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Mathematics and Mechanics of Solids

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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

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Article

Elastic disturbance due to a

nanoparticle near a free surface

Mathematics and Mechanics of Solids

1–14

©The Author(s) 2013

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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:miseu.edu.cn

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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.

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Mi and Kouris 3

¯

G,¯ν

O

a

G,ν

¯

O

TT

y

z

x

d

A

B

C

D

λ

0

,μ

0

,τ

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

,φ

4

and λ

3

are scalar harmonic functions and {φ

1

,φ

2

,φ

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,

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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

rθ

= −

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)

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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

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6 Mathematics and Mechanics of Solids

were developed with regard to the interfacial projections of bulk displacements

θθ

= τ

0

+

(λ

0

+2μ

0

)

a

1

1 −μ

2

∂u

θ

∂θ

+μu

ϕ

+u

R

+

(λ

0

+τ

0

)

a

u

R

−

1 −μ

2

∂u

ϕ

∂μ

,

ϕϕ

= τ

0

+

(λ

0

+τ

0

)

a

1

1 −μ

2

∂u

θ

∂θ

+μu

ϕ

+u

R

+

(λ

0

+2μ

0

)

a

u

R

−

1 −μ

2

∂u

ϕ

∂μ

,

θϕ

=

μ

0

a

1 −μ

2

∂u

ϕ

∂θ

−μu

θ

−

(μ

0

−τ

0

)

a

1 −μ

2

∂u

θ

∂μ

,

ϕθ

=

(μ

0

−τ

0

)

a

1 −μ

2

∂u

ϕ

∂θ

−μu

θ

−

μ

0

a

1 −μ

2

∂u

θ

∂μ

,

Rθ

= τ

0

∂u

θ

∂R

,

Rϕ

= 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

∂

Rθ

∂θ

−

1 −μ

2

∂

Rϕ

∂μ

+μ

Rϕ

−

θθ

+

ϕϕ

a

,

(∇

S

· )

θ

=

1

a

1 −μ

2

∂

θθ

∂θ

−

1 −μ

2

∂

θϕ

∂μ

+μ

θϕ

+

ϕθ

+

Rθ

a

,

(∇

S

· )

ϕ

=

1

a

1 −μ

2

∂

ϕθ

∂θ

−

1 −μ

2

∂

ϕϕ

∂μ

+μ

ϕϕ

−

θθ

+

Rϕ

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

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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

,λ

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

+χ

2

,(18)

¯σ

θθ

T = −χ

1

+χ

2

,(19)

¯σ

ϕϕ

T = −3χ

1

cos

2

ϕ

+2χ

1

+χ

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)

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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

,χ

2

,ω

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

(τ

0

,λ

0

,μ

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

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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,

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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.

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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.

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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))

,χ

2

= −

4

(

1 + ¯ν

)

3(2(1 −2¯ν) +(1 + ¯ν))

,

ω

1

=

2

(7 −5ν +2(4 −5ν))

,ω

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

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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

,λ

0

,μ

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.

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