Nano Mechanics and Materials: Theory, Multiscale Methods and Applications

1. Nano Mechanics and Materials:Theory, Multiscale Methods and Applications by Wing Kam Liu, Eduard G. Karpov, Harold S. Park

2. 9. Multiscale Methods for Material Design

3. Why is Multi-scale Material Design Important? Environmental factors lead to fracture of a gas pipeline Fatigue Fracture of firefighter’s ladder Ductile to Brittle Transformation causes a hull to fracture http://www.engr.sjsu.edu/WofMatE/FailureAnaly.htm Material response (Ductility, Strength, Corrosion & Fatigue Resistance) is controlled by nano and micro mechanisms

4. Why is Multi-scale Material Design Important? • We can usually design against failure mechanisms in some structural components by • Using more material • Improving material design • Problem: It is not possible to increase mass in most transport applications, e.g. aero, railway, automotive because weight and cost are important. Fatigue is rarely improved by increasing mass. Option (a) is unacceptable.

5. Why is Multi-scale Material Design Important? A macroscale demonstration Courtesy of Yip Wah Chung Slide taken from his opening remark on the Short course, entitled “Nano scale design of materials” given at the NSF Summer Institute on Nano Mechanics and Materials Northwestern University, August 25, 2003.

6. Example of a Micro Mechanism in an Alloy Ductile fracture by void nucleation, growth and coalescence. Nucleation of voids occurs due to (a) particle fracture or (b) debonding of the particle matrix interface. This depends on : particle size, shape, temperature, spacing, distribution, chemical composition, interfacial strength, coherency, stress state. Void growth is followed by voidcoalescence which occurs by (a) void impingement or (b) a void sheet mechanism. This depends on the stress state, presence of secondary particles, and those factors listed above. Nucleation at secondary particles along a shear band Horstemeyer et al. 2000

7. Goals of Virtual Design of Multiscale Materials To improve the engineering design cycle using simulations and computational tools • Connect macroscopic continuum response with driving meso- , micro- & nano-scale behavior • Understand continuum response due to underlying atomsitic structural response • Generate multiple scale governing equations and material laws for concurrent calculations • Produce methods to determine material constants based on each length scale • Create methods by which to effectively simulate the complex response of these coupled systems • Provide tools for the design/virtual testing of engineered materials

8. Performances Goals/means Properties Structure Cause and effects Processing Integration of Nanoscale Science and Engineering: From Atoms to Continuum • Old paradigm: separate manufacturing with design • New paradigm: consider all environments in manufacturing processing through the life cycle performance of a component/system • Gap Closer: models that relate structure to properties (Olson 1997)

9. 9.1 Multiresolution Continuum Analysis

10. Multi-scale Theory for Three-Scale Material Multi-scale decomposition of material • Statistically homogeneous structure => unit cell at each scale (smallest representative element) • Expansion of velocity in unit cells => characteristic rate of deformation of a unit cell Sub-micro unit cell Micro unit cell Macroscopic domain v2 x2 Mathematical domains v1 x1 x0 v0 Physical domains Deformation measure

11. Total macro and micro stresses : Macro RVE Micro unit cell Stress and internal power decomposition for a two-scale material • Decomposition of the deformation and stress measure in the micro cell: Due to macro deformation Due to micro deformation The internal power of a unit cell is: Homogenized internal power?

12. Linear variation of where Averaging operation • Homogenized internal power is the average over a domain : • Averaging domain captures microstructure interactions • Linear expansion of the micro deformation in this domain

13. Micro component Sub-micro component Macro component Generalization • For a three-scale material: • Three stresses : conjugate to • Two averaging domains and Where the stresses are defined as follows: => Good for cell models

14. Constitutive relation • Define generalized stress and strain: • Constitutive relation Plasticity / damage Hypo-elasticity Generalized Yield function/plastic potential Internal variables How to find the constitutive relation and material constants?

15. Sub-micro velocity gradient Micro velocity gradient = micro spin Macro velocity gradient Example: Granular material • Internal power density Cosserat material

16. Granular material: constitutive relation Elasticity At the micro-scale Average in the averaging domain Plasticity Generalized Yield function/plastic potential : Generalized J2 flow theory Material constants

17. Granular material: Material constants Goal : Determination of the constants Is defined as an average of the slip s(x+y) measure in Using the linear variation of in the averaging domain Perform averages and get material constants Kadowaki(2004) Remark:. In this analytical derivation, we chose (still empirical but can be determined by a more accurate physical model)

18. Example 2 : Deformation theory of Strain Gradient Plasticity Micro strain Sub-micro strain Macro velocity gradient Assumption : only gradients play a role in the internal power : Set the macro averaging domain equals to the micro averaging domain • Internal power density

19. Strain gradient plasticity: constitutive relation • Taylor relation at the micro scale where is the stress in the averaging domain. The same equation can written in the form • From mechanistic models (bending , torsion, void growth), Gao found an expression for the • equivalent strain gradient • The constitutive relation at small scale follows the deformation • theory of plasticity:

20. Strain gradient plasticity : constitutive relation Linear variation of the micro strain field in the averaging domain The microscopic constitutive relation is averaged in : => Same result as mechanism based strain gradient plasticity (Gao & al)

21. Cell Modeling Micro stresses are averages over averaging domains: Cell model of the averaging domain at each scale Apply strain boundary conditions Total micro stress Curve fitting of the generalized potential Determination of the elastic matrix for each scale • Periodic BC We ensure that periodicity is preserved

22. Cell modeling - Successes • Industrial Applications : • Prediction of edge cracking during rolling • Prediction of central bursting during extrusion Simulation Experimental Rolling: Edge cracks Extrusion: Central Bursting

23. Computer based material law Macro cell model Strain boundary conditions macro stress Micro cell model Fitting of material Constants in  and C Strain boundary conditions Total micro stress Sub-micro cell model Strain boundary conditions Total micro stress

24. Softening in Pure Shear Interaction of primary particles with secondary particles • Debonding around primary particles • increased local strain field close to particles • higher triaxility • debonding and softening at the sub-micro scale • Void sheet forms Shear bands forming during a ballistic impact ( Cowie, Azrin, Olson 1988) Continuum accounting for damage from secondary particles Periodic boundary conditions Fracture by void sheet Primary particles

25. instability Shear stress Shear strain Softening in Pure shear Results • 1) shear stress/strain curve in shown below (softening occurs) • 2) dependence of the shear strain at instability is plotted as a function of pressure ( agrees with experimental results) simulation Experiment

26. 9.2 Multiscale Constitutive Modeling of Steels

27. TiN Review of Multiscale Structure of Steel b) sub-micro scale TiC Micrograph of high strength steel • quantum scale d) macro-scale c) micro-scale

28. Ultra High Strength Steels Microstructure of steel • Two levels of particles : primary and secondary ( three-scale micromorphic material) • primary particles • secondary particles Micro scale Macro scale Sub-micro scale • Deformation of microstructure at each scale is important in the fracture process (see fracture surface) • Need a general multi-scale continuum theory for materials that accounts for microstructure deformation and interactions Fracture surface

29. Multi-scale Nature of a Steel Alloy Multi-level decomposition of the structure of steel Secondary Particles Primary Particles Dislocations Macro-scale Matrix/particle bonding Micro-scale Scales considered for concurrent model Sub micro-scale with theromodynamics and mathematical model uncertainties Quantum scalewith atomic lattice uncertainties scale

30. Predictive Multiscale Mathematical Models Goals: • Develop a predictive multiscale mathematical model • Integrate materials design at the atomic scale into virtual manufacturing, at the continuum scale • Use probabilistic optimization to address uncertainties in processing and modeling

31. Quantum Theory: (e.g. One particle Schödinger Eqt) m: mass; V: potential yi: ith eigenfunction Ei:ith eigenvalue ? Continuum mechanics Force and displacement boundary condition Why Start from the Atomic-Electronic Scale? Thomas-Fermi Model electron gas Ti N,C nuclei

32. TSD Diagram for Steel Design (Toughness-Strength-Decohesion Energy Diagram) 2 COD MgS TiN Ti2CS TiC

33. Cybersteel: Cell Modeling Macro velocity gradient Sub-micro velocity gradient Micro velocity gradient • Internal power density Micro Macro Sub-micro

34. Cybersteel: Constitutive relation Elasticity Elastic constants To be determined

35. Plasticity Generalized Yield function/plastic potential : Multiscale Gurson model Material constants 13 constants + equation of evolution of void volume fraction F with stress and strain Goal => determination of these material constants through cell modeling a each scale

36. Vision The next generation of CAE software will integrate nano and micro structures into traditional CAE software for design and manufacturing We propose five key new developments: (1) Concurrent multi-field variational FEM equations that couple nano and micro structures and continuum. (2) A predictive multiscale constitutive law that bridges nano and micro structures with the continuum concurrently via statistical averaging and monitoring the microstructure/defect evolutions (i.e., manufacturing processes). (3) Bridging scale mechanics for the hierarchical and concurrent analysis of (1) and (2). (4) Models for joints, welds and fracture, etc., that embody the above. (5) Probabilistic simulation-based design techniques enabling the integration of all of the above.

37. 9.3 Bio-Inspired Materials

38. Bio-Inspired Self Healing Materials – Multiscale Nature Background “The day may come when cracks in buildings or in aircraft structures close up on their own, and dents in car bodies spring back into their original shape,” SRIC-BI (2004). Origin: Biomimesis - the study and design of high-tech products that mimic biological systems Goals: • Reducing maintenance requirements • Increasing safety and product lifetime • Autonomous devices • medical implants, sensors, space vehicles that • applications where repair is impossible or impractical • e.g. implanted medical devices, electronic circuit boards, aerospace/space systems.

39. Bio-Inspired Self Healing Materials • Self-healing structural composite: • Matrix with an encapsulated healing agent • Catalyst particles embedded in matrix • Crack penetrates capsule • Healing agent reacts with catalyst and polymerizes • Polymerized agent seals crack White S.R., et al., Nature 409, 2001.

40. Bio-Inspired Self Healing Materials • Bioinspired SMA self-healing composite with bone shaped SMA inclusions: • Composite with SMA bone shaped inclusions • Crack propagation, inclusion transformation, interfacial debonding, crack halting and energy dissipation Loading Heating Prof. Olson group on SH composite • Healed composite with some change of chrystallography of affected inclusions and crack closure.

41. Shape Memory Alloys - Basics • Metal alloys that recover apparent permanent strains when they are heated above a certain temperature • Key effects are pseudoelasticity and shape memory effect • Atomic level - Two stable phases low-temperature phase martensite • high-temp phase • austenite twinned detwinned www.msm.cam.ac.uk/phase-trans/2002/memory.movies.html http://smart.tamu.edu/overview/smaintro/simple/pseudoelastic.html Cubic Crystal Monoclinic Crystal

42. Phase Transformation (Temp only) • Phase transformation occurs between these two phases upon heating/cooling NO SHAPE CHANGE http://smart.tamu.edu/overview/smaintro/simple/pseudoelastic.html

43. 2. Unload – deformation remains 3. Heat – Reverse transformation Phase Transformation (Temp + Load) • 1. Apply a load to TWINNED martensite – get DETWINNED martensite (SHAPE CHANGE)

44. Phase Transformation (Temp + Load) Assuming a linear relationship between applied load and transformation temperature

45. Phase Transformation (Load) 1. Apply a pure mechanical load 2. Get detwinned martensite AND very large strains 3. Complete shape recovery is observed upon unloading – pseudoelasticity

46. A M SMA transformations M A As Af 1-D SMA Constitutive Law b a Fraction of Martensite = f (a/b) Flow stress = g ( Fraction of Martensite) *1-d constitutive law from Prof. Brinson’s Group at Northwestern

47. Brittle Matrix SMA inclusion Bone Shaped Inclusions Bridging Strong bonding is not effective Weak bonding • Crack energy dissipated through anchoring effect of BRIDGING inclusions • Inclusions are stretched – phase transformation occurs (A-M) • Heat, (M-A) original shape regained. Crack closes • Use pre-strained inclusions – significant detwinned martensite • Clamping at high temp – partial re-welding of fracture surface

48. X t A homogenized continuum approx for wave velocity is, Deformation is on the order of the spacing, scale effects arise – wave dispersion Constitutive behavior changes with scale of deformation Conventional continuum theory is no longer a good approx Validation and Example X • Apply a deformation ‘wave’ to the rod. • Wave propagates along bar. Composite theory (continuum) used to find homogenized modulus Long Wave – homogenized Long Wave with microstructure ZOOM Short Wave – homogenized Short Wave with microstructure ZOOM

49. v1 x1 DOI l1 MICRO Application to SMA composites Maths x0 v0 MACRO Physics Theoretical Material!

50. Motivation for Multi Scale Approach to SH Materials • Capture important microscopic failure and healing mechanisms • Failure of conventional continuum approach – localized micro effects are averaged out