Handbook of Materials Modeling pp 1-26 | Cite as

# Surface Energy and Nanoscale Mechanics

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## Abstract

The mechanical response of nanostructures, or materials with characteristic features at the nanoscale, differs from their coarser counterparts. An important physical reason for this size-dependent phenomenology is that surface or interface properties are different than those of the bulk material and acquire significant prominence due to an increased surface-to-volume ratio at the nanoscale. In this chapter, we provide an introductory tutorial on the continuum approach to incorporate the effect of surface energy, stress, and elasticity and address the size-dependent elastic response at the nanoscale. We present some simple illustrative examples that underscore both the physics underpinning the capillary phenomenon in solids as well as a guide to the use of the continuum theory of surface energy.

## 1 Introduction

In a manner of speaking, a free surface or an interface is a “defect.” We will use the word “surface” to imply both a free surface as well as an interface separating two materials. It represents a drastic interruption in the symmetry of the material – much like the more conventionally known defects such as dislocations. Imagine the atoms on a free surface. They have a different coordination number, charge distribution, possible dangling bonds, and many other attributes that distinguish them from atoms further away in the bulk of the material (Ibach 1997; Cammarata 2009). It is therefore hardly surprising that the surface of a material should have mechanical (or in general other physical) properties that differ from the bulk of the material. In other words, as much as conventional defects (such as dislocations) impact the physical response of materials, so do surfaces. However, for coarse-sized structures, the surface-to-volume ratio is negligible, and so even though surfaces do have different properties, they hardly matter in terms of the overall response of the structure. This situation changes dramatically at the nanoscale. As a rather extreme example, 2 nm cube of Copper has nearly 50*%* of its atoms on the surface. What length scale is “small enough” for surface effects to become noticeable arguably depends on the strength of the surface properties. For hard crystalline materials, this length scale is certainly below 50 nm and often only of significant importance below 10 nm (Miller and Shenoy 2000). Ultra-soft materials (with elastic modulus in the 1–5 kPa range) are an interesting exception where even at micron scale, surface energy-related size effects may be observable (Style et al. 2013).

*excess*energy that is distinct from the bulk. The surface energy concept for solids encompasses the fact that surfaces appear to possess a residual “surface tension”-like effect known as surface stress and also an elastic response, termed “surface or superficial elasticity.” In the context of fluids, “capillarity” has long been studied, and the concept of surface tension is well-known (see Fig. 1). The situation for solid surfaces is somewhat more subtle than fluids in many ways since deformation is a rather important contributor to surface energy – which is not the case for simple liquids. For a simple liquid, surface energy, the so-called surface stress and surface tension are the same concept. This is not the case for solid surfaces. For further details, see a recent review article by Style et al. (2017). Given the existence of an extensive body of work on this subject, we avoid a detailed literature review and simply point to the following overview articles that the reader may consult (and the references therein): (Javili et al. 2013; Duan et al. 2009; Wang et al. 2010, 2011; Li and Wang 2008; Cammarata 2009; Ibach 1997; Müller and Saúl 2004). We will primarily follow the approach pioneered by Gurtin, Murdoch, Fried, and Huang (Gurtin and Murdoch 1975b; Biria et al. 2013; Huang and Sun 2007) and attempt to present a simplified tutorial on the continuum theory for surface energy. For the sake of brevity and with the stipulation that this chapter is merely meant to be a first step to understand surface elasticity, we avoid several complexities and subtleties that exist on this topic such as choice of reference state in the development of the continuum theory (Huang and Wang 2013; Javili et al. 2017), consistent linearization from a nonlinear framework and differences in the various linearized theories (Javili et al. 2017; Liu et al. 2017), curvature dependence of surface energy (Steigmann and Ogden 1997, 1999; Chhapadia et al. 2011; Fried and Todres 2005), and generalized interface models that may allow greater degree of freedom than just a “no slip” surface c.f. (Gurtin et al. 1998; Chatzigeorgiou et al. 2017). We also present three simple case studies or illustrative examples that both highlight the use of the theory and the physical consequence of surface energy effects. While the focus of the chapter is primarily on mechanics and, specifically, elasticity, the framework used in this chapter can be used as a starting point for applications outside mechanics. Indeed, surface energy effects are of significant interest to a variety of disciplines and permeate topics as diverse as fluid mechanics (de Gennes et al. 2004), sensors and resonators (Park 2008), catalysis (Müller and Saúl 2004; Haiss 2001; Pala and Liu 2004), self-assembly (Suo and Lu 2000), phase transformations (Fischer et al. 2008), biology (Liu et al. 2017), and composites (Duan et al. 2005), among others.

## 2 Preliminary Concepts

### 2.1 The Need for Surface Tensors

In the study of mechanics of surfaces, we have to contend with tensor fields that “live” on a surface. For example, we will need to define the strain field experienced by a surface. The general machinery of curvilinear tensor calculus then becomes necessary to describe surface mechanics which (at least for our taste) becomes somewhat cumbersome. A direct notation was developed by Gurtin and Murdoch (1975b) which we (prefer and) briefly motivate in this section.

*=*

**n**

**e**_{3}. On physical grounds, assuming that there is no “slip” between the surface and the underlying bulk material, the surface strain is simply the strain field of the bulk material

*at the spatial position of the surface*. In addition, intuitively, the normal components of the strain tensor ought not to exist for the zero-thickness surface, i.e.,

**E**_{s}(

*) =*

**x**

**E**^{bulk}(

*) projected on the tangent plane of the surface where*

**x***is on the surface. To make this idea more concrete, assume a body occupies a domain*

**x***Ω*

_{0}experiencing a strain \(\boldsymbol {E} : \varOmega _0 \to \mathbb {R}_{sym}^{3\times 3}\). We then expect the surface strain to be simply

Our intuition works well for flat surfaces, but we must generalize the mathematical framework to contend with general curved surfaces. This brings to fore the question of how we define surface tensors on arbitrarily curved surfaces. To understand this, we define next the surface projection tensor. More details on the mathematical preliminaries can be found in the work of Gurtin, Murdoch, and co-workers (Gurtin and Murdoch 1975b, 1978; Gurtin et al. 1998).

#### 2.1.1 Surface Projection Tensor

*(*

**n***) is defined as*

**x****I**is the second-order identity tensor, and “⊗” denotes the tensor product (or the dyadic product).

*=*

**n**

**e**_{3}in Cartesian coordinates with a positively oriented orthonormal basis {

**e**_{1},

**e**_{2},

**e**_{3}}. With the definition in (2), the projection tensor on the flat upper surface admits the form

*R*that can be represented by \(S=\{ \boldsymbol {x} \in \mathbb {R}^3 : \boldsymbol {x}\cdot \boldsymbol {x}-R^2=0 \}\). The outward unit normal to this surface is

*=*

**n**

**e**_{r}in spherical coordinates {

*r*,

*θ*,

*ϕ*} with basis {

**e**_{r},

**e**_{θ},

**e**_{ϕ}}. From the definition of the projection tensor in (2), we have

The projection tensor in (3b) can also be easily expressed in the Cartesian coordinates by using the identities \(\boldsymbol {e}_{\theta } = \cos \theta \cos \phi \boldsymbol {e}_1 + \cos \theta \sin \phi \boldsymbol {e}_2 - \sin \theta \boldsymbol {e}_3\) and \(\boldsymbol {e}_{\phi } = - \sin \phi \boldsymbol {e}_1 + \cos \phi \boldsymbol {e}_2\) as well as the tensor product between two vectors.

#### 2.1.2 Surface Vector and Tensor Fields

*(*

**v***) be a smooth vector field and*

**x***(*

**T***) be a smooth second-order tensor. Their projections*

**x**

**v**_{s}(

*) and*

**x**

**T**_{s}(

*) on a smooth surface with outward unit normal*

**x***(*

**n***), respectively, are*

**x***=*

**v***v*

_{i}

**e**_{i}and a second-order tensor field

*=*

**T***T*

_{ij}

**e**_{i}⊗

**e**_{j}in Cartesian coordinates with a positively oriented orthonormal basis {

**e**_{1},

**e**_{2},

**e**_{3}}. Take the upper surface with outward unit normal

*(*

**n***) =*

**x**

**e**_{3}. The projection tensor in (2) becomes \({\mathbb {P}}(\boldsymbol {x})= \mathbf {I} - \boldsymbol {e}_3 \otimes \boldsymbol {e}_3 = \boldsymbol {e}_1 \otimes \boldsymbol {e}_1 + \boldsymbol {e}_2 \otimes \boldsymbol {e}_2\). Thus, with (4), the projected vector and tensor on the upper surface, respectively, are

_{2}, the projection of the identity tensor

**I**on the surface with outward unit normal

*(*

**n***) can be obtained as*

**x***surface identity tensor*on the surface and is often used in the context of second-order tensors.

### 2.2 Differentiation and Integration on a Surface

#### 2.2.1 Surface Gradient, Normal Derivative, and Curvature Tensor

*Ω*

_{0}. By using the gradient operator ∇, their (three-dimensional) gradients are represented by ∇

*ϕ*(

*) and ∇*

**x***(*

**v***), respectively. In contrast, the surface gradient operator on a surface with unit normal*

**x***is denoted by ∇*

**n**_{s}, and together with the projection tensor \({\mathbb {P}}\) in (2), the

*surface gradients*of the two fields can be represented by (Gurtin et al. 1998)

*normal derivative*of fields on a surface with outward unit normal

*. Following the scalar field*

**n***ϕ*and the vector field

*in (7), we define their normal derivatives as*

**v***ϕ*in the direction

*. By (2) and (8), the surface derivatives (7) can be recast as*

**n**Equation (9) represents the relation between the surface gradient, the gradient, and the normal derivative. We note that Eqs. (7) and (9) are two (alternative but equivalent) ways to represent the surface gradient.

*curvature tensor*of a surface with outward unit normal

*is then defined as (Gurtin et al. 1998)*

**n**In order to enhance the understanding of these definitions, we now consider some simple examples by referring to Fig. 2. In Cartesian coordinates, the gradients of the scalar and vector fields, respectively, are \(\nabla \phi =\frac {\partial \phi }{\partial x_i} \boldsymbol {e}_i\) and \(\nabla \boldsymbol {v}= \frac {\partial \boldsymbol {v}}{\partial x_j}\otimes \boldsymbol {e}_j = \frac {\partial v_i}{\partial x_j}\boldsymbol {e}_i \otimes \boldsymbol {e}_j\). The outward unit normal to the flat upper surface is * n* =

**e**_{3}, and then its projection tensor is \(\mathbb P = \boldsymbol {e}_1 \otimes \boldsymbol {e}_1 + \boldsymbol {e}_2 \otimes \boldsymbol {e}_2\) which was mentioned in (3a).

*R*. In spherical coordinates {

*r*,

*θ*,

*ϕ*} with orthonormal basis {

**e**_{r},

**e**_{θ},

**e**_{ϕ}}, the outward unit normal to the spherical surface is

*=*

**n**

**e**_{r}, whose gradient is ∇

**e**_{r}=

*r*

^{−1}(

**e**_{θ}⊗

**e**_{θ}+

**e**_{ϕ}⊗

**e**_{ϕ}). By (7) and (10), the curvature tensor in this example is

#### 2.2.2 Surface Divergence, Trace, and Mean Curvature

*surface divergences*of a vector field

*(*

**v***) and a tensor field*

**x***(*

**T***) are defined as follows:*

**x***trace*and \(\boldsymbol {a} \in \mathbb {R}^3\) is an arbitrary constant vector. An important identity related to the surface divergence operator is

*mean curvature*is then simply

*=*

**T***T*

_{ij}

**e**_{i}⊗

**e**_{j}, its surface gradient is a vector, and its

*k*-th component, by (15)

_{2}, can be represented by

**T**^{T}

**e**_{k}=

*T*

_{ki}

**e**_{i}and the example result in (18) are used. Regarding the examples of the curvature tensors (13) and (14), their mean curvatures are 0 and −

*R*

^{−1}, respectively.

#### 2.2.3 Divergence Theorem for Surfaces

_{0}⊂

*∂Ω*

_{0}with a smooth boundary curve

*∂*S

_{0}. For a smooth vector

*that is tangential on the surface S*

**u**_{0}and a smooth tensor field

*, the divergence theorem is defined by Gurtin and Murdoch (1975b) and Gurtin et al. (1998)*

**T***is the outward unit normal to the boundary curve*

**v***∂*S

_{0}.

## 3 Theoretical Framework for Surface Mechanics

With the mathematical preliminaries necessary for surface mechanics described earlier, we can now proceed in a rather standard manner to derive the pertinent governing equations. The original theory by Gurtin and Murdoch (1975b) was derived by employing stress as the primitive concept. We (following Huang and co-workers Huang and Wang 2006; Huang and Sun 2007) favor a variational approach where we take the surface energy as the primitive concept.

### 3.1 Kinematics

*Ω*

_{0}in the reference configuration shown in Fig. 3. The boundary of the domain is denoted by

*∂Ω*

_{0}, which can be divided into two parts: the displacement boundary \(\partial \varOmega _0^u\) and the traction boundary \(\partial \varOmega _0^t\). Mathematically, \(\partial \varOmega _0^u \cup \partial \varOmega _0^t = \partial \varOmega _0\) and \(\partial \varOmega _0^u \cap \partial \varOmega _0^t = \varnothing \).

*∈*

**x***Ω*

_{0}. Consider a smooth mapping \(\boldsymbol {y}: \varOmega _0 \to \mathbb R^3\), that is,

*(*

**y***) =*

**x***+*

**x***(*

**u***). Here*

**x***is the displacement vector. The deformation gradient is defined as*

**u***= ∇*

**F***. The displacement and traction boundary conditions are*

**x**### 3.2 Energy Variation and Equations of Equilibrium

*Ψ*is the strain energy function per unit volume of the bulk,

*Γ*

_{s}is the surface energy function per unit area of the surface S

_{0}⊂

*∂Ω*

_{0}, and

**t**_{0}is the dead load applied on the traction boundary \(\partial \varOmega _0^t\) in the reference configuration. The reference configuration taken here is actually the initial configuration, which is neither subjected to any body force nor tractions. We note that in the reference configuration, there exists the surface stress, which is regarded as “residual.” The residual stress field can be described according to the surface energy function

*Γ*

_{s}. To further clarify the surface effects on the energy functional, Huang and Wang (2006, 2013) proposed an extra configuration, a “fictitious stress-free configuration.” For further details on such subtleties, the reader is referred to their work.

*Ω*

_{0}, and the displacement field must satisfy

*=*

**u**

**u**_{0}on \(\partial \varOmega _0^u\) in (21)

_{1}.

*to be the minimizer of the energy functional in (22), then by the principle of energy minimization (23), we have*

**u***+*

**u***𝜖*

**u**_{1}belongs to the set of all kinematically admissible deformations in the neighborhood of the deformation

*. We only consider small perturbations, so the norm \(\left \Vert \epsilon \boldsymbol {u}_1 \right \Vert \ll 1\). By the displacement boundary (21)*

**u**_{1}, the variation

**u**_{1}satisfies

*is the first Piola-Kirchhoff bulk stress tensor.*

**S**_{2}as well as the property of the surface tensor \({\mathbb {S}}\), that is, \({\mathbb {S}}\boldsymbol {n} = \boldsymbol {0}\), the integrand \({\mathbb {S}}\cdot \nabla \boldsymbol {u}_1\) in (28) can be recast as

_{0}(see their Eq.(37)) by assuming an arbitrary function \(\frac {\partial \boldsymbol {u}_1 }{ \partial n}\) on the surface S

_{0}. Here we obtain this identity through the definition of a surface tensor (4)

_{2}, that is, for an arbitrary surface tensor \(\boldsymbol {T}_s={\mathbb {P}} \boldsymbol {T}{\mathbb {P}}\), we have \(\boldsymbol {T}_s \boldsymbol {n}={\mathbb {P}} \boldsymbol {T}{\mathbb {P}} \boldsymbol {n} ={\mathbb {P}} \boldsymbol {T} \boldsymbol {0} = \boldsymbol {0}\) since \({\mathbb {P}} \boldsymbol {n} = (\mathbf I -\boldsymbol {n} \otimes \boldsymbol {n}) \boldsymbol {n} = \boldsymbol {0}\).

_{0}⊂

*∂Ω*

_{0}. However, the relation between S

_{0}and \(\partial \varOmega _0^t\) (or \(\partial \varOmega _0^u\)) is not given before. To simplify the discussion, we regard S

_{0}as a subset of \(\partial \varOmega _0^t\), that is, \({\mathsf S}_0 \subset \partial \varOmega _0^t\). Thus, the last integral in (34) can be reformulated as

**u**_{1}=

*on \(\partial \varOmega _0^u\) in (25), the second integral in (34) reduces to*

**0**

**u**_{1}in (37) is arbitrary, the vanishing of the first variation \(\delta \mathscr F [\boldsymbol {u}] = 0\) and the fundamental lemma of calculus of variations (Courant and Hilbert 1953) leads us to the following set of governing equations

Here we rewrite the equation \(\mathbb S \boldsymbol {n} = \boldsymbol {0}\) on S_{0} (see the statement above (30)) in (38). Equation (38), together with (21)_{1} and (29), forms a well-defined boundary value problem. For the readers convenience, we reiterate the notations here: * S* denotes the first Piola-Kirchhoff stress, \({\mathbb {S}}\) the first surface Piola-Kirchhoff stress,

*the outward unit normal to the surface,*

**n***the outward unit normal to the boundary curve, and*

**v**

**t**_{0}the applied dead load.

### 3.3 Constitutive Equations and Elastic Stress Tensors

*and the first Piola-Kirchhoff surface stress tensor \(\mathbb S\) through the partial derivative with respect to the displacement gradient ∇*

**S***. By the chain rule, these two first Piola-Kirchhoff stresses can also be defined as*

**u***= ∇(*

**F***+*

**x***) =*

**u****I**+ ∇

*and*

**u***frame indifference*in the strain energy functions (Gurtin et al. 2010) and the polar decomposition

*=*

**F***,*

**RU***∈Orth*

**R**^{+}= {all rotations}, and

*is the right stretch tensor, we have*

**U***Ψ*

^{∗}(

*) and*

**E***Γ*

_{s}

^{∗}(

*), such that*

**E**### 3.4 Linearized Bulk and Surface Stresses and Constitutive Choice

A peculiarity of considering surface effects is the perceived presence of residual stresses – the surface tension-like quantity in solids is precisely a residual stress state. This becomes evident if we linearize the energy functions around a reference configuration which is not stress-free.

*=*

**C**

**F**^{T}

*= (*

**F****I**+ ∇

*)*

**u**^{T}(

**I**+ ∇

*) can be reduced to*

**u***=*

**C****I**+ ∇

*+ ∇*

**u**

**u**^{T}, |∇

*|≪ 1, by dropping the higher-order terms*

**u***o*(|∇

*|). Thus, the strain tensor \(\boldsymbol {E} = \frac {1}{2} (\boldsymbol {C} - \mathbf I)\) in (44) may be approximated by the infinitesimal strain:*

**u**_{1}gives

*o*(|

*|) denotes the higher-order terms.*

**E***=*

**S***in (45)*

**FT**_{1}and the Cauchy stresses \(\boldsymbol {\sigma } = (\det \boldsymbol {F})^{-1} \boldsymbol {S} \boldsymbol {F}^{T}\) in (46)

_{1}of bulk materials at small deformations. By the relation (50) and the infinitesimal strain (47), it is easy to show that to a first order

The equivalence of the first P-K, the second P-K, and the Cauchy stresses in (51), however, does not hold for surface stresses due to the existence of the residual stress. This is rather important to note. In linearized elasticity (without residual stresses), it is usual to ignore the distinction between the various stress measures. For surface elasticity, even in the linearized case, we must take cognizance of the different interpretations of the various stress measures. The first P-K stress measure is the most useful since it represents the force per unit *referential* area and is likely to be the quantity controlled in traction-controlled experiments (as opposed to Cauchy traction).

_{2}and the first P-K surface stress \(\mathbb S\) in (45)

_{2}can be recast as

**E**_{s}, in contrast to the infinitesimal strain (47), is the infinitesimal surface strain

_{1}, for small deformation, we have

*Γ*

_{s}

^{∗}and

*Ψ*

^{∗}), we now have all the governing equations and can solve the pertinent boundary value problems of physical interest. Analytical solutions are rather hard to come by for the anisotropic case, and by far, most problems solved in the literature have been restricted to isotropic continua.

*λ*

_{0}and

*μ*

_{0}.

_{2}and (57), the first P-K surface stress \({\mathbb {S}}\) becomes

_{s}

*)tr(*

**u**

**E**_{s})| and |(∇

_{s}

*)*

**u**

**E**_{s}| are omitted. Here and henceforth, the difference between ∇

_{s}

*and \(\mathbb P \nabla _s\boldsymbol {u}\) is not specified for simplicity.*

**u**_{2}and a similar argument as (52), the Cauchy surface stress \({\mathbb \sigma }^s\) and the first P-K stress \({\mathbb {S}}\) in (58) have the relation

**E**_{s}),

**E**_{s}, ∇

_{s}

*, and ∇*

**u**_{s}

**u**^{T}.

*τ*

_{0}≠ 0, namely:

*λ*

_{s},

*μ*

_{s}) and (

*λ*

_{0},

*μ*

_{0}).

*λ*

_{s},

*μ*

_{s}) to zero in (62), we obtain

*λ*

_{0}=

*τ*

_{0}and

*μ*

_{0}= −

*τ*

_{0}, and then the result in (58) is what is often called the

*surface tension*(in the reference configuration) (Gurtin and Murdoch 1975a)

*λ*

_{s},

*μ*

_{s}) to zero will be in the form of isotropic “pressure”

## 4 Illustrative Examples

In this section we choose three illustrative examples that highlight both the use of the surface elasticity theory as well as provide insights into the physical consequences of surface energy at the nanoscale. These examples are inspired from Altenbach et al. (2013), Murdoch (2005), and Sharma et al. (2003) although, to be consistent with our own style (presented in the preceding sections), we have modified them slightly. Germane to the study of analytical study of nanostructures, we also note parallel developments in the literature on the so-called surface Cauchy-Born rule and numerical methods (Park et al. 2006).

### 4.1 Young’s Modulus of a Nano-rod Considering Surface Effects

*R*

_{0}whose axis coincides with the

**e**_{z}direction. To interrogate the elastic response, we assume that the rod is under uniaxial tension and the coordinate system used for this problem is cylindrical coordinate with basis (

**e**_{r},

**e**_{θ},

**e**_{z}) as shown in Fig. 4.

*A*and

*B*are constants. The assumed displacement (66) can be used for either small or finite deformation.

*=*

**n**

**e**_{r}, the projection tensor is \({\mathbb {P}} = \boldsymbol {e}_{\theta }\otimes \boldsymbol {e}_{\theta }+\boldsymbol {e}_{z}\otimes \boldsymbol {e}_{z}\). By (66), the displacement gradient ∇

*, the strain tensor \(\boldsymbol {E} = \frac {1}{2} (\nabla \boldsymbol {u} + \nabla \boldsymbol {u}^T)\) for small deformation, the surface displacement gradient \(\nabla _s \boldsymbol {u} = (\nabla \boldsymbol {u}) {\mathbb {P}}\), and the surface strain tensor \(\boldsymbol {E}_s = {\mathbb {P}} \boldsymbol {E} {\mathbb {P}}\) for small deformation become*

**u**_{1}, we have

*λ*and

*μ*are Lamé constants of bulk materials and

*τ*

_{0}= 0, together with (67)

_{2}, we have

_{3}, without external load

**t**_{0}=

*on the radial surface (*

**0***r*=

*R*

_{0}) with unit normal

**e**_{r}, we have the equality \(\boldsymbol {S}\boldsymbol {e}_r = \text{div}_{\text{s}} {\mathbb {S}}\). By (68a), the identity (16), and \({\mathbb {S}}^T \boldsymbol {e}_r = \boldsymbol {0}\) in (38)

_{3}, we can obtain

*A*∕

*B*is given by (73). By the equilibrium of the rod in the axial direction (Altenbach et al. 2013), the effective Young’s modulus can be defined as \(E^{\mathrm {eff}} = (S_{zz} + \frac {2}{R_0} {\mathbb {S}}_{zz}) / E_{zz} = \lambda \left (1+2\frac {A}{B}\right ) + 2 \mu + \frac {2}{R_0} [\lambda _0 (1 + \frac {A}{B}) + 2\mu _0]\).

This simple example makes clear that the effective or apparent elastic response of nanostructures becomes size-dependent as a result of surface energy effects and that with smaller *R*_{0}, the effective elastic modulus may become significantly different than its bulk value. We remark that if surface effects are ignored, that is, *λ*_{0} = *μ*_{0} = 0 and the ratio *A*∕*B* = −*λ*∕(2*λ* + 2*μ*) in (73), then the Young modulus in (74) becomes *E*^{rod} = *μ*(3*λ* + 2*μ*)∕(*λ* + *μ*)–which is essentially the relation between bulk Young’s modulus and the Lamé constants.

### 4.2 Influence of Surface Effects on the Thermoelastic State of a Ball

We consider the equilibrium state of a spherical ball (radius *R*_{0}) in vacuum undergoing thermal expansion – following Murdoch (2005). Only traction boundary conditions need to be considered here; hence, (21)_{1} can be omitted. Moreover, (38)_{2} and (38)_{4} are also omitted since the entire surface of the sphere is considered, that is, \(\mathsf S_0 = \partial \varOmega _0 = \partial \varOmega _0^t\).

_{1}, (38)

_{3}, and (29). As before, we assume the ball material to be isotropic and that the ball is in its natural, stress-free state in the reference configuration. Hence, we assume constitutive equations for the bulk and surface as follows:

*is the first P-K stress field in the reference configuration,*

**S***λ*

_{b}and

*μ*

_{b}are Lamé constants for the ball material,

*ΔT*is the temperature difference,

*α*is the coefficient of thermal expansion, and

*α*

_{0}is the thermal expansion coefficient of the surface.

**e**_{r},

**e**_{θ},

**e**_{ϕ}) with the origin at the center of the ball. For the surface with outward unit normal

*=*

**n**

**e**_{r}, the projection tensor is \({\mathbb {P}} = \boldsymbol {e}_{\theta }\otimes \boldsymbol {e}_{\theta }+\boldsymbol {e}_{\phi }\otimes \boldsymbol {e}_{\phi }\). Given that the problem is spherically symmetric, the form of the displacement field can be assumed to be

_{1}into the equilibrium equation (38)

_{1}, together with the divergence of a tensor in spherical coordinates, we obtain

*u*

_{r}(

*r*) =

*Ar*+

*Br*

^{−2}, where

*A*and

*B*are constants to be determined by the boundary conditions. The displacement at the origin must vanish for the field to be bounded, i.e., and consequently

*B*= 0 and then

_{3}. Since the ball surface is traction-free, that is,

**t**_{0}=

*, (38)*

**0**_{3}can be reduced to \(\boldsymbol {S}\boldsymbol {e}_r = \text{div}_{\text{s}} {\mathbb {S}}\). By (82), we can further simplify this to

_{2}, (38)

_{3}, and (14), we finally obtain

*surface modulus*

*K*

_{s}as

*A*in the displacement (81), namely:

This example, as can be noted from (87), nicely shows how a positive surface residual stress *τ*_{0} may hinder the thermal expansion in a size-dependent manner.

### 4.3 Effect of Residual Stress of Surfaces on Elastic State of Spherical Inclusion

*R*

_{0}but incorporating surface effects. The solution here is a slight modification of the work by Sharma et al. (2003). We use spherical coordinates with basis (

**e**_{r},

**e**_{θ},

**e**_{ϕ}) with the origin at the center of the sphere in this problem. We assume that a far field stress is exerted on the matrix as follows:

*is the first Piola-Kirchhoff stress and*

**S***λ*

_{ν}and

*μ*

_{ν}are Lamé constants of the material corresponding to either the inclusion(I) or the matrix(M). For the surface we use the constitutive equation defined in (58), i.e.:

_{1}–(38)

_{3}becomes

_{1}and (93)

_{2}are

*A*,

*B*,

*C*, and

*D*are constant. Since

*u*

_{r}(0) = 0,

*B*= 0. Substituting (94)

_{2}into (92)

_{1}, the far field (93)

_{4}gives

*C*=

*S*

^{∞}∕(3

*K*

_{M}), where

*K*

_{M}=

*λ*

_{M}+ 2

*μ*

_{M}∕3. Using the continuity of the displacement at the interface, that is, Open image in new window at

*r*=

*R*

_{0}, and the Young-Laplace equation (93)

_{3}, we obtain two algebraic equations of

*A*and

*D*. The routine calculation is not shown here and we just list the final results. Thus, the displacement (94) is obtained as

*E*

^{∞}=

*S*

^{∞}∕(3

*K*

_{M}), \(\alpha :=\dfrac {(3K_M+4\mu _M)E^\infty -2\tau _0/R_0}{4\mu _M+3K_I+2K_s/R_0+2\tau _0/R_0} \),

*K*

_{I}=

*λ*

_{I}+ 2

*μ*

_{I}∕3 is the inclusion bulk modulus, and

*K*

_{s}= 2(

*λ*

_{s}+

*μ*

_{s}) is defined as the surface modulus.

*λ*

_{I}= 0 and

*μ*

_{I}= 0, we obtain the much-studied case of a void in a solid. The displacement (95)

_{2}for this special case is

*E*

^{∞}=

*S*

^{∞}∕(3

*K*

_{M}). Thus, the bulk stress (92)

_{1}in the matrix (

*r*>

*R*

_{0}) is

*S*

^{∞}=

*E*

^{∞}= 0, the bulk stress

*becomes*

**S***S*

^{∞}≠ 0. By (97) and (98), we may then define the stress concentration factor at

*r*→

*R*

_{0}as

*τ*

_{0}=

*K*

_{s}= 0 in (99), we obtain a stress concentration factor of 1.5 which is the well-known classical elasticity result for a spherical void under hydrostatic stress. We have graphically plotted the results in Fig. 5 that illustrates the qualitative behavior of how stress concentration on a void alters due to size and the surface elasticity modulus (Interestingly, the surface elasticity modulus can have negative values as shown via atomistic simulations by Miller and Shenoy 2000.).

## 5 Perspectives on Future Research

As well-motivated by Steigmann and Ogden (1999, 1997), under certain circumstances, the dependence of surface energy on curvature must be accounted for. This was recently explored by Fried and Todres (2005), who examined the effect of the curvature-dependent surface energy on the wrinkling of thin films, and Chhapadia et al. (2011) who (using both atomistics and a continuum approach) explained certain anomalies in the bending behavior of nanostructures. However, due to the complexity of the Steigmann-Ogden curvature-dependent surface elasticity, relatively few works exist on this topic.

Intriguing recent experiments by Style and co-workers (Style et al. 2013, 2017) on capillarity and liquid inclusions in soft solids have revealed that the pertinent size effects due to surface effects may be observed at

*micron*length scales (in contrast to the nanoscale for hard materials). This represents an important future direction and requires the use of*nonlinear*surface elasticity due to the need to account for the inevitable large deformations in soft matter. Arguably, the study of capillarity in soft matter will also require the development and use of numerical methods cf. Henann and Bertoldi (2014).Finally, the literature on coupling of capillarity with electrical and magnetic fields is quite sparse.

## Notes

### Acknowledgements

Support from the University of Houston and the M. D. Anderson Professorship is gratefully acknowledged.

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