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

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<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
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 and 14: 3.4 Micro-scale RVE boundary condit
Page 15 and 16: where the relation (with account fo
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: where the fourth-order tensor 4 C (

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