A multi-scale oddity : unifying localization and homogenization
Citation for published version (APA):Coenen, E. W. C., Kouznetsova, V., & Geers, M. G. D. (2010). A multi-scale oddity : unifying localization and homogenization. Poster session presented at Mate Poster Award 2010 : 15th Annual Poster Contest.
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Mechanics of Materials
A multi-scale oddity: Unifying
localization and homogenization
Erica Coenen, Varvara Kouznetsova, and Marc Geers
/department of mechanical engineering
Introduction
Macro Meso Micro Atomic
Fig. 1 The world of engineering is multi-scale.
Project Goal: The development of a two-scale
computa-tional framework, which correctly upscales the microscale damage towards macroscale fracture.
Multi-scale modelling
Classical computational homogenization schemes rely on Microstructural Volume Elements (MVE) which are locally representative for the microstructure. Strain localization inevitably limits the concept of homogenization.
bulk jump Macro Micro Deformation experiment Hom. stress
Lumped strain loc. Orientation crack
fracture balance: jump = lumped strain loc.
MVE deformation
(bulk and jump deformation)
Fig. 2 Localization enriched multi-scale framework.
The developed scheme reconciles this conflict by disen-tangling the bulk and the collective strain localization be-haviour.
MVE boundary conditions (BCs)
New BCs (called aligned) are proposed that provide a good estimate for the effective stiffness and simultaneously allow for a strain localization band to develop.
Localization band aligned boundary conditions
1 1.02 1.04 1.06 1.08 1.1 0 100 200 300 400 500 600 700 Deformation [-] S tr es s [M P a] Aligned Periodic Minimal Uniform 0.0 0.5 0.25
Aligned Periodic Minimal Uniform
T ot . eq . st ra in [-] (Def. ×1 2) Fig. 3 Influence of BC-type on strain localization.
Numerical example
Horizontal stretching of a heterogeneous plate (see Fig. 4) results in a pre-localization ◦and post-localization ◦fase.
The MVE crossed by the localization band (middle) con-tinues stretching, while the other MVEs (right and left) unload. 1.12 400 1 0 200 600 800 1.12 400 1 0 200 600 800 1.12 400 1 0 200 600 800 Macro Micro Deformation [-] Deformation [-] Deformation [-] S tre ss [M P a] S tre ss [M P a] S tre ss [M P a] 150mm 1mm Lumped str. loc. Ap p lie d d is p la ce me n t 550 166 182 eq . von M is es st re ss [M P a]
Fig. 4 Microstructural response of a heterogeneous plate under applied horizontal displacement.
The localization enriched scheme is well-regularized (no mesh dependency), the response is comparable to the ref-erence result (direct numerical simulation, DNS) and the computational costs are much smaller (±1/50).
0 2 4 6 8 0 200 400 600 800 1000 Applied disp. [mm] T ra ct ion [N /mm] Classical
û
0 2 4 6 8 0 200 400 600 800 1000 Applied disp. [mm] T ra ct ion [N /mm] Macroscale element size: 3×50mm 6×25mm 12×12.5mm reference Localization enrichedü
Fig. 5 Macroscale response obtained by the classical and the localization enriched multi-scale scheme.
Conclusion
The proposed multi-scale scheme in combination with the developed MVE boundary conditions comprises a versatile and powerful analysis tool for multi-scale problems involv-ing localization and damage.
