-sive properties of elementary particles (‘molecules’)5 depending on their mu-tual distance, as in the original idea of Newton, specified later by Boscovich,

5In these treatises the molecule, or atom, is perceived as ultimate particle inside which

no forces are accounted for (or are of smaller order than the intermolecular forces).

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Coulomb and others.6 With this mechanistic approach it was possible topredict the macroscopic constitutive behaviour on the basis of the defini-tion of microscopic laws for systems of molecules/particles, which interact inpairs depending on their mutual distance through forces directed along thisdistance (‘central–force’ scheme). Macroscopic quantities, like stress, elasticmoduli, etc., were then derived as averages of molecular material quanti-ties.7 However, this scheme led to experimental discrepancies concerningthe number of material constants needed to represent material symmetryclasses. Successively, Voigt (1887, 1900, 1910) and Poincare (1892) provideda refined description of the Cauchy molecular model that by–passed the ex-perimental discrepancies related to the central–force scheme. In particular,Voigt introduced potentials of force and moment interactions, which are ex-erted between pairs of rigid bodies, while Poincare proposed a ‘multibody’potential description.

In this Section a recapitulation of the main ideas contained in the workof early elasticians is presented in order to point out some basic issuesof homogenization processes that, in a sense, give rise to current multi-scale strategies.8 The original notation per Cartesian components has beenrewritten with a current tensor algebra notation.

6Following the presentation of the corpuscolar theory of ligth propagation by Newton,

elasticity was originally explained, by Newton itself, in terms of attractive properties

between atoms (Opticks or a treatise of the reflections, refractions, inflections and

colours of ligth, Queries, XXXI, 2nd ed. 1717 (1701)). van Mussenbroeck (Physicae

experimentales et geometricae dissertationes, Leiden, 1729) also interpreted the co-

herentia corporum as the effect of attracting inter–molecular actions (vires internae

attrahens). Boscovich (Philosophia Naturalis, Venezia, 1763) defined the intermolecu-

lar force as an action depending on the distance between molecules. This mechanistic

interpretation of interactions was successful for some time and extended to various field

in physics: the molecular theory of magnetism developed, between 1777 and 1787 by

C.-A. de Coulomb (Collection de Memoires Relatifs a la Physique, Tome 1: Memoires

de Coulomb, Societe Francaise de Physique. Paris, Gauthier–Villars, 1884), in which

the presence of a fluid was supposed; capillarity or surface tension in liquids explained

in terms of adhesion and cohesion at molecular level (P. S. Laplace, Annales de Chimie

et de Physique, Tome 12, Paris, Crochard, 1819); etc.7The averages are evaluated within a convenient volume, called ‘molecular sphere of

action’, outside which intermolecular forces are negligible.8Further detailed descriptions of the original molecular models, here reported and rein-

terpreted, can be found in (Trovalusci et al., 2009; Capecchi et al., 2010, 2011).

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2.1 Cauchy model for elasticity

In 1827 Cauchy presented a memoir at the Academie des Sciences inwhich all the basic concepts of continuum mechanics, the definition of stressand the constitutive relations, were derived on the basis of a molecularcentral–force approach (Cauchy, 1828).9 In the following, the basic as-sumptions of the Cauchy molecular model are described and reinterpreted.

In the Cauchy model the body is supposed to be made of material el-ementary particles without extension, called molecules.5 The molecules in-teract through a force directed along the line connecting their centres, de-pending on their mutual distance, r, and its variation ∆ r.

Let A and B be two molecules, of mass ma and mb, at the places A andB, respectively, with:

r =‖ A − B ‖ , n = (A − B)/r.

Denoting with w(A) and w(B) the displacement vectors of A and B, thecurrent positions of A and B respectively are:

a = A + w(A) , b = B + w(B) ,

with:r = r + ∆r =‖ a − b ‖ , n = (a − b)/r .

Let the vector ta = ta n (tb = tb n) be the force that the moleculeB (A) exerts on A (B), with ta (tb) scalar constant. The intermolecularforce between A and B, tab = ta = −tb, depends on a function of r, isproportional to the masses and is directed along n:

tab = ma mb f(r) n . (1)

The transition from the micro molecular model to the macro continuummodel is ensured by the correspondence map, connecting discrete and con-tinuum degrees of freedom, obtained under the hypothesis of regularity ofdisplacement fields on the whole Euclidean space E ; that is:

∆w = w(A) −w(B)

= ∇w(X)(A − B) +1

2[∇2 w(X)(A − B)](A − B) + o (A − B), (2)

where X ∈ C, and C ∈ E is the reference shape of the continuous body (with∇(·) = ∂(·)/∂X). By linearizing in the surrounding of X, the homogeniza-tion rule, called the Cauchy rule, is obtained. That is the homogeneous

9This memoir followed a memoir of 1821 on the molecular theory of elasticity presented

at the Academie des Sciences of Paris by Navier (1827).

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map:10

∆w ∼= ∇w(A − B) , (3)

Consequently, the variation of the intermolecular distance becomes:

∆r = ∆w · n ∼= ∆w · n = r∇w · n⊗ n = r E · n ⊗ n = r ε , (4)

where n ∼= n in this linearized frame and

E =1

2(∇w + ∇wT ) , ε = E · n ⊗ n .

Function f(r) is continuous to any order for all values of the distance r,which are negligible for distances greater than the radius of the so–called‘molecular sphere of action’.7 Then, only small discrete values of r are con-sidered and the expansion of quantities depending on r can be stopped atthe first order:

f(r) = α r + β r ε , (5)

where α and β are coefficients respectively depending on f and its derivativeevaluated at b. The intermolecular force (Equation 1) becomes:

tab = ma mb (α r + β r ε )n , (6)

thus satisfying a principle of local action.To define the stress measure Cauchy considers an infinitesimal surface of

area dAm, with outward unit normal m, and a control cylinder, having asits basis that surface and height l = r n · m. The diameter of the cylinderis greater than the radius of molecular activity. The stress vector tm actingover this surface of normal m, is defined as the resultant of the forces perunit surface exerted on all the pairs of molecules (A, B) of the cylindercontained in an half–space defined by m:

tm =1

dAm

∑

ab

ma mb f(r)n =γ

2

∑

a

maf(r) r (n ⊗ n)m = Tm , (7)

10This corresponds to assume for the particles, as it occurs in crystal elasticity and

in the molecular theory of elasticity (Ericksen, 1977, 1984), the same homogeneous

transformation of the corresponding continuum point (also called Cauchy–Born rule).

12

where γ is the mass density, mb = γ l dAm/N , N being the number ofmolecules within the cylinder, and T is the Cauchy stress tensor:11

T =γ

2

∑

a

maf(r) r (n ⊗ n) . (8)

The constitutive relations can be finally obtained by introducing Equa-tion (5) in Equation (8). Considering also Equation (4), being r = r + ∆ r,and by assuming that in the initial configuration, when E = 0 and thenε = 0, the stress is null, after some algebra, the following relation for thestress tensor T is obtained:

T =γ

2

∑

a

ma β r2 (n ⊗ n ⊗ n ⊗ n)E = CE , (9)

with the constitutive tensor defined as:

C =γ

2

∑

a

ma β r2 (n ⊗ n ⊗ n ⊗ n) . (10)

It can be observed that each coefficient of the elastic tensor C is a func-tion of the intermolecular distance r and the components of n, that is ofthe director cosines. Since the dependence on r is the same for all the co-efficients, each coefficient can be considered as a function of the directorcosines, only. In the general anisotropic case, the number of independentcoefficients is equal to the number N of combinations by repetition of thethree objects {n}i (i = 1 , 2 , 3) of Class 4: N= 6!

4!2!= 15. These coefficients

reduce to N= 6, in the case of orthotropy; to N= 3, in the case of identicalorthotropy and to N=1 in the isotropic case.

This circumstance leads to the experimental contradiction, known incrystal elasticity when studying triangular lattices with pair interactionsonly (Palla et al., 2007), for which all isotropic materials should have thesame Poisson’s coefficient (ν = 0.25). This model is called ‘rari–constant’ inopposition to the so–called ‘multi–constant’ model (Thodhunder and Pear-son, 1886–1889; Benvenuto, 1991), derived from the hypothesis by Green(1839) on the existence of a stored energy function corresponding to anexact differential form; as the quadratic strain energy in linear–elastic ma-terials.

11The summation is extended to all the molecules of the cylinder lying in a half-space

defined by m acting on a molecule B. Due to symmetry it becomes the summation

extended to all the molecules surrounding B, thus entailing the introduction of the

factor 1/2.

13

This experimental discrepancy was also found by 19th–century scientistsin measuring the Poisson’s coefficients of metallic materials12 and generateda long controversy particularly animated by Saint–Venant (Navier, 1864),who did not like to renounce a mechanical based molecular approach infavour of the merely mathematical Green’s (continuous) approach.

2.2 Voigt’s model with particle rotations

The possibility of clarifying the basic defects of the molecular theoryof elasticity was originally found outside the canonical studies on mechan-ics. In 1866, the publication of a work on crystallography by Bravais13

motivated Voigt (1887), a professor of theoretical physics, to improve theCauchy molecular model in order to circumvent the difficulties related tothe identification of the material constants.

His solution to the dispute on the elastic constants was based on the in-troduction of a molecular model made of oriented rigid particles interactingthrough forces and couples.14 Voigt also published various works on crys-tallography and elasticity (Voigt, 1887, 1900, 1910) reconciling the resultsof the mechanistic/corpuscular and of the energetistic/continuous theoriesof matter; the latter that was becoming increasingly important in relationto the development of conservative concepts in physics.

Basing on the hypotesis that the mechanism of crystal growth is gov-erned by an internal moment which gives orientation to the particles, Voigtproposed an hyperelastic model characterized by an intermolecular poten-tial of forces and couples from which derive, by stationariety, the balance

12Several experiments on metallic materials proved that the values of Poisson’s coefficient

ν could be different from 0.5 (G. Wertheim, Memoire sur l’equilibre des corps solides

homogenes, Annales de Chimie et de Physique, 23, 52-95, 1848) and was different

for various materials (G.R. Kirchhoff, Uber das Verhaltnis der Querkontraktion zur

Langendilatation bei Staben von federhartem Stahl, Annalen der Physik und Chemie,

108, 369–392, 1859).13 A. Bravais, Etudes cristallographiques, Gauthier–Villars, Paris, 1866.14The concept of force systems reducible to a force and a couple in mechanics was

introduced by L. Poinsot (Theorie nouvelle de la rotation des corps. Journal de

Mathematiques Pures et Appliquees, 16, 9–129, 289–336, Paris, 1851) who, when

studying the rigid motion of bodies, investigated how a system of forces acting on

a rigid body could be resolved into a single force and a couple. Successively, Lord

Kelvin (W. Thomson and P. G. Tait, Treatise on Natural Philosophy, I ed. Oxford,

1867; II ed. Cambridge 1879–1883) defined the continuum body as a material made

of points interacting also through forces and moments. The systematic treatise of the

Cosserat brothers on the deformable bodies with both the translational and rotational

degrees of freedom appeared on 1909 (Cosserat and Cosserat, 1909).

14

equations for the internal actions as well as the constitutive functions forthe stress measures. In this way he demonstrated that the Navier–Cauchymolecular theory failed only because it assumed a central–force scheme withforces only dependent on mutual distance between point–like particles andnot on their orientation.15

Let A and B be two interacting molecules perceived as rigid particlescentred at the positions a and b. The vectors w(a) and w(b) respectivelyrepresent the displacements of a and b, while the skew–symmetric tensorsW(a) and W(b) respectively represent the rotations of A and B. Thedisplacements of two points pa, on A, and pb, on B, are:

w(pa) = w(a) + W(A) (pa − a) , w(pb) = w(b) + W(B) (pb − b) . (11)

Let us denote with the vectors ta and tb and the skew–symmetric tensorCa and Cb, respectively, the force and couple that the molecule B (A)exerts on A (B) through pa (pb). Voigt defines the interacting potential asa function that takes into account the intermolecular elementary work ofboth forces and couples:

−dΠab = ta · dw(pa) + tb · dw(pb) +1

2[Ca · dW(A) + Cb · dW(B)] . (12)

By requiring dΠab = 0 for any rigid infinitesimal displacement, the bal-ance equations between pairs of molecules (A ,B) are obtained as:

ta + tb = 0 ,

Ca + Cb + (pa − pb) ⊗ ta − ta ⊗ (pa − pb) = 0 . (13)

Voigt then introduces a constraint acting as a strong constitutive pre-scription: the molecules have the same orientation within the so–calledmolecular sphere of action. That is, for any pair (A ,B) within this sphereit is:

W(A) = W(B) . (14)

Basing on this constraint of uniform rotations Voigt also assumes thatthe internal couples are equal:16

15Although the mathematical teory of elasticity was developing according to the Green

energetistic/continuous approach (e.g. (Love, 1906)), Voigt’s and Poincare’s molecular

models, the latter discussed in Subsection 2.3, were still well known at the beginning

of 20th century to be reported on textbooks as that by Marcolongo (1904).16To motivate this position Voigt (1910, p. 599) invokes symmetry considerations and

implicitly assumes linear constitutive relations between couples and rotations.

15

Ca = Cb = Cab . (15)

As a consequence, taking into account the balance equations (13) andputting:

tab = ta = −tb , (16)

Mab = (pa − pb) ⊗ ta − ta ⊗ (pa − pb) = 2Cab ,

by reducing the system of forces and moments of force to a forces system,the intermolecular potential becomes:

−dΠab = tab · d∆w , (17)

with ∆w = w(a) −w(b). The intermolecular force can be then derived as:

tab = − ∂Π

∂∆w. (18)

To define the stress measures Voigt recalls the Cauchy approach consid-ering a control cylinder elevated on a surface element of area dAm defined bythe outward unit vector m. The stress vector tm and the skew–symmetriccouple–stress tensor Cm are respectively defined as the summations, ex-tended to any molecule B within the cylinder interacting with a moleculeA in the half body outside the cylinder, of the interaction force and coupleper unit surface:

tm =1

dAm

∑

ab

tab , (19)

Cm =1

dAm

∑

ab

Mab .

Since the distance between molecules within the sphere of action is small incomparison with the diameter of the cylinder, and this diameter is infinites-imal, ‖ pa − pb ‖ is a higher order infinitesimal and, taking into accountEquation (16b), it is Mab = 0, for any pair (A ,B), and then Cm = 0.

Like for Cauchy, Voigt’s homogenization process resorts to a correspon-dence map, based on the hypotesys of regularity of both displacement androtation fields, that by linearization gives for any molecule A:

w(a) = w(x) + ∇w(a − x) , (20)

where x is a position of the continuous region occupied by the body andw(x) its homogeneous displacement (with ∇(·) = ∂(·)/∂x). Then by putting:

Θ =1

2(∇w −∇wT ) , E =

1

2(∇w + ∇wT ) ,

16

and, taking into account the constraint (14), it is:

W(A) = W(B) = Θ , ∀(A ,B) . (21)

Using Equations (20) and (11), when ‖ pa−pb ‖∼= 0 , the homogenizationrule is then obtained:

∆w = (∇w − Θ)(a− b) = E (a − b) . (22)

The constitutive response for the intermolecular force is a vector func-tion, f, of ∆w developable in series:

tab = f(∆w) ∼= f(∆w)|0 + ∇f(∆w)|0 ∆w + o(∆w) (23)

whose linear representation, in the natural state free of forces (f(∆w)|0 = 0),becomes:

tab = Kab ∆w , (24)

with Kab = ∇f(∆w)|0. Thus, taking into account Equation (20) and thesymmetry of the tensors Kab and E, the intermolecular force is:

tab = EKab (a − b) , (25)

in agreement with a principle of local action. Taking into account Equation(20) again, after some algebra, the intermolecular potential (17) becomes:

−dΠab = Kab(a− b) ⊗ (a − b)E · dE . (26)

The average potential of the molecular system is then defined consideringthe summation extended to all the pairs of interacting molecules within thesphere of action, supposed of volume V :

−dΠ = − 1

V

∑

ab

dΠab =1

V

∑

ab

[Kab(a − b) ⊗ (a− b)] E · dE . (27)

The stress–strain relationship are then derived as:17

T = −∂Π

∂E=

1

V

∑

ab

[Kab(a − b) ⊗ (a− b)] E

=1

V

∑

ab

(a − b)⊗ (a − b) ⊗Kab E = CE , (28)

17Equation (25) corresponds to the so–called virial stress formula of a particle system

under a deformation state E.

17

where T is the symmetric stress tensor and

C = − ∂2Π

∂E ⊗ ∂E=

1

V

∑

ab

(a− b)⊗ (a − b) ⊗Kab (29)

the elastic tensor. In the general anisotropic case the independent compo-nents of C,

[C]ijhk =1

V

∑

ab

[a− b]j [a− b]k [Kab]ih

(i, j, h, k = 1, 3), due to the major symmetries ([C]ijhk = [C]hkij), dΠ be-ing an exact potential, and to the minor symmetries ([C]ijhk = [C]jihk =[C]ijkh = [C]jikh), E and T being symmetric tensors, are 21. In the isotropiccase the elastic components reduces to 2. Thus providing a ‘multi–constant’model, like the Green hyperelastic model. Given its double nature, mecha-nically and energetically based, such a model can be defined as a mechanis-tic/energetistic model.

2.3 Poincare’s refined molecular model for elasticity

In those years Poincare (1892) presented a molecular approach in elastic-ity which also by–passed the intrinsic limitation of the central–force scheme,and then the problem of the identification of the correct number of materialconstants in linear elasticity. He proposed a lattice model refined by theintroduction of a potential energy function of point–like molecules, inter-acting not only in pairs; thus providing a description that in current termswould be called ‘multibody’ potential description.

Poincare’s model is made of molecules of small dimensions with respectto their distance and are conceived as material points interacting, in pairsand triplets, through forces depending on their mutual distance directedalong this distance.

Let us consider a system of N molecules occupying the places Xi in thereference configuration and Xi + wi after a deformation. The elementarwork of the internal forces, fi, of the system is an exact differential of ascalar function of any current position U = U(Xi + wi):

N∑

i=1

fi · dwi = dU , (30)

In this circumstance the internal forces are conservative:

fi =∂U

∂wi

. (31)

18

Let now M1 and M2 be two interacting molecules occupying the placesX and X+∆X, respectively. After a deformation the two molecules occupyrespectively the places (X + w) and (X + ∆X) + (w + ∆w); w being thedisplacement of M1 and ∆w the relative displacement between M1 andM2. We put r = ∆X and introduce, with Poincare, the scalar quantities:

R = r · r = r2 , (32)

R + ∆R = (r + ∆w) · (r + ∆w) = (r + ∆w)2 .

The scalar quantity:∆R = 2 r · ∆w + ∆w2 ,

can be then divided into a linear and a quadratic term in w:

∆R = ∆R1 + ∆R2 , ∆R1 = 2 r · ∆w , ∆R2 = ∆w2 . (33)

Poincare assumes that the potential energy, that he calls force func-tion, depends only on the quantities R + ∆R, evaluated for more thanone molecule interacting with a given molecule. This function can be ex-panded into Taylor’s series stopped at the second order in such a way that,tacking into account Equations (33) for any pair of molecules (Mi ,Mj),(Mi ,Mh) , ... (with i 6= j 6= h) within the molecular sphere of action, it is:

U = U((R + ∆R)ij, (R + ∆R)ih, ...) ∼= U0 + U1 + U2 , (34)

with

U0 = U((R)ij |0, (R)ih|0, ...) ,

U1 =∑

ij

∂U

∂(R)ij

|0 (∆R1)ij ,

U2 =∑

ij

∂U

∂(R)ij

|0 (∆R2)ij

+1

2

∑

ij

∂2U

∂(R)2ij|0 (∆R1)

2ij +

∑

ij,ih

∂2U

∂(R)ij∂(R)ih

|0 (∆R1)ij(∆R1)ih ,

where

(∆R)2ij∼= (∆R1)

2ij , (∆R)ij(∆R)ih

∼= (∆R1)ij(∆R1)ih .

The potential energy function of the assembly is then a (force) functionon any intermolecular distance not only of pairs but also of triplets of point–like molecules. In current terms this corresponds to a three–body potential

19

description.18 In Equation (34) U0 is an inessential constant; U1 and U2 arethe linear and quadratic term, respectively. It is worth noting that underthe hypothesis of natural state free of force,19 U1 = 0, while under thehypothesis of central–forces, that is when only pair interactions are takeninto account, it is:

∂2U

∂(R)ij∂(R)ih

= 0 . (35)

At this point Poincare, like Cauchy and Voigt, introduces an affine maplinking discrete and continuum variables which plays the role of homoge-nization rule:

∆w = ∇w r (36)

in such a way that, taking into account Equations (33),

∆R1 = 2∇w · r ⊗ r = 2E · r⊗ r , (37)

∆R2 = ∇wT∇w · r ⊗ r ,

with E = 12 (∇w + ∇wT ).

Then, the second order term in (34) is composed of the following three

18Since the product of two distances is reportable to the angle formed by three point–

molecules, the description corresponds to an angular potential description.19Poincare reports this hypothesis as the Lame hypothesis.

20

terms:

U ′

2 =∑

ij

∂U

∂(R)ij

|0 (∆R2)ij

=∑

ij

∂U

∂(R)ij

|0 rij ⊗ rij · ∇wT∇w

U ′′2 =

1

2

∑

ij

∂2U

∂(R)2ij|0 (∆R1)

2ij

= 2∑

ij

∂2U

∂(R)2ij|0 rij ⊗ rij ⊗ rij ⊗ rij · E⊗ E

U ′′′2 =

∑

ij,ih

∂2U

∂(R)ij∂(R)ih

|0 (∆R1)ij(∆R1)ih

= 4∑

ij,ih

∂2U

∂(R)ij∂(R)ih

|0 rij ⊗ rij ⊗ rih ⊗ rih · E⊗ E . (38)

It can be noted that the term U ′2, linear in the 6 components of ∇wT∇w

and requiring 6 independent coefficients, is null in the force–free naturalstate. U ′′

2 and U ′′′2 are the terms, quadratic in the infinitesimal strain E

and requiring 21 independent coefficients, accounting for pair and tripletinteractions, respectively. The potential function can be then written as aquadratic form in E:

U = U ′′

2 + U ′′′

2 =1

2CE · E , (39)

where C is the constitutive tensor of 21 coefficients.It can be finally shown that in the central force hypothesis (Equation

(35)), for which U ′′′2 = 0, six relations among the elastic coefficients holds

and the independent components of C become 15, thus returning to theCauchy ‘rari–constant’ description.20

2.4 Discrete–continuum theories: new perspectives

The 19th century was generally characterized by the attempt of mech-anistic interpretation of any physical phenomena, most of which were de-scribed as propagated by contact in space through a medium, like in elas–icity.6 The mechanistic perspective of the molecular theories of elasticity

20These relations are the so–called Cauchy–Poisson relations reported by Marcolongo

(1904); Stakgold (1949).

21

was then much appreciated by the scientists of the period, among these, asmentioned above, Saint–Venant (Navier, 1864), mostly because it preservedthe Newtonian interpretation of force as mechanical interaction betweenmaterial particles.

Indeed, molecular theories in mechanics, as well as in thermodynamicand in the electromagnetic frame, or more recently in statistical and quan-tum mechanics, were essentially based on the concept of the discrete natureof matter. By taking into account the actual discrete nature of matter,the risks regarding metaphysical implications, always inherent to NaturalPhilosophy when the ‘causes’ of physical phenomena are sought, were cir-cumvented. Despite the experimental support derived from the first dis-coveries of the existence of atoms,21 the introduction of the concept of dis-creteness in physics, and of the corresponding ‘simple mathematics’ (matrixalgebra), was difficult and opposed for a long time. The main difficultiesconsisted in removing the Laplacian principle of ‘continuum indispensabil-ity’ and the related differential mathematics (Drago, 2004). On the otherhand, in agreement with the predominant tradition of Lagrange’s analyticalmechanics,22 the aim of the elasticians of the period, although they usedcorpuscular descriptions to derive constitutive relations, was the construc-tion of continuous models in order to obtain field equations and use theapparatus of differential mathematics. Therefore, in a scientific frameworkdominated by the analytical mechanics, it does not seem strange that, in thepresence of experimental discrepancies, the molecular approach was rapidlyabandoned. Even if the significant works of Voigt and Poincare providedenriched corpuscular–continuum constitutive models, which gave results inagreement with the experimental data, the theory of potential by Green(1839), based on the mathematical assumption that the work of any in-ternal force system is an exact differential, also confirmed by experiments,were definitively adopted in elasticity.23

At present, the determination of structure-property relationships for ma-terials with microstructure often requires the material description at vari-ous length scales, and models based on discrete–continuous descriptions en-

21W. Prout, On the relation between the specific gravities of bodies in their gaseous state

and the weights of their atoms, Annals of Philosophy, 6, 321-330, 1815; R. Brown, A

brief account of microscopical observations made in the months of June, July and

August, 1827, on the particles contained in the pollen of plants; and on the general

existence of active molecules in organic and inorganic bodies, Philosophical Magazine,

4, 161-173, 1828; J. J. Thomson, Cathode rays, Philosophical Magazine, 44, 1897.22G. L. Lagrange, Mecanique Analytique, Academie Royale des Sciences, Paris, 1788.23E.g. Love (1906). For a general retrospect of continuummechanics on the 20th century,

focusing the basic concepts and their recent developments see (Maugin, 2013).

22

counter renewed interest. Globally speaking, coarse–graining homogeniza-tion resorting to explicit link between the fine and coarse scale behaviourscan increase our understanding of the microstructural origin of deformationmechanisms of the tailored materials widely used nowadays. Thus, sugges-tions for constructing physically motivated constitutive models for complexmaterials are provided.

Not very differently than in the past, in current atomistic modelling aswell as in the mechanics of composite materials, there are many circum-stances in which the inadequacy of Cauchy’s hypotheses:

(i) homogeneity of lattice–continuum deformations (‘harmonic approxi-mation’);

(ii) particles interacting in pair depending on their mutual distance (‘cen-tral force scheme’);

call for the need of improved hyperelastic models, as for instance to by–pass experimental discrepancies related to phenomena dominated by themicrostructure size (plasticity, damage, fracture, long range interactions,etc.). To this aim the strategies proposed are essentially of two kinds.

• One strategy is to remove the hypothesis of homogeneous deforma-tions (i), developing hyperelastic material models with non–convexenergies (Zanzotto, 1996; Friesecke and Theil, 2002; Del Piero andTruskinovsky, 2009; Braides and Gelli, 2000).

• Other approaches remove the local character of the description bymodifying the central–force scheme (ii), obtaining continua with ad-ditional degrees of freedom or multi–body potential descriptions.

On the track of Voigt and Poincare, this last approach is based on the refine-ment of the lattice models either by endowing the particles with extension,by adding degrees of freedom, or by considering multiple interactions amonglattice points. In the former case, the equivalent continua derived belong tothe family of multifield continua (Section 3). The latter is the case of con-tinua originated by multi–body potential descriptions, widely explored inthe field of condensed matter physics. These models are developed in orderto calculate chemical–physical properties, important in the design of highperformance complex materials, avoiding having to determine empirical po-tentials, which are necessary when ‘non–ideal’ materials are accounted for(Pansianot and Savino, 1993; Delph, 2005; Palla et al., 2007).

In both Voigt’s and Poincare’s models the hypothesis of homogeneous de-formations, which is related to a principle of local action, is maintained andthe description is enriched by angular potentials.18 It can be noticed thatEquations (21) or (35) act as internal constraints leading to simple continua

23

that can be seen as continua with ‘latent microstructure’ (Capriz, 1985). Ifno constraints were posed, the derived homogeneous continua would showthe dependence on material length parameters, thus belonging to the classof ‘implicit’ non–local continua. Otherwise, if non–homogeneous deforma-tions were taken into account the equivalent continua would be classifiedas ‘explicit’ non–local media. These concepts will be specified in Section 3.The conceptual framework of Voigt’s coarse–graining process can be appliedto materials described, at a conventional micro–level, as assemblage of in-teracting structured molecules, which in a wide meaning represent variouskinds of internal phases (fibres, microcracks, voids, etc.). The constitu-tive multifield model for composite materials with fibres and flaws, derivedin Section 4, provides an example of the current effectiveness of such anapproach.

3 Multifield continua, basics synoptic

Macroscopic description of the internal structure for materials made of partsof significant size, avoiding physical inadequacies and theoretical as well ascomputational problems, always concerns non–local modelling. In agree-ment with the definition by Kunin (1982), we call non–local continua ormicrocontinua any continuum retaining a memory of the fine material struc-ture through internal length and dispersion properties. In agreement withthe definition of complex continua by Capriz and Podio-Guidugli (2004),such a continuum can be also called non–simple continuum. Internal lengthparameters can be the distance between particles in a lattice, grain or cellsize, correlation radius of at–a–distance force, etc. Spatial dispersion isinstead related to the dependency of wave–velocities on wave–length or fre-quency. The classical continuum (Grade 1) is lacking in both the abovementioned features.

The presence of internal lengths and dispersion parameters in the fieldequations is ensured both in the ‘explicitly’ or ‘strongly’ (Kroner, 1967;Eringen, 1983, 2002) non–local models and in the ‘implicitly’ or ‘weakly’non–local descriptions.24 For all of these non–local models an extendedvirtual power framework, with standard and non–standard primal fieldscoupled with dual standard and/or non–standard fields, is considered. Thecircumstances below are distinguished.

(a) The standard equations of motions contain derivatives in space or timeof the standard primal field (macrovelocity) of order different than the

24The former pair of opposite definitions is by Kunin (1982); Eringen (1999) and the

latter by Maugin (1979, 2010).

24

second.(b) The equations of motion contain non–standard primal fields and:

(b1) they can be derived from the standard frame invariance axioms,as in the classical case (Noll, 1963; Green and Rivlin, 1964a; Podio-Guidugli, 1997);(b2) there are additional equations of motion which contain non stan-dard primal fields and, usually but not necessarily, their derivatives ofvarious order. The derivation of these equations requires a differentaxiomatic framework than the invariance of power under changes ofobserver (Germain, 1973; Gurtin and Podio-Guidugli, 1992; Di Carlo,1996).

Examples of case (a) are higher grade theories or rate–type materials(Needelman, 1988; Muhlhaus and Aifantis, 1991; de Borst and Muhlhaus,1992; Sluys et al., 1993; Bassani et al., 2001; Aifantis, 2010). These theo-ries are also related to explicit non–local descriptions (Bazant et al., 1984;Pijaudier-Cabot and Bazant, 1987), as it has been known for some time(Beran and McCoy, 1970; Muhlhaus and Aifantis, 1991).

Models of kind (b) are continua with additional degrees of freedom withrespect to the classical continua, here called multifield continua. These con-tinua are implicitly (weakly) non–local, always because of the presence ofinternal length scales and dispersion properties. Among multifield continuawe further distinguish the categories: (b1) micromorphic continua (Eringen,1964a, 1999; Mindlin, 1964) and (b2) other non–classical continua with ma-terial or configurational forces (Capriz, 1989; Maugin, 1993, 2011; Gurtin,1995, 2000). Note that case (b1) can be seen as a special case of (b2). More-over, as detailed below,the higher gradient models of the Mindlin (1965) type(Mindlin and Eshel, 1968) can be seen as belonging to the category (b1).

Theories of kind (a) represent an effective way to overcome the inade-quacies of simple material models (Pijaudier-Cabot and Bazant, 1987; Sluyset al., 1993). However, difficulties arise when a thermomechanical processis considered.25 In the following we focus on continua of kind (b), whichnaturally are thermodinamically compatible.26 Some basic features of mul-tifield continua are recalled, only to emphasize the main common ideas that

25For example Gurtin (1965) showed that the constitutive assumption of a continuum

of grade N > 1 is incompatible with the second law of thermodynamics, unless ad-

ditional quantities or additional (dual) variables were introduced. Successively, Dunn

and Serrin (1985) introduced an additional quantity, referred as interstitial working,

in the energy balance which preserve the validity of the Clausius–Duhem inequality.

From balance considerations, Capriz (1985) proved that this quantity corresponds to

introducing in this inequality the internal additional density power of a multifield con-

tinuum.26Also internal variable models, originating from the work of Coleman and Gurtin (1966),

25

underlie the various formulations proposed in literature. A rigorous expo-sition concerning non–classical continua within a generalized virtual powerframework is referred to Chapter 2, by Del Piero.

Origin of the theories of continua with additional degrees of freedom wasthe work of Voigt, described in Section 2, who introduced continua with par-ticle rotations and couple interactions deriving constitutive functions fromthe requirement of stationarity of potentials of both forces and couples. Suc-cessively, he also introduced constraints on rotations and couples obtaininga classical model that, as mentioned above, can be seen as continuum withlatent microstructure (Capriz, 1985).

At the beginning of 20th century the work of the Cosserat brothers, devel-oped within a structured variational context, was published (Cosserat andCosserat, 1909). This general treatise for continua with rotational degreesof freedom and moment interactions was destined to definitively influencethe mechanics of continua with additional degrees of freedom introduced toaccount for the physical or geometrical microstructure of materials. Thatis to account for the discontinuous and heterogeneous nature of the matteror to overcome the limitations of reduced dimension structural models.

In the 50’s, the concept of additional degrees of freedom began to be re-covered, first to study reduced dimension problems (Mindlin, 1951), devel-oping into a theory of continua made of particles endowed with deformabledirectors (Ericksen and Truesdell, 1958). At the beginning of 60’s, startingfrom some ideas of Truesdell and Toupin,27 the concept of continua madeof particles endowed with deformable directors developed in the theory ofgeneralized velocity and stress (Toupin, 1962; Mindlin, 1963; Green andRivlin, 1964b), also with focus on hyperelastic materials with strain energydepending on the first and higher order deformation gradients (Grioli, 1960;Toupin and Gazis, 1963).

The matter physics point of view was proposed by Gunter (1958) andSchaefer (1965) who used the Cosserat continuum to account for the pres-ence of materials dislocations. Then it was developed by Aero and Kuvshin-skii (1961); Kroner (1963, 1968); Kunin (1968, 1982, 1983) who particularlyfocused on materials with internal length, that is on scale parameters con-nected with geometry or long–range interaction forces.

present additional, local and/or non–local, variables and provide solutions to the prob-

lem of thermodynamic compatibility with the introduction of terms related to internal

power dissipation (see for instance Chapter 6, by Tarleja). These terms are non observ-

able variables satisfying kinetics which do not have the meaning of balance equations

and are not included in the classification proposed here.27Truesdell and Toupin (1960), Sections 166, 205, 262.

26

With the basic paper by Mindlin (1964) the need for continua with ad-ditional degrees was clearly related to the presence of material microstruc-ture. In this work microstructure is represented by a unit cell interpreted asa molecule of polymer, a crystallite of a policrystal or a grain of a granularmaterial. The mathematical model of the cell is a linear version of Ericksenand Truesdell’s deformable directors’ model, which in the case of rigid di-rectors becomes a Cosserat’s model. In the same years, and those following,the systematic works of Eringen and Suhubi (1964); Eringen (1964a,b, 1965,1999), who defined the microcontinua within a hierarchy which includes mi-cromorphic, microstretch, micropolar (Cosserat) and classical models, ap-peared.

Those were years in which composite materials were beginning to belargely used in aerospace and naval engineering for instance, and the in-terest in developing theoretical, mathematical and computational tools forthe modelling of their behaviour became a new challenge for continuum me-chanics and its relevant applications. To illustrate the practical interest ofthese non–classical theories a mention also goes to a few explicit solutionsthat can be found in literature since the early 60’s, with particular refer-ence to micropolar elasticity adopted for treating problems in the presenceof load or geometrical singularities (Mindlin and Tiersten, 1962; Sternbergand Muky, 1965, 1967; Bogy and Sternberg, 1968; Nowacki, 1970; Cowin,1970; Stojanovic, 1972; Sokolowski, 1972).

Studies involving more additional independent fields, such as kinematicaladditional degrees of freedom or independent state variables, were widelydeveloped until recently, several in a general mechanical and thermome-chanical framework (Green and Rivlin, 1964b; Kafadar and Eringen, 1971;Eringen and Kafadar, 1976; Nowacki and Olszak, 1972; Green and Naghdi,1995; Bertram and Forest, 2014), and many others oriented to engineeringapplications (Forest, 2009; Forest and Trinh, 2011; Altenbach and Eremeyev,2013; Forest, 2013; Trovalusci and Pau, 2014). Among these, some basictreatises particularly address more general descriptions of microstructure,accounting for every kind of heterogeneities such as voids, defects, bub-bles, rigid inclusions, etc. (Capriz, 1989; Maugin, 1993, 2011; Gurtin, 1995,2000; Kienzler and Herrmann, 2000; Kienzler, 2001; Svendsen, 2001; Svend-sen et al., 2009). These models can be traced back to the pioneering work ofEshelby (1951), concerning the internal actions related to the changes of thematerial configurations (defects), and can be structured within a general-ized virtual power approach (Germain, 1973; Maugin, 1980; Di Carlo, 1996;Fremond and Nedjar, 1996; Fried and Gurtin, 2006; Podio-Guidugli, 2009;Del Piero, 2009, 2014; Podio-Guidugli and Vianello, 2010; Fried and Gurtin,2011; Maugin, 2013), crucial for the derivation of the equations of motion

27

when, as mentioned above, the standard invariance axioms fail (Gurtin andPodio-Guidugli, 1992).

According to Capriz (1989) a continuum with microstructure (b2) isa mathematical model of material bodies endowed with some sort of mi-croscopic order preserving the classical scheme of the continuum descrip-tion. Each material point, P , corresponds to its geometrical position,X ∈ E , E being the three–dimensional Euclidean space, and to ν parame-ters, Ξα (α = 1, 2, ..., ν), representing the inner structure. These so–calledorder parameters belong to an appropriate manifold, M, of a given dimen-sion. A complete placement for P is: P (X, Ξα) (α = 1, ν). The parametersand eventually their gradients, included in the internal power formula asprimal variables, define the Grade of the micromodel: 0, 1, ...,N . They canbe of a different nature, not only kinematic. This scheme is compatiblewith the picture of a material point endowed with physical or geometricalproperties, such as mass, porosity, extension, orientation, electric field, etc.

In the description provided by Eringen (1999), but also Mindlin (1964),a microcontinuum (b1) is a continuous medium perceived as a collection ofmaterial particles endowed with additional degrees of freedom representingthe material sub–structure (particles can be called sub-continua). Theseadditional parameters have kinematical meaning and are represented asvectors (M ≡ V, V being the translation space of E).

Let us consider the case in which α = 1 and denote with (C , C) ⊂ E thereference and deformed shape of a body (B = E × M), with X ∈ C. Themacro and microtransplacement fields can be defined as:

x : X → x ∈ C , µ : (X, M) → µ ∈ M .

The macro and microtransplacement gradients can be introduced:

F =∂x

∂X= ∇x , Fµ =

∂µ

∂X= ∇µ , (40)

together, but not necessarily, with higer order gradients:

Fn =∂nx

∂X ⊗ ...⊗ ∂X= ∇nx , Fn

µ =∂nµ

∂X ⊗ ...⊗ ∂X= ∇nµ , (41)

with n=2,...,N.Eringen defines as micromorphic a microcontinuum (b1) for which the

microtransplacement field can be linearized in M :

µ ∼= 0 + ∇µ|0 M = ΨM ,

28

where Ψ is the microdeformation parameter. In this way:28

Fµ = Ψ , F2µ = ∇Ψ . (42)

In the following we consider a linearized framework, in which veloci-ties/angular velocities stand for infinitesimal displacements/rotations andthe power for rate of work. Let w and wµ be the macro and microveloc-ity vectors and Ψ the linearized microdeformation measure of a two–fieldsmicrocontinuum.

In agreement also with Mindlin’s (1964) description, a micromorphiccontinuum is a continuum for which Ψ = ∇wµ. Accordingly, we here extendthe Eringen (1999) classification, obtaining special cases of micromorphiccontinua by introducing internal constraints.29 For example: Ψ = ∇w

leads to a second gradient continuum; Ψ = W, with W ∈ Skw30 an angularvelocity, leads to a micropolar continuum (Cosserat).

Table 1. Micromorphic continua: kinematic and dynamic descriptors

MACRO RELATIVE MICRO

Micromorphic Ψ = ∇wµ

strain E = 12(∇w + ∇wT ) Γ = ∇w −∇wµ ∇Ψ = ∇2 wµ

rotation Θ = 12(∇w −∇w

T ) Θµ = 12(∇wµ −∇w

Tµ )

stress T A S

Second Gradient Ψ = ∇w

strain E = 12(∇w + ∇w

T ) Γ = 0 ∇Ψ = ∇2w

rotation Θ = 12(∇w −∇w

T ) Θµ = 12(∇w −∇w

T )stress T S

Micropolar Ψ = W

strain E = 12(∇w + ∇wT ) Γ = ∇w − W ∇Ψ = ∇W

rotation Θ = 12(∇w −∇w

T ) Θµ = W

stress T A S

In Table 1 the kinematical and dynamical (macro, relative, micro) de-scriptors of a micromorphic continuum are reported. With reference to the

28It can be noted that Eringen defines the grade of the microcontinuum as the number

of parameters α. In this sense a micromorphic continuum is a material of grade α =

1. Referring to the definition of the material grade recalled above, this definition is

coherent with the introduction of the single microdeformation gradient ∇Ψ. However,

since Ψ is interpreted as a gradient, the micromorphic continum is a continuum of

Grade 2. The linearized case reported illustrates this assertion.29See also (Sulem et al., 1995; Forest, 2009, 2013).30Sym and Skw indicate the set of the symmetric and skew–symmetric second order

tensors, respectively. In the following the operators sym and skw respectively select

the symmetric and the skew–symmetric part of a second order tensor.

29

strain and stress measures defined in this table, the structure of a micromor-phic continuum can be then encoded in its internal power density formula,with these measures as primal, kinematic, and dual, dynamic, fields:

πint = T ·E + A · Γ + S · ∇Ψ = S · ∇w −A ·Ψ + S · ∇Ψ , (43)

where S = T + A, with T ∈ Sym and A ∈ Skw.31

It can be observed that the second gradient model perceived as a con-strained micromorphic model, so that the microdeformation equals themacrovelocity gradient, differs from the so–called second grade models ofkind (a) because of the presence of a microstress measure (S) power conju-gated to higher order gradients of velocity (∇Ψ = ∇2 w).32 The micropolarmodel, perceived as micromorphic model in which the microdeformation isconstrained to be an angular velocity (microrotation), is a model with rigidlocal structure.

Let us now consider a two–field continuum characterized by the macrov-elocity vector field w and an independent micro vector field wµ which canbe a rate of a field of a different nature other than kinematic (b2).33 Consid-ering these fields and their gradients up to the N th order as strain (primal)fields the internal power formula can be symbolically expressed as:

πint = s · w + sµ · wµ + S · ∇w + Sµ · ∇wµ (44)

+ S2 · ∇2 w + S2µ· ∇2 wµ + ... + SN · ∇N w + SNµ

· ∇N wµ,

where the stress (dual) fields are also present from the zero up to the N th

order. It can be shown that the standard invariance axioms yield the localbalance of macroactions but the derivation of the local balance of microac-tions requires an extended virtual work setting (Germain, 1973; Di Carlo,1996).34This circumstance is shown in the case study of Section 4.3.

31In the case of micropolar media, since the term ∇Ψ and its dual S are related to a

skew–symmetric tensor and its dual, their inner product in Equation (43) should be

multiplied by a factor 1/2. To highlight the common structure of the micromorphic

continua power formula, this factor, only related to the tensor representation, is not

reported therein.32In these models the microactions by-contact can depend not only on the local unit

normal but also on the curvature or on the edges of a selected control volume (Toupin,

1962; Mindlin, 1965; Mindlin and Eshel, 1968; Germain, 1973; Dell’Isola and Seppecher,

1995; Fried and Gurtin, 2006; Podio-Guidugli and Vianello, 2010).33 Here the case in which M is a vector space is represented. For a general treatment of

continua with scalar or tensorial microstructure see (Capriz, 1989; Capriz and Podio-

Guidugli, 2004).34Gurtin (2000) in his basic work on materials with configurational forces asserts that

30

The detailed description of these assertions is beyond the scope of thischapter, but it is worth noticing that in the presence of a rigid transplace-ment the internal power density of a micromorphic medium (Equation 43)is null. Differently, in the case of multifield media the rigid microtransplace-ment is in general distinguished from the rigid macrotransplacement, thenthe additional strain measures and the internal power density (Equation44) can be non zero under a rigid macrotransplacement. Coherently, theaxiomatic scheme recalled here (Di Carlo, 1996) imposes the condition thatπint = 0 under any macro and micro rigid velocity fields. This requirementleads to additional balance equations, which act as constitutive prescrip-tions for the multifield continuum.35 This circumstance is also shown in thecase study of Section 4.3. Another feature that can be finally highlighted isthat, differently from continua of higher grade and standard stress measures(a), in multifield continua (b) to each strain measure a power conjugatedcounterpart is associated. This makes multifield continua naturally compat-ible with the thermodynamical Clausius–Duhem restriction (Gurtin, 1965;Capriz, 1985).25

The theory of multifield continua is now sufficiently developed withina theoretical framework and its validity for studying coarse scale problemsdominated by scale effects, without incurring in physical inconsistencies,with ill–conditioning in the field equations and the related mesh–dependencyin numerical solution, is widely recognized. Classical computational homog-enization schemes based on the principle of separation of scales are in factconsistent with the standard local continuum mechanics concept, that es-sentially recalls the principle of local action and homogeneous deformation.When classical simple models fail, internal length scales and dispersion prop-erties casted within enhanced homogenization procedures are required. Thisis for example the case of size dependent problems, as the widely investi-gated case of strain localization and damage–fracture detection (Muhlhausand Vardoulakis, 1990; de Borst, 1992; Sluys et al., 1993; Su, 1994).

The applicability of non–classical continuum theories, however, is strictlyconnected to the development of constitutive models and numerical proce-

additional fields need independent observers to measure their generalized velocities (mi-

crovelocities) and consequently additional independent balance laws. Following Gurtin

and Podio-Guidugli (1992), we can also recognize that the microfield wµ, differently

from the classical fields w, can be invariant under Galilean changes of frames and then

the balance of microforces cannot be a consequence of standard Galilean invariance.35It can be easily shown that this requirement imposes to the zero–order term s in the

power formula (44) to be null, differently from sµ which appears in the microforce

balance.

31

dures, which for several years is undergoing a big propulsive boost expeciallywithin the framework of a non–linear computational multiscale approach(Trovalusci and Masiani, 2003, 2005; Forest, 2009; Geers et al., 2007; Liuet al., 2009; Sansalone and Trovalusci, 2010; Addessi et al., 2010; Bellis andAddessi, 2011; Coenen et al., 2012a,b; Addessi and Sacco, 2012). Movingfrom Voigt and Poincare suggestions, non–classical multiscale analyses, thatovercome the principle of scale separation and adequately take into accountthe materials internal lengths at the coarse level, become decisive. Thus,provided by constitutive information and specific strategies for numericalsolutions (enhanced multiscale approaches), the microntinuum theoreticalframework appears as one of the main promising challenges in the studyof any kind of complex material. This also addresses the production ofmaterials with artificial innovative properties, intended to have broader ap-plications in various fields of engineering (Geers et al., 2010; Nguyen et al.,2012).

4 A molecular–multifield model for composites

The basic intention of this study is to point out that discrete modelling ofmaterials, crucial in the past for constructing constitutive theories for solids,can still be of help in deriving physically plausible constitutive models forcomplex materials, avoiding the complex and uncertain determination ofmaterial response functions directly at the coarse scale. While for compos-ite materials the necessity of a gross modelling is generally ascertained, theeffectiveness of this description is conditioned by the choice of the specificcontinuum model which retains a memory of the material internal struc-ture. The procedure here proposed indicates one way to select the kind ofmacroscopic multifield model equivalent to discontinuous physically–baseddescriptions of materials at a conventional micro–level.

The most significant suggestion derived by Voigt and Poincare enrichedmodels is the idea of developing a non–local internal power formula forcontinua energetically equivalent to discrete systems to be defined case bycase. The key ideas of this approach can be resumed as follows:

• a material can be described as an assemblage of interacting ‘molecules’endowed with extension, which in a wider meaning represent materialphases (fibres, microcracks, voids, etc.);

• inter–molecular actions can depend not only on the mutual particlesdistance but also on the particle orientation or other (no central–forcesscheme);

• the local character of the description can be removed without re-sorting to non–homogeneous discrete–continuum mapping (weak non–

32

locality);

• ‘a priory’ general principles governing macroscopic phenomena are notnecessary.

As a sample problem we consider the derivation of a constitutive modelfor a composite material made of inclusions of two types: stiff fibres em-bedded in a more deformable matrix (glass, carbon, etc.) and flaws due tomanufacturing defects or lack of cohesion (microcracks, pores, etc.). Thismodel can be representative of a wide class of quasi–brittle composites ma-terials, such as fibre–reinforced ceramic (C/SiC), porous ceramic composite(CMC like Al2O3/ZrO2) or metal-ceramic composites (MCM like WC/Co,TiC-Mo2C/Ni), but also geomaterials, jointed or porous rocks and masonry.The architecture of the considered composites consists of a polycrystallinestructure made of rigid particles, whose shape is polygonal, interactingthrough interfaces (grain boundaries, joints, etc.), sometimes filled by moredeformable material (cobalt, nichel, mortar, etc.) The pores also have animportant role and it is assumed that they are localized at the interfaces.

The lattice model is made of rigid particles, representing the fibres, muchstiffer than the matrix in which they are embedded, and by a distributionof flaws, representing the microcracks or pores. Assuming that a represen-tative volume element of the discrete microscopic model can be defined, theequivalent macromodel is determined by the requirement of the preservationof power in the transition between discrete and continuum description, forany admissible deformation rate field. This continuum directly derives fromthe selected map, that is a generalization of the Cauchy–Voigt–Poincaremap,36 which relates discrete and continuum degrees of freedom. In theframework of a non–simple theory, a formula for the mechanical power ofthe multifield continuum power, which encodes a rigid (fibres) and affine(flaws) local structure, is then derived. The balance equations for the, stan-dard and non–standard, actions of the multifield continuum are successivelyderived from the axiom of vanishing power and of invariance of power un-der changes of observers (Di Carlo, 1996). It is worth noting that the microto macro transition casted within an integral non–variational setting givesresults which apply regardless of the material response. Once the responsefunctions for the internal actions of the lattice models are selected, thehomogenized constitutive functions are then derived in terms of materialconstants, shape, size, orientation and texture of the constituent phases.37

36As already mentioned, this map corresponds to the so–called Cauchy–Born rule in

crystal physics.37 A general description of the presented model, that also addresses to the derivation of

the external, inertial and non–inertial, actions, is reported in Trovalusci et al. (2010).

33

4.1 Lattice model of a material with fibres and flaws

The discrete model adopted for the fine description of the reference com-posite material is made of a kind of structured molecules broadly repre-senting the internal phases of the material: the fibres, described as rigidparticles of polygonal shape, and the flaws, perceived as slits of arbitraryshape and a predominant dimension. The slits are considered opened, sta-tionary and with blunt edges (no tip effects accounted for). The particlesinteract in pairs through forces and couples while the slits interact throughforces directed along the line connecting their centres. Particles and slitsalso interact each other by forces. The slits must be considered as devicesto transmit to the matrix additional forces due to the presence of defects.In this sense they represent the microcracks/pores. Their stiffness dependson the surrounding elastic field. In this paragraph we limited the analysisin the framework of a linearized theory, so that the velocity fields stand fordisplacement fields and the power for work.

Let A and B be two particles centred respectively at a and b, and Hand K two slits with centres h and k respectively. The vectors wa and wb

denote the velocity of a and b respectively, and the skew–symmetric tensorsWa and Wb denote the angular velocities of the two particles respectively.For each pair ith of adjacent particles we define as strain measures of thelattice:

wi = wap − wb

p = wa − wb + Wa(pa − a) − Wb(pb − b) ,

Wi = Wa −Wb , (45)

where pa and pb are two test points, on A and B respectively, throughwhich the particles interact, and wa

p and wbp their velocities, respectively.

Considering the direction nh (nh · nh = 1) of the major axis of a slit H,we assume that the slit is deformed only in the plane normal to nh. Furtherlattice strain measures for each slit H, each pair jth of interacting slits (H,K) and each pair lth of interacting particle–slit (A, H) are:

dh , dj = dh − dk , wl = wap − (wh + dh) , (46)

where: the vector dh represents in a smeared sense the half–crack openingdisplacement of H and wh is the velocity vector of h.

The forces and the couples that B (A) exerts on A (B) are representedrespectively by the vector ta (tb) and the skew–symmetric tensor Ca(Cb).The force due to dh on H is represented by the vector zh

o . Due to thedisplacement jump dh, the slit interacts with the adjacent particles and theneighbouring slits. The vector zh (zk) is the action that K (H) exerts on H

34

(K), while the vector rh (ra) represents the action transmitted by H (A) toA (H). Considering the directions:

nj = (h − k)/ ‖ h − k ‖ , nl = (a − h)/ ‖ a − h ‖ ,

we put the following prescriptions on internal lattice actions:

(nh ⊗ nh) zho = 0 , (I− nj ⊗ nj) z

h = 0 , (I − nl ⊗ nl) rh = 0 , (47)

where I is the identity tensor. Note that Equations (47b,c) corresponds toa central–force hypothesis for the slit and particle–slit interactions.

If the material can be considered periodic, or at least statistically homo-geneous, a representative volume element Mµ, referred as the module, canbe individuated. The balance equations for each pair (A,B), (H,K), (A,H)in Mµ are:

ta + tb = 0 , Ca + Cb + [(pa − pb) ⊗ ta − ta ⊗ (pa − pb)]

+[(pa − h) ⊗ rh − rh ⊗ (pa − h)] = 0 ,

zh + zh = 0 , ra + rh = 0 . (48)

Then by putting: ta = −tb = ti , zh = −zk = zj , rh = −ra = rl ,

Ca = −Cb+[(pa−pb)⊗ ta−ta⊗ (pa−pb)]+[(pa−h)⊗ rh−rh⊗ (pa−h)] = Ci,

the mean power of the internal actions over the volume V (Mµ) of themodule can be written:

πintµ = 1

V (Mµ) {P

∑

i=1

{ti · [wi − Wa(pa − pb)] +1

2Ci · Wi}

+

N∑

h=1

zho · dh +

M∑

j=1

zj · dj

+

L∑

l=1

{rl · [wl −Wa(pa − h)]}} , (49)

where P is the number of the pairs of interacting particles, N the number ofthe slits, M the number of interacting slits, and L the number of the pairsof interacting particle–slits in Mµ.

We select linear elastic response functions for the interactions betweenparticles and for the forces due to the crack opening displacements. For theinteractions between slits, according to the Baremblatt’s theory (e.g. Lan-dau and Lifsit (1979)) which regards fractures as continuous distributions of

35

edge dislocations having Burgers’ vectors parallel to the opening directions,we assume non linear elastic constitutive functions. Finally, for the interac-tions between slits and particles, we consider a phenomenological relationbetween microcracks and hard fibers embedded in an elastic matrix:38

ti = Ki[wi −Wa(pa − pb)] i = 1, P ,

Ci = KiWi , i = 1, P ,

zho = Dhdh , h = 1, N ,

zj = Dj

‖ dh ‖‖ dk ‖‖ h − k ‖ nj , j = 1, M ,

rl =f1(a)f2(h)

‖ a − h ‖ nl , l = 1, L . (50)

The second order tensors Ki, Dh and the fourth order tensor Ki have com-ponents depending the elastic constants of the matrix and the geometryof the two kind of inclusions, as have been identified in Trovalusci et al.(2010). The constant Dj and the scalar functions f1 and f2 have not beenyet identified. It can be noted however, that in this context the selected re-sponse functions have only paradigmatic meaning and they can be modifiedin order to define a more refined physically–based micromodel accountingfor the size, shape, disposition and orientation of the material inclusions.

4.2 Micro–macro transition

In order to identify the equivalent continuum model, hypotheses of reg-ularity of the kinematical descriptors introduced are given. According todiscrete–continuum coarse–graining approaches described in Section 2 letus now introduce a kinematical map relating discrete–to–continuous kine-matical fields given by Taylor expansions up to the second order of the

38The constitutive relation (50d) is of the kind proposed by Mattoni et al. (2004) for

the interaction between a hard inclusion and a microcrack in fibre reinforced silicon

carbide (β–SiC), with f1 and f2 two approximating Gaussian functions describing the

local force around a particle A and a slit H.

36

independent velocity w(x), d(x) and angular velocity W(x) ∈ Skw fields:

wa = w(x) + ∇w(x)(a − x) +1

2∇2 w(x)(a − x)(a − x) + o (a − x)

Wa = W(x) + ∇W(x)(a − x) +1

2∇2W(x)(a − x)(a − x) + o (a − x)

dh = d(x) + ∇d(x)(h− x) +1

2∇2 d(x)(h− x)(a − x) + o (h− x) ,

(51)

where x is the centre of the module (∇(·) = ∂(·)/∂x). Assuming that acontinuous neighborhood M of x, occupying the same Euclidean region ofMµ exists, this map imposes that the continuum locally undergoes the samedeformations as the lattice system. Equations (51) provide a generalizationof the Cauchy (3), Voigt (22) or Poincare (36) homogenization rule recalledin Section (2). From now on, the explicit dependence of any field on x willbe undertaken.

Based on the map (51) various kinds of continua can be identified, thatare in general multifield continua.39 In the example reported here we con-sider only the first order continuum approximation with ∇w, ∇W and ∇d

constant. It is worth noting that, also in this case of homogeneous defor-mations, the presence of the fields ∇W and d, guarantees the non–localcharacter of the description.

Using Equations (51) the strain measures of the lattice (45), (46) canbe expressed in terms of the smooth fields ∇w − W, ∇W, d and ∇d.After some algebra, the mean power of the module can be then expressedas function of these strain fields:

39 By expanding the series up to higher orders refined descriptions allowing to take into

account long–range interactions can be obtained (Bardenhagen and Triantafyllidis,

1994; Stefanou et al., 2010).

37

πintµ =

1

V (Mµ){

P∑

i=1

ti ⊗ (a− b) +L

∑

i=1

rl ⊗ (a − h)} · (∇w −W)

+1

2V (Mµ){

P∑

i=1

{2ti ⊗ [(p− a) ⊗ (a − x) − (p − b)⊗ (b − x)]

+ Ci ⊗ (a − b)} · ∇W +1

V (Mµ){

N∑

h=1

zho +

L∑

l=1

rl} · d (52)

+1

V (Mµ){

N∑

h=1

zho ⊗ (h− x) +

M∑

j=1

zj ⊗ (h − k) +

L∑

l=1

rl ⊗ (h − x)} · ∇d.

Let us now consider a continuum scalar field representing the internalpower density of a multifield continuum having the strain fields ∇w − W,∇W, d, ∇d as primal fields:

πint(∇w −W, d,∇d) = S · (∇w −W) +1

2S · ∇W + z · d + Z · ∇d, (53)

while the second order tensor S, the third order tensor S, the vector z

and the second order tensor Z, are the dual stress fields. Based on therequirement of power preservation in the transition from the fine to the grossdescription, and resorting to the localization theorem, this power density ismade to coincide with the mean internal power of the module (53):

πintµ (∇w −W, d,∇d) = πint(∇w − W, d,∇d) . (54)

Requiring that Equation (54) is verified for any ∇w−W, ∇W, d and ∇d,the continuum stress measures are identified as functions of the internalactions and of the fabric vector and tensors of the module (i.e. size, shape

38

and disposition of inclusions):

S =1

V (Mµ){

P∑

i=1

ti ⊗ (a − b) +

L∑

i=1

rl ⊗ (a− h)} ,

S =1

V (Mµ){

P∑

i=1

2ti ⊗ [(p− a) ⊗ (a− x) − (p − b) ⊗ (b − x)]

+ Ci ⊗ (a− b)} ,

z =1

V (Mµ){

N∑

h=1

zho +

L∑

l=1

rl} ,

Z =1

V (Mµ){

N∑

h=1

zho ⊗ (h − x) +

M∑

j=1

zj ⊗ (h − k) +

L∑

l=1

rl ⊗ (h − x)} .

(55)

The lattice system described in Section 4.1 can be then replaced by a struc-tured continuum that admits the fields ∇w − W, ∇W, d and ∇d as lin-earized strain measures to each of which the power conjugated stress mea-sure counterpart, identified through Equations (55), correspond. In thepower formula (53) in particular we can recognize the encoded structure ofa continuum with a rigid local structure (Cosserat), with the primal/dualfields: (∇w −W)/S, ∇W/S, plus a deformable (affine) structure (Capriz,1989), with the primal/dual fields: d/z, ∇d/Z.40

If a continuum characterized by different kinematical descriptors wasadopted, the power equivalence with the lattice model of Section 4.1 wouldnot be obtained unless internal constraints or restrictions on the componentsof the constitutive tensors introduced were given. Examples of such con-stitutive prescriptions have been shown in (Masiani and Trovalusci, 1996;Pau and Trovalusci, 2012). A useful consequence of this assertion is thatspecific continuous models can be derived by imposing proper internal con-straints obtaining, as in the cases studied by Voigt and Poincare, con-tinua with latent microstructure (Capriz, 1985). In particular, it can beshown that for d = 0, a Cosserat continuum is identified. Coherently, thismultifield–micropolar model corresponds to an assembly of rigid particleswhich undergo homogeneous displacements and rotations, independent ofeach other, interacting through forces and couples. Otherwise, when d = 0

and W = skw∇w = Θ, it is ∇w − W = sym∇w = E and, consider-ing the map (51): if ∇2 w 6= 0, a second gradient continuum is identified;

40See Equations (43) and (44).

39

if ∇2 w = 0, a classical continuum is obtained. Referring to the originallattice system, these last two cases correspond to a system without slitsand with particles constrained to undergo the same local rotation of thecontinuum, as in the Voigt’s model described in Section 2 (Equation 21)(Trovalusci and Pau, 2014).

In the virtual power setting delineated the results apply regardless of thematerial response. When the constitutive equations for the lattice systemare defined, for instance those of Equations (50), by also identifying theactual strain rates of the discrete model using again the map (51) in thecase of homogeneous deformations, the continuum constitutive relations forall the stress measures introduced are derived in the following form:

S = A(∇w − W) + B∇W + Cd + D∇d + ΨS(d2,∇d2, ‖ d ‖‖ ∇d ‖) ,

S = E(∇w −W) + F∇W ,

z = I(∇w − W) + Md + N∇d + Ψz(d2,∇d2, ‖ d ‖‖ ∇d ‖) ,

Z = O(∇w − W) + Qd + R∇d + ΨZ(d2,∇d2, ‖ d ‖‖ ∇d ‖) . (56)

In Equations (56) the constitutive tensors of the second (M), third (C, I, N,Q), fourth (A, D, O, R), fifth (B, E) and sixth (F) order have componentsdepending on the elastic constants of the matrix and on the geometry of theinclusions. The non–linear vector (Ψz) and second order tensor (ΨS , ΨZ)functions depend on the constitutive and geometrical parameters of thematerial phases.

If the discrete system is hyperelastic, also the equivalent continuum ishyperelastic and the following symmetry relations between constitutive ten-sors hold: BT · V = T · EV, for any third order tensor T and second ordertensor V; Cv · T = I · v ⊗ T, for any vector v and second order tensor T;DT · V = T · OV, for any second order tensor T and V; NT · v = T · Qv,for any second order tensor T and vector v. In particular, the tensors B,C, F, M, N and the corresponding transposed tensors defined by the aboverelations contain internal length parameters. If the material microstructureis arranged respecting the central symmetry, the odd order tensors B, C, N,and the corresponding transposed tensors are null. This is the case of themajority of materials with periodic microstructure.

4.3 Continuum with rigid and affine structure

It has been recognized that the multifield continuum equivalent to thelattice system of Section 4.1 and characterized by the linearized strain andstress measures introduced belongs to the class of continua endowed withboth rigid and affine structure (Capriz, 1989). As described in Section 3,these continua undergo microdeformations independent of the local macro-

40

scopic deformation and below a brief description of the basics is reported,starting from finite deformations.41

Let us denote E the Euclidean space, V the vector space of the transla-tions of E and SO(3) the proper orthogonal group.42 The continuous bodyB occupies the region C ⊂ E in such a way that B → E × SO(3) × V. Ateach material point P ∈ B is then associated the triplet:

X = X(P ) , R = R(P ) , v = v(P ) , (57)

with X ∈ C , R ∈ SO(3) , v ∈ V, which constitutes a ‘complete placement’for B. The microstructural fields R and v represent the rigid and thedeformable (affine) local structure, respectively: the former accounts forthe orientation of the fibers, the latter for the presence of the flaws.43

In the reference shape C : R(X) = I, I being the identity tensor, andv(X) = 0. After a deformation the material point P can be seen as occu-pying the place x + v in such a way that the overall displacement of thebody is vt = x − X + v, with x ∈ C, C being the actual shape of the body.Thus, the vector field v represents the difference between vt and the dis-placement of the flawless body. It can also be interpreted as a deformabledirector associated with the material particle (Eringen, 1999). The followingtransplacement gradients are introduced:

Ft = F + ∇v , with F = ∇x ,

F = ∇R , (58)

with ∇(·) = ∂(·)/∂X. The additive decomposition in Equation (58) corre-sponds to the multiplicative decomposition:

Ft = FFµ ,

withF = ∇x + I , Fµ = I + F−1∇v .

41Original versions of this model have been presented in (Trovalusci and Augusti, 1998;

Mariano and Trovalusci, 1999). The proposed approach is consistent with the one of

Nunziato and Cowin (1979); Cowin and Nunziato (1983). It can be also contextualized

within the more general framework described in (Fremond and Nedjar, 1996; Gurtin

and Podio-Guidugli, 1996).42That is the group of rotations: SO(3) = {R |RR

T = RTR = I , detR > 0}, R being

a second order tensor.43The term microstructure as used in this section refers to the local structure at which

a material point is attached. The fields defined on the microstructure are called mi-

crofields, accordingly.

41

We then introduce the strain measures:44

U = RTF − I , U = RT ◦ F ,

uµ = Rv , Uµ = R∇v , (59)

which at the non–linear level account for the coupling between the rotationof the fibers and the deformation of the flaws.

With reference to the polar decomposition:45

F = (psymF) (orth+F) ,

we say that the multifield continuum undergoes a rigid transplacementwhen:

psymF = I and R = orth+ F = Q ,

In this circumstance:

U∗ = 0 , U∗ = O ,

u∗µ = Qv , U∗

µ = Q∇v , (60)

where the superscript ‘∗’ stands for ‘rigid’. As mentioned in Section 3,differently from a micromorphic continuum, under a rigid macrotransplace-ment the strain measures representative of the affine microstructure can benon null.

By linearizing near the reference configuration C, considering a small realparameter ε we have:

x− X ∼= ε x + o(ε) ,

v ∼= ε v + o(ε) ,

RT ∼= I + ε RRT + o(ε) , (61)

where the superimposed dot denotes the derivative with respect to the pa-rameter ε evaluated for ε = 0.46 Then by putting:

w = x, W = −RRT , d = v ,

44In terms of components [RT ◦ F]ijk = [R]ih[F]hjk.45The operators psym and orth+ respectively select the symmetric positive definite and

the proper ortogonal part of a tensor.46 The assumptions: R(X) = I and v(X) = 0 for ε = 0 mean that in the reference shape

the orientation of the rigid local structure and the affine local structure are independent

of the point P at which they are attached.

42

from Equations (59) we obtain the linearized strain measures as:

U = ∇w − W , U = ∇W ,

uµ = d , Uµ = ∇d , (62)

It can be noticed that these quantities correspond to the strain measuresof the model identified in Subsection 4.2. Consistently with the underlyinglattice system described in the linear frame, at the infinitesimal level thetwo, rigid and deformable, microstructures do not interact.

The rigid velocity fields of the microstructured continuum are character-ized by the following equations:

w∗ = c(o) + Θ(x− o) , W∗ = Θ ,

d∗ = Θd , (63)

∀x , o ∈ C, where c is the velocity of o and Θ = skw∇w. Substituting intoEquations (62) we have:

U∗ = 0 , U∗ = O ,

u∗µ = Θd , U∗

µ = Θ∇d . (64)

Let us now consider the internal power over a control region P ⊆ C withsmooth boundary ∂P and outward normal n:

Πint(∇w−W, d,∇d) =

∫

P

[S · (∇w −W) +1

2S · ∇W + z · d + Z · ∇d] dV.

(65)The divergence theorem gives:

Πint(w, W, d) =

∫

P

[div S · w

+1

2(div S + 2 skwS) · W + (div Z − z) · d] dV

+

∫

∂P

(Sn ·w +1

2Sn ·W + Zn · d) dA , (66)

Then, let us denote: b the vector of the external volume forces; t andC ∈ Skw the vector and tensor of surface forces and couples on ∂P , re-spectively; p the vector of surface microforces exerted through ∂P. For thesake of simplicity, neither external volume couples nor microforces are con-sidered. The the external surface microforces instead can be experiencedas constitutive prescription through the standard tractions on ∂C (Capriz,

43

1989).47 The power equivalence between internal and external power re-quired for any w, W and d and any subset P ⊆ C allows us to identify thestructure of the external power:

Πest(w, W, d) =

∫

P

b · w dV +

∫

∂P

(t · w +1

2C · W + p · d) dA , (67)

together with the standard and non–standard balance equations for the bulk

divS + b = 0 ,

div S + 2 skwS = 0 , in PdivZ− z = 0 , (68)

and the contact

Sn = t , Sn = C , Zn = p , on ∂P , (69)

macro and microactions (tractions, surface couples, microtractions) wheren is the outward normal to ∂P.48

Equation (68a) expresses the classical linear momentum balance, (68b)the angular momentum balance and (68c) the micro linear momentum bal-ance. The virtual power equivalence Πint = Πest yields this last balanceequation, which is not obtainable via the standard invariance under Galileanchanges of observers (Gurtin and Podio-Guidugli, 1992).

It can be noted that the presence of the grade–zero term d in the internalpower formula (65) implies the presence of the (dual) volume action z; aswell as the grade–one terms ∇w,∇W, and ∇d imply the presence of the(dual) surface actions S, S and Z, depending on the direction n throughEquations (69). In Equations (68) and (69), S represents the second ordermacrostress tensor, S the third order couple–stress tensor, while z and Z

are respectively the vector of the internal microstructural actions and thesecond order microstress tensor. These last non–standard fields representthe additional state of stress on the body due to the presence of defects andto their interactions. The volume force z can be interpreted as an auto–force accounting for the internal changes of the material configurations dueto the presence of defects, while it can be shown that the stress tensor Z

47As an example, Trovalusci et al. (2010) identify these forces as function of displacement

jump distributions caused by external tractions on ∂C.48Dealing with non–standard continua the definition of surface non–standard tractions

is in general non trivial and depend on the choice of the control volume region. See for

instance the case of second gradient media (Fried and Gurtin, 2006; Podio-Guidugli

and Vianello, 2010) and the example reported in (Trovalusci and Pau, 2014).

44

is related to the so–called configurational tensor (Eshelby, 1951; Maugin,1993; Gurtin, 1995, 2000; Maugin, 2011) due to the relative deformationbetween defects.49

Since the microstrain measures are non–null under a rigid transplace-ment (Equations 64), according to the axiomatic description of Di Carlo(1996), we require that the internal power (65) is null for any rigid velocityfields as defined in Equations (63). Applying the divergence theorem wethen have:

Πint∗ =

∫

P

[S · (∇w∗ − W∗) +1

2S · ∇W∗ + z · d∗ + Z · ∇d∗] dV

=

∫

P

(z · Θd · +Z · Θ∇d)dV

= −∫

P

Θz · d dV +

∫

P

div (ΘZ) · d dV −∫

P

ΘZn · d dV

=

∫

P

Θ · (divZ − z) ⊗ d dV +

∫

∂P

Θ · Zn ⊗ d dA = 0 .

(70)

By requiring that Equation (70) is valid for any subset P ⊆ C and accountingfor the microforce balance, in the absence of external volume microforces,(Equation 68c) it is:

skw(Zn ⊗ d) = 0 , on ∂P . (71)

Equation (71) is a micromoment balance equation which plays the role ofa constitutive prescription.50 It imposes to the vector of non–standardtractions p (69c) to be parallel to the vector d.51

Due to the presence of the rigid microstructure, W and ∇W, the skew–symmetric part of the stress tensor, skwS, must only satisfy the balanceequation (68b). If only the rigid microstructure is present (d = 0), theinternal power is zero for any rigid velocity field, and no equation must be

49Damage is here described not as a reduction of the global stiffness, an in internal

variable models (e.g. Pijaudier-Cabot and Bazant (1987)), but as an additional state

of strain induced by additional autoforces z and stresses Z.50Di Carlo (1996) pointed out the difference between balance equations (68), which

must be regarded as selection rules for mechanical processes and equations obtained as

Equations (56), which should be valid for any mechanical process and then represent

selection of the rules, like any constitutive prescriptions.51Equation (71) holds also when the particles have constrained rotations (W = Θ), like

in Voigt’s model (21).

45

added to Equations (68). In this case the balance equations obtained usingthe principle of virtual power correspond to those of a micropolar contin-uum. Conversely, when only the affine microstructure is present (W = 0)Equation (71) becomes the moment balance equation of continua with vec-torial microstructure (Capriz, 1989) as reported in (Mariano and Trovalusci,1999):

skw(S + z⊗ d + ZT∇d) = 0 in P ,

leading, as in Equation (70), to Equation (71) and skwS = 0. Finally, whenW = Θ and d = 0, or when no additional fields are present (W = 0 , d =0), the field equations become those of a classical continuum.

5 Case study: a one–dimensional microcracked bar

A comprehensive study concerning the application of the proposed mod-elling is beyond the scope of this study. Several applications of the describedcoarse–graining approach have been presented and discussed in various ar-ticles, among these (Trovalusci et al., 2010; Trovalusci and Varano, 2011;Trovalusci and Pau, 2014).

In this section some simulations are reported by way of a one–dimensionalexample devised for highlighting the main features of non–local multifieldmodels described in Section 4. In particular, the ability of the equivalentmultifield continuum to reveal the presence of internal heterogeneities is in-vestigated by analyzing the relevant dispersive wave propagation propertiesin a microcracked bar with ‘frozen’ fibres.52 Scattering of travelling wavesis shown to be associated with the microcrack density per unit length.

In the linear elastic one–dimensional case the constitutive equations (56)become:

S = Aw′

+ Dd′

,

S = Fφ′

,

z = M d ,

Z = Dw′

+ R d′

, (72)

where S, S, z, Z respectively are the non null components of the stressmeasures S, S, z, Z, while w, φ, d respectively are the non null componentsof the kinematical fields w, W, d. The apex indicates the spatial derivative.

The material coefficients A, D, F, M, R have been identified for a moduleof periodic medium with internal inclusions distributed according to the

52This one–dimensional model is derived from the model described in Section 4.3 in

which the constraint W = Θ (Voigt’s constraint (21)) is posed.

46

orthotropic symmetry. Besides that on the elastic constants of the matrix,these coefficients depend on the number of microcracks per unit length(microcrack density ρµ), the microcrack size (lc) and arrangement.53

The case of a linear elastic bar with microcracks and constrained rota-tions is here considered in such a way that: φ = 0 and S = 0.54

From Equations (68) and (72), the equations of motion, written account-ing for inertial terms (also identified by Trovalusci et al. (2010)), become:

Aw′′

+ D d′′

= ρ w ,

D w′′ − M d + R d

′′

= ρµ d , (73)

where ρ is the mass density of the medium while ρµ is the mass densityrelative to the microstructure. Superposed dot indicates the time derivative.By putting α2 = A/ρ, β = D/ρ, γ = M/ρµ, δ = D/ρµ and ε2 = R/ρµ,Equations (73) can be written as:

w − α2 w′′ − β d

′′

= 0 ,

d + γ d − δ w′′ − ε2 d

′′

= 0 , (74)

Free oscillations analysis allows us to emphasize the specific non–local fea-ture of the multifield model that is spatial dispersion. Denoting with x thecoordinate of the bar axis and t the time variable, let us consider two waves,which propagate along the bar with angular frequency ω and different wavenumbers k , kµ. A general solution for w and d, respectively called macroand micro wave, is assumed:

w (x , t) = wo ei(kx−ωt) , d (x , t) = do ei(kµx−ωt) , (75)

where do and wo are constant. The substitution of Equations (75) into (74),providing that ω = c k, gives:

(Q− c2I)v = 0 , (76)

where:

{v} =

{

wo

do

}

, [Q] =

[

α2 β kkd

δ (ε2 + γ

kµ2 ) k2

k2µ

]

(77)

and I is the identity tensor. A non trivial solution of the system (76) existsif:

(α2 − c2)[(ε2 +γ

k2µ

)k2

k2µ

− c2] − β δk2

k2µ

= 0 . (78)

53Their explicit expressions are reported in (Trovalusci et al., 2010).54This corresponds to put W = 0.52

47

Tensor Q plays the role of acoustic tensor of the multifield body; then byputting:

a =γ

k2µ

, b =k2

k2µ

, f = β δ , (79)

the solutions of the characteristic equation (78) give the eigenvalues:

c21,2 =

1

2ab+α2+bε2+

−

√

a2b2 + 4f − 2abα2 + α4 + 2ab2ε2 − 2bα2ε2 + b2ε4) ,

(80)whose positive square roots are the phase velocities of the system. In gen-eral, both these velocities depend on the wave numbers and the system isdispersive.

Note that the dispersion is due to the presence of the coefficient, γ, of theterm d which is not a space nor time derivative of order two of the variableand contains internal length. In the case in which β = δ = 0 the system(74) is decoupled. This case corresponds to a material in which no particle–slit interactions are accounted for (D = 0). In this case macrowaves, w,propagate with constant phase velocity, cw, while, due to the dispersionterm γ, microwaves, d, propagate with phase velocity, cd, depending on thewave number or frequency:

c1 = cw = α , c2 = cd =

√

ε2 +γ

k2µ

=ω ε√ω − γ

. (81)

As a consequence, the resulting travelling wave, w +d, propagates alongthe axis, x, of the microcracked bar with variation in amplitude and distor-tion in shape. Figure 5 reports the superposition of the two waves w (thinline) and d (dashed line), normalized to the value wo, propagating (at a givent = to) in a bar characterized by different levels of microcrack density perunit length, ρµ. The phase macrovelocity is cw = 2000 msec−1, while thephase microvelocity is (for β = δ = 0): cd = 1500 msec−1, for ρµ = 20 m−1;cd = 900 msec−1, for ρµ = 100 m−1; cd = 200 msec−1, for ρµ = 1000 m−1.The resulting wave, w + d (thick line), appears as a modulation of a shortand a long wave carrier term, propagating with group velocity differentfrom the average velocity. It can be noticed that the peak–to–peak distancereduces when the microcrack density increases, to become a distributed dis-turbance in which the resulting wave is completely carriered by the elasticwave, w. Figure 2 shows the dispersion diagrams (phase velocity, c, versuswave–number, k) plotted for different values of ρµ (20 ; 100 ; 1000 m−1) andfor different values of the microcracks size, lc (5× 10−3 ; 10× 10−3 m). Thehorizontal lines correspond to the velocity of the bar without microcracks,

48

0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.5

1.0

1.5

0.2 0.4 0.6 0.8 1.0

-1.0

-0.5

0.5

1.0

1.5

0.2 0.4 0.6 0.8 1.0

-1.0

-0.5

0.5

1.0

0.05 0.10 0.15 0.20

-1.0

-0.5

0.5

1.0

Figure 1. Wave propagating along the axis of a microcracked bar (x mm) fordifferent value of microcrack density. Top: (left) low (ρµ = 20 m−1); (right)medium (ρµ = 100 m−1). Bottom: (left) high (ρµ = 1000 m−1); (right) detailof the left side. Thin lines: macrowave (elastic), w; dashed lines: microwave, d;thick lines: resulting wave, w + d. Signals are normalized to the value wo.

cw. It can be observed that there is a critical wave–number (or wave–length)beyond which waves propagates with constant velocity: no–dispersion ef-fect are present and local theories are suitable. The higher the microcrackdensity the higher the extension of the dispersive zone in which the effectsin changing of velocity cannot be neglected.

The same microcracked bar, of length L (x ∈ [0, L]) and cross sectionA, has been also studied under the effect of forced sinusoidal oscillationF (t) = Fo sin(ω t), with Fo constant, under the mixed boundary conditions:

w(0 , t) = 0 , d(0 , t) = 0 , (82)

S(L , t) = F (t)/A , Z(0 , t) = λF (t)/A ,

where, in accordance with the description in Capriz (1989), the microtrac-tion Z at the boundary (x = L) has been identified in terms of the forceF .55

Figure 3 reports the results in terms of travelling waves obtained in thecase of decoupled constitutive equations (72) with D = 0 for different levels

55As detailed in (Trovalusci et al., 2010), λ is a non null coefficient that can be identified

in terms of material parameters; in this case λ = ρµ lc π/2.

49

Figure 2. Phase velocity, c (msec−1), versus wave number, k (rad m−1), fordifferent microcrack density ρµ: low (20 m−1), dashed line; medium (100 m−1),thin line; high (1000 m−1), thick line; and different microcrack size lc: (left)5 × 10−3m; (b) 10 × 10−3m. The horizontal lines correspond to the velocity ofthe elastic bar.

Figure 3. Forced waves in an elastic (thin lines) and microcracked (thick lines)bar (x axis (m)) for different values of microcrack density ρµ (10 ; 100 ; 200 m−1).Uncoupled case (D = 0).

of ρµ (10 ; 100; 200 m−1). The problem has been also solved in the caseof coupled constitutive equations by parametrically varying the values ofthe material coupling coefficient D as a percentage of the coefficient A. InFigure 4 it can be observed that in both cases waves propagate with changesin amplitude and in shape depending on the microcrack density per unitlength, and these changes increase when the coupling term increases.56

Overall, the results show that the presence of microcracks is revealedas a disturbance spread along the bar which alters the shape, amplitudeand velocity of the travelling waves, and that these changes depend on themicrocrack density per unit length.

56The solution of the forced oscillation problem has been obtained numerically using

COMSOL Multiphysicsr code. In particular the results of Figure 3 are reported in

(Trovalusci et al., 2010).

50

Figure 4. Forced waves in a microcracked bar (ρµ = 200 m−1) along x axis (m)for different values of the coupling term D (from left to right: 0; 10−2A ; 10−1 A).Macrowaves (thin lines), resulting waves (thick lines).

6 Final remarks

Aim of this study is to recognise the current validity of significant old ideasfor the formulation of new models for material behaviours. Exploiting thesuggestions of the molecular models of elasticity, originally developed inthe 19th century, it has been shown that starting from properly refinedlattice models multifield (non–simple) continuous formulations can be de-rived, which retain a memory of the internal material structure by meansof additional field descriptors.

In particular, taking into account a discrete ensemble made of ‘struc-tured molecules’ and based on the choice of a correspondence map, relatingthe finite number of degrees of freedom of the discrete model to the con-tinuum kinematical fields,57it is possible to select a multifield model withthe appropriate, standard and non–standard, field descriptors. This modelis the continuum obtained by requiring the virtual power equivalence, withthe generalized lattice system, without introducing any internal constraint.

This approach, avoiding the arbitrariness in assuming the kind of con-tinuum a priori, provides a useful guidance on the choice of continuumapproximations for heterogeneous media. Moreover, the virtual power prin-ciple allows us to derive all the classical and non–classical balance equationsof the selected multifield continuum pointing out the mechanical meaningof any non–standard at–a–distance or by–contact internal actions.58 This isachieved without resorting to the classical frame invariance axioms, whichin the case of multifield materials can fail (Gurtin and Podio-Guidugli, 1992;

57It has been shown that this map is a generalization of the Cauchy (3), Voigt (22)

or Poincare (36) rule, widely used in crystal elasticity and in the classical molecular

theory of elasticity (Ericksen, 1977).58See Chapter 2, by Del Piero.

51

Di Carlo, 1996).Within such a theoretical framework, any kind of thermomechanically

coherent (Gurtin, 1965) continuum with additional degrees of freedom canbe constructed. However, such a multifield model acquires relevance whenprovided by a constitutive characterization, and becomes challenging if itcan be cast within an efficient multiscale computational scheme. In thiscase, microcontinuum theories can give a very powerful frame for the grossmechanical description of complex material behaviours, avoiding the restric-tions of classical coarse–graining approaches. Restrictions that are relatedto the principle of scale separation, which is violated in the presence ofphenomena dependent on the microstructural size. Thus, a major issue ofthe described approach becomes the definition of the physically–based ma-terial response functions for the internal actions at the fine scale as well asthe constitutive characterization of the non–standard external actions forthe solution of boundary value problems. And this is currently an openchallenge.

Acknowledgements This research has been partially supported by theItalian ‘Ministero dell’Universita e della Ricerca Scientifica’ (Research fund:MIUR Prin 2010–11; Sapienza 2011, 2013). The author would like to thankdr. Annamaria Pau and Alessandro Fascetti for patiently re–reading andchecking the text.

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