Effect of residual surface stress and surface elasticity on deformation of nanometer spherical inclusions in an elastic matrix

R.V. Goldstein, V.A. Gorodtsov and K.B. Ustinov / Physical Mesomechanics 13 5–6 (2010) 318–328

318

Effect of residual surface stress and surface elasticity on deformation of nanometer spherical inclusions in an elastic matrix R.V. Goldstein*, V.A. Gorodtsov and K.B. Ustinov Ishlinsky Institute for Problems in Mechanics RAS, Moscow, 119526, Russia The analytical solution of the Eshelby problem, which describes the deformation of an elastic medium inside and outside a spherical inclusion with uniform internal eigenstrain and specified remote stress, is generalized taking into account both surface elasticity and residual surface stress. Expressions are derived for the internal and external Eshelby tensors and stress concentration tensors with regard to the above effects. A characteristic strain field inhomogeneity and its dependence on the inclusion diameter in the nanometer range (the scale effect) are found. It is shown that under certain conditions, the effect of residual surface stress surpasses that of surface elasticity. Keywords: inclusion, elastic matrix, Eshelby problem, Eshelby tensor, stress concentration tensor, surface strain, eigenstrain, residual surface stress

1. Introduction As the size of deformed solids is decreased down to nanometers, scale effects of their physical properties come into play. The classical theory of elasticity lacks characteristics of a medium with length dimension such that this theory fails to describe the scale effect. Describing the observed scale effect of mechanical behavior of nanoobjects, such as nanotubes, nanowhiskers (nanowires), nanoinclusions, thin films, atomic clusters, nanoislands, etc., requires one or another generalization of the theory of elasticity. A possible explanation for the arising scale effect is the impossibility to apply the continuum approximation to nanosizes, i.e., of importance on these scales is the discrete atomic structure of material. The governing factor, in this case, can be peculiar features of the atomic structure of near-surface layers and near-interface regions. The role of these relatively narrow regions can greatly increase in importance where the number of near-surface atoms is no longer too small compared to that of atoms in the rest material. Numerous ab initio calculations and semiempirical molecular simulations, which take into account the atomic structure of materials, and experimental studies confirm the scale effect in nanoobjects of size from fractions to tens of na-

* Corresponding author Prof. Robert V. Goldstein, e-mail: [email protected] Copyright © 2010 ISPMS, Siberian Branch of the RAS. Published by Elsevier BV. All rights reserved. doi:10.1016/j.physme.2010.11.012

nometers. This behavior is found, in particular, in simplified discrete film models (e.g., [1]) and generalized nanotube elasticity models based on molecular simulation (e.g., [2]). In recent years, there has been a dominant trend toward describing the mechanical behavior of nanoobjects in the framework of generalized theory of elasticity in which a nonstandard characteristic is introduced only for surfaces and interfaces of material, while its bulk is treated using the classical theory. In so doing, anomalous surface elasticity is described by various constitutive relations that supplement ordinary Hooke’s law. A theoretical estimate of the role of surface elasticity peculiarities of nanoparticles is rather easy to obtain using the well-known Eshelby problem [3]. This problem consists in determining the stress-strain state of an infinite elastic medium with a spherical inclusion that differs in material from the matrix and experiences uniform eigenstrain. The eigenstrain can be induced by thermal expansion, phase transformation, incompatible atomic lattices of the matrix and inclusion, residual stress, plastic flow, twinning, etc. The analytical solution of the Eshelby problem with regard to additional surface strains described by two-dimensional Hooke’s law and generalized surface elastic moduli is analyzed in detail in [4, 5]. However, the researchers take no account of the so important factor like residual surface stress. In our work, this parameter is taken into account and, using available theoretical estimates of surface elastic moduli and residual stress, it is shown that the latter can be of greater

R.V. Goldstein, V.A. Gorodtsov and K.B. Ustinov / Physical Mesomechanics 13 5–6 (2010) 318–328

significance than the former. We begin our paper with general relations and solutions of the Eshelby problem for a spherical inclusion with internal eigenstrain and surface effects and analyze kinematics, statics and constitutive equations for its spherical boundary. These data are reported in Sect. 2, 3. In Sect. 4, we present a dependence of the strain fields induced by the spherical inclusion on its eigenstrain with regard to surface effects (Eshelby tensor). This section considers in detail only the internal Eshelby tensor components for the strain field inside the inclusion; certain of cumbersome coefficient functions entered in the solution are given in Appendix. A distinctive feature of the derived solution is a nonuniform strain field and a scale effect (the dependence of the solution on the inclusion diameter). Section 5 provides a solution of the more general problem on the stress-strain state inside and outside the inclusion with specified internal eigenstrain and specified remote stress fields with regard to elastic strain and residual stress at the “elastic matrix – inclusion” interface. Finally, Section 7 discusses the association between the interface constitutive equations used and the known, by and large, more general Gurtin–Murdoch constitutive equations [6, 7]. The effects under consideration, in our viewpoint, are important for the further advances in physical mesomechanics [8, 9] whose main approaches concern, among other things, the description of the processes occurring on various scales at the meso-to-nanolevel transition. 2. Relations for the medium inside and outside the inclusion In the subsequent discussion we assume that the medium under study can be considered as a piecewise-homogeneous medium in which every subregion is described by equations of the linear theory of elasticity with possible eigenstrain (initial or residual). In this context, due to the problem linearity, the strain H for the k-th homogeneous region can be expressed as the sum of elastic and inelastic components: HijkT

Hijk  Hijk 0.

(1)

Hereinafter the second upper index T stands for total strain, and the second upper index 0 for eigenstrain. If no second upper index is used, it is for elastic strain. The first upper index k characterizes the region under study; this index can go with e (matrix) and i (inclusion). The lower indices denote components of tensor quantities. In the problem, which is similar to the known Eshelby problem, we assume that the infinite elastic matrix is everywhere free from eigenstrain, and the inclusion experiences homogeneous eigenstrain. The assumption of homogeneity is sufficient to resolve both the strain and the displacements into elastic and inelastic components:

HeT ij

Hije , HiT ij

Hiij  Hi0 ij ,

319

(2) U ieT U ie, U iiT U ii  U ii0. Here the indices for U are similar to those for the strain. In view of the problem linearity, it is sufficient to further consider uniaxial eigenstrain, whereupon we can arrive at a general solution by simple superposition of the solutions corresponding to eigenstrain in different directions. The elastic displacement field inside and outside the spherical inclusion can be represented in spherical coordinates r, T, M as follows (see, e.g., [10], Sect. IV, formulae (1.8), (1.10)):

U ri

N

¦ ª¬ An (n  1)(n  2  4Qi )r n 1  Bn nr n 1 º¼ u

n 0

u Pn (cos T), r d R , U Ti

N

¦ ª¬ An ( n  5  4Qi ) r n 1  Bn r n 1 º¼ u

n 0

dP (cos T) , r d R, u n dT U re

N

¦ ª¬Cn n( n  3  4Qe ) r  n  Dn ( n  1) r  n2 º¼ u n 0

(3)

u Pn (cos T), r t R , U Te

N

¦ ª¬Cn (n  4  4Qe ) r  n  Dn r  n2 º¼ u

n 0

dPn (cos T) , r t R. dT Here R is the inclusion radius; Pn ( x ) is the Legendre polynomials; Qi, Qe are Poisson’s ratios for the inclusion and matrix, respectively; An , Bn , Cn , Dn are the coefficients to be determined. The discussion below makes clear that for the problem under study we suffice to retain the lowest terms with n d 2, while the terms with n = 1 should be omitted because they are related to the resultant force vector, which is absent in the problem. In the inclusion problem with eigenstrain, the total displacements inside the inclusion are elastic displacements (3) and eigendisplacements due to the uniform tensile eigenstrain Hi0 along the z axis: u

U ri0

H i0 r cos 2 T,

(4) H i0 r sin T cos T. The elastic strain inside and outside the inclusion is expressed in terms of the elastic displacements in the ordinary way:

U Ti0

H krr k HMM

H krT

1 wU Tk U rk wU rk k  , , HTT wr r wT r U Tk Uk ctg T  r , r r k 1 § wU T U Tk 1 wU Tk ·   ¨ ¸. 2 ¨© wr r r wT ¸¹

(5)

320

R.V. Goldstein, V.A. Gorodtsov and K.B. Ustinov / Physical Mesomechanics 13 5–6 (2010) 318–328

For elastic isotropy of the matrix and inclusion, the stress is related to the elastic strain as follows:

Vkrr

k k k 2P k Hrr  Ok (Hrr  HkTT  HMM ),

VkTT

k k k 2P k HTT ),  O k (Hkrr  HTT  H MM

VkMM

2Pk H kMM

Vkr T

2Pk H rkT . k

O

k

(Hkrr

 H kTT

k ),  H MM

(6)

Here O k , P are the Lamé constants of the k-th phase. In the problem posed, finding the stress and displacement fields inside and outside the inclusion requires additional boundary conditions for the derived relations; these conditions are relations for surface (interface) elasticity to which we now pass on. 3. Relations for the interface Like bulk elasticity, surface elasticity is described by two groups of variables and three groups of equations. The two groups of variables are kinematic variables for displacements and strains, and static variables for surface stress; the three groups of equations are kinematic equations for components of surface and bulk displacements and strains, static equations for components of surface and bulk stresses, and constitutive equations for the two groups of variables. 3.1. Surface kinematics Let us restrict out consideration to a bilaterally coherent interface, i.e., we assume that the total surface displacements U isT coincide with the total bulk displacements of the two volumes (inclusion and surrounding matrix): (7) U isT U ieT U iiT, i.e., the total displacement vector is a continuous function of coordinates. The total surface strain is related to the total displacements by ordinary formulae. The continuity of the normal and tangential displacement vector components (with respect to the surface) requires the absence of surface strain tensor components with indices of the normal, i.e., sT HsT ni and H nn ). The total surface strain tensor becomes a projection of the total bulk strain tensor onto a plane tangential to the interface. Hence, each of the tensor indices of surface strain can take on values from 1 to 2, rather than from 1 to 3 like in the spatial case. This decrease in the dimensionality of the total surface strain tensor corresponds to the continuity of total displacements (7). The continuity of the total displacements along the interface involves continuity of all derivatives of the total displacements with respect to coordinates tangential to the interface such that the continuity condition for the total strain components with tensor indices 1 and 2, like that for the total displacements, is bound to hold: eT iT . HsT HDE HDE (8) DE For the axisymmetrically deformed sphere under consideration, the relation between the total surface strains and

the total displacements are similar to the second and third relations in (5); we are just to replace the index k for the matrix and inclusion by the index s for the interface and to add the index T for the total displacements and strains:

1 wU TeT U reT U TeT U eT sT (9) ctg T  r . , HMM  r wT r r r Note that the above relation for surface and bulk kinematics is not unique [11–13]. In this work, we assume the validity of continuity relations (7) – (9), (1) – (3); however, other forms of relations, e.g., for interface sliding, interface reconstruction, and other elastic and inelastic interactions between the surface and bulk components, must not be ruled out. HsT TT

 

 

 

 

3.2. Surface statics Let us assume that surface equilibrium is described by the generalized Young–Laplace law [4 –7]:  

s

i

e

’s V  [V ]n 0, [V] V  V , (10) s where V is the surface stress tensor. This law can be derived with resort to the force balance in coordinates related to a surface element, i.e., it is an analogue of equilibrium equations. The surface gradient ’ s can thus be expressed in the form [4, 5]: ’sós 

§ Vs Vs  ¨¨ 11  22 © R1 R2

· e1 ¸¸ n  h h 1 2 ¹

ª w s  h2V11 « wD ¬ 1

wh s wh2 s º w  V22 »  h1Vs21  1 V12 wD 2 wD 2 wD1 ¼

ª wh1 s w s «  wD V11  wD h2 V12  ¬ 2 1 º w wh  h1Vs22  2 Vs21 » . wD1 wD1 ¼ 

e2 h1h2

(11)

Here e1 , e 2 are two mutually orthogonal basis vectors in a plane tangential to the interface; n is the normal unit vector; D1 , D 2 are two parameters that determine the interface such that the curves with D1 const, D 2 const specify two mutually orthogonal families on the surface; h1 , h2 are the corresponding metric coefficients; R1 , R2 are the principal curvature radii; Vsij is the surface stress tensor components. In the particular case of a spherical surface, formulae (10), (11) take the form:

Verr  Virr 

VsMM  VsTT R

0,

(12) VsTT  VsMM 1 w s 0.  VTT  R tg T wT R These expressions are also derived from equilibrium equations of a spherical shell under axisymmetric load (see [14],

VerT

 Vri T

R.V. Goldstein, V.A. Gorodtsov and K.B. Ustinov / Physical Mesomechanics 13 5–6 (2010) 318–328

§105) and correspond to the equilibrium equations written in spherical coordinates. 3.3. Constitutive equations for the surface For the system of equations to be supplemented with boundary conditions, we are to derive constitutive equations for the surface stress tensor components and strain tensor components. Now, many forms of the constitutive equations are available. Without going into detail, we would like to make some remarks on these equations. The constitutive equations can be nonlinear and contain initial (residual) surface stress components or surface eigenstrain components. The elastic parameters entered into the constitutive equations, in general, correspond to an anisotropic solid. The elastic surface parameters and the initial surface stress can both depend on the crystallographic surface orientation, i.e., on the position of a selected point and local basis orientation at the inclusion surface. The theoretical constraints on the values of surface moduli are rather weak, because one should take into account not only the surface, but also its adjacent volumes such that the requirement of positive elastic energy is applicable only to the solid as a whole, and not to its surface portion. Therefore, unlike purely bulk elasticity, surface elasticity can have many negative moduli, as evidenced by the theoretical calculations of elasticity of crystalline materials made in [15]. In our further analysis, we use constitutive equations of the particular form: VsTT

s s sT s sT Vs0 TT  (O  2P ) H TT  O H MM s0 s sT s0 (O s  2P s )(HsT TT  H TT )  O (H MM  H MM ),

VsMM

s sT s s sT Vs0 MM  O H TT  ( O  2P ) H MM

(13)

s0 s s sT s0 O s ( HsT TT  H TT )  ( O  2P )( H MM  H MM ).

Here O s, Ps are surface elastic moduli similar to the Lamé s0 constants for bulk isotropic elasticity; Vs0 TT , VMM are the residual surface stress components. Because these constitutive equations take into account the residual (initial) stress, they are more general than those used in [4, 5]. The first equalities in (13) describe the operating surface stress through s0 the residual stress Vs0 TT , VMM and total surface strains, and the second equalities through the surface strain and surface s0 eigenstrain Hs0 TT , H MM . The relation between the residual stress and the eigenstrain is thus apparent:

Vs0 TT

s s0 (Os  2Ps )Hs0 TT  O HMM ,

Vs0 MM

Os Hs0 TT

s

s

 (O  2 P

)Hs0 MM .

(14)

We can also derive constitutive equations (13) using the expression for the energy density and surface strain:

F

Os  2Ps sT s0 2 Os  2Ps sT s0 2 (H11  H11 )  (H22  H 22 )  2 2 sT s0 s0 s sT s0 2 )(HsT  Os (H11  H11 22  H22 )  2P (H12  H12 ) .

(15)

321

The assumption of infinitesimal of both the elastic strain s0 Hik and the surface eigenstrain Hi0 ik , Hik allows us to restrict the consideration of the problem to small strains. In the constitutive equations derived in [4, 5], the terms corresponding to the residual surface stress (surface eigenstrain) were discarded assuming their infinitesimal. However, according to the atomic calculations of the surface elasticity characteristics for cubic crystals [15], this is not quite correct. In particular, the authors of [15] found that the surface elastic moduli are ~10 N/m and are a mere order of magnitude greater than the residual surface stress (Vs0 ~ 1 N/m). In the range of linear strain H d 0.01, this residual stress is a major contributor to the surface stress (the stress corresponding to the elastic surface strain is Vs Os H ~ 0.1 N/m). Therefore, our constitutive equations take into account the residual surface stress alongside the contribution to deformation by Hooke’s law. An additional originality to the used constitutive equations is imparted by the assumption of anisotropy of residual surface stress that partially reflects anisotropy of interface structural reconstruction. According to the calculations made in [15], the difference in normal residual stress components for certain crystallographic planes can reach tens of percents of the average. In our further calculations, we assume that the surface eigenstrains are compatible such that there exists a surface displacement vector that can be represented in a form similar to (4): U rs0

Hs0 R cos 2 T,

U Ts0

Hs0 R sin T cos T,

VsTT

e s e s0 (Os  2Ps )(HTT  Hs0 TT )  O (HMM  H MM )

(16)

where Hs0 is the effective uniaxial strain responsible for the above surface displacements. Note that constitutive equations (13) can be rewritten in terms of the elastic strain differing for the internal and external interface regions. Substitution (8) and (2) in (13) gives s0 (Os  2Ps )(HiTT  Hi0 TT  H TT )  s0  Os (HiMM  Hi0 MM  HMM ),

VsMM

(17)

s s e s0 Os (HeTT  Hs0 TT )  (O  2P )(H MM  H MM ) s0 s s i i0 s0 Os (HiTT  Hi0 TT  H TT )  (O  2P )(H MM  H MM  H MM ).

In the particular case Hs0 Hi0, we again arrive at the constitutive relation in which the unstressed state of the surface coincides with the unstressed state of the inclusion [4, 5]. In another particular case Hs0 0, the unstressed state of the surface coincides with the unstressed state of the matrix. Then, according to (17), the surface stress can be expressed in terms of either internal elastic strain

322

R.V. Goldstein, V.A. Gorodtsov and K.B. Ustinov / Physical Mesomechanics 13 5–6 (2010) 318–328

VsTT

s i i0 (Os  2Ps )(HiTT  Hi0 TT )  O (H MM  HMM ) s0 i i , VTT  (Os  2Ps )H TT  O sHMM

VsMM

(18)

s s i i0 Os (HiTT  Hi0 TT )  (O  2P )(H MM  H MM ) s i s s i Vs0 MM  O H TT  ( O  2P ) H MM ,

or external elastic strain

VsTT

 

e (Os  2Ps )HeTT  Os HMM ,

(19) VsMM Os HeTT  (Os  2Ps )HeMM . These relations are very cumbersome; it is more convenient to write the constitutive equations in terms of the total strain of the interface. 4. Eshelby tensor 4.1. Displacement field inside and outside the spherical inclusion with uniaxial eigenstrain Taking into account surface effects (13), the solution of the problem of finding displacement, strain and stress fields in a solid with a spherical heterogeneity that experiences uniaxial eigenstrain is reduced to finding the coefficients A0 , A2 , B2 , C2 , D0 , D2 in (3). The coefficients B0 , C0 , as evidenced from expressions (3), have no effect on the results and can be discarded. The other coefficients are found from conjugation of the solutions inside and outside the sphere. Two of the conjugation conditions are the continuity conditions for the radial and tangential surface displacement components:

U ri ( R, T)  U ri0 ( R, T)

U re ( R, T),

(20)

U Ti ( R, T)  U Ti0 ( R, T) U Te ( R, T). (21) Another two conditions are obtained from (12) if we substitute into the this expression the surface stress expressed in terms of the total surface strain coincident, according to (8), with the total bulk strain. The total bulk strain, in turn (see (2)), coincides with the elastic strain in the matrix: (Verr  Virr ) R  2(Os  Ps ) u e e s0 u (HTT  H MM  HTT  Hs0 MM )

0 for r = R,

(22)

2Ps e (H  H e  tgT TT MM w ª s s0 e s0 (O  2Ps )(HTT ) Hs0  HTT TT  HMM )  wT ¬

(VerT  VirT ) R 

O

s

e (HMM

º  Hs0 MM ) ¼

The coefficients of Pn (cos T) with n > 2 disappear. In the particular case Hs0 H0, expressions (A1)–(A6) coincide to an accuracy of notation with the expressions derived in [4, 5]. Taking into account other possibilities for Hs0 adds little complexity to the derived expressions. The solution for the case with no surface stress is obtained through the passage to the limit O s o 0, Ps o 0 (or R o f). For O e O i, P e P i the solution coincides with the wellknown solution obtained by a different method in [3].

0 for r

(23)

R.

Equating the coefficients with like Pn (cos T) in (20) and (22) with regard to (5), (6) gives us six equations (two at a time from (20) and (22), and one at a time from (21) and (23)) in six unknown coefficients A0 , A2 , B2 , C2 , D0 , D2 . A solution of the system is found from formulae (A1) – (A6) presented in Appendix.  

 

4.2. Eshelby tensor components Using the obtained solution, we write the components of the internal and external Eshelby tensors — forth-rank tensors relating the total strain tensor and the eigenstrain tensor in the inclusion: HiT ij

i eT Hi0 Sijkl kl , H ij

e i0 H kl . Sijkl

(24)

Certain of the above components are followed directly from (3) when passing to the Cartesian coordinate system. These components of the internal Eshelby tensor have the form: i 1  2(1  2Qi ) A0c  2B2c  S3333

 3 A2c ª¬ 7( x 2  y 2 )  6Qi ( x 2  y 2  2 z 2 )º¼ , i S1133

2(1  2Qi ) A0c  B2c 

 3 A2c ª¬ (7  8Qi ) z 2  2Qi (3x 2  y 2 )º¼ , i S 2233

2(1  2Qi ) A0c  B2c 

(25)

 3 A2c ª¬ (7  8Q ) z  2Q (x  3 y )º¼ , i

i S1233

12Qi A2c xy ,

i S1333

6Qi A2c xz ,

2

i

2

2

i 6Qi A2c yz. S 2333 The quantities A0c , A2c , B2c , C2c , D0c , D2c are determined by formulae (A2) – (A6) in Appendix. Here the solution is found dependent on coordinates, as is the case in [4, 5]. For the component containing unlike indices in their second pair, the solution can be obtained by superposition of the solutions corresponding to the tensile eigenstrains along the x axis and compressive eigenstrains along the y axis with subsequent rotation of the resulting superposition through an angle S 4 about the z axis. The corresponding components are thus  

 

i S1112

i S 2212

3(7  16 Qi ) A2c xy,

i S1212

1 B2c 3 A2c ª(7  2Qi ) r 2  6Qi z 2 º ,   ¼ 2 2 2 ¬

i S1312

i 9Qi A2c yz, S 2312

(26)

9Qi A2c xz,

i 3(7  10Qi ) A2c xy . S3312 The rest of the components are obtained by circular permui tation of the indices. The derived expressions for Sijkl co-

R.V. Goldstein, V.A. Gorodtsov and K.B. Ustinov / Physical Mesomechanics 13 5–6 (2010) 318–328

incide to an accuracy of notation with the expressions found in [4, 5, 16]. Expressions for the external Eshelby tensor components can be obtained in a similar way; they are derived below. 5. Elastic heterogeneity under specified remote load. Stress concentration tensors Let us now turn to the solution of the more general problem on the stress-strain state of a medium with a spherical inclusion due to its uniform internal eigenstrains and remote stress field. Like in the previous problem, the solution can be simplified using the superposition principle. Namely, as a preliminary we suffice to consider the problem on an infinite elastic solid subjected to uniaxial tensile stress Vfzz far from its spherical inclusion (at infinity) that differs in material from the solid and experiences internal eigenstrains Hi0zz . We assume that the medium inside and outside the inclusion is homogeneous and isotropic and is characterized by elastic constants O e , Pe , Oi , Pi in the matrix and inclusion, respectively. The behavior of the surface is described by relations (7) – (13) as before. The problem on displacement, strain and stress fields is also as before reduced to finding the coefficients A0 , A2 , B2 , C2 , D0 , D2 in (3) from sphere conjugation conditions (20) – (23). The only difference is that the matrix displacements U re , U Te are now meant to be the quantities:  

 

 

U re

2

¦ ª¬Cn n( n  3  4Qe ) r  n  Dn ( n  1) r n 2 º¼ u Vf33 § 2 Qe cos T  r ¨ 2Pe ¨© 1  Qe

2

· ¸¸ , ¹

(27)

dPn (cos T) Vf  33e r sin T cos T, dT 2P that have, unlike the previous expressions, additional terms corresponding to uniform tension along the z axis. The expressions for the internal displacements U ri , U Ti remain the same as (3). Equating the coefficients with like Pn (cos T) in (20) – (23) gives us six equations in six unknown coefficients A0 , A2 , B2 , C2 , D0 , D2 . The linearity of the system of equations makes it possible to represent its solution as the sum of three solutions, s0 f each having only one nonzero value among Hi0 zz , H zz , Vzz :

A2i Hi0  A2s Hs0  A2V Vf ,

B2

B2i Hi0  B2s Hs0  B2V Vf ,

C2

C2i Hi0  C2sHs0  C2VV f ,

D0 D2

ii i0 is s0 iV f Hkl  Tijkl Hkl  Tijkl Vkl , Tijkl

Vije

ei i0 es s0 eV f Vfkl  Tijkl H kl  Tijkl H kl  Tijkl Vkl .

(29)

mi ms mV , Tijkl , Tijkl characterize the The tensor components Tijkl values of the ij-th stress tensor component (for m = i inside and for m = e outside the inclusion) at a unit value of the kl-th component of the inclusion eigenstrain tensor Hi0 kl , surface eigenstrain tensor Hs0 and stress tensor at infinity Vfkl . The kl mV tensor Tijkl is termed a stress concentration tensor [4, 5], mi ms the tensors Tijkl can be termed stress-strain concen, Tijkl tration cross tensors. in en Expressions for the components Tij 33 , Tij 33 (n = i, s, V) are derived through successive substitution of (28) in (3), (5), (6). For the internal tensor components Tijin33 (n = i, s, V) we have

2Pi ª¬ 2(1  Q i )A0n  3(z 2 (7  6Qi ) 

in T1133

 ( x 2  5 y 2 ) Qi ) A2n  B2n º¼ , 2P i ª¬ 2(1  Q i ) A0n  3(z 2 (7  6Q i ) 

2P i ª¬ 2(1  Q i ) A0n  3(2Q i z 2 

in T3333

2

 ( x  y )(7  Q

u

A2

 

Viij

2

n 0

A0i Hi0  A0s Hs0  A0VV f ,

 

 (5 x 2  y 2 ) Q i ) A2n  B2n º¼ ,

¦ ª¬Cn (  n  4  4 Q e ) r  n  Dn r  n 2 º¼ u

A0

Expressions (A3) – (A7) derived from the solution of the system are presented in Appendix. Expressions for the displacements are obtained by substituting (28) in (3), whereupon expressions for the strain and stress by substituting the displacements in (5) and strain in (6). The solution for the stress inside and outside the inclusion is convenient to represent as the expansions:

in T2233

n 0

u Pn (cos T)  U Te

 

323

in T1233

i

)) A2n

24 P i Qi xy A2n ,

in 12 Pi Qi xz A2n , T1333 in 12 Pi Qi yz A2n . T2333

Here A0n , A2n , B2n , C2n , D0n , D2n with n = i, s, V are given by formulae (A3) – (A7) in Appendix. For the components with unlike indices in their second pair, the solution can be obtained through superposition of the solutions corresponding to the tensile eigenstrains along the x axis and compressive eigenstrains along the z axis with subsequent rotation of the resulting superposition through an angle S 4 about the y axis. The corresponding components are thus  

 

in T1113

 6Pi (7  5Qi ) xz A2n ,

in T2213

 6Pi (7  11Qi ) xz A2n ,

D0i Hi0  D0s Hs0  D0VV f ,

in T3313

 6Pi (7  5Qi ) xz A2n ,

D2i Hi0  D2s Hs0  D2V Vf.

in 18 Pi Qi yz A2n , T1213

(28)

(30)

 2 B2n º¼ ,

324

R.V. Goldstein, V.A. Gorodtsov and K.B. Ustinov / Physical Mesomechanics 13 5–6 (2010) 318–328 in 18PiQ i xy A2n , T2313

(31)

in 3 Pi [((7  2Qi ) r 2  6Qi y 2 ) A2n  B2n ]. T1313 The rest of the components are obtained by circular permutation of the indices. Expressions for the external stress tensor components are obtained in the same way. These expressions are very cumbersome, and are presented as formulae (A8) in Appendix. The solution for the strain is convenient represent as the expansions:

Hiij

ii i0 is s0 iV f H kl  Sijkl H kl  Sijkl V kl , Sijkl

Heij

ei i0 es s0 eV f H kl  Sijkl H kl  Sijkl Vkl . Sijkl

(32)

mi ms mV The tensor components Sijkl give values of , Sijkl , Sijkl the ij-th strain tensor components (for m = i inside and for m = e outside the inclusion) at a unit value of the kl-th component of the inclusion eigenstrain tensor Hi0 kl , surface eigenf and stress tensor V at infinity, respecstrain tensor Hs0 kl kl mi tively. The tensors Sijkl can be considered as a variant of generalization of the internal and external Eshelby tensors for the case with surface stress. Note that the authors of [4, 5] considers generalization of the Eshelby tensor in the presence of surface stress as a quantity which is written in our mi ms . As indicated above, the difference notation as Sijkl  Sijkl is due to the difference in the reference point for calculations of the surface strain: in our case, it is the unstressed matrix, and in [4, 5] it is the unstressed inclusion. The tenmi ms sors Sijkl , Sijkl can be termed strain concentration tensors, mV and the tensors Sijkl can be termed strain-stress concentration cross tensors. Using (25), (26) as the base, we can rewrite the internal tensor components in the form: in S3333

Gin  2(1  2Qi ) A0n  2 B2n  2(1  2Qi ) A0n  B2n 

 3 A2n ª¬ (7  8Qi ) z 2  2Qi (3x 2  y 2 )º¼ , in S 2233

2(1  2Qi ) A0n  B2n 

(33)

 3 A2n ª¬ (7  8Qi ) z 2  2Qi ( x 2  3 y 2 )º¼ , in S1233

in 12Qi A2n xy , S1333

in S 2333

6Qi A2n yz ,

in S1112

in S 2212

in S1212

1 in B2n 3 A2n ª (7  2Qi )r 2  6Qi z 2 º , G   ¼ 2 2 2 ¬

in S1312

in 9Qi A2n yz , S 2312

6Qi A2n xz ,

in S3312

3(7  10Qi ) A2n xy .

3(7  16Q i ) A2n xy ,

9Qi A2n xz ,

 

 

6. Estimation of the role of surface effects In this section, we estimate the role of surface effects associated with elastic interface anomalies using the derived solutions in the simple situation of a spherical inclusion with uniform tensile eigenstrain H0 and surface eigenstrain Hs0. The particular case Hs0 H 0 is considered in [4, 5, 16]. For this case, the unstressed surface state corresponds to a deformed inclusion (the inclusion surface is unstressed in response to the eigenstrains), and for Hs0 0, the unstressed surface state corresponds to an undeformed inclusion (the matrix surface is not stressed in response to the eigenstrain). An expression for the total strains in the inclusion can be derived as a superposition of the solutions corresponding to three orthogonal eigenstrains. Substitution of (A3) – (A9) in (25) and (24) gives  

HiT

 

3K i  2 K s R H0  i e s 3K  4P  2 K R

2K s R Hs0 , (35) e s 3K  4P  2 K R where K i is the uniform compression modulus of the inclusion; K s 2(Os  P s ) is the surface elastic modulus (an analogue of the uniform compression modulus) that can go negative; R is the inclusion radius. Power series expansion of expression (35) in terms of the dimensionless parameter 

i

2K s (36) RPe which is small for inclusions of rather large radius, with only zero- and first-order terms left for the small parameter gives · 3K i § 4Pe2 Pe 1 HiT H 0 G  KG ¸¸ . (37) i e¨ i i e i 3K  4P ©¨ 3K (3K  4P ) 3K ¹ Here, we also use the dimensionless ratio of the surface eigenstrain to the inclusion eigenstrain: K { Hs0 H0 . In the particular case Hs0 H0, the formula for the strain is simplified as follows: · 3K i § Pe 1 (38) HiT H0  G ¸, ¨ i e¨ i e ¸ 3K  4P © 3K  4P ¹ and in the particular case Hs0 0, it is reduced to the form: G

 3 A2n ª¬ 7( x 2  y 2 )  6Qi ( x 2  y 2  2 z 2 )º¼ , in S1133

Here Gin 1 for n = i, and Gin 0 for n = s, V. The quantities A0n , A2n , B2n , C2n , D0n , D2n for n = i, s, V are given by formulae (A3) – (A7) in Appendix. The rest of the components are obtained by circular permutation of the indices. Expressions for the external concentration tensor components are obtained in the same way. They are cumbersome and are represented as formulae (A9) in Appendix.

(34)

ª º 3K i 4P 2 1  G». (39) « i e i i e 3K  4P ¬ 3K (3K  4P ) ¼ Note that in these particular cases, the corrections from the surface effect are opposite in sign. HiT

H0

R.V. Goldstein, V.A. Gorodtsov and K.B. Ustinov / Physical Mesomechanics 13 5–6 (2010) 318–328

Formulae (37) – (39) for symmetric spherical deformation (expansion) of the inclusion allow rather simple estimation of the effect (including relative) of surface elastics0 ity and surface eigenstrain Hs0 (residual surface stress V ) on deformation. For many hard materials (e.g., metals), the elastic coefficients of cubic expansion are typically K | 2.5, P| 1011 N/m2. The desired surface characteristics K s and Vs0 can be estimated from the results of theoretical calculations for cubic metals obtained in [15]. By order of magnitude, we have K s ~ Os ~ Ps ~  10 N/m and Vs0 ~ 1 N/m, Hs0 ~ 0.1. The latter estimate of the surface eigenstrain is much higher than typical bulk eigenstrains in the inclusion material; they can also differ in sign. The above estimates for the bulk and surface parameters give the estimate for the dimensionless parameter of their relative effect:  

 

10 9 m (40) . R According to this estimate, the key parameter in the examined problem becomes small where the inclusion diameter is greater than tens of nanometers. Here we should note that according to (37), the effect of surface eigenstrain (residual surface stress) on the strain fields of materials depends on the product of the “small” dimensionless complex G and dimensionless ratio K { Hs0 H 0, whose probable value from the previous estimates is not so small. Thus in the presence of residual surface stress, the variation in the bulk strain field is mainly determined by this rather large product of the dimensionless parameters: G|

Hs0 9 10 ì. (41) H0 R At the same time, with no residual stress the effect of surface elasticity alone, according to (39), is determined by a somewhat smaller parameter G. The foregoing analysis makes fairly clear that the constitutive equations for the surface effect on the stress-strain state of the inclusion should take into account the residual surface stress along with the surface elasticity. Moreover, the role of the residual stress can be found decisive. KG |

7. Comments on the association of the used constitutive equations for the surface effect with the Gurtin–Murdoch constitutive equations Generally speaking, the linearized constitutive equations for surface elasticity derived in [6, 7] differ greatly from (13). Their componentwise form with the residual stress Vs0 is s S11 Vs0 (1  H22 )  (O0  2P0 )H11  O0 H22 , s S 22

Vs0 (1  H11 )  O 0 H11  (O0  2P0 )H22 ,

s S12

2P0 H12  Vs0 (u1, 2  H12 ),

s S 21

2P0 H12  Vs0 (u1, 2  H12 ),

(42)

325

where Sijs is the Piola–Kirchhoff surface stress tensor components determined as the force to area ratio for the reference configuration; Hij is the surface strain tensor components; O 0 , P 0 are the Lamé surface constants that by and large differ from the constants O s , Ps introduced by us; ui is the surface displacement vector components. The authors of [6, 7] emphasize that the form of constitutive equations (42) for the surface stress and strain was dictated by the desired constancy of the Cauchy stress, i.e., the stress in the actual state, for zero surface elastic moduli O 0 , P 0. In the particular case of liquid media, it is reasonable to assume that O0 P0 0, and the Cauchy stress (the force to s area ratio for the current configuration) is V11 s s0 s s s0 S11 (1  H22 ) V , V22 S22 (1  H11 ) V . It can be seen from (42) that with O0 P0 0, the Piola–Kirchhoff stress components Sijs increase with an increase in surface area. However, for solids where a new surface is not formed and all strains (and, hence, an increase in area) are due to tension of the existing surface, the absence of elasticity (zero surface elastic moduli) is more natural to relate to strainindependent Piola–Kirchhoff stress. It is these moduli, i.e., the moduli whose vanishing results in a constant Piola– Kirchhoff stress tensor, that we denote as Os , Ps . The assumed condition is written as follows: s S11

s S 22

Vs0 for Os

(43) Ps 0. Let us find a relation between the moduli O s , Ps and O 0, P 0 . For this purpose, we introduce the constant coefficients A, B according to 0 s s0 (44) O 0 Os  AVs0 , P P  BV . These coefficients are subject to determination from condition (43). Substitution of expressions (44) in constitutive equations (42), and then in Piola–Kirchhoff stress constancy condition (43) gives the constants A = –1, B 1 2. After substitution of the values in expressions (44) and constitutive equations (42), the latter are transformed to the form:

s S11

Vs0  (O s  2Ps )H11  Os H 22 ,

s S 22

Vs0  Os H11  (Os  2Ps )H22 ,

s S12

2Ps H12  Vs0 (u1,2  u2,1 ) 2,

s S 21

2Ps H12  Vs0 (u1,2  u 2,1 ) 2.

(45)

The derived relations for the normal stress coincide with introduced constitutive equations (13) at constant residual surface stress. For nonconstant (variable with coordinates) residual stress, the situation is complicated: from (44) it follows that either Os , Ps or O0 , P0 (or both) become functions of coordinates or strains. However, this dependence is of the order 1  Vs0 O 0 | 1  H s0 and is negligible if only the eigenstrains (the strains corresponding to the residual stress) are small (as we can see even from formulae (42)).

326

R.V. Goldstein, V.A. Gorodtsov and K.B. Ustinov / Physical Mesomechanics 13 5–6 (2010) 318–328

Appendix The coefficients entered in the earlier derived expression for the displacements, strains, and stresses have the following form:

A0

A0c Hi0, A2

A2c Hi0, B2

B2c Hi0,

C2

C2c Hi0, D0

D0c Hi0, D2

D2c Hi0;

A0c

A0i  A0s Hs0 Hi0, A2c

A2i  A2sH s0 Hi0,

B2c

B2i  B2s Hs0 Hi0 , C2c

C2i  C2s Hs0 Hi0,

D0c Here

D0i  D0sH s0 H i0, D2c

D2i  D2s Hs0 Hi0.

A0i

Os  Ps  RP e , A0s '0

D0i

R 4Pi (1  Qi ) , D0s  '0

A2i



A2s B2i

D2s

4Pe (7  10Qi ))]  R(O s  2Ps ) u Pi (119  107Qi  16Qe (7  10Q i )))),

(A2)

'0

(A3) 2 R3 (Os  Ps )(1  2Qi ) ,  '0

'2

7Pi (4  5Qe )(1  Qi ))  2Pi (4  5Qe )))(O s (14  20Q i )) 

(A4)

 R(Pe (28  40Qi )  Pi (7  5Qi )))),

s

B2V

1 (15 R 2 (1  Qe )(56Ps (1  Q i )  2'2 4Pe ( 7  10Qi )))),

R (2(Os  2Ps )2 (4  5Q e )(7  10Q i )  '2

C2V

i

e

s

D0V

 R (Pi (7  5Qi )  4Pe (7  10Qi )))), C2i

4Pe (7  10Qi ))))),

i

O (4  5Q )( 2O (7  10Q )  R5Pi (7(5R (4P e  Pi )  24Os  28Ps )  2' 2 i

s

s

e

(1  2Qi )( RPe (1  Q e ) 

5Q (48O  56P  5R (8P  P ))), C2s

s

D2V

s

i

i

i

P (20P (7  10Q )  7 RP (7  5Q )  e

i

28RP (7  10Q ))), D2i

2Ps ( 1  2Qe )  Os ( 2  4Q e )))),

4

 RPi (7  5Qi )  20Ps (7  10Qi )) 

7 i

R P (2(Os  2Ps )(1  4Qe )(7  10Qi )  '2

1 ( R 3 (Pe ) 1 u 2(1  Qe )' 0 u( RPi (1  2Qe )(1  Q i ) 

i

R (O s (4 RP e (7  10Qi )  2'2

1 (5 R 4 (Pe )1 (4Ps (Os  Ps )( 7  10Q i )  4'2

 R (49PiPs (1  Qi )  Os (Pi (35  47Qi ) 

e

2P (7  16Q  (11  20Q )Q ))  s



 R 2 (P e  Pi )(Pi (7  5Qi )  4P e (7  10Q i )) 

7 R (Os  2Ps )(Pi (4  5Qe )(1  Qi )  i



6 Os ( 7  8Qi )  R(Pi (7  5Qi ) 

i

 R (P (7  5Q )  4P (7  10Q )))),

e

15(Os  2Ps )(1  Qe ) , '2



i

e

3 R(1  Qe ) , A2V 4(1  Q e )' 0

A0V

O (4  5Q ))(2O (7  10Q ) 

B2s

(A6)

7 R (Os  2Ps )(6Pe (1  Qe )(7  10Qi ) 

6Pe (1  Qe )(7  10Q i ))  (RPe (7  5Q e )  i

3R (2(Os  2Ps )2 (4  5Q e )(7  10Q i ) 

(Os (4  5Qe )  R(Pe ( 7  5Q e ) 

7 R (Os  2Ps )(Pi (4  5Qe )(1  Q i ) 

i

3 RPi (1  Qi )  6(Os  Ps  RPe )(1  2Qi ),

2Pi (4  5Qi )  Pe (7  5Q e ) , '2

e

R6 (6(Os  2Ps )2 (7  10Qi )  2'2

u(4P e (7  2Q e )(7  10Q i ) 

Os  Ps ,  '0

R (2(Os  2Ps )2 (4  5Q e )(7  10Q i )  '2

s

(A5)

3Os [2Os (7  10Qi )  R (Pi (7  5Qi ) 

(A1)

4Pi (O s  2Ps )(4  5Qi ) , '2

2(Os  2Ps )

3( RPi (7  5Qi )  2(Os  2 RPe )(7  10Q i ))),



1 (3 R 6 (P e ) 1 ( 4Ps (Os  P s )(7  10Q i )  2'2

 R 2 (P e  Pi )(Pi (7  5Qi )  4Pe (7  10Qi ))  R (Os (Pi (35  47Q i )  4Pe Q e (7  10Q i ))  Ps ( 49Pi (1  Qi )  8P e (1  Q e )(7  10Q i ))))).

(A7)

R.V. Goldstein, V.A. Gorodtsov and K.B. Ustinov / Physical Mesomechanics 13 5–6 (2010) 318–328

The external stress concentration tensor components are

1 (P(2 r 2 (r 4 (5  4Q )  r 9 (1  2Q)

en T1133 2

2

2

en T2213

 r (4 y (4  5Q )Q  z (9  30Q  24Q ) 

uC2n  5( x 2  6 y 2  z 2 )(1  2Q ) D2n )),

 x 2 (15  26Q  4Q 2 ))  5z 2 (2( y 2  z 2 ) u u Q(5  4Q )  x (9  8Q  8Q

2

)))C2n 2

1 (3 xzP(6r 2 ( z 2  4 z 2 Q  2r 2 Q2  r (1  2Q ) 9

2 z 2Q 2  2 y 2 (2  3Q  Q 2 )  x 2 (1  4Q  2Q 2 )) u

2

2

en T3313



1 (3 xzP (6r 2 ( x 2 (1  4Q  2Q 2 )  r (1  2Q ) 9

 2r 4 (3x 2 (1  Q )  3( y 2  z 2 )Q  r (1  Q ))D0n 

 y 2 (1  4Q  2Q 2 )  2(r 2 ( 1 Q )Q 

 3(r 4  7 z 2 ( x 2 (5  8Q)  2( y 2  z 2 )Q ) 

 z 2 (2  3Q  Q 2 )))C2n 

 r 2 ( 2 y 2 Q  z 2 (5  6Q)  x 2 (5  8Q))) D2n )), 1 en (P(2 r 2 (r 4 (5  4Q )  r 2 (4x 2 (4  5Q )Q  T2233 9 r (1  2Q)  5 z 2 (2( x 2  z 2 )Q (5  4Q )  y 2 (9  8Q  8Q 2 )))C2n  2

2

2

 2r (r (1  Q)  3( y (1  Q )  (x  z 4

2

2

2

2

)Q ))D0n



en T1213

en T1313

2

 3( r  7 z ( y (5  8Q)  2( x  z )Q ) 



1 (P( 2 r 2 (2r 4 Q (5  4Q )  9 r (1  2Q)

2

2

2

2

2

 r ( x (1  6Q  4Q )  y (1  6Q  4Q )  6 z 2 (5  6Q  4Q 2 ))  5z 2 (2 z 2 (5  9Q  4Q 2 )   x 2 (1  8Q2 )  y 2 (1  8Q 2 )))C2n  2r 4 ( r 2 (1  Q )  3( z 2 (1  Q )  ( x 2  y 2 )Q )) D0n  3(2r 4Q  7 z 2 (2 z 2 ( 1  Q )  x 2 ( 3  8Q )   y 2 (3  8Q ))  r 2 (2 z 2 (3  2Q )  x 2 ( 3  8Q )   y 2 (3  8Q ))) D2n )), en T1233

en T1333

1

1 ( xzP(2 r 2 (5( x 2 (1  8Q )  y 2 (1  8Q )  9 2r  z 2 (19  8Q ))  7 r 2 (5  4Q )C2n  3( 4r 4 D0n  

en T2313

1 ( yzP(2r 2 (5( x 2 (1  8Q)  y 2 (1  8Q )  9 2r  z 2 (19  8Q ))  7r 2 (5  4Q ))C2n  3( 4r 4 D0n  

(9r 2  7(3x 2  3 y 2  7 z 2 ))D2n ))), en T1113



1 r 9 (1  2Q)

(3xzP ( 6r 2 ( z 2  2 z 2Q  2r 2Q 2 

(3 yzP(6r 2 ( x 2 ( 5  Q)  ( y 2  z 2 )Q)C2n 

r 5(6 x 2  y 2  z 2 ) D2n ), 1 9

(3P(2r 2 ( x 4  y 4  z 4  r 2 z 2 Q 

r 2 y 2 ( z 2  r 2 Q )  x 2 (2 y 2  13z 2  r 2 Q )) u

1

(3xyP (6r 2 ( z 2 (5  Q )  ( x 2  y 2 )Q )C2n  r9 5( x 2  y 2  6 z 2 ) D2n ). Here for brevity, the upper indices e of the shear moduli P e and Poisson’s ratio Q e of the matrix are omitted. The external strain concentration tensor components have the form: 1 en (2r 2 (45x 2 z 2  r 4 (5  4Qe )  S1133 2r 9 r 2 (9 z 2  3x 2 (5  4Qe )))C 2n  2 r 4 ( r 2  3 x 2 )D0n  3(r 4  35x 2 z 2  5r 2 ( x 2  z 2 )) D2n ), en S 2233

1 (2r 2 ( 45 y 2 z 2  r 4 (5  4Qe )  2r 9 r 2 (9 z 2  3 y 2 (5  4Qe )))C 2n  2r 4 (r 2  3 y 2 )D0n  3(r 4  35 y 2 z 2 

(9 r 2  7(3 x 2  3 y 2  7 z 2 )) D2n ))), en T2333

9

3 x 2 ( y 2  9 z 2 )) D2n )),

(3 xyP(2r 2 (15z 2  r 2 ( 5  4Q ))C2n  r9 2r 4 D0n  5(r 2  7 z 2 )D2n )), 

1

uC2n  (4 x 4  y 4  3 y 2 z 2  4 z 4 

 r 2 (2 x2 Q  z 2 (5  6Q)  y 2 (5  8Q))) D2n )), en T3333

(A8)

5(3 x 2  3 y 2  4 z 2 )(1  2Q )D2n )),

 z 2 (9  30Q  24Q 2 )  y 2 (15  26Q  4Q 2 ))  4

327

5r 2 ( y 2  z 2 ))D2n ), en S3333

1 (2r 2 ( 5z 2 (2 z 2 (5  4Qe )  2r 9  x 2 (1  8Qe )  y 2 ( 1  8Q e ))   r 2 (6 z 2 (5  4Q e )  x 2 (1  8Q e )   y 2 (1  8Q e )))C2n  2r 4 (r 2  3z 2 ) u

2 z 2 Q2  y 2 (1  2Q  2Q 2 )  2x 2 (2  4Q  2Q 2 ))C2n 

u D0n  3(3r 2 ( x 2  y 2  2 z 2 ) 

5(4 x 2  3( y 2  z 2 ))(1  2Q )D2n )),

7 z 2 (3 x 2  3 y 2  2 z 2 ))D2n ),

328

R.V. Goldstein, V.A. Gorodtsov and K.B. Ustinov / Physical Mesomechanics 13 5–6 (2010) 318–328 en S1233

en S1333

1 (3 xy (2r 2 (15z 2  r 2 (5  4Q e ))C2n  2r 9 2 r 4 D0n  5( r 2  7 z 2 ) D2n ),

The work was performed under Fundamental Research Program No. 22 of the Presidium of RAS.

1  9 ( xz (2r 2 (7r 2 (5  4Qe )  4r 5( x 2  y 2  19 z 2  8 x 2Q e 

[1] A.M. Krivtsov and N.F. Morozov, Anomalies in mechanical characteristics of nanometer-size objects, Dokl. Phys., 46, No. 11 (2001) 825. [2] R.V. Goldstein, V.A. Gorodtsov, A.V. Chentsov, S.V. Starikov, V.V. Stegailov and G.E. Norman, To description of mechanical properties of nanotubes. Tube wall thickness problem. Size effect / Preprint No. 937, Inst. Probl. Mech., Moscow, 2010. [3] J.D. Eshelby, The determination of the elastic field of an ellipsoidal inclusion, and related problems, Proc. Roy. Soc. L. A, 241, No. 1226 (1957) 376. [4] H.L. Duan, J. Wang, Z.P. Huang and B.L. Karihaloo, Eshelby formalism for nanoinhomogeneities, Proc. Roy. Soc. L. A, 461, No. 2062 (2005) 3335. [5] H.L. Duan, J. Wang and B.L. Karihaloo, Theory of elasticity at the nanoscale, Adv. Appl. Mech., 42 (2008) 1. [6] M.E. Gurtin and A.I. Murdoch, A continuum theory of elastic material surfaces, Arch. Ration. Mech. Anal., 57, No. 4 (1975) 291; V. 59, 389. [7] A.I. Murdoch, Some fundamental aspects of surface modeling, J. Elast., 80 (2005) 33. [8] Physical Mesomechanics of Heterogeneous Media and ComputerAided Design of Materials, Ed. by V.E. Panin, Cambridge Interscience Publishing, Cambridge, 1998. [9] V.E. Panin, Synergetic principles of physical mesomechanics, Phys. Mesomech, 3, No. 6 (2000) 5. [10] V.Z. Parton and P.I. Perlin, Methods of Mathematical Theory of Elasticity, Nauka, Moscow, 1981 (in Russian). [11] R.C. Cammarata, K. Sieradzki and F. Spaepen, Simple model for interface stresses with application to misfit dislocation generation in epitaxial thin films, J. Appl. Phys., 87, No. 3 (2000) 1227. [12] R.C. Cammarata, Surface and interface stress effects in thin films, Progr. Surf. Sci., 46, No. 1 (1994) 1. [13] J.W. Cahn and F. Larche, Surface stress and the chemical equilibrium of small crystals. II. Solid particles embedded in a solid matrix, Acta Met., 30, No. 1 (1982) 51. [14] S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, McGraw-Hill, New York, 1959. [15] V.B. Shenoy, Atomic calculations of elastic properties of metallic fcc crystal surfaces, Phys. Rev. B, 71, No. 9 (2005) 94. [16] P. Sharma, S. Ganti and N. Bhate, Effect of surfaces on the sizedependent elastic state of nano-inhomogeneities, Appl. Phys. Lett., 82, No. 4 (2003) 535.



8 y 2Q e  8 z 2 Qe ))C2n  3(4r 4 D0n  en S1113

(9 r 2  21x 2  21y 2  49 z 2 )D2n ))), 1 3xz (6r 2 (2 x 2 ( 2  Q e )  9 2r ( y 2  z 2 )(1  2Qe ))C2n 

(A9)

5(4 x 2  3( y 2  z 2 ))D2n ), en S 2213

en S3313

en S1213

en S1313

1 (3 xz (6r 2 ( x 2  4 y 2  z 2 )C2n  2r 9 5( x 2  6 y 2  z 2 ) D2n )), 1 3 xz (6r 2 ( x 2  y 2  4 z 2  2r 2Q e )C2n  2r 9 5(3 x 2  3 y 2  4 z 2 ) D2n ), 1 3 yz (6r 2 (( y 2  z 2 )Qe  x 2 ( 5  Q e ))C2n  2r 9 5(6 x 2  y 2  z 2 ) D2n ), 1 3(2r 2 (x 4  y 4  z 4  r 2 z 2 Qe  2r 9 2 y 2 ( z 2  r 2Q e )  x 2 (2 y 2  13z 2  r 2Q e )) u uC2n  (4 x 4  y 4  3 y 2 z 2  4 z 4  3 x 2 ( y 2  9 z 2 )) D2n )),

en S 2313

r2

1 3 xy (6r 2 (( x 2  y 2 )Qe  z 2 ( 5  Q e ))C2n  9 2r 5( x 2  y 2  6 z 2 ) D2n ), x 2  y 2  z 2.

(A10)

References