Nano Mechanics and Materials:
Theory, Multiscale Methods and Applications
by
Wing Kam Liu, Eduard G. Karpov, Harold S. Park
9. Multiscale Methods for Material Design
Why is Multi-scale Material Design Important?
Fatigue Fracture of
Ductile to Brittle Transformation causes a firefighter’s ladder
hull to fracture
http://www.engr.sjsu.edu/WofMatE/FailureAnaly.htm
Environmental
factors lead to
fracture of a gas
pipeline
Material response (Ductility, Strength, Corrosion & Fatigue
Resistance) is controlled by nano and micro mechanisms
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.
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.
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 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.
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
Performances
Goals/means
Properties
Structure
Cause and effects
Processing
(Olson 1997)
9.1 Multiresolution Continuum Analysis
Multi-scale Theory for Three-Scale Material
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
Sub-micro unit cell
Macroscopic domain
Mathematical
domains
Micro unit cell
1
x1 v
x0
v0
1
2
x2 v
2
Physical
domains
Deformation
measure
L  vi / x j
0
ij
0
L1ij
L2ij
Stress and internal power decomposition
for a two-scale material
• Total macro and micro0 stresses
:
0
σ ,L
σ1 , L1
Macro RVE
x0
Micro unit cell

Decomposition of the deformation and stress measure in the micro cell:
L1  L0   L1  L0 
σ1  σ 0  β1
Due to micro deformation
Due to macro deformation
0
0
1
1
0
The internal power of a unit cell is: pint  σ : L  β :  L  L 
Homogenized internal power?
Averaging operation
• Homogenized internal power is the average over a domain
pint  pint
1
:
1
• Averaging domain captures microstructure interactions
y2
Linear variation of L1
1

y1
x
L1
1
• Linear expansion of the micro deformation in this domain
 L1 
L  L 
y
 x 
1
1
 L 
pint  σ : L  β :  L  L   β 


x


β1  β1
1
0
0
1
1
0
1
where
1
β1  β1  y
1
Generalization

For a three-scale material:
0
1
2
 Three stresses : σ , β , β
conjugate to

 pint
Two averaging domains

1
and

D0 , L1  L0 , L2  L1
2
 L1 
  σ :  D d  β :   L  L  d  β  
 d



x


0
0
Macro
component
1
1
0
1
 L2
  β :  L  L  d   β  
 x
2
2
1
2

 d

Micro
component
Sub-micro
component
Where the stresses are defined as follows:
β1  β1
β 2  β2

β1  β1  y
1
1
=> Good for cell models
2
β 2  β2  y
2
Constitutive relation
Define generalized stress and strain:

Σ  σ0

β1 β1 β 2
   D0
L1  L0
β2 

 pint  Σ  
 L / x 
L2  L1
1
• Constitutive relation Σ  Cep : 
 L / x 
2
  e   p
Plasticity / damage
Hypo-elasticity
Σ   Ce :  e
0 
 σ   
 1 
β   0
 1  0
β  
 β2   0
 2 
β   0
 
0

0
0
0
0
0

0
0

0
0
0
0
0  D 


0   L1  L0 
0   L1 / x 


0   L2  L1 
   L2 / x 
0
e
Generalized Yield function/plastic potential
  , Q   0
Q Internal variables
How to find the constitutive relation and material constants?
Example: Granular material
L0ij  vi 0 / x j
Macro velocity gradient

L2ij  L1ij
L1ij  Wij1
Sub-micro velocity
gradient
Micro velocity gradient
= micro spin
Internal power density
 pint
 W1 
  σ :  D d  β :   W  W  d   β  
 d


 x 
0
0
1
Cosserat material
1
0
1
Granular material: constitutive relation
Elasticity
At the micro-scale

σ 0  x   C0 : D0
 σ0   C 0
 1 
c
β  
 1 
β  
Average in the
 1  x + y   c W 1  W 0 
averaging domain
Bij  yi y j
Plasticity
 D0 
 1
0
W

W


1

Bc 
 W / x 
e
1
Generalized Yield function/plastic potential : Generalized J2 flow theory
  ,    3 J 2   y     0
J 2  a1s 0 : s 0  a2 β1 : β1  a3 
  b1L : L  b2  W  W
2
0
0
1
0

1 2
Material constants
β1 β1
 : W
1
W
0

b3
 
1 2
W1 W1
x
x
Granular material: Material constants
Goal : Determination of the constants
a1 , a2 , a3 , b1 , b2 , b3
 Is defined as an average of the slip s(x+y) measure in 1
s  x + y   s  D0  x  , W1  x + y  
Using the linear variation of
W1 in the averaging domain
1

W
W1  x  y   W1  x  
x y
x
Perform averages and get material constants
b1  2
b2  4
b3  4 B
3
3
3
a1  1 a2  1 a3  1
2
4
4B
Kadowaki(2004)
Remark:. In this analytical derivation, we chose
determined by a more accurate physical model)
1
Bij  yi y j
1
 10 R  5 1(still empirical but can be
Example 2 : Deformation theory of
Strain Gradient Plasticity
 ij1
L0ij  vi 0 / x j
 ij2   ij1
Micro strain
Macro velocity gradient
Sub-micro strain
Assumption : only gradients play a role in the internal power :
Set the macro averaging domain equals to the micro averaging domain

Internal power density
 pint 
σ
0
:  ε d
0
 ε1 
 β  
 d
 x 
1
ε1  ε0
Strain gradient plasticity: constitutive relation
•Taylor relation at the micro scale
where σ1  σ1  x + y  is the stress in the averaging domain. The same equation can
written in the form
ijk 
 ij1
x
•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:
Strain gradient plasticity : constitutive relation
Linear variation of the micro strain field in the averaging domain
1

ε
ε1  x  y   ε 1  x    x  y
x
The microscopic constitutive relation is averaged in
1
:
=> Same result as mechanism based strain gradient plasticity (Gao & al)
Cell Modeling
β1  β1
Micro stresses are averages over averaging domains:
n
Cell model of the averaging domain at each scale 
1
β1  β1  y
Apply strain boundary conditions
Total micro stress
1
L
σ1  β1
 L1 


 x 
σ1  y  β1

Curve fitting of the generalized potential  , , F
Determination of the elastic matrix for each scale
• Periodic BC
We ensure that periodicity is preserved

1
Cell modeling - Successes
Industrial Applications :
•Prediction of edge cracking during rolling
•Prediction of central bursting during extrusion
Experimental
Rolling: Edge cracks
Extrusion: Central Bursting
Simulation
Computer based material law
Macro cell model
Strain boundary
conditions
0
L
macro stress
 0 σ0 
σ  σ0
Micro cell model
Strain boundary conditions
Total micro stress
L1
σ1  β1
 L1 


 x 
σ1  y  β1
Sub-micro cell model
Strain boundary conditions
Total micro stress
L2
σ2  β2
 L2 


 x 
σ2  y  β2
Fitting of material
Constants in  and
C
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
Primary
particles
Shear bands forming during a
ballistic impact ( Cowie, Azrin, Olson 1988)
Continuum
accounting for
damage from
secondary particles
Fracture
by void sheet
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)
Shear stress
instability
Shear strain
simulation
Experiment
9.2 Multiscale Constitutive Modeling of Steels
Review of Multiscale Structure of Steel
 ijmic , Eijmic
 ij , ij
b) sub-micro scale
TiC
Micrograph of high strength steel


E ijmic   m  ijmic ,...
ij , Eij


TiN
Eij   ij ,...
d) macro-scale
a) quantum scale
 ijmic ,
c) micro-scale
Eijmic
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
Multi-scale Nature of a Steel Alloy
Multi-level decomposition
of the structure of steel
Secondary
Particles
Primary
Particles
Dislocations
Macro-scale
Micro-scale
Scales considered
for concurrent model
Quantum scale with
atomic lattice uncertainties
1010 m
Sub micro-scale with
theromodynamics and
mathematical model uncertainties
107  106 m
scale
106  10 5 m
 103 m
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
Why Start from the Atomic-Electronic Scale?
Thomas-Fermi Model
electron gas
Ti
N,C
Quantum Theory:
(e.g. One particle Schödinger Eqt)
 2 2

 
  V  i  Ei i
 2m

m: mass; V: potential
i: ith eigenfunction
Ei:ith eigenvalue
nuclei
?
Continuum mechanics
 ij, j  bi
Force and displacement
boundary condition
TSD Diagram for Steel Design
(Toughness-Strength-Decohesion Energy Diagram)
2
MgS
TiN
Ti2CS
TiC
COD
Cybersteel: Cell Modeling
L0ij  vi 0 / x j
L2ij
L1ij
Macro velocity gradient
Micro velocity gradient
Sub-micro velocity gradient
• Internal power density
 pint
 L1 
  σ :  D d  β :   L  L  d  β  
 d



x


0
Macro
0
1
1
0
1
 L2
  β :  L  L  d   β  
 x
2
2
1
2

 d

Micro
Sub-micro
Cybersteel: Constitutive relation
Elasticity
0 
σ 
 1
β 
 1
β 
 β2 
 2
β 
 
  C0 

e



  D0 

 1 0 
C1 

 L L 
  L1 / x 
B1 


 2 1 
2

 L L 
C 

  L2 / x 



B 2  


Elastic constants
To be determined
Plasticity
Generalized Yield function/plastic potential : Multiscale Gurson model
2
 3J 2 
 0 
m
  , , F   
  1  F cosh 
 F2  0
 y   
 y   




J 2  a1s0 : s0  a2 β1 : β1 
a3
 
1 2
β1 β1  a4 β 2 : β 2 
  b1L : L  b2  L  L  :  L  L   b3 
0
0
1
0
1
0
F  c1 f 0  c2  f 1  f 0   c3  f 2  f 1 

1 2
a5
Material constants
 
2 2
β2 β2
L1 L1
 b4  L2  L1  :  L2  L1   b5 
x x

1 2
L2 L2
x x
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
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.
9.3 Bio-Inspired Materials
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.
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.
Bio-Inspired Self Healing Materials
Bioinspired SMA self-healing composite
with bone shaped SMA inclusions:
• Composite with SMA bone shaped
inclusions
Loading
Heating
Prof. Olson group on SH composite
• Crack propagation,
inclusion
transformation,
interfacial debonding,
crack halting and
energy dissipation
• Healed composite with some change
of chrystallography of affected
inclusions and crack closure.
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
high-temp
phase
austenite
low-temperature phase
martensite
twinned
detwinned
www.msm.cam.ac.uk/phasetrans/2002/memory.movies.h
tml
http://smart.tamu.edu/over
view/smaintro/simple/pseu
doelastic.html
Cubic Crystal
Monoclinic Crystal
Phase Transformation (Temp only)
• Phase transformation occurs between these two phases upon
heating/cooling
NO SHAPE CHANGE
http://smart.tamu.edu/overview/sm
aintro/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

A
SMA
transformations
M
M
A
b
a
As
Af
T
Fraction of Martensite = f (a/b)
Flow stress = g ( Fraction of Martensite)
*1-d constitutive law from Prof. Brinson’s Group at Northwestern
Bone Shaped Inclusions
Bridging
Brittle Matrix
SMA inclusion
Strong bonding
is not effective
Weak bonding
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
Validation and Example
X
X
•Apply a deformation ‘wave’ to the rod.
t
•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
A homogenized continuum approx for wave velocity is, vw  E / 
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
Application to SMA composites
Maths
x1
x0
v1
v0
l1
MACRO
Physics
MICRO
Theoretical Material!
DOI
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