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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
Page 1 and 2: Scale tran
Page 3 and 4: 1 Introduction1.1 Mechanics across
Page 5 and 6: assumptions on the fin</str
Page 7 and 8: Figure 1: Macroscopic conti
Page 9 and 10: enization technique defin</
Page 11 and 12: local equilibrium and compatibility
Page 13: 3.4 Micro-scale RVE boundary condit
Page 17 and 18: The volume average of the microscop
Page 19 and 20: whereby⃗u p =(F M − I)· ⃗X p
Page 21 and 22: Each constraint re
Page 23 and 24: Next, system (60) is further split,
Page 25 and 26: which allows to obtain</str
Page 27 and 28: (a)(b)Figure 7: Microstructural cel
Page 29 and 30: leads to much smaller RVE sizes tha
Page 31 and 32: (a)(b)Figure 11: Unit cell with one
Page 33 and 34: 100908070Axial stress, MPa605040302
Page 35 and 36: tensile test with the same material
Page 37 and 38: 8.2 Two-scale higher-order k<strong
Page 39 and 40: averaging theorem
Page 41 and 42: Elaboration of this equation leads
Page 43 and 44: where the fourth-order tensor 4 C (
Page 45: in figure 21. The

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