Scale transitions in solid mechanics based on computational ... - Cism

<strong>Scale</strong> <strong>transitions</strong> <strong>in</strong> <strong>solid</strong> <strong>mechanics</strong><strong>based</strong> on computational homogenizationMarc G.D. Geers, V.G. KouznetsovaDepartment of Mechanical Eng<strong>in</strong>eer<strong>in</strong>gE<strong>in</strong>dhoven University of TechnologyP.O. Box 513, 5600 MB E<strong>in</strong>dhoven, The NetherlandsE-mail: m.g.d.geers@tue.nlAbstractThese lecture notes address basics and advanced topics on the computational homogenizationof the <strong>mechanics</strong> of highly non-l<strong>in</strong>ear <strong>solid</strong>s with (possibly evolv<strong>in</strong>g)microstructure under complex non-l<strong>in</strong>ear load<strong>in</strong>g conditions. The key componentsof the computational homogenization scheme, i.e. the formulation of the microstructuralboundary value problem and the coupl<strong>in</strong>g between the micro and macrolevel<strong>based</strong> on the averag<strong>in</strong>g theorems, are addressed. The numerical implementationof the framework, particularly the computation of the macroscopic stress tensorand extraction of the macroscopic consistent tangent operator <strong>based</strong> on the totalmicrostructural stiffness, are treated <strong>in</strong> detail. The application of the method isillustrated by the simulation of pure bend<strong>in</strong>g of porous alum<strong>in</strong>um. The classical notionof a representative volume element is <strong>in</strong>troduced and the <strong>in</strong>fluence of the spatialdistribution of heterogeneities on the overall macroscopic behaviour is <strong>in</strong>vestigatedby compar<strong>in</strong>g the results of multi-scale modell<strong>in</strong>g for regular and random structures.F<strong>in</strong>ally, an extension of the classical computational homogenization schemeto a framework suitable for multi-scale modell<strong>in</strong>g of macroscopic localization andsize effects is briefly discussed.These lecture notes constitute the expanded version of a number of journal publicationsand comprise l<strong>in</strong>ks to other relevant material. For direct cit<strong>in</strong>g to theselecture notes, please make use of its associated references <strong>in</strong>stead:• V. G. Kouznetsova, W. A. M. Brekelmans, and F. P. T. Baaijens. An approachto micro-macro model<strong>in</strong>g of heterogeneous materials. ComputationalMechanics, 27:37–48, 2001• M. G. D. Geers, V. G. Kouznetsova, and W. A. M. Brekelmans. Gradientenhancedcomputational homogenization for the micro-macro scale transition.Journal de Physique IV, 11(5):5145–5152, 2001• V. G. Kouznetsova, M. G. D. Geers, and W. A. M. Brekelmans. Advancedconstitutive model<strong>in</strong>g of heterogeneous materials with a gradient-enhancedcomputational homogenization scheme. International Journal for NumericalMethods <strong>in</strong> Eng<strong>in</strong>eer<strong>in</strong>g, 54:1235–1260, 2002• M.G.D. Geers, V.G. Kouznetsova, and W.A.M. Brekelmans. Multi-scale secondordercomputational homogenization of microstructures towards cont<strong>in</strong>ua. InternationalJournal for Multiscale Computational Eng<strong>in</strong>eer<strong>in</strong>g, 1(4):371–386,2003

• V.G. Kouznetsova, M.G.D. Geers, and W.A.M. Brekelmans. Size of a representativevolume element <strong>in</strong> a second-order computational homogenizationframework. International Journal for Multiscale Computational Eng<strong>in</strong>eer<strong>in</strong>g,2(4):575–598, 2004• V.G. Kouznetsova, M.G.D. Geers, and W.A.M. Brekelmans. Multi-scale secondordercomputational homogenization of multi-phase materials: a nested f<strong>in</strong>iteelement solution strategy. Computer Methods <strong>in</strong> Applied Mechanics and Eng<strong>in</strong>eer<strong>in</strong>g,193:5525–5550, 2004• T.J. Massart, R.H.J. Peerl<strong>in</strong>gs, and M.G.D. Geers. An enhanced multi-scaleapproach for masonry wall computations with localization of damage. InternationalJournal for Numerical Methods <strong>in</strong> Eng<strong>in</strong>eer<strong>in</strong>g, 69(5):1022–1059, 2007• T.J. Massart, R.H.J. Peerl<strong>in</strong>gs, and M.G.D. Geers. Structural damage analysisof masonry walls us<strong>in</strong>g computationa homogenization. International Journalof Damage Mechanics, 16:199–226, 2007• M. G. D. Geers, E. W. C. Coenen, and V. G. Kouznetsova. Multi-scale computationalhomogenization of structured th<strong>in</strong> sheets. Modell<strong>in</strong>g Simul. Mater.Sci. Eng., 15:S393–S404, 2007• I. Özdemir, W. A. M. Brekelmans, and M. G. D. Geers. Computational homogenizationfor heat conduction <strong>in</strong> heterogeneous <strong>solid</strong>s. Int. J. Numer. MethodsEngrg., 73:185–204, 2008• I. Özdemir, W. A. M. Brekelmans, and M. G. D. Geers. FE2 computationalhomogenization for the thermo-mechanical analysis of heterogeneous <strong>solid</strong>s.Computer Methods <strong>in</strong> Applied Mechanics and Eng<strong>in</strong>eer<strong>in</strong>g, 198:602–613, 2008• I. Özdemir, W. A. M. Brekelmans, and M. G. D. Geers. Model<strong>in</strong>g thermalshock damage <strong>in</strong> refractory materials via direct numerical simulation (dns).Journal of the European Ceramic Society, 30(7):1585–1597, 2009• I. Özdemir, W. A. M. Brekelmans, and M. G. D. Geers. A thermo-mechanicalcohesive zone model. Computational Mechanics, 2010. Accepted, onl<strong>in</strong>e availablesoon• E. W. C. Coenen, V. G. Kouznetsova, and M. G. D. Geers. Computationalhomogeneization for heterogeneous th<strong>in</strong> sheets. International Journal for NumericalMethods <strong>in</strong> Eng<strong>in</strong>eer<strong>in</strong>g, 2010. Published onl<strong>in</strong>e <strong>in</strong> Early View• M. G. D. Geers, V. G. Kouznetsova, and W. A. M. Brekelmans. Multi-scalecomputational homogenization: trends & challenges. Journal of Computationaland Applied Mathematics, 234(7):2175–2182, 2010Note that many other papers from different contributors <strong>in</strong> the scientific communityare directly cited <strong>in</strong> these lecture notes as well, for which the citation details are<strong>in</strong>cluded <strong>in</strong> the list of references.2

1 Introduction1.1 Mechanics across the scalesThe <strong>in</strong>tr<strong>in</strong>sic role of different scales <strong>in</strong> <strong>mechanics</strong> of materials is nowadays well recognized.At the material level, it is the typical scale at which many heterogeneities can be identifiedthat matters. The <strong>mechanics</strong> and physics of these multi-phase heterogeneous microstructuresis recognized as the ma<strong>in</strong> driver for the macroscopic eng<strong>in</strong>eer<strong>in</strong>g response of a materialupon mechanical load<strong>in</strong>g, up to the po<strong>in</strong>t of failure. The proper understand<strong>in</strong>g of thebehaviour, evolution and mechanical response of materials at this scale is certa<strong>in</strong>ly one ofthe key factors at the micro scale. Over time, it has become clear that even smaller scalesand th<strong>in</strong> <strong>in</strong>terfaces may have a pronounced <strong>in</strong>fluence on the micron scale. In this sense,multi-scale methods have emerged which l<strong>in</strong>k up this scale to smaller and large scales. Asecond characteristic of this multi-discipl<strong>in</strong>ary field, is the emphasis which is put on themechanical aspects, cover<strong>in</strong>g the role of stress, stra<strong>in</strong>, deformation and degradation. Generally,this goes hand <strong>in</strong> hand with the material synthesis and microstructure evolution,s<strong>in</strong>ce <strong>in</strong>ternal stress fields are an <strong>in</strong>tr<strong>in</strong>sic characteristic of heterogeneous microstructures.It is obvious that this cannot be trivially separated from the govern<strong>in</strong>g physics. Mechanicalaspects generally represent a source of <strong>in</strong>ternal (stra<strong>in</strong>) energy, which is an essential<strong>in</strong>gredient of the underly<strong>in</strong>g thermodynamics. Vice verse, other physical mechanisms (e.g.diffusion, dislocation motion) will have a pronounced <strong>in</strong>fluence on the relaxation of these<strong>in</strong>ternal stresses, and consequently on the overall mechanical response. In other words,multi-scale <strong>mechanics</strong> nowadays covers (i) what <strong>mechanics</strong> contributes to physics acrossthe scales and (ii) how physics at the smallest scales contributes to the <strong>mechanics</strong> at themacro scale. This holds <strong>in</strong> particular when failure mechanisms come <strong>in</strong>to play.1.2 Some historical notes on multi-scale <strong>mechanics</strong>The past years have been marked by a significant <strong>in</strong>terest <strong>in</strong> the various length scales thatgovern the <strong>mechanics</strong> of materials. The ma<strong>in</strong> issue consists <strong>in</strong> identify<strong>in</strong>g the relationshipsthat bridge those various scales, i.e. multi-scale <strong>mechanics</strong>. The multi-scale methodologyaims to predict, describe, quantify or qualify the ’macroscopic’ behaviour of eng<strong>in</strong>eer<strong>in</strong>gmaterials through the consistent modell<strong>in</strong>g of the <strong>mechanics</strong> and physics of the heterogeneous,multi-phase, anisotropic, discrete microstructure. Various techniques have beenproposed to contribute to this challeng<strong>in</strong>g task. Among them, a large class of homogenizationtechniques exists, also called coarse gra<strong>in</strong><strong>in</strong>g <strong>in</strong> the physics community [16, 17, 18].Early steps towards multi-scale or micro-scale modell<strong>in</strong>g were taken long ago, when the <strong>in</strong>terestfor the <strong>mechanics</strong> of heterogeneous materials became more pronounced. Prelim<strong>in</strong>arysteps go back to the 19th century, where the rule of mixtures was first <strong>in</strong>troduced (Voigt,1887), followed by the Sachs model (1928) [19], Reuss estimate (1929) and the frequentlyused Taylor model (1938) [20]. Whereas Voigt and Reuss focused more on composite systems,Taylor and Sachs were typically derived for polycrystals. The grow<strong>in</strong>g <strong>in</strong>terest forcomposite materials constituted the ma<strong>in</strong> trigger for new developments. The best-knownearly contribution of this type is probably the work of Eshelby (1957) [21]. Still today,these first steps have a pronounced impact, giv<strong>in</strong>g rise to alternative cont<strong>in</strong>uum <strong>mechanics</strong>frameworks (Eshelbian <strong>mechanics</strong>; materials forces). The field of ’cont<strong>in</strong>uum micro<strong>mechanics</strong>’,which was formally established by Hill (1965) [22], extended tremendously s<strong>in</strong>ce3

then. A survey of activities over the past forty years is given <strong>in</strong> [23].Homogenization techniques were first developed with<strong>in</strong> the framework of elasticity, as anexcellent tool to predict the effective or apparent l<strong>in</strong>ear elastic properties of heterogeneousmaterials. Several closed-form homogenization techniques have been proposed <strong>in</strong> this context,e.g. the Voigt-Reuss-Hill bounds, the Hash<strong>in</strong>-Shtrikman variational pr<strong>in</strong>ciple, theself-consistent method, etc., see [24] for an overview. Asymptotic or mathematical homogenizationschemes have been used frequently to assess effective properties of elasticheterogeneous materials [25, 26]. Extensions towards higher-order and nonlocal constitutiveequations have been considered as well, e.g. developments <strong>in</strong>clud<strong>in</strong>g Cosserat media[27], couple stress theory [28], nonlocal effective cont<strong>in</strong>ua [29] or higher-order gradienthomogenized elastic materials [30, 31, 32]. Interest<strong>in</strong>g approaches towards the analysisof random (physically nonl<strong>in</strong>ear) microstructures [33, 34, 35] are the Taylor-Bishop-Hillestimates, several generalizations of self-consistent schemes. The mathematical asymptotichomogenization theory [36, 37] applies an asymptotic expansion of displacementand stress fields on a “natural scale parameter”, which is the ratio of a characteristicsize of the heterogeneities and a measure of the macrostructure [38, 39, 40, 41, 42]. Theasymptotic homogenization approach provides effective overall properties as well as localstress and stra<strong>in</strong> values. However, usually the considerations are restricted to verysimple microscopic geometries and simple material models, mostly at small stra<strong>in</strong>s. Acomprehensive overview of different homogenization methods may be found <strong>in</strong> a book byNemat-Nasser and Hori. [24]. Homogenization of <strong>solid</strong>s <strong>in</strong> a geometrically and physicallynonl<strong>in</strong>ear regime is clearly more cumbersome.The <strong>in</strong>creas<strong>in</strong>g complexity of microstructural mechanical and physical behaviour, alongwith the development of computational methods, made the class of so-called unit cellmethods attractive. These approaches have been used <strong>in</strong> a great number of differentapplications [43, 44, 45, 46, 47, 48, 49]. A selection of examples <strong>in</strong> the field of metalmatrix composites has been collected, for example, <strong>in</strong> a book by Suresh et al.[50] Theunit cell methods serve a twofold purpose: they provide valuable <strong>in</strong>formation on the localmicrostructural fields as well as the effective material properties. These properties aregenerally determ<strong>in</strong>ed by fitt<strong>in</strong>g the averaged microscopical stress-stra<strong>in</strong> fields, result<strong>in</strong>gfrom the analysis of a microstructural representative cell subjected to a certa<strong>in</strong> load<strong>in</strong>gpath, on macroscopic closed-form phenomenological constitutive equations <strong>in</strong> a formatestablished a priori. Once the constitutive behaviour becomes non-l<strong>in</strong>ear (geometrically,physically or both), it becomes <strong>in</strong>tr<strong>in</strong>sically difficult to make a well-motivated assumptionon a suitable macroscopic constitutive format. For example, McHugh et al. [51] havedemonstrated that, when a composite is characterized by power-law slip system harden<strong>in</strong>g,the power-law harden<strong>in</strong>g behaviour is not preserved at the macroscale. Hence, most ofthe known homogenization techniques are not suitable for large deformations nor complexload<strong>in</strong>g paths, neither do they account for the geometrical and physical changes of themicrostructure (which is relevant, for example, when deal<strong>in</strong>g with phase <strong>transitions</strong>).Closed-form homogenization towards constitutive material frameworks or effective (orrather apparent) material properties of composites turns out to be really cumbersomeif one wishes to take <strong>in</strong>to account more complex physics, geometrical nonl<strong>in</strong>earities ordamage and localization.Another class of hierarchical techniques are generally known as variational multi-scalemethods [52, 53]. In here, the weak form of the govern<strong>in</strong>g equations is the po<strong>in</strong>t ofdeparture, which can be separated <strong>in</strong> a coarse and a f<strong>in</strong>e scale part on the basis of suitable4

assumptions on the f<strong>in</strong>e scale field. The key issue resides <strong>in</strong> the elim<strong>in</strong>ation of the f<strong>in</strong>escale from the obta<strong>in</strong>ed formulation. Though promis<strong>in</strong>g, the method relies strongly onthe assumptions made on the f<strong>in</strong>e scale and the restrictions that apply to enable theelim<strong>in</strong>ation <strong>in</strong> practice. Well-known f<strong>in</strong>e scale patterns, e.g. displacement discont<strong>in</strong>uitiesmodelled by Heaviside functions, can be easily implemented. The obta<strong>in</strong>ed method thenshows considerable similarities with the extended f<strong>in</strong>ite element method [54, 55].S<strong>in</strong>ce a few years, substantial progress has been made <strong>in</strong> the two-scale computationalhomogenization of complex multi-phase <strong>solid</strong>s. This technique is essentially <strong>based</strong> on thesolution of nested boundary value problems, one for each scale. If attention is focusedon the nonl<strong>in</strong>ear characteristics of the material behaviour, this technique proves to bea valuable tool <strong>in</strong> retriev<strong>in</strong>g the constitutive response. First-order (i.e. <strong>in</strong>clud<strong>in</strong>g firstordergradients of the macroscopic displacement field only) computational homogenizationschemes fit entirely <strong>in</strong> a standard cont<strong>in</strong>uum <strong>mechanics</strong> framework (pr<strong>in</strong>ciple of localaction) and are now readily available <strong>in</strong> literature [1, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65,66]. Ma<strong>in</strong> characteristics of this solution strategy are• The constitutive response at the macro scale is a priori undeterm<strong>in</strong>ed. No explicitassumptions are required at that level, s<strong>in</strong>ce the macroscopic constitutive behaviourensues from the solution of the micro scale boundary value problem.• The method deals with large displacements (large stra<strong>in</strong>s and rotations) <strong>in</strong> a trivialway under the condition that the microstructural constituents are modelled adequately.• The different phases <strong>in</strong> the microstructure can be modelled with arbitrary nonl<strong>in</strong>earand time-dependent constitutive models.• The <strong>in</strong>fluence of the evolution of the microstructure (as described on the micro-scale)can be assessed directly on the macro-scale.• The micro scale problem is a classical boundary value problem, for which any appropriatesolution strategy can be used, e.g. F<strong>in</strong>ite Element Method [1, 58, 61, 62],the boundary element method[67], the Voronoi cell method [57, 68], a crystal plasticityframework [59, 60] or numerical methods <strong>based</strong> on Fast Fourier Transforms[69, 70]and Transformation Field Analysis[71].• Macroscopic constitutive tangent operators can be obta<strong>in</strong>ed from the microscopicoverall stiffness tensor through static condensation. Consistency is preserved throughthis scale transition.In spite of the huge computational cost of a nested two-scale solution problem, efficiencycan be achieved by solv<strong>in</strong>g the problem through parallel computations ([61, 72]). Anotheroption is selective usage, where non-critical regions are modelled by cont<strong>in</strong>uumclosed-form homogenized constitutive relations or by the constitutive tangents obta<strong>in</strong>edfrom the microstructural analysis but kept constant <strong>in</strong> the elastic doma<strong>in</strong>, while <strong>in</strong> thecritical regions the multi-scale analysis of the microstructure is fully performed ([63]).Despite the required computational efforts the computational homogenization techniquehas proven to be a valuable tool to establish non-l<strong>in</strong>ear micro-macro structure-propertyrelations, especially <strong>in</strong> the cases where the complexity of the mechanical and geometrical5

microstructural properties and the evolv<strong>in</strong>g character prohibit the use of other homogenizationmethods. The first-order technique is by now well-established and widely used<strong>in</strong> the scientific and eng<strong>in</strong>eer<strong>in</strong>g community [73, 74, 75, 76, 77, 78]. The available paperscan be roughly classified <strong>in</strong> the follow<strong>in</strong>g categories [15]:• First-order computational homogenization• Second-order computational homogenization: to resolve some <strong>in</strong>tr<strong>in</strong>sic shortcom<strong>in</strong>gsof the first-order scheme, <strong>in</strong>corporat<strong>in</strong>g the size of the underly<strong>in</strong>g microstructure.• Cont<strong>in</strong>uous-discont<strong>in</strong>uous multi-scale approach for damage: the coarse scale is modelleddiscretely or with a discrete band (weak discont<strong>in</strong>uity), whereas the f<strong>in</strong>e scaleis modelled with a cont<strong>in</strong>uum.• Computational homogenization of multi-physics problems, focus<strong>in</strong>g on the homogenizationof the thermal (heat conduction) problem, and its coupl<strong>in</strong>g to a mechanicalhomogenization scheme.• Computational homogenization of structured th<strong>in</strong> sheets and shells: application ofsecond-order homogenization pr<strong>in</strong>ciples to through-thickness representative volumeelements, enabl<strong>in</strong>g its application to shell-type cont<strong>in</strong>ua.• Computational homogenization of <strong>in</strong>terface problems, which is now emerg<strong>in</strong>g.These lecture notes focus on the basics underly<strong>in</strong>g each of these categories, with an outreachto some of the extensions listed above. Note that there is a vast amount of recentliterature on other multi-scale (and multi-physics) methods [79, 80, 81, 82, 83, 84], oftenpartially connected to the subjects addressed <strong>in</strong> these lecture notes.Cartesian tensors and associated tensor products will be used throughout these lecturenotes, mak<strong>in</strong>g use of a Cartesian vector basis {⃗e 1 ,⃗e 2 ,⃗e 3 }. Us<strong>in</strong>g the E<strong>in</strong>ste<strong>in</strong> summationrule for repeated <strong>in</strong>dices, the follow<strong>in</strong>g conventions are used <strong>in</strong> the notations of well-knowntensor productsC = ⃗a ⃗ b = a i b j ⃗e i ⃗e jC = A·B = A ij B jk ⃗e i ⃗e kC = 4 A : B = A ijkl B lk ⃗e i ⃗e jC = 4 A . 4 B = A iklm B mlkj ⃗e i ⃗e j2 Underly<strong>in</strong>g pr<strong>in</strong>ciples and assumptions2.1 <strong>Scale</strong> separationAt the macro-scale, the material is considered as a homogeneous cont<strong>in</strong>uum, whereas atthe micro level it is generally heterogeneous (the morphology consists of dist<strong>in</strong>guishablecomponents or phases, i.e. <strong>in</strong>clusions, gra<strong>in</strong>s, <strong>in</strong>terfaces, cavities, etc.). This is schematicallyillustrated <strong>in</strong> figure 1. The microscopic length scale is much larger than the moleculardimensions l discrete , so that a cont<strong>in</strong>uum approach is justified for every constituent. At thesame time, <strong>in</strong> the context of the pr<strong>in</strong>ciple of separation of scales, the microscopic length6

Figure 1: Macroscopic cont<strong>in</strong>uum po<strong>in</strong>t representation (M) <strong>in</strong> relation to its underly<strong>in</strong>gheterogeneous microstructure.scale l micro is assumed to be much smaller than the characteristic length l macro over whichthe size of the macroscopic load<strong>in</strong>g varies <strong>in</strong> space, i.e.l discrete ≪ l micro ≪ l macro (1)Note that it is not the size of the macroscopic doma<strong>in</strong> which is important, but rather thespatial variation of the k<strong>in</strong>ematic fields and stress fields with<strong>in</strong> that doma<strong>in</strong>.2.2 Local periodicityMost of the homogenization approaches rely on the assumption of global periodicity of themicrostructure, imply<strong>in</strong>g that the whole macroscopic doma<strong>in</strong> consists of spatially repeatedunit cells. In a computational homogenization approach, a more realistic assumption ismade, which is commonly denoted by local periodicity. Accord<strong>in</strong>g to this assumption,the microstructure can have different morphologies correspond<strong>in</strong>g to different macroscopicpo<strong>in</strong>ts, whereas it repeats itself only <strong>in</strong> a small vic<strong>in</strong>ity of each <strong>in</strong>dividual macroscopicpo<strong>in</strong>t. The concepts of local and global periodicity are schematically illustrated <strong>in</strong> figure 2.The assumption of local periodicity adopted <strong>in</strong> the computational homogenization allowsto <strong>in</strong>corporate a non-uniform distribution of the microstructure at the macroscopic level(e.g. <strong>in</strong> functionally graded materials). Note that the local periodicity assumption is(a) local periodicity(b) global periodicityFigure 2: Local periodicity (a) versus global periodicity (b).directly l<strong>in</strong>ked to the pr<strong>in</strong>ciple of separation of scales.7

2.3 Homogenization pr<strong>in</strong>ciplesThe basic pr<strong>in</strong>ciples of computational homogenization have gradually evolved from theconcepts employed <strong>in</strong> other homogenization methods and well fit <strong>in</strong>to the four-step homogenizationscheme established by Suquet [56]:1. def<strong>in</strong>ition of a microstructural representative volume element (RVE), of which theconstitutive behaviour of <strong>in</strong>dividual constituents is assumed to be known;2. formulation of the microscopic boundary conditions from the macroscopic <strong>in</strong>putvariables and their application on the RVE (macro-to-micro transition);3. calculation of the macroscopic output variables from the analysis of the deformedmicrostructural RVE (micro-to-macro transition);4. obta<strong>in</strong><strong>in</strong>g the (numerical) relation between the macroscopic <strong>in</strong>put and output variables.The ma<strong>in</strong> ideas of the first-order computational homogenization have been established <strong>in</strong>[40, 56, 57, 68, 85] and further developed and improved <strong>in</strong> more recent works [1, 58, 59,61, 63, 64, 65, 69, 72].2.4 Computational homogenization schemeA computational homogenization generally departs from the computation of a macroscopicdeformation (gradient) tensor F M , which is calculated for every material po<strong>in</strong>t ofthe macrostructure (e.g. the <strong>in</strong>tegration po<strong>in</strong>ts with<strong>in</strong> a macroscopic f<strong>in</strong>ite element doma<strong>in</strong>).Here and <strong>in</strong> the follow<strong>in</strong>g the subscript “M” refers to a macroscopic quantity,while the subscript “m” will denote a microscopic quantity. The deformation tensor F Mfor a macroscopic po<strong>in</strong>t is next used to formulate the boundary conditions to be imposedon the RVE that is assigned to this po<strong>in</strong>t. Upon the solution of the boundaryvalue problem for the RVE, the macroscopic stress tensor P M is obta<strong>in</strong>ed by averag<strong>in</strong>gthe result<strong>in</strong>g RVE stress field over the volume of the RVE. As a result, the (numerical)stress-deformation relationship at the macroscopic po<strong>in</strong>t is readily available. Additionally,the local macroscopic consistent tangent is extracted from the microstructural stiffness.The entire framework is schematically illustrated <strong>in</strong> figure 3. The computational homog-Figure 3: Computational homogenization scheme.8

enization technique def<strong>in</strong>ed <strong>in</strong> this sense, is entirely consistent with the pr<strong>in</strong>ciple of localaction <strong>in</strong> cont<strong>in</strong>uum <strong>mechanics</strong>. Therefore, the response at a (macroscopic) material po<strong>in</strong>tdepends only on the first gradient of the displacement field. This macroscopically localcomputational homogenization framework may therefore be categorized as a “first-order”approach.2.5 K<strong>in</strong>ematically driven multi-scale schemeThe multi-scale procedure outl<strong>in</strong>ed <strong>in</strong> this framework is “deformation driven”. The po<strong>in</strong>tof departure is thereby the macroscopic deformation gradient tensor F M , which is usedto determ<strong>in</strong>e the stress P M and the constitutive tangent, <strong>based</strong> on the response of theunderly<strong>in</strong>g microstructure. A “stress driven” procedure (given a local macroscopic stress,obta<strong>in</strong> the deformation) is also possible. However, such a procedure does not directly fit<strong>in</strong>to a standard displacement-<strong>based</strong> f<strong>in</strong>ite element framework, which will be here employedto solve the macroscopic boundary value problem. Moreover, <strong>in</strong> case of large deformationsthe macroscopic rotational effects have to be added to the stress tensor <strong>in</strong> order touniquely determ<strong>in</strong>e the deformation gradient tensor, thus complicat<strong>in</strong>g the implementation.Therefore, the “stress driven” approach, which is often used <strong>in</strong> the analysis of s<strong>in</strong>gleunit cells, is generally not adopted <strong>in</strong> coupled multi-scale computational homogenizationstrategies of the type described here.3 The micro-scale problem3.1 The representative volume elementThe physical and geometrical properties of the microstructure are identified by a representativevolume element (RVE) [29, 86]. An example of a typical two-dimensional RVEis depicted <strong>in</strong> figure 4. The actual choice of the RVE is a rather delicate task. The RVEshould be large enough to represent the microstructure, without <strong>in</strong>troduc<strong>in</strong>g non-exist<strong>in</strong>gproperties (e.g. undesired anisotropy) and at the same time it should be small enough toallow efficient computational modell<strong>in</strong>g. Some issues related to the concept of a representativecell are discussed furtheron, <strong>in</strong> section 7. Here it is supposed that an appropriateRVE has been already selected. Then the problem on the RVE level can be formulatedas a standard problem <strong>in</strong> quasi-static cont<strong>in</strong>uum <strong>solid</strong> <strong>mechanics</strong>.3.2 Micro-scale characterization & equilibriumThe RVE deformation field <strong>in</strong> a po<strong>in</strong>t with the <strong>in</strong>itial position vector X ⃗ (<strong>in</strong> the referencedoma<strong>in</strong> V 0 ) and the actual position vector ⃗x (<strong>in</strong> the current doma<strong>in</strong> V ) is described by themicrostructural deformation gradient tensor Fm =( ∇ ⃗ 0,m ⃗x) c , where the gradient operator⃗∇ 0,m is taken with respect to the reference microstructural configuration; the superscript“c” <strong>in</strong>dicates conjugation.The RVE is <strong>in</strong> a state of equilibrium. This is mathematically expressed through thestandard equilibrium equation <strong>in</strong> terms of the Cauchy stress tensor σ m or, alternatively,<strong>in</strong> terms of the first Piola-Kirchhoff stress tensor P m =det(F m )σ m·(F c m )−1 accord<strong>in</strong>g to9

Figure 4: Schematic picture of a typical two-dimensional representative volume element(RVE).(<strong>in</strong> the absence of body forces)⃗∇ m·σ m = ⃗0 <strong>in</strong> V, or ⃗ ∇0,m·P c m = ⃗0 <strong>in</strong> V 0 , (2)where ∇ ⃗ m is the the gradient operator with respect to the current configuration at themicro-scale.The mechanical characterization of the microstructural components are described by theirconstitutive laws, specify<strong>in</strong>g a time and history dependent stress-deformation relationshipfor every microstructural constituentσ (α)m (t) =F (α)σ {F (α)m (τ), τ∈ [0,t]}, or P (α)m (t) =F (α)P{F(α) m (τ), τ∈ [0,t]}, (3)where t denotes the current time; α = 1,N, with N the number of microstructuralconstituents to be dist<strong>in</strong>guished (e.g. matrix, <strong>in</strong>clusions, etc.). Note that the knowledgeof the separate constitutive laws for each of the <strong>in</strong>dividual phases is essential.3.3 The macro-micro scale transitionThe macro-micro scale transition requires a method to impose the macroscopic deformationgradient tensor F M or stress tensor P M on the microstructural RVE. Classicalsimplified methods to do this are:• by impos<strong>in</strong>g that all the microstructural constituents undergo a constant deformationidentical to the macroscopic one (the Taylor or Voigt assumption).• by impos<strong>in</strong>g an identical constant stress (and additionally identical rotation) to allthe components (the Sachs or Reuss) assumption).• by <strong>in</strong>termediate procedures, where the Taylor and Sachs assumptions are appliedonly to certa<strong>in</strong> components of the deformation and stress tensors.These simplified procedures do not satisfy all local static equilibrium and compatibilityconditions and generally provide only rough estimates of the overall material properties.They are therefore not well-suited <strong>in</strong> complex non-l<strong>in</strong>ear deformation regimes. The Taylorassumption usually overestimates the overall stiffness, while the Sachs assumption leads toan underestimation of the stiffness. A computational homogenization scheme does enforce10

local equilibrium and compatibility between phases, and therefore necessitates a differentmacro-micro scale transition method.The first-order scheme naturally departs from the classical l<strong>in</strong>earization of the macroscopicnonl<strong>in</strong>ear deformation map, ⃗x = φ( X), ⃗ applied to a f<strong>in</strong>ite material vector Δ⃗x <strong>in</strong> thedeformed state:Δ⃗x = F M·ΔX ⃗ + O(Δ X ⃗ )2 , (4)with ⃗x and X ⃗ associated position vectors <strong>in</strong> the deformed and reference state, respectively,and<strong>in</strong>whichF M =( ∇ ⃗ 0,M ⃗x) T is the macroscopic deformation gradient tensor. Consider<strong>in</strong>gan undeformed volume V 0 of material with its centre positioned at X ⃗ c , permits to writethe deformed position of any po<strong>in</strong>t of this volume (with respect to the centre of thatvolume) as the sum of a macroscopic (or coarse scale) and a microscopic (or f<strong>in</strong>e scale)contribution:⃗x − ⃗x c = F M·( X ⃗ − X ⃗ c )+⃗w (5)The f<strong>in</strong>e scale contribution is here represented by the microfluctuation field ⃗w. Thevector ⃗x c is the actual position of the reference RVE center X ⃗ c . Obviously, rigid bodydisplacements have to be elim<strong>in</strong>ated to uniquely determ<strong>in</strong>e ⃗x. An arbitrary boundarypo<strong>in</strong>t may be fixed to this purpose, e.g. for a po<strong>in</strong>t with label 1 (see figure 4) by impos<strong>in</strong>g⃗x 1 = X ⃗ 1 . Substitut<strong>in</strong>g this <strong>in</strong> (5) leads to⃗x = ⃗c + F M·( ⃗ X − ⃗ X c )+(⃗w − ⃗w 1 ) (6)where ⃗w 1 is the microfluctuation <strong>in</strong> po<strong>in</strong>t 1 and where vector ⃗c is determ<strong>in</strong>ed from ⃗ X 1 ,be<strong>in</strong>g <strong>in</strong>dependent of the f<strong>in</strong>e scale field⃗c = ⃗ X 1 − F M·( ⃗ X 1 − ⃗ X c ) (7)The deformed position ⃗x c of the reference centre ⃗ X c is then (us<strong>in</strong>g the trivial relation⃗w c = ⃗0) given by⃗x c = ⃗c − ⃗w 1 (8)Note that this deformed position is unknown and implicitly depends on the f<strong>in</strong>e scalefield. The scale transition between the k<strong>in</strong>ematics at the f<strong>in</strong>e and the coarse scale typically<strong>in</strong>volves the volume average ¯F m of the f<strong>in</strong>e scale deformations tensors, i.e.¯F m = 1 V 0∫V 0F m dV 0 (9)(10)This volume <strong>in</strong>tegral can be rewritten to the boundary Γ 0 of the RVE by mak<strong>in</strong>g use ofthe divergence theorem¯F m = 1 ∫F m dV 0 = 1 ∫V 0 V 0( ) c⃗∇0,m·(I⃗x) dV0=V 0 V 0∫ ( ) c⃗N·(I⃗x) dΓ0V 0Γ 0=Γ 0∫⃗x NdΓV 0011

where I represents the second-order unit tensor, Γ 0 the external boundary of the undeformedRVE V 0 and N ⃗ the outward unit normal on that boundary.Comput<strong>in</strong>g the f<strong>in</strong>e scale deformation gradient tensor F m by tak<strong>in</strong>g the f<strong>in</strong>e scale spatialgradient of the position vector given <strong>in</strong> equation (6) results <strong>in</strong>F m =( ⃗ ∇ 0,m ⃗x) c = F M +( ⃗ ∇ 0,m (⃗w − ⃗w 1 )) c = F M +( ⃗ ∇ 0,m ⃗w) c (11)Alternatively, mak<strong>in</strong>g use of the micro-fluctuation field, ¯F m can be expanded to¯F m = F M + 1 ∫[ ∇V ⃗ 0,m (⃗w − ⃗w 1 )] c dV 0 = F M + 1 ∫[ ∇0 V ⃗ 0,m ⃗w] c dV 00V 0 V 0= F M + 1 ∫(⃗w − ⃗w 1 ) NdΓΓ ⃗ 0 = F M + 1 ∫(12)⃗w NdΓ0 Γ ⃗ 00Γ 0 Γ 0InthecasewhereF M is known and displacements at the RVE boundary are to be prescribedare constra<strong>in</strong>ed, use is made of a scale transition relation that enforces the macroscopicdeformation gradient F M to equal the volume average of its microscopic counterparts¯F m ,F M = ¯F m (13)Enforc<strong>in</strong>g the scale transition relation (13) clearly leads to a constra<strong>in</strong>t <strong>in</strong> the form of aboundary <strong>in</strong>tegral ∫Γ 0∫Γ 0(⃗w − ⃗w 1 ) NdΓ ⃗ 0 = 0 = ⃗w NdΓ ⃗ 0 (14)The boundary <strong>in</strong>tegral (14) is the necessary condition that enforces the averag<strong>in</strong>g theorem(13), which will be used <strong>in</strong> the scale transition, see also [87]. Stronger conditions areobta<strong>in</strong>ed by mak<strong>in</strong>g specific choices for ⃗w that enforce this boundary <strong>in</strong>tegral to vanish.A few possible choices for these boundary conditions are discussed further on.The follow<strong>in</strong>g remarks can be made with respect to the macro-micro scale transition:• From equation (6) and (14) it appears that the microfluctuation field only enters thek<strong>in</strong>ematics relative to ⃗w 1 <strong>in</strong> po<strong>in</strong>t 1, i.e. through ⃗w− ⃗w 1 . Tak<strong>in</strong>g the microfluctuationfield <strong>in</strong> this po<strong>in</strong>t ⃗w 1 equal to zero will not <strong>in</strong>fluence the obta<strong>in</strong>ed solution, s<strong>in</strong>ce theaverag<strong>in</strong>g theorem rema<strong>in</strong>s valid. The only difference resides <strong>in</strong> the result<strong>in</strong>g vector⃗x c , which is entirely determ<strong>in</strong>ed from the coarse scale, i.e. ⃗x c = ⃗c see equation (8).Clearly, ⃗x c no longer represents the deformed position of the orig<strong>in</strong>al RVE centre⃗X c , s<strong>in</strong>ce it is translated with respect to this position. The choice ⃗w 1 = ⃗0 isoftenmade <strong>in</strong> practical implementations of the first-order homogenization scheme, s<strong>in</strong>ceit leads to the correct solution <strong>in</strong> a practical way.• For the first-order case, any base po<strong>in</strong>t could have been taken to expand ⃗x accord<strong>in</strong>gto (4) <strong>in</strong>to the RVE, lead<strong>in</strong>g to the same solution as the specific choice made here(the RVE center ⃗ X c ).• Logically, the solution does not depend on the po<strong>in</strong>t that was fixed at the boundary.A po<strong>in</strong>t <strong>in</strong>side the volume V 0 can be taken as well to elim<strong>in</strong>ate rigid body displacements<strong>in</strong> (6). Aga<strong>in</strong> the deformed shape of the RVE and the stress state extractedfrom it, rema<strong>in</strong> the same.12

3.4 Micro-scale RVE boundary conditionsAs emphasized <strong>in</strong> the previous section, possible RVE boundary conditions naturally resultfrom the constra<strong>in</strong>t (14) imposed by the scale transition. Among the various choicespossible, only three particular cases will be considered hereafter <strong>in</strong> more detail. Note thatthe Taylor assumption trivially satisfies (14) s<strong>in</strong>ce the microfluctuation field is then zero<strong>in</strong> the entire volume V 0 and hence also at its boundary Γ 0 .3.4.1 Displacement boundary conditionsThe first case considered is def<strong>in</strong>ed by constra<strong>in</strong><strong>in</strong>g each po<strong>in</strong>t at the RVE boundarythrough the macroscopic deformation by⃗x = F M· ⃗X with ⃗ X on Γ0 , (15)This simply implies that the micro-fluctuation field ⃗w is zero at the boundary Γ 0 ,whichtrivially satisfies (14). The position of all po<strong>in</strong>ts at the boundary are determ<strong>in</strong>ed throughthe macroscopic deformation only, lead<strong>in</strong>g to a l<strong>in</strong>ear mapp<strong>in</strong>g of the RVE boundary. Theboundary will therefore reproduce typical stretch (tension/compression) and shear modesonly.3.4.2 Traction boundary conditionsThis case departs from the assumption that P M is to be prescribed to the RVE. Theboundary conditions are then def<strong>in</strong>ed by constra<strong>in</strong><strong>in</strong>g all tractions at the RVE boundaryto the macroscopic stress tensor by⃗t = ⃗n·σ M on Γ, or ⃗p = ⃗ N·P c M on Γ 0, (16)with ⃗n the normal to the current (Γ) RVE boundary. Note that the traction boundaryconditions (16) do not completely def<strong>in</strong>e the microstructural boundary value problem,s<strong>in</strong>ce rotations are yet undeterm<strong>in</strong>ed. As emphasized earlier, these boundary conditionsare a priori not appropriate <strong>in</strong> a deformation driven procedure as pursued <strong>in</strong> the presentcomputational homogenization scheme. The <strong>in</strong>terested reader is referred to the work of[88, 89], where it is shown that the traction boundary condition is the weakest conditionto enforce (14). From a practical po<strong>in</strong>t of view, these boundary conditions generallyyield unsatisfactory results. Therefore, the RVE traction boundary conditions will not beexplored further <strong>in</strong> these lecture notes.3.4.3 Periodic boundary conditionsMak<strong>in</strong>g use of the earlier <strong>in</strong>troduced concept of local periodicity, periodic boundary conditionsare next <strong>in</strong>troduced. The periodicity conditions for the microstructural RVE arewritten <strong>in</strong> a general format asor formulated <strong>in</strong> terms of the micro-fluctuation fields⃗x + − ⃗x − = F M·( ⃗ X + − ⃗ X − ), (17)⃗w + = ⃗w − (18)13

Deformations are periodic s<strong>in</strong>ce micro-fluctuations on opposite sides are identical. Herethe (opposite) parts of the RVE boundary Γ − 0 and Γ + 0 are def<strong>in</strong>ed such that N ⃗ − = −N ⃗ +at correspond<strong>in</strong>g po<strong>in</strong>ts on Γ − 0 and Γ + 0 , see figure 4. The periodicity condition (17),be<strong>in</strong>g prescribed on an <strong>in</strong>itially periodic RVE, preserves the periodicity of the RVE <strong>in</strong> thedeformed state.The periodic boundary conditions (17) clearly satisfy the constra<strong>in</strong>t (14). This is easilyobserved by splitt<strong>in</strong>g the RVE boundary <strong>in</strong>to the parts Γ + 0 and Γ − 0∫∫∫⃗w NdΓ ⃗ 0 = ⃗w + N ⃗ + dΓ + 0 + ⃗w − N ⃗ − dΓ − 0Γ 0 Γ + 0Γ − 0∫∫= ⃗w + N ⃗ + dΓ + 0 − ⃗w + N ⃗ + dΓ − (19)0Γ + 0= 0Γ − 0Note that as a result of microstructural equilibrium, tractions will be anti-periodic onopposite sides:⃗p + = −⃗p − , (20)Note that, as has been observed by several authors (e.g. [62, 90]), periodic boundaryconditions provide a better estimation of the overall properties, than the prescribed displacementor prescribed traction boundary conditions.4 The macro-scale problem4.1 The micro-macro scale transitionOnce the micro-scale problem has been solved, macroscopic quantities have to be extractedfrom the obta<strong>in</strong>ed results. Whereas deformation averag<strong>in</strong>g was the key assumptionfor the macro-micro transition, energy averag<strong>in</strong>g constitutes the key assumption forthe reverse transition. This energy averag<strong>in</strong>g theorem, known <strong>in</strong> the literature as theHill-Mandel condition or macro-homogeneity condition [56, 86], requires that the macroscopicvolume average of the variation of work performed on the RVE is equal to the localvariation of the work on the macro-scale, i.e.δW 0M = δW 0m (21)Formulated <strong>in</strong> terms of a work conjugated set, i.e. the deformation gradient tensor andthe first Piola-Kirchhoff stress tensor, the Hill-Mandel condition readsP M : δF c M} {{ }δW 0M= 1 V 0∫P m : δF c m dV 0(22)V}0{{ }δW 0mThe averaged microstructural work <strong>in</strong> the right-hand side of (22) may be expressed <strong>in</strong>terms of RVE boundary quantitiesδW 0m = 1 ∫P m : δF c mV dV 0 = 1 ∫⃗p ·δ⃗x dΓ 0 , (23)0 V 0Γ 0V 014

where the relation (with account for microstructural equilibrium)P m : ∇ 0m δ⃗x = ∇ 0m·(P c m·δ⃗x) − (∇ 0m·P c m)·δ⃗x = ∇ 0m·(P c m·δ⃗x),and the divergence theorem have been used.As will be shown next, an important result of postulat<strong>in</strong>g the Hill-Mandel condition for anRVE with k<strong>in</strong>ematic boundary conditions (fully prescribed or periodically tied), is the factthat the macroscopic stress tensor P M equals the volume average ¯P m of the microscopicstress tensors. To this purpose, it is convenient to establish the boundary relation for themean RVE stress ¯P m , i.e.¯P m = 1 ∫P m dV 0V 0V 0= 1 ∫V 0(24)=V 0 ∫V 0N·(P ⃗ cm X)dΓ0 ⃗∇0·(P ⃗ c ⃗ m X)dV 0Γ 0=Γ 0∫⃗p XV dΓ 004.1.1 Displacement boundary conditionsIn case of fully prescribed boundary displacements (15), substitution of the variation ofthe boundary position vectors δ⃗x = δF M· ⃗X <strong>in</strong>to the expression for the averaged microwork(23) with <strong>in</strong>corporation of (33) gives⎡⎤δW 0m = 1 ∫⃗p·(δF M·V ⃗X) dΓ 0 = ⎣ 1 ∫⃗p X0 V ⃗ dΓ 0⎦ : δF c M = ¯P m : δF c M (25)0Γ 0Enforc<strong>in</strong>g the Hill-Mandel condition (22) thus implies thatΓ 0P M = ¯P m (26)4.1.2 Traction boundary conditionsSubstitution of the traction boundary condition (16) <strong>in</strong>to (23), with account for the variationof the average of the microscopic deformation gradient tensor obta<strong>in</strong>ed by vary<strong>in</strong>grelation (9), leads to⎡⎤δW 0m = 1 ∫( N·PV ⃗ c M )·δ⃗x dΓ 0 = P M : ⎣ 1 ∫Nδ⃗x ⃗ dΓ0 ⎦ = P M : δ0 V ¯F c m . (27)0Γ 0In this case, the Hill-Mandel condition (22) enforces the result<strong>in</strong>g macroscopic deformationgradient to be taken as the volume average of the microscopic deformation gradients, i.e..F M = ¯F m = 1 V 0∫V 0F m dV 0 (28)15Γ 0

This implies that the traction boundary conditions, complemented by the Hill-Mandelconditions, constitute the weakest k<strong>in</strong>ematic constra<strong>in</strong>t for the boundary displacements,i.e. equation (14). Comput<strong>in</strong>g the volume average of the micro-scale RVE stresses fromequation (24) now yields¯P m = 1 ∫⃗p XV dΓ 00=Γ 0 ∫( N·PV M ) X ⃗ dΓ 00Γ 0⎡= P M· ⎣ 1 ∫V 0⃗ N ⃗ X dΓ0⎤⎦(29)Γ[0] 1= P M· V 0 IV 0= P MAga<strong>in</strong>, the macroscopic stress equals the volume average of the microscopic stress, butthis time this conclusion does not result from the Hill-Mandel condition.4.1.3 Periodic boundary conditionsFor the periodic boundary conditions (17) and the result<strong>in</strong>g anti-periodic tractions (20)δW 0m = 1 {∫⃗p +·δ⃗x ∫+ dΓ 0 + ⃗p −·δ⃗x }− dΓ 0 = 1 ∫⃗p +·(δ⃗x + − δ⃗x − )dΓ + 0V 0 V 0Γ + 0⎡⎢= ⎣ 1 ∫V 0Γ + 0Γ − 0⎤ ⎡⃗p + ( X ⃗ + − X ⃗ − )dΓ + ⎥0 ⎦ : δF c M = ⎣ 1 ∫V 0Γ 0Γ + 0⃗p ⃗ X dΓ 0⎤⎦ : δF c M(30)= ¯P m : δF c MEnforc<strong>in</strong>g the Hill-Mandel condition (22) aga<strong>in</strong> implies thatP M = ¯P m (31)4.2 Macroscopic stress tensorsS<strong>in</strong>ce the scale transition implies stress averag<strong>in</strong>g for all considered boundary conditions,the macroscopic stress tensor is given byP M = 1 ∫P m dV 0 (32)V 0=V 0 ∫⃗p XV dΓ 0 (33)0Γ 016

The volume average of the microscopic Cauchy stress tensor σ m over the current RVEvolume V can be elaborated similarly to (33)σ ∗ M = 1 ∫σ m dV = 1 ∫⃗t⃗x dΓ. (34)VVVΓJust as it is the case for k<strong>in</strong>ematic quantities, the usual pull-back push-forward relationsbetween stress measures (e.g. the Cauchy and the first Piola-Kirchhoff stress tensors)are, <strong>in</strong> general, not valid for the volume averages of the microstructural counterpartsσ ∗ M ≠ P M·F c M / det(F M). If the averag<strong>in</strong>g is <strong>based</strong> on P M , the Cauchy stress tensor on themacrolevel should be def<strong>in</strong>ed asσ M =1det(F M ) P M·F c M. (35)Clearly, there is some arbitrar<strong>in</strong>ess <strong>in</strong> the choice of the govern<strong>in</strong>g deformation and stresstensors, whose macroscopic measures are equal to the volume average of their microscopiccounterparts (through the imposed scale transition relations). Macroscopic measures def<strong>in</strong>edon another configuration are then expressed <strong>in</strong> terms of the govern<strong>in</strong>g averagedquantities us<strong>in</strong>g the standard pull-back push-forward relations. The specific selectionmade here is ma<strong>in</strong>ly <strong>based</strong> on its ease of implementation. The actual choice of the “primary”averag<strong>in</strong>g measures used here, i.e. the deformation gradient tensor F and the firstPiola-Kirchhoff stress tensor P (and their rates), has been advocated <strong>in</strong> [59, 91, 92] (<strong>in</strong>the last two references the nom<strong>in</strong>al stress S N =det(F)F −1·σ = P c has been used). Thisparticular choice is motivated by the fact that these two measures are work conjugated,comb<strong>in</strong>ed with the observation that their volume averages can exclusively be def<strong>in</strong>ed <strong>in</strong>terms of the microstructural quantities of the undeformed RVE boundary.5 Two-scale numerical solution strategyOnce the boundary conditions have been properly def<strong>in</strong>ed through one of the methodsoutl<strong>in</strong>e above and once all phases <strong>in</strong> the microstructure have been characterized, a standardboundary value problem (BVP) has been obta<strong>in</strong>ed. The solution of this BVP followsstandard procedures. In the present computational homogenization method, it will be assumedthat the f<strong>in</strong>ite element method has been used to this purpose. The solution ofthis BVP problem automatically leads to the proper determ<strong>in</strong>ation of all position vectors<strong>in</strong> the RVE and all tractions along its boundary. The analysis is further restricted tok<strong>in</strong>ematic RVE boundary conditions only.5.1 RVE boundary value problemThe RVE problem to be solved is a standard non-l<strong>in</strong>ear quasi-static boundary value problemwith k<strong>in</strong>ematic boundary conditions. Follow<strong>in</strong>g the standard f<strong>in</strong>ite element procedurefor the microlevel RVE, after discretization, the weak form of equilibrium (2) with accountfor the constitutive relations (3) leads to a system of non-l<strong>in</strong>ear algebraic equations <strong>in</strong> theunknown nodal displacements ũ˜<strong>in</strong>t f ( ũ) =f˜ext, (36)17

express<strong>in</strong>g the balance of <strong>in</strong>ternal and external nodal forces. This system has to becompleted by the govern<strong>in</strong>g boundary conditions. To this purpose, the earlier <strong>in</strong>troducedk<strong>in</strong>ematic boundary conditions (15) or (17) will be elaborated <strong>in</strong> more detail.5.1.1 Fully prescribed boundary displacementsIn the case of the fully prescribed displacement boundary conditions (15), the displacementsof all nodes on the boundary is simply given by⃗u p =(F M − I)· ⃗X p , p = 1,N p (37)where N p is the number of prescribed nodes, which <strong>in</strong> this case simply equals to the numberof boundary nodes. The boundary conditions (37) are simply added to the system (36)<strong>in</strong> a standard manner by static condensation, Lagrange multipliers or penalty functions.5.1.2 Periodic boundary conditionsPrior to the <strong>in</strong>corporation of the periodic boundary conditions (17), they have to berewritten <strong>in</strong>to a format that is more suitable for a f<strong>in</strong>ite element framework. Consider atwo-dimensional periodic RVE schematically depicted <strong>in</strong> figure 4. The boundary of thisRVE can be split <strong>in</strong>to four parts, here denoted as “T” top, “B” bottom, “R” right and“L” left. To ease application of the periodicity constra<strong>in</strong>t, a f<strong>in</strong>ite element discretization isnext considered which has a periodic distribution of nodes on opposite edges. Exploit<strong>in</strong>gthe <strong>in</strong>itial periodicity of the RVE (<strong>in</strong> its reference configuration) allows to write for everyrespective pair of nodes on the top-bottom and right-left boundaries:⃗X T − X ⃗ B = X ⃗ 4 − X ⃗ 1 ,⃗X R − X ⃗ L = X ⃗ 2 − X ⃗ 1 , (38)where ⃗ X p , p = 1, 2, 4 are the position vectors of the corner nodes 1, 2 and 4 <strong>in</strong> theundeformed state. Consider<strong>in</strong>g pairs of periodic nodes on opposite boundaries, allows toexpress (17) as⃗x T − ⃗x B = F M·( X ⃗ 4 − X ⃗ 1 ),⃗x R − ⃗x L = F M·( X ⃗ 2 − X ⃗ 1 ). (39)Apply<strong>in</strong>g these relations to the four corner nodes, permits to conclude that the positionvectors of the corner nodes <strong>in</strong> the deformed state are <strong>in</strong> fact prescribed accord<strong>in</strong>g to⃗x p = F M· ⃗X p , p =1, 2, 4 (40)The periodic boundary conditions may f<strong>in</strong>ally be rewritten as⃗x T = ⃗x B + ⃗x 4 − ⃗x 1 ,⃗x R = ⃗x L + ⃗x 2 − ⃗x 1 . (41)S<strong>in</strong>ce these conditions are trivially satisfied <strong>in</strong> the undeformed configuration, they maybe formulated <strong>in</strong> terms of displacements⃗u T = ⃗u B + ⃗u 4 − ⃗u 1 ,⃗u R = ⃗u L + ⃗u 2 − ⃗u 1 , (42)18

whereby⃗u p =(F M − I)· ⃗X p , p =1, 2, 4 (43)In a discretized format the relations (42) lead to a set of homogeneous constra<strong>in</strong>ts of thetypeC a ũ a , (44)=0˜with C a a matrix conta<strong>in</strong><strong>in</strong>g coefficients <strong>in</strong> the constra<strong>in</strong>t relations and ũ a a column withthe degrees of freedom <strong>in</strong>volved <strong>in</strong> the constra<strong>in</strong>ts. Procedures for impos<strong>in</strong>g constra<strong>in</strong>ts(44) <strong>in</strong>clude the direct elim<strong>in</strong>ation of the dependent degrees of freedom from the system ofequations, or the use of Lagrange multipliers or penalty functions. In the follow<strong>in</strong>g, constra<strong>in</strong>ts(44) are enforced by elim<strong>in</strong>ation of the dependent degrees of freedom. Althoughsuch a procedure may be found <strong>in</strong> many textbooks on f<strong>in</strong>ite elements (e.g. [93]), it is heresummarized for the sake of clarity and completeness, s<strong>in</strong>ce it will be applied <strong>in</strong> section 5.3for the derivation of the macroscopic tangent stiffness.To this purpose, the homogeneous constra<strong>in</strong>t relations (44) are partitioned accord<strong>in</strong>g to[ ]ũi[C i C d ] , (45)u=0˜˜dwhere ũ i are the <strong>in</strong>dependent degrees of freedom (to be reta<strong>in</strong>ed <strong>in</strong> the system) and ũ d arethe dependent degrees of freedom (to be elim<strong>in</strong>ated from the system). Because there areas many dependent degrees of freedom ũ d as there are <strong>in</strong>dependent constra<strong>in</strong>t equations<strong>in</strong> (45), matrix C d is square and non-s<strong>in</strong>gular. Solution for ũ d yieldsu = C di ũ i , with C di = −C˜d −1d C i. (46)This relation may be further rewritten as[ ]ũi= T ũui , with T =˜d[IC di], (47)where I is a unit matrix of size [N i × N i ], with N i the number of the <strong>in</strong>dependent degreesof freedom.With the transformation matrix T def<strong>in</strong>ed such that = T d d˜ ′ , the common transformationsr ′˜˜ = T T and K r˜ ′ = T T KT can be applied to a l<strong>in</strong>ear system of equations of the formK = , lead<strong>in</strong>g to a new system K d˜r˜ ′ d ′˜ = r′˜ .The standard l<strong>in</strong>earization of the non-l<strong>in</strong>ear system of equations (36) leads to a l<strong>in</strong>earsystem <strong>in</strong> the iterative corrections δũ to the current estimate ũ. This system may bepartitioned as [ ][ ] [Kii K id δũi= δr˜iK di K dd δũ d δr˜dwith the residual nodal forces at the right-hand side.], (48)Not<strong>in</strong>g that all the constra<strong>in</strong>tequations considered above are l<strong>in</strong>ear, and thus their l<strong>in</strong>earization is straightforward,application of the transformation (47) to the system (48) gives[Kii + K id C di + C T di K di + C T di K ]ddC di δũi = [ ]δr˜i + C T di δr˜d . (49)Note that the boundary conditions (43) prescrib<strong>in</strong>g displacements of the corner nodeshave not yet been applied. The column of “<strong>in</strong>dependent” degrees of freedom ũ i <strong>in</strong>cludesthe prescribed corner nodes ũ p among other nodes. The boundary conditions (43) shouldbe applied to the system (49) is a standard manner.The condition of antiperiodic tractions (20) will be addressed <strong>in</strong> section 5.2.2.19

5.2 Extraction of the macroscopic stressAfter the analysis of a microstructural RVE is completed, the RVE averaged stress have tobe extracted. Of course, the macroscopic stress tensor can be calculated by numericallyevaluat<strong>in</strong>g the volume <strong>in</strong>tegral (32). However it is computationally more efficient tocompute the boundary <strong>in</strong>tegral (33), which can be further simplified for the case of theperiodic boundary conditions.5.2.1 Fully prescribed boundary displacementsFor the case of prescribed displacement boundary conditions the boundary <strong>in</strong>tegral (33)simply leads toP M = 1 V 0N∑ pp=1⃗f p⃗ Xp , (50)where ⃗ f p are the result<strong>in</strong>g external forces at the boundary nodes and ⃗ X p the positionvectors of these nodes <strong>in</strong> the undeformed state; N p is the number of the nodes on theboundary.5.2.2 Periodic boundary conditionsIn order to simplify the boundary <strong>in</strong>tegral (33) for the case of periodic boundary conditions,consider all the forces act<strong>in</strong>g on the RVE boundary subjected to the boundaryconditions accord<strong>in</strong>g to (42)–(43). At the three prescribed corner nodes the result<strong>in</strong>g externalforces f ⃗ p,p=1, e 2, 4 act. Additionally, there are forces <strong>in</strong>volved <strong>in</strong> every constra<strong>in</strong>t(ty<strong>in</strong>g) relation (42). For example, for each constra<strong>in</strong>t relation between pairs of the nodeson the bottom-top boundaries there is a ty<strong>in</strong>g force at the node on the bottom boundary⃗p t B , a ty<strong>in</strong>g force at the node on the top boundary ⃗p t T and ty<strong>in</strong>g forces at the cornernodes 1 and 4, ⃗p t B1 and ⃗p t B4 , respectively. Similarly there are forces ⃗p t L , ⃗p t R , ⃗p t L1 and ⃗p t L2correspond<strong>in</strong>g to the left-right constra<strong>in</strong>ts. All these forces are schematically shown <strong>in</strong>figure 5.fe4p t Tp t B4p t Rp t Lp t B1p t L1fe1p t Bp t L2fe2Figure 5: Tractions act<strong>in</strong>g on the boundary of a two-dimensional RVE subjected to periodicboundary conditions.20

Each constra<strong>in</strong>t relation satisfies the condition of zero virtual work, i.e.⃗p t B·δ⃗x B + ⃗p t T·δ⃗x T + ⃗p t B1 ·δ⃗x 1 + ⃗p t B4 ·δ⃗x 4 =0,⃗p t L·δ⃗x L + ⃗p t R·δ⃗x R + ⃗p t L1 ·δ⃗x 1 + ⃗p t L2 ·δ⃗x 2 =0. (51)Substitution of the variation of the constra<strong>in</strong>ts (41) <strong>in</strong>to (51) gives(⃗p t B + ⃗p t T )·δ⃗x B +(⃗p t B1 − ⃗p t T )·δ⃗x 1 +(⃗p t T + ⃗p t B4 )·δ⃗x 4 =0,(⃗p t L + ⃗p t R)·δ⃗x L +(⃗p t L1 − ⃗p t R)·δ⃗x 1 +(⃗p t R + ⃗p t L2 )·δ⃗x 2 =0. (52)These relations should hold for any δ⃗x B , δ⃗x L , δ⃗x 1 , δ⃗x 2 , δ⃗x 4 , therefore⃗p t B = −⃗p t T = −⃗p t B1 = ⃗p t B4 ,⃗p t L = −⃗p t R = −⃗p t L1 = ⃗p t L2 . (53)Equation (53) reflects the antiperiodicity of ty<strong>in</strong>g forces on the opposite boundaries, whichhas been <strong>in</strong>troduced previously <strong>in</strong> equation (20).With account for all forces act<strong>in</strong>g on the RVE boundary, the boundary <strong>in</strong>tegral (33) iswritten asP M = 1 (f ⃗1 e XV ⃗ 1 + f ⃗ 2 e X ⃗ 2 + f ⃗ 4 e X ⃗ 4 +0∫∫∫∫⃗p t ⃗ BX B dΓ 0 + ⃗p t TX ⃗ T dΓ 0 + ⃗p t LX ⃗ L dΓ 0 + ⃗p t RX ⃗ R dΓ 0 +Γ 0B Γ 0T Γ 0L Γ 0R( ∫ )⃗p t B1 dΓ ⃗X1 0 + (∫ )⃗p t L1 dΓ ⃗X1 0 + (∫ )⃗p t B4 dΓ ⃗X4 0 + (∫ )⃗p t L2 dΓ ⃗X2 0).Γ 0BΓ 0BΓ 0LMak<strong>in</strong>g use of the relation between ty<strong>in</strong>g forces (53) givesP M = 1 ( ∑∫∫⃗f p e XV ⃗ p + ⃗p t B ( X ⃗ B − X ⃗ T )dΓ 0 + ⃗p t L ( X ⃗ L − X ⃗ R )dΓ 0 +0p=1,2,4Γ 0B Γ 0L( ∫ )⃗p t B1 dΓ ⃗X1 0 + (∫ )⃗p t L1 dΓ ⃗X1 0 + (∫ )⃗p t B4 dΓ ⃗X4 0 + (∫ )⃗p t L2 dΓ ⃗X2 0).Γ 0LInsert<strong>in</strong>g the conditions of the <strong>in</strong>itial periodicity of the RVE (38) results <strong>in</strong>P M = 1 ( ∑∫∫⃗f p e XV ⃗ p + (⃗p t B + ⃗p t B1 ) X ⃗ 1 dΓ 0 + (⃗p t L + ⃗p t L1 ) X ⃗ 1 dΓ 0 +0p=1,2,4Γ 0B Γ 0L∫∫(⃗p t B4 − ⃗p t B) X ⃗ 4 dΓ 0 + (⃗p t L2 − ⃗p t L) X ⃗ 2 dΓ 0),Γ 0B Γ 0Lwhich after substitution of the rema<strong>in</strong><strong>in</strong>g relations between ty<strong>in</strong>g forces (53) givesP M = 1 ∑⃗f p e XV ⃗ p . (57)0p=1,2,4Therefore, when the periodic boundary conditions are used, all the terms with forces<strong>in</strong>volved <strong>in</strong>to the periodicity constra<strong>in</strong>ts cancel out from the boundary <strong>in</strong>tegral (33) andthe only contribution left is by the external forces at the three prescribed corner nodes.Γ 0BΓ 0BΓ 0LΓ 0L(54)(55)(56)21

5.3 Extraction of the macroscopic tangent operatorWhen the micro-macro approach is implemented with<strong>in</strong> the framework of a non-l<strong>in</strong>earf<strong>in</strong>ite element code, the stiffness matrix at every macroscopic <strong>in</strong>tegration po<strong>in</strong>t is required.Because <strong>in</strong> the computational homogenization approach there is no explicit form of theconstitutive behaviour on the macrolevel assumed a priori, the stiffness matrix has to bedeterm<strong>in</strong>ed numerically from the relation between variations of the macroscopic stressand variations of the macroscopic deformation at such a po<strong>in</strong>t. This may be realized bynumerical differentiation of the numerical macroscopic stress-stra<strong>in</strong> relation, for exampleus<strong>in</strong>g a forward difference approximation as has been suggested <strong>in</strong> [94]. Another approachis to condense the microstructural stiffness to the local macroscopic stiffness. This isachieved by reduc<strong>in</strong>g the total RVE system of equations to the relation between theforces act<strong>in</strong>g on the RVE boundary and the associated boundary displacements. Sucha procedure <strong>in</strong> comb<strong>in</strong>ation with the Lagrange multiplier method to impose boundaryconstra<strong>in</strong>ts has been elaborated <strong>in</strong> [65]. Here an alternative scheme, which employs thedirect condensation of the constra<strong>in</strong>ed degrees of freedom, as has been presented <strong>in</strong> [1,72] will be considered. After the condensed microscopic stiffness relat<strong>in</strong>g the prescribeddisplacement and force variations is obta<strong>in</strong>ed, it needs to be transformed to arrive at anexpression relat<strong>in</strong>g variations of the macroscopic stress and deformation tensors, typicallyused <strong>in</strong> the f<strong>in</strong>ite element codes. These two steps are elaborated <strong>in</strong> the follow<strong>in</strong>g.5.3.1 Condensation of the microscopic stiffness matrix:fully prescribed boundary displacementsFirst the total microstructural system of equations (<strong>in</strong> its l<strong>in</strong>earized form) is partitionedas [ ][ ] [ ]Kpp K pf δũp= δf˜p, (58)K fp K ff δũ f 0˜where δũ p and δf˜p are the columns with iterative displacements and external forces of theboundary nodes, respectively, and δũ f the column with the iterative displacements of therema<strong>in</strong><strong>in</strong>g (<strong>in</strong>terior) nodes; K pp , K pf , K fp and K ff are the correspond<strong>in</strong>g partitions ofthe total RVE stiffness matrix. The stiffness matrix <strong>in</strong> the formulation (58) is taken atthe end of a microstructural <strong>in</strong>crement, where a converged state is reached. Elim<strong>in</strong>ationof δũ f from (58) leads to the reduced stiffness matrix K M relat<strong>in</strong>g boundary displacementvariations to boundary force variationsK M δũ p = δf˜p, with K M = K pp − K pf (K ff ) −1 K fp . (59)5.3.2 Condensation of the microscopic stiffness matrix:periodic boundary conditionsIn the case of the periodic boundary conditions the po<strong>in</strong>t of departure is the microscopicsystem of equations (49) from which the dependent degrees of freedom have been elim<strong>in</strong>ated(as described <strong>in</strong> section 5.1.2)K ⋆ δũ i = δr˜⋆ , (60)with K ⋆ = K ii + K id C di + C T di K di + C T di K ddC di ,δr˜⋆ = δr˜i + C T di δr˜d.22

Next, system (60) is further split, similarly to (58), <strong>in</strong>to the parts correspond<strong>in</strong>g to thevariations of the prescribed degrees of freedom δũ p (which <strong>in</strong> this case are the variedpositions of the three corner nodes prescribed accord<strong>in</strong>g to (43)), variations of the external⋆forces at these prescribed nodes denoted by p , and the rema<strong>in</strong><strong>in</strong>g (free) displacementvariations δũ f :δf˜[ ][ ] [ ]K⋆pp K ⋆ ⋆pf δũpK ⋆ fp K ⋆ p=ff δũ fδf˜ . (61)0˜Then the reduced stiffness matrix K ⋆ M <strong>in</strong> case of periodic boundary conditions is obta<strong>in</strong>edasK ⋆ M δũ ⋆p = p δf˜, with K⋆ M = K⋆ pp − K⋆ pf (K⋆ ff )−1 K ⋆ fp . (62)Note that K ⋆ M is [6 × 6] matrix only (<strong>in</strong> the two-dimensional case).5.3.3 F<strong>in</strong>al macroscopic tangentF<strong>in</strong>ally, the result<strong>in</strong>g relation between displacement and force variations (relation (59)if prescribed displacement boundary conditions are used, or relation (62) if periodicityconditions are employed) needs to be transformed to arrive at an expression relat<strong>in</strong>gvariations of the macroscopic stress and deformation tensorsδP M = 4 C P M : δFc M , (63)where the fourth order tensor 4 C P M represents the required consistent tangent stiffness atthe macroscopic <strong>in</strong>tegration po<strong>in</strong>t level.In order to obta<strong>in</strong> this constitutive tangent from the reduced stiffness matrix K M (or K ⋆ M ),first relations (59) and (62) are rewritten <strong>in</strong> a specific vector/tensor format∑K (ij)M ·δ⃗u (j) = δf ⃗ (i) , (64)jwhere <strong>in</strong>dices i and j take the values i, j = 1,N p for prescribed displacement boundaryconditions (N p is the number of boundary nodes) and i, j =1, 2, 4 for the periodicboundary conditions. In (64) the components of the tensors K (ij)Mare simply found <strong>in</strong>the tangent matrix K M (for displacement boundary conditions) or <strong>in</strong> the matrix K ⋆ M (forperiodic boundary conditions) at the rows and columns of the degrees of freedom <strong>in</strong> thenodes i and j. For example, for the case of the periodic boundary conditions the totalmatrix K ⋆ M has the format⎡ [] [] [] ⎤K (11)11 K (11)12 K (12)K (11)21 K (11)11 K (12)12 K (14)22 K (12)21 K (12)11 K (14)1222 K (14)21 K (14)22[] [] []K ⋆ M = K (21)11 K (21)12 K (22)K (21)21 K (21)11 K (22)12 K (24)22 K (22)21 K (22)11 K (24)1222 K (24)21 K (24), (65)22⎢[] [] []⎣ K (41)11 K (41)12 K (42)K (41)21 K (41)11 K (42)12 K (44)22 K (42)21 K (42)11 K (44) ⎥12 ⎦22 K (44)21 K (44)22where the superscripts <strong>in</strong> round brackets refer to the nodes and the subscripts to thedegrees of freedom at those nodes. Then each submatrix <strong>in</strong> (65) may be considered as therepresentation of a second-order tensor K (ij)M .23

Next, the expression for the variation of the nodal forces (64) is substituted <strong>in</strong>to therelation for the variation of the macroscopic stress follow<strong>in</strong>g from (50) or (57)δP M = 1 ∑ ∑(K (ij)MV ·δ⃗u (j)) X ⃗ (i) . (66)0ijSubstitution of the equation δ⃗u (j) = X ⃗ (j)·δF c M <strong>in</strong>to (66) givesδP M = 1 ∑ ∑( XV ⃗ (i) K (ij)MX ⃗ (j) ) LC : δF c M , (67)0ijwhere the superscript LC denotes left conjugation, which for a fourth-order tensor 4 T isdef<strong>in</strong>ed as Tijkl LC = T jikl. F<strong>in</strong>ally, by compar<strong>in</strong>g (67) with (63) the consistent constitutivetangent is identified as4 C P M = 1 ∑ ∑( XV ⃗ (i) K (ij)MX ⃗ (j) ) LC . (68)0ijIf the macroscopic f<strong>in</strong>ite element scheme requires the constitutive tangent relat<strong>in</strong>g thevariation of the macroscopic Cauchy stress to the variation of the macroscopic deformationgradient tensor accord<strong>in</strong>g toδσ M = 4 C σ M : δFc M , (69)this tangent may be obta<strong>in</strong>ed by vary<strong>in</strong>g the def<strong>in</strong>ition equation of the macroscopic Cauchystress tensor (35), followed by substitution of (50) (or (57)) and (67). This gives[ 1 ∑ ∑δσ M =(⃗x (i) K (ij)MXV⃗ (j) ) LC + 1 ∑]⃗f (i) I XV⃗ (i) − σ M F −cM: δF c M , (70)ijwhere the expression <strong>in</strong> square brackets is identified as the required tangent stiffness tensor4 C σ M . In the derivation of (70) it has been used that <strong>in</strong> case of prescribed displacements ofthe RVE boundary (15) or of periodic boundary conditions (17), the <strong>in</strong>itial and currentvolumes of an RVE are related accord<strong>in</strong>g to J M =det(F M )=V/V 0 .i5.4 Nested solution strategyBased on the above developments the actual implementation of the computational homogenizationstrategy may be described by the follow<strong>in</strong>g subsequent steps.The macroscopic structure to be analyzed is discretized by f<strong>in</strong>ite elements. The externalload is applied by an <strong>in</strong>cremental procedure. Increments can be associated with discretetime steps. The solution of the macroscopic non-l<strong>in</strong>ear system of equations is performed<strong>in</strong> a standard iterative manner. To each macroscopic <strong>in</strong>tegration po<strong>in</strong>t a discretized RVEis assigned. The geometry of the RVE is <strong>based</strong> on the microstructural morphology of thematerial under consideration.For each macroscopic <strong>in</strong>tegration po<strong>in</strong>t the local macroscopic deformation gradient tensorF M is computed from the iterative macroscopic nodal displacements (dur<strong>in</strong>g the <strong>in</strong>itializationstep, zero deformation is assumed throughout the macroscopic structure, i.e. F M = I,24

which allows to obta<strong>in</strong> the <strong>in</strong>itial macroscopic constitutive tangent). The macroscopic deformationgradient tensor is used to formulate the boundary conditions accord<strong>in</strong>g to (37)or (42)–(43) to be applied on the correspond<strong>in</strong>g representative cell.The solution of the RVE boundary value problem employ<strong>in</strong>g a f<strong>in</strong>e scale f<strong>in</strong>ite elementprocedure, provides the result<strong>in</strong>g stress and stra<strong>in</strong> distributions <strong>in</strong> the microstructural cell.Us<strong>in</strong>g the result<strong>in</strong>g forces at the prescribed nodes, the RVE averaged first Piola-Kirchhoffstress tensor P M is computed accord<strong>in</strong>g to (50) or (57) and returned to the macroscopic<strong>in</strong>tegration po<strong>in</strong>t as a local macroscopic stress. From the global RVE stiffness matrix thelocal macroscopic consistent tangent 4 C P M is obta<strong>in</strong>ed accord<strong>in</strong>g to (68).When the analysis of all microstructural RVEs is f<strong>in</strong>ished, the stress tensor is available atevery macroscopic <strong>in</strong>tegration po<strong>in</strong>t. Thus, the <strong>in</strong>ternal macroscopic forces can be calculated.If these forces are <strong>in</strong> balance with the external load, <strong>in</strong>cremental convergence hasbeen achieved and the next time <strong>in</strong>crement can be evaluated. If there is no convergence,the procedure is cont<strong>in</strong>ued to achieve an updated estimation of the macroscopic nodal displacements.The macroscopic stiffness matrix is assembled us<strong>in</strong>g the constitutive tangentsavailable at every macroscopic <strong>in</strong>tegration po<strong>in</strong>t from the RVE analysis. The solution ofthe macroscopic system of equations leads to an updated estimation of the macroscopicdisplacement field. The solution scheme is summarized <strong>in</strong> Table 1. It is remarked thatthe two-level scheme outl<strong>in</strong>ed above can be used selectively depend<strong>in</strong>g on the macroscopicdeformation, e.g. <strong>in</strong> the elastic doma<strong>in</strong> the macroscopic constitutive tangents do not haveto be updated at every macroscopic load<strong>in</strong>g step.6 Example: two-scale coupled analysis <strong>in</strong> bend<strong>in</strong>gAs an example, the computational homogenization approach is applied to pure bend<strong>in</strong>gof a rectangular strip under plane stra<strong>in</strong> conditions. Both the length and the heightof the sample equal 0.2 m, the thickness is taken 1 m. The macromesh is composed of5 quadrilateral 8 node plane stra<strong>in</strong> reduced <strong>in</strong>tegration elements. The undeformed anddeformed geometries of the macromesh are schematically depicted <strong>in</strong> figure 6. At theleft side the strip is fixed <strong>in</strong> axial (horizontal) direction, the displacement <strong>in</strong> transverse(vertical) direction is left free. At the right side the rotation of the cross section isprescribed. As pure bend<strong>in</strong>g is considered the behaviour of the strip is uniform <strong>in</strong> axialdirection and, therefore, a s<strong>in</strong>gle layer of elements on the macrolevel suffices to simulatethe situation.(a)(b)Figure 6: Schematic representation of the undeformed (a) and deformed (b) configurationsof the macroscopically bended specimen.25

Table 1: Incremental-iterative nested multi-scale solution scheme for the computationalhomogenization.MACRO1. Initialization⊲ <strong>in</strong>itialize the macroscopic model⊲ assign an RVE to every <strong>in</strong>tegrationpo<strong>in</strong>t⊲ loop over all <strong>in</strong>tegration po<strong>in</strong>tsset F M = Istore the tangent⊲ end <strong>in</strong>tegration po<strong>in</strong>t loop2. Next <strong>in</strong>crement⊲ apply <strong>in</strong>crement of the macro load3. Next iteration⊲ assemble the macroscopic tangent stiffness⊲ solve the macroscopic system⊲ loop over all <strong>in</strong>tegration po<strong>in</strong>tscalculate F Mstore P Mstore the tangent⊲ end <strong>in</strong>tegration po<strong>in</strong>t loop⊲ assemble the macroscopic <strong>in</strong>ternalforces4. Check for convergence⊲ if not converged ⇒ step 3⊲ else ⇒ step 2F M−−−−−−−−→tangent←−−−−−−−−−FM−−−−−−−−→P M←−−−−−−−−−tangent←−−−−−−−−−MICROInitialization RVE analysis⊲ prescribe boundary conditions⊲ assemble the RVE stiffness⊲ calculate the tangent 4 C P MRVE analysis⊲ prescribe boundary conditions⊲ assemble the RVE stiffness⊲ solve the RVE problem⊲ calculate P M⊲ calculate the tangent 4 C P MIn this example two heterogeneous microstructures consist<strong>in</strong>g of a homogeneous matrixmaterial with <strong>in</strong>itially 12% and 30% volume fractions of voids are studied. To generatea random distribution of cavities <strong>in</strong> the matrix with a prescribed volume fraction,maximum diameter of holes and m<strong>in</strong>imum distance between two neighbour<strong>in</strong>g holes, for atwo-dimensional RVE, the procedure from [95] and [96] has been adopted. The microstructuralcells used <strong>in</strong> the calculations are presented <strong>in</strong> figure 7. It is worth mention<strong>in</strong>g thatthe absolute size of the microstructure is irrelevant for the first-order computational homogenizationanalysis (see also discussion <strong>in</strong> section 8).The matrix material behaviour has been described by a modified elasto-visco-plasticBodner-Partom model [97]. This choice is motivated by the <strong>in</strong>tention to demonstrate thatthe method is well-suited for complex microstructural material behaviour, e.g. non-l<strong>in</strong>earhistory and stra<strong>in</strong> rate dependent at large stra<strong>in</strong>s. The material parameters for annealedalum<strong>in</strong>um AA 1050 determ<strong>in</strong>ed <strong>in</strong> [97] have been used; elastic parameters: shear modulusG =2.6 × 10 4 MPa, bulk modulus K =7.8 × 10 4 MPa and viscosity parameters:Γ 0 =10 8 s −2 , m =13.8, n =3.4, Z 0 =81.4 MPa,Z 1 = 170 MPa.Micro-macro calculations for the heterogeneous structure, represented by the RVEs shown26

(a)(b)Figure 7: Microstructural cells used <strong>in</strong> the calculations with 12% voids (a) and 30%voids (b).<strong>in</strong> figure 7 have been carried out, simulat<strong>in</strong>g pure bend<strong>in</strong>g at a prescribed moment rateequal to 5 × 10 5 Nms −1 . Figure 8 shows the distribution plots of the effective plasticstra<strong>in</strong> for the case of the RVE with 12% volume fraction voids at an applied momentequal to 6.8 × 10 5 N m <strong>in</strong> the deformed macrostructure and <strong>in</strong> three deformed, <strong>in</strong>itiallyidentical RVEs at different locations <strong>in</strong> the macrostructure. Each hole acts as a plasticstra<strong>in</strong> concentrator and causes higher stra<strong>in</strong>s <strong>in</strong> the RVE than those occurr<strong>in</strong>g <strong>in</strong> thehomogenized macrostructure. In the present calculations the maximum effective plasticstra<strong>in</strong> <strong>in</strong> the macrostructure is about 25%, whereas at RVE level this stra<strong>in</strong> reaches 50%.It is obvious from the deformed geometry of the holes <strong>in</strong> figure 8 that the RVE <strong>in</strong> theupper part of the bended strip is subjected to tension and the RVE <strong>in</strong> the lower partto compression, while the RVE <strong>in</strong> the vic<strong>in</strong>ity of the neutral axis is loaded considerablymilder than the other RVEs. This confirms the conclusion that the method realisticallydescribes the deformation modes of the microstructure.Figure 8: Distribution of the effective plastic stra<strong>in</strong> <strong>in</strong> the deformed macrostructure and<strong>in</strong> three deformed RVEs, correspond<strong>in</strong>g to different po<strong>in</strong>ts of the macrostructure.In figure 9 the moment-curvature (curvature def<strong>in</strong>ed for the bottom edge of the specimen)diagram result<strong>in</strong>g from the computational homogenization approach is presented. To givean impression of the <strong>in</strong>fluence of the holes also the response of a homogeneous configuration27

987Moment, N m654310 x 105 Curvature,1/mhomogeneous30% voids12% voids2100 0.2 0.4 0.6 0.8 1Figure 9: Moment-curvature diagram result<strong>in</strong>g from the first-order computational homogenizationanalysis.(without cavities) is shown. It can be concluded that even the presence of 12% voids<strong>in</strong>duces a reduction of the bend<strong>in</strong>g moment (at a certa<strong>in</strong> curvature) of more than 25% <strong>in</strong>the plastic regime. This significant reduction <strong>in</strong> the bend<strong>in</strong>g moment may be attributedto the formation of microstructural shear bands, which are clearly observed <strong>in</strong> figure 8.This <strong>in</strong>dicates that <strong>in</strong> order to capture such an effect a detailed microstructural analysisis required. A straightforward application of, for example, the rule of mixtures would leadto erroneous results.7 The RVE <strong>in</strong> first-order computational homogenization7.1 General concept of an RVEThe computational homogenization approach, as well as most of other homogenizationtechniques, are <strong>based</strong> on the concept of a representative volume element (RVE). An RVE isa model of a material microstructure to be used to obta<strong>in</strong> the response of the correspond<strong>in</strong>ghomogenized macroscopic cont<strong>in</strong>uum <strong>in</strong> a macroscopic material po<strong>in</strong>t. Thus, the properchoice of the RVE largely determ<strong>in</strong>es the accuracy of the modell<strong>in</strong>g of a heterogeneousmaterial.There appear to be two significantly different ways to def<strong>in</strong>e a representative volume element[29]. The first def<strong>in</strong>ition requires an RVE to be a statistically representative sampleof the microstructure, i.e. to <strong>in</strong>clude virtually a sampl<strong>in</strong>g of all possible microstructuralconfigurations that occur <strong>in</strong> the composite. Clearly, <strong>in</strong> the case of a non-regular and nonuniformmicrostructure such a def<strong>in</strong>ition leads to a considerably large RVE. Therefore,RVEs that rigorously satisfy this def<strong>in</strong>ition are rarely used <strong>in</strong> actual homogenization analyses.This concept is usually employed when a computer model of the microstructure isbe<strong>in</strong>g constructed <strong>based</strong> on experimentally obta<strong>in</strong>ed statistical <strong>in</strong>formation (see e.g. [98]).Another def<strong>in</strong>ition characterizes an RVE as the smallest microstructural volume that sufficientlyaccurately represents the overall macroscopic properties of <strong>in</strong>terest. This usually28

leads to much smaller RVE sizes than the statistical def<strong>in</strong>ition described above. However,<strong>in</strong> this case the m<strong>in</strong>imum required RVE size also depends on the type of materialbehaviour (e.g. for elastic behaviour usually much smaller RVEs suffice than for plasticbehaviour), macroscopic load<strong>in</strong>g path and difference of properties between heterogeneities.Moreover, the m<strong>in</strong>imum RVE size, that results <strong>in</strong> a good approximation of the overall materialproperties, does not always lead to adequate distributions of the microfields with<strong>in</strong>the RVE. This may be important if, for example, microstructural damage <strong>in</strong>itiation orevolv<strong>in</strong>g microstructures are of <strong>in</strong>terest.The latter def<strong>in</strong>ition of an RVE is closely related to the one established by Hill [86], whoargued that an RVE is well-def<strong>in</strong>ed if it reflects the material microstructure and if theresponses under uniform displacement and traction boundary conditions co<strong>in</strong>cide. If amicrostructural cell does not conta<strong>in</strong> sufficient microstructural <strong>in</strong>formation, its overallresponses under uniform displacement and traction boundary conditions will differ. Thehomogenized properties determ<strong>in</strong>ed <strong>in</strong> this way are called “apparent”, a notion <strong>in</strong>troducedby Huet [99]. The apparent properties obta<strong>in</strong>ed by application of uniform displacementboundary conditions on a microstructural cell usually overestimate the real effective properties,while the uniform traction boundary conditions lead to underestimation. As hasbeen verified by a number of authors [62, 90], for a given microstructural cell size, theperiodic boundary conditions provide a better estimation of the overall properties, thanthe uniform displacement and uniform traction boundary conditions. This conclusion alsoholds if the microstructure does not really possess geometrical periodicity [62]. Increas<strong>in</strong>gthe size of the microstructural cell leads to a better estimation of the overall properties,and, f<strong>in</strong>ally, to a “convergence” of the results obta<strong>in</strong>ed with the different boundary conditionsto the real effective properties of the composite material, as schematically illustrated<strong>in</strong> figure 10. The convergence of the apparent properties towards the effective ones at <strong>in</strong>creas<strong>in</strong>gsize of the microstructural cell has been <strong>in</strong>vestigated <strong>in</strong> [62, 99, 100, 101, 102, 103].displacement b.c.apparent propertyperiodic b.c.effective valuetraction b.c.microstructural cell size(a)(b)Figure 10: (a) Several microstructural cells of different sizes. (b) Convergence of theapparent properties to the effective values with <strong>in</strong>creas<strong>in</strong>g microstructural cell size fordifferent types of boundary conditions.7.2 Unit cells versus RVEsIn practice, <strong>in</strong>stead of a representative volume element, a unit cell is often used as amicrostructural model, s<strong>in</strong>ce it requires substantially less computational effort. This sectionexam<strong>in</strong>es the possible error, which is made <strong>in</strong> the obta<strong>in</strong>ed overall response of a29

multi-phase material, if the analysis is performed on a unit cell <strong>in</strong>stead of an RVE.As the simplest unit cell, a piece (for example a square or cube) of the matrix materialconta<strong>in</strong><strong>in</strong>g a s<strong>in</strong>gle heterogeneity (e.g. <strong>in</strong>clusion or void) could be suggested. The useof such a unit cell implicitly assumes a regular arrangement of the heterogeneities <strong>in</strong> thematrix, which contradicts the observations that almost all materials have a non-periodicor even spatially random microstructural composition. Examples are precipitates <strong>in</strong> metalalloys arranged randomly by their nature and artificial fiber re<strong>in</strong>forced composites, possess<strong>in</strong>ga non-regular distribution of the fibers due to the production process. At thesame time, several experimental evidences exist show<strong>in</strong>g that the spatial variability <strong>in</strong> themicrostructure significantly <strong>in</strong>fluences the overall behaviour and particularly the fracturecharacteristics of composites, as reported <strong>in</strong> [104, 105].Different authors, e.g. [46, 47, 57, 70], have performed a comparison of the overall compositeresponses result<strong>in</strong>g from the modell<strong>in</strong>g of regular and random structures. Theyhave found a significant response difference <strong>in</strong> the plastic regime, while there is almost nodeviation <strong>in</strong> elastic regime. Also it has been shown [106], that soften<strong>in</strong>g behaviour of aregularly composed structure may change to harden<strong>in</strong>g <strong>in</strong> the case of a random composition.Most of these considerations, except for the latter, have been performed for smalldeformations, very simple elasto-plastic behaviour and relatively stiff <strong>in</strong>clusions (fibers).In this section the overall behaviour of regular and random structures is compared at largedeformations, non-l<strong>in</strong>ear history dependent material behaviour, for voided material (anappropriate approximation for material with soft <strong>in</strong>clusions). Apart from the calculationson the microstructural cell (tensile configuration), also a full multi-scale analysis (purebend<strong>in</strong>g) of both regular and random structures is presented.A material with a 12% volume fraction of voids is considered. The regularly stackedstructure is modelled by a square unit cell conta<strong>in</strong><strong>in</strong>g a s<strong>in</strong>gle hole (figure 11a). For themodell<strong>in</strong>g of a random structure 10 different unit cells with non-regular arrangements ofvoids with a distribution of void sizes have been generated (figure 11b). The averagedbehaviour of these 10 unit cells is expected to be representative for the real randomstructure with a given volume fraction of heterogeneities. Us<strong>in</strong>g several small non-regularunit cells <strong>in</strong>stead of one larger RVE also allows to estimate the amount of deviation ofthe apparent properties obta<strong>in</strong>ed by the unit cell modell<strong>in</strong>g, from the effective values fordifferent types of material models and load<strong>in</strong>g histories.In the subsequent sections a comparison is performed for three different constitutive modelsof the matrix material: hyper-elastic, elasto-visco-plastic with harden<strong>in</strong>g and elastovisco-plasticwith <strong>in</strong>tr<strong>in</strong>sic soften<strong>in</strong>g. First uniaxial extension (under plane stra<strong>in</strong> conditions)of a macroscopic sample is considered. Because <strong>in</strong> this case the macroscopicdeformation field is homogeneous a full micro-macro modell<strong>in</strong>g is not necessary and ananalysis of an isolated unit cell with adequate boundary conditions (periodic) suffices.In the last section the results of a micro-macro simulation of bend<strong>in</strong>g us<strong>in</strong>g random andregular microstructures are compared.Elastic behaviour, tensionFirst, a comparison of the overall behaviour of regular and random structures is carriedout for the case of hyper-elastic behaviour of the matrix material, modelled as a compressibleNeo-Hookean material. The material parameters used <strong>in</strong> the calculations are30

(a)(b)Figure 11: Unit cell with one hole (a), represent<strong>in</strong>g a regular structure, and 10 randomlycomposed unit cells (b).K = 2667 MPa, G = 889 MPa.Figure 12 shows the stress-stra<strong>in</strong> curves for the unit cells with regular and random voidstack<strong>in</strong>g. For small deformations there is almost no difference <strong>in</strong> the responses orig<strong>in</strong>at<strong>in</strong>gfrom the regular and random void distributions. This result is <strong>in</strong> agreement with theexperiences reported <strong>in</strong> the literature for small deformations, see, e.g. [46, 47, 70]. Forlarge deformations the stiffer behaviour of the regular structure becomes a little bit morepronounced, however, the deviations rema<strong>in</strong> small. The difference between the responseof the regular structure and the response averaged over the random unit cells does notexceed 2%. This small deviation is expla<strong>in</strong>ed by figure 13, present<strong>in</strong>g the distribution ofthe equivalent von Mises stress <strong>in</strong> the regular unit cell and <strong>in</strong> a random unit cell for 20%macroscopic stra<strong>in</strong>. The stress field around any hole of the random structure is almostthe same as around the hole of the regular structure, which <strong>in</strong>dicates little <strong>in</strong>teractionbetween voids. If only the averaged elastic constants are of <strong>in</strong>terest, it is concluded thatcalculations performed on the simplest regular unit cell usually provide an answer with<strong>in</strong>an acceptable tolerance.Elasto-visco-plastic behaviour with harden<strong>in</strong>g, tensionThe <strong>in</strong>fluence of the randomness of the microstructure on the macroscopic response becomesmore significant when plastic yield<strong>in</strong>g of one or more constituents occurs. Thissection <strong>in</strong>vestigates the responses of the regular and random unit cells under tensile load-31

400350300Axial stress, MPa25020015010050unit cell with one holerandom void stackaveraged for random void stack00 5 10 15 20L<strong>in</strong>ear axial stra<strong>in</strong>, %Figure 12: Tensile stress-stra<strong>in</strong> responses (unit cell averages) of the regular and randomstructures <strong>in</strong> a voided hyper-elastic matrix material.(a)(b)0 200 400 600 800Figure 13: Distribution of the equivalent von Mises stress (MPa) <strong>in</strong> the deformed regular(a) and random (b) structures <strong>in</strong> a voided hyper-elastic matrix material.<strong>in</strong>g when the matrix material exhibits elasto-visco-plastic behaviour with harden<strong>in</strong>g. Theconstitutive description is given by the Bodner-Partom model [97]. The material parametersare the same as those used <strong>in</strong> section 6. The unit cells are subjected to uniaxialtension at a constant stra<strong>in</strong> rate of 0.5s −1 .In figure 14 the stress-stra<strong>in</strong> curves are presented. In this case the difference betweenthe overall response of the regular structure and the averaged response of the randomstructures reaches 10%. The rather large scatter<strong>in</strong>g <strong>in</strong> the responses of different randomcells is due to the small number of voids <strong>in</strong>cluded. As has been demonstrated <strong>in</strong> [96], thescatter<strong>in</strong>g is significantly reduced if microstructural cells conta<strong>in</strong> more heterogeneities.The averaged response is, however, hardly affected, provided that a sufficient number ofrandom realizations has been considered.The fundamental mechanism that governs the difference between the response of theregular structure and the averaged response of the random structures is illustrated <strong>in</strong>figure 15, where the distribution of the effective plastic stra<strong>in</strong> <strong>in</strong> the deformed regular andrandom unit cells at 15% applied macroscopic stra<strong>in</strong> is presented. In the regular unit cellthe ligaments yield simultaneously rather than sequentially with <strong>in</strong>creas<strong>in</strong>g macroscopicstra<strong>in</strong>, which is the case for the random unit cell. As a result, at the same value ofthe macroscopic stra<strong>in</strong> the regular unit cell is deformed relatively smoothly, while some32

100908070Axial stress, MPa605040302010unit cell with one holerandom void stackaveraged for random void stack00 5 10 15L<strong>in</strong>ear axial stra<strong>in</strong>, %Figure 14: Tensile stress-stra<strong>in</strong> responses (unit cell averages) of the regular and randomstructures for an elasto-visco-plastic matrix material with harden<strong>in</strong>g.ligaments <strong>in</strong> the random unit cell have already accumulated a significant amount of plasticstra<strong>in</strong>. Consequently, the regular unit cell (<strong>in</strong> fact a structure with a periodic stack<strong>in</strong>g ofheterogeneities) has a larger overall stiffness than a random configuration.(a)(b)0 20 40 60 80 100Figure 15: Distribution of the effective plastic stra<strong>in</strong> <strong>in</strong> the deformed regular (a) andrandom (b) structures for an elasto-visco-plastic matrix material with harden<strong>in</strong>g.Elasto-visco-plastic behaviour with soften<strong>in</strong>g, tensionThe difference <strong>in</strong> yield<strong>in</strong>g mechanisms for regular and random microstructures outl<strong>in</strong>ed<strong>in</strong> the previous section causes not only a quantitative deviation <strong>in</strong> the responses of thesestructures (as illustrated by figure 14), but <strong>in</strong> some cases also the qualitative characterchanges, as has been shown <strong>in</strong> [106]. For example, such a phenomenon can be observedwhen the matrix material is described by a generalized compressible Leonov model with<strong>in</strong>tr<strong>in</strong>sic soften<strong>in</strong>g and subsequent harden<strong>in</strong>g. The model is designed for the plastic deformationof polymers and <strong>in</strong>corporates a stress dependent Eyr<strong>in</strong>g viscosity extended bypressure dependence and <strong>in</strong>tr<strong>in</strong>sic soften<strong>in</strong>g effects. Details of this model can be found <strong>in</strong>[107, 108, 109].The result<strong>in</strong>g stress-stra<strong>in</strong> curves for uniaxial tension of polycarbonate at a constant stra<strong>in</strong>rate of 0.01 s −1 are given <strong>in</strong> figure 16. The overall behaviour of the regular structure <strong>in</strong>the plastic regime exhibits some <strong>in</strong>itial soften<strong>in</strong>g followed by harden<strong>in</strong>g. The response33

of the regular structure is, <strong>in</strong> fact, similar to the response of one s<strong>in</strong>gle ligament, thatsoftens accord<strong>in</strong>g to the <strong>in</strong>tr<strong>in</strong>sic material behaviour. A completely different response canbe observed for the random configurations. Although some of the random unit cells alsodemonstrate some soften<strong>in</strong>g behaviour, orig<strong>in</strong>at<strong>in</strong>g from the relatively simple compositionof the unit cells used <strong>in</strong> the calculations, the average response of the random unit cells doesnot show any soften<strong>in</strong>g but exhibits cont<strong>in</strong>uous harden<strong>in</strong>g. This is caused by the sequentialappearance of elastic, soften<strong>in</strong>g and harden<strong>in</strong>g zones with<strong>in</strong> the random microstructure.6050Axial stress, MPa40302010unit cell with one holerandom void stackaveraged for random void stack00 5 10 15L<strong>in</strong>ear axial stra<strong>in</strong>, %Figure 16: Tensile stress-stra<strong>in</strong> responses (RVE averages) of the regular and random structuresfor an elasto-visco-plastic matrix material with <strong>in</strong>tr<strong>in</strong>sic soften<strong>in</strong>g and subsequentharden<strong>in</strong>g.This example illustrates that the overall response of heterogeneous materials, when detersi3joda!m<strong>in</strong>edfrom a modell<strong>in</strong>g by a regular structure, should be <strong>in</strong>terpreted with greatcare, particularly <strong>in</strong> the case of complex material behaviour (e.g. <strong>in</strong> case of soften<strong>in</strong>g followedby harden<strong>in</strong>g or vice versa).Elasto-visco-plastic behaviour with harden<strong>in</strong>g, bend<strong>in</strong>gThe comparison of the overall behaviour of the regular and random microstructures performedabove has been <strong>based</strong> on the averaged behaviour of a s<strong>in</strong>gle unit cell subjected toa particular load<strong>in</strong>g history (uniaxial tension). The question rema<strong>in</strong>s how the randomnessof the microstructure does <strong>in</strong>fluence the overall behaviour when a macroscopic sample isdeformed heterogeneously, so that potentially every material po<strong>in</strong>t of the sample is subjectedto a different load<strong>in</strong>g history. In order to <strong>in</strong>vestigate this item the computationalhomogenization approach is a helpful tool.As an example the <strong>in</strong>fluence of the spatial composition of the microstructure on the overallmoment-curvature response of the voided material under pure bend<strong>in</strong>g is studied. Thebehaviour of the matrix material is described by the Bodner-Partom elasto-visco-plasticmodel with harden<strong>in</strong>g. The macrogeometry and the material parameters are the same asthese used <strong>in</strong> section 6.Figure 17 shows the moment-curvature diagram result<strong>in</strong>g from the full micro-macro analysisof pure bend<strong>in</strong>g of the material us<strong>in</strong>g the regular and the random microstructures.Aga<strong>in</strong>, the regular structure exhibits a stiffer response than the averaged random result,while the maximum deviation is only about 5%, which is considerably less than for the34

tensile test with the same material behaviour (figure 14). This smaller deviation orig<strong>in</strong>atesfrom the fact that <strong>in</strong> case of bend<strong>in</strong>g all the unit cells assigned to the various macroscopicpo<strong>in</strong>ts over the height of the bended strip are loaded differently, see figure 8. The unit cellat the top of the bended strip experiences tension, so that the observations dealt with <strong>in</strong>the previous examples apply. At the same time, there are also unit cells that are stretchedless or still are <strong>in</strong> elastic regime, like for example the one <strong>in</strong> the vic<strong>in</strong>ity of the neutrall<strong>in</strong>e, so that <strong>in</strong> average for the whole bend<strong>in</strong>g process the <strong>in</strong>fluence of randomness can beexpected to be smaller than for uniaxial extension.8 x 105 Curvature,1/m76Moment, N m54321unit cell with one holerandom void stackaveraged for random void stack00 0.2 0.4 0.6 0.8 1Figure 17: Moment-curvature responses of the regular and random structures for anelasto-visco-plastic matrix material with harden<strong>in</strong>g.8 Second-order computational homogenizationIn spite of the attractive characteristics listed above, there are a few important limitationsof the first-order framework, which can be summarized as follows• The pr<strong>in</strong>ciple of separation of scales must be respected. Hence, the characteristiclength that characterizes the spatial variations of the macroscopic load<strong>in</strong>g must bevery large with respect to the size of the microstructure. As a consequence, onlysimple first-order deformation modes (tension, compression, shear or comb<strong>in</strong>ationsthereof) of the microstructure can be retrieved. The case shown <strong>in</strong> figure 8, whichis a typical bend<strong>in</strong>g mode, which from a physical po<strong>in</strong>t of view should appear forsmall, but f<strong>in</strong>ite, microstructural cells, cannot be found.• The framework is completely <strong>in</strong>sensitive to the absolute size of the microstructuralconstituents (scale <strong>in</strong>dependent). Size effects emanat<strong>in</strong>g from the absolute size atthe micro scale cannot be dealt with properly.• Macroscopic gradients must rema<strong>in</strong> very small with respect to the micro scale. Localizationproblems, where non-uniform macroscopic deformations arise, cannot besolved properly.Whenever strong gradients appear at the macro-level (localization, size effects) care mustbe taken <strong>in</strong> us<strong>in</strong>g a first-order scheme. In all other cases, one should cont<strong>in</strong>ue us<strong>in</strong>g it35

Figure 18: Second-order computational homogenizationand not jump to a second-order scheme for which an additional price <strong>in</strong> complexity andcomputational costs is to be paid.In order to overcome these shortcom<strong>in</strong>gs, the computational homogenization methodologyhas been extended recently to higher-order cont<strong>in</strong>ua [2, 3, 4, 5, 6, 72, 110]. In this course,the methodology and the essential parts of the multi-scale k<strong>in</strong>ematics and statics will beoutl<strong>in</strong>ed briefly, whereas more details can be found <strong>in</strong> the cited references. The method isnext applied to heterogeneous multi-phase microstructures, as typically the case <strong>in</strong> mostmetals, polymer blends and composites. Some comments on the parallel implementationof the multi-scale technique are given and an illustrative example is used to scrut<strong>in</strong>ize theadded value of the second-order framework <strong>in</strong> relation to the more standard first-orderscheme.8.1 Pr<strong>in</strong>ciplesThe second-order case, which may be considered as a generalization of the classical firstorderscheme, departs from a Taylor series expansion of the classical nonl<strong>in</strong>ear deformationmap, ⃗x = φ( X), ⃗ applied to a f<strong>in</strong>ite material vector Δ⃗x <strong>in</strong> the deformed state:Δ⃗x = F M·ΔX ⃗ + 1Δ X·3G ⃗2 M·ΔX ⃗ + O(Δ X ⃗ )3 (71)Us<strong>in</strong>g this Taylor series expansion, the macroscopic (coarse scale) k<strong>in</strong>ematics is determ<strong>in</strong>edthrough the deformation gradient tensor F M and its Lagrangian gradient 3 G M = ∇ ⃗ 0,M F M .The key po<strong>in</strong>t <strong>in</strong> the second-order two-scale framework, resides <strong>in</strong> apply<strong>in</strong>g relation (71) toa representative part of the microstructure, such that a classical boundary value problemis obta<strong>in</strong>ed at the micro scale (or f<strong>in</strong>e scale). The scale bridg<strong>in</strong>g is then realized throughthe application of averag<strong>in</strong>g theorems. This is schematically depicted <strong>in</strong> figure 18. Notethat the tensor 3 G M has a m<strong>in</strong>or symmetry, 3 G M = 3 G T M (or G Mijk = G Mkji <strong>in</strong> <strong>in</strong>dexnotation), which is used throughout these lecture notes.36

8.2 Two-scale higher-order k<strong>in</strong>ematicsIn order to apply equation (71) to the f<strong>in</strong>e scale, all higher-order terms (represented byO(Δ ⃗ X 3 )) are condensed <strong>in</strong>to an unknown microfluctuation field ⃗w, which represents thef<strong>in</strong>e scale contribution <strong>in</strong> the k<strong>in</strong>ematics. Hence,Δ⃗x = F M·Δ ⃗ X + 1 2 Δ ⃗ X·3G M·Δ ⃗ X + ⃗w (72)Apply<strong>in</strong>g this to an undeformed volume V 0 (the RVE) with a geometrical center ⃗ X c thatis located <strong>in</strong> ⃗x c after deformation gives (notice the similarity and differences with theelaboration <strong>in</strong> the previous section, equation (5)).⃗x − ⃗x c = F M·( ⃗ X − ⃗ X c )+ 1 2 ( ⃗ X − ⃗ X c )·3G M·( ⃗ X − ⃗ X c )+⃗w (73)Elim<strong>in</strong>at<strong>in</strong>g rigid body displacements like for the first-order case (e.g. by fix<strong>in</strong>g a boundarypo<strong>in</strong>t1)thenleadstowith⃗x = ⃗c + F M·( ⃗ X − ⃗ X c )+ 1 2 ( ⃗ X − ⃗ X c )·3G M·( ⃗ X − ⃗ X c )+(⃗w − ⃗w 1 ) (74)⃗c = X ⃗ 1 + F M·( X ⃗ 1 − X ⃗ c )+ 1( X ⃗ 2 1 − X ⃗ c )·3G M·( X ⃗ 1 − X ⃗ c ) (75)⃗x c = ⃗c − ⃗w 1 (76)The microscopic deformation gradient tensor F m is easily reconstructed asF m=( ⃗ ∇ 0,m ⃗x) T= F M +( ⃗ X − ⃗ X c )·3GM +( ⃗ ∇ 0,m (⃗w − ⃗w 1 )) T (77)Apply<strong>in</strong>g the earlier <strong>in</strong>troduced scale transition relation (13) with respect to equation (77)then leads to two k<strong>in</strong>ematical constra<strong>in</strong>ts (to be imposed on the RVE)∫1( XV ⃗ − X ⃗ c )dV 0 = ⃗0 (78)0V∫01( ∇V ⃗ 0,m (⃗w − ⃗w 1 )) T dV 0 = 1 ∫(⃗w − ⃗w 1 ) NdΓ0 V ⃗ 0 = 0 (79)0V 0 Γ 0where the divergence theorem was used to derive the latter relation. Equation (78) isclearly satisfied here, s<strong>in</strong>ce the Taylor series has been expanded with respect to the geometricalcentre ⃗ X c <strong>in</strong> equation (73). This appears to be a necessary condition <strong>in</strong> thesecond-order case, which deviates from the first-order scheme where any po<strong>in</strong>t to develop(4) around (<strong>in</strong>stead of ⃗ X c ) gives the same result. The second constra<strong>in</strong>t (79) appliesto the unknown fluctuation field. Logically, the <strong>in</strong>tegral <strong>in</strong>volves ( ⃗w − ⃗w 1 ), which impliesa constra<strong>in</strong>t on the boundary position vectors ⃗x through (74). There are various ways tomake this boundary <strong>in</strong>tegral zero, e.g. by constra<strong>in</strong><strong>in</strong>g (⃗w − ⃗w 1 )=⃗0 for all po<strong>in</strong>ts of theRVE (Taylor/Voigt), or by constra<strong>in</strong><strong>in</strong>g (⃗w − ⃗w 1 )=⃗0 at the boundary of the RVE only(displacement or k<strong>in</strong>ematic boundary condition), or through the application of periodicboundary conditions on the microfluctuation field (the macroscopic field is generally notperiodic <strong>in</strong> the second-order case!). The latter conditions are used here, lead<strong>in</strong>g to the37

Figure 19: Lagrangian undeformed 2D reference RVEfollow<strong>in</strong>g microperiodicity equations valid between the left(L)-right(R) and bottom(B)-top(T) boundaries of a two-dimensional rectangular RVE as shown <strong>in</strong> figure 19.⃗w L = ⃗w R ⃗w B = ⃗w T (80)Note that aga<strong>in</strong> all equations <strong>in</strong>volve the microfluctuation field with respect to ⃗w 1 . Anychoice for ⃗w 1 will then lead to the same solution (except for ⃗x c ). This is also obviousfrom the constra<strong>in</strong>t relation (79), which can be easily elaborated to a format <strong>in</strong> which thecontribution of ⃗w 1 vanishes, i.e.∫∫(⃗w − ⃗w 1 ) NdΓ ⃗ 0 = ⃗w NdΓ ⃗ 0 = 0 (81)Γ 0 Γ 0It is easy to show that the micro scale problem def<strong>in</strong>ed by the equations (72), (78), (80),applied to the rectangular 2D RVE depicted <strong>in</strong> figure 19 with periodic microfluctuations,fully determ<strong>in</strong>es the k<strong>in</strong>ematics of the four corner po<strong>in</strong>ts [2, 3]. This set of equationsimposes 8 macroscopic degrees-of-freedom to the 2D microstructure, whereas the fullmacroscopic k<strong>in</strong>ematics consists of 12 degrees-of-freedom (2 rigid body displacement, 4degrees-of-freedom <strong>in</strong> F M and 6 degrees-of-freedom <strong>in</strong> the m<strong>in</strong>or-symmetric 3 G M ). Themiss<strong>in</strong>g k<strong>in</strong>ematical quantities appear to be the stretch gradients [2, 3], i.e. so far an RVEwith 8 macroscopic degrees-of-freedom has been established, where the displacements areprescribed through the four corner nodes. This is a typical example of couple stresshomogenization.In order to <strong>in</strong>corporate the entire gradient field, the set of averag<strong>in</strong>g relations needs to becompleted <strong>in</strong> order to account for the miss<strong>in</strong>g stretch gradient degrees-of-freedom of 3 G M .On the basis of the Taylor series expansion (72), it is easy to show that the follow<strong>in</strong>g38

averag<strong>in</strong>g theorem can be derived (by means of some manipulations of the equationsgiven <strong>in</strong> [3]), relat<strong>in</strong>g 3 G M to the position vectors ⃗x (implicitly <strong>in</strong>corporat<strong>in</strong>g the f<strong>in</strong>e scalecontribution through (74)) of all material po<strong>in</strong>ts <strong>in</strong> the the square RVE with <strong>in</strong>itial sizeW and volume V 0 :W 212[3 G M + 1 2( ) ]II : 3 G RT RTM +(I⃗c) RT = 1 ∫2V 0(⃗∇0,m (⃗x X)+[ ⃗ ∇ ⃗ 0,m (⃗x ⃗ )X)] T dV 0 (82)V 0In here, I is the second-order unit tensor, and RT stands for the right transpose, i.e.TijkRT = T ikj <strong>in</strong> <strong>in</strong>dex notation. The third term <strong>in</strong> the left-hand side of this equation ispresent to account for the deformed position ⃗x c of the center of the undeformed RVE,which is generally no longer the center of the deformed RVE. Comput<strong>in</strong>g the <strong>in</strong>tegral <strong>in</strong>the right-hand side of the latter equation through substitution of equation (74), reveals∫1(⃗∇0,m (⃗x X)+[2V ⃗ ∇ ⃗ 0,m (⃗x ⃗ )X)] T dV 0 =0V 0W 23 G M + W 212 24( )II : 3 G RT RTM +[I⃗c] RT + 1 ∫2V 0Γ 0(⃗X(⃗w − ⃗w1 ) N ⃗ + N(⃗w ⃗ − ⃗w 1 ) X ⃗ )dΓ 0(83)Enforc<strong>in</strong>g the averag<strong>in</strong>g relation (82) requires that the last <strong>in</strong>tegral <strong>in</strong> (83) should vanish,which leads to a new constra<strong>in</strong>t on the microfluctuation field.∫ (⃗X(⃗w − ⃗w1 ) N ⃗ + N(⃗w ⃗ − ⃗w 1 ) X ⃗ )dΓ 0 = 3 0 (84)Γ 0This boundary <strong>in</strong>tegral clearly <strong>in</strong>corporates (⃗w− ⃗w 1 ), which constra<strong>in</strong>s the position vectors⃗x of the boundary po<strong>in</strong>ts through (74). The microfluctuation ⃗w 1 (<strong>in</strong> the fixed boundarypo<strong>in</strong>t) cannot be elim<strong>in</strong>ated <strong>in</strong> general as done <strong>in</strong> the previously <strong>in</strong>troduced boundary<strong>in</strong>tegral (81). If constra<strong>in</strong>t (84) is enforced, it is easy to rewrite the averag<strong>in</strong>g equation (82)as a boundary <strong>in</strong>tegral( ) 3 G M + 1 2 II : 3 G RT RT 12M +W 2 [I⃗c]RT = 6 ∫ ( )⃗X⃗x N ⃗ + N⃗x ⃗ X ⃗ dΓV 0 W 2 0 (85)Γ 0Equation (85) typically illustrates that 3 G M is imposed on the RVE boundary, which isnecessary to construct a classical boundary value problem at the micro scale.For an <strong>in</strong>itially square RVE (H = W <strong>in</strong> figure 19), on which the microperiodicity equations(80) for the microfluctuation field hold, the constra<strong>in</strong>t (84) can be simplified to∫(⃗w L − ⃗w 1 )dΓ 0 = ⃗0Γ∫0L(86)(⃗w B − ⃗w 1 )dΓ 0 = ⃗0Γ 0BClearly, these two conditions enforce the shape of a part of the boundary to be equal<strong>in</strong> average to the shape ensu<strong>in</strong>g from the macroscopic field. Prescrib<strong>in</strong>g (86) at the39

oundaries can be done by generalized displacement constra<strong>in</strong>ts (non-homogeneous ty<strong>in</strong>grelations), see [3, 72] for more details on this topic. Aga<strong>in</strong>, it is obvious that impos<strong>in</strong>g⃗w 1 = ⃗0 does not <strong>in</strong>fluence the solution for the two-scale homogenization.Note that the macroscopic 3 G M is not the volume average of the microscopic 3 G m =⃗∇ 0,m F m . This not possible if one wants to construct a classical boundary value problemat the micro scale. The scale transition is here driven by boundary <strong>in</strong>tegrals <strong>in</strong>volv<strong>in</strong>gdisplacements of boundary po<strong>in</strong>ts of the RVE only. Enforc<strong>in</strong>g 3 G M to be the volumeaverage of 3 G m would lead to higher-order boundary conditions on the microstructuralfluctuation field, which would make the f<strong>in</strong>e scale problem second-order as well.8.3 Extract<strong>in</strong>g stress tensorsThe macroscopic stress quantities are next extracted from the analysis of the deformedRVE by equat<strong>in</strong>g the macroscopic work per unit of volume to the average work performedon the RVE (Hill-Mandel or macrohomogeneity condition). For the second-order case,this condition reads∫1P m : δF T mV dV 0 = P M : δF T M + 3 Q M . δ 3 G M (87)0V 0In here, P M is the macroscopic first Piola-Kirchhoff stress tensor, P m its microstructuralcounterpart and 3 Q M the higher-order stress tensor which is work-conjugated to 3 G M .Note that equation (87) <strong>in</strong> fact def<strong>in</strong>es the two macroscopic stress tensors P M and 3 Q M .The microstructural work (per unit of volume <strong>in</strong> the reference state) can be written asδW 0M = 1 ∫P m : δF c mV dV 0 = 1 ∫⃗p·δΔ⃗x dΓ 0 , (88)0 V 0Γ 0V 0where use has been made of the divergence theorem and the static equilibrium equation<strong>in</strong> the microstructure (2). Tak<strong>in</strong>g the variation of the position vector δΔ⃗x accord<strong>in</strong>g to(72) leads toδΔ⃗x = δF M· ⃗X + 1 2 ⃗ X·δ 3 G M· ⃗X + δΔ⃗w, (89)which after substitution <strong>in</strong> equation (88) yieldsδW 0M = 1 ∫⃗p XV ⃗ dΓ 0 : δF c M + 1 ∫X⃗p ⃗ X ⃗ dΓ0 . δ 3 G M + 1 ∫0 2V 0 V 0Γ 0 Γ 0Γ 0⃗p·δΔ⃗w dΓ 0 . (90)S<strong>in</strong>ce the boundary constra<strong>in</strong>ts (80) do not contribute to the total work and account<strong>in</strong>gfor (86), the last term <strong>in</strong> (90) can be proven to disappear∫Γ 0⃗p·δΔ⃗w dΓ 0 =0, (91)manifest<strong>in</strong>g the fact that the microstructural fluctuation field does not affect the averagevariation of the microscopic work.40

Elaboration of this equation leads to two boundary <strong>in</strong>tegrals that permit to compute thestress tensors P M and 3 Q M :P M = 1 ∫⃗p XdΓV 0 (92)0 3 Q M =Γ 0 ∫X⃗p ⃗ X ⃗ dΓ0 (93)2V 0Γ 0Both stress tensors can be easily computed once the boundary value problem on the microscale has been solved.The above formulas relate the macroscopic stress tensor and the macroscopic higher-orderstress tensor to microstructural variables def<strong>in</strong>ed on the RVE boundary. The relations (92)and (93) can also be transformed <strong>in</strong>to volume <strong>in</strong>tegrals, allow<strong>in</strong>g the macroscopic stressmeasures to be expressed <strong>in</strong> terms of volume averages of microstructural quantities. Themacroscopic stress tensor P M aga<strong>in</strong> equals the volume average of the microscopic stresstensor P mP M = 1 ∫P m dV 0 (94)V 0V 0The proof of this equation is identical to that for the first-order framework (for the derivationsee (32)-(33)).The derivation for the higher-order stress tensor 3 Q M followsthesameprocedure. Apply<strong>in</strong>gthe divergence theorem to transform the boundary <strong>in</strong>tegral <strong>in</strong> (93) to a volume<strong>in</strong>tegral gives3 Q M = 1 ∫∫2V 0 2V 0= 12V 0∫Γ 0⃗ X⃗p ⃗ XdΓ0 = 1V 0(∇ 0m·(P c m ⃗ X ⃗ X)) LC dV 0 ,Γ 0(( ⃗ N·P c m) ⃗ X ⃗ X) LC dΓ 0(95)where the superscript LC denotes left conjugation, T LCijk= T jik. F<strong>in</strong>ally us<strong>in</strong>g the equality∇ 0m·(P c m ⃗ X ⃗ X)=(∇ 0m·P c m ) ⃗ X ⃗ X + P m·(∇ 0m⃗ X) ⃗ X +( ⃗ XPm·(∇ 0m⃗ X))LC= P m⃗ X +( ⃗ XPm ) LC ,(96)where equilibrium has been exploited, the relation between the macroscopic higher-orderstress tensor and microstructural quantities is obta<strong>in</strong>ed3 Q M = 1 ∫2V 0V 0(P T ⃗ mX + XP ⃗ )m dV 0 (97)Note that the macroscopic higher-order stress tensor 3 Q M does not equal the volumeaverage of its microscopic counterpart ∇ ⃗ 0,m P m .Likefor 3 G M this is due to the fact thatthe micro scale problem is formulated as a classical boundary value problem. It is clearfrom (97) that 3 Q M can be <strong>in</strong>terpreted as the first moment (with respect to the RVE center)of the microscopic first Piola-Kirchhoff stress tensor P m over the <strong>in</strong>itial RVE volume V 0 .41

8.4 Two-scale computational solution strategyThe boundary constra<strong>in</strong>ts (86) can be explicitly written <strong>in</strong> terms of the displacementvectors of the boundary po<strong>in</strong>ts <strong>in</strong> the form∫∫⃗u L dΓ 0 = ⃗u L ∗(F M , 3 G M ), ⃗u B dΓ 0 = ⃗u B ∗(F M , 3 G M ) (98)Γ 0L Γ 0Bwhere ⃗u L ∗ and ⃗u B ∗ solely depend on the given F M and 3 G M and RVE geometry as apparentfrom their def<strong>in</strong>itions∫∫⃗u L ∗ =(F M − I)· ( X ⃗ L − X ⃗ 1 )dΓ 0 + 1 3 G LC2 M : ( X ⃗ LXL ⃗ − X ⃗ 1X1 ⃗ )dΓ 0Γ 0L Γ∫0L∫⃗u B ∗ =(F M − I)· ( X ⃗ B − X ⃗ 1 )dΓ 0 + 1 3 G LC2 M : ( X ⃗ BXB ⃗ − X ⃗ 1X1 ⃗ )dΓ 0 (99)Γ 0B Γ 0BOnce the BVP associated to the microstructural RVE problem is def<strong>in</strong>ed (boundary conditions,constitutive equations) the micro-problem can be solved with a standard f<strong>in</strong>iteelement method. On the basis of the result<strong>in</strong>g boundary tractions, the RVE averagedstress tensors are extracted (see equations (94), (97))) and transported to the correspond<strong>in</strong>gmacroscopic material po<strong>in</strong>t.For the f<strong>in</strong>ite element solution of the macroscopic problem a stiffness matrix at everymacroscopic <strong>in</strong>tegration po<strong>in</strong>t is required. As emphasized earlier, <strong>in</strong> computational homogenizationschemes there is no explicit form of the macroscopic constitutive behaviourassumed a priori. Like for the first-order case, the tangent operator is determ<strong>in</strong>ed numericallyby condensation of the microscopic stiffness matrix. For this, first the elaboratedconstra<strong>in</strong>t relations between boundary nodes (equations (80), (86)) are applied to thetotal assembled stiffness matrix of the RVE follow<strong>in</strong>g a similar procedure as presented forthe first-order case. Details for the second-order case are given <strong>in</strong> [5, 6, 72]. This results<strong>in</strong> the elim<strong>in</strong>ation of the dependent degrees of freedom from the system of equations. Thenext step is to partition the rema<strong>in</strong><strong>in</strong>g system of equations as[ ][ ] [ ]Kpp K pf δũp= δf˜p(100)K fp K ff δũ f 0˜where the subscript p refers to “prescribed” degrees of freedom (degrees of freedom throughwhich the macroscopic tensors F M and 3 G M are imposed on the RVE). In the presentframework these are the degrees of freedom correspond<strong>in</strong>g to the four corner nodes of theRVE (⃗u i ,i = 1, 4) and to the degrees of freedom enter<strong>in</strong>g the RVE system of equationsthrough the boundary constra<strong>in</strong>ts (98). The subscript f <strong>in</strong> (100) refers to all rema<strong>in</strong><strong>in</strong>g“free” nodes. Elim<strong>in</strong>ation of δũ f from the system (100) then leads to the reduced stiffnessmatrix K M that relates the variations of the prescribed degrees of freedom to the variationsof the associated forcesK M δũ p = δf˜p, with K M = K pp − K pf (K ff ) −1 K fp (101)The l<strong>in</strong>earized constitutive relations for the second gradient cont<strong>in</strong>uum can be written asδP M = 4 C (1)M : δFc M + 5 C (2)Mδ 3 Q M = 5 C (3)M : δFc M + 6 C (4)M. δ 3 G RCM (102). δ 3 G RCM (103)42

where the fourth-order tensor 4 C (1)M , the fifth-order tensors 5 C (2)M and 5 C (3)Mand the sixthordertensor 6 C (4)Mare the macroscopic consistent tangents. Us<strong>in</strong>g the RVE reduced stiffnessmatrix K M rewritten <strong>in</strong> a tensor format such that∑jK (ij)M ·δ⃗u (j) = δ ⃗ f (i) , i,j =1, 2, 3, 4,L ∗ ,B ∗ (104)permits to extract the macroscopic consistent tangents <strong>in</strong> the follow<strong>in</strong>g format (see [6] forthe derivation)4 C (1)M= 1 ∑ ∑( XV ⃗ (i)K ⋆ (ij)MX ⃗ (j)) ⋆ LC ,05 C (3)M = 1 ∑ ∑2V 0iijj(Y ⋆ (i)K (ij)M ⃗ X ⋆ (j)) C 23,5 C (2)M = 12V 0∑6 C (4)M = 14V 0∑ii∑j∑j( ⃗ X ⋆ (i)K (ij)M(Y ⋆ (i)K (ij)Y⋆ (j)) LCM Y⋆ (j)) C 23(105)with the superscript C 23 <strong>in</strong>dicat<strong>in</strong>g conjugation on the second and third <strong>in</strong>dices and⎧⃗X (i) − X ⃗ (1) , for i =1, 2, 3, 4,⎪⎨ ∫⃗X (i) ⋆ ( X=⃗ L − X ⃗ (1) )dΓ 0 , for i = L ∗ ,Γ 0L(106)∫⎪⎩ ( X ⃗ B − X ⃗ (1) )dΓ 0 , for i = B ∗Γ 0B⎧⃗X (i) X(i) ⃗ − X ⃗ (1) X(1) ⃗ , for i =1, 2, 3, 4,⎪⎨ ∫Y(i) ⋆ ( X=⃗ LXL ⃗ − X ⃗ (1) X(1) ⃗ )dΓ 0 , for i = L ∗ ,Γ 0L(107)∫⎪⎩ ( X ⃗ BXB ⃗ − X ⃗ (1) X(1) ⃗ )dΓ 0 , for i = B ∗Γ 0BIn the second-order computational homogenization framework the macroscopic problemrepresents a full second gradient cont<strong>in</strong>uum [111, 112, 113]. For such a second gradientcont<strong>in</strong>uum the local equilibrium equation (<strong>in</strong> the absence of body forces and bodymoments) is written as∇ (P0M· c M − (∇ 0M·3QM ) c) = ⃗0 (108)The natural boundary conditions associated with this system of partial differential equationsare expressed <strong>in</strong> (i) the surface traction ⃗t M⃗t M = ⃗ N M· (P c M − (∇ 0M·3QM ) c) +(∇ s 0M· ⃗N M ) ⃗ N M·( ⃗ N M·3QM ) c −∇ s 0M·( ⃗ N M·3QM ) c (109)where the surface gradient operator is def<strong>in</strong>ed as ∇ s 0M =(I− ⃗ N M⃗ NM )·∇ 0M with ⃗ N M the unitoutward normal on the surface of the macroscopic body <strong>in</strong> the undeformed configurationand (ii) the double stress traction ⃗r M⃗r M = ⃗ N M·3Q M· ⃗N M (110)In the case of a non-smooth surface of the body (with edges) also an additional l<strong>in</strong>eload appears. The k<strong>in</strong>ematic boundary conditions for the second gradient cont<strong>in</strong>uum43

<strong>in</strong>clude prescribed displacements ⃗u M and normal gradients of displacements D 0M ⃗u M withD 0M = N ⃗ M·∇ 0M .The constitutive equations relat<strong>in</strong>g the macroscopic first Piola-Kirchhoff stress tensor P Mand the higher-order stress tensor 3 Q M to the history of the macroscopic deformationtensor F M and its gradient 3 G M are thus obta<strong>in</strong>ed numerically, whereas their variationsare obta<strong>in</strong>ed <strong>in</strong> the l<strong>in</strong>earized form (102)–(103) with the macroscopic consistent tangentscalculated from the condensed microscopic stiffness matrix accord<strong>in</strong>g to (105).8.5 Parallel solution of the multi-scale nested boundary valueproblemsIn spite of the large computational effort required by a computational homogenizationscheme, it is well possible to make an efficient analysis if optimal use is made of the <strong>in</strong>herentparallel nature of this multi-scale framework. Whenever microstructural constitutive<strong>in</strong>formation is needed <strong>in</strong> a macroscopic (<strong>in</strong>tegration) po<strong>in</strong>t, a separate subrout<strong>in</strong>e can bestarted on the RVE-level that solves the requested boundary value problem. This can bedone <strong>in</strong> parallel <strong>in</strong> as many <strong>in</strong>tegration po<strong>in</strong>ts as available processors. Us<strong>in</strong>g PVM (ParallelVirtual Mach<strong>in</strong>e) or MPI (Message Pass<strong>in</strong>g Interface), it is relatively easy to constructsuch a parallel implementation for this type of problems, as schematically depicted <strong>in</strong>figure 20. Evidently, this procedure drastically reduces the total calculation time.Figure 20: Schematic overview of the parallel solution of the multi-scale nested BVPs9 Higher-order issues9.1 First-order versus second-orderThe first example concerns the comparison of the mechanical and k<strong>in</strong>ematical responseof a heterogeneous microstructure for the first-order and the second-order scale transition.To this purpose, an RVE is considered, which is depicted <strong>in</strong> its undeformed state44

<strong>in</strong> figure 21. The material considered is a metal with very weak <strong>in</strong>clusions, which have aFigure 21: Undeformed two-dimensional RVE of a voided metalnegligible mechanical contribution (e.g. voids). The matrix material is elasto-viscoplastic,constitutively prescribed by a Bodner-Partom viscosity function. Follow<strong>in</strong>g the conventionalmultiplicative split of F, the elastic response is modelled by a classical (isotropic)Neo-Hookean relationship, where the Kirchhoff stress tensor is given byτ = K(J − 1)I + G¯b d e (111)In here, J is the volume change ratio, K is the bulk modulus, G is the shear modulus, ¯b d eis the deviatoric part of the isochoric elastic left Cauchy-Green deformation tensor. Theplastic part is determ<strong>in</strong>ed through the plastic deformation rate tensor D pD p= σd2η(112)where the viscosity η is related to the von Mises equivalent stress σ e and the effectiveplastic stra<strong>in</strong> ɛ p by( [ ] ) 2nσ e 1 Zη = √ exp 12Γ0 2 σ e (113)Z = Z 1 +(Z 0 − Z 1 )e −mɛpwith Γ 0 , n, Z 0 , Z 1 and m material constants. In the present analysis, an alum<strong>in</strong>ummatrix (AA 1050) has been considered for which the material parameters are given byG =2.6·10 4 MPa, K =7.8·10 4 MPa, Γ 0 =10 8 s −2 , m =13.8, n =3.4, Z 0 =81.4 MPa,Z 1 = 170 MPa.The comparison between the first and second-order formulation is next made for a microstructurewith a second phase (12% volume fraction of voids <strong>in</strong> this case) with an averagesize of about 6.6 μm. The macroscopic deformation history of a specific material po<strong>in</strong>trepresent<strong>in</strong>g bend<strong>in</strong>g with superimposed tension is extracted. This history is imposed tothe RVE, after which the micro scale BVP can be solved. The deformed microstructuresshown <strong>in</strong> figure 22 are then obta<strong>in</strong>ed for the considered po<strong>in</strong>t (with the same macroscopicdeformation history!). The deformation modes obta<strong>in</strong>ed and the small scale stra<strong>in</strong> fieldsare obviously different, which reflects the k<strong>in</strong>ematical enrichment of the second-order approximation.Note that the RVE is clearly bend<strong>in</strong>g <strong>in</strong> the second-order case, which is theresult of the presence of the higher-order deformation modes that properly account forthe size of the microstructure. The periodicity of the microfluctuation field can also benoticed. The macro field however, is no longer periodic for the second-order case.45