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

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