1 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie,

Präsentation zum Thema: "1 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie,"—  Präsentation transkript:

1 1 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: Wolfgang.H.Mueller@tu-berlin.de, 2008 A multi-component theory of solid mixtures with higher gradients and its application to binary alloys by A. Brandmair, 1) T. Böhme, 1),2) W. Dreyer, 3) W.H. Müller 1) STAMM 2008 Symposium on Trends in Applications of Mathematics to Mechanics Levico Italy, September 24, 2008 1) Technische Universität Berlin Institut für Mechanik - LKM Einsteinufer 5 Einsteinufer 5 D-10587 Berlin D-10587 Berlin German Federal Environmental Foundation 2) ThyssenKrupp Steel AG Werkstoffkompetenzzentrum Werkstoffkompetenzzentrum Physikalische Technik Physikalische Technik Kaiser-Wilhelm-Straße 100 Kaiser-Wilhelm-Straße 100 D-47166 Duisburg D-47166 Duisburg 3) Weierstraß-Institut für Angewandte Analysis und Stochastik Angewandte Analysis und Stochastik Mohrenstr. 39 Mohrenstr. 39 D-10117 Berlin D-10117 Berlin

2 2 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: Wolfgang.H.Mueller@tu-berlin.de, 2008 Outline Introduction and motivation: Three types of microstructural change An experimental investigation of spinodal decomposition and coarsening Constitutive equations for diffusion flux and stress Some continuum theory: Entropy principle Classical theory of mixtures: w/o higher gradients Theory of mixtures for heterogeneous solids (with higher gradients) Reduction to the case of binary mixtures Numerical simulation of spinodal decomposition and coarsening Comparison with the experiment Homogenization and effective properties Conclusions and outlook

3 3 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: Wolfgang.H.Mueller@tu-berlin.de, 2008 Outline Introduction and motivation: Three types of microstructural change An experimental investigation of spinodal decomposition and coarsening Constitutive equations for diffusion flux and stress Some continuum theory: Entropy principle Classical theory of mixtures: w/o higher gradients Theory of mixtures for heterogeneous solids (with higher gradients) Reduction to the case of binary mixtures Numerical simulation of spinodal decomposition and coarsening Comparison with the experiment Homogenization and effective properties Conclusions and outlook

4 4 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: Wolfgang.H.Mueller@tu-berlin.de, 2008 SMT SnPb solder joints: formation of interface cracks spinodal decomposition microstructural coarsening Ball Grid Arrays and solder ball before and after 4000 temperature cycles aging at RT after (a) 2h, (b) 17d and (c) 63d (a) after solidification, (b) 3h and (c) 300 h at 125°C MELF miniature resistor and solder joint before and after 3000 temperature cycles Microstructural changes in solids I Will and, if so, how will the microstructural change influence the material properties ? Will and, if so, how will the microstructural change influence the material properties ?

5 5 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: Wolfgang.H.Mueller@tu-berlin.de, 2008 Introduction: Microstructural changes in solids II after solidification after 2h after 20h after 40h Ag-Cu: aging at 1000 Kelvin Ag-rich Cu-rich cracks along the phase boundaries SnPb solder balls with lead-free bumps decomposition + coarsening in the bulk Cu Leadfree solder, e.g., AgCu28: Formation of Inter-Metallic Compounds IMCs (Cu 6 Sn 5, Ag 3 Sn) at the interface and in the bulk Leadfree solder, e.g., SnAg3.8, SnAg3.8Cu0.7: thor.inemi.org/webdownload/newsroom/Articles/Lead-Free_Watch_Series/Oct05.pdf Will and, if so, how will the microstructural change influence the material properties ? Will and, if so, how will the microstructural change influence the material properties ?

6 6 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: Wolfgang.H.Mueller@tu-berlin.de, 2008 Triggered by different diffusion coefficients Cu atoms move to pile up IMCs. Kirkendall voids appear, e.g., between Cu substrate and thin Cu 3 Sn layer due to migration of Cu atoms from Cu 3 Sn to Cu 6 Sn 5, which is much faster than Sn-diffusion from Cu 6 Sn 5 towards Cu 3 Sn. Unbalanced Cu-Sn interdiffusion generates atomic vacancies at lattice sites which coalesce to voids. Model: vacancy diffusion Introduction: Microstructural changes in solids III Will and, if so, how will the microstructural change influence the material properties ? Will and, if so, how will the microstructural change influence the material properties ?

7 7 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: Wolfgang.H.Mueller@tu-berlin.de, 2008 Outline Introduction and motivation: Three types of microstructural change An experimental investigation of spinodal decomposition and coarsening Constitutive equations for diffusion flux and stress Some continuum theory: Entropy principle Classical theory of mixtures: w/o higher gradients Theory of mixtures for heterogeneous solids (with higher gradients) Reduction to the case of binary mixtures Numerical simulation of spinodal decomposition and coarsening Comparison with the experiment Homogenization and effective properties Conclusions and outlook

8 8 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: Wolfgang.H.Mueller@tu-berlin.de, 2008 Experiments I: Setup & realization material eutectic Ag-Cu temperature 970 K aging time 0 - 40 h etching (for Ag) solution of NH 3 -H 2 O 2

9 9 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: Wolfgang.H.Mueller@tu-berlin.de, 2008 instantaneous decomposition coarsening (Ostwald ripening) light: α-phase (Ag-rich), dark: β-phase (Cu-rich) after solidification after 2h aging after 20h aging after 40h aging Experiments II: Micrographs

10 10 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: Wolfgang.H.Mueller@tu-berlin.de, 2008 Experiments III: Image analysis determination of the β-areas and the total number of β-precipitates N by means of Digital Image Analysis (DHS TM ) Attention: 2D analysis of a 3D problem (observation of one cross-section) Solution: statistical averaging sufficient large areas of intersection analysis of various micrographs at each coarsening stage (individual photo) (individual stage) (spherical phases) (oblate spheroids) (cf., Underwood, 1970)

11 11 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: Wolfgang.H.Mueller@tu-berlin.de, 2008 faster coarsening rate for oblate spheroids with and t 1/3 - dependence well-known from LSW-theories Experiments IV: Coarsening rates 1/3

12 12 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: Wolfgang.H.Mueller@tu-berlin.de, 2008 Outline Introduction and motivation: Three types of microstructural change An experimental investigation of spinodal decomposition and coarsening Constitutive equations for diffusion flux and stress Some continuum theory: Entropy principle Classical theory of mixtures: w/o higher gradients Theory of mixtures for heterogeneous solids (with higher gradients) Reduction to the case of binary mixtures Numerical simulation of spinodal decomposition and coarsening Comparison with the experiment Homogenization and effective properties Conclusions and outlook

13 13 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: Wolfgang.H.Mueller@tu-berlin.de, 2008 Nomenclature I motion displacements, velocity, deformation gradient strains and stresses

14 14 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: Wolfgang.H.Mueller@tu-berlin.de, 2008 Nomenclature II Primary variables: Variables determined by partial mass balances (w/o chemical reactions), and balance of momentum and internal energy: total mass balance (no external forces) constitutive equations for sought in a situation where solids separated by phase boundaries move toward equilibrium higher gradient theories which, again, is characterized by a boundary higher gradient theories required !

15 15 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: Wolfgang.H.Mueller@tu-berlin.de, 2008 Constitutive relations for the diffusion flux (w/o thermo-diffusion) w/o gradientswith density gradients (w/o strain gradients) ( : mobility) chemical potential Helmholtz free energy density functional derivative:

16 16 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: Wolfgang.H.Mueller@tu-berlin.de, 2008 Constitutive relations for the stress tensor without gradients Castiglianos 2nd theorem specific Helmholtz free energy with density gradients (no strain gradients) pressure

17 17 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: Wolfgang.H.Mueller@tu-berlin.de, 2008 Outline Introduction and motivation: Three types of microstructural change An experimental investigation of spinodal decomposition and coarsening Constitutive equations for diffusion flux and stress Some continuum theory: Entropy principle Classical theory of mixtures: w/o higher gradients Theory of mixtures for heterogeneous solids (with higher gradients) Reduction to the case of binary mixtures Numerical simulation of spinodal decomposition and coarsening Comparison with the experiment Homogenization and effective properties Conclusions and outlook

18 18 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: Wolfgang.H.Mueller@tu-berlin.de, 2008 Entropy principle I: Historic remarks E ntropy principles: a) Clausius & Duhem (18th century) b) Coleman-Noll (1963) c) Green-Nagdhi (1967) d) Müller (1968), Liu (1972) Shortcomings of the above principles: a) Entropy flux relation, application to mixtures of solids ? b) Entropy balance global / local / principle for every constituent (too strong) ? c) Lagrange multipliers in order to consider the balances as constraints (some class of materials requires additional constraints, e.g., materials with gradients as variables) Therefore: Attempt to formulate a strategy based on commonly accepted points of the aforementioned principles

19 19 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: Wolfgang.H.Mueller@tu-berlin.de, 2008 Entropy principle I: Historic remarks E ntropy principles: a) Clausius & Duhem (18th century) b) Coleman-Noll (1963) c) Green-Nagdhi (1967) d) Müller (1968), Liu (1972) Shortcomings of the above principles: a) Entropy flux relation, application to mixtures of solids ? b) Entropy balance global / local / principle for every constituent (too strong) ? c) Lagrange multipliers in order to consider the balances as constraints (some class of materials requires additional constraints, e.g., materials with gradients as variables) Therefore: Attempt to formulate a strategy based on commonly accepted points of the aforementioned principles R. Clausius P. M. M. Duhem

20 20 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: Wolfgang.H.Mueller@tu-berlin.de, 2008 E ntropy principles: a) Clausius & Duhem (18th century) b) Coleman-Noll (1963) c) Green-Nagdhi (1967) d) Müller (1968), Liu (1972) Shortcomings of the above principles: a) Entropy flux relation, application to mixtures of solids ? b) Entropy balance global / local / principle for every constituent (too strong) ? c) Lagrange multipliers in order to consider the balances as constraints (some class of materials requires additional constraints, e.g., materials with gradients as variables) Therefore: Attempt to formulate a strategy based on commonly accepted points of the aforementioned principles Entropy principle I: Historic remarks B.D. Coleman W. Noll A.E. Green P.M. Nagdhi I. Müller I-S. Liu

21 21 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: Wolfgang.H.Mueller@tu-berlin.de, 2008 E ntropy principles: a) Clausius & Duhem (18th century) b) Coleman-Noll (1963) c) Green-Nagdhi (1967) d) Müller (1968), Liu (1972) Shortcomings of the above principles: a) Entropy flux relation, application to mixtures of solids ? b) Entropy balance global / local / principle for every constituent (too strong) ? c) Lagrange multipliers in order to consider the balances as constraints (some class of materials requires additional constraints, e.g., materials with gradients as variables) Therefore: Attempt to formulate a strategy based on commonly accepted points of the aforementioned principles Entropy principle I: Historic remarks Duhem: radiation: ideal gas: mixtures:

22 22 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: Wolfgang.H.Mueller@tu-berlin.de, 2008 Entropy principle I: Historic remarks Clausius: E ntropy principles: a) Clausius & Duhem (18th century) b) Coleman-Noll (1963) c) Green-Nagdhi (1967) d) Müller (1968), Liu (1972) Shortcomings of the above principles: a) Entropy flux relation, application to mixtures of solids ? b) Entropy balance global / local / principle for every constituent (too strong) ? c) Lagrange multipliers in order to consider the balances as constraints (some class of materials requires additional constraints, e.g., materials with gradients as variables) Therefore: Attempt to formulate a strategy based on commonly accepted points of the aforementioned principles

23 23 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: Wolfgang.H.Mueller@tu-berlin.de, 2008 Entropy principle I: Historic remarks E ntropy principles: a) Clausius & Duhem (18th century) b) Coleman-Noll (1963) c) Green-Nagdhi (1967) d) Müller (1968), Liu (1972) Shortcomings of the above principles: a) Entropy flux relation, application to mixtures of solids ? b) Entropy balance global / local / principle for every constituent (too strong) ? c) Lagrange multipliers in order to consider the balances as constraints (some class of materials requires additional constraints, e.g., materials with gradients as variables) Therefore: Attempt to formulate a strategy based on commonly accepted points of the aforementioned principles

24 24 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: Wolfgang.H.Mueller@tu-berlin.de, 2008 Entropy principle I: Historic remarks E ntropy principles: a) Clausius & Duhem (18th century) b) Coleman-Noll (1963) c) Green-Nagdhi (1967) d) Müller (1968), Liu (1972) Shortcomings of the above principles: a) Entropy flux relation, application to mixtures of solids ? b) Entropy balance global / local / principle for every constituent (too strong) ? c) Lagrange multipliers in order to consider the balances as constraints (some class of materials requires additional constraints, e.g., materials with gradients as variables) Therefore: Attempt to formulate a strategy based on commonly accepted points of the aforementioned principles

25 25 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: Wolfgang.H.Mueller@tu-berlin.de, 2008 Entropy principle II: Formulation Two constitutive quantities: entropy density and entropy flux. Constitutive relation of the entropy density: Local balance for entropy density: Entropy production positive definite & of the form fluxes x driving forces: (Absolute) temperature defined as follows: Stress tensor decomposed into an elastic and a dissipative part: 2 nd law

26 26 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: Wolfgang.H.Mueller@tu-berlin.de, 2008 Entropy principle III: The evaluation procedure H.W. Alt, I. Pawlow: On the entropy principle of phase transition models with a conserved order parameter. Advances in Mathematical Science and Applications, 6(1), pp. 291–376, 1996: Balances interpreted as evolution equations for variables: Balances viewed as a system of algebraic equations, i.e., choose right hand sides arbitrarily & calculate left hand sides. Right hand sides of the balances + chain rule list of arbitrary terms: Construct constitutive relations such that 2nd law is identically satisfied.

27 27 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: Wolfgang.H.Mueller@tu-berlin.de, 2008 Outline Introduction and motivation: Three types of microstructural change An experimental investigation of spinodal decomposition and coarsening Constitutive equations for diffusion flux and stress Some continuum theory: Entropy principle Classical theory of mixtures: w/o higher gradients Theory of mixtures for heterogeneous solids (with higher gradients) Reduction to the case of binary mixtures Numerical simulation of spinodal decomposition and coarsening Comparison with the experiment Homogenization and effective properties Conclusions and outlook

28 28 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: Wolfgang.H.Mueller@tu-berlin.de, 2008 Mixtures w/o higher gradients I: Choice of variables Simple -component mixture without reactions / viscous effects: Different representations of entropy density: with a constant number of variables, e.g.: Different representations useful under different circumstances, e.g.:

29 29 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: Wolfgang.H.Mueller@tu-berlin.de, 2008 Mixtures w/o higher gradients II: Entropy production Local entropy balance: Chain rule for Kinematic relation right hand side of balances (note: balance of momentum not required for exploitation, since not in ), chain rule kinematic relation

30 30 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: Wolfgang.H.Mueller@tu-berlin.de, 2008 Mixtures w/o higher gradients III: Entropy production Substitution and rearrangement: a) separate terms into flux / force: b) separate terms linear in (cf., arbitrary terms) Definition entropy-flux (first parenthesis): Residual inequality: Q P terms linear in drop out Galileian invariance

31 31 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: Wolfgang.H.Mueller@tu-berlin.de, 2008 Mixtures w/o higher gradients IV: Heat & diffusion flux Helmholtz free energy density experimentally inconvenient quantity (chemical potential) Legendre Transformation Constraint (Fouriers law of heat conduction) ( : mobility) No coupling, quadratic form Q-bit

32 32 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: Wolfgang.H.Mueller@tu-berlin.de, 2008 Mixtures w/o higher gradients V : Selected results Pressure Legendre transform applied to P-bit Gibbs-Duhem relation 2 nd Piola-Kirchhoff stress tensor

33 33 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: Wolfgang.H.Mueller@tu-berlin.de, 2008 Outline Introduction and motivation: Three types of microstructural change An experimental investigation of spinodal decomposition and coarsening Constitutive equations for diffusion flux and stress Some continuum theory: Entropy principle Classical theory of mixtures: w/o higher gradients Theory of mixtures for heterogeneous solids (with higher gradients) Reduction to the case of binary mixtures Numerical simulation of spinodal decomposition and coarsening Comparison with the experiment Homogenization and effective properties Conclusions and outlook

34 34 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: Wolfgang.H.Mueller@tu-berlin.de, 2008 Mixtures with higher gradients I: Functional representations functional representation of the entropy density Note:- Choice of Higher Gradients depends on the problem - Present choice: Convenient for diffusion problems & for definition of the chemical potential

35 35 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: Wolfgang.H.Mueller@tu-berlin.de, 2008 Mixtures with higher grad. II: Specialization to binary alloy Now: Binary alloy A-B: Relating difference of chem. pot. to derivative of Helmholtz free energy density: with Helmholtz free energy density diffusion flux for a binary alloy: (isothermal case)

36 36 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: Wolfgang.H.Mueller@tu-berlin.de, 2008 Transformation to the reference configuration / re-definition of mobility Problem with free energy density (more general case): Decomposition & Taylor expansion HGC can be identified & calculated by microscopic atomistic theories (e.g., EAM) it follows (Böhme et al., 2007): (periodic arrangement of the lattice) Approach for elastic contributions phase diagram ? ? leading term Mixtures with higher grad. III: Specialization to binary alloy

37 37 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: Wolfgang.H.Mueller@tu-berlin.de, 2008 Partial mass balance & mass concentration : Material parameters: Mixtures with higher grad. IV: Specialization to binary alloy

38 38 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: Wolfgang.H.Mueller@tu-berlin.de, 2008 Outline Introduction and motivation: Three types of microstructural change An experimental investigation of spinodal decomposition and coarsening Constitutive equations for diffusion flux and stress Some continuum theory: Entropy principle Classical theory of mixtures: w/o higher gradients Theory of mixtures for heterogeneous solids (with higher gradients) Reduction to the case of binary mixtures Numerical simulation of spinodal decomposition and coarsening Comparison with the experiment Homogenization and effective properties Conclusions and outlook

39 39 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: Wolfgang.H.Mueller@tu-berlin.de, 2008 Application: Spinodal decomposition in AgCu I: Simulations in 1D (no external stress) Study concentration development along a line

40 40 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: Wolfgang.H.Mueller@tu-berlin.de, 2008 Application: Spinodal decomposition in AgCu II: Simulations in 1D (no external stress) Fortran 95, FFTPack, explicit Euler scheme

41 41 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: Wolfgang.H.Mueller@tu-berlin.de, 2008 Application: Spinodal decomposition in Ag-Cu II: Simulations in 1D (tensile stress: 10 3 MPa) Fortran 95, FFTPack, explicit Euler scheme stresses accelerate coarsening

42 42 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: Wolfgang.H.Mueller@tu-berlin.de, 2008 Application: Spinodal Decomposition in Ag-Cu III: 2D (no external stress) initial 3000 time loops 6000 loops 60 000 loops Cu-rich phase Ag-rich phase

43 43 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: Wolfgang.H.Mueller@tu-berlin.de, 2008 Outline Introduction and motivation: Three types of microstructural change An experimental investigation of spinodal decomposition and coarsening Constitutive equations for diffusion flux and stress Some continuum theory: Entropy principle Classical theory of mixtures: w/o higher gradients Theory of mixtures for heterogeneous solids (with higher gradients) Reduction to the case of binary mixtures Numerical simulation of spinodal decomposition and coarsening Comparison with the experiment Homogenization and effective properties Conclusions and outlook

44 44 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: Wolfgang.H.Mueller@tu-berlin.de, 2008 Experiment and simulation: Coarsening rates By image analysis of experiments and computer generated microstructural evolution 3D simulations required ?

45 45 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: Wolfgang.H.Mueller@tu-berlin.de, 2008 Outline Introduction and motivation: Three types of microstructural change An experimental investigation of spinodal decomposition and coarsening Constitutive equations for diffusion flux and stress Some continuum theory: Entropy principle Classical theory of mixtures: w/o higher gradients Theory of mixtures for heterogeneous solids (with higher gradients) Reduction to the case of binary mixtures Numerical simulation of spinodal decomposition and coarsening Comparison with the experiment Homogenization and effective properties Conclusions and outlook

46 46 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: Wolfgang.H.Mueller@tu-berlin.de, 2008 Average elastic properties by homogenization I Homogenization performed by S.V. Sheshenin, M. Savenkova (FE-program Elast) Boundary value problem (plane strain analysis) for a given micrograph (= RVE): Loading sequences applied, e.g.: etc.

47 47 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: Wolfgang.H.Mueller@tu-berlin.de, 2008 Average elastic properties by homogenization II AgCu28 simulated temporal development of microstucture 1 2 3 4 5 6 Conclusion material is cubic, just like its constituents Ag and Cu changing microstructure leads to no change in elastic coefficients

48 48 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: Wolfgang.H.Mueller@tu-berlin.de, 2008 Average elastic properties by homogenization III SnPb37 effective elastic moduli of simulated microstucture Conclusions composite material is less tetragonal due to the slight difference between C 1111 and C 2222 for Sn and the presence of the cubic Pb laminate theory gives similar results: Attention lead (Pb) is cubic = 3 elastic constants, tin (Sn) is tetragonal = 6 elastic constants 2D excerpt:

49 49 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: Wolfgang.H.Mueller@tu-berlin.de, 2008 Outline Introduction and motivation: Three types of microstructural change An experimental investigation of spinodal decomposition and coarsening Constitutive equations for diffusion flux and stress Some continuum theory: Entropy principle Classical theory of mixtures: w/o higher gradients Theory of mixtures for heterogeneous solids (with higher gradients) Reduction to the case of binary mixtures Numerical simulation of spinodal decomposition and coarsening Comparison with the experiment Homogenization and effective properties Conclusions and outlook

50 50 Technische Universität Berlin Fakultät für Verkehrs- und Maschinensysteme, Institut für Mechanik Lehrstuhl für Kontinuumsmechanik und Materialtheorie, Prof. W.H. Müller Copyright © Prof. Dr. rer. nat. W.H. Müller, e-mail: Wolfgang.H.Mueller@tu-berlin.de, 2008 Conclusions and Outlook Phase field / higher gradient models to be used for microstructural changes in multi- component alloys should not simply be postulated but rather based on balance laws for physical quantities. Material theory should and can be used to derive the corresponding field equations. Material parameters in these relations should not simply be guessed, rather they should be obtained from experiments that are independent of the to-be-described phenomenon and, eventually, also be obtained from atomic methods (e.g., embedded atom methchnique). The spinodal decomposition observed in some solder/welding materials as well as the subsequent process of coarsening can be modeled quantitatively using such a strategy. Homogenized elastic properties for experimentally observed as well as predicted micrographs showing microstructural change have been obtained. Non-linear homogenized material properties were not obtained yet.