Nano Mechanics and Materials: Theory, Multiscale Methods and Applications
description
Transcript of Nano Mechanics and Materials: Theory, Multiscale Methods and Applications
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