2006 Fiscal Year Final Research Report Summary
Formulation of homogenization theory for multiscale inelastic analysis based on implicit method
| Project/Area Number |
17360049
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Materials/Mechanics of materials
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| Research Institution | Nagoya University |
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Principal Investigator |
OHNO Nobutada Nagoya University, Graduate School of Engineering, Professor, 大学院工学研究科, 教授 (30115539)
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| Co-Investigator(Kenkyū-buntansha) |
OKUMURA Dai Nagoya University, Graduate School of Engineering, Assistant Professor, 大学院工学研究科, 助手 (70362283)
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| Project Period (FY) |
2005 – 2006
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| Keywords | Multiscale analysis / Homogenization theory / Implicit method / Periodic solids / Inelastic deformation |
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| Research Abstract |
In this study, to perform inelastic two-scale analysis of macro-structures with periodic micro-structures, a fully implicit incremental formulation was developed for the homogenization problem of unit cells. The formulation was done using the virtual work equation of Hill (1967), along with a linearized constitutive relation and a micro/macro-kinematic relation. A boundary value problem and a computational algorithm were thus derived to iteratively determine perturbed displacement increment fields in unit cells. It was shown that the computational algorithm developed is versatile for setting initial strain increment field Δε^<(0)>_<n+1>, in unit cells. It was also shown that the algorithm reduces to those of Terada and Kikuchi (2001) and Miehe (2002), if Δε^<(0)>_<n+1> is taken to be equal to the macro-strain increment ΔE_<n+1>.
As a numerical example, elastic-plastic and elastic-viscoplastic tensile deformations of a holed plate with a rectangular micro-structure were analyzed by assum
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ing three settings for Δε^<(0)>_<n+1>, providing the following findings: The iterations to solve the homogenization problem, called local iterations, converged well, if the micro-strain increment field determined in the last step, Δεn, was chosen as Δε^<(0)>_<n+1>. The local iterations converged fairly well by taking Δε^<(0)>_<n+1> = 0. In contrast, initial setting Δε^<(0)>_<n+1>=ΔE_<n+1> led to bad convergences of local iterations in the course of tensile loading. Initial settings Δε^<(0)>_<n+1>=Δε_n and Δε^<(0)>_<n+1>=0 were therefore very effective for the convergence of local iterations when compared to Δε^<(0)>_<n+1>=ΔE_<n+1>.
The success attained by setting Δε^<(0)>_<n+1> = Δε_n, can be expected, if the variation from Δε_n to Δε_<n+1> is relatively small as in the present numerical example. Taking Δε^<(0)>_<n+1> = 0 is generally effective, because this allows the first estimate of Δε_<n+1> to be obtained by solving Eq. (35) based on the forward Euler method. It is emphasized that the fully implicit incremental formulation developed in the present study has enabled choosing initial settings Δε^<(0)>_<n+1> = Δε_n and Δε^<(0)>_<n+1> = 0 instead of Δε^<(0)>_<n+1> = ΔE_<n+1>. Less
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