Scale effect in mechanics of nanostructure objects

1
R.V. Goldstein , V.A.Gorodtsov, N.F.Morozov
Scale effect in mechanics of nanostructure
objects ?????????? ?????? ? ????????
???????????????
A.Yu. Ishlinsky Institute for Problems in
Mechanics, Russian Academy of Sciences,St-Peter
sburg State University goldst_at_ipmnet.ru
Moscow, 2009
2
Contents

• Introduction
• Multiscale nature of atomic structures- Size
  scaling for carbon nanotubes- Single-walled
  nanotubes deformation at tension and compression.
  - Size scaling for atomic layered structures
• Surface stresses and surface elasticity.Scale
  effect
• Size scaling for thin films
• Scale interaction at deformation and fracture-
  Nonlocal elasticity- Cohesive crack models.-
  Macrocrack. Scale separation. Intermediate
  asymptotics- The generalized atomic model of a
  Thomson type crack
• Discrete-continuum approach for scale effects
  modeling
• Conclusion

3
Multiscale nature of atomic structures(Carbon
tube structures)
TEM images of CNTs. Scale a) 200 nm b) 50
nm.(H. Xie. JAP. 2003. V.94)
Carbon nanotubes. Scale 3nm(S.Iijima. Nature.
1991. V.354)
Carbon nanotubes. Scale gt 100 nm (?. ?.
?????????? ? ?. ?. ??????????. ???. 1952. 26. 88
1952)
Carbon nanotube. Scale 25 µm (G.G.Tibbets. APL.
1983. V.42. N.8)
4
Size scaling for carbon nanotubes
d -nanotube diameter
Necessary conditions of CM applicability
Scale effect
Polygonization and collapse of layered
multi-wall nanotubes
Polygonization 1
Full collapse 2
1. WuF.Y., Cheng H.M. J.Phys. 2005. D38.
4302-4307.2. Xiao J. et al. Nanotechnology.
2007. 18. 395703 (7).
5
Single-walled nanotubes deformation at tension
and compression.
Scale effect modeling by molecular dynamics
Interatomic potential - Brenner-Stuart

• generalized Youngs modulus and Poissons ratio
• S surface area of a nanotube, r nanotube
  radius

zigzag
armchair
zigzag
armchair
Starikov S.V., Stegailov V.V., Goldstein R.V.,
Gorodtsov V.A., Chentsov A.V. II Vserossiiskaya
Konferenciya Mnogomasshtabnoe modelirovanie
processov i struktur v nanotehnologiyah. 27 -
29 may, 2009. MIFI, Moscow Sb. Tezisov dokladov.
P.391-392.
6
Single-walled nanotubes deformation at tension
and compression. Molecular dynamics modeling
Nonlinear deformation of single-walled carbon
nanotubes
Different values of the generalized Young modulus
at tension and compression
armchair
zigzag
generalized Youngsmodulus
7
Size scaling for atomic layered structures
Dependencies of the Young modulus and Poisson
ratio for a strip of monocrystal material with
hexagonal close-packed lattice on atomic layers
number (accounting for pairwise atomic
interactions)
Dependence of the Young modulus and Poisson ratio
on atomic layers number
Two-dimensional monocrystal strip
Krivtsov A.M., Morozov N.F. Reps. RAS. 2001.
V.381. N.3. P.825-827.
8
Surface stresses and surface elasticity.Scale
effect
Surface stresses and surface elasticity
Dimensionality difference of bulk and surface
stresses
Characteristic length scale
Scale effect

• Surface stresses are characterized by the surface
  energy and its variation -
  deformation components
• Shuttleworth-Herring formula (an interrelation
  between surface stresses and surface energy)

9
Surface stresses and surface elasticity.Scale
effect
Scale effect
The effective elastic moduli of a finite body are
determined by bulk and surface elastic
characteristics
Example. Elastic isotropic body
- elastic moduli of the body with negligible role
of surface effects
- surface elasticity characteristics
- characteristic size of the body
In dimensionless form for an effective elastic
modulus, E, we have
For relatively large bodies, , the
following expansion is valid
The expansion describes the scale effect, caused
by surface elasticity
10
Surface stresses and surface elasticity.Scale
effect
Scale effect
Similar formulae were obtained for tension,
bending, and torsion of plates and beams of
silicon and aluminum monocrystals Miller R.E.,
Shenoy V.B. (2000) Nanotechnology 11,
139-147, Shenoy V.B. (2002) Int. J. Solids
Struct. 39, 4039-4052.
- ordinary and bending stiffnesses with
and without surface effects
were estimated based on continuum and
atomistic calculations
11
Surface stresses and surface elasticity.Scale
effect
Surface stresses and surface elasticityEffective
stiffness of nanoplates
z
Hookes laws (bulk and surface)
y
- tangential deformation tensor
x
Averaging through the plate thickness h
Effective tensor of moment
Torsion tensor
Vector of rotation
Eremeev V. A., Altenbach, H., Morozov N. F. //
Reps. RAS. 2009. V. 424. N. 5 P. 618620.
12
Size scaling for thin films
Scale effect of deformation characteristics for
thin films
A boundary layer model with the mixture rule
boundary layer
film
-yield stress of the film
substrate

• Assumptions
• A boundary layer with the mechanical properties
  essentially differing from ones of the rest film
  is formed near the film substrate interface at
  deposition.
• The thickness of the boundary layer, , is a
  constant for the given system of film and
  substrate materials (as well as deposition
  technology).
• Estimate of the dependence - mixture
  rule
• (1)
• - values of the yield stress
  for the boundary layer and the rest part of the
  film
• (2)

13
Size scaling for thin films
Scale effect of deformation characteristics for
thin films
A boundary layer model with the mixture rule

• Experimental data for Cu film
  
• Curve
  approximation of the data by Eq. (2)
• with coefficients
  A666 MPa, B h1 27846 MPa nm calculated by
• the least square
  method
• Experimental data for AlCu film
• Curve
  approximation of the data by Eq. (2)
• A329 MPa, B
  h164923 MPa nm

• - Niu R.M., Liu G. et al. Apll. Phys. Lett.
  2007, v.90, p.161907

• - Macionczyk F., Bruckner W. J. Appl. Phys.
  1999, v.86, N9, p.4922-4929

14
Size scaling for thin films
Scale effect of deformation characteristics for
thin films
Comparison with other models of size scaling in
thin films

• Boundary layer
• mixture rule
• Strain gradient plasticity
• (Hutchinson J.W. In
• Materials Science for
  ( - scale value close to the bulk yield
  stress
• the 21st Century. Soc. of
  - characteristic length in the theory
  of strain
• Materials Sci. of Japan,
  gradient plasticity)
• Kyoto, Japan, 2001) v.A, p.307)
• Boundary layer with moduli
• variable through the film
• thickness (Bazant Z., Gruo Z., et al.
  ( h0 length
  parameter
• J. Appl. Phys. 2005, v.97,
  m empirical shape factor
• 073506)
  z - coordinate transverse to and
  referenced from
• 
  the interface)

15
Scale interaction at deformation and fracture
Nonlocal elasticity
(Aifantis E.C.. In Atluri S.N., Fitszerald J.E.
(eds). NSF Workshop on Mechanics of Damage and
Fracture, Georgia Tech. Atlanta. 1982. pp. 1-12.
Aifantis E.C. J. Mater. Engng. Technol. 1984.
106, 326-330.)

• A generalized expression for the potential energy

• Ratios of new stiffness coefficients to regular
  ones reflect characteristic length scale
  parameters and nonlocality of elastic behavior in
  small scales

• Estimates of the critical length scale of
  nonlocal elastic behavior of materials

• Nonlocal (gradient) elasticity is essential in
  the scale 1 nm for crystalline metals and
  semiconductors in scale 10-100 nm for polymers
  (because of an influence of their supramolecular
  structures)

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Scale interaction at deformation and fracture
Cohesive crack models.Macrocrack. Scale
separation. Intermediate asymptotics
d size of the crack end zone with cohesion
forces q(x)
- crack halflength
Condition for critical crack length calculation
- stress intensity factor for mode I crack

• critical stress intensity factor for mode I crack

17
Scale interaction at deformation and fracture
The generalized atomic model of a Thomson type
crack
I-crack zone, II-cohesion zone, III-zone of
linear elastic deformation of the bonds Atomic
interaction within each chain is characterized by
springs having bending stiffness while atomic
interaction between the adjacent chains is
modeled by bilinear tensile springs.
Goldstein R.V., Shatalov G.A. Mechanics of
Solids. 2006. N.4. P.151-164.
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Scale interaction at deformation and fracture
The generalized atomic model of a Thomson type
crack
Here, m the number of broken bonds (the crack
length) s the number of the bonds in the
nonlinear deformation state (the cohesive zone
length) . Evaluation of the cohesive zone size
in dependence on the parameters and the crack
length m510, sm, m1520, s0.10.2 m.
Conclusion Cohesion type crack models of the
linear fracture mechanics can be used for
modeling of nanocracks behavior
Goldstein R.V., Shatalov G.A. Mechanics of
Solids. 2006. N.4. P.151-164.
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Scale interaction at deformation and fracture
Modeling of macrostrength and structure
interconnection. Porous medium
local tension.stable growth of microcracks
unstable coalescence of adjacent microcracks
Goldstein R.V., Ladygin B.M., Osipenko N.M.
Journal of Mining Science. 1974. N.1. P.3-13.
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Scale interaction at deformation and fracture
Nanoscale and scale effect
Nano-viscous and micro-brittle fracture of glasses
Nanopores in the microcrack end zone
Glass Breaks like Metal, but at the Nanometer
Scale Celarie F., et al. Phys. Rev. Lett. 2003.
075504.
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Discrete-continuum approach for scale effects
modeling
Discrete-continuum model of nanotubes and their
systems
A method for accounting for nonlinear interatomic
interactions of Lennard-Joness type and
formation of rod systems, equivalent to nanotubes
and their systems by the deformation energy value
Transition from atomic to rod model
Goldstein R.V., Chentsov A.V. Mechanics of
solids. 2005. V.40. P.4. P. 45-59.
22
Discrete-continuum approach for scale effects
modeling
Development and modeling of experimental schemes
for testing deformation and strength properties
of nano- and microscale objects
Scheme of a tensile testing of carbon nanotube
embedded in the polymer matrix
Outer problem
First inner problem
Second inner problem
Scheme of nanotube loading at its pulling out of
the polymer matrix
Continuum modeling
Discrete modeling
Goldstein R.V., Osipenko N.M., Chentsov A.V.
Mechanics of Solids. 2008. ?3
23
Discrete-continuum approach for scale effects
modeling
Development and modeling of experimental schemes
for testing deformation and strength properties
of nano- and microscale objects
Scheme of a tensile testing of carbon nanotube
embedded in the polymer matrix
The initial state of the nanotube, partially
inserted into a matrix, and its shape after the
loading has been applied are given in the figure.
One can note, that in the zone of nanotube-matrix
interaction the nanotube has greatest diameter in
the face section of the inserted part and this
influence of adhesive bonding extends onto a free
part of the nanotube, adjoining to the matrix
surface. The opposite effect is observed in the
zone of other face section where loading is
applied.
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• Conclusion
• Scale effects of deformation characteristics of
  thin-walled tubular structures and thin films was
  demonstrated
• The role of the scale effect in a transition
  from a discrete to continuum description of
  nanostructured objects was confirmed
• Inapplicability of traditional elasticity
  characteristics for description of thin
  nanostructures was shown
• Leading role of the nanoscale in formation of
  fracture toughness of nanostructured objects was
  clarified
• A possibility of scale effect description within
  the framework of elasticity with surface stresses
  was analyzed. The role of the characteristic
  length scale determined by the ratio of surface
  and bulk Youngs moduli was demonstrated.
  Universality of the dependences of the mechanical
  characteristics scaling on the ratio of the
  characteristic nanoscale to the scale being
  analyzed was emphasized
• An influence of the surface and interface
  boundary layers in the scale effects was analyzed
  
• A discrete continuum modeling of scale effects
  in mechanical behavior of nanostructures was
  suggested