CM3 Mechanics of Materials www.ltas-cm3.ulg.ac.be Multi-scale - PowerPoint PPT Presentation

Computational & Multiscale CM3 Mechanics of Materials www.ltas-cm3.ulg.ac.be Multi-scale modelling Multiscale models Ludovic Noels for composite failure G. Becker, S. Mulay, V.-D. Nguyen, V. Péron-Lührs, L. Wu FE2 computations of foamed structures DG-based fracture framework Stiction failure in a MEMS sensor (picture Sandia National QC method for grain-boundary sliding Laboratories ) 2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science CM3

Content • Introduction – Multi-scale modelling: Why? – Multi-scale modelling: How? • Mean-Field-Homogenization with non-local damage • Conclusions CM3 2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science - 2

Multi-scale modelling: Why? • Limitations of one-scale models – Physics at the micro-scale is too complex to be modelled by a simple material law at the macro-scale • Engineered materials • Multi-physics/scale problems • …. • See next slides – Lack of information of the micro-scale state during macro-scale deformations • Required to predict failure • ….. – Effect of the micro-structure on the macro- structure response • Grain-size effect in metals • … • Solution: multi-scale models CM3 2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science - 3

Multi-scale modelling: Why? • Examples of multi-scale problems – Different physics at the different scales – Stiction (adhesion of MEMS) Continuum solid t mechanics g Stiction failure in a MEMS sensor ( Jeremy A.Walraven Sandia National s Laboratories. Albuquerque, NM USA) l F n z 0 2 a Statistical representation of 2 c rough surfaces Van der Waals/capillary/Hertz forces at the asperity level CM3 2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science - 4

Multi-scale modelling: Why? • Examples of multi-scale problems (2) – Continuum solid mechanics at the different scales – Non-linear response of [-45 2 /45 2 ] S composites CM3 2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science - 5

Multi-scale modelling: How? • Principle – 2 problems are solved concurrently • The macro-scale problem • The micro-scale problem (Representative Volume Element) – Scale transitions coupling the two scales • Downscaling: transfer of macro-scale quantities (e.g. strain) to the micro-scale to determine the equilibrium state of the Boundary Value Problem • Upscaling: constitutive law (e.g. stress) for the macro-scale problem is determined from the micro-scale problem resolution Material Extraction of a RVE response Macroscale Assumptions: L macro >> L RVE >> L micro BVP CM3 2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science - 6

Multi-scale modelling: How? • Computational technique: FE 2 σ – ε Macro-scale ? • FE model At one integration point e is know, s is sought • CM3 2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science - 7

Multi-scale modelling: How? • Computational technique: FE 2 σ – ε Macro-scale ? • FE model At one integration point e is know, s is sought • – Micro-scale • Usual 3D finite elements • Periodic boundary conditions CM3 2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science - 8

Multi-scale modelling: How? • Computational technique: FE 2 – Macro-scale • FE model At one integration point e is know, s is sought • – Transition Downscaling: e is used to define the BCs • Upscaling: s is known from the reaction forces •  σ σ ε  ε – Micro-scale • Usual 3D finite elements • Periodic boundary conditions CM3 2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science - 9

Multi-scale modelling: How? • Computational technique: FE 2 – Macro-scale • FE model At one integration point e is know, s is sought • – Transition Downscaling: e is used to define the BCs • Upscaling: s is known from the reaction forces •  σ σ ε  ε – Micro-scale • Usual 3D finite elements • Periodic boundary conditions – Advantages • Accuracy • Generality – Drawback • Computational time Ghosh S et al. 95, Kouznetsova et al. 2002, Geers et al. 2010, … CM3 2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science - 10

Multi-scale modelling: How? • Mean-Field Homogenization σ – ε Macro-scale ? • FE model At one integration point e is know, s is sought • CM3 2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science - 11

Multi-scale modelling: How? • Mean-Field Homogenization σ – ε Macro-scale ? • FE model At one integration point e is know, s is sought • – Micro-scale • Semi-analytical model • Predict composite meso-scale response • From components material models w I w 0 CM3 2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science - 12

Multi-scale modelling: How? • Mean-Field Homogenization σ – ε Macro-scale ? • FE model At one integration point e is know, s is sought • – Transition Downscaling: e is used as input of the MFH model • Upscaling: s is the output of the MFH model •  σ σ ε  ε – Micro-scale • Semi-analytical model • Predict composite meso-scale response • From components material models w I w 0 CM3 2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science - 13

Multi-scale modelling: How? • Mean-Field Homogenization σ – ε Macro-scale ? • FE model At one integration point e is know, s is sought • – Transition Downscaling: e is used as input of the MFH model • Upscaling: s is the output of the MFH model •  σ σ ε  ε – Micro-scale • Semi-analytical model • Predict composite meso-scale response • From components material models – Advantages • Computationally efficient w I • Easy to integrate in a FE code (material model) w 0 – Drawbacks • Difficult to formulate in an accurate way – Geometry complexity Mori and Tanaka 73, Hill 65, Ponte Casta ñ eda 91, Suquet – Material behaviours complexity 95, Doghri et al 03, Lahellec et al. 11, Brassart et al. 12, … CM3 2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science - 14

Multi-scale modelling: How? • Semi analytical Mean-Field Homogenization – Based on the averaging of the fields w I σ 1 inclusions   a a ( X ) d V w 0 V V – Meso-response  v  • composite ? v 1 From the volume ratios ( ) 0 I      σ σ σ σ σ σ v v v v 0 w I w 0 0 I I 0 I      ε ε ε ε ε ε v v v v matrix w w 0 I 0 0 I I 0 I ε • One more equation required  e ε : ε B I 0 – Difficulty: find the adequate relations    σ ε f I I   e  σ ε ? B f 0 0 e  ε : ε B I 0 CM3 2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science - 15

Multi-scale modelling: How? • Mean-Field Homogenization for linear materials – System of equations w I  v  • v 1 σ From the volume ratios ( ) 0 I inclusions w 0   σ σ σ v v 0 0 I I   ε ε ε v v 0 0 I I composite ? • Assume linear behaviours σ  : ε C I I I σ  : ε matrix C 0 0 0  e ε : ε ε B • Relation between average strains I 0 – Single inclusion problem from Eshelby tensor S e   e ε H e      H C C ( I , , ) : 1 1 • I S C C C [ : : ( )] with I 0 I 0 1 0 • Results from a phase transformation analysis – Multiple inclusions problem   e  ε ε B C C I , , : • I 0 I 0   B  e e ε ε • H Mori-Tanaka assumption 0 CM3 2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science - 16

Computational & Multiscale CM3 Mechanics of Materials www.ltas-cm3.ulg.ac.be Non-local damage-enhanced mean-field-homogenization L. Wu (ULg), L. Noels (ULg), L. Adam (e-Xstream), I. Doghri (UCL) SIMUCOMP The research has been funded by the Walloon Region under the agreement no 1017232 (CT-EUC 2010- 10-12) in the context of the ERA-NET +, Matera + framework. 2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science CM3