Multi-scale modelling of localization and damage
Citation for published version (APA):
Coenen, E. W. C., Kouznetsova, V., & Geers, M. G. D. (2009). Multi-scale modelling of localization and damage. Poster session presented at Mate Poster Award 2009 : 14th Annual Poster Contest.
Document status and date: Published: 01/01/2009 Document Version:
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Mechanics of Materials
Introduction
Damage and strain localization analyses are necessary in many structural or functional material designs. Ad-vanced technological developments rise a demand for coupled damage analysis at different scales.
Macro Meso Micro Atomic
Figure 1: Upscaling of the microstructural response. Project Goal: The development of a two-scale computa-tional framework, which correctly upscales the effect of microscale damage on macroscale fracture.
Multi-scale modelling
Classical computational homogenization schemes are based on the solution of two nested boundary value problems. They rely on Microstructural Volume Ele-ments (MVEs) which are locally representative for the mi-crostructure. Strain localization inevitably limits the con-cept of homogenization.
Force equilibrium (FEM) Crack modelling (XFEM)
Macro
Micro
Bulk deformation Lumped strain localization Homogenized Crack opening Microscale deformation experiment stressFigure 2: Developed multi-scale framework.
The innovative multi-scale scheme presented here splits the microscopic deformation into a bulk and localization type of deformation. The macroscopic continuum is en-riched with a cohesive discrete crack, which lumps the microstructural strain localization and the residual load carrying capacity. This overcomes the limitation of clas-sical schemes.
Results
Straining a voided material (see Figure 3) results in a pre-localization (1-7) and post-localization (8-12) fase. Pre-localization: the MVE is representative for the corre-sponding macroscopic material point. Post-localization: a strain localization band has developed that needs to be accounted for.
Figure 3: Strain localization within a voided elasto-plastic material. Stress Homogenization:The definition for the stress ho-mogenization is classical and is defined as the volume average of the microscale stress field.
Lumping of strain localization: The distinction between “strain localization” and “bulk deformation” is defined by a minimization of the microfluctuation field in a least square sense. Strain Homogenized stress 1 2 3 4 56 7 8 9 10 1112 Position Bulk deformation 9 Displacement (a) (b) Lumped strain localization
Figure 4: (a) Homogenized stress-strain response and (b) calculation of the lumped strain localization (frame 9).
Conclusion
Upscaling of the microstructural material response after the point of strain localization has been developed. This is a versatile tool for multi-scale analysis of damage.
