Nano Mechanics and Materials:Theory, Multiscale Methods and Applications

byWing Kam Liu, Eduard G. Karpov, Harold S. Park

9. Multiscale Methods for Material Design

Material response (Ductility, Strength, Corrosion & Fatigue Resistance) is controlled by nano and micro mechanisms

Ductile to Brittle Transformation causes a hull to fracture

Fatigue Fracture of firefighter’s ladder

Environmental factors lead to fracture of a gas pipeline

Why is Multi-scale Material Design Important?

http://www.engr.sjsu.edu/WofMatE/FailureAnaly.htm

Why is Multi-scale Material Design Important?

We can usually design against failure mechanisms in some structural components by

(a) Using more material

(b) 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.

A macroscale demonstrationCourtesy of Yip Wah ChungSlide taken from his opening remark on theShort course, entitled “Nano scale design of materials”given at the NSF Summer Institute on Nano Mechanics and MaterialsNorthwestern University, August 25, 2003.

Why is Multi-scale Material Design Important?

Example of a Micro Mechanism in an Alloy

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 void coalescence 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.

Ductile fracture by void nucleation, growth and coalescence.

Nucleation at secondary particles along a shear band

Horstemeyer et al. 2000

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

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

Processing

Structure

Properties

Performances

Cause and effects

Goals/means

(Olson 1997)

9.1 Multiresolution Continuum Analysis

Multi-scale Theory for Three-Scale Material

0 0 /ij i jL v x 1ijL 2

ijL

Physicaldomains

x0

v0

x1 v1

1

x2 v2

2

Mathematicaldomains

Micro unit cellSub-micro unit cell

Macroscopic domain

Deformation measure

Multi-scale decomposition of material

1. Statistically homogeneous structure => unit cell at each scale (smallest representative element)

2. Expansion of velocity in unit cells => characteristic rate of deformation of a unit cell

Decomposition of the deformation and stress measure in the micro cell:

Stress and internal power decomposition for a two-scale material

0 0 1 1 0int : :p σ L β L L

• Total macro and micro stresses :0 0 , σ L

Macro RVE

0 x

1 0 1 σ σ β

Due to macro deformationDue to micro deformation

Homogenized internal power?

1 0 1 0 L L L L

The internal power of a unit cell is:

Micro unit cell

1 1 , σ L

• Homogenized internal power is the average over a domain :

Averaging operation

1int intp p

1

1

1 1

β β

1

1 1

β β y

where 1

0 0 1 1 0 1int : :p

Lσ L β L L β

x

1

x

1

11 1

LL L y

x

• Linear expansion of the micro deformation in this domain

1L

Linear variation of 1L

1y

2y• Averaging domain captures microstructure interactions

For a three-scale material: Three stresses : conjugate to

Two averaging domains and

Generalization

0 0int : d p σ D 1 1 0 : d β L L

11 d

L

βx

2 2 1 : d β L L2

2 d

Lβ

x

Microcomponent

Sub-microcomponent

Macrocomponent

20 1σ ,β ,β

2

1

1 1

β β

1

1 1

β β y

2

2 2

β β

2

2 2

β β y

=> Good for cell models

1 0 2 1 0D ,L L ,L L

1

Where the stresses are defined as follows:

Constitutive relation Define generalized stress and strain:

0 1 1 2 2 Σ σ β β β β

0 1 0 1 2 1 2/ / D L L L x L L L x

:ep Σ C

Hypo-elasticity Plasticity / damage

:e e Σ C

, 0Q

Generalized Yield function/plastic potential

0 0

1 1 0

1 1

2 2 1

22

0 0 0 0

0 0 0 0

0 0 0 0 /

0 0 0 0

0 0 0 0 /

e

σ Dβ L L

β L x

β L L

L xβ

Q Internal variables

How to find the constitutive relation and material constants?

e p

intp Σ

• Constitutive relation

Example: Granular material

Internal power density

0 0 /ij i jL v x 1 1ij ijL W 2 1

ij ijL L

Macro velocity gradient Micro velocity gradient = micro spin

Sub-micro velocity gradient

0 0int : d p σ D 1 1 0 : d β W W

11 d

W

βx

Cosserat material

Granular material: constitutive relation

Plasticity

2, 3 0yJ

Generalized Yield function/plastic potential : Generalized J2 flow theory

0 0 0

1 1 0

11 /

e

c

c

C

B

σ D

β W W

W xβ

20 0 1 1 1 1 12 1 2 3

1 12 0 0 1 0 1 0 3

21 2 1

: :

: :

J a a a

bb b

s s β β β β

W WL L W W W W

x x

Material constants

Elasticity

0 0 0

1 1 0

:

c W W

σ x C D

x + y

At the micro-scale

Average in the

averaging domain

1ij i jB y y

Goal : Determination of the constants 1 2 3 1 2 3, , , , ,a a a b b b

Is defined as an average of the slip s(x+y) measure in

1

0 1,s sx + y D x W x + y

1

1 1

W

W x y W x x yx

Using the linear variation of in the averaging domain

1W

1 2 3

1 2 3

2 4 43 3 3

1 1 12 4 4

Bb b b

a a a B

Perform averages and get material constants

Remark:. In this analytical derivation, we chose (still empirical but can be determined by a more accurate physical model)

Granular material: Material constants

1ij i jB y y

Kadowaki(2004)

1 110 5R

Example 2 : Deformation theory of Strain Gradient Plasticity

Internal power density

0 0 /ij i jL v x 1ij 2 1

ij ij

Macro velocity gradient Micro strain Sub-micro strain

0 0int : d p σ ε

11 d

ε

βx

Assumption : only gradients play a role in the internal power :Set the macro averaging domain equals to the micro averaging domain 1 0ε ε

Strain gradient plasticity: constitutive relation•Taylor relation at the micro scale

1 1σ σ x + y where is the stress in the averaging domain. The same equation can written in the form

1ij

ijk x

•From mechanistic models (bending , torsion, void growth), Gao found an expression for theequivalent strain gradient

•The constitutive relation at small scale follows the deformation theory of plasticity:

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)

Strain gradient plasticity : constitutive relation

1

1 1

ε

ε x y ε x x yx

1

Cell Modeling1

1 1

β β

1

1 1

β β y

1 1σ β

1 1 σ y β

Apply strain boundary conditions

1

1

L

L

x

Total micro stress

Micro stresses are averages over averaging domains:Cell model of the averaging domain at each scale

Curve fitting of the generalized potential Determination of the elastic matrix for each scale

, , F

• Periodic BC

We ensure that periodicity is preserved

n

Cell modeling - Successes

Rolling: Edge cracks

Extrusion: Central Bursting

ExperimentalSimulation

Industrial Applications :

•Prediction of edge cracking during rolling

•Prediction of central bursting during extrusion

Computer based material law

Fitting of material Constants in and C

Micro cell model

1 1σ β

1 1 σ y β

Strain boundary conditions1

1

L

L

x

Total micro stress

Macro cell model

0 0 σ

Strain boundary conditions

0L

macro stress

0σ σ

Sub-micro cell model

2 2σ β

2 2 σ y β

Strain boundary conditions2

2

L

L

x

Total micro stress

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

Periodic boundary conditions

Continuum accounting for damage from

secondary particles

Primary particles

Fractureby void sheet

Shear bands forming during a ballistic impact ( Cowie, Azrin, Olson 1988)

Shear strain

She

ar s

tres

s

instability

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

9.2 Multiscale Constitutive Modeling of Steels

a) quantum scale

b) sub-micro scale

d) macro-scale c) micro-scale

TiC

ijij ,

micij

micij E,

,...E micijm

micij

ijij E,

,...E ijij TiN

micij

micij E,

Review of Multiscale Structure of Steel

Micrograph of high strength steel

Ultra High Strength SteelsMicrostructure of steel Two levels of particles : primary and secondary ( three-scale micromorphic

material)

Macro scale

Micro scale

Sub-micro scale

primary particles

secondary particles

Fracture surface

• 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

Multi-scale Nature of a Steel Alloy

6 510 10 m 1010 m7 610 10 m

310 m

Macro-scale

Micro-scale

Sub micro-scale withtheromodynamics and mathematical model uncertainties

Quantum scale with atomic lattice uncertainties

Primary Particles

Dislocations

scale

Scales consideredfor concurrent model

Mat

rix/

part

icle

bon

ding

Multi-level decomposition of the structure of steel Secondary

Particles

Predictive Multiscale Mathematical Models

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

Goals:

Why Start from the Atomic-Electronic Scale?Thomas-Fermi ModelThomas-Fermi Model

N,CTi

nuclei

electron gas

Quantum Theory: Quantum Theory: (e.g. One particle Schödinger Eqt)(e.g. One particle Schödinger Eqt)

iii EVm

2

2

2

m: mass; V: potentiali: ith eigenfunction Ei:ith eigenvalue

Continuum mechanicsContinuum mechanics

ijij b,

Force and displacementboundary condition

?

COD

TSD Diagram for Steel Design (Toughness-Strength-Decohesion Energy Diagram)

TiN

Ti2CS

TiC

2MgS

Cybersteel: Cell Modeling

0 0 /ij i jL v x 1ijL

2ijL

Macro velocity gradient Micro velocity gradient Sub-micro velocity gradient

0 0int : d p σ D 1 1 0 : d β L L

11 d

L

βx

2 2 1 : d β L L2

2 d

Lβ

x

• Internal power density

Macro

Micro

Sub-micro

Cybersteel: Constitutive relation

Elasticity

00 0

11 1 0

11 1

2 2 12

222

/

/

e

Cσ D

Cβ L L

Bβ L x

β L LCL xβ B

Elastic constantsTo be determined

Plasticity

20

2 23, , 1 cosh 0m

y y

JF F F

Generalized Yield function/plastic potential : Multiscale Gurson model

Material constants

Goal => determination of these material constants through cell modeling a each scale

13 constants

+ equation of evolution of void volume fraction F with stress and strain

0 0 1 1 1 1 2 2 2 23 52 22 1 2 41 2

1 1 2 22 20 0 1 0 1 0 1 2 1 2 1 1

1 2 3 4 5

0 1 0 2 11 2 3

: : :

: : :

a aJ a a a

b b b b b

F c f c f f c f f

s s β β β β β β β β

L L L LL L L L L L L L L L

x x x x

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.

Vision

9.3 Bio-Inspired Materials

Bio-Inspired Self Healing Materials – Multiscale Nature

Origin: Biomimesis - the study and design of high-tech products that mimic biological systems

Goals:

“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).

• 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.

Background

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.

Bioinspired SMA self-healing composite with bone shaped SMA inclusions:

• Composite with SMA bone shaped inclusions

Loading

Heating

• Crack propagation, inclusion transformation, interfacial debonding, crack halting and energy dissipation

Bio-Inspired Self Healing Materials

Prof. Olson group on SH composite

• Healed composite with some change of chrystallography of affected inclusions and crack closure.

Shape Memory Alloys - Basics

www.msm.cam.ac.uk/phase-trans/2002/memory.movies.html

• 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

high-temp phase

austenite

low-temperature phasemartensite

twinned detwinned

Cubic Crystal Monoclinic Crystal

http://smart.tamu.edu/overview/smaintro/simple/pseudoelastic.html

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

Phase Transformation (Temp + Load)

1. Apply a load to TWINNED martensite – get DETWINNED martensite (SHAPE CHANGE)

2. Unload – deformation remains

3. Heat – Reverse transformation

Phase Transformation (Temp + Load)

Assuming a linear relationship between applied load and transformation temperature

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

1-D SMA Constitutive Law

M A

A MSMA transformations

As AfT

a

b

Flow stress = g ( Fraction of Martensite)

Fraction of Martensite = f (a/b)

*1-d constitutive law from Prof. Brinson’s Group at Northwestern

Bone Shaped Inclusions

Weak bonding

Strong bonding is not effective

1. Crack energy dissipated through anchoring effect of BRIDGING inclusions

2. Inclusions are stretched – phase transformation occurs (A-M)

3. Heat, (M-A) original shape regained. Crack closes

4. Use pre-strained inclusions – significant detwinned martensite

5. Clamping at high temp – partial re-welding of fracture surface

SMA inclusion

Brittle Matrix

Bridging

Validation and Example

X

t

•Apply a deformation ‘wave’ to the rod.

•Wave propagates along bar.

Long Wave – homogenized

Short Wave – homogenized

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

A homogenized continuum approx for wave velocity is, /Evw

Long Wave with microstructure

Short Wave with microstructure

ZOOM

ZOOM

X

Composite theory (continuum) used to find homogenized modulus

Application to SMA composites

x0v0

Maths

Physics

MACRO

x1 v1

l1

MICRO

DOI

Theoretical Material!

• Capture important microscopic failure and healing mechanisms

• Failure of conventional continuum approach – localized micro effects are averaged out

Motivation for Multi Scale Approach to SH Materials