The size effect of nanoporous materials is generally believed to be caused by the large ratio of surface area to volume, so that it is also called surface effect. Based on a recently developed elastic theory, in which the surface effect of nanomaterials is characterized by the surface energy density, combined with two micromechanical models of composite materials, the surface effect of nanoporous materials is investigated. Closed-form solutions of both the effective bulk modulus and the effective shear one of nanoporous materials are achieved, which are related to the surface energy density of corresponding bulk materials and the surface relaxation parameter of nanomaterials, rather than the surface elastic constants in previous theories. An important finding is that the enhancement of mechanical properties of nanoporous materials mainly results from the compressive strain induced by nanovoid's surface relaxation. With a fixed volume fraction of nanovoids, the smaller the void size, the harder the nanoporous material will be. The results in this paper should give some insights for the design of nanodevices with advanced porous materials or structures.

References

References
1.
Lew
,
K. K.
, and
Redwing
,
J. M.
,
2003
, “
Growth Characteristics of Silicon Nanowires Synthesized by Vapor–Liquid–Solid Growth in Nanoporous Alumina Templates
,”
J. Cryst. Growth
,
254
(
1–2
), pp.
14
22
.
2.
Wittstock
,
A.
,
Biener
,
J.
, and
Baumer
,
M.
,
2010
, “
Nanoporous Gold: A New Material for Catalytic and Sensor Applications
,”
Phys. Chem. Chem. Phys.
,
12
(
40
), pp.
12919
12930
.
3.
Gowda
,
S. R.
,
Reddy
,
A. L. M.
,
Zhan
,
X. B.
,
Jafry
,
H. R.
, and
Ajayan
,
P. M.
,
2012
, “
3D Nanoporous Nanowire Current Collectors for Thin Film Microbatteries
,”
Nano Lett.
,
12
(
3
), pp.
1198
1202
.
4.
Xia
,
R.
,
Li
,
X. D.
,
Qin
,
Q. H.
,
Liu
,
J. L.
, and
Feng
,
X. Q.
,
2011
, “
Surface Effects on the Mechanical Properties of Nanoporous Materials
,”
Nanotechnology
,
22
(
26
), p.
265714
.
5.
Biener
,
J.
,
Hodge
,
A. M.
,
Hayes
,
J. R.
,
Volkert
,
C. A.
,
Ruiz
,
L. A. Z.
,
Hamza
,
A. V.
, and
Abraham
,
F. F.
,
2006
, “
Size Effects on the Mechanical Behavior of Nanoporous Au
,”
Nano Lett.
,
6
(
10
), pp.
2379
2382
.
6.
Goudarzi
,
T.
,
Avazmohammadi
,
R.
, and
Naghdabadi
,
R.
,
2010
, “
Surface Energy Effects on the Yield Strength of Nanoporous Materials Containing Nanoscale Cylindrical Voids
,”
Mech. Mater.
,
42
(
9
), pp.
852
862
.
7.
Wang
,
J. X.
,
Huang
,
Z. P.
,
Duan
,
H. L.
,
Yu
,
S. W.
,
Feng
,
X. Q.
,
Wang
,
G. F.
,
Zhang
,
W. X.
, and
Wang
,
T. J.
,
2011
, “
Surface Stress Effect in Mechanics of Nanostructured Materials
,”
Acta. Mech. Solida Sin.
,
24
(
1
), pp.
52
82
.
8.
Cheng
,
I. C.
, and
Hodge
,
A. M.
,
2013
, “
Strength Scale Behavior of Nanoporous Ag, Pd and Cu Foams
,”
Scr. Mater.
,
69
(
4
), pp.
295
298
.
9.
Mathur
,
A.
, and
Erlebacher
,
J.
,
2007
, “
Size Dependence of Effective Young's Modulus of Nanoporous Gold
,”
Appl. Phys. Lett.
,
90
(
6
), p.
061910
.
10.
Luhrs
,
L.
,
Soyarslan
,
C.
,
Markmann
,
J.
,
Bargmann
,
S.
, and
Weissmuller
,
J.
,
2016
, “
Elastic and Plastic Poisson's Ratios of Nanoporous Gold
,”
Scr. Mater.
,
110
, pp.
65
69
.
11.
Gurtin
,
M. E.
, and
Murdoch
,
A. I.
,
1975
, “
A Continuum Theory of Elastic Material Surfaces
,”
Arch. Ration. Mech. Anal.
,
57
(
4
), pp.
291
323
.
12.
Feng
,
X. Q.
,
Xia
,
R.
,
Li
,
X. D.
, and
Li
,
B.
,
2009
, “
Surface Effects on the Elastic Modulus of Nanoporous Materials
,”
Appl. Phys. Lett.
,
94
(
1
), p.
011916
.
13.
Liu
,
D. J.
,
Xie
,
Y. M.
,
Li
,
Q.
,
Huang
,
X. D.
, and
Zhou
,
S. W.
,
2014
, “
Towards Ultra-Stiff Materials: Surface Effects on Nanoporous Materials
,”
Appl. Phys. Lett.
,
105
(
10
), p.
101903
.
14.
Sun
,
X. Y.
,
Xu
,
G. K.
,
Li
,
X. Y.
,
Feng
,
X. Q.
, and
Gao
,
H. J.
,
2013
, “
Mechanical Properties and Scaling Laws of Nanoporous Gold
,”
J. Appl. Phys.
,
113
(
2
), p.
023505
.
15.
Yang
,
F. Q.
,
2004
, “
Size-Dependent Effective Modulus of Elastic Composite Materials: Spherical Nanocavities at Dilute Concentrations
,”
J. Appl. Phys.
,
95
(
7
), pp.
3516
3520
.
16.
Duan
,
H. L.
,
Wang
,
J.
,
Huang
,
Z. P.
, and
Karihaloo
,
B. L.
,
2005
, “
Size-Dependent Effective Elastic Constants of Solids Containing Nano-Inhomogeneities With Interface Stress
,”
J. Mech. Phys. Solids
,
53
(
7
), pp.
1574
1596
.
17.
Duan
,
H. L.
,
Wang
,
J.
,
Karihaloo
,
B. L.
, and
Huang
,
Z. P.
,
2006
, “
Nanoporous Materials Can Be Made Stiffer Than Non-Porous Counterparts by Surface Modification
,”
Acta Mater.
,
54
(
11
), pp.
2983
2990
.
18.
Ouyang
,
G.
,
Yang
,
G. W.
,
Sun
,
C. Q.
, and
Zhu
,
W. G.
,
2008
, “
Nanoporous Structures: Smaller is Stronger
,”
Small
,
4
(
9
), pp.
1359
1362
.
19.
Mogilevskaya
,
S. G.
,
Crouch
,
S. L.
,
Grotta
,
A. L.
, and
Stolarski
,
H. K.
,
2010
, “
The Effects of Surface Elasticity and Surface Tension on the Transverse Overall Elastic Behavior of Unidirectional Nanocomposites
,”
Compos. Sci. Technol.
,
70
(
3
), pp.
427
434
.
20.
Kushch
,
V. I.
,
Mogilevskaya
,
S. G.
,
Stolarski
,
H. K.
, and
Crouch
,
S. L.
,
2013
, “
Elastic Fields and Effective Moduli of Particulate Nanocomposites With the Gurtin–Murdoch Model of Interfaces
,”
Int. J. Solids Struct.
,
50
(
7–8
), pp.
1141
1153
.
21.
Qiang
,
F. W.
, and
Wei
,
P. J.
,
2015
, “
Effective Dynamic Properties of Random Nanoporous Materials With Consideration of Surface Effects
,”
Acta Mech.
,
226
(
4
), pp.
1201
1212
.
22.
Miller
,
R. E.
, and
Shenoy
,
V. B.
,
2000
, “
Size-Dependent Elastic Properties of Nanosized Structural Elements
,”
Nanotechnology
,
11
(
3
), pp.
139
147
.
23.
Shenoy
,
V. B.
,
2005
, “
Atomistic Calculations of Elastic Properties of Metallic FCC Crystal Surfaces
,”
Phys. Rev. B
,
71
(
9
), p.
094104
.
24.
Mi
,
C. W.
,
Jun
,
S.
,
Kouris
,
D. A.
, and
Kim
,
S. Y.
,
2008
, “
Atomistic Calculations of Interface Elastic Properties in Noncoherent Metallic Bilayers
,”
Phys. Rev. B
,
77
(
7
), p.
075425
.
25.
Chen
,
S. H.
, and
Yao
,
Y.
,
2014
, “
Elastic Theory of Nanomaterials Based on Surface Energy Density
,”
ASME J. Appl. Mech.
,
81
(
12
), p.
121002
.
26.
Yao
,
Y.
, and
Chen
,
S. H.
,
2015
, “
Surface Effect on Resonant Properties of Nanowires Predicted by an Elastic Theory for Nanomaterials
,”
J. Appl. Phys.
,
118
(
4
), p.
044303
.
27.
Yao
,
Y.
,
Wei
,
Y. C.
, and
Chen
,
S. H.
,
2015
, “
Size Effect of the Surface Energy Density of Nanoparticles
,”
Surf. Sci.
,
636
, pp.
19
24
.
28.
Yao
,
Y.
, and
Chen
,
S. H.
,
2016
, “
Surface Effect in the Bending of Nanowires
,”
Mech. Mater.
,
100
, pp.
12
21
.
29.
Yao
,
Y.
, and
Chen
,
S. H.
,
2016
, “
Buckling Behavior of Nanowires Predicted by a New Surface Energy Density Model
,”
Acta Mech.
,
227
(
7
), pp.
1799
1811
.
30.
Nix
,
W. D.
, and
Gao
,
H.
,
1998
, “
An Atomic Interpretation of Interface Stress
,”
Scr. Mater.
,
39
(
12
), pp.
1653
1661
.
31.
Sun
,
C. Q.
,
2003
, “
Oxidation Electronics: Bond–Band–Barrier Correlation and Its Applications
,”
Prog. Mater. Sci.
,
48
(
6
), pp.
521
685
.
32.
Shi
,
M. X.
,
Liu
,
B.
,
Zhang
,
Z. Q.
,
Zhang
,
Y. W.
, and
Gao
,
H. J.
,
2012
, “
Direct Influence of Residual Stress on the Bending Stiffness of Cantilever Beams
,”
Proc. R. Soc. A
,
468
(
2145
), pp.
2595
2613
.
33.
Benveniste
,
Y.
,
1985
, “
The Effective Mechanical Behavior of Composite Materials With Imperfect Contact Between the Constituents
,”
Mech. Mater.
,
4
(
2
), pp.
197
208
.
34.
Shen
,
G. L.
, and
Hu
,
G. K.
,
2006
,
Mechanics of Composite Materials
,
Tsinghua University Press
,
Beijing, China
.
35.
Chen
,
T. Y.
,
Chiu
,
M. S.
, and
Weng
,
C. N.
,
2006
, “
Derivation of the Generalized Young–Laplace Equation of Curved Interfaces in Nanoscaled Solids
,”
J. Appl. Phys.
,
100
(
7
), p.
074308
.
36.
Huang
,
Z. P.
, and
Sun
,
L.
,
2007
, “
Size-Dependent Effective Properties of a Heterogeneous Material With Interface Energy Effect: From Finite Deformation Theory to Infinitesimal Strain Analysis
,”
Acta Mech.
,
190
(
1–4
), pp.
151
163
.
37.
Benveniste
,
Y.
,
1987
, “
A New Approach to the Application of Mori–Tanaka's Theory in Composite Materials
,”
Mech. Mater.
,
6
(
2
), pp.
147
157
.
38.
Christensen
,
R. M.
, and
Lo
,
K. H.
,
1979
, “
Solutions for Effective Shear Properties in Three Phase Sphere and Cylinder Models
,”
J. Mech. Phys. Solids
,
27
(
4
), pp.
315
330
.
39.
Medasani
,
B.
, and
Vasiliev
,
I.
,
2009
, “
Computational Study of the Surface Properties of Aluminum Nanoparticles
,”
Surf. Sci.
,
603
(
13
), pp.
2042
2046
.
40.
Wolfer
,
W. G.
,
2011
, “
Elastic Properties of Surfaces on Nanoparticles
,”
Acta Mater.
,
59
(
20
), pp.
7736
7743
.
41.
Ouyang
,
G.
,
Wang
,
C. X.
, and
Yang
,
G. W.
,
2009
, “
Surface Energy of Nanostructural Materials With Negative Curvature and Related Size Effects
,”
Chem. Rev.
,
109
(
9
), pp.
4221
4247
.
42.
Eringen
,
A. C.
,
1980
,
Mechanics of Continua
,
Robert E. Krieger Publishing
,
New York
.
43.
Sharma
,
P.
, and
Dasgupta
,
A.
,
2002
, “
Average Elastic Fields and Scale-Dependent Overall Properties of Heterogeneous Micropolar Materials Containing Spherical and Cylindrical Inhomogeneities
,”
Phys. Rev. B
,
66
(
22
), p.
224110
.
44.
Horn
,
R. A.
, and
Johnson
,
C. R.
,
2013
,
Matrix Analysis
,
2nd ed.
,
Cambridge University Press
,
New York
.
45.
Gao
,
W.
,
Yu
,
S. W.
, and
Huang
,
G. Y.
,
2006
, “
Finite Element Characterization of the Size-Dependent Mechanical Behaviour in Nanosystems
,”
Nanotechnology
,
17
(
4
), pp.
1118
1122
.
46.
Vitos
,
L.
,
Ruban
,
A. V.
,
Skriver
,
H. L.
, and
Kollar
,
J.
,
1998
, “
The Surface Energy of Metals
,”
Surf. Sci.
,
411
(
1–2
), pp.
186
202
.
47.
Sheng
,
H. W.
,
Kramer
,
M. J.
,
Cadien
,
A.
,
Fujita
,
T.
, and
Chen
,
M. W.
,
2011
, “
Highly Optimized Embedded-Atom-Method Potentials for Fourteen FCC Metals
,”
Phys. Rev. B
,
83
(
13
), p.
134118
.
48.
Sharma
,
P.
,
Ganti
,
S.
, and
Bhate
,
N.
,
2003
, “
Effect of Surfaces on the Size-Dependent Elastic State of Nano-Inhomogeneities
,”
Appl. Phys. Lett.
,
82
(
4
), pp.
535
537
.
49.
Zhang
,
C.
,
Yao
,
Y.
, and
Chen
,
S. H.
,
2014
, “
Size-Dependent Surface Energy Density of Typically FCC Metallic Nanomaterials
,”
Comput. Mater. Sci.
,
82
(
1
), pp.
372
377
.
50.
Hashin
,
Z.
,
1983
, “
Analysis of Composite Materials—A Survey
,”
ASME J. Appl. Mech.
,
50
(
3
), pp.
481
505
.
51.
Hashin
,
Z.
,
1991
, “
Thermoelastic Properties of Particulate Composites With Imperfect Interface
,”
J. Mech. Phys. Solids
,
39
(
6
), pp.
745
762
.
You do not currently have access to this content.