mechanics of 2d materials - unitrento...exercise 4 derive the maximal graphene curvature. for...
TRANSCRIPT
MECHANICS OF 2D
MATERIALS
Nicola Pugno Cambridge
February 23rd, 2015
Outline
โข Stretching Stress
Strain
Stress-Strain curve
โข Mechanical Properties Youngโs modulus
Strength
Ultimate strain
Toughness modulus
โข Size effects on energy dissipated
2
โข Linear Elastic Fracture Mechanics (LEFM) Stress-intensity factor
Energy release rate
Fracture toughness
Size-effect on fracture strength
โข Quantized Fracture Mechanics (QFM) Strength of graphene (and related materials)
โข Bending Flexibility
Bending stiffness
Elastic line equation
Elastic plate equation
3
Exercises by me
1. Apply QFM for deriving the strength of realistic thus defective
graphene (and related 2D materials);
2. Apply LEFM for deriving the peeling force of graphene (and
related 2D materials).
Exercises by you
1. Apply LEFM for measuring the fracture toughness of a sheet
of paper;
2. Apply the Bending theory for calculating the maximal
curvature before fracture.
4
Stretching Fiber under tension
A ๐
F
F
โ๐
2
โ๐
2
Stress: ๐ =๐น
๐ด
Strain: ํ =โ๐
๐
5
Stress-strain curve:
Signature of the material and main tool for deriving its
mechanical properties
ฯ
ฮต
A
C
D B
x
x x
x = failure
If the curve is monotonic, a
force-control is sufficient.
You can derive AB in
displacement control, CD in
crack-opening. The dashed
area is the kinetic energy
released per unit volume
under displacement control.
6
Mechanical properties ฯ
ฮต
x
๐๐๐๐ฅ
ํ๐ข
1
๐ธ ๐ธ
๐
๐๐
๐๐ 0
โก ๐ธ โก Youngโs modulus
๐๐๐๐ฅ = maximum stress โก Strength
ํ๐ข = ultimate strain
๐๐ํ =๐ธ
๐=
๐๐ข
0 Energy dissipated per unit volume or Toughness modulus.
๐ธ = ๐น๐๐ ๐ = ๐ด๐
๐ธ
๐=
๐น
๐ด
๐๐
๐= ๐๐ํ
The toughness modulus is a material
property only for ductile materials.
7
The post-critical behaviour is size-dependent especially for brittle
materials.
ฯ
ฮต
๐ 0
โ
๐ โ 0 ductile
๐ โ โ brittle
Size-effects
8
๐ธ = ๐บ๐๐ด
๐บ๐ โก fracture energy or energy
dissipated per unit area
๐๐ํ โก๐ธ
๐=
๐บ๐๐ด
๐๐ด=
๐บ๐
๐
For brittle materials ๐ธ
๐ has no meaning: instead of the toughness modulus we
have to use the fracture energy.
Fracture
The reality is in between: energy dissipated on a fractal domain:
๐ธ
๐โ ๐ ๐ทโ3 ๐ = ๐
3
D is the fractal exponent, 2 โค ๐ท โค 3
9
10
Fracture Mechanics
ฯ
๐๐ก๐๐
2a r
๐พ๐ผ โก Stress intensity factor
๐๐ก๐๐~๐พ๐ผ
2๐๐
constant
Elastic solution is singular:
We cannot say
๐๐: ๐๐ก๐๐ = ๐๐๐๐ฅ
๐๐ stress at fracture
11
Linear Elastic Fracture Mechanics
(LEFM) Griffithโs approach
๐บ = โ๐๐
๐๐ด ๐ = ๐ธ โ ๐ฟ
Energy release
rate Crack surface
area
Total potential
energy
Stored elastic
energy
External work
Crack propagation criterion:
๐ฎ = ๐ฎ๐ช ๐บ๐ โก fracture energy
๐บ =๐พ๐ผ
๐ธ Elastic solution
๐พ๐ผ only for a function of external load and geometry
๐ฒ๐ฐ = ๐ฒ๐ฐ๐ช ๐พ๐ผ๐ถ = ๐บ๐ถ๐ธ Fracture toughness
12
๐พ๐ผ values reported in stress-intensity factor Handbooks
E.g., infinite plate (i.e. width โซ crack length ~ graphene)
๐พ๐ผ = ๐ ๐๐
Strength of graphene with a crack of length 2๐
๐พ๐ผ = ๐ ๐๐ = ๐พ๐ผ๐ถ
๐๐ =๐พ๐ผ๐ถ
๐๐
Assuming statistically ๐ โ ๐ โก structural size:
๐๐ โ ๐ โ1
2
Size effect on a fracture strength larger is weaker
problem of the scaling upโฆ.
13
Paradox ๐๐ โ โ ๐ โ 0
With graphene pioneer Rod Ruoff we invented Quantized Fracture
Mechanics (2004).
The hypothesis of the continuous crack growth is removed: existence of
fracture quanta due to the discrete nature of matter.
14
Related papers:
N. M. Pugno, โDynamic quantized fracture mechanicsโ, Int. J. Fract, 2006
N. M. Pugno, โNew quantized failure criteria: application to nanotubes and
nanowiresโ, Int. J. Fract, 2006
15
Quantized Fracture Mechanics (QFM)
๐บโ = โโ๐
โ๐ด
Quantized energy
release rate
โ๐ด = fracture quantum of surface area = ๐๐ก
Plate
thickness
Fracture
quantum
of length
๐ฎโ = ๐ฎ๐ช
๐บ = โ๐๐
๐๐ด โ๐ = โ ๐บ๐๐ด = โ
๐พ๐ผ2
๐ธ๐๐ด
๐บโ = โโ๐
โ๐ด=
๐พ๐ผ
2
๐ธ ๐๐ด
โ๐ด ๐บโ = ๐บ๐
1
โ๐ด
๐พ๐ผ2
๐ธ๐๐ด =
๐พ๐ผ๐ถ2
๐ธ
Quantized stress-intensity factor (generalized): ๐พ๐ผโ =
1
โ๐ด ๐พ๐ผ
2 ๐๐ด = ๐พ๐ผ๐ถ
16
Monolayer graphene and examples
Tennis racket made of
graphene (Head ยฉ)
Youngโs modulus โก ๐ธ = 1 TPa
Intrinsic strength ๐๐๐๐ก = 130 GPa
Ultimate strain ํ๐ข = 25 %
C. Lee, X. Wei, J.W. Kysar, J. Hone,
โMeasurement of the elastic properties and intrinsic strength of monolayer grapheneโ, Science, 2008 :
Monolayer graphene hanging on a
silicon substrate (scale bar: 50ยตm) Tensile test on macro samples
of graphene composites
17
Carbon nanotubes
The MWCNTs breaks in the outermost layer
(โsword-in-sheathโ failure),
Youngโs modulus โก ๐ธ =250 to 950 GPa
Tensile strength ฯ = 11 to 63 GPa
Ultimate strain ํ๐ข = 12 %
Min-Feng Yu, Oleg Lourie, Mark J. Dyer, Katerina Moloni, Thomas F. Kelly, Rodney S. Ruoff, โStrength and
Breaking Mechanism of Multiwalled Carbon Nanotubes Under Tensile Loadโ, Science, 2000:
Multiwalled carbon nanotubes (MWCNTs)
The singlewalled carbon nanotubes
(SWCNTs) presents:
Youngโs modulus โก ๐ธ =1000 GPa
Tensile strength ฯ = 300 GPa
Ultimate strain ํ๐ข = 30 %
18
Exercise 1
E.g., infinite plate
๐พ๐ผ = ๐ ๐๐
๐พ๐ผโ =
1
๐๐ก ๐2๐๐๐ก ๐๐
๐+๐
๐
= ๐๐
2๐๐ + ๐ 2 โ ๐2 = ๐ ๐ ๐ +
๐
2
Strength of graphene ๐พ๐ผโ = ๐พ๐ผ๐ถ
๐๐ =๐พ๐ผ๐ถ
๐ ๐ +๐2
๐ โ ๐๐ โ
Unstable crack growth
19
๐ โ 0 Correspondence Principle ๐๐น๐ โ ๐ฟ๐ธ๐น๐
๐ โ 0 ๐๐ = ๐๐๐๐๐๐
๐๐๐๐๐๐ =๐พ๐ผ๐ถ
๐๐2
โน ๐ =2
๐
๐พ๐ผ๐ถ2
๐๐๐๐๐๐2
๐
๐๐๐๐๐๐ ~ 100 GPa
๐๐ =๐พ๐ผ๐ถ
๐ ๐ +๐2
๐พ๐ผ๐ถ ~ 3MPa m
(QFM 2004)
Very close to predictions
by MD and DFT
Very close to experimental
measurement (2014)
๐๐๐๐๐๐ ~ ๐ธ
10 ๐ธ~1 TPa
P. Zhang, L. Ma, F. Fan, Z. Zeng, C. Peng, P. E. Loya, Z. Liu, Y. Gong, J. Zhang, X. Zhang, P. M. Ajayan, T.
Zhu & J. Lou, Fracture toughness of graphene, Nature, 2014
20
Exercise 2 Peeling of graphene
๐
๐ฟ
๐
๐น
๐ธ โ โ ๐ธ โ 0
๐ฟ = ๐ โ ๐ cos ๐
๐บ = โ๐๐
๐๐ด=
๐๐ฟ
๐๐ด ๐ฟ = ๐น๐ ๐๐ด = ๐๐๐
๐บ =๐
๐๐๐๐น๐ 1 โ cos ๐ =
๐น 1 โ cos ๐
๐
๐บ = ๐บ๐ถ โน ๐น๐ถ =๐๐บ๐ถ
1 โ cos ๐
๐น๐ถ strongly dependent on ๐
N. M. Pugno, โThe theory of multiple peelingโ, Int. J. Fract, 2011
21
Exercise 3
Apply LEFM for measuring the fracture
toughness of a sheet of paper with a
crack of a certain length in the middle:
22
Bending theory Bending of beams
โ
๐
๐๐ง
๐ ๐
๐๐๐
๐๐ง
๐๐ฅ ๐๐ฅ
Bending Moment
๐๐๐
2
๐ง ๐ฆ
ํ๐๐ง
2โ ๐ฆ
๐๐๐
2
๐๐ง =๐๐ฅ
๐ผ๐ฅ๐ฆ โ ๐พ๐ฆ ๐๐ฅ = ๐๐ง๐ฆ๐๐๐ฆ
โ2
โโ2
= ๐พ๐ผ๐ฅ
๐๐ฅ =1
๐ ๐ฅ=
๐๐๐ฅ
๐๐ง=
ํ๐ง
๐ฆ=
๐๐ง
๐ธ๐ฆ=
๐๐ฅ
๐ผ๐ฅ๐ธ
๐๐๐๐ฅ =ํ๐ง,๐๐๐ฅ
โ 2
Moment of inertia
๐ผ๐ฅ =๐โ3
12
23
Exercise 4 Derive the maximal graphene curvature.
For graphene โ = ๐ก โ 0.34 ๐๐ โ ๐๐๐๐ฅhuge โ flexibility is more a structural
than a material property
๐๐ฅ = โ๐๐ฃ
๐๐ง โน
๐2๐ฃ
๐๐ง2 = โ๐๐ฅ
๐ผ๐ฅ๐ธ
๐๐ง
M M
q T T
N N
๐๐
๐๐ง= โ๐
๐๐
๐๐ง= ๐
โน ๐4๐ฃ
๐๐ง4 =๐
๐ผ๐ฅ๐ธ
Elastic line equation
Load per unit length
For plates: ๐ป4๐ =๐
๐ท
Elastic plane equation
Load per unit area
Bending stiffness ๐ท =๐ธ๐ก3
12 1 โ ๐2