Mathematics and Mechanics of Solids

baconossifiedΜηχανική

29 Οκτ 2013 (πριν από 3 χρόνια και 7 μήνες)

71 εμφανίσεις


http://mms.sagepub.com/
Mathematics and Mechanics of Solids
http://mms.sagepub.com/content/early/2013/07/01/1081286513491761
The online version of this article can be found at:
 
DOI: 10.1177/1081286513491761
published online 1 July 2013Mathematics and Mechanics of Solids
Changwen Mi and Demitris Kouris
Elastic disturbance due to a nanoparticle near a free surface
 
 
Published by:

http://www.sagepublications.com
can be found at:Mathematics and Mechanics of SolidsAdditional services and information for
 
 
 
 

http://mms.sagepub.com/cgi/alertsEmail Alerts:
 

http://mms.sagepub.com/subscriptionsSubscriptions:
 

http://www.sagepub.com/journalsReprints.navReprints:
 

http://www.sagepub.com/journalsPermissions.navPermissions:
 
What is This?
 
- Jul 1, 2013OnlineFirst Version of Record >>
at Southeast University on July 3, 2013mms.sagepub.comDownloaded from
Article
Elastic disturbance due to a
nanoparticle near a free surface
Mathematics and Mechanics of Solids
1–14
©The Author(s) 2013
Reprints and permissions:
sagepub.co.uk/journalsPermissions.nav
DOI:10.1177/1081286513491761
mms.sagepub.com
Changwen Mi
Jiangsu Key Laboratory of Engineering Mechanics,Department of Engineering Mechanics,Southeast
University,Nanjing,Jiangsu,China
Demitris Kouris
College of Science and Engineering,Texas Christian University,Fort Worth,TX,USA
Received 06 April 2013;accepted 06 May 2013
Abstract
The presence of nanoparticles in an elastic solid introduces disturbances that can vary significantly from the ones pre-
dicted by classical elasticity.In this study,we were able to determine that nanoscale effects introduced via a coherent
interface model can indeed result in complex,non-local displacement and stress fields near the free surface of a sub-
strate.The specific geometry is defined by a spherical nanoparticle near a straight boundary.The system is loaded either
through a far-field uniaxial tension or a transformation strain (eigenstrain) in the particle itself.The elastic field can
be fully determined using a three-dimensional displacement formulation that incorporates a well-established interface
model.
Keywords
Interface effect,nanoparticle,half-space,uniaxial load
1.Introduction
Atomic imperfections such as interstitials,dislocations,voids,and heterogeneous particles are sources of stress
concentration in crystalline solids.Such internal defects play a crucial role in the macroscopic mechanical
behavior of a material.This is because deformation and stress depend not only on external loads but also on the
microstructural details of a solid [1,2].
Within the framework of the continuumtheory of elasticity,defects in solids are often modeled as inclusions
or inhomogeneities,regardless of their specific source.When the size of these heterogeneous particles is at the
microscale or higher,only the volume and shape of the particles influence the elastic field [1].However,the
role of the interfaces that separate different bulk phases becomes prominent when the size of particles reaches
sub-microscale or nanoscale levels [3,4].The continuing efforts to fabricate ordered nanostructures [5] could
benefit froman improved understanding of their mechanical behavior.
Several models in the literature have attempted to address the elastic disturbance introduced by matrix–
nanoparticle interfaces [6–11].Among these models,perhaps the most established is the coherent interface
model proposed by Gurtin and Murdoch [7].The major assumption of this model lies in the conservation of
interface coherency.No relative displacements,i.e.stretching,slipping or twisting,are permitted between the
Corresponding author:
Changwen Mi,Jiangsu Key Laboratory of Engineering Mechanics,Department of Engineering Mechanics,Southeast University,Nanjing,Jiangsu 210096,
China.
Email:miseu.edu.cn
at Southeast University on July 3, 2013mms.sagepub.comDownloaded from
2 Mathematics and Mechanics of Solids
two bulk phases separated by the interface.As a result,an average tangential strain is chosen to represent the
interface strain.Aresidual interface stress and a linearly elastic response are superposed to collectively describe
the mechanical behavior of the interfaces.
Interface mechanics enter into the physics of nanoparticles by affecting the force balance across the interface.
An interface divergence of the interface stress tensor is introduced to balance the discontinuity experienced by
the stress vectors of the two abutting bulk phases.The new balance condition is analogous to the balance
of pressure drop across a fluid–fluid interface,when the deformation of a solid–solid interface is taken into
account [12].
Following this line of research,a number of classical micromechanical problems have been revisited
since the 1980s.Most of these studies involve nanoparticles or nanofibers embedded in either two- or three-
dimensional infinite domains and focus on the elastic displacement and stress fields [3,4,13–22].Another
group of studies focused on developing equivalent material properties of fibrous or particulate reinforced
nanocomposites [23–26].
Nonetheless,it should be noted that no studies in the latter group used the complete Gurtin and Murdoch
model.For example,Chen et al.[23] and Duan et al.[24,25] did not account for the residual interface stress.
Yang [26] included this term but still used an incomplete (symmetric and tangential) surface constitutive rela-
tion.Moreover,Yang [26] erroneously concluded that effective properties of nanocomposites depend on applied
strains.These issues were resolved by Mogilevskaya et al.[27] by using the complete Gurtin and Murdoch
model and a two-step evaluation strategy.
Nanocomposites involving substrates with finite domains are rarely considered,perhaps as a result of the
problem’s mathematical complexity.Stresses around circular nanoinhomogeneities that are present in an elastic
half-plane were studied by Avazmohammadi et al.[28] and Jammes et al.[29].
Mi and Kouris [30] attempted to determine the elastic field around a spherical nanoparticle embedded in
a three-dimensional half-space using displacement potentials.An all-around far-field stress applied parallel to
the half-space surface was considered.Recently,they extended the displacement potentials approach to the
problem of a nanovoid near the same free surface subjected to a unidirectional remote load [31].Nonetheless,
the important problemof uniaxial loading for the case of a nanoparticle remains unresolved and is the primary
goal of the present study.
Anovel feature of the present study is the implementation of the full-version of Gurtin and Murdoch’s coher-
ent interface model [7].Many previous studies have deliberately or unconsciously implemented an incomplete
coherent interface model,as summarized in detail by Kushch et al.[17] and Mogilevskaya et al.[19].One of the
incomplete versions excludes the normal components of interface stress.Although these normal terms would
have resulted in only negligible difference in stresses,the exclusion was often due to overlooking the superficial
nature of the interface stress tensor.It was erroneously treated as a tangential tensor,a property possessed by
interface strain.
The incorporation of the coherent interface model significantly alters the elastic fields in the vicinity of
the nanoparticle.For a given stiffness ratio,the “disturbance trend” is opposite in the nanoparticle and the
matrix.This result could provide a mechanismto regulate the equivalent stiffness of nanocomposites.The most
significant effects are observed in the case of nanoporous materials [25,30].
Given the available material constants for interfaces [32],residual interface stress turns out to be much more
important than interface elasticity.Their relative influence is roughly proportional to the ratio of the shear mod-
ulus and the external load.One recent study on rough surfaces,however,reports that interface elasticity may
be significantly affected by the microstructural detail of interfaces,both in sign and magnitude [9].The resid-
ual interface stress,in contrast,is almost unchanged.The relative importance of residual stress and interface
elasticity is thus bound to change with interfacial structure.
In addition to the uniaxial far-field load,we also consider a uniaxial eigenstrain prescribed inside the
nanoparticle.The tedious mathematical derivations have been consciously omitted since the solution procedure
has been previously detailed in Mi and Kouris [18].
The rest of this paper is structured as follows:in Section 2 the solution methodology that will yield dis-
placements and stresses is outlined in detail.In Section 3 we report numerical experiments that illustrate the
influence of the coherent interface model and finally,in Section 4,we discuss our conclusions and present some
thoughts on future research.
at Southeast University on July 3, 2013mms.sagepub.comDownloaded from
Mi and Kouris 3
¯
G,¯ν
O
a
G,ν
¯
O
TT
y
z
x
d
A
B
C
D
λ
0

0

0
ϕ
θ
R
P
Figure 1.A spherical nanoparticle embedded near the straight boundary of a substrate.
2.Method of solution
Figure 1 shows a spherical nanoparticle of radius a embedded at a distance d from the straight boundary of
an elastic substrate.The center of the particle is chosen as the coordinate origin.Given the geometry of the
system,Cartesian,cylindrical,and spherical coordinates are utilized.The substrate is subjected to a uniform
far-field stress σ
xx
= T.In addition,a uniform eigenstrain ε

xx
= ε

is specified inside the particle.The elastic
properties of the two bulk phases are represented by the shear modulus (G) and Poisson ratio (ν).An additional
overhead bar denotes the quantities of the particle.Within the context of coherent interface mechanics [7],three
constants,i.e.the residual interface stress (τ
0
) and two interface Lamé constants (λ
0
and μ
0
),are sufficient to
characterize the matrix–particle interface.
In the absence of body forces,the equilibrium equations in terms of displacements can be expressed with
the indicial notation as [2]:
1
(1 −2ν)

2
u
j
∂x
i
∂x
j
+

2
u
i
∂x
j
∂x
j
= 0,(1)
where u
i
refers to the displacement components.Einstein’s summation from1 to 3 over repeated Roman indices
is applicable unless otherwise stated.
In terms of cylindrical coordinates,one general solution to equation (1) was given by Tsuchida and Nakahara
[33] by appropriately combining the Boussinesq,Neuber and Dougall displacement potentials [18]:
2Gu
r
=
∂φ
0
∂r
−(3 −4ν) (cos θφ
1
+sinθφ
2
) +r cos θ
∂φ
1
∂r
(2)
+r sinθ
∂φ
2
∂r
+z
∂φ
3
∂r
+r
∂φ
4
∂z
+
2
r
∂λ
3
∂θ
,
2Gu
θ
=
1
r
∂φ
0
∂θ
+(3 −4ν) (sinθφ
1
−cos θφ
2
) +cos θ
∂φ
1
∂θ
+sinθ
∂φ
2
∂θ
+
z
r
∂φ
3
∂θ
−2
∂λ
3
∂r
,
2Gu
z
=
∂φ
0
∂z
+r cos θ
∂φ
1
∂z
+r sinθ
∂φ
2
∂z
−(3 −4ν) φ
3
+z
∂φ
3
∂z
−4(1 −ν) φ
4
−r
∂φ
4
∂r
,
where φ
0

4
and λ
3
are scalar harmonic functions and {φ
1

2

3
} denote a vector harmonic function.The
corresponding spherical displacements are derived with the aid of the relations r = Rsinϕ and z = Rcos ϕ as
well as the directional cosine
￿
u
R
u
ϕ
￿
=
￿
cos ϕ sinϕ
−sinϕ cos ϕ
￿￿
u
z
u
r
￿
.(3)
The next task is to construct appropriate groups of displacement potentials that are compatible with both the
geometry and the loading of the present problem.For a uniaxial far-field stress,two displacement potentials,
at Southeast University on July 3, 2013mms.sagepub.comDownloaded from
4 Mathematics and Mechanics of Solids
i.e.φ
0
and φ
3
,are sufficient
φ
0
=
(1 −ν)
4(1 +ν)
￿
r
2
−2z
2
￿
T +
1
4
cos 2θr
2
T,φ
3
= −
1
2(1 +ν)
zT.(4)
Equation (4) yields σ
xx
= T and zero for the other stress components.The remote boundary conditions are thus
automatically satisfied.In the formof cylindrical coordinates,the corresponding stress components are
σ
rr
=
1
2
T +
1
2
cos 2θT,σ
θθ
=
1
2
T −
1
2
cos 2θT,σ

= −
1
2
sin2θT.(5)
Each stress component in equation (5) is composed of two parts,corresponding to the axial symmetric and
the anti-symmetric component of the uniaxial far-field load,respectively.In viewof this fact,the solution to the
present problemcan be subsequently divided into two parts.
We propose a solution to the axially symmetric problemfirst.The “matrix” consists of a semi-infinite elastic
medium containing a spherical void in the vicinity of the straight boundary.To accommodate this geometry,
two groups of displacement potentials were chosen
φ
0
= G

￿
n=0
A
n
d
n+3
R
n+1
P
n
(μ),φ
3
= G

￿
n=0
B
n
d
n+2
R
n+1
P
n
(μ),(6)
φ
0
=
￿

0
ψ
1
(λ)J
0
(λr) e
−λz
dλ,φ
3
=
￿

0
λψ
2
(λ)J
0
(λr) e
−λz
dλ,(7)
where P
n
(μ) and J
n
(λr) are the Legendre and Bessel functions of the first kind;A
n
and B
n
are dimensionless
constants;and ψ
1
and ψ
2
are unknown functions of the integral variable λ.The argument of P
n
(μ) is related
to the spherical polar angle through μ = cos ϕ.The potential set of equations (6) and (7) represents general
solutions to spherical harmonic equations defined for R > a and cylindrical harmonic equations defined for z >
−d,respectively.They act in concert to represent the axially symmetric solution.For the spherical nanoparticle
only one group of displacement potentials is necessary
φ
0
=
¯
G

￿
n=0
¯
A
n
R
n
d
n−2
P
n
(μ),φ
3
=
¯
G

￿
n=0
¯
B
n
R
n
d
n−1
P
n
(μ),(8)
where the bars over the shear modulus (G) and dimensionless constants (A
n
& B
n
) signify that these quantities
belong to the nanoparticle.This potential set satisfies the spherical harmonic equations defined for a sphere
R ≤ a.
Similar strategies were employed for the anti-symmetric far-field loading,i.e.σ
xx
= −σ
yy
= T/2.Two
groups of displacement potentials for the matrix are
φ
0
= Gcos 2θ

￿
n=2
C
n
d
n+3
R
n+1
P
2
n
(μ),φ
1
= Gcos θ

￿
n=1
D
n
d
n+2
R
n+1
P
1
n
(μ),
φ
2
= −Gsinθ

￿
n=1
D
n
d
n+2
R
n+1
P
1
n
(μ),φ
3
= Gcos 2θ

￿
n=2
E
n
d
n+2
R
n+1
P
2
n
(μ),
φ
4
= −Gcos 2θ

￿
n=2
D
n
(n −1)
d
n+2
R
n+1
P
2
n
(μ),(9)
φ
0
= cos 2θ
￿

0
ψ
3
(λ) J
2
(λr) e
−λz
dλ,φ
1
= cos θ
￿

0
ψ
4
(λ) J
1
(λr) e
−λz
dλ,
φ
2
= −sinθ
￿

0
ψ
4
(λ) J
1
(λr) e
−λz
dλ,φ
3
= cos 2θ
￿

0
λψ
5
(λ) J
2
(λr) e
−λz
dλ,
λ
3
= sin2θ
￿

0
ψ
6
(λ) J
2
(λr) e
−λz
dλ,(10)
at Southeast University on July 3, 2013mms.sagepub.comDownloaded from
Mi and Kouris 5
where P
n
m
(μ) is the associated Legendre function of order n and degree m [34].The dimensionless coeffi-
cients and unknown functions preceding the spherical and cylindrical harmonics assume similar interpretations
to those in equations (6) and (7).The single group of displacement potentials necessary for the spherical
nanoparticle is
φ
0
=
¯
Gcos 2θ

￿
n=2
¯
C
n
R
n
d
n−2
P
2
n
(μ),φ
1
=
¯
Gcos θ

￿
n=1
¯
D
n
R
n
d
n−1
P
1
n
(μ),
φ
2
= −
¯
Gsinθ

￿
n=1
¯
D
n
R
n
d
n−1
P
1
n
(μ),φ
3
=
¯
Gcos 2θ

￿
n=2
¯
E
n
R
n
d
n−1
P
2
n
(μ),
φ
4
=
¯
Gcos 2θ

￿
n=2
¯
D
n
(n +2)
R
n
d
n−1
P
2
n
(
μ
)
.(11)
For the case of a uniformaxial eigenstrain (ε

xx
= ε

) specified inside the nanoparticle,the total displacement
field within the nanoparticle becomes the sumof elastic and non-elastic contributions.The elastic displacements
are clearly due to the potential groups in equations (8) and (11).The non-elastic displacements due to the
eigenstrain load are given by
u

R
=
1
3
R(P
0
(μ) −P
2
(μ)) ε

+
1
6
cos 2θRP
2
2
(μ) ε

,
u

θ
= −
1
6
sin2θR
P
2
2
(μ)
￿
1 −μ
2
ε

,
u

ϕ
=
1
6
R
￿
1 −μ
2
P
2

(μ) ε


1
12
cos 2θR
￿
1 −μ
2
P
2
2

(μ) ε

,(12)
where the prime in P
n

(μ) and P
m
n

(μ) denotes differentiation with respect to their argument μ.
The zero-traction conditions at the straight boundary z = −d were subsequently enforced in order to deter-
mine the unknown functions ψ
1
(λ) – ψ
6
(λ),as functions of the dimensionless coefficients A
n
– E
n
.This is a
tedious but straightforward procedure,following Mi and Kouris [30] and Tsuchida and Nakahara [33].
The dimensionless coefficients A
n
– E
n
are determined by satisfying the boundary conditions at the matrix–
nanoparticle interface.Following Gurtin and Murdoch [7],a solid–solid coherent interface can be modeled as an
ultrathin filmwith neither stretching nor slipping between the abutting bulk phases.As a result,the displacement
field remains continuous across the interface
(u
i
)
R=a
=
￿
¯u
i
+u

i
￿
R=a
(i = R,θ,ϕ).(13)
Furthermore,for the sake of coherency,interfacial deformations are limited to the case of equal tangential strain
in both bulk phases.The average of this tangential strain is the natural candidate of interface strain [7,8]:
E
αβ
=
1
2
￿
(∇
S
u)
αβ
+(∇
S
u)
βα
￿
.(14)
Here ∇
S
u represents the interface gradient of the displacement field.It is a superficial tensor field defined on
the separating interface although only its tangential components enter into the formulation of interface strain.
The Greek subscripts may denote either the azimuthal (θ) or polar (ϕ) coordinate.
In contrast to its bulk counterpart,the interface stress is non-zero and assumes a residual value when the
bulk is unstrained [7].In addition,the interface gradient ∇
S
u also enters into the definition thus making the
interface stress superficial too.To facilitate the calculation,explicit expressions of interface stress components
at Southeast University on July 3, 2013mms.sagepub.comDownloaded from
6 Mathematics and Mechanics of Solids
were developed with regard to the interfacial projections of bulk displacements

θθ
= τ
0
+

0
+2μ
0
)
a
￿
1
￿
1 −μ
2
￿
∂u
θ
∂θ
+μu
ϕ
￿
+u
R
￿
+

0

0
)
a
￿
u
R

￿
1 −μ
2
∂u
ϕ
∂μ
￿
,

ϕϕ
= τ
0
+

0

0
)
a
￿
1
￿
1 −μ
2
￿
∂u
θ
∂θ
+μu
ϕ
￿
+u
R
￿
+

0
+2μ
0
)
a
￿
u
R

￿
1 −μ
2
∂u
ϕ
∂μ
￿
,

θϕ
=
μ
0
a
￿
1 −μ
2
￿
∂u
ϕ
∂θ
−μu
θ
￿


0
−τ
0
)
a
￿
1 −μ
2
￿
∂u
θ
∂μ
￿
,

ϕθ
=

0
−τ
0
)
a
￿
1 −μ
2
￿
∂u
ϕ
∂θ
−μu
θ
￿

μ
0
a
￿
1 −μ
2
￿
∂u
θ
∂μ
￿
,


= τ
0
￿
∂u
θ
∂R
￿
,

= τ
0
￿
∂u
ϕ
∂R
￿
,(15)
where the chevrons   represent the projecting and averaging operation.Two characteristics of the interface
stress tensor should be noted from equation (15):asymmetric (
θϕ
= 
ϕθ
) and superficial (due to the normal
components of ∇
S
u).To finalize the governing equations of a coherent interface,a traction balance condition
must be taken into account [7,35]
￿
σ
ij
￿
n
j
= −(∇
S
· )
i
(i,j = R,θ,ϕ),(16)
where n
j
represent the components of the unit normal vector to the interface and [σ
ij
] denotes the stress dis-
continuity across the interface along the same orientation as the interface normal;∇
S
·  is the interface
divergence of the interface stress tensor,due to elastic as well as non-elastic contributions.In reference to
spherical coordinates,explicit expressions of its three components are given by
(∇
S
· )
R
=
1
a
￿
1 −μ
2
￿
∂

∂θ

￿
1 −μ
2
￿
∂

∂μ
+μ

￿

￿

θθ
+
ϕϕ
￿
a
,
(∇
S
· )
θ
=
1
a
￿
1 −μ
2
￿
∂
θθ
∂θ

￿
1 −μ
2
￿
∂
θϕ
∂μ

￿

θϕ
+
ϕθ
￿
￿
+


a
,
(∇
S
· )
ϕ
=
1
a
￿
1 −μ
2
￿
∂
ϕθ
∂θ

￿
1 −μ
2
￿
∂
ϕϕ
∂μ

￿

ϕϕ
−
θθ
￿
￿
+


a
.(17)
The interface divergence of interface stress can therefore be readily expressed in terms of displacements with
the aid of equation (15).
At this point,the derivation of displacements,stresses,and the interface divergence of interface stress at the
matrix–nanoparticle interface becomes straightforward.The total displacement field in the matrix is obtained
by superposing the contributions due to equations (4),(6),(7),(9) and (10) while that in the nanoparticle can be
obained by superposing the contributions due to equations (8),(11) and (12).Further derivation of the stress and
at Southeast University on July 3, 2013mms.sagepub.comDownloaded from
Mi and Kouris 7
strain fields is possible via the displacement–strain relationship and bulk constitutive law formulated in either
cylindrical or spherical coordinates.
One should be cautious when addressing the total interface divergence of interface stress defined on the
matrix–nanoparticle interface.All contributions should be included in its derivation,both elastic and non-
elastic.In particular,the contribution due to the eigenstrain load,equation (12),plays its role through the
interface constitutive relationship,equation (15),and the evaluation formulae of the interface divergence,equa-
tion (17).For brevity,the explicit expressions of displacements,stresses,and interface divergence of interface
displacements have been omitted.
After the displacements,stresses,and interface divergence vector were derived,we were able to enforce both
the displacement continuity,equation (13),and the traction balance condition,equation (16).These boundary
conditions were eventually transformed into 10 sets of equations in series form,i.e.4 for the axial symmetric
component of the present problem and 6 for the anti-symmetric one.Equating the coefficients preceding the
Legendre and associated Legendre functions as well as their derivatives in these equations,we obtained 10 sets
of linear algebraic equations leading to the dimensionless coefficients A
n
– E
n
and
¯
A
n

¯
E
n
.Solving the resultant
(10n) linear equations for the (10n) unknowns,the problemcan be deemed as completely solved.
3.Results and discussion
Several numerical experiments were performed to investigate the elastic field in the vicinity of the nanoparticle
as well as of the half-space surface.Governing parameters of the solution include the sign and magnitude of
the loads (T and ε

),the material constants of both bulk phases (G,ν,
¯
G and ¯ν),the material properties of the
matrix–nanoparticle interface (τ
0

0
and μ
0
),the size of the nanoparticle (a),and its distance fromthe straight
boundary (d).
The material properties of the matrix were chosen for nickel with G = 76GPa and ν = 0.31 [36],a
moderately strong metal.Without loss of generality,the Poisson ratio of the nanoparticle was also set as ¯ν =
0.31.The shear modulus of the nanoparticle
¯
G,on the other hand,was allowed to vary in order to simulate
nanoparticles of different strength.
Previous investigations have adopted interface material constants fromatomistically informed properties of
crystalline surfaces or interfaces with specific orientations [15,19,21,30],due to the lack of isotropic values.
Given the reported values of orientation-dependent surface and interface properties [32,27],we decided to use
the nominal values τ
0
= λ
0
= μ
0
= 1N/m in our numerical studies.Other values can be implemented in the
proposed formulation with no additional difficulty.This choice seems adequate for our effort to develop a trend
analysis of the disturbance due to interface effects [17,18].
We first examined the stress disturbance resulting from the presence of both a soft and a hard nanoparticle,
i.e.with shear moduli ratio  =
¯
G/G = 0.5 and 2.0.Only a uniaxial tensile load of T = 100 MPa was taken
into account (Figure 1).A spherical particle of radius 10nm,embedded 20 nm beneath the free surface was
considered.Figure 2 highlights the distribution of radial stresses along the line segment ABC (θ = 0
0
) on the
spherical interface R = a.The radial stresses belonging to both the nanoparticle ( ¯σ
RR
) and the matrix (σ
RR
)
were plotted.As expected,the radial stress remained continuous across the interface in the absence of interface
effects.
Should the nanoparticle be embedded in an infinite substrate,a closed-form solution can be found for the
classical stress components [18]
¯σ
RR
￿
T = 3χ
1
cos
2
ϕ

−χ
1

2
,(18)
¯σ
θθ
￿
T = −χ
1

2
,(19)
¯σ
ϕϕ
￿
T = −3χ
1
cos
2
ϕ

+2χ
1

2
,(20)
σ
RR
￿
T =
￿
18ω
1
a
5
R
5
−5(5 −ν) ω
1
a
3
R
3
+1
￿
cos
2
ϕ

−6ω
1
a
5
R
5
+
5
3
(5 −ν) ω
1
a
3
R
3

2
3
ω
2
a
3
R
3
,(21)
at Southeast University on July 3, 2013mms.sagepub.comDownloaded from
8 Mathematics and Mechanics of Solids
Polar Angle (degrees)
RadialStressσRR
/T&σRR/T
0 30 60 90 120 150 18
0
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
interface
Γ=2.0:
classical
Γ=0.5:
interfaceclassical
Figure 2.Variation of ¯σ
RR
/T and σ
RR
/T along the line segment ABC (Figure 1) on the spherical interface for two shear moduli
ratios: = 0.5 and 2.The stress curves corresponding to the nanoparticle and matrix are drawn with black open and red filled
symbols,respectively.
σ
θθ
￿
T =
￿

15
2
ω
1
a
5
R
5
+
15
2
(1 −2ν) ω
1
a
3
R
3
￿
cos
2
ϕ

+
3
2
ω
1
a
5
R
5

25
6
(1 −2ν) ω
1
a
3
R
3
+
1
3
ω
2
a
3
R
3
,(22)
σ
ϕϕ
￿
T =
￿

21
2
ω
1
a
5
R
5
+
5
2
(1 −2ν) ω
1
a
3
R
3
−1
￿
cos
2
ϕ

+
9
2
ω
1
a
5
R
5
+
5
6
(1 −2ν) ω
1
a
3
R
3
+
1
3
ω
2
a
3
R
3
+1,(23)
where χ
1

2

1
and ω
2
are all dimensionless constants of the material properties and are defined by
χ
1
=
5(1 −ν)
(7 −5ν +2(4 −5ν))
,
χ
2
=
(1 −ν) (1 + ¯ν)
(1 +ν) (2(1 −2¯ν) +(1 + ¯ν))
,
ω
1
=
(1 −)
(7 −5ν +2(4 −5ν))
,
ω
2
=
((1 +ν) (1 −2¯ν) −(1 −2ν) (1 + ¯ν))
(
1 +ν
) (
2
(
1 −2¯ν
)
+
(
1 + ¯ν
))
.(24)
Note also that here ϕ

= tan
−1
￿
y
2
+z
2
/x represents circular cones symmetric about the x-axis.
The values of ¯σ
RR
and σ
RR
at the pole A (ϕ = 0 and ϕ

= π/2) are reasonably represented by equations (18)
and (21) because A is the most distant fromthe free surface.The corresponding values at the points that are far
away fromA,on the other hand,deviate fromequations (18) and (21) due to the disturbance of the free surface.
The magnitude of deviation,however,is minor for the present numerical scenarios and is expected to become
more prominent for large ratios of a to d.
A significant discontinuity was found between ¯σ
RR
and σ
RR
when the effects of residual interface stress and
the interface elasticity,equation (15),were considered.Mi and Kouris [18] recently identified that the distur-
bances of residual interface stress and interface elasticity are approximately of the order of magnitude τ
0
/Ta and

0

0

0
)/Ga,respectively.Given the relative magnitudes of ultimate strength and shear modulus of common
engineering materials,the residual interface stress τ
0
plays a far more important role.As an approximation,it
at Southeast University on July 3, 2013mms.sagepub.comDownloaded from
Mi and Kouris 9
Polar Angle (degrees)
HoopStressσθθ/T&σθθ/T
0 30 60 90 120 150 18
0
-0.5
0
0.5
1
1.5
interface
Γ=2.0:
classical interface
Γ=0.5:
classical
interfaceΓ=0.5:
classical
interface
classical
Figure 3.Variation of ¯σ
θθ
/T and σ
θθ
/T along the line segment ADC (Figure 1) on the spherical interface for two shear moduli
ratios: = 0.5 and 2.The stress curves corresponding to the nanoparticle and matrix are drawn with black open and red filled
symbols,respectively.
seems reasonable to take only τ
0
into account.FromMi and Kouris [18],the stress components due to τ
0
alone
for the case of the infinite matrix can be derived
¯σ
RR
= ¯σ
θθ
= ¯σ
ϕϕ
= −
2(1 + ¯ν)
(2(1 −2¯ν) +(1 + ¯ν))
τ
0
a
,(25)
σ
RR
= −2σ
θθ
= −2σ
ϕϕ
=
4(1 −2¯ν)
(2(1 −2¯ν) +(1 + ¯ν))
a
3
R
3
τ
0
a
.(26)
The above equation obviously represents a jump of 2τ
0
/a across the interface,as dictated by the first part of
equation (17).As seen from Figure 2,the impact of the half-space surface on the modified solution shows
similar characteristics to those of the classical solution.The radial stresses at A are closest to the superposition
of equation (18) with equation (25) and equation (21) with equation (26).The greater the polar angle ϕ,the
larger the separation becomes.This separation reaches its maximumat the pole C.
Figures 3–7 report the hoop stresses ¯σ
θθ
and σ
θθ
along several line segments in the vicinity of the nanoparti-
cle and the half-space surface.The hoop stress along the line segment ADC (Figure 1) is described in Figure 3.
For the case of an infinite matrix,¯σ
θθ
and σ
θθ
along ADC are represented by equations (20) and (23) after
setting ϕ

= π/2.Equations (25) and (26) play a role when the residual interface stress is included.When the
interface elasticity is excluded,both solutions yield constant values.This invariance is inevitably violated due to
the presence of the substrate surface and because of the inclusion of interface elasticity,although the influence
of both factors is marginal.
Figures 4–6 showthe variation of hoop stress along three radial directions:the positive z-axis,the y-axis,and
the negative z-axis up to the half-space surface.Since the positive z-axis (ϕ = 0 and ϕ

= π/2) points away from
the half-space surface,the hoop stress in the nanoparticle and matrix are well represented by the superposition
of equation (20) with equation (25),and equation (23) with equation (25),respectively.For the given ϕ

the
hoop stress ¯σ
θθ
becomes constant.Although interface elasticity introduces an additional dependence on the
radial coordinate R,the magnitude of variation proves to be very small for T/G
1 [18].This is confirmed by
the modified solutions of ¯σ
θθ
shown in Figures 4–6.
Another interesting aspect worthy of note from Figures 4–6 is the rough overlapping of ¯σ
θθ
for both shear
moduli ratios.The numerical values of ¯σ
θθ
are approximately −0.247 and −0.240,respectively.Atrend analysis
of ¯σ
θθ
on  reported that ¯σ
θθ
reaches its minimumvalue (ca.−0.266) for a nanoinclusion ( = 1) and gradually
approaches toward zero as  deviates from unity to either zero or infinity.The incorporation of the coherent
interface model helps to decrease the largest possible variation of ¯σ
θθ
from∼ 2 in the classical case to ∼ 0.266.
For the hoop stress σ
θθ
in the matrix,the dependence on the radial coordinate is very localized along the
positive z-axis and y-axis (Figures 4 and 5).The deviation between the classical and modified solutions essen-
tially disappears when R becomes greater than five times the nanoparticle radius a.For the negative z-axis,
at Southeast University on July 3, 2013mms.sagepub.comDownloaded from
10 Mathematics and Mechanics of Solids
Normalized Distance z/d
HoopStressσθθ/T&σθθ/T
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
-0.5
0
0.5
1
1.5
interface Γ=0.5
2
classical Γ=0.5
2
Figure 4.Variation of ¯σ
θθ
/T and σ
θθ
/T along positive z axis (θ = 90
0
and ϕ = 0) as a function of the normalized distance z/d for
two shear moduli ratios: = 0.5 and 2.
Normalized Distance R/a
HoopStressσθθ/T&σθθ/T
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
-0.5
0
0.5
1
1.5
interface Γ=0.5
2classical Γ=0.5
2
Figure 5.Variation of ¯σ
θθ
/T and σ
θθ
/T along the y-axis (θ = 90
0
and ϕ = 90
0
) as a function of the normalized distance R/a for
two shear moduli ratios: = 0.5 and 2.
Normalized Distance r/d
HoopStressσθθ/T
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
0.5
0.6
0.7
0.8
0.9
1.0
1.1
interface Γ=0.5
2classical Γ=0.5
2
Figure 6.Variation of ¯σ
θθ
/T and σ
θθ
/T along the negative z-axis (θ = 90
0
and ϕ = π) as a function of the normalized distance z/d
for two shear moduli ratios: = 0.5 and 2.
however,this deviation remains prominent all the way from the pole C to the half-space surface
¯
O (Figure 6).
The smallest separation occurs at half-way between the ends.
at Southeast University on July 3, 2013mms.sagepub.comDownloaded from
Mi and Kouris 11
Normalized Distance r/d
HoopStressσθθ/T
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
0.5
0.6
0.7
0.8
0.9
1.0
1.1
interface Γ=0.5
2classical Γ=0.5
2
Figure 7.Variation of σ
θθ
along one line segment on the straight boundary (z = d and θ = 90
0
) as a function of the normalized
distance r/d for two shear moduli ratios: = 0.5 and 2.
Polar Angle (degrees)
PolarStressσϕϕ/T&σϕϕ/T
0 30 60 90 120 150 18
0
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
interface
a/d=0.8:
classical
a/d=0.2:
interfaceclassical
Figure 8.Variation of ¯σ
ϕϕ
/T and σ
ϕϕ
/T along the line segment ABC (Figure 1) on the spherical interface for  = 0.25 and two
ratios:a/d = 0.2 and 0.8.For all cases,the distance between the nanoparticle center and the free surface was fixed at d = 20nm.
The stress curves corresponding to the nanoparticle and matrix are drawn with black open and red filled symbols,respectively.
The prominent separation between the classical and modified solutions is the result of the collective distur-
bances of the coherent interface model and the substrate surface.This trend of large separation continues on
the half-space surface in the area close to
¯
O but the magnitude of separation rapidly decays with the cylindrical
coordinate r and roughly disappears for r ≥ 4a (Figure 7).
The classical elastic field in both domains depends only on the ratio between two characteristic lengths a/d
[30,33].When the coherent interface model is taken into account,separate dependencies on both characteristic
lengths were introduced.Figures 8 and 9 show the variation of tangential stresses ¯σ
ϕϕ
and σ
ϕϕ
for two ratios
a/d = 0.2 and 0.8.In addition,the shear moduli ratio was fixed at  = 0.25.The only difference is that the
embedding distance d was fixed a 20nmin Figure 8 whereas the particle radius a was fixed at 10nmin Figure 9.
The common observation that can be made for both Figures 8 and 9 is that the asymmetry of stress dis-
tribution due to the presence of the substrate surface is larger for particles that are closer to the free surface.
However,for the same ratio a/d the asymmetry in Figure 9 is much stronger than that in Figure 8.Both the
stress distribution and magnitude of the modified solution are significantly affected.
Numerical experiments were also performed to investigate the possible interaction between the coherent
interface model and a uniaxial eigenstrain (ε

x
= ε

) prescribed inside the nanoparticle.The magnitude of the
uniaxial eigenstrain was chosen so that its product with the shear modulus of the matrix Gε

was equal to
100MPa.Figure 10 shows the variation of ¯σ
RR
and σ
RR
.The governing parameters assume the same values as
those used for Figures 2–7,except for the loading condition.
at Southeast University on July 3, 2013mms.sagepub.comDownloaded from
12 Mathematics and Mechanics of Solids
Polar Angle (degrees)
PolarStressσϕϕ/T&σϕϕ/T
0 30 60 90 120 150 18
0
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
interface
a/d=0.8:
classical
a/d=0.2:interface
classical
Figure 9.Variation of ¯σ
ϕϕ
/T and σ
ϕϕ
/T along the line segment ABC (Figure 1) on the spherical interface for  = 0.25 and two
ratios:a/d = 0.2 and 0.8.For all cases,the radius of the spherical nanoparticle is fixed as a = 10 nm.The stress curves corresponding
to the nanoparticle and matrix are drawn with black open and red filled symbols,respectively.
Polar Angle (degrees)
RadialStressσRR/Gε*
0 30 60 90 120 150 18
0
-4
-3
-2
-1
0
1
interfaceΓ=2.0:classical
Γ=0.5:interfaceclassical
Figure 10.Variation of ¯σ
RR
/Gε

and σ
RR
/Gε

along the line segment ABC (Figure 1) on the spherical interface for two shear
moduli ratios: = 0.5 and 2.The stress curves corresponding to the nanoparticle and matrix are drawn with black open and red
filled symbols,respectively.
If the nanoparticle was present in a full-space the classical stresses would be formally very similar to equa-
tions (19)–(23).Only three modifications would need to be made:zeroing the three terms corresponding to
the uniaxial far-field tension in equations (21) and (23),changing the normalization factor from T to Gε

,and
redefining the four dimensionless constants in equation (24)
χ
1
= −
2
(
7 −5ν
)
3(7 −5ν +2(4 −5ν))

2
= −
4
(
1 + ¯ν
)
3(2(1 −2¯ν) +(1 + ¯ν))
,
ω
1
=
2
(7 −5ν +2(4 −5ν))

2
=
2(1 + ¯ν)
(2(1 −2¯ν) +(1 + ¯ν))
.(27)
Because of the same functional form shared between the stress distributions due to a uniaxial far-field load
and an eigenstrain,some common characteristics can be observed.As seen fromFigure 10,the classical values
of ¯σ
RR
and σ
RR
at points close to A can be reasonably approximated by equations (18) and (21),respectively.
Additional superpositions with equations (25) and (26) result approximately in the modified solution,pro-
vided that the effects of interface elasticity are marginal.Again,deviation fromthe full-space solution becomes
appreciable for points close to the north pole C.
Both the sign and magnitude of the dimensionless constants,as defined in equations (24) and (27),are
different nonetheless.The uniaxial far-field tension produces the maximum radial stresses along the tensile
at Southeast University on July 3, 2013mms.sagepub.comDownloaded from
Mi and Kouris 13
direction.For the uniaxial eigenstrain load,however,the maximum radial stresses occur along perpendicular
directions.This behavior is explained by the constraint of the matrix with respect to the uniaxial eigenstrain,
which results in the most negative eigenstress field along the expansion direction.In addition,the separation
between the same stress components,e.g.¯σ
RR
or σ
RR
,due to a soft and a hard nanoparticle becomes more
evident.No crossovers occur for either the classical or the modified solution.The relative magnitudes of the
dimensionless constants defined in equations (24) and (27) explain this difference.
4.Concluding remarks
We have analyzed the problemof an elastic substrate containing a spherical nanoparticle by using a displacement
formulation.It was analytically solved by clearing the traction vector at the substrate surface and enforcing a
coherent interface model [7] across the spherical interface.Based on numerical evaluations of the elastic field
in the vicinity of the nanoparticle and the half-space surface,a few conclusions can be drawn:
• When the characteristic length ratio was reasonably small,e.g.a/d ≤ 0.5,the elastic field around
the farthest pole from the half-space surface was well approximated by the corresponding full-space
solution.The elastic field in regions close to the substrate surface was obviously disturbed.
• Inside the nanoparticle,hard inhomogeneities ( > 1) produced larger separations between the clas-
sical and modified solutions than the soft ones.This conclusion was reversed for stress separations in
the matrix.These trends hold for both loading types,as evidenced by equations (25) and (26).
• For the cases of same ratio a/d,the smaller the absolute values of the two characteristic lengths the
more strongly the elastic field was affected by the substrate surface.The impact was reflected in both
the distribution and magnitude of the stress.
• For reported values of bulk and interface material properties for common engineering materials,the
residual interface stress alone plays a much more important role than the interface elasticity.Their net
disturbances are proportional to τ
0
/a(T,Gε

) and (τ
0

0

0
)/a(T,Gε

),respectively.Both the resultant
magnitude and sign of these combinations are important.
Despite the completeness of this work in the framework of linear,isotropic elasticity,many issues remain
unresolved.Future work should attempt to address:(1) the incorporation of incoherent interface models;(2) the
atomistic calculations of relevant interface material constants;and (3) the interaction of more practical interface
models with low-dimensional domains.
Funding
This work was supported by the PhDPrograms Foundation of the Ministry of Education of China (grant number 20110092120018) and
the National Natural Science Foundation of China (grant numbers 11202050 and 11202051).
Conflicting interest
None declared.
References
[1] Mura,T.Micromechanics of defects in solids.Dordrecht:Martinus Nijhoff,1987.
[2] Sadd,MH.Elasticity:theory,applications,and numerics.Oxford:Elsevier,2005.
[3] Cahn,J,and Lärché,F.Surface stress and the chemical-equilibrium of small crystals.II.Solid particles embedded in a solid
matrix.Acta Metall 1982;30:51–56.
[4] Sharma,P,Ganti,S,and Bhate,N.Effect of surfaces on the size-dependent elastic state of nano-inhomogeneities.Appl Phys Lett
2003;82:535–537.
[5] Shevchenko,EV,Talapin,DV,Kotov,NA,O’Brien,S,and Murray,CB.Structural diversity in binary nanoparticle superlattices.
Nature 2006;439:55–59.
[6] Benveniste,Y.A general interface model for a three-dimensional curved thin anisotropic interphase between two anisotropic
media.J Mech Phys Solids 2006;54:708–734.
[7] Gurtin,M,and Murdoch,A.Surface stress in solids.Int J Solids Struct 1978;14:431–440.
[8] Gurtin,M,Weissmüller,J and Lärché,F.A general theory of curved deformable interfaces in solids at equilibrium.Philos Mag
A 1998;78:1093–1109.
at Southeast University on July 3, 2013mms.sagepub.comDownloaded from
14 Mathematics and Mechanics of Solids
[9] Mohammadi,P,Liu,LP,Sharma,P,and Kukta,RV.Surface energy,elasticity and the homogenization of rough surfaces.J Mech
Phys Solids 2013;61:325–340.
[10] Povstenko,Y.Theoretical investigation of phenomena caused by heterogeneous surface tension in solids.J Mech Phys Solids
1993;41:1499–1514.
[11] Steigmann,D,and Ogden,R.Elastic surface-substrate interactions.Proc R Soc London Ser A;455:437–474.
[12] Adamson,A,and Gast,A.Physical chemistry of surfaces.New York:Wiley,1997.
[13] Duan,H,Wang,J,Huang,Z and Karihaloo,BL.Eshelby formalism for nano-inhomogeneities.Proc R Soc A 2005;461:3335–
3353.
[14] Duan,H,Wang,J,Huang,Z,and Luo,Z.Stress concentration tensors of inhomogeneities with interface effects.Mech Mater
2005;37:723–736.
[15] He,L,and Li,Z.Impact of surface stress on stress concentration.Int J Solids Struct 2006;43:6208–6219.
[16] Lim,C,Li,Z,and He,L.Size dependent,non-uniformelastic field inside a nano-scale spherical inclusion due to interface stress.
Int J Solids Struct 2006;43:5055–5065.
[17] Kushch,V,Mogilevskaya,S,Stolarski,H,and Crouch,S.Elastic interaction of spherical nanoinhomogeneities with Gurtin–
Murdoch type interfaces.J Mech Phys Solids 2011;59:1702–1716.
[18] Mi,C,and Kouris,D.On the significance of coherent interface effects for embedded nanoparticles.Math Mech Solids 2012.
DOI:10.1177/1081286512465426.
[19] Mogilevskaya,SG,Crouch,SL,and Stolarski,HK.Multiple interacting circular nano-inhomogeneities with surface/interface
effects.J Mech Phys Solids 2008;56:2298–2327.
[20] Sharma,P,and Ganti,S.Interfacial elasticity corrections to size-dependent strain-state of embedded quantum dots.Phys Status
Solidi B 2002;234:R10–R12.
[21] Sharma,P,and Ganti,S.Size-dependent Eshelby’s tensor for embedded nano-inclusions incorporating surface/interface energies.
J Appl Mech 2004;71:663–671.
[22] Tian,L,and Rajapakse,RKND.Analytical solution for size-dependent elastic field of a nanoscale circular inhomogeneity.J Appl
Mech 2007;74:568–574.
[23] Chen,T,Dvorak,GJ,and Yu,CC.Size-dependent elastic properties of unidirectional nano-composites with interface stresses.
Acta Mech 2007;188:39–54.
[24] Duan,HL,Wang,J,Huang,ZP,and Karihaloo,BL.Size-dependent effective elastic constants of solids containing nano-
inhomogeneities with interface stress.J Mech Phys Solids 2005;53:1574–1596.
[25] Duan,HL,Wang,J,Karihaloo,BL,and Huang,ZP.Nanoporous materials can be made stiffer than non-porous counterparts by
surface modification.Acta Mater 2006;54:2983–2990.
[26] Yang,F.Size-dependent effective modulus of elastic composite materials:Spherical nanocavities at dilute concentrations.J Appl
Phys 2004;95:3516–3520.
[27] Mogilevskaya,SG,Crouch,SL,La Grotta,A,and Stolarski,HK.The effects of surface elasticity and surface tension on the
transverse overall elastic behavior of unidirectional nano-composites.Compos Sci Technol 2010;70:427–434.
[28] Avazmohammadi,R,Yang,F,and Abbasion,S.Effect of interface stresses on the elastic deformation of an elastic half-plane
containing an elastic inclusion.Int J Solids Struct 2009;46:2897–2906.
[29] Jammes,M,Mogilevskaya,S,and Crouch,S.Multiple circular nano-inhomogeneities and/or nano-pores in one of two joined
isotropic elastic half-planes.Eng Anal Boundary Elem 2009;33:233–248.
[30] Mi,C,and Kouris,D.Nanoparticles under the influence of surface/interface elasticity.J Mech Mater Struct 2006;1:763–791.
[31] Mi,C,and Kouris,D.Stress concentration around a nanovoid near the surface of an elastic half-space.Int J Solids Struct 2013.
DOI:10.1016/j.ijsolstr.2013.04.029.
[32] Mi,C,Jun,S,Kouris,D,and Kim,S.Atomistic calculations of interface elastic properties in noncoherent metallic bilayers.Phys
Rev B 2008;77:075425.
[33] Tsuchida,E,and Nakahara,I.Stress-concentration around a spherical cavity in a semi-infinite elastic body under uniaxial tension.
Bull JSME 1974;17:1207–1217.
[34] Arfken,G,and Weber,H.Mathematical methods for physicists.New York:Elsevier,2005.
[35] Gurtin,M,and Murdoch,A.Continuumtheory of elastic-material surfaces.Arch Ration Mech Anal 1975;57:291–323.
[36] Callister,W.Materials science and engineering:an introduction.New York:John Wiley &Sons,2007.
[37] Shenoy,V.Atomistic calculations of elastic properties of metallic fcc crystal surfaces.Phys Rev B 2005;71:094104.
at Southeast University on July 3, 2013mms.sagepub.comDownloaded from